mathematics meets finance

MATHEMATICS MEETS FINANCE

A Non-Profit Story on A Profitable Matter PRELIMINARY VERSION, 4th edition Lecture at the University of Konstanz, Department of Mathematics and Statistics, Interdisciplinary Program On Mathematical Finance 2001-2011

c ⃝Michael Kohlmann, University of Konstanz Department of Mathematics and Statistics email: [email protected] http://www.math.uni-konstanz.de/∼kohlmann/

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To Evi and Benjamin

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NOTICE: THIS IS WORK IN PROGRESS copyrights: michael [email protected] Note that an extended version of the first chapter is available as separate scripts on arbitrage and portfolio optimization

WHO CAN DOES

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Contents 1 The General Market Model 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Simple Self-Financing Hedging Strategies . . . . . . . . . . . 1.2.1 Change of Numéraire . . . . . . . . . . . . . . . . . . 1.2.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Arbitrage, an exercise in Binary Markets . . . . . . . 1.3 Simple Lp -Hedging Strategies . . . . . . . . . . . . . . . . . . 1.3.1 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . 1.4 Signed Local Martingale Measures . . . . . . . . . . . . . . . 1.4.1 The Law of One Price . . . . . . . . . . . . . . . . . . 1.5 Typical Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Theories of utility . . . . . . . . . . . . . . . . . . . . 1.6.2 Utility Functions . . . . . . . . . . . . . . . . . . . . . 1.6.3 Conjugate Young Functions . . . . . . . . . . . . . . . 1.6.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . 1.6.5 Optimal Portfolios and Local Martingale Measures . . 1.6.6 A Worked Example: The Merton Portfolio . . . . . . 1.6.7 Mean-Variance Hedging . . . . . . . . . . . . . . . . . 1.6.8 Utility Indifference Prices . . . . . . . . . . . . . . . . 1.7 Mean-Variance Efficiency . . . . . . . . . . . . . . . . . . . . 1.7.1 The Generalized Sharpe-Ratio . . . . . . . . . . . . . 1.7.2 The Intertemporal Price for Risk . . . . . . . . . . . . 1.7.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Economical Consideration . . . . . . . . . . . . . . . . 1.7.5 Mean-Variance Efficient Hedging . . . . . . . . . . . . 1.7.6 The Risk Premium Indifference Price for Information

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2 Classical Theory 2.1 A Discrete Example . . . . . . . . . . . . . . . . . . 2.2 The Continuous Case: The Market . . . . . . . . . . 2.3 The Equivalent Martingale Measure . . . . . . . . . 2.4 European Options, B-S-Formula, American Options 2.4.1 The complete case . . . . . . . . . . . . . . . 2.4.2 All Kinds of Options . . . . . . . . . . . . . . 2.4.3 American Contingent Claim, Complete Case

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2.5

2.6

2.7

2.4.4 Hedging in Incomplete Markets . . . . . . . . . . . . . . . 85 The BSDE-Approach To The Classical Theory . . . . . . . . . . 87 2.5.1 Pricing of Contingent Claims, a New View . . . . . . . . . 87 2.5.2 The Model Revised . . . . . . . . . . . . . . . . . . . . . . 88 2.5.3 Admissible Portfolios . . . . . . . . . . . . . . . . . . . . . 91 2.5.4 Pricing in Complete Markets . . . . . . . . . . . . . . . . 95 2.5.5 Black-Scholes-Type equations . . . . . . . . . . . . . . . . 99 2.5.6 Superhedging . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.5.7 The New Tools . . . . . . . . . . . . . . . . . . . . . . . . 101 2.5.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.5.9 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . 111 2.5.10 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 The Partially Informed Agent . . . . . . . . . . . . . . . . . . . . 115 2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.6.2 An existence result for a BSDE under additional information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.6.3 Setting the hedging problem . . . . . . . . . . . . . . . . 117 2.6.4 Available Information . . . . . . . . . . . . . . . . . . . . 119 2.6.5 The Föllmer-Schweizer uninformed agent . . . . . . . . . 123 2.6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Neyman-Pearson hedging . . . . . . . . . . . . . . . . . . . . . . 127 2.7.1 The market tools . . . . . . . . . . . . . . . . . . . . . . . 129 2.7.2 Pricing a contingent claim and mean variance hedging . . 131 2.7.3 Superhedging and the upper price in an incomplete market133 2.7.4 Measuring risk and Neyman-Pearson hedging . . . . . . . 137 2.7.5 The Neyman-Pearson hedge . . . . . . . . . . . . . . . . . 138 2.7.6 Uncertain real world . . . . . . . . . . . . . . . . . . . . . 143 2.7.7 Testing and Optimization . . . . . . . . . . . . . . . . . . 144 2.7.8 Incomplete case . . . . . . . . . . . . . . . . . . . . . . . . 148 2.7.9 The American contingent claim . . . . . . . . . . . . . . . 149 2.7.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 151

3 Mean Variance Hedging 3.1 The Toolbox . . . . . . . . . . . . 3.2 Main Results . . . . . . . . . . . . 3.2.1 The Proof of Theorem 3.2.1 3.2.2 Decreasing Drifts . . . . . . 3.2.3 The Positivity of K j . . . . 3.2.4 Uniform Boundedness . . . 3.2.5 Existence . . . . . . . . . . 3.2.6 Feynman-Kac . . . . . . . . 3.2.7 A Remark . . . . . . . . . . 3.2.8 Proof . . . . . . . . . . . . 3.3 LQ-Problem . . . . . . . . . . . . . 3.4 MV-Hedging . . . . . . . . . . . . 3.4.1 The Market . . . . . . . . . 3.4.2 Formulation of the problem

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3.4.3 Complete Solution . . . . . . . . . . . . . . . . . . . . . . 174 3.4.4 Markovian Market . . . . . . . . . . . . . . . . . . . . . . 177 3.4.5 Modified Model . . . . . . . . . . . . . . . . . . . . . . . . 178 3.4.6 Exponential Hedging, work in progress . . . . . . . . . . . 185 3.4.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Markovitz and Sons . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . 198 3.5.2 An Auxiliary Problem . . . . . . . . . . . . . . . . . . . . 201 3.5.3 Solutions to General LQ Problems . . . . . . . . . . . . . 202 3.5.4 Solution to The Auxiliary Problem . . . . . . . . . . . . . 205 3.5.5 Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . 207 3.5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 211 The Partially Observed Case . . . . . . . . . . . . . . . . . . . . 211 3.6.1 Formulation of the Problem . . . . . . . . . . . . . . . . . 213 3.6.2 An Equivalent Formulation . . . . . . . . . . . . . . . . . 216 3.6.3 Existence and Uniqueness of an Optimal Hedging Policy . 218 3.6.4 The Partially Observed Maximum Principle for Optimal Hedging Policies . . . . . . . . . . . . . . . . . . . . . . . 219 3.6.5 Optimal Hedging Policy and Approximate Pricing Equation220 3.6.6 The Separated Problem . . . . . . . . . . . . . . . . . . . 223

4 Bonds and Interest Rate Derivatives 4.1 Zero Coupon Bonds . . . . . . . . . . . . . . . 4.2 Forward and Future Price . . . . . . . . . . . . 4.2.1 Forwards and Futures, an interpretation 4.2.2 Change of Numeraire, revisited . . . . . 4.3 Term Structure . . . . . . . . . . . . . . . . . . 4.3.1 The Vasicek Model . . . . . . . . . . . . 4.3.2 The Hull-White Model . . . . . . . . . . 4.3.3 The Cox-Ingersoll-Ross Model . . . . . 4.3.4 The Heath-Jarrow-Morton Model . . . . 4.3.5 Remarks . . . . . . . . . . . . . . . . . .

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225 . 225 . 227 . 230 . 231 . 232 . 233 . 234 . 235 . 237 . 239

5 Continuous Time CAPM 5.1 Conditional Markets . . . . . . . . . . . . . . . . . . . . . 5.2 Conditional Utility Maximization . . . . . . . . . . . . . . 5.2.1 Extended Conditional Expectations . . . . . . . . 5.3 Lq0 -integrable Martingale Measures . . . . . . . . . . . . . 5.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Consistency of Lp -optimal Martingale Measures . . 5.4 E-Martingale Measures . . . . . . . . . . . . . . . . . . . . 5.4.1 Reverse Hölder Conditions . . . . . . . . . . . . . 5.4.2 The Stochastic Logarithm of Z opt . . . . . . . . . . 5.4.3 The Lq -Optimal E-Martingale Measure . . . . . . 5.4.4 Existence of the Lq -Optimal E-Martingale Measure 5.4.5 Counterexamples . . . . . . . . . . . . . . . . . . . 5.4.6 Example 1 . . . . . . . . . . . . . . . . . . . . . . .

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5.5 5.6 5.7

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5.10 5.11

5.12

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5.14 5.15

5.4.7 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . Conditional Mean-Variance Efficiency . . . . . . . . . . . . . . 5.5.1 The Conditional Intertemporal Price for Risk . . . . . . The Modified Market . . . . . . . . . . . . . . . . . . . . . . . . Semimartingale Market Models . . . . . . . . . . . . . . . . . . 5.7.1 Self-financing Hedging Strategies . . . . . . . . . . . . . 5.7.2 Exponential Hedging Strategies . . . . . . . . . . . . . . E[N ]-Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Properties of E[N ]-Martingales . . . . . . . . . . . . . . 5.8.2 Convergence of E[N ]-Martingales . . . . . . . . . . . . . 5.8.3 Doob Maximal Inequalities under a Change of Measure 5.8.4 Burkholder-Davis-Gundy Inequalities . . . . . . . . . . . Lp -integrable Hedging Strategies . . . . . . . . . . . . . . . . . 5.9.1 Limits of Simple Hedging Strategies . . . . . . . . . . . 5.9.2 The Space of Lp -integrable Hedging Strategies . . . . . 5.9.3 Lp -Optimal Hedging Strategies . . . . . . . . . . . . . . 5.9.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . The Exponential Hedging Numéraire . . . . . . . . . . . . . . . Continuous Semimartingale Markets . . . . . . . . . . . . . . . 5.11.1 Equivalent Local Martingale Measures . . . . . . . . . . 5.11.2 Increasing Utility Functions . . . . . . . . . . . . . . . . 5.11.3 Optimal Pairs . . . . . . . . . . . . . . . . . . . . . . . . 5.11.4 The Optimal Local Martingale . . . . . . . . . . . . . . Locally Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . 5.12.1 The Instantaneous Price for Risk . . . . . . . . . . . . . 5.12.2 Totally Unhedgeable Instantaneous Price for Risk . . . 5.12.3 Deterministic Instantaneous Price for Risk . . . . . . . . 5.12.4 Totally Hedgeable Instantaneous Price for Risk . . . . . Globally Efficient Portfolios . . . . . . . . . . . . . . . . . . . . 5.13.1 The BSDE Approach . . . . . . . . . . . . . . . . . . . . 5.13.2 Markovian Market Models . . . . . . . . . . . . . . . . . 5.13.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-discounted Markets . . . . . . . . . . . . . . . . . . . . . . 5.14.1 The Price for Risk in a Non-discounted Market . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .

6 FrBM-Markets 6.1 Introduction . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . 6.2.1 FrBM Construction . . . . . . . . . . . 6.2.2 Integration . . . . . . . . . . . . . . . . 6.2.3 Fractional Integration - continued . . . 6.2.4 Fractional Itô and the Hitsuda-Skorohod 6.3 An Itô Formula . . . . . . . . . . . . . . . . . . 6.4 GFrBM . . . . . . . . . . . . . . . . . . . . . . 6.5 Linear FrBSDEs . . . . . . . . . . . . . . . . . 6.5.1 Reduction to a PDE . . . . . . . . . . .

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6.5.2 Solvability of the PDE . . . . . . . . . 6.5.3 LFrBSDEs, Solutions . . . . . . . . . . Pricing of Derivatives in FrBSM . . . . . . . 6.6.1 The Model . . . . . . . . . . . . . . . 6.6.2 The Pricing of Derivatives . . . . . . . Change of Measure . . . . . . . . . . . . . . . 6.7.1 Expectation under Change of Measure 6.7.2 A Girsanov Theorem . . . . . . . . . . 6.7.3 Removal of a Drift . . . . . . . . . . . Absence of Arbitrage . . . . . . . . . . . . . . 6.8.1 The Model . . . . . . . . . . . . . . . 6.8.2 Absence of Arbitrage . . . . . . . . . . 6.8.3 Remarks . . . . . . . . . . . . . . . . .

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7 Conclusion

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8 Appendix BSDEs 8.1 General Notation . . . . . . . . . . . . . . . . . . . 8.1.1 Equivalence classes and appropriate spaces 8.2 The Problem . . . . . . . . . . . . . . . . . . . . . 8.2.1 Existence and Uniqueness . . . . . . . . . . 8.2.2 Linear BSDEs . . . . . . . . . . . . . . . . 8.2.3 Markovian BSDEs . . . . . . . . . . . . . . 8.2.4 BSDEs and partial BSDEs . . . . . . . . . 8.2.5 Remarks . . . . . . . . . . . . . . . . . . . . 8.3 Appendix (a, β)-Theory . . . . . . . . . . . . . . . 8.3.1 Formulation of The Main Result . . . . . . 8.3.2 ”‘Simple”’ BSDEs . . . . . . . . . . . . . . 8.3.3 Proof of The Main Result . . . . . . . . . . 8.3.4 Comparison Theorem . . . . . . . . . . . . 8.3.5 One-Dimensional Linear BSDEs . . . . . . 8.3.6 Remarks . . . . . . . . . . . . . . . . . . . .

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9 Bibliographies General Bibliography References on BSDEs . . . . . . . . . . . . . . . References on Neyman-Pearson-hedging . . . . . References on Grossisement de Filtration . . . . References on LQ-Control and MV-Hedging . . . References on the Markovitz Problem . . . . . . References Partially Observed MV-Hedging . . . References Fractional Brownian Motion Markets

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1

Introduction Let us first not talk about King Arthur, then the whole story started towards the end of the nineteenth century, when H. Poincare got some of his ph.d. students to work on dynamic systems which underlie random influences. Of course random dynamic systems had been studied earlier when mathematicians treated different kinds of games, like playing dies, but now questions from real world asked for mathematical models for continuous time random dynamical systems. As before also here random influences just mean that the deterministic influences on a system are so complex that they are not accessible by mathematical treatment. Bachelier studied economical models, Langevin was concerned with problems in physics. Bachelier looked at the problem of the so-called ”rentes” at the Bourse de Paris and Langevin studied a phenomenon which had been observed by Captain Cook’s botanist Brown. Bachelier’s thesis vanished in the cellars of a University library, and Langevin’s work was noticed by Einstein and it was used to develop a theory of what is known as Brownian motion today. It turned out that this mathematical model not only described the motion of particles by the molecular motion of he surrounding material. This model became the base for models in all kinds of sciences (see the famous books by Nelson and Wiener), and so the theory was developed further and further: We should mention Kolmogorov for his basic models in probability theory, then Ito’s and Wiener’s results on Brownian motion as a stochastic process in today’s sense, we must remind the reader of the works at the Seminaire de Probabilites in Strasbourg, the works of Feynman and Kac, up to the famous work of Malliavin. So, there was a well developed theory, called stochastic analysis now, when Bachelier was reborn: After 1950 Markovitz started work on problems related to Bacheliers work, and especially after 1972 Merton, Black and Scholes improved the results to derive solutions to portfolio management problems and the famous option pricing. This famous formula was the first mathematical formula which made it onto the front page of the New York Times when Merton and Scholes were awarded the Nobel Prize (in Economics) in 1997. The problems treated by mathematicians and economists in what is now called mathematical finance are very well described in a very old tale from the King Arthur time: The Peredur ap Efrawg, a knight in the medieval Welsh tales, was confronted a flock of 10000 white and 10000 black sheep. When a black sheep bleaked a white sheep turned black and the other way round. So at time zero the magical transitions start and the dynamical system evolves. Let us assume that the price for one sheep is determined according to the ratio of black and white sheep, and let the starting price at time zero be 1 Euro for both a black or a white sheep. So what would you pay now for the right to receive a white sheep for the strike price at maturity time T, say after 100 transitions. This is a typical question from option pricing. The other problem of how to interfere with the system by taking away certain numbers of black and-or white sheep with the goal to maximize the number of black sheep may be interpreted as a portfolio selection

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problem. By the way optimal selection will give you 19824 black sheep on average. Mathematical finance developed very fast during the last 20 years and one reason for this fast development certainly was the fact that stochastic analysis turned out to be tailor-made for applications in mathematical finance. Meanwhile the early interdisciplinary development of mathematical finance has ended in some sense: The mathematical theory has become a subclass of mathematics and what formerly was considered as a problem in mathematical finance has become so mathematically complex that they are considered to be theoretical mathematical problems. On the other hand, economists working in banks and insurance companies often do not have the mathematical knowledge to apply the advanced techniques from theoretical mathematics. This was the reason why we introduced a new study, Mathematische Finanzökonomie, at the University of Konstanz in 1998, after a first try ten years earlier failed due to the suggestions of a wise man from economics. This book is in the state of being born. So you will find a lot of misprints, even mistakes. Please take this as an invitation to work on our common project, namely Mathematische Finanzökonomie at the University of Konstanz. Konstanz, in January 2002, michael kohlmann On the second β-edition: This version of MMF is a revised and extended version. We replaced a lot of misprints and mistakes by other mistakes and misprints and added many remarks and comments to make the first version more readable. Especially, we added a new chapter on market models basing on fractional Brownian motions. These models turned out to be of interest because of their long range dependence which is observed in reality. I would like to thank several students in the Mathematics department for suggestions and contributions. Unfortunately those students who were intended to be the main group of readers were no great help in improving the script. They were not able to contribute anything to both the theory and the economical background. University of Konstanz. Konstanz, in June 2002, michael kohlmann On the third β-edition: Meanwhile the first students got their diplomas in mathematical finance at my University. These students contributed a lot to these lecture notes and to my understanding of the subject. So here I would like to thank these students as well as my ph.d. students in the Math department. University of Konstanz. Konstanz, in March 2004, michael kohlmann

CONTENTS

3

4

CONTENTS

Chapter 1

The General Market Model 1.1

Preliminaries ( ) ¯ F, (Fs )s≥0 , P , Ω := Ω,

satisfying the usual conditions be given. ∨ s≥0 Fs to equal F∞ := F.

For simplicity we assume F∞− :=

For d ≥ 1 denote by Ad (Ω) the space of adapted Rd -valued processes on Ω and by Sd (Ω) the space of Rd -valued semimartingales on Ω. To be precise, we work with equivalence classes of indistinguishable processes and writing X = Y for two adapted processes is to be interpreted as {X ̸= Y } is an evanescent set. Denote the set of adapted Rd -valued uniformly integrable, resp. local, martingales on Ω by U˜d (Ω), resp. by L˜d (Ω). Denote the set of RCLL Rd -valued uniformly integrable, resp. local, martingales on Ω by Ud (Ω), resp. by Ld (Ω). Since F satisfies the usual conditions we can always find for Z ∈ U˜d (Ω), resp. Z ∈ L˜d (Ω) an RCLL-modification Z˜ ∈ Ud (Ω), resp. Z˜ ∈ Ld (Ω), see [51], Chapter 1. Denote by Nd (Ω) 5

6

CHAPTER 1. THE GENERAL MARKET MODEL

the space of semimartingales which are Rd -valued processes of finite variation and by Pd (Ω), resp. Pdb (Ω) the space of predictable processes, resp. bounded predictable processes. For 1 ≤ p ≤ ∞ set Lp := Lp (P ) := Lp (Ω) and denote the usual norm on by ∥ · ∥Lp , resp. ∥ · ∥ := ∥ · ∥L2 . We equip L∞ with the weak-star topology σ(L∞ , L1 ). L∞ is then a metrizable locally convex linear space. In general locally convex linear spaces are not complete and their topology might not be a norm topology for any norm. However, the Hahn-Banach theorem still holds and this is all we need, see [58]. Furthermore, the dual of Lp is isomorphic as a topological linear space to Lq for all 1 < p ≤ ∞, the dual of L1 is L∞ equipped with the usual norm topology. For 1 < p < ∞, the dual of Lp is norm-isomorphic to Lq and the spaces Lp are reflexive Banach spaces, see [31]. In general L1 and L∞ are not reflexive, therefore we will not be able to solve convex optimization problems in these cases. Lp

For 1 ≤ p ≤ ∞, denote by U˜dq (Ω), resp. Udp (Ω) the elements of Z ∈ U˜d (Ω), resp. Z ∈ Ud (Ω), such that Z∞ ∈ Lp ,

Denote by (p)

Ad (Ω) the space of processes in Ad (Ω) which are uniformly Lp -integrable in the following sense. Define (p) Ad (Ω) (p)

by X ∈ Ad (Ω) iff X ∈ Ad (Ω) and {Xτ |τ stopping time} is bounded in Lp and uniformly integrable in the case p = 1, see [23]. A∗p d (Ω)

1.1. PRELIMINARIES

7

is defined as the space of processes X ∈ Ad (Ω), such that the process Xt∗ := sup0≤s≤t |Xs | is Lp -integrable for all t ≥ 0. We will work with different subspaces of Sd (Ω). For 1 ≤ p ≤ ∞ denote by Sdp (Ω) the space of Lp -integrable Rd -valued semimartingales on Ω, see [28] and [34] for p = ∞. Sdp (Ω) is a subspace of Sd∗p (Ω) := Sd (Ω) ∩ A∗p d (Ω), which is itself (p) (p) a subspace of Sd (Ω) := Sd (Ω) ∩ Ad (Ω). We will often drop the dimension index or the probability space from the notation of the above spaces and write e.g. A, Ad or A(Ω) instead of Ad (Ω). For a stopping time τ and a process X ∈ A, we denote the process X stopped at τ by X τ . For X ∈ S, we define X τ − := X1[0,τ ) + Xτ − 1[τ,∞) ∈ S. Localized classes are denoted by the index loc. As a general reference for unexplained notation we cite [48]. For d ≥ 1 denote the set of Rd -valued predictable processes which are integrable with respect to X ∈ Sd (Ω) by L(X), see [15], [47] and [93], Chapter VII, for a summary on stochastic vector integration. Denoting by Lvar (A) the space of predictable Rd -valued processes which are Riemann-Stieltjes integrable with respect to A ∈ Nd (Ω) and by L1loc (M ) the space of predictable Rd -valued processes which are integrable with respect to M ∈ Ld , L(X) is the space of predictable Rd -valued processes H such that there exists a (not necessarily unique) semimartingale decomposition X = X0 + A + M, where A ∈ Nd (Ω) with A0 = 0, M ∈ Ld with M0 = 0 and H ∈ Lvar (A) ∩ L1loc (M ). For H ∈ L(X) denote the stochastic integral of H with respect to X by H · X, ∫t where (H · X)t := 0 Hs dXs for all 0 ≤ t < ∞. We will often restrict the time parameter to stochastic intervals. For stopping times τ, τ˜, we therefore introduce the following filtered probability spaces (again satisfying the usual conditions), ( ) ¯ F, (τ Fs )s≥0 , P Ω, τ Ω := ( ) ¯ Fτ˜ , (F τ˜ )s≥0 , P|F Ωτ˜ := Ω, (1.1.1) s τ ˜ ( ) τ˜− τ ˜ − ¯ Fτ˜− , (F )s≥0 , P|F Ω := Ω, s τ ˜− τΩ

τ˜

:= (τ Ω)τ˜ ,

8


τ˜ , for all where τ Fs := Fτ ∨s , Fsτ˜ := Fs∧˜τ and Fsτ˜− := F¯s+ for F¯s := Fs− 0 ≤ s < ∞. If e.g. X ∈ Ad (Ω), then τ X ∈ Ad (τ Ω), where for 0 ≤ s < ∞, τ˜ τ Xs := Xτ ∨s 1{τ 0, we define the restriction of Ω to A as ( ) P|F∩A Ω|A := A, F ∩ A, (Fs ∩ A)s≥0 , , (1.1.2) P (A)

and τ Ω|B := (τ Ω)|B for B ∈ Fτ with P (B) > 0. ¯ →R For an F∞ -measurable random variable X, the restriction of X : Ω to A is the F∞ ∩ A-measurable random variable X|A : A → R. If not causing ambiguities we will often write X instead of X|A . The stochastic exponential of a semimartingale X is denoted as E(X) and we set τE(X) := E(τ X), E τ˜(X) := E(X)τ˜ and τE τ˜ (X) := E(τ X)τ˜ . Note that (τ E(X))τ˜ = E(X)τ τE τ˜(X). For X ∈ A, writing X∞ we implicitly assume limt→∞ Xt to exist almost surely. For Z ∈ U, limt→∞ Zt exists a.s. and is denoted as Z∞ . We will identify subsets D of U with the spaces of their terminal values {Z∞ |Z ∈ D} ⊆ Lq if Z∞ ∈ Lq , 1 ≤ q ≤ ∞ for all Z ∈ D.

1.2

Simple Self-Financing Hedging Strategies

Let S ∈ Ad (Ω), d ≥ 1 be given. M := (Ω, S) is a model for a market, where S describes the price process of d assets. We will often consider such a market on a stochastic interval. For stopping times τ, τ˜ and A ∈ F0 , resp. B ∈ Fτ , we define ( ) M|A := Ω|A , S|A , τM τ˜

M

τM

τ˜

τ M|B

:= (τ M)|B

:= (τ Ω, τ S) , ( ) := Ωτ˜ , S τ˜ ,

(1.2.1)

:= (τ M) , τ˜

,

˜ Mτ|A := (Mτ˜ )|A .

We want to model the economic activity of investing money into a portfolio of assets and changing the number of assets held over time according to a certain hedging strategy. This is achieved with the following definition: Definition 1.2.1. A general hedging strategy in M is a predictable Rd -valued process. For H ∈ Pd , the corresponding value process ∑ V H ∈ A is defined as V H := HS. A simple hedging strategy is a process H = ni=0 Ki , n ∈ N, where K0 = k0 1[0] for an Rd -valued F0 -measurable random variable k0 , Ki , 1 ≤ i ≤ n are processes of the form Ki = ki 1]τi−1 ,τi ] , where 0 =: τ0 ≤ τi−1 ≤ τi ≤ τn := ∞ are stopping times for 1 ≤ i ≤ n, and ki 1{τi−1 0.

10

1.2.2


Arbitrage

There exist a large number of different notions of arbitrage. We will define here only the most basic type of arbitrage. Definition 1.2.5. Let G ⊆ P. We call an H ∈ G a G-arbitrage, if V0H = 0, H ≥ 0 and P (V H > 0) > 0. If there exists no G-arbitrage, then G is called V∞ ∞ arbitrage-free. In probabilistic theories of financial markets allowing to trade at an infinitely large number of instances of time, in general, one has to exclude certain selffinancing hedging strategies, e.g. doubling strategies, in order to avoid arbitrage opportunities, see [30], [50]. We will therefore introduce in the following several spaces of self-financing hedging strategies and give conditions ensuring these spaces to be arbitrage-free.

1.2.3

Arbitrage, an exercise in Binary Markets

In order to make the above definitions a bit more transparent we will shortly look at a kind of most simple market which were recently studied by Sottinen (FBM, Random Walks and Binary Market Models, preprint to appear in FINASTO) and Bender (Binary Market Models aand discrete Wick products, preprint UK to appear in FINASTO): For n ∈ N let AN be the power set of ΩN = {−1, 1}N and let PN be the discrete uniform probability, so that ξi (ω) = ωi is a family of independent binary variables with PN (ξi = 1) = PN (ξi = −1). We consider a pair of L2 (ΩN , PN )-random vectors (B, S) with Bn = 1 and Sn Fn = σ(ξ1 , · · · , ξn )-measurable. Especially choose Sn = Sn−1 (1 + µn + Xn )), S0 = s > 0. The function µ = µ1 , · · · , µN is interpreted as the drift and the volatility is given by Xn which will be assumed to be given by Xn =

n ∑

x(n, i)ξi

1

with real numbers x(n, i). Note that in short X may be written using the triangular arrays considered in Stochastik I:    ξ11 x11 0 0 0 x21 x22 0 0 ξ21    X= x31 x32 x33 0 ξ31  · · · · · Assume Xn + µn > −1

1.3. SIMPLE LP -HEDGING STRATEGIES

11

to make the stock price positive. A portfolio then is a pair of L2 (ΩN , AN )random vectors π = (F, G), where F and G are Fn−1 -measurable. So the value of the portfolio is given by Vn (π) = Fn + Gn Sn and it is self-financing if Vn (π) = Vn−1 (π) + Gn (Sn − Sn−1 ), 1 ≤ n ≤ N. From above we repeat the notion of arbitrage: Definition 1.2.6. π is said to be an arbitrage opportunity if there is a time step n such that V0 (π) = 0, Vn (π)(ω) ≥ 0 for all ω ∈ Ω and Vn (π)(η) > 0 for at least one η ∈ Ω. Try to prove the following Theorem (Sottinen, Bender): Theorem 1.2.7. If for some n n−1 ∑

|x(n, i)| > |x(n, n)|

1

then there is an arbitrage opportunity in the class of self-financing one-step buy-and-hold strategies. A strategy π = (F, G) is called a one-step buy-and-hold strategy if there is an n such that Gk = 0 for all n ̸= k. The reason for this phenomenon in this simple model seems to be that the random input into the market depends on the whole past and not only on the present. These examples were studied in the context of fractional Brownian motion markets where this dependence on the past was proposed to better describe the reality. One then tried to reinterpret the model in a different way by introducing the so-called Wick-product ♢ and replaced the above model by Sn = Sn−1 ♢(1 + µn + Xn )), S0 = s > 0. This leads to the result that the drift of Sn only depends on µ which is different in the first model (e.g. compute E(S2 )). But the Wick product formulation leads to extra terms which are hard to interpret in economical terms (Bender, ph.d. thesis, to appear).

1.3

Simple Lp -Hedging Strategies

We a discounted market M := (Ω, (S, 1)), where from now on S ∈ ( consider ) (p) Ad , 1 ≤ d, 1 ≤ p ≤ ∞. loc

We will now introduce several spaces of simple self-financing hedging strategies, following closely the ideas of [23], but we consider the more general situation, where S is not assumed to be a semimartingale. Denote the conjugate

12


exponent of p by q such that p1 + 1q = 1. Denote by Hdp (M) the set of Rd -valued processes which are a linear combination of processes of the form K = k1]τ1 ,τ2 ] , where τ1 ≤ τ2 < ∞ are stopping times such that S τ2 ∈ A(p) , and k1{τ1 0} and its Lp closure Daq := {Z ∈ Dq |Z∞ ≥ 0} play an important role in the theory of mathematical finance. The fundamental theorem of asset pricing states in its most general version that Deq ̸= ∅ is equivalent to the NFLVR-no-arbitrage condition, see [21], [22], [26]. See also [75] for a systematic application of equivalent local martingale methods to the theory of pricing of financial derivatives. As we will see in the next section, optimization problems and duality properties lead quite naturally to the space of signed local martingale measures. In the following chapters we will see, that Lp -convergence properties of self-financing hedging strategies are related to the properties of the Lq -norm optimal signed local martingale measures. The notion of a signed local martingale measures goes back to [74]. We therefore generalize the approach in [23], where Deq and Daq were considered, and will work with Dq instead.

14


Lemma 1.4.2. V p is stable under stopping. For V ∈ V p , V Z ∈ U˜ for all Z ∈ Vp◦ . Proof. For H = k1]τ1 ,τ2 ] ∈ Hp and a stopping time τ we have (H · S)τ = ˜ · S, where H ˜ := k1]τ ∧τ,τ ∧τ ] . Since k1{τ ∧τ K q . Thus for K → ∞, sup

P (|ab| > K) ≤ sup P (|a|p > K) + sup P (|b|q > K) → 0, a∈A

a∈A,b∈B

b∈B

since A is bounded in Lp and B p is bounded in L1 . Hence for K → ∞, [ ] sup E |ab|1{|ab|>K} ≤ sup ∥a∥Lp sup ∥b1{|ab|>K} ∥Lq → 0, a∈A

a∈A,b∈B

a∈A,b∈B

by the uniform integrability of B q .

Proposition 1.4.4. For S ∈ Aloc , we have Dq = Vp◦ . (p)

Proof. Let Z ∈ Dq . We want to show that V Z ∈ U˜ for all V ∈ V p which ˜ we have to implies E[V∞ Z∞ ] = E[V0 Z0 ] = 0, hence Z ∈ Vp◦ . Since V Z ∈ L, show that V Z is uniformly integrable. For this, it suffices to show that for every stopping time τ˜ such that S τ˜ ∈ A(p) , we have E[k(Sτ2 − Sτ1 )Z∞ ] = 0 for all k1]τ1 ,τ2 ] ∈ Hp with τ2 ≤ τ˜. Since S τ˜ Z ∈ L˜ we only have to show uniform integrability of the family {Sττ˜ Zτ |τ stopping time }. This follows from Lemma 1.4.3 applied to A := {Sττ˜ |τ stopping time } and B := {Zτ |τ stopping time }. Conversely, for Z ∈ V ◦ , Lemma 1.4.2 implies S τn Z−S0 Z0 = (1]0,τ ] ·S)Z ∈ U˜ p

for a localizing sequence of stopping times (τn )n∈N for S ∈ Hp . Hence Z ∈ Dq .

n

(p) Aloc

since 1]0,τn ] ∈

1.4. SIGNED LOCAL MARTINGALE MEASURES

15

Corollary 1.4.5. For 1 ≤ p ≤ ∞, Dq ̸= ∅ if and only if 1 ̸∈ V¯ p . If Dq ̸= ∅, then V¯ p = (Dq )◦ . Proof. This follows from Proposition 1.4.4 and Lemma 1.3.4, 4. and 5. In the next chapter we will need the following results. Lemma 1.4.6. For V ∈ v + V¯ p , V n ∈ vn + V p , with F0 -measurable v, vn ∈ Lp , n → V , and Z ∈ D q , we have for any stopping time τ , V n Z → such that V∞ τ τ E[V Z|Fτ ] in L1 . In particular, E[V Z|F0 ] = vZ0 and on {Z0 ̸= 0}, vn → v in probability. Proof. We have for n → ∞,

] [ n − V )Z∞ |Fτ ] E[|Vτn Zτ − E[V Z∞ |Fτ ]|] = E E[(V∞ ]] [ [ n − V )Z∞ |Fτ ≤ E E (V∞ n = E [|V∞ − V ||Z∞ |] → 0.

Lemma 1.4.7. Let λ be a bounded F0 -measurable random variable and V, W ∈ V¯ p . Then λV + (1 − λ)W ∈ V¯ p . n → V , resp. W n → W in Lp , Proof. For sequences V n , W n ∈ V p such that V∞ ∞ n n n p n we have X := λV + (1 − λ)W ∈ V and X → λV + (1 − λ)W in Lp .

¯ = ∪N Ai be an F0 -measurable partition. If for all 1 ≤ Lemma 1.4.8. Let Ω i=0 ∑N ¯p i ≤ N with P (Ai ) > 0, Vi ∈ V¯ p (M|Ai ), then V := i=0 Vi 1Ai ∈ V (M). p p Conversely for V ∈ V¯ (M) we have 1A V ∈ V¯ (M|A ) for all A ∈ F0 with P (A) > 0. Proof. For 1 ≤ i ≤ N such that P (Ai ) > 0, let Vin ∈ V p (M|Ai ) be a sequence ∑ n converging to Vi in Lp (Ω|Ai ). For P (Ai ) = 0 set Vin = 0. Then N i=0 Vi 1Ai ∈ p p p ¯ V (M) converges to V in L . Conversely, for V ∈ V (M) there exists a sequence V n ∈ V p (M) converging to V in Lp . For A ∈ F0 with P (A) > 0, we have 1A V n ∈ V p (M|A ) and 1A V n → 1A V in Lp (Ω|A ).

1.4.1

The Law of One Price

We still consider the discounted market M = (Ω, (S, 1)) and assume S ∈ (p) Aloc , 1 ≤ p ≤ ∞. Definition 1.4.9. 1. We will say that the law of one price, (see [77]), holds in M for simple Lp -integrable self-financing hedging strategies, abbrevip ated as LPp , if V ∈ vi + V∞ for constants vi , i = 1, 2, implies v1 = v2 . 2. We say that the strong law of one price, holds in M, for Lp -integrable strategies, (SLPp ), if the following property is satisfied: If V ∈ v + V¯ p for a constant v and there exists a sequence Vn ∈ vn + V¯ p for constants vn such that Vn → V , then vn → v.

16


The law of one price for simple Lp -integrable self-financing hedging strategies is a kind of minimum requirement for a market to be useful. If it does H i ∈ v + V p , i = 1, 2 for constants v > v and simple not hold, and say V∞ ∞ i 1 2 self-financing hedging strategies H i , then an investor could sell the hedging strategy H = H 1 − H 2 , earning v1 − v2 > 0 at time t = 0, without bearing any H = 0 almost surely. The strong law of one price is a stronger conrisk, since V∞ dition, ensuring that an instantaneous risk-free gain can not be approximated with respect to Lp -norm. p Proposition 1.4.10. The LPp holds in M iff 1 ̸∈ V∞ and the following assertions are equivalent:

1. The SLPp holds in M. 2. 1 ̸∈ V¯ p . 3. Dp ̸= ∅. p p Proof. If 1 ∈ V∞ , then the LPp does not hold, since 1 ∈ 1+V∞ too. Conversely, p if the LPp does not hold, then there exists a V ∈ vi + V∞ , i = 1, 2 for constants −v2 ) p v1 ̸= v2 and we have 1 = (V −vv12)−(V ∈ V∞ . If 1 ∈ V¯ p , then the SLPp does −v1 p p not hold, since there exists a sequence Vn ∈ V∞ converging to 1 ∈ 1 + V∞ . p ¯ Conversely, if the SLPp does not hold, there exists a sequence Vn ∈ vn + V for constants vn such that Vn → V ∈ v + V¯ p and vn ̸→ v. There are two cases, first, if the sequence vn has an unbounded subsequence, we can choose a subsequence Vn vnj , such that 0 ̸= |vnj | → ∞. Then 1 − vn j ∈ V¯ p converges to 1. (For p = ∞, j

this follows from the properties of the locally convex topology σ(L∞ , L1 ), which imply for Xn , X ∈ L∞ that Xn → X in L∞ iff l(Xn ) → l(X), ∀ l ∈ L1 ). If the sequence vn is bounded, then there exists a converging subsequence Vnj such (v−V )−(vnj −Vnj ) that vnj → w ̸= v. Then ∈ V¯ p converges to 1. The remaining v−w equivalence was proved in Corollary 1.4.5. The importance of the strong law of one price will become clear in the following sections. It will turn out, that the solution to a certain class of expected terminal utility maximization problems exists uniquely, in the sense that it can be approximated by simple self-financing hedging strategies in Lp norm, if and only if Dq ̸= ∅. For a wide class of problems the conditions Deq ̸= ∅ or Daq ̸= ∅ are not necessary. The SLPp is compatible with restrictions: Corollary 1.4.11. For B ∈ F0 with P (B) > 0, we have equivalence between, 1. The SLPp (M|B ) holds, 2. Dq (M|B ) = ∅ and 3. 1B ∈ V¯ p . Proof. By Proposition 1.4.10, we have for P (B) > 0, Dq (M|B ) = ∅ iff 1B ∈ V¯ p (M|B ), which is by Lemma 1.4.8 equivalent to 1B ∈ V¯ p .

1.5. TYPICAL PROBLEMS

1.5

17

Typical Problems

Up to here we have described on a filtered probability space (Ω, F, Ft , P )t≤T (here T may be finite or infinite) a simple market M together with some rules SF how to use this market and we mentioned already some desirable properties of the market. This is -let us say: the toolbox- and before we go into the details of working with these tools we will shortly describe some basic problems which we will treat in this book. For this rough description we refrain from giving assumptions and we simplify the notation to make this overview more easily readable: So we are given a market M simply given by a stochastic process S and a set of trading strategies H ∈ SF v , the wealth VtH is the sum of the initial wealth v and the gains from using the strategy GH t : H VtH = V0H + GH t = v + Gt .

We have already some idea about what kind of reasonable properties we have to assume on the market, but this should not worry at the moment. The first class of problems is called the class of • goal problems which simply means that we want to use the market tools to reach a certain pre-given goal. More specifically we are pre-given an FT -measurable random variable ξ. T may here be finite or infinite. The set of reachable goals was defined as VT (SF 0 ) = VT (in abbreviated notation, as we do not go into details here), the set of terminal values of the value processes with respect to H ∈ SF 0 . So we may formulate the goal problem in a very general way as the problem of finding out whether – ∃v ξ ∈ v + VT or – ∀v ξ ∈ / v + VT . If the first case holds for a class Ξ of pre-given goals the market will be called complete, if the second case is true for at least one ξ ∈ Ξ it is called incomplete. So from here on whenever we consider a market (M, SF) we have to consider the two cases (in general) separately. Let us first consider a complete market: • The next problem then consists in finding a strategy H which leads us from the initial wealth v to ξ in the sense that VTH = ξ. The solution of this problem depends on the initial v and so the next question is

18

CHAPTER 1. THE GENERAL MARKET MODEL • to determine the set of initial wealths for which this is possible in order to • find the smallest such initial wealth v0 which is known as the (pricing problem), • and to give a computational rule how to determine v0 and the hedging strategy.

In an incomplete market the last questions are obsolete. As it will not be possible to perfectly reach the goal we try to get as near as possible. This leads in incomplete markets to • the class of mean-variance hedging problems which means that we try to reach ξ in such a way that the distance between VTH and ξ becomes small: E[(VTH − ξ)2 ] = min !. And again here we have the following problems: ∗

– find the strategy H ∗ which leads us to ξ in the sense that E[(VTH − ξ)2 ] = min !, – to determine the set of initial wealths for which this is possible in order to – find the smallest such initial wealth (approximate pricing problem). • The mean-variance hedging has the great disadvantage that both situations of missing the goal ∗

– (VTH − ξ) ≥ 0 and ∗

– (VTH − ξ) ≤ 0 are treated in the same way. So we will try to minimize the probability of ∗ a default P ((VTH − ξ) < 0) = min !. This is what we call the NeymanPearson hedging. • A different market property which leads to incompleteness is a lack of information for the investor. Let ξ be measurable w.r.t. a G ⊃ FT . It is clear that in general this ξ cannot be reached. Also if the investor has less information than Ft or if seller and buyer of a financial derivative have different information about the market, this will lead to incompleteness and the incomplete markets have to be considered under special restrictions on the portfolios, the wealth etc.. Another important class of problems is concerned with the maximization of the utility of a terminal wealth corresponding to a certain strategy: Let U be a utility function and consider the problem E(U (VTH )) = sup !H∈SF . Again we have the related problems, namely

1.6. UTILITY MAXIMIZATION

19

• to find out whether the supremum exists, • whether there is an admissible strategy H ∗ such that ∗

E(U (VTH )) = maxH∈SF , • and to characterize this optimal H ∗ . This will be extensively studied in the sections on capital asset pricing models CAPM. This includes the Merton portfolio selection problem. • The above problems ignore the risk inherent in the investment strategy. So a related problem will be to maximize a utility and to minimize the inherent risk. Of course this cannot be done at the same time, instead we get a relation between the utility and the risk, called the efficient frontier, and the investor has to choose his strategy according to his preferences. In all these problems we have ignored how the interest rate behaves. So a related problem will be studied to show how interest rates may be modeled by term structure models. This will be described in Chapter 4. Not all these problems can be solved in the generality of the last sections. That is why from time to time we make special assumptions on the market to achieve results with reasonable efforts. The simplest model to be considered in the following will be the standard B-S-market where we assume that the wealth from the investment in one bond with interest rate r and one stock with appreciation rate b and volatility σ is given by dxt = (r(t)xt )dt + (b(t) − r(t))π(t)dt + π(t)σt dwt ,

(1.5.1)

for non-random coefficients. π here is the investment strategy. And from this basic model we will consider markets with increasing complexity up to the case where the evolution of the market is described by a local semimartingale.

1.6

Utility Maximization

In this section we formulate the utility maximization problem which we are interested in, as an optimization problem on the reflexive Banach spaces Lp , 1 < p < ∞. Before we go into the mathematical problems some remarks on utility from the economist’s point of view -as far as I understand these explanations which I got from a colleague years ago and whose name I do not remember. So the following is reported with many thanks to an anonymous colleague with the warning to economists that the presentation might not be completely correct. (I always consider a utility function to be a function for which the Merton problem as a control problem is solvable.)

20

1.6.1


Theories of utility

The analysis of price and value may be divided into two parts: the supply part and the demand part. If cost can be said to build the basis of the supply relationship that determines price, the demand side must be taken to reflect consumer tastes and preferences. ”Utility” is a concept that has been used to describe these subjective tastes. In mathematical terms it assigns a certain utility U (x) to each wealth x. Obviously, the cost-of-production analysis of value must take care of the fact that cost itself depends on the quantity produced. The cost analysis, moreover, applies only to commodities the production of which can be expanded and contracted. The price of a first-folio Goethe has no relation to cost of production; it must depend in some sense on its utility to purchasers as it affects their bids. Marginal utility The classical economists suggest that this leads to a paradox as that utility cannot explain the relative price of one kilogram of diamonds and one kilogram of bread, because the latter is for many consumers essential to life, and hence its utility must surely be greater than that of diamonds. Yet the price of bread is far lower than that of diamonds. The theory of marginal utility that began to grow towards the end of the 19th century supplied the key to the paradox and provided the basis for today’s analysis of demand. Marginal utility was defined as the value to the consumer of an additional unit of some commodity. If, for example, the consumer is offered a choice between 19 and 20 slices of bread for his family, marginal utility measures how much more valuable 20 slices are than 19. It is clear that the magnitude of the marginal utility varies with the magnitude of, say, the smaller of the alternatives. That is, for a family of four, the difference between seven and eight slices of bread per day can be substantial, if the family will still be hungry in either case. But the difference in value between 30 and 31 slices may be negligible. If 30 slices offer enough for everyone to fill his stomach, a 31nd slice may be worth very little. Moreover, the difference in value between 1220 and 1221 slices may be negative–a 1221st slice may just add to the family’s garbage problem. These observations lead directly to the reasonable notion that marginal utility in some sense diminishes with the base from which one starts the calculation. With only seven or eight slices the marginal utility (incremental value) of an eighth slice is high. With 30 or 31 slices it is lower, and so on. The less rare a commodity, the lower is its marginal utility, because its owner in any case will have enough to satisfy his most urgent uses for it, and an increment in his holdings will only permit him to satisfy, in addition, desires of lower priority. In mathematical terms the marginal utility is therefore defined as △U (x) . △x The consumer will be motivated to adjust his purchases so that the price of each and every good will be approximately equal to its marginal utility (that


21

is, to the amount of money he is willing to pay for an additional unit). If the price of an item is P dollars, for example, and the consumer is considering buying, say, 10 units, at which point the marginal utility of the good to him is M (M > P ), the consumer will be better off if he purchases 11 rather than 10 units, since the additional unit costs him P dollars. He will keep revising his purchase plans upward until he reaches the point where the marginal utility of the item falls to P dollars. In sum, the consumer’s self-interest will lead him (without conscious calculation) to purchase an amount such that the marginal utility is as close as possible to market price. As long as the consumer selects a bundle of purchases that gives him the most benefit for his money, he must end up with quantities such that the marginal utility of each commodity in the bundle is approximately equal to its price. It now becomes easy to explain the paradox underlying the relationship between the prices of diamonds and bread. Because a piece of diamond is a rare object, its marginal utility is high, and consumers are willing to pay comparatively high prices for it. The explanation is perfectly consistent with a utility analysis of demand, so long as one relates price to the marginal utility of the item rather than to its total utility. A family’s bread may be very valuable to it, but, if it has enough, the marginal utility of the bread will be small, and this will be reflected in its low price. The relationship between price and marginal utility is important not because it explains issues like the diamond-bread paradox but because it enables one to analyze the relationship between prices and quantities demanded. It also, as a practical matter, permits one to judge how well any portion of the price mechanism is working as a device to secure the efficient satisfaction of the desires of the public, within the limits set by available resources. The conclusion that at any price the consumer will purchase the quantity at which marginal utility is equal to price makes it possible to draw a demand curve showing–to a reasonable degree of approximation–how the amount demanded will vary with price. A curve based on the previous example of bread consumption is given in Figure 1 This shows that if the family gets 10 slices per day the marginal utility of bread will be nine cents (point A). One may reverse the question and ask how much the family would purchase at any particular price, say three cents. The graph indicates that at this price the quantity would be 30 slices, because only at that quantity is marginal utility equal to the three-cent price (point B). Thus the curve in Figure 1 , to a reasonable degree of approximation, may be able to do double duty: it may serve as a marginal-utility curve relating marginal utility to quantity and, at the same time, as a demand curve relating quantity demanded to price. Utility measurement and ordinal utility As originally conceived, utility was taken to be a subjective measure of strength of feeling. An item that might be described as worth ”40 utils” was to be interpreted to yield ”twice as much pleasure” as one valued at 20 utils. It was not long before the usefulness of this concept was questioned. It was criticized

22


for its subjectivity and the difficulty (if not impossibility) of quantifying it. An alternative line of analysis developed that was able to accomplish most of the same purposes but without as many assumptions. First introduced by the economists F.Y. Edgeworth in England (1881) and Vilfredo Pareto in Italy (1896-97), it was brought to realization by Eugen Slutsky in Russia (1915) and J.R. Hicks and R.D.G. Allen in Great Britain (1934). The idea was that to analyze consumer choice between, say, two bundles of commodities, A and B, given their costs, one need know only that one is preferred to another. This may at first seem a trivial observation, but it is not as simple as it sounds. In the following discussion, it is assumed for simplicity that there are only two commodities in the world. (see Figure 2) Figure 2 is a graph in which the axes measure the quantities of two commodities, X and Y. Thus, point A represents a bundle composed of seven units of commodity X and five units of commodity Y. The assumption is made that the consumer prefers to own more of either or both commodities. That means he must prefer bundle C to bundle A, because C lies directly to the right of A and hence contains more of X and no less of Y. Similarly, B must be preferred to A. But one cannot say, in general, whether A is preferred to D or vice versa, since one offers more of X and the other more of Y. The consumer may in fact not care whether he receives A or D–that is, he may be indifferent(see Figure 3). Assuming that there is some continuity in his preferences, there will be a locus connecting A and D, any point on which (E or A or D) represents bundles of commodities of equal interest to this consumer. This locus (the line I-I’ in Figure 3 ) is called an indifference curve. It represents the consumer’s subjective trade off between the two commodities–how much more of one he will have to get to make up for the loss of a given amount of another. That is, one may treat the choice between bundle D and bundle E as involving the comparison of the gain of quantity FD of X with the loss of FE of Y. If the consumer is indifferent between D and E, the gain and loss just offset one another; hence, they indicate the proportion in which he is willing to exchange the two commodities. In mathematical terms, FE divided by FD represents the average slope of the indifference curve over arc ED; it is called the marginal rate of substitution between X and Y. Figure 3 also contains other indifference curves, some representing combinations preferred to A (curves lying above and to the right of A) and some representing combinations to which A is preferred. These are like contour lines on a map, each such line being a locus of combinations that the consumer considers equally desirable. Conceptually, through every point in the diagram there is an indifference curve. Figure 3 , with its family of indifference curves, is called an indifference map. This map obviously does no more than rank the available possibilities; it indicates whether one point is preferred to another but not by how much it is preferred. It is easy to show that at any point such as E the slope of the indifference curve, roughly FE divided by ED, equals the ratio of the marginal utility of X to the marginal utility of Y for the corresponding quantities. For in moving from E to D the consumer gives up FE of Y, a loss valued, by definition,


23

at approximately FE multiplied by the marginal utility of Y, and he gains FD of X, a gain worth FD multiplied by the marginal utility of X. Relative marginal utilities can be measured in this way because their ratio does not measure subjective quantities–rather, it represents a rate of exchange of two commodities. The marginal utility of X measured in money terms tells one how much of the commodity used as money the consumer is willing to give for more of the commodity X but not what psychic pleasure the consumer gains. A very important measure in connection with the marginal utility is the elasticity E of a utility function U . This elasticity relates the average utility U (x) U ′ (x) ′ x to the marginal utility U (x): E = U (x) . The interpretation is obvious. x

24



1.6.2

25

Utility Functions

For the moment we are interested in a special class of utility functions, which we define in this subsection. The class will be designed in such a way that we can uniquely solve the utility maximization problem considered in Subsection 1.6.4. Define C as the set of R-valued strictly convex functions defined on R. c ∈ C is continuous and has increasing right and left derivatives c′− ≤ c′+ and the inequality c(x) ≥ c(y) + k(x − y), ∀ k ∈ [c′− (y), c′+ (y)] (1.6.1) holds for all x, y ∈ R. For x ̸= y, (1.6.1) becomes a proper inequality, see [83], Section 10, Section 24 and Section 25. For c ∈ C set c(±∞) := limx→±∞ c(x) ∈ ¯ resp. c′ (±∞) := limx→±∞ c′ (x) = limx→±∞ c′ (x) =: c′ (±∞) ∈ R, ¯ and R, + + − − ′ ′ cˆ(x) := c(−x). Note that cˆ ∈ C and cˆ+ (x) = −c− (−x). For a reason to become clear later (see the proof of Proposition 1.6.11), we are interested in the following subset of C C0 := {c ∈ C|c(0) = 0, c′− (0) ≤ 0 ≤ c′+ (0)}.

(1.6.2)

Remark 1.6.1. For c ∈ C0 , the restriction of c, resp. cˆ, to [0, ∞) are so-called Young functions, see e.g. [28], Chapter VI, No. 97. For c ∈ C0 set c′± (x) := c′+ (x), resp. c′± (x) := c′− (x), for x ∈ (0, ∞), resp. x ∈ (−∞, 0) and c′± (0) = 0. In order to be able to solve the type of optimization problem we are interested in, we will later in this subsection define for 1 < p < ∞ subsets C p ⊆ C0 of convex functions, satisfying sufficient conditions guaranteeing the unique solvability of the corresponding optimization problem over closed convex subsets of Lp , 1 < p < ∞. We need some technical preparations. We first collect some easy facts about convex functions which can be found in [83]. Lemma 1.6.2. Let c ∈ C. The following properties hold 1. c(∞) = ∞ if and only if c′+ (∞) > 0. 2. c(∞) = −∞ implies c′+ < 0 and −∞ < c′+ (∞) ≤ 0. 3. c(∞) ∈ R implies c′+ < 0 and c′+ (∞) = 0. 4. c′+ (∞) < 0 implies c(∞) = −∞. 5. If c′+ (x) ≥ 0 for x ∈ R, then limy→∞ c(y) = ∞. 6. If c′− (x) ≤ 0 for x ∈ R, then limy→−∞ c(y) = ∞.

26

CHAPTER 1. THE GENERAL MARKET MODEL 7. supy∈R c(y) = ∞.

Lemma 1.6.3. Let c ∈ C0 . The following properties hold 1. c is strictly increasing on [0, ∞), resp. strictly decreasing on (−∞, 0]. 2. c(∞) = ∞ and c′+ (∞) > 0, resp. c(−∞) = ∞ and c′+ (−∞) < 0. Let L be a reflexive Banach space, denote its dual by L∗ , and let F : L → R be a convex functional. If F is bounded from above locally at some point in L, then F is known to be locally Lipschitz continuous, see e.g. [18], Proposition 2.4.1. In particular, F is then weakly sequentially lower semicontinuous, see e.g. [32], Corollary I.2.2. If F is in addition coercive, i.e. F (V ) → ∞ for ∥V ∥ → ∞, then the minimization problem over a closed convex set W ̸= ∅, F (W min ) = min F (W ), W ∈W

(1.6.3)

for W min ∈ W, has a solution, which is unique if F is strictly convex, see [32], Proposition II.1.2. The study of this type of optimization problem is facilitated by considering a certain dual problem in L∗ and relating the solution of both problems to each other. For locally Lipschitz continuous F , the directional derivatives F ′ (z, v) at z ∈ L in direction v ∈ L and the generalized gradient ∂F (z) ⊆ L∗ exist. ∂F (z) is a non-empty, convex weak-star compact set. We have ∂F (z) = {ζ ∈ L∗ | F (z + y) − F (z) ≥ ⟨ζ, y⟩ ∀ y ∈ L} = {ζ ∈ L∗ | F ′ (z, y) ≥ ⟨ζ, y⟩ ∀ y ∈ L}, see [18], Proposition 2.1.5 and Proposition 2.4.3. If W = x + V¯ for a closed subspace V¯ ⊆ L, we have F (W min + λy) − F (W min ) ≥ λ⟨ζ, y⟩ for all ζ ∈ ∂F (W min ) and all y ∈ L. Since for v ∈ V¯ and λ ∈ R, by optimality F (W min + λv) − F (W min ) ≥ 0, this implies ⟨ζ, v⟩ = 0 ¯ Intuitively speaking, at W min , the gradient for all ζ ∈ ∂F (W min ) and v ∈ V. min of F in W vanishes on the tangent space V¯ of W in W min . Assume now F ≥ F (0) = 0. If ⟨ζ, x⟩ = 0 for all ζ ∈ ∂F (W min ), then F ′ (W min , v + λx) ≥ ⟨ζ, v+λx⟩ = 0 for all v ∈ V¯ and λ ∈ R, hence W min is a local minimum and thus ¯ x} implying x ∈ V. ¯ Hence, if x ̸∈ V, ¯ a global minimum of F restricted to span{V, ◦ min ¯ then V ∩ ∂F (W ) ̸= ∅, where ◦ denotes the annulator with respect to x. In particular, if F is Fréchet differentiable, (or weaker Hadamard differentiable), then ∂F (W min ) is a singleton, see [18], Proposition 2.3.1 and Problem 2.9.1. In this case, F determines a unique element in V¯ ◦ . For L = Lp (P ), x = 1, and V¯ = V¯ p , we have thus determined a signed local martingale measure. This is basically the idea behind stochastic duality theory, which goes back to [11], [12]. See also [52], [44], [19] and [50]. Economists express this idea in the following way. (Signed) local martingale measures can be interpreted as state price densities, since V0 = E[V∞ Z∞ ] for all V ∈ V0 + V p with deterministic V0 and all Z ∈ Dq . Measuring terminal


27

expected utility of a portfolio V ∈ 1 + V p by −F (V∞ ), ∂F (V ) is called the marginal utility. For the optimal W min ∈ 1 + V¯ p , the marginal utility defines a ∂F (W min ) . state price density E[∂F (W min )] We now define a class of convex functionals, for which an element in V¯ ◦ can explicitly be constructed from W min , as we will see in Subsection 1.6.4. Furthermore, it will turn out that this element solves a corresponding dual optimization problem. Definition 1.6.4. Let 1 < p < ∞. The space C p is defined as the set of functions c ∈ C0 satisfying the following conditions: 1. c(Lp ) ⊆ L1 . 2. c′± (Lp ) ⊆ Lq . 3. The convex functional defined by F (V ) := E[c(V )], ∀ V ∈ Lp ,

(1.6.4)

is coercive. Proposition 1.6.5. For c ∈ C p and F defined as in (1.6.4), the minimization problem (1.6.3) as a unique solution. Proof. By [32], Proposition II.1.2, we only have to show that F is sequentially lower semicontinuous. Since ⟨c′± (V ), ·⟩ ∈ (Lp )∗ ∼ = Lq , we find for Vn ⇀ V , F (Vn ) = E[c(V + Vn − V )] ≥ E[c(V ) + c′± (V )(Vn − V )] ≥ E[c(V )] + E[c′± (V )(Vn − V )] → F (V ),

hence F (V ) ≤ lim inf n→∞ F (Vn ) holds. Note also that ⟨c′± (V ), ·⟩ ∈ ∂F (V ) for all V ∈ Lp . The following utility functions will be studied later, see [78]: Definition 1.6.6. The isoelastic utility functions are defined as  |x|r : x∈R , r>1  − r xr Ur (x) := : x ∈ [0, ∞) , 0 ̸= r < 1 r  ln(x) : x ∈ (0, ∞) , r = 0.

(1.6.5)

Note that for p > 1, −Up ∈ C p . The following result shows that C p contains all convex functions behaving asymptotically as −Up . Proposition 1.6.7. Let c ∈ C0 and assume that there exist constants 0 < k ≤ K < ∞ and h > 0, such that k|x|p ≤ c(x) ≤ K|x|p holds for all |x| ≥ h. Then c ∈ Cp.

28


Proof. We have to show c′± (Lp ) ⊆ Lq . For |x| ≥ h and y ∈ R with sgn(x)y > 0, p −c(x) we have c(x) + c′± (x)y ≤ c(x + y) ≤ K|x + y|p , hence |c′± (x)| ≤ K|x+y| ≤ |y| K|x+y|p |y|

K|1+ y |p

x = |x|p−1 . Since y x and the assertion follows.

y x

> 0, we find |c′± (x)| ≤ K|x|p−1 inf y>0

|1+y|p y

We now define the set of utility functions, with which we will work in the sequel: Definition 1.6.8. A function U such that −U ∈ C p for 1 < p < ∞ is called a p-moderate utility function. Note that Up is p-moderate for 1 < p < ∞. This type of utility function can be used to define a measure of quality for approximations, e.g. E[U2 (·)] is used for the mean-variance approximation. The problem of maximizing the expected utility of terminal wealth, e.g. for Ur , with r < 1, requires different methods and will be considered later. Before solving the above minimization problem we introduce the convex conjugate function for a convex function. This function will be used to pose a minimization problem in L∗ , being dual to the original problem.

1.6.3

Conjugate Young Functions

In this subsection we define for a given c ∈ C0 a conjugate function cˇ ∈ C0 . We follow the approach in [28], Chapter VI, No. 97. We have chosen this approach, which is in the one-dimensional case equivalent to the LegendreFenchel approach, since it is well-suited to the type of utility functions we are interested in. For c ∈ C0 define the right continuous inverse of c′+ on [0, ∞) by Ic (t) := inf{s : c′+ (s) > t}, see [28], Chapter VI, 97.6. If c is continuously differentiable, then (c′ )−1 = Ic on [0, c′+ (∞)), in general we have Ic (c′+ ) = Id on [0, ∞), but only c′− (Ic ) ≤ Id ≤ c′+ (Ic ) on [0, c′+ (∞)). For s ≥ 0, we have Ic (t) = c′+ (s) for t ∈ [c′− (s), c′+ (s)]. Define for x ≥ 0 the conjugate Young function of c as ∫ x cˇ(x) := Ic (s)ds, (1.6.6) 0

resp.

∫ cˆˇ(x) = cˇ(−x) := cˇˆ(x) =

x

Icˆ(s)ds.

(1.6.7)

0

In the following lemma we collect some properties of the conjugate Young functions.


29

Lemma 1.6.9. For c ∈ C0 , we have 1. cˇ is a [0, ∞]-valued, not necessarily strictly convex function. If c′+ (∞) = ∞ and c′+ (−∞) = ∞, then cˇ is [0, ∞)-valued. 2. cˇ′+ = Ic on [0, ∞), resp. cˇ′− = −Icˆ(−Id) on (−∞, 0]. 3. cˇ(0) = 0, cˇ′− (0) ≤ 0 ≤ cˇ′+ (0). 4. cˇ is strictly convex if and only if c is continuously differentiable. 5. cˇ ∈ C0 if and only if c is continuously differentiable, c′+ (∞) = ∞ and c′+ (−∞) = −∞. 6. cˇ = c. 7. xc′± (x) = c(x) + cˇ(c′± (x)), resp. xˇ c′± (x) = cˇ(x) + c(ˇ c′± (x)). 8. cˇ(y) = supx∈R {xy − c(x)}, resp. c(y) = supx∈R {xy − cˇ(x)}. Proof. See [28], Chapter VI, No. 97, especially equations (97.9), (97.10). Lemma 1.6.10. We have cˇ′± (c′± ) = Id. Proof. The proof is an elementary calculation. From now on we write c′ instead of c′± .

1.6.4

Optimization

Let V be a closed linear proper subspace of Lp , 1 < p < ∞ and fix an element x ∈ Lp \ V. Let ◦ denote the annulator with respect to x, set W := x + V. Consider the following minimization problem for c ∈ C p : E[c(W min )] = inf E[c(W )], W ∈W

(1.6.8)

for a W min ∈ W and the corresponding dual optimization problem: E[ˇ c(Z min )] = inf ◦ E[ˇ c(Z)], Z∈yV

(1.6.9)

for a Z min ∈ yV ◦ , and a constant y ̸= 0. The solutions of the two optimization problems are related in the following way: Proposition 1.6.11. The solution W min to problem (1.6.8) exists uniquely and defines with y := E[c′ (W min )x] > 0 an element Z min := c′ (W min ) ∈ yV ◦ , solving (1.6.9), and doing so uniquely if c is continuously differentiable. Conversely, if Z ∈ yV ◦ for y := E[Zx] admits a representation Z = c′ (W ) for some W ∈ x + V, then Z = Z min and W = W min . Furthermore, y = E[Z min W min ] = E[c(W min )] + E[ˇ c(Z min )] > 0.

(1.6.10)

30


Proof. W min exists uniquely by Proposition 1.6.5. By definition of C p , we have c′ (W min ) ∈ Lq . Since E[c(W min + V )] ≤ E[c(W min ) + c′ (W min )V ] for all V ∈ V, we have E[c′ (W min )V ] = 0 for all V ∈ V. In particular for V min := W min − x ∈ V, we find 0 = E[c′ (W min )V min ] = E[c′ (W min )W min ] − E[c′ (W min )x].

(1.6.11)

Since x ̸∈ V, we have W min ̸= 0. Furthermore, by c′ (W min )W min ≥ 0 and c′ (W min )W min = 0 iff W min = 0 for c ∈ C0 , we find E[c′ (W min )x] > 0, hence Z min ∈ yV ◦ . Conversely, by Lemma 1.6.9, 7., we find cˇ(Z) ∈ L1 . Now, for Z˜ ∈ yV ◦ we ˜ ≥ cˇ(Z) + cˇ′ (Z)(Z˜ − Z) = cˇ(Z) + W (Z˜ − Z) ∈ L1 . have (see Lemma 1.6.10), cˇ(Z) ˜ ˜ ≥ E[ˇ Since E[W (Z − Z)] = 0, we find E[ˇ c(Z)] c(Z)]. By Lemma 1.6.9, 4., cˇ is strictly convex if c is continuously differentiable, implying uniqueness of the solution to (1.6.9). For V ∈ V we have W + V ∈ W and c(W + V ) ≥ c(W ) + c′ (W )V = c(W ) + ZV , hence E[c(W + V )] ≥ E[c(W )] and we find W = W min and thus Z = Z min . Equation (1.6.10) follows from (1.6.11) and Lemma 1.6.9, 7.

1.6.5

Optimal Portfolios and Local Martingale Measures

ˇ the conjugate Young Let c ∈ C p , 1 < p < ∞. Set U := −c and denote by U (p) q function of c. Assume D ̸= ∅ and S ∈ Aloc . We now apply the results of the previous subsection to the following expected terminal utility maximization problem: U (x, V¯ p ) := E[U (V max )] =

sup E[U (V )],

(1.6.12)

¯p V ∈x+V

for a V max ∈ x + V¯ p , and an x ∈ Lp \ V¯ p . Note that V¯ p is a closed subspace of the reflexive Banach-space of the Lp -integrable R-valued random variables on Ω, see [31]. The corresponding dual problem is ˇ (Z min )] = inf E[U ˇ (Z)], E[U q Z∈yD

(1.6.13)

for a Z min ∈ yDq and a constant y ̸= 0. Proposition 1.6.5 can be applied directly to these problems. For later reference we explicitly state the result for the special case of the isoelastic utility functions Up : Corollary 1.6.12. For x = 1, the problems (1.6.12), resp. (1.6.13), with respect to Up have unique solutions V opt ∈ 1 + V¯ p , resp. Z min =

sgn(V opt )|V opt |p−1 ∈ Dq . y

where y := E[sgn(V opt )|V opt |p−1 ] > 0. Furthermore, ∥V opt ∥Lp ∥Z opt ∥Lq = 1, and ∥V opt ∥Lp ≤ 1, resp. ∥Z min ∥Lq ≥ 1.

(1.6.14)


31

Proof. By Proposition 1.4.10 we have 1 ̸∈ V¯ p . Hence all the assertions, except the last equation follow directly from Proposition 1.6.11. Since for x = 1, we opt p have y > 0, we find |Z min |q = |V yq | . We also have E[|V opt |p ] = E[V opt yZ min ] = yE[E[V opt Z min |F0 ]] = yE[V0opt Z0min ] = yE[Z0min ] = y, and therefore E[|Z min |q ] =

E[|V opt |p ] = y 1−q yq − pq

= (E[|V opt |p ])

opt ∥ p ≤ 1. ¯p hence ∥Z min ∥Lq = ∥V opt ∥−1 L Lp . 1 ∈ 1 + V implies ∥V

Remark 1.6.13. V¯ p is a space of random variables, which can be approximated by the terminal values of the processes in the space V p . At this point we H = V opt for a selfcan in general not determine a process V H such that V∞ financing hedging strategy H. We will achieve this only later, when we study the semimartingale market model. In this case V opt is for p = 2 known as the terminal value of the hedging numéraire, see [24], [41], [42]. The process Z min is the variance optimal martingale measure for p = 2, resp. the Lq -optimal martingale measure, see [24], [42].

1.6.6

A Worked Example: The Merton Portfolio

Exercise 1.6.14. (Merton’s Portfolio Selection) An interesting one-dimensional example is the by now classical Merton portfolio selection theorem which was treated in the Stochastik script. Nevertheless it might be a good exercise to go through it again before we treat more general approaches to similar problems which deal with arranging a wealth among two assets such that the terminal utility is maximized. The methods used here are completely different from the methods of the preceding sections as we are using the Bellman approach here. This approach is restricted to Markovian settings and so is less general than the stochastic duality approach above. The advantage is that we get more explicit results. We shortly sketch the results from the control theory part of the Stochastik script. We are given two investment possibilities, a risky investment possibility given by the SDE: dk1 (t) = k1 (t)a dt + k1 (t)α dwt

α > 0,

(1.6.15)

and a non-risky investment which behaves like a bank account: dk2 (t) = k2 (t)b dt.

(1.6.16)

We assume that b < a, as in general the investor will only accept a risky investment if the return is higher than in the non-risky investment. Let us

32


assume that at any time the investor can decide what part u of his wealth ξt he invests into the rslky investment. Then his wealth develops according to the following SDE for the wealth ξt = ξtu : dξtu = u(t)ξtu a dt + u(t)ξtu α dwt + (1 − u(t))ξtu b dt = ξtu (au(t) + b(1 − u(t))) dt + αξtu u(t) dwt ξ0u = x0 . Let x0 > 0 be the initial wealth at time 0. The investor aims at maximizing his wealth at time T . It is not allowed that in the time interval the wealth becomes zero or less, so ξt ≥ 0 for all t ∈ [0, T ]. We are given a utility function c : [0, ∞) → [0, ∞), c(0) = 0 which usually is increasing and concave. The problem is to find a Markov control u∗ = u∗ (t, x) such that: M (t, x, u∗ ) = sup{M (t, x, u)|u M arkov control, 0 ≤ u ≤ 1} u

u (x))]. E[c(ξt,T

with M (t, x, u) = Let J(t, x, u) = −M (t, x, u), so one has a cost criterion as in chapter 7 of the Stochastik script, and we have to minimize this cost criterion. We have the following HJB-equation for the problem: inf (Au H)(t, x) = 0 u

H(T, x) = −c(x)

(1.6.17) (1.6.18)

with (Au H)(t, x) =

∂H ∂H 1 ∂2H + x(au + b(1 − u)) + α2 u2 x2 2 (1.6.19) ∂t ∂x 2 ∂x

or sup (Au G)(t, x) = 0

(1.6.20)

u

G(T, x) = c(x)

(1.6.21)

where G(t, x) = −H(t, x) = − inf u J(t, x, u) = supu M (t, x, u). In other words we try to maximize the function: η(u) = Au G ∂G ∂G 1 2 2 2 ∂ 2 G = + x(b + (a − b)u) + α u x . ∂t ∂x 2 ∂x2 With Gt =

∂G ∂t ,

Gx =

∂G ∂x

and Gxx =

∂2G ∂x2

(1.6.22) (1.6.23)

the solution is:

u = u(t, x) = −

(a − b)Gx . xα2 Gxx

(1.6.24)

By putting this into the HJB-equation we arrive at the following nonlinear terminal value problem for G: Gt + bxGx −

(a − b)2 G2x = 0 for t ∈ [0, T ), x > 0 2α2 Gxx G(T, x) = c(x)

(1.6.25) (1.6.26)


33

It is difficult to solve this problem for general cost function c. So let us specify c. Let us choose the important class of functions having potential form c(x) = xr , 0 < r < 1, so we find a solution having the form: G(t, x) = h(t)xr . This leads to: ( x

r

( )) ∂h(t) (a − b)2 r + h(t) br − 2 = 0 t ∈ [0, T ) ∂t 2α (r − 1) G(T, x) = xr

(1.6.27) (1.6.28)

and G(t, x) = eλ(T −t) xr with λ = br +

(a−b)2 r . 2α2 (1−r)

Put this into (1.6.24) to get: u∗ (t, x) =

(a − b) . α2 (1 − r)

(1.6.29)

If α(a−b) 2 (1−r) ∈ [0, 1], then we have found the solution of the problem. We see that this u∗ is constant in (t, x).

In this way we found an optimal strategy by making use of the HJB-equation. This is possible as we found the in general rare solution for V ∈ C 1,2 . If V ∈ / C 1,2 , then we have to find other ways: We can try to find more general solutions of the HJB-equation, like viscosity solutions (see Stochastik script) and if the underlying process is no longer Markovian, we ahve to go into the general optimization theory to apply completely different methods as we will see later on. Exercise 1.6.15. In earlier versions we here gave an exercise on the duality solution of the Merton problem. Patrick Breil wrote the following text which describes the connection between the HJB approach and the stochastic duality approach to the Merton problem: Given are the following stochastic processes to model a non-risky asset k1 (t) and a risky asset k2 (t) in a complete market, where r, µ, σ ∈ R with µ > r and σ > 0. dk1 (t) = rk1 (t)dt k1 (0) = 1 dk2 (t) = µk2 (t)dt + σk2 (t)dwt k2 (0) = ?

(1.6.30) (1.6.31) (1.6.32) (1.6.33)

Let π(t) denote the investment strategy, i.e. the proportion of the investment in the risky asset k2 and 1 − π(t) the proportion of the investment in the risk

34


free asset k1 , where Xtπ represents the portfolio process with investment strategy π. dXtπ = r(1 − π(t))Xtπ dt + µπ(t)Xtπ dt + σπ(t)Xtπ dwt X0π

= x

With θ(t) := θ :=

(1.6.35)

µ−r σ ,

the market price of risk, this yields

dXtπ = (rXtπ + θσπ(t)Xtπ )dt + σπ(t)Xtπ dwt X0π

(1.6.34)

= x.

(1.6.36) (1.6.37)

Since θ is independent of∫ time it is progressively measurable and also fulfills T the Novikov condition, i.e. 0 ||θ(t)||2 dt < ∞ a.s. The overall aim is to determine the optimal portfolio process Xt∗ and the optimal control π ∗ . We further assume a utility function of the form U (x) = 1δ xδ where δ ∈ (0, 1). Remark 1.6.16. Since −U (x) ∈ / C p we cannot (yet) use the results of section 1.6.5 on Optimal Portfolios and Local Martingale Measures (Stochastic Script III), but these also hold in principle under some technical changes for our assumed utility function as will be seen in a subsequent chapter. In terms of the duality the two control problems can be expressed by x,π V (s, x) = max E[U (XsT )] π

(1.6.38)

with the original value function V , over the time horizon s until T , with initial wealth x and portfolio strategy π. The trivial dual problem can be summarized V˜ (s, y) =

˜ (z y )] min E[U sT

y zsT ∈yDq

y dzst = ydzst y zss

= y

with y > 0

(1.6.39) (1.6.40) (1.6.41)

with dual value function V˜ and its deflator process zst with dzst = −rzst dt − θzst dwt zss = 1.

(1.6.42) (1.6.43)

It is not obvious yet, why the deflator process should be of this form, but it is according to the definition of the annulator so that X becomes a martingale under z as will be seen shortly. However, the risk free asset process is not discounted nor is the risky asset process driftless. The need to deal with these disliked properties leads to the


35

application of the following methods. The drift can be removed with the Girsanov functional ∫ t ∫ ( 1 t 2 ) θdws − θ ds h0t = exp − 2 0 0 h0t hst := h0s hss = 1,

(1.6.44) (1.6.45) (1.6.46)

making use of the Novikov condition in −θ which yields E[h0T ] = 1, we define the new measure dQ := h0T (ω)dP

(1.6.47)

and a new Brownian motion w ˜ dw ˜t := θdt + dwt .

(1.6.48)

Substitution of the original Brownian motion in (1.6.36) removes the drift from the risky asset in the portfolio process dXtπ = rXtπ dt + σπ(t)Xtπ dw ˜t

(1.6.49)

and by applying Itˆ o’s Lemma to ln(Xtπ ) one can state an explicit solution for the portfolio process by integration ∫ t ∫ ) ( t σ 2 π 2 (s) σπ(s)dw ˜s . )ds + Xtπ = x exp (r − (1.6.50) 2 0 0 Hint: d ln(Xtπ ) =

1 π Xtπ dXt

−

σ 2 π 2 (t) dt 2

Let the discount process β0t be defined by ∫ t ) ( β0t := exp − rds

(1.6.51)

0

=

βst βss

exp(−rt) β0t := β0s = 1.

Discounting of the solution (1.6.50) yields ∫ ∫ ) ( t 1 t 2 2 π σ π (s)ds β0t Xt = x exp σπ(s)dw ˜s − 2 0 0 ∫ t = x+ β0s Xsπ σπ(s)dw ˜s .

(1.6.52) (1.6.53) (1.6.54)

(1.6.55) (1.6.56)

0

Itˆ o’s Lemma applied to the product of β0t Xtπ and h0 (t) results in ∫ t π β0t h0t Xt = x + β0s h0s Xsπ (σπ(s) − θ)dws . 0

(1.6.57)

36


where β0 h0 X is recognized to be a non-negative local martingale. In fact it can be shown that z0t zst

=

β0t h0t z0t := z0s

(1.6.58) (1.6.59)

is the solution for the deflator process in (1.6.42). Proof. The statement is easily verified by the application of Itô’s Lemma to the function f (t, hst ) := exp(−r(t − s))hst = zst : 1 ∂ 2 f (t, hst ) ∂f (t, hst ) ∂f (t, hst ) dhst + dt + d < hs > (1.6.60) t ∂t ∂hst 2 ∂h2st = −r exp(−r(t − s))hst dt + exp(−r(t − s))dhst + 0 (1.6.61)

df (t, hst ) = dzst

= −r exp(−r(t − s))hst dt + exp(−r(t − s))(−θhst dwt )(1.6.62) = −rzst dt − θzst dwt

(1.6.63)

zss = 1

(1.6.64) 2

y y With zst = yzst and zss = y the complete solution to the dual problem is established and one thereby also solved the original problem. Before returning to the original problem, however, let us investigate some properties of the dual value function. Since Dq is a singleton the dual maximization reduces to:

˜ (z y )] V˜ (s, y) = E[U sT 1 1 − δ y 1−δ = E[ (zsT ) ]. δ

(1.6.65) (1.6.66)

Let us further define functions X , Y and S where Y is the inverse function 1 of X and I(y) = y δ−1 is the inverse of U ′ (x): y X (s, y) := E[zsT I(zsT )]

(1.6.67)

S(s, y) := yX (s, y)

(1.6.68)

= =

y y E[zsT I(zsT )] δ y δ−1 E[(zsT ) ]

(1.6.69) (1.6.70)

The dual value function can now be expressed as V˜ (s, y) = G(s, y) − S(s, y),

(1.6.71)

y G(s, y) = E[U (I(zsT ))] δ 1 y δ−1 ) ]. = E[ (zsT δ

(1.6.72)

where G equates to

(1.6.73)


37

y In this context be reminded that I(zsT ) = XT (ω).

A result from the fact that the dual value function V˜ is the Legendretransform of V is that one can calculate the original value function V by calculating G for optimal y ∗ := Y(s, x). V (s, x) = G(s, Y(s, x))

(1.6.74)

Furthermore it can easily be verified that the following properties hold: V˜y (s, y)y = −S(s, y)

(1.6.75)

= −yX (s, y)

(1.6.76)

Vx (s, x) = Y(s, x).

(1.6.77)

Let us now focus on deriving the optimal trading strategy π ∗ . Consider the positive martingale m defined by m(t) := E[β0T h0T XT∗ |Ft ] XT∗

with

:= I(Y(T, x)β0T h0T ).

(1.6.78) (1.6.79)

The martingale representation theorem allows an integral representation for ∫T m with integrand ψ, where the latter is Ft adapted and suffices 0 |ψ(s)|2 ds < ∞ a.s. ∫ t m(t) = x + ψ(s)dws (1.6.80) 0

Itˆ o’s Lemma applied to the function g(x, y) := xy with x = m(t) and y = h0t results in: ( m(t) ) 1 d = (ψ(t) + m(t)θ)dw ˜t (1.6.81) h0t h0t Hint: d( h10t ) =

θ ˜t h0t dw

Integration over the full time horizon leads to the optimal discounted portfolio process in T : β0T XT∗

m(T ) h0T ∫ = x+

=

0

(1.6.82) T

1 (ψ(s) + m(s)θ)dw ˜s h0s

(1.6.83)

Define the optimal portfolio process for the intermediate points in time by: ∫ t ) 1 ( 1 ∗ Xt := x+ (ψ(s) + m(s)θ)dw ˜s (1.6.84) β0t 0 h0s This is derived by making use of the Bayes Formula, note that: E Q [XT |Ft ] = E[ztT XT |Ft ] = E[htT βtT XT |Ft ] m(t) = h0t β0t

(1.6.85) (1.6.86) (1.6.87)

38

CHAPTER 1. THE GENERAL MARKET MODEL Comparing integrands with (1.6.56) delivers the unique solution for π ∗ ψ(t) + m(t)θ σβ0t h0t Xt∗

(1.6.88)

δ µ−r β0t h0t Xt∗ . 1 − δ σ2

(1.6.89)

π ∗ (t) = and the integrand ψ equates to ψ(t) =

Deriving XTπ under the original Brownian motion using (1.6.50) gives us the first representation XTπ = x exp

(

∫

T

(

r+

0

σπ(s)(2θ − σπ(s)) ) ds + 2

∫

T

) σπ(s)dws (1.6.90) ,

0

where calculating XT∗ from the dual approach using (1.6.79) leads to the second representation XT∗

= x exp

(

∫ 0

T

(

1 − 2δ 2 ) r+ θ ds + 2(1 − δ)2

∫

T

0

) θ dws . 1−δ

δθ Hint: 1−δ also suffices the Novikov condition, i.e. ~0t := exp ( ) ) ∫ 2 t δθ 1 2 0 1−δ ds is a martingale with E[~0T ] = 1.

(∫t

(1.6.91)

δθ 0 1−δ dws

−

Comparing integrands in both expressions leads to the optimal solution for π θ 1−δ

= σπ(t)

π ∗ (t) = =

(1.6.92)

θ σ(1 − r) µ−r σ 2 (1 − δ)

(1.6.93) (1.6.94)

which equals the result derived from the Hamilton-Jacobi-Bellmann approach. Coming straight away from the dual approach the same results can be derived by: Xt∗

=

X (t, η s,x (t)) Y(s,x)

with

η s,x (t) := zst µ − r Y(t, Xt∗ ) π ∗ (t) = − 2 σ Yx (t, Xt∗ )Xt∗ µ − r Vx (t, Xt∗ ) = − 2 . σ Vxx (t, Xt∗ )Xt∗

(1.6.95) (1.6.96) (1.6.97) (1.6.98)

Remark 1.6.17. Stochastic duality more and more turns out to be the most powerful tool when considering certain portfolio optimization and hedging problems.


39

In his 2004 thesis Volker Buerkel completely solved the Lp -control/hedging problem which consists in solving the problem E|xt − ξ|p = min! By stochastic duality this problem is related to the Lq -minimal measure problem. These measures converge to the minimal relative entropy measure, so we may say that the dual problems converge. By duality it is shown that also the primal problems converge, and they converge to the exponential hedging problem which again is dual to the minimal entropy problem. As the primal problems are explicitly solved this gives an efficient way to derive the solution of the exponential hedging problem explicitly. These works are described in the recent literature and are found in the appropriate list from my homepage. Remark 1.6.18. There are some rather puzzling examples of special problems from utility maximization theory which shed a light on the complexity of the theory: • (Gan-Lin Xu: Zero investment in a high yield asset can be optimal, Mathematics of OR, 14,3 (1989), 457-461) Though the expected return of a stock is higher than the risk-free return the optimal investment strategy prefers the riskfree asset: Let the risk free asset be constant P0 (t) = 1 ∫

and let

∫

t

P1 (t) = 1 +

θs P1 (s)ds + 0

t

P1 (s)dws 0

on t ∈ [0, 4]. For a sufficiently integrable portfolio π let the wealth process be dxt = πθdt + πdw, x0 = e3 . The return is given by 2 3 J(π) = Ex43 . 2 ∫t ∫t Let as = signws and define Mt = 0 = 0 as dws . Let τ1 , τ2 be the crossing times of −1 and 1 by Mt and Mt + 12 < M >t . Finally let At = at [t < τ1 ∧ τ2 ] and Bt = √32 [τ1 < t ∧ τ2 ] and

θ = At + Bt . Note then that on {(t, ω)|t < τ (ω)} θt takes both values +1 and −1 but the optimal portfolio is identically zero on this set. • (W. Schachermayer: A supermartingale property of the optimal portfolio process, FINASTO 7, 43-456, (2003)) Schachermayer proves the following result:

40

CHAPTER 1. THE GENERAL MARKET MODEL Proposition 1.6.19. There are processes S 1 and S 2 defined on and adapted to (Ω, F, (Fn )n ∈ N0 , P ) and a probability measure Q ∼ P on F with the following properties: – H(Q|P ) = EQ [ln( dQ dP )] < ∞. – S 1 is a martingale under Q and P which is uniformly integrable under Q, but not u.i. under P . – S 2 is a martingale under Q but not under P . – Denote by S l the R2 -valued process (S 1 , S 2 ). Q is the unique equivalent local martingale measure for S l . The equality 1 S∞ = − ln(

dQ ) dP

holds true and therefore Sn1 equals the investment process starting at S01 which is optimal with respect to the utility function U (x) = exp(−x). – Denoting by S s the R-valued process Sn1 , P is a martingale measure for S s . Hence the optimal investment process with initial endowment S01 now is the constant process S01 . The optimality pertains to U (x). – So: In the large market it is optimal to invest in S 1 and by taking away the unused second stock in the small market we would invest into the bond.

1.6.7

Mean-Variance Hedging

For U = Up , p = 2, the optimization problem (1.6.12) is known as the intensively studied mean-variance hedging problem, which goes back to [37], [38], [41]. See also [92] for an overview. We develop some well known results in our setting: Definition 1.6.20. A non-attainable contingent claim is an Lp -integrable F∞ measurable random variable C, such that C ̸∈ z + V¯ p for all z ∈ R. Let D2 ̸= ∅ and let C be an L2 -integrable non-attainable contingent claim. For z ∈ R and x := C − z, the optimization problem (1.6.12) has a unique solution Wzmin . For U = U2 , the corresponding solution to the dual problem (1.6.13) equals Wzmin by Proposition 1.6.11. We want to determine the optimal initial price zˆ ∈ R, such that ∥Wzˆmin ∥ = min ∥Wzmin ∥. z∈R

(1.6.99)

Let V opt denote the terminal value of the hedging numéraire and Z min = the variance optimal martingale measure. Proposition 1.6.21. We have zˆ = E[CZ min ] and for all z ∈ R, Wzmin = Wzˆmin − (z − zˆ)V opt .

V opt E[V opt ]


41

Proof. By Proposition 1.6.5, applied to W = R+ V¯ 2 , there exists a solution zˆ for the problem (1.6.99). We have ∥Wzˆmin +kV opt ∥2 = ∥Wzˆmin ∥2 +2kE[Wzˆmin V opt ]+ k 2 ∥V opt ∥2 = ∥Wzˆmin ∥2 + 2kE[Wzˆmin ] + k 2 ∥V opt ∥2 for all k ∈ R, since Wzˆmin ∈ (V¯ 2 )◦ and V opt ∈ 1 + V¯ 2 , hence E[Wzˆmin ] = 0 by optimality of zˆ. Since Wzˆmin = C − zˆ + V˜ for a V˜ ∈ V¯ 2 , we find 0 = E[Wzˆmin V opt ] = E[(C − zˆ + V˜ )Z min ] = E[CZ min ] − zˆ. This proves the first assertion. With V := Wzmin − Wzˆmin + (z − zˆ)V opt ∈ V¯ 2 , we find ∥Wzmin ∥2 = ∥Wzˆmin − (z − zˆ)V opt + V ∥2 ≥ ∥Wzˆmin − (z − zˆ)V opt ∥2 + 2E[(Wzˆmin − (z − zˆ)V opt )V ] = ∥Wzˆmin − (z − zˆ)V opt ∥2 . Remark 1.6.22. For Wzmin = C − Vzopt , z ∈ R, we therefore have Vzopt = Vzôpt + (z − zˆ)V opt ∈ z + V¯ 2 , i.e. the mean-variance optimal hedging strategy approximating C for initial capital z is just to buy the optimal strategy for initial capital zˆ and to invest the remaining capital z − zˆ into the portfolio (z − zˆ)V opt such that the L2 -norm of its terminal value is minimal.

1.6.8

Utility Indifference Prices

In this subsection we develop a pricing method based on a utility indifference argument, see [20] and [91] for similar approaches. (p)

The problem is the following. Let M = (Ω, S), S ∈ Aloc be a general market model with Dq ̸= ∅ and let U be a p-moderate utility function, c := −U . We want to introduce a price for a non-attainable contingent claim, given by an Lp -integrable F∞ -measurable random variable C. Since we have not developed a theory that can deal with restricted trading it is not enough to find a price for C at time t = 0, we rather have to find a process X ∈ A with X∞ = C, representing the price process of the claim. We define the market extended by X as MX := (Ω, (S, X)). The idea of utility indifference pricing is to find a process X, such that the maximal expected terminal utility in M, resp. MX are equal for a fixed constant x ̸= 0: ( ) ( ) U x, V¯ p (M) = U x, V¯ p (MX ) . (1.6.100) ( ) ( ) (p) Obviously we have U x, V¯ p (M) ≤ U x, V¯ p (MX ) for any X ∈ Aloc . Denote the solution of the maximization problem (1.6.8) with respect to x + V¯ p (M), resp. x + V¯ p (MX ) by V max , resp. by V max,X , and the solution to the corresponding dual problem (1.6.9), Z min := c′ (V max ) ∈ y V¯ p (M)◦ , resp. Z min,X := c′ (V max,X ) ∈ y X V¯ p (MX )◦ , where y := E[c′ (V max )x], resp. y X := E[c′ (V max,X )x] and ◦ denotes the annulator with respect to x. Note that we have by Proposition 1.6.11 that y, y X > 0, hence Z min ∈ xy Dq (M), resp. Z min,X ∈ placed by

yX q X x D (M ) y yX x , resp. x .

and Z min , resp. Z min,X , solves (1.6.13), where y is re-

Proposition 1.6.23. Let X ∈ A(p) with X∞ = C. Then (1.6.100) holds iff min ∞ |Ft ] ˜ In particular Xt = E[CZmin XZ min ∈ U. a.s. on {Ztmin ̸= 0} for all t ≥ 0. Z t

Proof. Assume (1.6.100) to hold. Then V max = V max,X by the strict concavity of U and by Proposition 1.6.11 Z min,X := Z min is a solution to (1.6.9) with

42


X respect to y V¯ p (MX )◦ . Since X ∈ A(p) , we find X − X0 ∈ V(SF s,p 0 (M )). By ˜ Conversely, if XZ min ∈ U, ˜ then Z min ∈ Lemma 1.4.2 we find XZ min ∈ U. p X ◦ max ¯ ¯ y V (M ) . Again by Proposition 1.6.11 and since V ∈ V p (MX ) we find max,X max V =V , hence (1.6.100) holds.

Exercise 1.6.24. • For U = Up the utility indifference price for C does not depend on x. • Try to compute the utility indifference price for C in the framework of exercise 1.6.15. Remark 1.6.25. If C is Fτ -measurable for a stopping time τ such that {C ̸= 0} ⊆ {Zτmin = 0} ∩ {τ < ∞}, then the process X := C1[τ,∞) satisfies X∞ = C and XZ min = 0 ∈ U˜ since Z min = 0 on [τ, ∞). Hence utility indifference pricing might lead to arbitrage opportunities, see also Remark 5.4.23.

1.7

Mean-Variance Efficiency

We now apply the results of the previous section to the problem of finding mean-variance efficient portfolios. The problem goes back to [68] and [86]. See also [69] and [87].

1.7.1

The Generalized Sharpe-Ratio

Let V¯ ⊆ L2 be a closed linear space, set ∥ · ∥ := ∥ · ∥L2 for the moment and E[V ] := ⟨V, 1⟩, resp. Var(V ) := ∥V ∥2 − E[V ]2 , for V ∈ L2 . Fix x ∈ L2 , let ◦ denote the annulator with respect to x and consider the constraint minimization problem F (e) := ∥V e ∥2 = inf ∥V ∥2 , (1.7.1) ¯ V ∈x+V

under the constraint E[V ] = e, ¯ In general, problem (1.7.1) might not have a for e ∈ R and V e ∈ x + V. solution for every e ∈ R. However, the solution V min of the unconstrained problem exists always and V min solves (1.7.1) for e = E[V min ] =: eˆ. Assume ¯ ̸= {ˆ ¯ = R. Since (x + V) ¯ ∩ {V ∈ that E[x + V] e} which is equivalent to E[x + V] 2 e L |E[V ] = e} is closed and convex, the unique solution V of (1.7.1) exists for all e ∈ R. Proposition 1.7.1. Let e¯ ̸= eˆ and e = (1 − λ)ˆ e + λ¯ e for a λ ∈ R, then V e = (1 − λ)V min + λV e¯. Furthermore, there exists a constant γ > 0 such that for all e ∈ R F (e) = F (ˆ e) + γ(e − eˆ)2 , holds.

1.7. MEAN-VARIANCE EFFICIENCY

43

Proof. Set Vλ := (1 − λ)V min + λV e¯. We have E[Vλ ] = e and [( )2 ] ∥Vλ ∥2 = E (1 − λ)V min + λV e¯ [ ] = (1 − λ)2 ∥V min ∥2 + λ2 ∥V e¯∥2 + 2λ(1 − λ)E V min V e¯ ( ) [ ] = 1 − λ2 ∥V min ∥2 + λ2 ∥V e¯∥2 + 2λ(1 − λ)E V min (V e¯ − V min ) ( ) = 1 − λ2 ∥V min ∥2 + λ2 ∥V e¯∥2 , [ ] ˜ with eˆ ̸= since V e¯ − V min ∈ V¯ implies E V min (V e¯ − V min ) = 0. Now fix λ ˜ e + λ¯ ˜ e ̸= e¯, define λ ¯ by e¯ = (1 − λ)ˆ ¯ e + λ˜ ¯ e and Fe¯(λ) := ∥Vλ ∥2 = e˜ := (1 − λ)ˆ 2 min 2 2 e ¯ 2 (1 − λ )∥V ∥ + λ ∥V ∥ . We have ˜ Fe˜(1) = F (˜ e) ≤ Fe¯(λ).

(1.7.2)

˜ ≤ F (˜ We want to show that Fe¯(λ) e). Interchanging e¯ with e˜ in equation (1.7.2), ¯ ¯ = e¯−ê , we find the equation we also have Fe¯(1) = F (¯ e) ≤ Fe˜(λ). Since λ e˜−ˆ e ) ( ) ( ) ( e¯ − eˆ 2 e¯ − eˆ 2 min 2 e¯ 2 ∥V ∥ + ∥V e˜∥2 , ∥V ∥ ≤ 1 − e˜ − eˆ e˜ − eˆ ˜2 = which leads, after multiplication with λ (

( 1−

e˜ − eˆ e¯ − eˆ

)2 )

( ∥V

∥ +

min 2

(

e˜−ˆ e e¯−ˆ e

e˜ − eˆ e¯ − eˆ

)2 and rearranging terms, to

)2 ∥V e¯∥2 ≤ ∥V e˜∥2 ,

˜ ≤ F (˜ but this is just Fe¯(λ) e), hence ˜ 2 (F (¯ F (˜ e) = F (ˆ e) + λ e) − F (ˆ e)), or F (˜ e) = F (ˆ e) + With γ :=

F (¯ e)−F (ˆ e) (¯ e−ˆ e)2

F (¯ e) − F (ˆ e) (˜ e − eˆ)2 . 2 (¯ e − eˆ)

> 0 the assertion follows.

Lemma 1.7.2. We have γ ≥ 1 and γ = 1 implies eˆ = 0. Proof. We have Var(V ) ≥ 0 for all V ∈ L2 and Var(V ) = 0 iff V = ∥V ∥. This implies Var(V e ) = F (e) − e2 ≥ 0. Since F (e) − e2 is a polynomial of at most second order, we must have γ ≥ 1 and γ = 1 implies eˆ = 0.

Now consider the following variance optimization problem: G(e) := Var(V e ) =

inf Var(V ),

¯ V ∈x+V

(1.7.3)

under the constraint E[V ] = e, ¯ This problem is trivially solved uniquely by the for e ∈ R and V e ∈ x + V. solutions of (1.7.1):

44


Lemma 1.7.3. If γ > 1, then we have for all e ∈ R G(e) = F (e) − e2 ≥ F (ˆ e) − where f :=

γ ˆ. γ−1 e

γ 2 eˆ = G(f ) ≥ 0, γ−1

Furthermore, (

)2 γ G(e) − G(f ) = (γ − 1) e − eˆ . γ−1

(1.7.4)

For γ = 1, we have G(e) = F (ˆ e), for all e ∈ R. Proof. The proof is an elementary calculation. Definition 1.7.4. Define for γ > 1 the generalized Sharpe-ratio for x and V¯ as √ 1 ¯ := βx := βx (V) . (1.7.5) γ−1 ¯ ̸= {ˆ ¯ = {ˆ So far we have assumed E[V] e}. If E[x + V] e}, we set βx := 0 and f := eˆ. Proposition 1.7.5. Assume γ > 1. For all V ∈ x + V¯ the following inequality holds: √ √ −βx Var(V ) − Var(V f ) ≤ E[V ] − E[V f ] ≤ βx Var(V ) − Var(V f ). (1.7.6) Furthermore, if V ̸= V f , then E[V ] − E[V f ] √ ≤ βx , Var(V ) − Var(V f )

(1.7.7)

and equality holds if and only if V = V e for some e ̸= f . ¯ = {ˆ ¯ ̸= {ˆ Proof. For E[x + V] e}, (1.7.6) holds trivially. For E[x + V] e} and e e V = V , (1.7.6) follows from (1.7.4). Since Var(V ) ≤ Var(V ) for all V ∈ x + V¯ ¯ with E[V ] = e, (1.7.6) holds for all V ∈ x + V. We will apply this result in the next two subsections to the problem of finding mean-variance-efficient portfolios, resp. the problem of finding meanvariance-efficient hedging strategies for a non-attainable contingent claim. ¯ we have for all V ∈ 1 + V¯ Corollary 1.7.6. For x = 1 ̸∈ V, √ √ −β1 Var(V ) ≤ E[V ] − 1 ≤ β1 Var(V ), resp. if V ̸= 1, then

|E[V ] − 1| √ ≤ β1 , Var(V )

(1.7.8)

(1.7.9)


45

and equality holds if and only if V = V e for some e ̸= 1. Furthermore, √ β1 = min Var(Z). (1.7.10) ¯◦ Z∈V

¯ ̸= {1}, then we have 0 < ∥V min ∥2 = eˆ ̸= eˆ2 , f = 1 = V f , If E[1 + V] γ= resp.

∥V min ∥2 > 1, ∥V min ∥2 − eˆ2 √

β1 =

∥V min ∥2 − 1 > 0. eˆ2

(1.7.11)

(1.7.12)

¯ = {1} iff 1 ∈ V¯ ◦ . E[1 + V] 2

Proof. By Proposition 1.6.11 applied to c(x) = x2 , we find E[V min x] = eˆ > 0, hence γ > 1. Since trivially 1 ∈ 1 + V¯ and Var(1) = 1, we have f = 1 = V f , hence (1.7.8) and (1.7.9) follow directly from Proposition 1.7.5. Now, ∥V min ∥2 = ¯ eˆ2 = E[V min ]2 implies V min to be constant. V min = k ̸= 1, would imply 1 ∈ V, min ¯ ¯ hence V = 1. But this implies E[V ] = 0 for all V ∈ V, hence E[1+ V] = {1}. γ ¯ ̸= {1}, we find by Lemma 1.7.3, 0 = G(f ) = F (ˆ Therefore for E[1+ V] e)− γ−1 eˆ2 ¯ we have V¯ ◦ ̸= ∅. By and this implies (1.7.11), resp. (1.7.12). Since 1 ̸∈ V, 2 Equation (1.6.11) applied to c(x) = x2 = cˇ(x), we find ∥V min ∥2 = eˆ and min Z min = V eˆ ∈ V¯ ◦ solves the dual optimization problem (1.6.9) for y = 1. min 2 1 ¯ = {1}, Hence γ−1 = ∥V eˆ2 ∥ − 1 = Var(Z min ) and (1.7.10) holds. For E[1 + V] we trivially have 1 ∈ V¯ ◦ , hence (1.7.10) holds in this case too. In the next subsection we will apply this result to the space V¯ 2 (M).

46


1.7.2

The Intertemporal Price for Risk (2)

We now assume S ∈ Aloc . |E[V ]−1| 2 with Var(V ) ̸= 0, the ratio √ For V ∈ 1 + V∞ , of the excess expected Var(V ) 2 , and the risk return of the portfolio V over the risk-free portfolio 1 ∈ 1 + V∞ of the portfolio V , measured by the standard deviation of V , can be used as a performance measure for self-financing portfolios. This fundamental idea goes back to Markowitz and Sharpe, see [68] and [86].

Definition 1.7.7. Define the intertemporal price for risk in M as { } √ 2 β(M) := inf δ |E[V ] − 1| ≤ δ Var(V ), ∀ V ∈ 1 + V∞ , .

(1.7.13)

An alternative interpretation of β(M) is that of a risk premium: Taking on a risk, measured by the standard deviation of the terminal value of the portfolio V , is at best rewarded with an expected excess return √ over the risk free portfolio 1, equal to risk premium×risk : E[V ] − 1 = β(M) Var(V ). Remark 1.7.8. The mean-variance efficiency problem for a discrete time, multiperiod market has been investigated by [66]. Using BSDE methods and LQTheory, in [67] and [100], it was shown for special cases, that the efficient frontier or market line is a straight line. In addition to Proposition 1.4.10, we have: Theorem 1.7.9. The following conditions are equivalent: 1. β(M) < ∞. 2. The SLP2 holds in M. 3. 1 ̸∈ V¯ 2 . 4. D2 ̸= ∅. Furthermore, β(M) =

inf

Z∈D2 (M)

Var(Z),

(1.7.14)

and β(M) = β1 (V¯ 2 ) for 1 ̸∈ V¯ 2 . Proof. By Proposition 1.4.10 and Corollary 1.7.6, we only have to show that 1. implies 3. We do this by showing that if 3. does not hold, then 1. does not 2 converging to hold: Assume 1 ∈ V¯ 2 . Then there exists a sequence Vn ∈ 1 + V∞ 2 implies Var(V ) > 0 for n large enough. Since 0, hence Var(Vn ) → 0. 1 ̸∈ V∞ n |E[Vn ]−1| 2 implies β(M) = ∞ too. E[Vn ] → 0, we have √ → ∞. 1 ∈ V∞ Var(Vn ) Equation (1.7.14) holds by Corollary 1.7.6 if 3. holds. If 4. does not hold, then β(M) = ∞ and (1.7.14) holds too.


1.7.3

47

Example

We show by a simple example, that the intertemporal price for risk can be finite in a market that is not arbitrage free in the sense of Definition 1.2.5. Let S = 1 on [0, 1) and S = 3 + z on [1, ∞), where z is an F1 -adapted random 2 = V ¯ 2 and Dq = {Z}, variable with P (z = 1) = 12 = P (z = −1). Then 1 ̸∈ V∞ where Z = −1 on {z = 1}, Z = 3 on {z = −1}, and we find β(M) = 2. For H > 0, i.e. there even exists a dominant H = (1, −1) we have V0H = 0 and V∞ trading strategy, see [77].

1.7.4

Economical Consideration

The importance of an arbitrage-free market stems from the idea, that it should not be possible to make a risk-free gain, since otherwise investors would excessively buy such a risk-free portfolio and the market would not be in an equilibrium and moving towards its equilibrium, the arbitrage should soon disappear. Our results show, that in a market, where investors invest either in mean-variance efficient portfolios or try to approximate non-attainable claims according e.g. the mean-variance criterion, the presence of arbitrage does not effect the existence of optimal strategies, in particular it is not optimal for an investor with a strictly positive initial capital to buy higher and higher multiples of portfolios leading to an arbitrage opportunity, which is at first somewhat counterintuitive. However, for expected terminal utility maximization with respect to an increasing utility function, e.g. Ur , 0 ̸= r < 1, the existence of an arbitrage would cause the optimization problem to have no solution, since buying an additional positive multiple of an arbitrage opportunity will always increase the expected terminal utility. (This of course depends on the space of admissible self-financing hedging strategies. In Section 5.11.1, we shall work with a set of self-financing hedging strategies SF + such that the corresponding value processes are non-negative. For H ∈ SF + and a self-financing hedging ˜ generating an arbitrage, H + H ˜ might not be an element of SF + .) strategy H Define the set of L2 -arbitrage opportunities by A := {a ∈ V¯ 2 |a ≥ 0, a ̸= 0}. For a ∈ A, we have γa ∈ A for all γ > 0. Set A1 := {a ∈ A|E[a] = 1} and consider for a ∈ A1 and γ > 0 the following elements aγ := 1 + γa ∈ 1 + V¯ 2 . We E[a ]−1 have E[aγ ] = 1 + γ and E[(aγ )2 ] = 1 + γ 2 E[a] + 2γ, hence Varγ(a ) = Var1 (a) . γ

1 By Corollary 1.7.6, we find Var(a) ≥ β(M) 2 for all a ∈ A, if β(M) > 0. For β(M) = 0 we have A = ∅ since then P ∈ D2 .

1.7.5

Mean-Variance Efficient Hedging

We now apply Proposition 1.7.1 to the mean-variance hedging problem. For a square integrable non-attainable claim C, set x = C − x0 for x0 ∈ R. F (ˆ e) can be interpreted as the maximal lower bound for the quadratic error we can achieve with a self-financing hedging strategy in SF s,2 x0 . F (e) can be interpreted as the maximal lower bound for the quadratic error we can achieve with a selffinancing hedging strategy in SF s,2 x0 , such that the expected payoff equals e.

48


Proposition 1.7.10. Assume E[x + V¯ 2 ] ̸= {ˆ e} and eˆ = E[V min ] ̸= 0. Then 2 βx ˆ for δ := 1+β 2 and for all e ̸= e x

√

|e − eˆ| F (e) − F (ˆ e)

≤ δ,

holds. Proof. This follows directly from Proposition 1.7.1 and Lemma 1.7.2.

1.7.6

The Risk Premium Indifference Price for Information

We close this chapter with some considerations about the price of information. Besides the idealization of a transaction cost free market, we also have been working with the assumption that the information structure of the market, described by the filtration (Ft )0≤t , is available to all participants without charge. Information is a good that is different from normal assets and there occur some difficulties if one tries to build a theory of the value of information. For example, information is often easily duplicated, and can be (re-)sold several times. We shall consider a very simple situation in order to avoid these problems. In the literature one finds quite a large number of papers on this topic, we cite a few: [95], [85], [88], [33], [2], [35], [99], [9], [55], [3], [4], [45]. We think of the filtration (Ft )0≤t as the information that is in principle available in our market M := (Ω, (S, 1)), where we assume S to be an adapted ¯ F, (Ft )0≤t , P ). Denote by (F S )0≤t the completed RCLL process on Ω = (Ω, t filtration generated by S, which satisfies the usual assumptions. Then S is ( ) S S S ¯ an adapted process on Ω := Ω, F∞ , (Ft )0≤t , P|F∞ as well. Set Mmin := S (ΩS , (S, 1)). The market Mmin can be thought of as the market perceived by a participant who can only observe the actual prices S and knows their history, i.e. the information structure given by (FtS )0≤t . This information is assumed to be available without charge. Now assume that a market participant can buy the information structure (Ft )0≤t for some price I > 0. We have the following situation in mind: We assume that there exists an information service, that charges a fee of I for the access to a source of information, e.g. a newspaper where estimations or forecasts of economical indicators are published. A model for such a market could be a markovian process (S, S ′ ) given by a SDE, where S ′ is a non-traded state variable which can be observed. If accurate observation of S ′ involves expensive data collection and computing, one can ask whether it pays off for an investor, to buy this information from a provider. In a second type of model an initially enlarged filtration can be used to represent some insider information, e.g. the knowledge of the outcome of a random variable. We now take a look at a simple version of this problem. min ) ⊆ V 2 (M) implies E[Z|F S ] ∈ Assume D2 (M) ̸= ∅. Note that V 2 (M ∞

2 S for all Z ∈ D (M). E[Z|F∞ ] L2 ≤ ∥Z∥L2 for Z ∈ D2 (M) together S ] ⊆ D 2 (Mmin ) implies β(Mmin ) ≤ β(M) < ∞. This can with E[D2 (M)|F∞

D2 (Mmin )


49

be interpreted as access to additional information increases the risk premium prevailing in a market. See [33] for a class of examples, where β(Mmin ) and β(M) can be calculated explicitly. Assume 0 < β(Mmin ). We consider an investor, endowed with an initial capital of v > 0, who wants to buy a mean-variance efficient portfolio with an expected return of e > v. For simplicity we assume the access to the information structure (Ft )0≤t only to be offered at time t = 0 and it is not possible the sell the information later, so that there is only a choice between two alternatives: The investor can either work in the market Mmin , where the optimal portfolio e is vV v , where V e¯ for e¯ ∈ R denotes the mean-variance efficient portfolio in Mmin with initial price 1 and expected terminal value e¯, or the investor can irreversibly invest the amount I < v to get access to the information structure e (Ft )0≤t and invest the remaining capital in the optimal portfolio (v − I)V˜ v−I , where V˜ e¯ for e¯ ∈ R denotes the mean-variance efficient portfolio in M with initial price 1 and expected terminal value e¯. We want to determine the risk premium indifference price for these alternatives, i.e. we want to know for which access price 0 < I < v the resulting Sharpe ratios of the two alternatives are equal: We calculate, using Corollary 1.7.6, [ ] e E (v − I)V˜ v−I − v β(Mmin ) =! √ ( ) e Var (v − I)V˜ v−I [ e ] E V˜ v−I − 1 I √ = √ ( e )− ( e ) Var V˜ v−I (v − I) Var V˜ v−I ( I [ = β(M) − = β(M) 1 − e ] (v − I) e−v = β(M) . e−v+I

E V˜

v−I

−1

I e−v+I

)

β(M)

(1.7.15)

Solving for I we find as the risk premium indifference price Ie−v for such an investor, β(M) − β(Mmin ) Ie−v := (e − v). (1.7.16) β(Mmin ) This means, that for an investor with expected excess return δ := e − v, the optimal strategy in the mean-variance efficiency sense is to buy access to the information structure (Ft )0≤t , if min{v, Iδ } > I holds. For Iδ β(Mmin ), β(Mmin ) δ max we have 0 < Iδ < v iff 0 < vδ < β(M)−β(M min ) =: v . This can be interpreted max in the following way: The closer δ < δ is to δ max , i.e. the more risk an investor is willing to take, the higher is the risk premium indifference price. Hence Iδ is the supremum of the still acceptable access prices to the additional information contained in (Ft )0≤t for a mean-variance efficient investor.

50


This model can be extended in the following way: Assume a family of filtrations {(Ftγ )0≤t | 0 ≤ γ ≤ 1} satisfying the usual assumptions to be given, such that (FtS )0≤t = (Ft0 )0≤t and (Ft )0≤t = (Ft1 )0≤t and Ftγ ⊆ Ftγ˜ for all t ≥ 0 and γ ≤ γ˜ . From this we can construct ( the family of market ) models γ γ γ γ γ ¯ γ {M := (Ω , (S, 1))| 0 ≤ γ ≤ 1}, where Ω = Ω, F∞ , (Ft )0≤t , P|F∞ . As an example, for the above mentioned markovian model with additional non-traded S′ state variables, one can define Ftγ := FtS ∨ F0∨(t−ϵ(1−γ)) for some ϵ > 0. This can be interpreted as an overall market model, where information structures of different quality are available. Now assume the increasing function γ 7→ β(Mγ ) to be strictly increasing, continuous and β(Mmin ) > 0. After a change of parameterization, we can assume β(Mγ ) = β(Mmin ) + γ(β(M) − β(Mmin )) = β + γ∆β, where ∆β := β(M) − β(Mmin ) and β := β(Mmin ). We denote the access price to the information structure (Ftγ )0≤t by I(γ). We want to determine the optimal amount of money to be invested into access to information in the mean variance efficiency sense. For simplicity, we make some assumptions on the access price function I: We assume I to be continuously differentiable, I ′ to be strictly increasing and I(0) = I ′ (0) = 0. Convexity of I means that it gets increasingly costly to further improve the quality of the additional information. Denote by Iδγ the risk premium indifference price for the information (Ftγ )0≤t . β By (1.7.16), we have for 0 < γ ≤ 1 and 0 < vδ < γ∆β , Iδγ =

∆β β(M) − β(Mmin ) γδ = γδ < v. min β(M ) β

β Assume vδ < ∆β , so that Iδγ is defined for all 0 < γ ≤ 1. Optimality in the mean-variance efficiency sense means, that for an initial capital of v and a desired expected return of e > v, we want to detere opt ) mine γ opt such that the Sharpe ratio for the portfolio (v − I(γ opt ))V˜ v−I(γ opt γ

is maximal, where V˜γe¯ for e¯ ∈ R and 0 ≤ γ ≤ 1 denotes the mean-variance efficient portfolio in the market Mγ with initial value 1 and expected terminal value e¯. By (1.7.15), this problem is equivalent to maximizing the expression δ F (γ) := β(Mγ ) δ+I(γ) , where δ := v − e. Proposition 1.7.11. The following alternative holds: Either F ′ (1) ≥ 0 and then γ opt = 1, or F ′ (1) < 0 and then γ opt is the unique solution to the equation dβ(Mγ ) dγ ′ I (γ)

=

∆β F (γ) = , ′ I (γ) δ

and 0 < γ opt < 1 holds. Proof. Note that we can extend I to a strictly convex, continuously differentiable function I˜ : [0, ∞) → [0, ∞), such that limx→∞ I˜′ (x) = ∞. Consider the

1.7. MEAN-VARIANCE EFFICIENCY function F˜ (γ) =

γ∆β+β 1+

˜ I(γ) δ

51

, which coincides with F on [0, 1]. We have

F˜ ′ (γ) =

∆β 1+

˜ I(γ) δ

γ∆β + β I˜′ (γ) −( . )2 ˜ δ I(γ) 1+ δ

Since F˜ (0) = β, F˜ ′ (0) = ∆β > 0 and limγ→∞ F˜ (γ) = 0, there exists a γ˜ opt such that F˜ (γ) ≤ F˜ (˜ γ opt ) for all γ ∈ [0, ∞) and F˜ ′ (˜ γ opt ) = 0. We will now show, that F˜ ′ (¯ γ ) = 0 for γ¯ ∈ [0, ∞) implies F˜ to have a global maximum at γ¯ . From this it follows that γ¯ = γ˜ opt , since F˜ can not have a local minimum at any γ ∈ (0, 1). Set C := F˜ (¯ γ ) = γ¯∆β+β By elementary transformations the following ˜ γ) . I(¯ 1+

equivalence is shown:

δ

F˜ ′ (γ) = 0 ⇐⇒

∆β I˜′ (γ) = . δ F˜ (γ)

(1.7.17)

We now determine the unique function J : [0, ∞) → [0, ∞) such that ˜ γ ) + (γ−¯γ )∆βδ , hence for all γ ∈ [0, ∞). We easily find J(γ) = I(¯ C

γ∆β+β J(γ) = C 1+ δ J ′ (¯ γ ) = ∆βδ C

which equals I˜′ (¯ γ ) by (1.7.17). Since I˜ is strictly convex, we find ˜ I(γ) > J(γ) for γ ̸= γ¯ , hence F˜ (γ) < F˜ (¯ γ ) for γ ̸= γ¯ . By (1.7.17) the assertion follows. Obviously γ opt depends on δ. For δ such that 0 < γ opt < 1, we find by a simple calculation opt

dF (γ opt ) β(Mγ )I(γ opt ) = > 0. dδ (δ + I(γ opt ))2 This means, that the Sharpe ratio, i.e. the risk premium, for the mean-variance efficient portfolio with expected excess return of δ, is no longer constant but increases in δ. In other words, taking on risk is rewarded with a super linearly increased expected outcome, provided an appropriately increased percentage of the initial capital is spent on information. For a qualitatively similar finding see [94]. Remark 1.7.12. In a more elaborate model one would assign to an increasing process K, representing the accumulated amount of money spent on information, a filtration (FtK )0≤t , representing the information now available for hedging. One has to optimize then over all self-financing strategies and such processes K. Well, this is rather a research project in its own.

52


Chapter 2

Classical Theory: Hedging of Contingent Claims, Options, Black Scholes 2.1

A Discrete Example

Before we go on to consider the general model of the preceeding section we have a short look at the classical model based on the geometric Brownian motion as a model for the stock price. To motivate this model which was extensively studied by Karatzas and Shreve in [[50]] we go even beyond this and present a discrete example: Consider the situation where an investor has two possibilities of investing, namely a secure investment (bond) with nonrandom interest rate and an investment with uncertain gain (stock). For a fixed time horizon N ∈ N the values of the investment may change at times {1, 2, · · · , N }. For n ∈ {1, · · · , N } denote by : 1. Pn0 = (1 + r)n · P00

(2.1.1)

the value of one unit of the bond in the open time interval (n, n + 1), 2. Pn1

(2.1.2)

the value of one unit of the stock in the open time interval (n, n + 1). The game starts just after time 0 with initial amount of money x, which is composed by N00 units in the secure and N01 units in the bond, so that X0

=

N01 P01

+

N00 P00

=

x.

(2.1.3)

Between the times 0 and 1 it is allowed to rearrange the portfolio, that is to rearrange the amounts in bond and stock, so that just before time (t = 1) the investor has N11 units of shares and N10 units in the secure investment. As we 53

54

CHAPTER 2. CLASSICAL THEORY

assume that the whole amount of initial money is invested, the portfolio before the first game is described by X0

N11 P01

=

N10 P00

+

=

x.

(2.1.4)

We may view (N11 , N10 ) as a bet on the first game. 1 , N0 ) Just after (n−1) the investor’s wealth is described by the portfolio (Nn−1 n−1 with value Xn−1

1 1 Nn−1 Pn−1

=

0 0 Nn−1 Pn−1 .

+

(2.1.5)

Between (n − 1) and n the portfolio is rearranged again, so that 1. just before time n: Xn−1

=

1 Nn1 Pn−1

=

Nn1 Pn1

0 + Nn0 Pn−1

(2.1.6)

Nn0 Pn0 .

(2.1.7)

2. and just after this time: Xn

+

The change of wealth is then given by: Xn − Xn−1

=

1 Nn1 (Pn1 − Pn−1 )

+

0 Nn0 (Pn0 − Pn−1 ).

(2.1.8)

With the interest rate r for the bond and a presumed random interest rate Rn of the stock at time n the wealth satisfies: Xn − Xn−1

=

rXn−1

+

Nn1 Pn1 (Rn − r).

(2.1.9)

Let Zn = (1 + r)−n Xn be the discounted value of the investor’s wealth then this may be described by the equation Zn − Zn−1

=

1 1 (1 + r)−n Nn−1 Pn−1 (Rn − r).

(2.1.10)

Now we have to model Rn : Let ϵ1 , · · · , ϵN be i.i.d random variables with P (ϵi = 1)

=

r−a b−a

=

1 − P (ϵi = −1)

(2.1.11)

r−a . b−a

(2.1.12)

with b, a ∈ (−1, ∞), a < r < b. On a suitable probability space define Yn

=

n ∑ k=1

(ϵk − 2p + 1)

with p =

This process obviously is a martingale wth respect to σ(ϵ1 , · · · , ϵn ) =: Fn . Now define a+b b−a + ϵn , (2.1.13) Rn = 2 2

2.1. A DISCRETE EXAMPLE so that Rn − r

=

55

1 (b − a)(Yn − Yn−1 ) ∈ {a, b}, 2

(2.1.14)

and Zn − Zn−1

=

1 1 (1 + r)−n (b − a)Nn1 Pn−1 (Yn − Yn−1 ). 2

(2.1.15)

This means that Zn is written as a stochastic integral w.r.t. Yn . A European option is a contract just after time 0, which allows to buy one unit of stock at time N at the price K. This time is called maturity time and the price fixed at time 0 is the so-called strike price. Just after time N the option is exercised which means that the investor receives the stock from the agent and sells the stock to gain the amount PN1 − K if PN1 > K. If PN1 ≤ K the option is obsolete, nothing is done. Hence at time N the value of the option is (PN1 − K)+ . The relevant question now is, what should be paid for the option at time 0? Black-Scholes give the answer to this question which is based on the concept of a hedging strategy: A hedging strategy for the initial value x for the above described option is a portfolio, that is a scheme of arrangements (Nn1 , Nn0 ) consisting of natural processes, so that for Xn = Nn1 Pn1 +Nn0 Pn0 ,

1 0 Xn −Xn−1 = Nn1 (Pn1 −Pn−1 )+Nn0 (Pn0 −Pn−1 ) (2.1.16)

the following holds: 1. X0 = x 2. Xn ≥ 0 3. XN = (PN1 − K)+ . This means that by using a hedging strategy, that is by using an appropriate portfolio management, we may achieve the same gain at time N as by using the option, without going bankrupt. Then the following theorem holds: Theorem 2.1.1. A hedging strategy for the initial value x exists if and only if x = x0

=:

E[(1 + r)−N (PN1 − K)+ ].

(2.1.17)

There is exactly one hedging strategy for x0 and Nn1 is never negative. Proof. 1. Let (P 0 , P 1 , X, Z) be the processes associated with a hedging strategy for x. We know that Zn is a P -martingale w.r.t. Y with Z0 = x, and the following equation holds ZN = (1 + r)−N (PN1 − K)+ , as (N 0 , N 1 ) is a hedging strategy. So x = x0 .

56

CHAPTER 2. CLASSICAL THEORY 2. Define the martingale Mn = E(ZN |Fn ). It is a nice exercise to prove that a process Nn1 exists with 1 Zn − Zn−1 = (1 + r)−n Nn1 Pn−1 (Rn − r).

(2.1.18)

This result is known as a martingale representation theorem, the result here is easily proved without further techniques from stochastic analysis. Define : Xn

=

(1 + r)n Zn

and Nn0

(Xn − Nn1 Pn1 ) . Pn0

=

(2.1.19)

Then (N0 , N1 ) obviously is a hedging strategy. As 1 1 E((Pn1 −K)+ |Pn−1 , Pn1 = (1+b)Pn−1 )

≥

1 E((Pn1 −K)+ |Pn−1 , Pn1 = (1+a)Pn−1 ), (2.1.20)

so obviously Nn1 ≥ 0. The reader is now invited to think about what a fair price of the option might be.

2.2

The Continuous Case: The Market

We consider a market M with (d + 1) investment possibilities, which are traded continuously until the fixed finite horizon T , 0 ≤ T < ∞, is reached. One investment is described by dP0 (t) = r(t)P0 (t)dt,

P0 (0) = P0 ,

(2.2.1)

which means that it evolves in a deterministic safe way (bond). The other d financial instruments (stocks) are described by: dPi (t) = bi (t)Pi (t)dt + Pi (t)

∑

σij (t)dwtj ,

Pi (0) = Pi .

(2.2.2)

We make the usual assumptions to make sure that on a probability space (Ω, F, P ) which carries a n-dimensional Brownian motion (wt , (Ftw )) strong solutions of the above equations exist. (So, the coefficients (r, b, σ) of the market M are assumed to be progressively measurable w.r.t. (Ft ) and sufficiently integrable [see Stochastik Skriptum].) Also note that if the coefficients are deterministic, then the processes are Markov processes. The given initial capital x ≥ 0 is invested into the bond and the d stocks. Let Ni (t) be the number of investments of type i, owned by the investor at time t. The investor’s wealth at time t is then described by: ∑ x0 = x = di=0 Ni (0)Pi ∑d xt = i=0 Ni (t)Pi (t).

(2.2.3)

2.2. THE CONTINUOUS CASE: THE MARKET

57

If at times t and t + h trading take place without withdrawing or adding fonds, then ∑ xt+h − xt = Ni (t)[Pi (t + h) − Pi (t)]. (2.2.4) If at time t+h the investor consumes the amount hct+h , then his wealth reduces and (2.2.4) must be replaced by ∑ xt+h − xt = Ni [Pi (t + h) − Pi (t)] − hct+h . (2.2.5) The continuous analogue then is: ∑ dxt = Ni (t)dPi (t) − ct dt.

(2.2.6)

The equations (2.2.1) -(2.2.2)-(2.2.3) then take the form (πi (t) = Ni (t)Pi (t)) dxt = (r(t)xt − ct )dt +

d ∑

(bi (t) − r(t))πi (t)dt +

∑∑ i

i=1

πi (t)σij dwtj , (2.2.7)

j

where: Definition 2.2.1. 1. the predictable process (π(t) = (πi (t))i=1,...,d , Ft ) is a portfolio-process with ∫ 0

T

||π ′ (t)σ||2 dt +

∫

T

|π ′ (t)(b(t) − r(t)1)|dt < ∞

(2.2.8)

0

a.s., whose i-th component gives the amount invested in the i-th stock. || · || denotes the usual Euclidean norm. ∫T 2. (ct , Ft ) is a consumption process with CT := 0 ct dt < ∞ a.s.. Instead of (ct , Ft ) we often consider the cummulative consumption process (Ct , Ft ), which is a right-continuous, increasing process with C(0) = 0, C(T ) < ∞. This process might then not be absolutely continuous. 3. We can make the model a bit more general by also considering an endowment process (Et , Ft ) with the same properties as the cumulative consumption process. It has the interpretation of a cumulative income. The wealth process then takes the following form: dxt = (r(t)xt dt − dCt + dEt +

d ∑

(bi (t) − r(t))πi (t)dt +

i=1

∑∑ i

πi (t)σij dwtj ,

j

(2.2.9) Note that each component of π is allowed to be negative what we interpret as a short selling of the stock. Also ∑ π0 (t) := xt − πi (t) (2.2.10) i=1

is allowed to be negative which means that we are allowed to borrow money from the bank at the given interest rate.

58


The solution of the wealth equation is the strong solution on (Ω, F, Ft , w) given by { } ∫ t ∫ ∫ t ∫ ∫t [ ] − 0s rdu T − 0s rdu T 0 rds xx,π,c = e x + e π(s) (b(s) − r(s) · 1) − c ds + e π σdw s s , t 0



0

(2.2.11)



1  ..  where 1 =  .  ∈ Rd . 1 Remark 2.2.2. We now must think about which strategies we allow to be admissible. For the following considerations it is useful to introduce the so-called discounted gains process Gπ (t) associated with π: ∫ t Gπ (t) := P0−1 (s)π ′ (s)[σdw + (b(s) − r(s)1)ds] = P0−1 (t)x0,π,0 . (2.2.12) t 0

Consider the following simple example due to S. Shreve: Let a market be given with n = d = 1, σ = 1, r = b = 0 so that the gains process corresponding to the portfolio π(t) = √T1−t satisfies ∫

t

Gπ (t) = 0

dw √ s ,0≤t 0) > 0

(2.2.15)

is called an arbitrage opportunity (or free lunch). The market M is said to be arbitrage free, if no such portfolios exist. The condition says that it is not possible to make a positive gain with positive probability while there is no exposure to risk. The next example due to A. V. Skorokhod will show that nds-admissible strategies exist which allow arbitrage: Exercise 2.2.7. Consider the market given by the data (Ω, F, P ), n = d = 1 , where R(t) is the d = 3-dimensional Bessel process 1, r = 0 σ = 1, b(t) := R(t) with drift (see Stochastik Script): dR(t) = (

1 − 2)dt + dw(t)), R(0) = 1. R(t)

(2.2.16)

• Show that w ˜t = wt − 2t is a Brownian motion under the measure P˜ given by the density exp(2(w1 − 1)) • Show that {R(t) > 0, 0 ≤ t ≤ 1} has probability one under both measures. • The gains process

∫

Gπ (t) = wt + 0

t

ds = 2t − 1 + R(t) R(s)

for the constant portfolio π = 1 is nds-admissible and the market is not arbitrage free: P (Gπ (1) = 1 + R(1) > 1) = 1. Exercise 2.2.8. In order to put the results of this chapter into the context of the preceding chapter the reader is asked to work out the following questions and problems: • Write the market model of this chapter in terms of the model of chapter 1. What is in this context (Ω, S), (2.2.17) (Ω, (S, 1)) etc.

(2.2.18)

60

CHAPTER 2. CLASSICAL THEORY • Compare the no-arbitrage condition here to the one in 1.2.5 • Give some arguments, whether the law of one price holds in the context of the model here.

2.3

The Equivalent Martingale Measure

The coefficient in the stochastic integral in (2.2.11) is predictable and integrable. So this stochastic integral is a local martingale and hence ∫ t ∫ ∫ s − 0t rds mt := xt e −x+ e− 0 rdu cs ds (2.3.1) 0

is a local semimartingale under P . So under P˜ (A) = E(zT IA ), where (zt ) with   ∫  ∑∫ t  1 zt = exp − θsj dwsj − ||θs ||2 ds , θt := σ −1 (b − r · 1) (2.3.2)   2 0 j

is the corresponding Girsanov funktional (see Stochastik Script), ∫ t ∫ ∫ t ∫ ∫t s s ˜ where dw ˜ = dw +θs ds. e− 0 rdu π T σdw, e− 0 rdu cs ds = x+ xt e− 0 rds + 0

0

(2.3.3) Here we assume that θ is well defined (σ is assumed to be invertible) and good enough so that the Girsanov functional is a martingale. Further on in this section we will specify properties of θ to gain existence and certain desirable properties of the market tools. To make things first a bit simpler let us for the moment forbid bankruptcy in the sense that the temporary assumption holds At : xt ≥ 0, for the initial x in order to find out whether an admissible strategy exists. As we have not specified properties of θ this makes the following considerations simpler. The reader is invited to find more general conditions on the basis of the nds-property. For an nds-portfolio and a consumption process (π, c) the left hand side of 2.3.3 is nonnegative and the right hand side is a P˜ -local martingale. So the left hand ∫ side is a nonnegative supermartingale and hence xt e− rds is a nonnegative supermartingale. Finally we have ˜ T e− E[x

∫T 0

∫ rds

+

T

e−

∫t 0

rdu

ct dt] ≤ x.

(2.3.4)

0

This gives a necessary condition for admissibility: ∫ T ∫s ˜ E[ e− 0 rdu cs ds] ≤ x. 0

The reader should give an intuitive interpretation of this result. More exactly we have:

(2.3.5)

2.3. THE EQUIVALENT MARTINGALE MEASURE

61

Theorem 2.3.1. Let x ≥ 0 and let the above condition for admissibility of the consumption process be fulfilled for the initial condition x. Then there is an nds-admissible portfolio process π, so that (π, c) is admissible for x in the sense of condition At . Proof. Apply the martingale representation theorem to ∫ T ˜ E( cs P0−1 (s)ds|Ft )

(2.3.6)

0

under P˜ to get vector process ϕ(t) as integrand. It is easily seen that the portfolio is given by π(t) = P0 (t)σ ′−1 (t)ϕ(t). (2.3.7) Note however that the martingale representation theorem must not be applied without further technical interpretations: The augmentation of the filtration w.r.t. P˜ might be different from the one w.r.t. P . So the problem has to be modified to make the representation result applicable. A nice exercise, or see the proof of theorem 2.4.5. To arrive at this result we assumed without notice in 2.3.2 that the inverse of σ exists. Exercise 2.3.2. • Remember that the Girsanov functional defines a new measure, if θ satisfies ∫ 1 T E(exp( ||θ(t)||2 dt) < ∞, (2.3.8) 2 0 (see Stochastik Script) which again is true if it satisfies the Novikov condition. • Derive that under P˜ the discounted stock prices P0 (t)−1 Pi (t) are local martingales. • P and P˜ are equivalent. • Under P˜ the gains process for an nds-portfolio is a local martingale with expectation less or equal to zero for all Ft -stopping times. From here on we give up the temporary assumption At again. Definition 2.3.3. Let the following hold for the market under consideration: • d ≤ n, (the number of stocks must not exceed the number of random sources) • there exists a progressively measurable process θ with b(t) − r(t)1 = σ(t)θ(t) such that

∫

T

|θi (t)|2 dt < ∞ a.s.,

0

and the corresponding Girsanov functional has expectation 1.

(2.3.9)

(2.3.10)

62


Then the market is called standard. So, for a standard market M the above reasoning is correct. Remark 2.3.4. • From the above exercise we see that under the new measure P˜ the stock appreciation rates bi are replaced by r, as the price of the i-th stock under P˜ is given by dPi = Pi (rdt + σdw).

(2.3.11)

This motivates the terminology ”yield equating” or ”risk neutral” (equivalent) martingale measure for P˜ . • The property d ≤ n in the above definition is no real restriction, as if this is not true in a standard market we can reduce the number of stocks by building mutual funds. Note that if σ in 2.3.2 has full rank θ may be taken to be θ(t) = σ ′ (t)(σ(t)σ ′ (t))−1 [b(t) − r(t)1]. Exercise 2.3.5. Compare the above construction and results to the discrete case: What or where is the risk neutral martingale measure in the discrete case. Now we can state the classical result on arbitrage opportunities in a market M: Theorem 2.3.6.

• If M is arbitrage free, then there exists a process θ with: b − r1 = σθ.

(2.3.12)

θ is called the market price of risk or relative risk process or risk premium process. • If θ exists with all properties above (in other words: Let M be standard), then M is arbitrage free. Proof. Let π be an nds-strategy. • If M is standard, then a predictable process θ with all the above properties exists. Assume π is an arbitrage opportunity, then Gπ (T ) ≥ 0 a.s. and from 2.3.2 the gains process is nonpositive under P˜ . So Gπ (T ) = 0 a.s. which contradicts the property of being an arbitrage opportunity. • Let A ⊆ [0, T ] × Ω with positive product measure, such that – σ ′ π = 0 (no exposure to risk) – π ′ (b − r1) ̸= 0 (non-zero rate) on A. For k > 0 let π ˆ = ksign(π ′ (b − r1))π1A

(2.3.13)

2.3. THE EQUIVALENT MARTINGALE MEASURE

63

then Gπˆ > 0 on a set B ∈ FT with positive probability. So π ˆ is an arbitrage opportunity. To make this construction impossible we must have, that every vector in ker σ ′ (t, ω) should be orthogonal to b(t, ω) − r(t, ω)1 for almost all (t, ω). In other words b(t, ω) − r(t, ω)1 should be in (ker σ ′ (t, ω))⊥ = range(σ). This however is exactly the condition given in the theorem. It is then possible to choose θ to be predictable.

This theorem gives an easily verifiable result on no-arbitrage properties of a standard market. The reader should notice here already that this result is closely connected to the question of existence of equivalent (local) martingale measures (see section 1.4). This will be made more rigorous in the subsequent chapters. A complete solution of the problem for general semimartingale models is found in [21]. The above theorem connects the question of existence of θ satisfying b − r1 = σθ

(2.3.14)

with the property of the market to be arbitrage-free. When we face the problem of solving b − r1 = σθ (2.3.15) for θ we have the following possibilities: • there is exactly one solution, • there are many solutions, • there is no solution. So if there is no solution (which has the consequence that the Girsanov functional cannot be defined which we used to define an equivalent martingale measure) our market is not arbitrage-free. The following considerations will be concerned with the other two possibilities and will connect the uniqueness of the solution to another important property of the market, namely to be complete. To this end let us now assume that we are in a standard market.

Definition 2.3.7. able ξ with

• A contingent claim is an FT -measurable random vari˜ −1 (T )ξ) < ∞. u0 = E(P 0

(2.3.16)

• A contingent claim ξ is attainable if there is an nds-admissible portfolio π such that xu0 ,π,0 (T ) = ξ. (2.3.17) • The standard market is called complete if any contingent claim is attainable.

64


The typical contingent claim is a Borel-function of the vector of one or more stocks at time T : ξ = ϕ(P1 (T ), · · · , Pd (T ). Special contingent claims are given by the options which we will get to know in the next section. Exercise 2.3.8. • In definition 2.3.7 the property of being nds-admissible for π may be replaced by Gπ (t) is a P˜ − martingale.

(2.3.18)

(note that the discounted gains process is a P˜ -supermartingale for ndsadmissible portfolios) • For x < u0 there can be no nds-admissible portfolio π with xx,π,0 (T ) ≥ ξ.

(2.3.19)

• For x > u0 there can be no nds-admissible portfolio π with xx,π,0 (T ) = ξ.

(2.3.20)

and Gπ being a martingale. Theorem 2.3.9. A standard market M is complete if and only if • n = d, and • σ(t, ω) is invertible for almost all (t, ω). Proof. The necessary part of the proof relies on the following result from linear algebra: Let L be the space of linear functions Rn → Rd . There exists a bounded measurable function f from L to Rn , such that: ∀σ ∈ L, f (σ) ∈ ker(σ), and f (σ) ̸= 0, if ker(σ) ̸= {0} . Now let the market be complete and let ψ(t) = f (σ(t)) with f from above. De∫T compose the contingent claim ξ = P0 (T )(1 + 0 ψdw ˜ = ξ + − ξ − . Then under P˜ −1 −1 ˜ P0 (T )|ξ| has finite expectation and E(P 0 (T )ξ) = 1. So both the positive and the negative part of ξ are attainable and so there exist portfolios π +,− such that the corresponding gains processes are P˜ -martingales. Then for π = π + − π − + − the process Gπ = Gπ − Gπ is a P˜ -martingale and ∫ T Gπ (t) = ψ ′ dw. ˜ (2.3.21) 0

∫.

Now 0 ψ ′ dw ˜ is a martingale and equals nally we have

∫t 0

P0−1 (s)π ′ (s)σdw ˜s almost surely. Fi-

ψ(t, ω) = P0−1 (t, ω)σ ′ (t, ω)π(t, ω) ∈ range(σ ′ (t, ω)) = (ker(σ(t, ω)))⊥ (2.3.22) ∩ a.e. on [0, T ] × Ω. Now ψ ∈ (ker(σ(t, ω)))⊥ ker(σ(t, ω)) = {0} and by construction ψ ̸= 0 if ker(σ) ̸= {0}. So ker(σ) = {0} and the result follows.

2.4. EUROPEAN OPTIONS, B-S-FORMULA, AMERICAN OPTIONS

65

The sufficiency is a simple consequence of the martingale representation theorem (see Stochastik Script): Consider for any contingent claim ξ the martingale ˜ −1 (T )ξ|Ft ) =: m E(P ˜t 0

(2.3.23)

with representation w.r.t. P˜ and predictable integrand ϕ. Then it is immediate that π ˜ (t) := P0 (t)ϕ′ (t)σ −1 (t) (2.3.24) is a portfolio in the sense of definition 2.2.1 and the following relation holds m ˜ t = P0−1 (t)xu0 ,˜π,0 (t).

(2.3.25)

So ξ is attainable and obviously the P˜ -martingale Gπ˜ (t) is bounded from below, so that π ˜ is nds-admissible. Remark 2.3.10. The theorem says that for M to be complete we must have the same number of stocks and random sources which must be in the driving terms of all stocks in a nonsingular way. Exercise 2.3.11. • Recall the above used representation result from the Stochastik Script. • Does the result of the preceding theorem hold for coefficients depending on past and present of the stocks, that is for coefficients which are σ(Pi (s)|0 ≤ s ≤ t, i = 1, · · · , d) measurable?

2.4

European Options, B-S-Formula, American Options

Let us consider the following contract between two agents called buyer and seller: The seller offers at time t = 0 (now) to deliver a random amount ξ > 0 at time T > 0 (in the future), where ξ is a contingent claim. For this commitment of the seller the buyer pays at time t = 0 an amount x. The question is, what the correct price x might be. Let us consider the situation of the seller: The price x in his eyes should be high enough to find a portfolio-consumption strategy π, C to make good on his commitment, that is to hedge the contingent claim at time t = T . So, his requirement on the price is given by xx,π,C − ξ ≥ 0 a.s.. T

(2.4.1)

Definition 2.4.1. The smallest amount x that allows to achieve this is called the upper hedging price: pup = inf{x ≥ 0|∃(π, C), π nds-admissible, xx,π,C − ξ ≥ 0 a.s.}. T

(2.4.2)

Thus the upper hedging price is the smallest payment the seller can accept to cover the obligation without risk. The buyer starts with the debt −x and tries to find a portfolio-consumption

66


strategy (π, C) so that the payment ξ which he receives at time T makes it possible to cover the debt from time t = 0 by purchasing the contingent claim: x−x,π,C + ξ ≥ 0 a.s.. T

(2.4.3)

Definition 2.4.2. The largest amount x to achieve this is the lower hedging price: plow = sup{x ≥ 0|∃(π, C) π nds-admissible and x−x,π,C + ξ ≥ 0 a.s.}. (2.4.4) T Exercise 2.4.3.

˜ −1 (T )ξ) the following inequality holds: • For u0 = E(P 0 0 ≤ plow ≤ u0 ≤ pup ≤ ∞.

(2.4.5)

(use 2.3.2) • Show that plow in 2.4.2 may be described by plow = sup{x ≥ 0|∃(π, C) Gπ is a P˜ −martingale and x−x,π,C +ξ ≥ 0 a.s.} (2.4.6) • Every price outside [plow , pup ] leads to an arbitrage opportunity. For a price within the interval no arbitrage occurs, why we call this interval the arbitrage-free interval. • Use the martingale representation based on stochastic flows (see Stochastik script) to derive a more explicit expression for the hedging portfolio.

2.4.1

The complete case

We assume to be in a standard complete market and for simplicity we assume that n = d = 1. Most of the results carry over to the case of arbitrary n = d Assume that at time t = 0 a contract between the buyer and the seller is signed: The buyer buys at time t = 0 the option to receive from the seller at the later maturity time T the share P1 (T ) for the exercise price q which is fixed at time t = 0. If at the maturity time T P1 (T ) < q then the option is worthless, if however P1 (T ) > q then the buyer makes use of the option: At time T he buys the share at the price q from the seller. He immediately sells the stock and gains (P1 (T ) − q)+ . (2.4.7) The option so is equivalent to a payment (P1 (T ) − q)+ in the eyes of the seller and a gain (P1 (T ) − q)+ from the buyers point of view. We here use a slightly more general definition of contingent claim than in 2.3.7 to be able to handle the consumption process. So we repeat the definitions: Definition 2.4.4.

1. A contingent claim is

(a) a payoff rate (gt , Ft , 0 ≤ t ≤ T ) and (b) a terminal payoff fT bei maturity.

2.4. EUROPEAN OPTIONS, B-S-FORMULA, AMERICAN OPTIONS Here g ≥ 0 is measurable, adapted, fT ≥ 0 FT -measurable and [∫ T ]µ E gt dt + fT < ∞, for a µ > 1.

67

(2.4.8)

0

An option again is a special case of a contingent claim with g ≡ 0 und fT = (P1 (T ) − q)+ .

(2.4.9)

2. Let x ≥ 0 and (π, c) admissible, f, x given data. The pair (π, c) is called a hedging strategy against the contingent claim (gt , fT ), if (a) ct = gt for 0 ≤ t ≤ T (b) xT = fT a.s., where (xt ) is the wealth process associated with (π, c) and initial x0 = x. Theorem 2.4.5. Consider the contingent claim (g, fT ) and let −

Q=e

∫T 0

∫ rdu

T

fT +

e−

∫s 0

rdu

gs ds.

(2.4.10)

0

then the

fair

price of the contingent claim (g, fT ) is given by ˜ EQ.

(2.4.11)

˜ There is a hedging strategy with initial x = EQ. Proof. Define ∫t

ηt := e

0

[ rds

∫ ˜ + mt − EQ

t

−

e

∫s 0

] rdu

gs ds

(2.4.12)

0

˜ ˜ with mt := E(Q|F ) − EQ. ∫t t rds T Let π(t) = e 0 (σ )−1 φ(t), where we have to find φ(t). (mt ) is a P˜ -martingale and Nt := E[QzT |Ft ] is a P -martingale, so that mt =

Nt − E(QzT ). zt

(2.4.13)

Note that (zt ) is the above defined Girsanov functional for the measure transformation. From the martingale representation we have ∫ Nt = E[QzT ] + y(s)dws (2.4.14) for an Ft predictable process yt = (yt1 . . . ytn ) P-a.s. Furthermore we have mt = u(Nt , zt ) − E(QzT ),

(2.4.15)

68


where u(x, y) = xy , so that from Itô’s-formula ∑∫ (y(t) + N (t)θt ) mt = φj (t)dw ˜j , φ(t) = . zt

(2.4.16)

˜ we get by 2.3.3 Putting this into the equation for π(t), defining x = EQ, [ ] ∫ T ∫ ∫ − tT rdu − ts rdu ˜ ηt = E (e fT + (2.4.17) e gs ds)|Ft , 0 ≤ t ≤ T. t

This however is the wealth process xt with xT = fT , gs = cs . Note that we ˜ have xt ≥ 0. So we have a hedging strategy with initial x = EQ. Exercise 2.4.6. • The above constructed hedging strategy is a.s. unique ˜ for the initial x = EQ. • The above constructed wealth process is the unique wealth process for the fair price x. • plow = u0 = x = pup . • Derive from the last property that x is the fair price. Black-Scholes consider an option as a contingent claim with: g ≡ 0, fT = (P1 (T ) − q)+ .

(2.4.18)

This contingent claim is called a European Call option. For simplicity let again n = d = 1, r(t) ≡ r = const, σ11 (t) = σ > 0. The price process for the contingent claim is given by [ ] [ ] ˜ e−r(T −t) (P1 (T ) − q)+ |Ft = E ˜ (t,P1 (t)) e−r(T −t) (P1 (T ) − q)+ xt = E (2.4.19) where we make use of the Markov property.. Apply Itô’s formula to er(T −s) ν(s, P1 (s))

(2.4.20)

for a good function ν, so ∫

T

∂ν +As ν}ds+a martingale term, ∂t t (2.4.21) 1 2 2 ∂2 ∂ where As is the differential operator As = rP1 ∂x + 2 σ P1 ∂x2 . We now make informal use of the Feynman Kac procedures: Let ν be a solution of the partial differential equation (of Cauchy type): r(T −t)

ν(T, P1 (T ))−e

rν −

ν(t, P1 (t)) =

er(T −s) {−rν+

∂ν = As ν with terminal condition ν(T, x) = (x − q)+ ∂t

(2.4.22)

on [0, T ] × (0, ∞), so we obviously have ˜ t,P1 (t) [ν(T, P1 (T ))] − er(T −t) ν(t, P1 (t)) = 0. E

(2.4.23)


69

From Markov’s property ˜ t,P1 (t) [ν(T, P1 (T ))] = E[ν(T, ˜ E P1 (T ))|Ft ] = er(T −t) ν(t, P1 (t)).

(2.4.24)

˜ ˜ 1 (T ) − q)+ |t, P1 (t)] = xt e−r(T −t) , E(ν(T, P1 (T )|Ft ) = E[(P

(2.4.25)

As

so ν(t, P1 (t)) = xt .

(2.4.26)

If we have a unique solution of the Cauchy p.d.e (2.4.22) we so have an explicit formula for the current value of the option at time t given by the current price of the share P1 (t), the time up to maturity T − t and the exercise price q. The solution of (2.4.22) is explicitly given by: { xΦ(φ+ (T − t, x)) − qe−r(T −t) Φ(φ− (T − t, x)) 0 ≤ t < T, x ≥ 0 ν(t, x) = (x − q)+ t = T, x ≥ 0 (2.4.27) ∫ x − u2 2 x σ 1 1 where φ± (t, x) = σ√t [log q + t(r ± 2 )], and Φ(x) = √2π −∞ e 2 du. With this the fair price is explicitly determined and computable. So: Good Luck! Remark 2.4.7. Let us consider again the Black-Scholes market above. For a C 2 -function h with |h(s)| ≤ const(1 + |s|)β for β > 0 let ξ(T ) = h(P1 (T )). We have seen above that from the Feynman-Kac-formula (see Stochastik script) the price of this claim ˜ −r(T −t) h(P1 (T ))|Ft ) pt = E(e is given by the solution of the corresponding p.d.e. pt = ν(t, P1 (t)), where ν satisfies

∂ν − rν + As ν = 0, ∀x 0 ≤ t < T ∂s ν(T, x) = h(x).

Obviously, from the representation of the martingale term in 2.4.21 dmt = σt P1 (t)

∂ν dw ˜t = σt πdw ˜t ∂x

the optimal hedging portfolio is given by π ˆ=

∂ν (t, P1 (t))P1 (t). ∂x

The last expression contains one of the local characteristics of the contingent claim that are known as the Greeks although the last one is more a star than a Greek letter:

70

CHAPTER 2. CLASSICAL THEORY ∂ν (t, P1 (t)) Delta ∆ = ∂x ∂2ν Gamma Γ = ∂x2 (t,P 1 (t)) Theta Θ = ∂ν (t, P 1 (t)) ∂t Rho ρ = ∂ν (t, P 1 (t)) ∂r ∂ν Vega ∂σ (t, P1 (t)).

These Greeks are used as instantaneous characteristics of the contingent claim. Lacoste [[61]] gave a very nice interpretation of the first two Greeks by making use of the chaos decomposition which sheds light on the mathematical meaning. Exercise 2.4.8. • Look at the chaos decomposition of the option price (see Stochastik script) and derive the relation between the n-th chaos, n = 1, 2 and the Delta and Gamma. • Following the proof above derive the price of the European put: ξ = (q − P1 (T ))+ .

(2.4.28)

(note that P1 (t)e−rt is a P˜ -martingale.) Derive a formula for the relation between put and call the so-called call-put-parity: price of the put = price of the call - price of the stock at time 0 + discounted exercise price.(hint: (q − P1 (T ))+ = (P1 (T ) − q)+ − P1 (T ) + K. • Then show that the European Call optimal hedging portfolio always borrows: π ˆ (t) > x ˆ(t). (hint: note that for ψ(p) = (p − q)+ an inequality holds between ψ ′ (p)p and ψ(p). • Derive the price of the Barrier option: ξ(t) = (P1 (T ) − q)1τa ≤T

(2.4.29)

where τa = inf{t ≥ 0|P1 (t) ≥ a} for a market with n = d = 1, 0 < P1 (0), q < a < ∞, σ = 1. • Derive the price of the Asian option ∫ T ξ=( P1 (t)f (t)dt − q)+

(2.4.30)

0

with f being H¨ older continuous with ∫inTthe setting of the preceding exercise 1 −r(T −t) M ˜ f (t)dt = 1 (e.g. take f ≡ . (hint: write x ˆ = E(ξ|F t t) = e t T 0 where Mt = P1 (t)ϕ(t, Λ(t)) is a P˜ -martingale. Show that ϕ satisfies the numerically easily solved Cauchy problem ∂ϕ σ 2 2 2 ∂ 2 ϕ ∂ϕ + x Λ − (f (t) + rx) + rϕ = 0 2 ∂t 2 ∂x ∂x ϕ(T, x) = x− )

(2.4.31)

(2.4.32)


2.4.2

71

All Kinds of Options

There are all kinds of options (bears, bulls, straddle, strip etc.) and it is nearly impossible (and certainly useless) to give a complete overview of what originates from financial intuition/fantasy. Here we again enter the sales/business/finance department of the bank (see the fancy names above) and leave the scientific division: So be careful when you see the threat: ”Welcome to Wonderland” and keep your wallet closed, better at home, when now we shortly look at the most common types of sold options: Let the market be described by a stochastic process St . • European Call Let K be a price fixed at time 0 for which the agent provides the financial instrument (share) at a fixed time T in the future called the maturity time. K is called the strike price and CT is the price we have to pay at time 0 for the obligation to deliver ST at the strike price K. The buyer’s net profit then is: V (ST ) = (ST − K)+ − CT • European Put Turn around the situation above in an obvious way. So the agent receives a share at the strike price K at maturity, so V (ST ) = (K − ST )+ − CT • Combinations – Straddle V (ST ) = |K − ST | − CT – Strangle V (ST ) = (K1 − ST )+ (ST − K2 )+ − CT – Strip V (ST ) = |K1 − ST |1ST K1 − CT • Bull Spread V (ST ) = |K2 − K1 |1ST ≥K1 + |K1 − ST |1K2 >ST >K1 − CT • Bear Spread V (ST ) = −|K2 − K1 |1ST ≥K2 + |K1 − ST |1K2 >ST >K1 − CT

72


Here the graphs show V (ST ) as a function of the price ST .


73

American-type options are such that in the above examples the strike time T is replaced by a stopping time τ for which different restrictions are agreed. • American Put sup{V (Sτ )/0 ≤ τ ≤ T } where V (Sτ ) = (Sτ − K)+ − Cτ • Russian Options sup{V (Sσ )/σ ∈ [0, T ]}

74

CHAPTER 2. CLASSICAL THEORY where V (Sσ ) = (Sσ − min(Sτ /τ ∈ [0, σ]))+ − CT • Asian Options sup{V (Sσ )/σ ∈ [0, T ]} ∫

where V (Sσ ) = (σSσ −

σ

Su du)+ 0

2.4.3

American Contingent Claim, Complete Case

Again we are in a standard complete market M. Definition 2.4.9. An American contingent claim is an (F)t -adapted random variable ξ : [0, T ] × Ω → [0, ∞] (2.4.33) −1 ˜ and E(sup (t)ξ(t))) < ∞. 0≤t≤T (P 0

In a completely similar way we now define the behaviour of buyer/seller except that now we allow the buyer to act at every time τ ∈ [0, ∞]. So to make things a bit more interesting we consider the slightly different model in 2.2.1 where we allow consumption and endowment. The first definitions easily carry over from the preceding section: • The upper price of the American contingent claim is

Definition 2.4.10. given by

Pup :=

(2.4.34) inf{x ≥ 0∥∃(π, C), π nds-admissible, and xx,π,C (τ ) ≥ ξ(τ ) ∀τ Ft − stopping time}

• The lower price is given by Plow :=

(2.4.35) sup{x ≥ 0∥∃(π, E), π nds-admissible, and E(τ ) = 0, x−x,π,−E (τ ) ≥ −ξ(τ ) for some τ Ft − stopping time}

where we interpret the negative consumption E as an endowment process (see definition 2.2.3). We then have a formally similar result: Theorem 2.4.11.

• Let ˜ −1 (τ )ξ(τ )} v(0) = supτ stopping time {E(P 0

(2.4.36)

ξ(0) ≤ Plow ≤ v(0) ≤ Pup .

(2.4.37)

then


75

• In a standard complete market Plow and Pup are attained and equal. Furthermore, there exist an nds-portfolio π and a consumption process C such that ˜ −1 (τ )ξ(τ )|Ft ) =: x xv(0),π,C (t) = −x−v(0),−π,−E (t) = P0 (t)esssupτ ∈St,T E(P ˆt ≥ ξ(t) 0 (2.4.38) for 0 ≤ t ≤ T and x ˆ(T ) = ξ(T ) (2.4.39) a.s..

1

• The supremum is attained in τˆ = inf{t ∈ [0, T ]|ˆ xt = ξ(t)} ∧ T

(2.4.40)

and we have E(ˆ τ ) = 0, x−v(0),−π,−E (ˆ τ ) = x−v(0),−π,0 (ˆ τ ) = −ξ(ˆ τ ).

(2.4.41)

To prove this theorem we need some facts about optimal stopping. These are stated here without proof, the reader is referred to [Karatzas, I., Lectures on optimal stopping and stochastic control, Department of Statistics, Columbia University, (1993)]: Theorem 2.4.12. Some Facts on Optimal Stopping There exists a nonnegative (Ft , P˜ )-supermartingale Z(t) with RCLL paths, called the Snell envelope of P0−1 (t)ξ(t) under P˜ such that: • Z is the smallest supermartingale that dominates P0−1 (t)ξ(t). • For all t the following relations hold: ˜ −1 (τ )ξ(τ )|Ft ) – Z(t) = esssupτ ∈St,T E(P 0 −1 ˜ −1 (τ )ξ(τ )) = E((P ˜ (τt )ξ(τt ))), where – E(Zt ) = v(t) := supτ ∈S E(P t,T

0

0

τt := inf{θ ∈ [t, T ]|Z(θ) = P0−1 (θ)ξ(θ)}∧T is a stopping time in St,θ , and the process Z(· ∧ τt ) is a P˜ -martingale

• The P˜ -supermartingale is of class D (see Stochastik scriptum), so that the Doob-Meyer decomposition applies: Z(t) = v(0) + m(t) − A(t), 0 ≤ t ≤ T,

(2.4.42)

where M is a uniformly integrable martingale and A is a continuous, ˜ T ) < ∞and A0 = M0 = 0. increasing, adapted process with E(A 1

The essential supremum of a family of random variables was defined in the Stochastik Script: Definition: Let X be a family of nonnegative random variables on a probability space. The esssupX is a random variable X ∗ satisfying – ∀X ∈ X , X ≤ X ∗ and – if Y is a random variable with X ≤ Y for all X ∈ X , then X ∗ ≤ Y . (the inequalities hold a.s., of course.) The following theorem is the basis for working with the new definition: Theorem: Let X be a nonempty set of random variables. Then esssupX exists, if X is closed under pairwise maximization (X, Y ∈ X ⇒ X ∨ Y ∈ X .)

76

CHAPTER 2. CLASSICAL THEORY • The process A is flat off the set {0 ≤ t ≤ T |Z(t) = P0−1 (t)ξ(t)}, which means that ∫ T (Z(t) − P0−1 (t)ξ(t))dAt = 0, a.s.. (2.4.43) 0

• A(τt ) = A(t) a.s., 0 ≤ t ≤ T Exercise 2.4.13.

• Prove that

Z(t) = P0−1 (t)ˆ x(t) ≥ P0−1 (t)ξ(t), τ0 = τˆ, x ˆ(ˆ τ ) = ξ(ˆ τ ).

(2.4.44)

Give an interpretation of A(t) as a regret for not having stopped by time t. • Compare the optimal stopping time τˆ to the stopping time τ˜ = inf{t|At > 0}. Obviously we have τˆ ≤ τ˜. For an intuitive interpretation look at the stopping problem sup E(ξτ ) τ

and give τˆ and τ˜ in terms of optimality of the stopping time. Why then is the fair price defined in a complete market? Proof. of 2.4.11: To prove the first item let us assume that the respective sets under consideration are nonempty to exclude trivial cases. Now let x be in the set defining Pup . Then from 2.3.4 and by definition ˜ −1 (τ )xx,π,C (τ )] ≥ E[P ˜ −1 ξ(τ )]], ∀τ stopping time x ≥ E[P 0 0

(2.4.45)

and this gives x ≥ v(0) andPup ≥ v(0).

(2.4.46)

As ξ(0) is in the set defining Plow so ξ(0) ≤ Plow . For an arbitrary x in the set we find from 2.3.3 ∫ −1 −x,π,−E ˜ ˜ −1 (τ )ξ(τ )]. (2.4.47) −x ≥ E[P0 (τ )x (τ ) − P0−1 (t)dE(t)] ≥ −E[P 0 (0,τ ]

˜ −1 (τ )ξ(τ )] ≤ v(0). This implies the desired result x ≤ E[P 0 To prove the other items it suffices to construct • an nds-admissible portfolio π such that Gπ (t) is a P˜ -martingale, • and a consumption process C such that ˜ −1 (τ )ξ(τ )) =: x xv(0),π,C (t) = −x−v(0),−π,−E (t) = P0 (t)esssupτ ∈St,T E(P ˆt ≥ ξ(t), 0 for 0 ≤ t ≤ T , E(ˆ τ ) = 0, x−v(0),−π,−E (ˆ τ ) = x−v(0),−π,0 (ˆ τ ) = −ξ(ˆ τ ), and E = C and τˆ = inf{t ∈ [0, T ]|ˆ xt = ξ(t)} ∧ T


77

The martingale in the Doob-Meyer decomposition in theorem 2.4.12 may be represented as ∫ t M (t) = P0−1 (s)π ′ (s)σ(s)dw ˜s (2.4.48) 0

for some ∫ nds-admissible portfolio π. From exercise 2.4.13 we find that with ˜ = t P0 (t)dAs : C(t) 0 P0−1 (t)ˆ x(t) = v(0) +

∫

t

0

P0−1 (s)π ′ (s)σ(s)dw ˜s −

∫ 0

t

P0−1 (s)dC˜s , 0 ≤ t ≤ T (2.4.49)

As obviously we have

x ˆ(T ) = ξ(T )

(2.4.50)

˜ ˜ τ) = 0 xv(0),π,C (ˆ τ) = x ˆ(ˆ τ ) = ξ(ˆ τ ), and C(ˆ

(2.4.51)

the result follows. Exercise 2.4.14. • Compute the difference between the price of an American and a European contingent claim. This price is called the early exercise premium ⌉(t). • ⌉(t) = 0 if P0−1 (t)ξ(t) is a P˜ -submartingale. • Consider the American contingent claim ξ(t) = (P1 (t) − q)+ (American Call). Show that the American Call is exercised at time T , and so is the same as the European Call. (Is this intuitively clear?) (hint: Note that (P0−1 (t)P1 (t) − P0−1 (t)q) is a P˜ -submartingale and apply Jensen’s inequality.) So, in this case the problem is completely solved by the preceding discussion. ∫∞ • Show that if 0 r(s)ds = ∞ then for T → ∞ the value of the American Call converges to the price of the underlying stock. Exercise 2.4.15. Let us assume that ξ(t) = ξ0 + mt + at is a submartingale under P˜ . Further, for simplicity of notation, let P0 (t) = 1. So let us compute the snell envelope for this process which we guess to be Zt = ξ˜0 + m ˜t , ˜ T ) and m ˜ T − E(a ˜ T )|Ft ). In the discrete case where ξ˜0 = ξ0 + E(a ˜t = mt + E(a this is easily seen from ∨ ˜ t+1 |Ft ). Zt = ξt E(Z ˜ T − at |Ft ) ≥ 0 and hence the exercise time is t = T . It is Then Zt − ξt = E(a immediately seen that the early exercise premium is zero: ˜ ˜ ⌉(t) = sup E(ξ(τ )|Ft ) − E(ξ(T )|Ft ) = 0. τ ∈St,T

78


Let us now look at the wealth processes of seller (Pup ) and buyer (Plow ): v(0),π

xt

= Zt ≥ ξ(t) (seller)

and −v(0),−π

xt

−v(0),−π

xt

= −Zt ≤ −ξ(t) (buyer for t)

= −Zt ≥ −ξ(t) (buyer for t=T)

as ZT = ξ(T ).

wealth of seller

xS ≥ ξ

|

ξ(t)

−ξ(t)

xS T = ξ(T ) and xB ≥ −ξ(T )

xB ≤ −ξ wealth of buyer

The situation is completely different for the American Put which will be treated next. Let us first consider the case of an infinite horizon T = ∞ in a market with


79

n = d = 1, constant interest rate and strictly positive volatility. The price process is then given by ˜ −r(τ −t) (q − P1 (τ ))+ |Ft ] =: G(P1 (t)) x ˆt = ess supτ ∈St E[e

(2.4.52)

˜ −rτ (q − P1 (τ ))+ |P1 (0) = x], 0 < x < ∞, G(x) := supτ ∈S0 E[e

(2.4.53)

where

where St = St,∞ is the set of stopping times in [t, ∞). We will first look for the optimal stopping time among the hitting times τa := inf{t ≥ 0|P1 (t) ≤ a}, 0 < a < q and then show that it is optimal in S0 . As for P1 (0) = x > a √ ˜ −rτa ) = exp(νy − |y| ν 2 + 2r) = ( a )γ , x > a E(e x where ν :=

r σ

− σ2 , y :=

1 σ

log( xa ), and γ :=

√ (ν+ ν 2 +2r) . σ

(2.4.54)

(2.4.55)

So we have for 0 < a < q

˜ −rτa (q − P1 (τa ))+ |P1 (0) = x] = (q − a)( a )γ 1x>a + (q − x)1{0≤x≤a} ga (x) := E[e x (2.4.56) and qγ g(x) = sup0 (q − x)+ x>b g(x) = q − x 0 ≤ x ≤ b

(2.4.58) (2.4.59) (2.4.60) (2.4.61)

This easily follows from the Feynman-Kac procedure (apply the Itˆ o-formula to e−rt g(P (t))) applied to e−rt g(P (t)) separately for P (t) b. With this it is clear that G = g = ga , 2

It goes beyond the scope of these lecture notes to explain in detail this principle introduced by Benes-Shepp-Witsenhausen ([8]). The basic idea is that when we have a solution of a Bellman equation with a hard constraint on its derivative then the derivative must be C 2 on the constraint. A similar concept is the smooth pasting condition in optimal stopping. For details see F. Boetius’ diploma thesis.

80


and the supremum in 2.4.53 is achieved by the stopping time τˆ = τb := inf{t ≥ qγ 0|P1 (t) ≤ b} with b = (1+γ) . So for any stopping time τ ∈ S0 and x > 0 we have from 2.4.58, from dP1 = P [rdt + σdw] ˜ and Ito’s rule ∫ τ ∧n −r(τ ∧n) e g(P1 (τ ∧ n)) − g(x) ≤ σ e−r P1 (t)g ′ (P1 (t))dw(t) ˜ a..s. (2.4.62) 0

for all n. Taking the expectation w.r.t. P˜ we find for n → ∞ ˜ −rτb (q − P1 (τb ))+ ) = g(x) ≥ E(e ˜ −rτ (q − P1 (τ ))+ ) E(e

(2.4.63)

So with this we have found the price process x ˆ(t) = G(P1 (t)) and the optimal qγ exercise time (of the buyer) by τˆ = τb := inf{t ≥ 0|P1 (t) ≤ b} with b = (1+γ) . Finally the cash flow process is given by ∫ t ˜ C(t) = rq 1{P1 (u)b  from the fact that π ˆ = G′ (P1 )P1 . π ˆ=    −ˆx  t for xˆt ≤b • Verify 2.4.55. Note that here τa is the first passage time to the level a 1 σ log( x ) < 0 by a Brownian motion.


The principle of smooth fit

81

82


The continuation region


83

The graph shows the boundary of the continuation region b as a function of q = 1, r = 1(0.04), σ.

84


The pricing of an American contingent claim/American put-option becomes a bit more involved when we consider a finite horizon T . In the same setting as above the price process now takes the following form ˜ −r(τ −t) (q − P1 (τ ))+ |Ft ] =: G(T − t, P1 (t)) x ˆt = ess supτ ∈St,T E[e

(2.4.64)

where G given by ˜ −rτ (q − P1 (τ ))+ |P1 (0) = x], 0 < x < ∞, G(s, x) := supτ ∈S0,s E[e

(2.4.65)

is • increasing in s, • decreasing, convex of class C 1 in x, • x + G(s, x) is increasing in x. Consider the continuation region C = {(s, x) ∈ (0, ∞)2 |G(s, x) > (q − x)+ }, then the s-section Cs of C takes the form (b(s), ∞) for some decreasing continuous function b(s) taking values in (0, q) with b(0+) = q, b(∞) = b = qγ 1+γ , G(∞, x) = g(x) = gb (x). Furthermore G is the solution of the following moving boundary problem which should be compared to 2.4.58 ∂G 1 2 2 ′′ + σ x G (x) + rxG′ (x) − rG(x) = 0 in C (2.4.66) ∂s 2 ∂G 1 2 2 ′′ + σ x G (x) + rxG′ (x) − rG(x) = −rq < 0 in (0, b(s) (2.4.67) − ∂s 2 G(s, x) > (q − x)+ in C (2.4.68) −

G(s, x) = q − x

x ≥ b(s) (2.4.69)

Unfortunately this finite horizon problem can only be solved numerically to yield that b is the unique solution of an integral equation and G can be represented in terms of b: ∫ s −rs q−x = qe Φ(−µ− (s, x, q))−xΦ(−µ+ (s, x, q))+rq e−rs Φ(−µ− (s, x, b(s−u)))du 0

(2.4.70) and −rs

G(s, x) = qe

∫ Φ(−µ− )−xΦ(−µ+ )+rq

e−ru Φ(−µ− (s, x, b(s−u)))du, x ≥ 0, s > 0

(2.4.71) and = q − x for x ≥ 0, s = 0. In analogy to the above we find the optimal exercise time of the buyer to be τˆ = inf{t ∈ [0, T )|P1 (t) ≤ b(T − t)} ∧ T (2.4.72) and for the optimal hedging portfolio of the seller π ˆ (t) = P1 (t)

∂G (T − t, P (t)). ∂x

(2.4.73)

2.4. EUROPEAN OPTIONS, B-S-FORMULA, AMERICAN OPTIONS Exercise 2.4.17. horizon put.

85

• Compute the early exercise price for the above finite

• Show that for the American Exchange Option ξ(t) = (P1 (t) − P2 (t))+ P0−1 (t)ξ(t) is a P˜ -submartingale and deduce that τˆ = T .

2.4.4

Hedging in Incomplete Markets

In theorem 2.3.9 we have seen a simply verified criterion of completeness which on the other hand shows that in general markets will be incomplete. There may be many reasons for is phenomenon, some of which will be discussed here: • there are more random sources than stocks. From the proof of theorem 2.3.9 the reader easily sees how to construct contingent claims which cannot be attainable. • We have considered nds-admissible self-financing strategies as admissible portfolios. When we restrict the set of portfolios by e.g. imposing constraints like positivity (prohibition of short-selling of stocks) or by not allowing some of the d stocks to be traded (-this results in cone constraints on the set of admissible strategies-) the market also becomes incomplete. The usual way to treat this problem is to embed the original constrained market in a family of unconstrained markets so that pup is the sup of the values of the unconstrained markets, and similar for plow the inf of the values of the unconstrained markets. • There are many other ways of arriving at incomplete markets. One of the more confusing situation bases on the famous representation theorem by Dudley : Any FT -measurable, a.s. finite random variable ξ can be represented in the form ∫ T ξ= ϕdw, 0

where ϕ is progressively measurable w.r.t. the given left-continuous filtration Ft for which a Wiener process (wt , Ft ) exists, with ∫

T

ϕ2 dt < ∞

0

a.s. P . So at first sight one might think the proof of theorem 2.3.9 might go through by making use of this ingeniously proved representation result. It is then of course easily seen that in this situation we had to choose a too large class of strategies which might fail the desirable properties of the market as e.g. to be arbitrage free. Using Dudley’s representation it is ∫ Teasily seen that any real number a, say a = 1, can be represented as 0 ϕdw = 1. So here we could choose ϕ as a portfolio in a market with one risky stock St1 = wt to make a certain gain with initial 0 what we

86

CHAPTER 2. CLASSICAL THEORY considered as an arbitrage opportunity. In this special case it can be shown that the representing function is not unique, so the hedging portfolio would not be unique and so the price for the claim would not be unique. We here have the typical situation of an incomplete market, where the prices (if they exist) fall into an interval [p− , p+ ]. In Dudley’s example this is a consequence of the too large class of strategies which allows for arbitrage. However, we have seen already that there are situations where we have many prices in an interval [plow , pup ] and any price within this interval is an arbitrage free price. So: the price is not unique. One can now think about which x ∈ [plow , pup ] gives the fair price and for a special situation the reader is referred to the work by Mark Davis. • In the first chapter we have taken the approach to consider a general market given just by a stochastic process S. Let us now assume that this process is a semimartingale. We considered simple strategies and defined simple self-financing strategies to model the way of trading in this market. The idea behind this was to find minimal assumptions on the market and sets of admissible strategies so that the desirable properties remain true. We then considered optimization problems to find (appropriate closures of) sets of terminal values. Let us now look at this from the other side: Consider such a terminal value as contingent claims, so the question arises whether this terminal value ξ may be written as ∫ ξ=

∞

ϕdS 0

for a ϕ which should again be admissible. From semimartingale theory (see Stochastik script) we know that the definition of the integral of a simple predictable function w.r.t. a semimartingale extends to all predictable procsses (plus some integrability properties) and the set of semimartingales is characterized by this property. So: there might be a good chance to be able to represent a contingent claim in the closure of terminal values by a good process in the closure of good simple processes. This will be pursued in the last chapters which base on the ph.d. thesis of my student J. Leitner. On the other hand this might lead to the idea that whenever we start with simple good functions and a general process which describes our market, the limiting market might not be good as a model for a market with the desirable properties. Let’ s say it more drastically: The only processes to be considered to describe a useful market should be semimartingales [see [25]]. However by redefining the integral e.g. w.r.t. a fractional Brownian motion (which indeed is not a semimartingale) it seems to be possible to model markets on more general processes. This will be worked out in C. Bender’s thesis.

2.5. THE BSDE-APPROACH TO THE CLASSICAL THEORY

87

We will now go on to study pricing / hedging in incomplete markets. However here we will follow a modern approach basing on the theory of backward stochastic differential equations. To make the access to this theory easier and to relate it to the result in the classical approach in this chapter, we repeat the classical results in complete markets in this new setting. We will also change the assumptions on the market data in order to make the BSDE results more easily applicable and to be able to consider more general markets. In particular the assumption of being nds-admissible for portfolios, what often is referred to as being tame in the literature, is replaced by a stronger condition. For details see remark 2.5.12. To make sure that the reader will be aware of this difference we change the notation of the market tools to: We will consider a market on a random time interval [0, τ ]. The bond is given by a nonnegative Ft -progressively measurable process r, the riskfree interest rate. We assume that ∫ τ |rt |dt < ∞; P -a.s. (2.4.74) 0

Then the bond is given on [0, τ ] by [∫

]

t

rs ds .

Bt := exp

(2.4.75)

0

It satisfies the differential equation dBt = rt Bt dt, B0 = 1

(2.4.76)

The n risky investments S i (i = 1, . . . , n) are given by   ∫ t d d ∫ t ∑ ∑ 1 σsi,j dWsj  . Sti := si0 exp  µis − (σsi,j )2 ds + 2 0 0 j=1

2.5 2.5.1

(2.4.77)

j=1

The BSDE-Approach To The Classical Theory Pricing of Contingent Claims, a New View

Again in this section we look at the pricing of contingent claims in a generalized Samuelson-Black-Scholes- Merton-model. In contrast to the techniques used in the previous section we here use the techniques from BSDE-theory to derive results for more general markets. The BSDE-theory has turned out to be a powerful, extremely useful tool in pricing. So here we can treat the case of unbounded risk premium processes and unbounded interest rate processes. The tools used from BSDE-theory here are collected in the appendix. The main new results here are generalizations of the Black-Scholes-formula to

88


the case of markets with random coefficients to derive an SPDE as a sort of stochastic Black-Scholes formula. Finally we consider the problem in incomplete markets what we aimed at in the end of the last section. We will see how the superhedging under constraints in complete markets can be used to treat American contingent claims. These results on superhedging will then be used to determine the seller’s price of a contingent claim in an incomplete market and to derive the Föllmer-Schweizerprice of a contingent claim. Most of these results were derived in C. Bender’s diploma thesis, on which we rely in the following section.

2.5.2

The Model Revised

Again we model the market M by one non-risky investment (say a bank account) and n risky investments, stocks. The model is a generalization of the model introduced by Samuelson [57], which was further developed by Black/Scholes [7] and Merton [47]. In the classical model the market data were given by constant coefficients. Here this is generalized to random, possibly unbounded market data. Let Wt = (Wt1 , . . . , Wtd )∗ be a d-dimensional standard Brownian motion and let Ft be the augmentation of the filtration generated by the Brownian motion. This filtration satisfies the usual conditions. The trading time is given by the stochastic interval [0, τ ], where τ is an Ft -stopping time. Let us first model the bond by a nonnegative Ft -progressively measurable process r, the risk free interest rate, Though it might be a random variable!. We assume that ∫ τ |rt |dt < ∞; P -a.s. (2.5.1) 0

Then the bond is given on [0, τ ] by

[∫

Bt := exp

]

t

rs ds .

(2.5.2)

0

It satisfies the differential equation dBt = rt Bt dt, B0 = 1

(2.5.3)

The n risky investments S i (i = 1, . . . , n) are given by   ∫ t d d ∫ t ∑ ∑ 1 Sti := si0 exp  µis − (σsi,j )2 ds + σsi,j dWsj  . 2 0 0 j=1

(2.5.4)

j=1

The processes µi , σ i,j are Ft -progressively measurable,and in order to make all integrals well-defined we assume: n ∑ d ∫ τ ∑ |µit | + |σti,j |2 dt < ∞; P -a.s. (2.5.5) i=1 j=1

0


89

By applying Itô’s formula we get: dSti = µit Sti dt +

d ∑

σti,j Sti dWtj ,

j=1

S0i

=

si0 .

(2.5.6)

We make the following assumptions an the market data: (M0) The non-negative interest rate r, as well as the coefficients σ i,j , µi are progressively measurable and satisfy the conditions (2.5.1) and (2.5.5). (M1) d ≥ n. i,j (M2) The diffusion matrix σt∧τ := (σt∧τ )i=1,...,n, j=1,...,d has P ⊗λ-a.s. maximal rank. With (M0) we guarantee the existence of the respective integrals used to model the market. The other conditions guarantee that at any time the number of independent random sources is greater or equal to the number of stocks. As we have seen in the last section, this of course again makes sure that the market is arbitrage free, if we choose the right class of self-financing strategies. As we are going to change the assumptions on the admissibility of the strategies, the reader should recall the exact assumptions we made in the last section (see below). The following sections will describe arbitrage-free markets and we will then address the problem of hedging in markets which are • complete in the sense that a certain class of contingent claims is reachable, • complete in principle in the sense above, but restrictions on the portfolios/price processes do not allow a perfect hedge of the contingent claims (so these markets could already be called incomplete!), • incomplete in the original sense that by definition of the market there are more random sources than stocks. Let Yt be the amount of money the investor has invested in the market at time t. The amount invested at time t in the i-th stock is denoted by πti and we call π := (π 1 , . . . , π n∑ )∗ a portfolio or an investment strategy. The rest of the investor’s wealth Yt − ni=1 πti is on the bank account. So, in particular, the investor does not keep any pocket money. Definition 2.5.1. A portfolio π is self-financing, if: dYt =

Yt −

∑n

i i=1 πt

Bt

n ∑ πti i dBt + dS . Si t i=1 t

(2.5.7)

Remark 2.5.2. Without notice, when speaking about portfolios we will mean self-financing ones. In particular we assume that there are no withdrawals and/or inputs of money into the market like amounts of money resulting from consumption, transaction costs etc..

90

CHAPTER 2. CLASSICAL THEORY For a given initial wealth Y0 = y and a portfolio π the wealth develops as: ] [ n d ∑ n ∑ ∑ i i σti,j πti dWtj , (µt − rt )πt dt + dYt = rt Yt + i=1

j=1 i=1

Y0 = y.

(2.5.8)

Yt is the wealth process corresponding to the portfolio π and the initial wealth y. We will now reformulate the pricing problem considered in the last section and relate it to the term of a BSDE: Definition 2.5.3. A non-negative Fτ -measurable random variable ξ is called a contingent claim. We consider a contingent claim as an obligation for the seller of the option which is exercised at the exercise time τ . We have already seen the European Call option as an example of a contingent claim: Example 2.5.4. A European Call option is a contract between two agents called buyer and seller. The buyer purchases the right to receive a stock from the seller at a deterministic future time T at a price x, the exercise price. Both time and price are fixed at time 0 and are part of the contract. Let St be the price of a stock at time t. IF ST > x the buyer will exercise his option. As he can immediately sell the stock at the market price ST he gains the amount ST − x. On the other hand if ST ≤ x the buyer will not exercise the option . So the seller has the obligation to pay ξ := [ST − x]+ . Now the question is: What is the fair price for such an option. The problem is called the pricing problem of a European Call option. We will now formulate the pricing problem for a general contingent claim ξ. Let M (ξ) be the set of initial wealths for which a portfolio π exists so that the corresponding wealth process satisfies: Yτ = ξ. So we are looking for pairs (y, π) with [ ] n d ∑ n ∑ ∑ dYt = rt Yt + σti,j πti dWtj , (µit − rt )πti dt + i=1

j=1 i=1

Y0 = y, Yτ = ξ.

(2.5.9)

For such a pair (y, π) we call π a (perfect) hedging portfolio for ξ and we say that ξ is reachable from the initial wealth y. If M (ξ) ̸= ∅ and if inf y∈M (ξ) y is attained then this minimum is the fair price of the contingent claim (see 2.4.5). This is the minimal initial wealth which exactly hedges the obligation ξ. This means that the obligation can be fulfilled by starting in inf y∈M (ξ) y and using the portfolio π. Let us look at (2.5.9) to derive the connection to the BSDE theory. Because of (M1)–(M2) the system of linear equations µt∧τ − rt∧τ 1 = σt∧τ θt∧τ ; P ⊗ λ-a.s.

(2.5.10)


91

has at least one Ft -progressively measurable solution θt . Here 1 is the row vector with all entries being 1 and µt := (µ1t , . . . , µnt )∗ . Let Θ be the set of all Ft -progressively measurable solutions of (2.5.10). For θ ∈ Θ we consider the BSDE dYt = [rt Yt + Zt θt ] dt + Zt dWt , Yτ

= ξ.

(2.5.11)

Let (Y, Z) be an (a, β)-solution of this BSDE for a suitable pair (a, β), and let the linear system ∗ ∗ σt∧τ πt∧τ = Zt∧τ ; P ⊗ λ-a.s. (2.5.12) have a solution πt . Then due to the assumptions (M1)–(M2) the solution is unique. Then the pair (Y0 , πt ) solves the system (2.5.9). So πt is a hedging portfolio for ξ and Y0 is an initial wealth from which ξ can be reached.

2.5.3

Admissible Portfolios

We now make some assumptions on the portfolios. First we need some technical mathematical assumptions to be able to formulate the pricing problem: Definition 2.5.5. A portfolio π is called feasible, if: (Po1) πt is Ft -progressively measurable and ∫ τ |πt∗ σt |2 + |πt∗ (µt − rt 1)|dt < ∞ P -a.s. 0

(Po1) guarantees that the integrals above are well defined. Furthermore the wealth process exists for such a portfolio: Theorem 2.5.6. Let π be a feasible portfolio, and let y ∈ R be an initial wealth. Then the SDE (2.5.8) which describes the wealth process associated with π and initial y has a unique strong solution Yt on the time interval [0, τ ]. Proof. Define ∫ t∧τ

{

∫

t∧τ

−

∫s

[πs∗ (µs

}

y+ e − rs 1)ds] 0 ∫ t∧τ ∫ ∫ t∧τ s +e 0 rs ds e− 0 ru du πs∗ σs dWs .

Yt := e

0

rs ds

0

ru du

0

Because of (M0) and the feasibility of π all integrals are well defined. Apply Itô’s formula to derive that Y is a solution of the SDE (2.5.8) on [0, τ ] (see Stochastik script). Let △ X be the difference of two solutions. Then it is the solution of d △ Xt = rt △ Xt dt, △ X0 = 0. Pathwise this is an which (under the assumptions on r) has a unique solution X ≡ 0.

92


It should be clear that the integrability condition (Po1) is just a technical assumption. The progressive measurability reflects the fact that the investor has to behave non-anticipatively, which means that he has no knowledge about future developments. We now show that we need further assumptions on the portfolio to make the market reasonable. Let n = d and for notational convenience d = 1. Then Theorem 2.5.7. Let n = d = ∫1 and τ = T deterministic and bounded. With T θt := σt−1 (µt − rt ) assume that 0 θt2 dt < ∞ P -a.s. A P -a.s. finite contingent claim ξ is given. Then for each initial wealth y ∈ R there is a feasible portfolio π which reaches ξ. Especially the price of ξ is not defined. Proof. We will make use of Dudley’s theorem 8.2.4. For details on the mathematical background the reader is referred to Proposition 8.2.5 in the Appendix. Define: { ∫ t∧T } ∫ t∧T 1 2 Φt := exp − rs + θs ds − θs dWs . 2 0 0 From Itô’s formula we get on [0, T ] : ∫ t ∫ t Φt = 1 − rs Φs ds − θs Φs dWs , 0 0 ∫ t ∫ t −1 −1 2 −1 θs Φ−1 Φt rs Φs + θs Φs ds + = 1+ s dWs . 0

0

Apply Dudley’s theorem to the random variable ΦT ξ − y for an arbitrary fixed y ∈ R. Then there is a progressively measurable process Z˜ with ∫ T Z˜s dWs ΦT ξ − y = 0

and

Let Y˜t := y +

∫ ∫ t∧T 0

T

|Z˜s |2 ds < ∞ P -a.s.

0

Z˜s dWs so that from Itô’s formula:

−1 ˜ −1 ˜ −1 ˜ −1 ˜ −1 ˜ ˜ d(Φ−1 t Yt ) = rt Φt Yt dt + θt [θt Φt Yt + Φt Zt ]dt + [θt Φt Yt + Φt Zt ]dWt , −1 ˜ ˜ Φ−1 0 ≤ t ≤ T. 0 Y0 = y, ΦT YT = ξ; −1 −1 ˜ −1 ˜ ˜ Define Yt := Φ−1 t Yt and πt := σt [θt Φt Yt + Φt Zt ], so by putting in θ:

dYt = rt Yt dt + (µt − rr )πt dt + σt πt dWt , Y0 = y,

YT = ξ.

So π is a hedging portfolio for ξ and initial y. We now have to prove that π is feasible: The continuity of Y˜ and Φ−1 gives: 2 sup |Y˜t |2 + sup |Φ−1 t | < ∞; P -a.s.

0≤t≤T

0≤t≤T

With this feasibility of π follows from the square integrability of Z˜ and θ.


93

If so we consider all feasible portfolios as admissible then the pricing problem (under the assumptions above) makes no sense. In particular we can make money without risk what we called an arbitrage possibility. We repeat the definition: Definition 2.5.8. (i) A feasible portfolio π is an arbitrage possibility, if there is a stopping time τ ∗ ≤ τ , so that the following holds for the wealth process of π with initial 0: Yτ ∗ ≥ 0 and P (Yτ ∗ > 0) > 0. (ii) A market M is arbitrage free, if there are no arbitrage opportunities. So to adjust the model to the reasonable market requirements we define: Definition 2.5.9. A feasible portfolio π is admissible for the initial y, if: (Po2) For the wealth process Y of y, π the following holds true: P (∀t∈[0,∞) Yt∧τ ≥ 0) = 1. In this way bankruptcy is excluded. We will see now that an additional further assumption on admissible portfolios guarantees that the market is arbitrage free: ∫τ Theorem 2.5.10. Let a solution θt of the linear system (2.5.10) satisfy 0 |θt |2 dt < ∞ P -a.s. If π is an admissible portfolio for initial 0, so π cannot be an arbitrage opportunity. Proof. Define { ∫ Φt := exp −

t∧τ

0

1 rs + |θs |2 ds − 2

∫

t∧τ

θs∗ dWs

}

0

and call Φ a deflator. Then: Φt∧τ Yt∧τ = Φ0 Y0 +

d ∑ n ∫ ∑ j=1 i=1

0

t∧τ

Φs σsi,j πsi dWtj −

d ∫ ∑ j=1

t∧τ

Φs θsj Ys dWsj .

0

As π is admissible the process Xt := Φt∧τ Yt∧τ is a local martingale, which is bounded below by 0, and so by Fatou’s lemma it is a supermartingale. As the supermartingale is of class D the optional sampling theorem gives for all stopping times τ ∗ ≤ τ : E [Φτ ∗ Yτ ∗ ] ≤ E [Φ0 Y0 ] = 0. As Φ is strictly positive we get: Yτ ∗ = 0 P -a.s. The following example shows that the integrability condition on θ is not dispensable. Example 2.5.11. For n = d ≥ 2 consider a market with coefficients rt := 0, µit := 1{t>0} (d − 1)/(2dRt ), σti,j := δi,j Wti /Rt on the deterministic time interval [0, T ]. Here R denotes the Bessel process and δ is Kronecker’s symbol. This definition makes sense as P -a.a. paths of the d-dimensional Brownian motion

94


for d ≥ 2 do not return to the origin. So σ fulfills (M2). (M0) is obviously satisfied for r und σ. The required conditions on µ are satisfied following the considerations in Karatzas/Shreve [18], pp. 159-161. However the integrability condition on θ does not hold. As the drifts of the stocks are arbitrarily large for times near 0 the obviously feasible portfolio πti = 1 should be an arbitrage opportunity. This as well as the admissibility of π follows if we can show that the corresponding wealth process is the Bessel process. For this we use the integral representation of the Bessel process: ∫

d ∫ t ∑ d−1 Wsi ds + dWsi 2R R s s 0 i=1 0 ) ∫ t( d d ∫ t ∑ ∑ i i = rs R s + (µs − rs )πs ds + σsi,j πsi dWsi t

Rt =

0

i,j=1 0

i=1

So R is the wealth process for π and initial 0. As R is P -a.s. positive for all t > 0 the portfolio π is admissible and an arbitrage opportunity. Remark 2.5.12. As noted before there are many ways of defining sets of admissible portfolios. The above characterization is used in e.g. Karatzas/Shreve [18], ch 5.8, or El Karoui/Peng/Quenez [9]. In the preceding chapter, however, we used a larger class of portfolios, the set of nds-admissible portfolios. The reader should notice that there is a neat interconnection between the assumptions on the market M and the assumptions on the set of admissible portfolios if we want to exclude arbitrage opportunities. As here the assumptions on the market are weaker it is not possible to follow the arguments of the last theorem for just nds-admissible portfolios. In the case of bounded market coefficients however this is possible as in this case an equivalent martingale measure exists. A characterization of arbitrage free markets for nds-admissible portfolios under the above assumptions on M with bounded terminal time and n = d is given in Levental/Skorohod [22]. Remark 2.5.13. From the early pioneering works of Harrison/Kreps [28] and Harrison/Pliska [14] the close connection between the property of the market being arbitrage free and the existence of an equivalent martingale measure for the process S is well known. Under very general assumptions on the market Delbaen/Schachermayer [7] prove that a slight generalization of the notion arbitrage free -namely the property not to allow free lunch with vanishing risk- is equivalent to the existence of an equivalent local martingale measure (ELMM). This very general result is not directly applicable in our situation as Delbaen and Schachermayer consider nds-admissible portfolios. In many cases an ELMM may be constructed in the following way: (i) Choose a solution of the linear system σt αt = µt , which exists by (M1)–(M2). ∫ t∧τ ∫ t∧τ (ii) Let Ψt := exp{−1/2 0 |αs |2 ds − 0 αs∗ dWs }.


95

(iii) If Ψ is a martingale, then an equivalent martingale measure is given by dQ := Ψτ dP . In the situation we consider the assumptions on the coefficients and the terminal time are too weak so that in general Ψ is not a martingale and the above construction of he ELMM fails. Nevertheless an ELMM may exist as an interesting example in Delbaen/Schachermeyer [12] shows.

2.5.4

Pricing in Complete Markets

In definition 2.3.7 we defined the completeness of a market and in theorem 2.3.9 we gave a simply verifiable condition for completeness. As we do not want to repeat all the considerations to adapt them to the new situation here we just call a market complete if n = d. Note that our assumptions on σ are not different from the assumptions in theorem 2.3.9. The reader should note however that by theorem 2.5.15 we can consider completeness in a similar definition as in definition 2.3.7 by considering contingent claims to be in L2τ,a,β (R) for appropriate a and β. We are now going to derive the price of a contingent claim in a suitable space L2τ,a,β (R) and the model allows for unbounded coefficients of the complete (i.e. here n = d) market. So we may summarize: (see theorem 2.3.9) Corollary 2.5.14. (i) σt is invertible. (ii) The set of solutions Θ of the linear system (2.5.10) has exactly one element θt , which we call again the risk premium process. (iii) For given Zt the linear system (2.5.12) has exactly one solution. The last condition on our market under consideration is an integrability condition which by the above considerations guarantees that the market is arbitrage free: ∫τ (M3) The risk premium process θ satisfies P -a.s.: 0 |θt |2 dt < ∞ and θt > 0. Let us consider the following class of contingent claims: { [ { ∫ τ } ] } 2 2 Ξβ := ξ; ξ contingent claim and E exp β |rt | + |θt | dt ξ < ∞ . 0

Then: Theorem 2.5.15. Let ξ ∈ Ξβ for a β > 2. Then there is an admissible hedging portfolio π for ξ with associated wealth process } ] [ { ∫ τ ∫ τ 1 ξ 2 ∗ (2.5.13) Yt∧τ = E exp − θs dWs ξ Ft . rs + |θs | ds − 2 t∧τ t∧τ Proof. Consider the BSDE dYt = [rt Yt + Zt θt ] dt + Zt dWt , Yτ

= ξ.

96


From theorem 8.3.27 this equation has a unique (a, β)-solution (Y ξ , Z ξ ) for a2t = |rt | + |θt |2 and Y ξ is of the form (2.5.13). π is given as the unique solution of the linear system (2.5.12) with right hand side (Z ξ )∗ . Admissibility of π follows from the representation of the corresponding wealth process Y ξ and the integrability of Z ξ . Next we show that Y ξ indeed is the fair price of the contingent claim: Theorem 2.5.16. Let ξ ∈ Ξβ for a β > 2. If π ˜ is an admissible hedging ξ portfolio for ξ with wealth process Y˜ then: P (∀t∈[0,∞) Y˜t∧τ ≥ Yt∧τ ) = 1. Proof. Let Φ be the deflator as in the proof of theorem 2.5.10. Then: Φt∧τ Y˜t∧τ = Φ0 Y˜0 +

d ∑ n ∫ ∑ j=1 i=1

0

t∧τ

Φs σsi,j π ˜si dWtj

−

d ∫ ∑ j=1

t∧τ

Φs θsj Y˜s dWsj .

0

Following the arguments in theorem 2.5.10 Φt∧τ Y˜t∧τ is a supermartingale. With the representation of Y ξ we then have: ξ Φt∧τ Y˜t∧τ ≥ E [ Φτ ξ| Ft ] = Φt∧τ Yt∧τ .

Remark 2.5.17. The last theorem has a straightforward generalization. Consider as class of admissible portfolios pairs of strategies (π, C) consisting of an admissible portfolio as above and a cumulative consumption process Ct . Such a strategy is called a superhedging strategy. After adapting the above notion to the situation of superhedging strategies in an obvious way the above result also holds true for all admissible superhedging strategies (see section 2.5.6). Next consider the mapping ∪ Π: Ξβ → Sτc,2 (R), ξ 7→ Y ξ . β>2

We define a pricing system by: Definition 2.5.18. Let X be a set of contingent claims. (i) A mapping Π, which associates with ξ ∈ X a progressively measurable process Π(ξ)t∧τ , called the pricing process von ξ, is called a pricing system, if: (a) For all t ∈ [0, ∞) : Π(ξ)t∧τ ≥ 0 P -a.s. (b) For all t ∈ [0, ∞) : Π(ξ)t∧τ is P -a.s. increasing in ξ. (ii) A pricing system is arbitrage free, if the following holds: If ξ 1 ≥ ξ 2 and if Π(ξ 1 )t∧τ and Π(ξ 2 )t∧τ are equal on a set A ∈ Ft , then ξ 1 and ξ 2 are P -a.s. equal on A. To see the meaning of the last condition assume that there is an arbitrage opportunity. Then at time t the investor buys the claim ξ 1 1A and sells the claim ξ 2 1A . Both claims have the same price, but at exercise time there is a set A ⊃ B with P (B) > 0, where the investor gets more money than he had


97

paid, and on B c the payments are equal. This is excluded by the no arbitrage condition above. We will use the following consequence of the comparison theorems 8.3.24 and 8.3.29 and of theorem 8.3.20: Theorem 2.5.19. (i) The mapping Π:

∪

Ξβ → Sτc,2 (R), ξ 7→ Y ξ

β>2

is an arbitrage free pricing system. This gives the fair price in a complete market. (ii) Let ξ 1 ∈ Ξβ 1 and ξ 2 ∈ Ξβ 2 . Then for 2 < β := β 1 ∧ β 2 there is a constant Cβ with [

{ ∫ t } ] ξ1 ξ1 2 2 E sup exp β |rs | + |θs | ds |Yt − Yt | 0≤t≤τ 0 [ { ∫ τ } ] 2 1 2 2 ≤ Cβ E exp β |rt | + |θt | dt |ξ − ξ | . 0

Remark 2.5.20. Note that (ii) is the continuity of the mapping c,2 Πβ : L2τ,a,β (R) ⊃ Ξβ → Sτ,a,β (R), ξ 7→ Y ξ .

Note however that this is not the continuity of the pricing system Π. We now consider the following special case: Example 2.5.21. Instead of (M0) we assume: (M0’) Let the exercise time τ = T be deterministic and finite, let the coefficients r, µ, σ be bounded and let the matrix σσ ∗ be positive definite for all t and almost all ω uniformly in t and ω. Obviously this assumption is essentially stronger than (M0) and it implies (M2). From the assumed completeness of the market σ and σ ∗ have bounded inverses, and also θ is bounded. Note that in this special case Ξβ consists for all β of exactly the square integrable contingent claims and the pricing system is continuous. A case of special interest is the case of deterministic market data: Example 2.5.22. Let τ = T < ∞ and the market coefficients be deterministic. Then: { ∫ τ } exp β |rt | + |θt |2 dt 0

is a constant depending only on β because of the local integrability properties r and θ. And again Ξβ is the set of all square integrable contingent claims for all β.

98


In the next section we will consider Black-Scholes-type representations of the fair price in special cases which allow for numerical solutions. There we will usually assume that the coefficients of the market are bounded. So -before we go into this- we will treat the problem of approximation of markets with unbounded coefficients by markets with bounded data: Theorem 2.5.23. Let (rtk ) and (θtk ) be sequences of progressively measurable processes (with: ) k k | ≤ |θ (i) ∀k∈N |rt∧τ | and |θt∧τ P ⊗ λ-a.s. t∧τ)| ( |rt∧τ | ≤ k (ii) P (limk→∞ rt∧τ = rt∧τ ; a.e.t ∈ [0, ∞) )= 1. k (iii) P limk→∞ θt∧τ = θt∧τ ; a.e.t ∈ [0, ∞) = 1. Then for ξ ∈ Ξβ , β > 2 and k → ∞: [ } ] 2 ] [ { ∫ τ ∫ τ ξ 1 E sup Yt − E exp − (θsk )∗ dWs ξ Ft → 0. rsk + |θsk |2 ds − 2 0≤t≤τ t∧τ t∧τ This means that the fair price of ξ may be approximated by the fair prices in fictitious markets with interest rate rk and risk premium vector θk . Proof. Consider the family of BSDEs dYt = [rtk Yt + Zt θtk ]dt + Zt dWt , Yτ

= ξ.

From assumption (i) the generators of all these BSDEs are (a, β)-standard generators with a2t := |rt | + |θt |2 . From theorem 8.3.27 we get that } ] [ { ∫ τ ∫ τ 1 k2 k,ξ k k ∗ rs + |θs | ds − (θs ) dWs ξ Ft Yt := E exp − 2 t∧τ t∧τ is the Y -part of the unique (a, β)-solution (Y k,ξ , Z k,ξ ) of this BSDE (depending on k). So we can apply theorem 8.3.20 to get for a constant Cβ depending only on β and not on k:

2

ξ

(Y − Y k,ξ , Z ξ − Z k,ξ ) s,a,β

[ ] 2

. ≤ Cβ a−1 (r − rk )Y ξ + Z ξ (θ − θk ) (2.5.14) 2,a,β

The integrand on the right hand side of (2.5.14) is dominated by: ( ) 2(d + 2)eβAt |aYtξ |2 + |Ztξ |2 , which follows from a similar argument as in the proof of Lemma 8.3.18. So the result follows by Lebesgue’s dominated convergence theorem for k → ∞ from (2.5.14), (ii) and (iii). Remark 2.5.24. The proof shows that the hedging portfolios in the fictitious markets converge to the hedging portfolio in the real market.


99

This result is quite remarkable as in the equivalent martingale measure approach this approximation is not at all obvious: In general the convergence of the real measures does not imply the convergence of the corresponding equivalent martingale measures (see the work by Hubalek and Schachermayer), and so the prices of a contingent claim in a e.g. discretized market might not converge to the price in the original continuous market and here lies one of the advantages of the BSDE approach to pricing. This approach does not make use of measure transformations but only relies on the basic physical measure thus circumventing problems in approximations. We finally give an example of how to approximate a general market by markets with bounded data: Example 2.5.25. For k ∈ N let rtk := rt ∧ k. Define θk in a similar way. Then for rk and θk the assumptions of the above theorem are fulfilled

2.5.5

Black-Scholes-Type equations

In this section we are going to generalize the results of section 2.4 to the case of random market data. Of course we cannot expect to end up with a completely solvable SPDE, but the new Black-Scholes formulae often allow for numerical solutions. We here follow the ideas of Ma und Yong [23, 46] and derive within our setting an SPDE as a new Black-Scholes formula which for deterministic market data reduces to the classical case [7]. For technical results used here the reader is again referred to section 8.2.4 in the appendix. For relations with the classical case it might e useful to look at section 8.2.3 in the appendix. Throughout this section we assume for notational simplicity that n = d = 1 and (M0’). For a contingent claim of the form g(ST ) we thus get the following system: dSt = µt St dt + σt St dWt , [ ] dYt = rt Yt + σt−1 (µt − rt )Zt dt + Zt dWt , S0 = s0 , YT = g(ST ).

(2.5.15)

We are going to derive the connection between the stock price S and the price process for the contingent claim ξ := g(St ). Notice that theorem 8.2.49 is not directly applicable as the coefficients of the FSDE are not bounded. To circumvent this difficulty we use the usual trick to replace the stock price by its logarithm Xt := log(St ), so that from Itô’s formula: [

dXt X0

] 1 2 = µt − σt dt + σt dWs , 2 = log(s0 ).

The BSDE then is of the form: [ ] dYt = rt Yt + σt−1 (µt − rt )Zt dt + Zt dWt , ( ) YT = g eXT .

(2.5.16)

(2.5.17)

100


If now g is sufficiently smooth the following BSPDE which we call the stochastic Black-Scholes formula has a unique classical solution by theorem 8.2.47: ( ) 1 2 1 σs ∂x,x u(s, x) + rs − σs ∂x u(s, x) − rs u(s, x)ds 2 t 2 ∫ T σs ∂x q(s, x) − σs−1 (µs − rs )q(s, x)ds − q(s, x)dWs . ∫

T

u(t, x) = g(ex ) + ∫

T

+ t

t

(2.5.18) And from theorem 8.2.49 we have: Yt = u(t, Xt ), πt = σt−1 Zt = σt−1 q(t, Xt ) + ∂x u(t, Xt )

(2.5.19)

This gives the hedging portfolio for ξ and the price process as a function of the present log-price of the stock. The special case of deterministic market data gives the result: If the PDE (

∂t u(t, x) =

−σt2 ∂x,x u(t, x)

) 1 − rt − σt ∂x u(t, x) + rt u(t, x), 2

u(T, x) = g(ex )

(2.5.20)

has a classical solution then (u, 0) is the classical solution of the BSPDE (2.5.18), and (2.5.19) reduces to: Yt = u(t, Xt ), πt = σt−1 Zt = ∂x u(t, Xt ).

(2.5.21)

Remark 2.5.26. Conditions for existence of solutions of the PDE (2.5.20) see Friedman [27], Theorem 1.12. Remark 2.5.27. To get the Black-Scholes formula of section 2.4 an Euler transformation shows that v(t, x) := u(t, ex ) on [0, T ] × (0, ∞) solves: 1 ∂t v(t, x) = − σt2 x2 ∂x,x v(t, x) − rt x∂x v(t, x) + rt v(t, x), 2 v(T, x) = g(x).

(2.5.22)

This is again the classical formula and (2.5.21) gets the form: Yt = v(t, St ), πt = σt−1 Zt = St ∂x v(t, St ). Remark 2.5.28. There are many advantages of these representations for the price process and the hedging portfolio. So for deterministic coefficients the PDEs (2.5.20), resp. (2.5.22) allow for numerical solutions (see Douglas/Ma/Protter [13]). Furthermore results on the robustness of the pricing formula can be derived (see El Karoui/Jeanblanc-Picqu/Shreve [18] and Ma/Yong [46], chapter 8.5).


2.5.6

101

Superhedging

We are now going to consider the case of an incomplete market. So here we are no longer able to find a perfect hedge of the contingent claim. In particular here we consider the situation where we have some constraints on the portfolio and/or the pricing process. As then obviously a perfect hedge might no longer be possible we consider a larger class of investment strategies by allowing for additional consumption. The hedging problem then is to find a superhedging strategy with minimal wealth process so that the constraint(s) are fulfilled. We shall see that this minimal wealth process (under appropriate assumptions) is derived as the limit of a sequence of nonlinear BSDEs. Throughout this section we assume the market to be complete in the sense that n = d and (M0’) to hold. So in particular all coefficients are bounded.

2.5.7

The New Tools

We first define the new investment policies: Definition 2.5.29. A pair (π, C) is called a superhedging strategy for a contingent claim ξ if: ∫T (i) π is an n-dimensional Ft -progressively measurable process with E 0 |σs∗ πs |2 ds < ∞. (ii) C is an Ft -progressively measurable increasing RCLL process with C0 = 0 and E[|CT |2 ] < ∞, the consumption process. (iii) There is an Ft -progressively measurable process Y , the wealth process corresponding to the superhedging strategy so that ] [ d ∑ n n ∑ ∑ σti,j πti dWtj , dYt = rt Yt + (µit − rt )πti dt − dCt + j=1 i=1

i=1

YT

= ξ [

and E

] sup |Yt |2 < ∞. 0≤t≤T

Remark 2.5.30. Ct describes the cumulative consumption until time t. Note that this process has to be RCLL which has the intuitive meaning that the investor first has to earn the money he wants to consume. From the purely mathematical point of view this has a dramatic consequence: In general the right continuity leads to the fact that the classical Doob-Meyer-decomposition is no longer possible as this decomposition requires a predictable bounded variation process. In the present situation we so need a Doob-Meyer-decomposition with an optional bounded variation process. The reader is referred to the work by Kramkov [39] for general results on the optional Doob-Meyer-decomposition. Remark 2.5.31. Obviously, all sufficiently integrable hedging portfolios are superhedging portfolios by adding the consumption process C ≡ 0. The additional integrability assumptions are just technical assumptions to allow the application of results on supersolutions of BSDEs.

102

CHAPTER 2. CLASSICAL THEORY We next describe the constraints:

Definition 2.5.32. (i) K := (Kt (ω))t∈[0,T ],ω∈Ω is called an admissible constraint if for all t ∈ [0, T ], ω ∈ Ω { } Kt (ω) = (y, p) ∈ R1+n ; ϕ(t, ω, y, p) = 0 for a nonnegative ϕ : [0, T ] × Ω × R × Rn → R, so that: (i) ϕ is Lipschitz in the two last components uniformly w.r.t the first two components and ϕ(·, ·, 0, 0) ∈ HT2 (R). (ii) A superhedging strategy (π, C) for ξ with wealth process Y fulfills the constraint K, if: (Yt (ω), πt (ω)) ∈ Kt (ω); P ⊗ λ[0,T ] -a.s. (iii) A superhedging strategy (π, C) for ξ with wealth process Y is optimal w.r.t. the constraint K, if for the wealth process Y˜ of any further superhedging strategy for ξ, which fulfills K the following holds: P (∀t∈[0,T ] : Y˜t ≥ Yt ) = 1. Remark 2.5.33. We identify the constraint K and the function ϕ defining K. Remark 2.5.34. Consider a superhedging strategy which is optimal (w.r.t. the constraint). Optimality here obviously means that the corresponding wealth process is the smallest such process which superhedges the claim under the constraint. This property makes the wealth process uniquely described. The same is true for the superhedging strategy, which will be proved in the next proposition. Before we do this look at the intuitive meaning of the wealth process corresponding to a superhedging strategy: As the seller consumes a certain amount of money when using a superhedging strategy, the wealth process does not describe the fair price of the contingent claim under the constraint. Indeed it is the lowest upper bound of the price and so it describes the price of the seller (see section 2.4.4 and Y is called the seller’s price. Proposition 2.5.35. An optimal superhedging strategy (π, C) for a claim ξ w.r.t. an admissible constraint ϕ is unique. Proof. We prove that for a given wealth process Y there is at most one super˜ with hedging strategy. Consider two superhedging strategies (π, C) and (˜ π , C) wealth process Y . Denote σ ∗ π and σ ∗ π ˜ by Z ∗ and Z˜ ∗ resp.. Then from Itô’s 2 formula applied to 0 = |Yt − Yt | , we get ∫ T E |Zt − Z˜t |2 dt ≤ 0. 0

So Z = Z˜ and from the invertibility of σ ∗ we have that π = π ˜ P -a.s.. From the ˜ right continuity of C and C˜ P -a.s. we get: C = C. With this uniqueness result we now consider the problem of finding an optimal superhedging strategy. The admissible constraint is given by ϕ and the contingent claim ξ is given. The idea behind our approach is to penalize violations of the constraint by additional payments.


103

Remember that the conditions on our complete market imply the boundedness of the risk premium process θ. Let us then consider the following sequence of BSDEs (k ∈ N): [ ] dYtk = rt Ytk + Ztk θt − kϕ(t, Ytk , (σt∗ )−1 (Ztk )∗ ) dt + Ztk dWt , YTk = ξ.

(2.5.23)

For all k this BSDE has a unique classical solution (Y k , Z k ) from PardouxPeng’s theorem. The additional term −kϕ(t, Ytk , (σt∗ )−1 (Ztk )∗ )dt here describes ′ the penalization . So it should be intuitively clear that: Y k ≤ Y k for k ≤ k ′ , what can immediately be verified from the comparison theorem 8.2.28. So with Y k we have an increasing sequence of stochastic processes. If this sequence is bounded above by a process X then the sequence Y k converges to a process Y ≤ X. We then have to prove that this limit process is the wealth process of an appropriate superhedging strategy. A positive answer is found from Peng [54], Theorem 2.4: Lemma 2.5.36. In a complete market consider a square integrable claim ξ and an admissible constraint ϕ. The Y -parts of the corresponding sequence of BSDEs (2.5.23) is assumed to be bounded from above by an Ft -progressively measurable process X with [ ] sup |Xt |2 < ∞.

E

0≤t≤T

Then there is a superhedging strategy (π, C) for ξ with wealth process Y := limk→∞ Y k P -a.s.. Here Ct is the weak limit of ∫ Ctk

t

ϕ(s, Ysk , (σt∗ )−1 (Zsk )∗ )ds

(2.5.24)

|Zs − Zsk |p ds = 0; 1 ≤ p < 2

(2.5.25)

:= k 0

in HT2 (R), and with Z given by ∫

T

lim E

k→∞

0

we have: Z ∗ = σ ∗ π. Proof. As Y ≤ X we get: [ E

] sup |Yt |2 < ∞. 0≤t≤T

Because of the boundedness of (σ ∗ )−1 the Lipschitz continuity also holds for ϕσ := ϕ(·, ·, (σ ∗ )−1 ·). So from Hölder’s inequality there is a constant K depending on T , k and the Lipschitz constant of ϕσ such that:

104


∫ T ] [ E (CTk )2 ≤ kT E |ϕσ (s, Ysk , (Zsk )∗ )|2 ds 0 ∫ T |ϕ(s, 0, 0)|2 + |Ysk |2 + |Zsk |2 ds < ∞. ≤ KE 0

With this all conditions in theorem 2.4 in Peng [54] are fulfilled and this theorem gives the following equality together with the existence of all limits involved: ∫

T

Yt = ξ −

∫ rs Ys + Zs θs ds + CT − Ct −

t

T

Zs dWs .

(2.5.26)

t

C is increasing as a limit of increasing processes. Replace Z by π ∗ σ and use the definition of the risk premium vector θ, and the assertion follows immediately. Remark 2.5.37. The proof of Theorem 2.4 in Peng [54] is quite technical why here we only give the idea of the proof for the situation considered in this section, namely the linear case: 1. Standard estimates show that for a constant K independent of k the following holds true: [ ] ∫ T k 2 k 2 E (CT ) + |Zt | dt ≤ K, 0

and we see, that the sequence E

∫T 0

|rt Ytk + Ztk θt |2 dt is bounded.

2. This implies the existence of weak limits of subsequences of C k , Z k and rt Ytk + Ztk θt , which we denote by C, Z and g b. (Weak limits are taken in HT2 -spaces, see Yosida [30] Theorem V.2.1.. With this: ∫ Yt = ξ − t

T

∫ gs ds + CT − Ct −

T

Zt dWt . t

From the uniqueness result for BSDEs we see that also the sequences Z k and C k converge weakly. 3. The most difficult part of the proof is the proof of (2.5.25) as Y and so also C might only be RCLL (and no longer continuous). So it is necessary to have a closer look at the behaviour of the jumps of C. It is possible to prove that the jumps of C are mainly in a finite number of stochastic intervals which together have an arbitrarily small length (see Lemma 2.3 in [54]). As the proof makes use of the predictability of jumps, an essential basic assumption is the assumption that the filtration under consideration is the one of the Brownian motion. With this lemma and Markov’s inequality the convergence in probability of Z k to Z can be derived what together with the boundedness of Z k finally implies (2.5.25).


105

4. From (2.5.25) we then get: g(t) = rt Yt + Zt θt . With this we constructed a superhedging strategy (π, C) and the corresponding wealth process Y . Next we show that this superhedging strategy fulfills the constraint ϕ. Lemma 2.5.38. We use the assumptions and notations of Lemma 2.5.36. Then obviously the superhedging strategy (π, C) fulfills the constraint ϕ. Proof. From (2.5.25) Z k converges to Z in L1 (Rn , P ⊗ λ[0,T ] ). From the P -a.s.convergence of Y k to Y we get from the fact that the Y k are dominated by Y the convergence in L1 (R, P ⊗ λ[0,T ] ). As ϕσ is Lipschitz continuous we get the L1 (R, P ⊗ λ[0,T ] )-convergence of ϕσ (·, Y k , (Z k )∗ ) to ϕσ (·, Y, Z ∗ ). Furthermore from [ (∫ )2 ] [ ] T k2 E ϕσ (t, Ytk , (Ztk )∗ )dt = E (CTk )2 ≤ const. 0

L1 (R, P

we derive the With this we have:

⊗ λ[0,T ] )-convergence of ϕσ (·, Y k , (Z k )∗ ) to 0.

ϕ(t, Yt , πt ) = ϕσ (t, Yt , Zt∗ ) = 0; P ⊗ λ[0,T ] -a.s. And so (π, C) fulfills the constraint ϕ. Finally we have to prove the optimality of this superhedging strategy: Theorem 2.5.39. In a complete market n = d we are given a square integrable contingent claim ξ and an admissible constraint ϕ. Let (Y k , Z k ) be the solutions of the sequence of BSDEs (2.5.23). Then the following assertions are equivalent: (i) There is a superhedging strategy for ξ which fulfills ϕ. (ii) There is an Ft -progressively measurable process X with [ ] E

sup |Xt |2 < ∞. 0≤t≤T

and Y k ≤ X for all k ∈ N. (iii) The sequence Y k is bounded in ST2 (R). (iv) The superhedging strategy (π, C) for ξ with corresponding wealth process Y from Lemma 2.5.36 is optimal w.r.t. the constraint ϕ. Proof. (iv) ⇒ (i) is trivial. ¯ (i) ⇒ (ii): Let X be the wealth process of the superhedging strategy (¯ π , C) k ∗ ∗ ¯ from (i). It suffices to show that X dominates the Y : Define Z := σ π ¯ . As the constraint is fulfilled we have for all k: ∫ T ∫ T Xt = ξ − rs Xs + Z¯s θs ds + C¯T − C¯t − Z¯s dWs t t ∫ T ∫ T ∗ ¯ ¯ ¯ ¯ = ξ− rs Xs + Zs θs − kϕσ (s, Xs , Zs )ds + CT − Ct − Z¯s dWs t

≥

Ytk .

t

106


The last inequality follows from the following theorem which generalizes the comparison theorem 8.2.28 by also taking care of the increasing process C. (ii) ⇒ (iii): As the sequence Y k is increasing and bounded above by X we see that ] [ ] [ ] [ E

sup |Ytk |2 ≤ E

0≤t≤T

sup |Yt0 |2 + E

0≤t≤T

sup |Xt |2 . 0≤t≤T

This proves the boundedness in ST2 (R). (iii) ⇒ (ii): From the boundedness of Y k in ST2 (R) we derive from Lebesgue’s theorem that [ ] E

sup |Yt |2 < ∞. 0≤t≤T

So we can choose X := Y . (ii) ⇒ (iv): From Lemma 2.5.36 we know that (π, C) is a superhedging strategy for ξ with wealth process Y and it fulfills ϕ by Lemma 2.5.38. It remains to prove optimality: Let Y˜ be the wealth process of another super˜ for ξ which fulfills ϕ. Using the same argument as in hedging strategy (˜ π , C) ”‘(i) ⇒ (ii)”’ we see that for all k: Y˜t ≥ Ytk . By taking the limit k → ∞ we finally get the optimality of (π, C). The following comparison theorem was used twice in the above proof: ˜ of standard data be given in the Theorem 2.5.40. Let two pairs (f, ξ) and (f˜, ξ) ˜ ˜ sense of definition 8.2.12. Let (Y , Z) be the solution to the BSDE corresponding ˜ The data fulfill the assumptions of theorem 8.2.28. Let for to the data (f˜, ξ). an increasing, progressively measurable process Ct with E[(CT )2 ] < ∞ the pair (Y, Z) be given by ∫

T

Yt = ξ +

∫ f (s, Ys , Zs )ds + CT − Ct −

t

T

Zs dWs , t

then: P (∀t∈[0,T ] Yt ≥ Y˜t ) = 1. Proof. Define the processes At , Bti as in theorem 8.2.28 and the process Φt as in theorem 8.2.24. Following the arguments in these theorems we get: [ ∫ T [ ] ] −1 ˜ + Yt − Y˜t = Φt E ΦT (ξ − ξ) Φs f (s, Y˜s , Z˜s ) − f˜(s, Y˜s , Z˜s ) ds Ft t ] [∫ T −1 +Φt E Φs dCs Ft . t

As Φ is positive and C is increasing so the term appearing additionally in comparison to the proof of theorem 8.2.28 nonnegative.

2.5.8

Examples

We are going to illustrate the above in some examples:


107

Constraints and closed sets From the definition 2.5.32 we immediately derive that an admissible constraint Kt (ω) is closed for all (t, ω). The following example shows that each closed set K defines an admissible constraint Kt (ω) := K. This together with theorem 2.5.39 generalizes a result by Cvitanic/Karatzas [9] on optimal superhedging strategies, as these authors let K be also convex. Example 2.5.41. Let K ⊂ R1+n be a closed set. Then: { } ′ ′ K = (y, p); ϕ(t, ω, y, p) := inf |(y, p) − (y , p )| = 0 , (y ′ ,p′ )∈K

and ϕ fulfills the assumptions in definition 2.5.32. So Kt (ω) := K is an admissible constraint and theorem 2.5.39 gives a criterion for the existence and the construction of optimal superhedging strategies under the constraint K. As an interesting special case the example covers the restriction of the socalled no short selling condition: Example 2.5.42. (i) When modeling the market we also allowed for short selling which means that the investor was not only allowed to borrow money from the bank but he could also hold a negative number of stocks as long as his total wealth is nonnegative. The no short selling condition is described by the constraint: (Yt , πt ) ∈ R × (R≥0 )n . (ii) In a similar way we can restrict trading to a certain number of stocks. Assume that only the first m stocks are really traded so this is described by the constraint (Yt , πt ) ∈ R × Rm × {0} × . . . × {0}. This is intimately connected to the superhedging in incomplete markets as we shall see further below, see section 2.5.9. We have the following generalization of example 2.5.41: Theorem 2.5.43. Consider a family of closed sets in R1+n (Kt (ω))t∈[0,T ],ω∈Ω . If inf (y,p)∈Kt (ω) |(y, p)| ∈ HT2 (R), then (Kt (ω))t,ω is an admissible constraint. Proof. Like in example 2.5.41 we define the distance ϕ(t, ω, y, p) :=

inf

(y ′ ,p′ )∈Kt (ω)

|(y, p) − (y ′ , p′ )|.

From the closedness of the sets Kt (ω) we have (y, p) ∈ Kt (ω) ⇔ ϕ(t, ω, y, p) = 0. The distance fulfills the Lipschitz condition of definition 2.5.32 with constant 1, and so we have: ϕ(t, ω, 0, 0) =

inf (y,p)∈Kt (ω)

|(y, p)| ∈ HT2 (R)

108


Let us illustrate this in an example where it is not allowed to trade fractions of a stock: Example 2.5.44. We consider the restriction that an investor is no longer allowed to buy/sell fractions of a stock. Only natural numbers of stocks are traded. So: (Yt , πt ) ∈ Kt (ω) := R × {kSt1 ; k ∈ Z} × . . . × {kStn ; k ∈ Z}. Here inf (y,p)∈Kt (ω) |(y, p)| = 0, as for all t ∈ [0, T ] and ω ∈ Ω we have: (0, 0) ∈ Kt (ω). From the last theorem this constraint is admissible. Note that this last case leads to a non-convex Kt (ω). So this case is not covered by the treatment in Cvitanic und Karatzas [9]. American Contingent Claims We consider again the situation of section 2.4.3: Definition 2.5.45. A nonnegative process ξt ∈ ST2 (R) is called an American contingent claim if ξt is P -a.s. continuous on [0, T ). We already know the example of an American contingent claim which is called an American put option: Example 2.5.46. An American put option is a contract between a seller and a buyer. The buyer gets the right to sell a stock at the price x fixed in the contract at any time t before the closing of the market T . If the buyer makes use of the option at time t ≤ T then the seller has at this time the obligation to pay ξt := [x − St ]+ . We formulate here now the aim of the seller to hedge his obligation at any time 0 ≤ t ≤ T by a constraint: ϕ(t, y, z) := [ξt − y]+ .

(2.5.27)

Using standard estimates (see El Karoui et al. [19], S. 719 ff.) we see that condition (iii) in theorem 2.5.39 is fulfilled. So the seller’s price of the American contingent claim may be derived as the limit of a sequence of nonlinear BSDEs: Theorem 2.5.47. The seller’s price of an American contingent claims ξt is given by Yt = limk→∞ Ytk P -a.s, where [ ] dYtk = rt Ytk + Ztk θt − k[ξt − Ytk ]+ dt + Ztk dWt , YTk = ξT .


109

Remark 2.5.48. We may see the American contingent claim as an obstacle. So the constraint (2.5.27) simply means that the wealth process has to stay above the obstacle. In the case of an American claim the optimal superhedging strategy (π, C) may be interpreted as being optimal in the following sense: C is continuous and increases only if Y hits the obstacle. So C only becomes active to reflect the wealth process away from the obstacle. With this we are in the theory of reflected BSDEs which are strongly connected to stopping problems and obstacle problems for parabolic PDEs. We here only give a rough idea of this extensively studied subject: Let τt be the stopping time t ≤ τ ≤ T which is optimal for the buyer after time t. Then at this time τt the seller’s wealth must cover the obligation ξτt . This hedging problem may be solved as in section 8.3. Let Ysτt ,ξ be the price

110


process of the problem. Then the wealth process Y of the optimal superhedging strategy for the American claim ξt is given by Yt = Ytτt ,ξ . For details and further interesting aspects of reflected BSDEs the reader is referred to El Karoui et al. [19], El Karoui/Quenez [23] und Kohlmann [17] and the reader is invited to prove some of these aspects in the following exercise: Exercise 2.5.49. To make the above results comparable to the classical approach prove the following relations • Let ξt be an American contingent claim, ξT = ξ. We call the BSDE a reflected BSDE RBSDE if it is of the form dYt = −f (t, Yt , Zt )dt − dCt + Zt dwt , YT = ξ, 0 ≤ t ≤ T,

(2.5.28)

where f is a ”’good”’ (give a set of assumptions!) generator. The triple (Y, Z, C) of twice integrable Ft -progressively measurable processes is called a solution of the RBSDE if the integral equation corresponding to (2.5.28) is fulfilled where – (Ct ) is a (here) continuous increasing process with C0 = 0, – Yt ≥ ξt and ∫T – 0 (Yt − ξt )dCt = 0. • Show that the above construction of a penalized BSDE leads to a RBSDE. • Relate the notion of RBSDE to the Skorokhod problem: Let x be a real valued continuous function on [0, ∞) with x0 = 0. There exists a unique pair (y, k) of functions on [0, ∞) such that – y =x+k – y is positive – and k is an increasing continuous function with k0 = 0 and 0.

∫∞ 0

yt dkt =

Show that the solution of Skorokhod’s problem is given by kt = sup x− s

(2.5.29)

s≤t

• Let (Y, Z, C) be a solution of the RBSDE. Use Skorokhod’s problem and its solution to show that ∫ T ∫ T CT − Ct = sup (ξ + f (s, Ys , Zs )ds − Zs dws − ξu )− (2.5.30) t≤u≤T

u

u

• Let (Y, Z, C) be a solution of the RBSDE. Show that ∫ τ Yt = ess supτ ∈Tt E( f (s, Ys , Zs )ds + ξτ |Ft ) t

where Tt is the set of stopping time taking values in [t, T ].

(2.5.31)


111

• Prove that the optimal stopping time is given by τt = inf{t ≤ u ≤ T |Yu = ξu }.

(2.5.32)

• Relate these results to the results in the classical treatment of the American contingent claim. • Relate the RBSDE to an obstacle problem for a nonlinear parabolic PDE.

2.5.9

Incomplete Markets

Up to now we only considered complete markets in the sense that every claim could be reached by a hedging strategy or in the above setting equivalently that the number of risky sources is equal to the number of stocks n = d. In incomplete markets in general there are more sources of risk than stocks. In mathematical terms this means that the information in the claim -which by definition is measurable w.r.t. the history of the sources of risk- is larger than the information in the history of the stocks. So in this situation we can no longer expect that the claim can be reached by a wealth process basing on the stocks. One possibility to circumvent the problem is to look for a hedging portfolio which minimizes the risk of missing the goal. This leads to the problem known as Föllmer-Schweizer-hedging . This will also be studied extensively in chapter 3. Let us start with a definition (see the comment at the beginning of the chapter): Definition 2.5.50. A market is incomplete if d > n. Furthermore we assume that (M0’) holds. Definition 2.5.51. We call a feasible (obviously non-self-financing) portfolio π a F¨ ollmer-Schweizer-portfolio (FS-portfolio for short) against the contingent claim ξ if: (i) There is a square integrable martingale Mt with EMt ≡ 0 and an Ft progressively measurable process V such that: [ ] n d ∑ n ∑ ∑ i i dVt = rt Vt + (µt − rt )πt dt + σti,j πti dWtj + dMt , i=1

VT

= ξ.

j=1 i=1

(2.5.33)

(ii) V ∈ STc,2 (R). ∫t (iii) Mt is orthogonal with 0 σs dWs . Mt is the difference between the value process of the portfolio πt and Vt . MT measures the discrepancy to the contingent claim ξ which is reached in expectation. Furthermore it can be shown that under appropriate assumptions on

112


the market the risk of missing the claim is minimized (see Föllmer/Sondermann [6], Föllmer/Schweizer [14], Schweizer [58, 59, 60]. Also see chapter 3 for the relation to the mean variance hedging problem. First we go into the existence problem where we first face the problem that the set Θ of solutions of the linear system (2.5.10) has more than one element. Lemma 2.5.52. Let Θ be as above. Then there is exactly one element θ¯ ∈ Θ with θ¯t ∈ im σt∗ . θ¯ is given by σ ∗ [σσ ∗ ]−1 (µ − r1). The proof uses similar results as theorem 2.3.9. θ¯ is called the minimal risk premium vector/process. It is bounded by (M0’). Consider the BSDE [ ] dVt = rt Vt + Zt θ¯t dt + Zt dWt , VT

= ξ.

(2.5.34)

Here ξ is a square integrable contingent claim. From theorem 8.2.24 this BSDE has a unique classical solution (V ξ , Z) given by: [ { ∫ T } ] ∫ T 1 ¯ 2 ξ ∗ ¯ Vt = E exp − rs + |θs | ds − θs dWs ξ Ft . (2.5.35) 2 t t In general the linear system

σt∗ πt = Zt∗

will have no solution. That is why we look for an approximative solution instead. From linear Algebra we know that σ ∗ has maximal rank and so there is a π with |σt∗ πt − Zt∗ | = min! and π is given by:

πt = [σt σt∗ ]−1 σt Zt∗ .

(2.5.36)

Define

Z˜t := Zt − πt∗ σt . Then Z˜t∗ ∈ ker σt . So Z˜t∗ is orthogonal with θ¯t . So we get: [ ] dVt = rt Vt + πt∗ σt θ¯t dt + πt∗ σt dWt + Z˜t dWt , VT

(2.5.37)

= ξ.

∫t From the integrability of Z Mt := 0 Z˜s dWs is a square integrable martingale and we have: ∫ . ∫ . ∫ t ˜ ⟨ Zs dWs , σs dWs ⟩t = σs Z˜s∗ ds = 0. 0

0

0

∫t So the martingale Mt is orthogonal (in the sense of Stochastics) with 0 σs dWs . This means that π is a FS-strategy. Let us summarize this in the next Theorem 2.5.53. Let ξ be a square integrable contingent claim. Then there is a FS-portfolio π against ξ where Vt = Vtξ is given by (2.5.35), and Mt is the Itˆ o’s integral of Z˜ in (2.5.37).


113

What about uniqueness? Let π ˜ be another FS-portfolio against the con˜ t . From the martingale tingent claim ξ with corresponding processes V˜t and M ˜ representation theorem 8.2.2 M may be written as ∫ Mt =

t

˜ s dWs . X

0

˜ t∗ is in the kernel of σt and so is orthogonal with θ¯t . So the pair (V˜ , π Then X ˜ ∗ σ+ ˜ solves (2.5.34). As this BSDE has a unique solution π X) ˜ must be equal with π and V ξ is the Föllmer-Schweizer-price of the contingent claims ξ. We now get the following result in the incomplete market which should be compared to the complete situation: Theorem 2.5.54. The mapping { } ˜ : ξ; ξ contingent claim with Eξ 2 < ∞ → S c,2 (R), ξ 7→ V ξ Π T is an arbitrage free pricing system. It gives the F¨ ollmer-Schweizer-price in an incomplete market. The Föllmer-Schweizer-hedging makes sure that the risk of missing the claim is minimal. There is nothing said about the fact that on bad paths we might end up far away from the claim. The exact shortfall might be immense. To overcome this problem we are now going to consider superhedging strategies for the contingent claim, and we try to find the seller’s price in this way. As we are going to use the results of the last section on superhedging we first must artificially complete the market. To this end we add d − n new risky investments S n+1 , . . . S d so that the market fulfills the general assumptions of the section on superhedging. Let the coefficients of the extended market ¯ and µ ¯. Here the first d − n rows of σ ¯ and µ ¯ are just σ and µ. As a be σ constraint/restriction we now let the new stocks not be traded. The constraint is given by: √ ϕ(t, y, p) :=

|pn+1 |2 + . . . + |pd |2 .

(2.5.38)

Here pi is the i-th component of the vector p. Obviously this constraint is admissible so that theorem 2.5.39 gives the following result: Theorem 2.5.55. Let ξ be a square integrable contingent claim which can be reached in an incomplete market by making use of a superhedging strategy. Then the seller’s price of ξ in the incomplete market is given by Yt := limk→∞ Ytk P a.s., where [ ] dYtk = rt Ytk + Ztk θ¯t − kϕ(t, Ytk , (¯ σt∗ )−1 (Ztk )∗ ) dt + Ztk dWt , YTk = ξ. Here σ ¯ is the diffusion matrix of the extended market, θ¯ is the risk premium vector in this market and ϕ is as in (2.5.38).

114


Exercise 2.5.56. • Each bounded square integrable contingent claim ξ fulfills the assumptions of the above theorem. Derive a superhedging strategy for this situation. • Derive the seller’s price of an American contingent claim in an incomplete market by making use of the methods of this section.

2.5.10

Remarks

on Section 2.5: • The market model in the above sections is a generalization of the model introduced by Samuelson [57]. Though we used only weak assumptions the model is not realistic under various aspects: So dividends as well as transaction costs can not be treated within this model. Furthermore one underlying assumption is the small investor’s hypothesis which says that the trading of the investor has no influence on the market data. Generalizations can be found in Karatzas [31] and Ma/Yong [46], chapter 8. • The construction of feasible hedging strategies for arbitrary initial wealth above goes back to El Karoui/Peng/Quenez [9]. There however the authors first construct a suicide strategy from Dudley’s theorem, that is a strategy that certainly annihilates a positive initial wealth. For more information on such strategies see Harrison/Pliska [14]. For general treatments of arbitrage opportunities see the recent works by Delbaen/Schachermayer, e.g. [7, 11] and Levental/Skorohod [22]. • The results on markets with unbounded random data are taken from C. Bender’s diploma thesis and generalize earlier results El Karoui/Peng/Quenez [9]: ”‘To suppose that the short rate r is uniformly bounded is an assumption rarely satisfied in a market. The same remark holds for the risk premium vector”’ (El Karoui/Huang [17]). • The Black-Scholes formula [7] has been developed further and further. Our approach is similar to Ma/Yong [23, 46]. • The results on superhedging rely on recent works by Peng [54]. The results derived here are more general than those in ElKaroui/Quenez [21] oder Cvitanic/Karatzas [9]. The main result on superhedging under constraints 2.5.39 is an adaptation of Theorem 4.2 in Peng [54] to the superhedging problem. This approach was first considered in C. Bender’s diploma thesis. • BSDEs were used to treat problems in incomplete markets by El Karoui/Quenez [22]. The approach here does not make use of the possibility of extending markets as we here rely on the approximative solution of linear systems.

2.6. THE PARTIALLY INFORMED AGENT

2.6 2.6.1

115

The Partially Informed Agent Introduction

The existence of an adapted solution to a backward stochastic differential equation has been proved when the filtration under consideration is generated by the underlying Brownian motion. In the first part of this article we use Girsanov’s theorem to derive an existence result for a larger filtration to which the Brownian motion is adapted. This result is applied to the pricing of a contingent claims by using forward-backward stochastic differential equations. This approach has several advantages in comparison to the classical techniques: There is no need to define an optimal hedging strategy and the whole system can be considered under the original measure. The power of this tool will be described here when we derive the price of a claim in both an incomplete and a complete market under different information structures available to the agent. Especially we are interested in the question when these prices are equal. The methods we use are described in [15], where techniques from control theory establish the relation between the price of a claim and the adjoint equation of a trivial control problem.

2.6.2

An existence result for a BSDE under additional information

From the results in [18], [19], [20] and [21] and the counterexamples in [1] it is common belief that a BSDE has an ordinary, that is adapted solution only if the underlying filtration is generated by the Brownian motion which is assumed to be augmented to satisfy the usual conditions in the sense of Meyer. In this section we shall establish an existence result for a BSDE where the filtration is greater than the one generated by the Brownian motion. Let (Ft )t∈[0,T ] be a filtration on a given probability space (Ω, F, P ) which carries a standard Brownian motion (wt ) adapted to this given filtration. Note that this filtration might be larger than (Ftw ) the one generated by (wt ) Ftw ⊂ Ft .

(2.6.1)

r : [0, T ] → ℜ

(2.6.2)

Let be a deterministic B (0, T )-measurable function and θ : [0, T ] × Ω → ℜ

(2.6.3)

be an (Ft )-progressively measurable function such that ∫ P(

T

θu2 du < ∞) = 1

(2.6.4)

0

and E(zT ) = 1

(2.6.5)

116


for (zt ) the solution of the Doléans-Dade equation dzt = −zt θt dwt

(2.6.6)

z0 = 1. We are given a BSDE dyt = − [yu r + Zu∗ θu ] du − Zu∗ dwu

(2.6.7)

yT

(2.6.8)

= ξ

on (Ω, F, Ft , P ), where ξ is an FT -measurable random variable. The problem now consists in finding an (Ft )-adapted pair (yt , Zt ) which satisfies integrability conditions such that the integral equation ∫

T

yt = ξ −

[yu r +

Zu∗ θu ] du

∫ −

t

T

Zu∗ dwu .

(2.6.9)

t

holds P-a.s.. Now consider the auxiliary problem eu∗ dw de yt = −e yu rdu − Z eu

(2.6.10)

yeT = ξ where (f wt )is the Girsanov transform of (wt ) with respect to the Girsanov functional associated with θ, i.e. ∫ w ft = wt +

T

θu du.

(2.6.11)

0

(w et ) is a standard Ft -Brownian motion with respect to Pe where dPe = zT dP (2.6.12) ( we ) is a probability measure on (Ω, F ). Let Ft be the filtration generated by w et . Clearly, ( ) Ftwe ⊂ (Ft ) . (2.6.13) The crucial assumption which at first sight might appear quite technical is the equality ( ) FTwe = (FT ) . (2.6.14) In the following section we will show that it is quite natural in an application to finance: (FT ) is the information available to an agent who has anticipative knowledge of the behavior of part of the market. This additional information is


117

given right from the beginning. A second agent has to gather this information as time goes on until in the end the additional information becomes useless: FTwe = FT .

(2.6.15)

Under this assumption we may now consider ∫ de yt = ξ −

T

∫ yeu rdu −

t

T

e ∗ dw Z u eu

(2.6.16)

t

( ) on Ω, F, Ftwe , Pe and from the results in [2] and [21] it is straightforward that ( ) ( ) et exists. an Ftwe -adapted solution yet , Z Theorem 2.6.1. The unique solution of (2.6.16) also solves (2.6.9). Proof. As ξ is FTwe = FT -measurable the following equation holds: ∫

T

yet = ξ − ∫

t

∫

t T

= ξ−

yeu rdu −

T

e ∗ dw Z u eu

∫ T e∗ dwu − Zeu∗ θu du Z u t t ∫ T [ ] eu∗ dwu eu∗ θu du − Z yeu r + Z ∫

yeu rdu −

T

t

t

(

(2.6.17)

t

T

= ξ−

∫

)

et solves (2.6.9). This proves that yet , Z This result will be applied to the case where an enlargement of filtration can be described by Girsanov’s theorem.

2.6.3

Setting the hedging problem

We consider - in the usual notation - a (d+1)-dimensional asset consisting of a bond dPt0 = r(t)Pt0 dt

(2.6.18)

dPti = µi (t, Pt )Pti dt + σij (t, Pt )Ptj dwtj ,

(2.6.19)

and d stocks

t ∈ [s, T ] , 1 ≤ i, j ≤ d with initial (1, p1 , ..., pd ).

(2.6.20)

118


Here (wt ) is a d-dimensional Brownian motion and (Ft ) the augmented filtration of (wt ). All processes are assumed to live on (Ω, F, Ft , P ) which is assumed to satisfy the usual condition in the sense of Meyer. The coefficients µi : [0, T ] × Rd → R

(2.6.21)

σij : [0, T ] × Rd → R.

(2.6.22)

and

are assumed to satisfy conditions such that a pathwise unique solution exists. Finally σ is assumed to be invertible. For details on the conditions, see [8] or [14]. The claim is an FT -measurable random variable ξ , which e.g. is of the form ξ = g(PT , ω)

(2.6.23)

where g is a B ⊗ FT -measurable function g : Rd × Ω → R

(2.6.24)

such that ξ is twice integrable. In [15] we showed that the pricing and hedging problem for the above claim and asset may be considered as a trivial control problem. To this end we define the risk premium process θ by σθ = µ − r1d

(2.6.25)

where σ = (σij ), µ = (µi ) and 1d is the d-dimensional vector whose every component is 1. We assume that θ exists as a d-dimensional bounded, predictable vector process, this ensuring the absence of arbitrage in our model (see [13], [14] ). For arbitrage considerations under change of filtrations, see [45]. Next consider the following trivial control problem with dynamics dzst = −zst [rdt + θt∗ dwt ]

(2.6.26)

zss = 1

(2.6.27)

J = E [zsT ξ] .

(2.6.28)

and cost criterion

The formal adjoint of this problem is given by ∫

T

yt = ξ −

[yu r + Zu∗ θu ] du −

t

∫

T

Zu∗ dwu .

(2.6.29)

t

Following [6], [7], [8], [9] the unique solution (yt , Zt ) of this forward-backward stochastic differential equation is the hedging price of the claim ξ ¯ [ξ | Ft ] yt = E

(2.6.30)


119

¯ denotes expectation with respect to the Girsanov measure associated where E with the Girsanov functional (zst ). The hedging portfolio for the risk premium process θ is given by πt = σ −1 Zt .

(2.6.31)

This may be written in a more familiar way when we consider a generalized pricing system, i.e. a predictable function u : [s, T ] × Rd × Ω → R

(2.6.32)

yt = u(t, Pt ).

(2.6.33)

such that

The pricing system satisfies the stochastic partial differential equation ] ∫ T 1 σuxx σ + (µ − σθ)ux − ru + σkx − kθ ds− k(s, x)dws . 2 t t (2.6.34) Obviously the hedging portfolio is then given by

∫ u(t, x) = g(x)+

T

[

πt = σ −1 Zt = ∇u(t, Pt ) + σ −1 k(t, Pt ).

(2.6.35)

The details of these results can be found in [15]. If in the above spde the coefficients are deterministic then the solution is given by (u, k) = (u, o) and the spde goes over into the form of the classical Black-Scholes formula for the optimal hedging problem. Note however here that for the pricing of the claim in this setting no change of measure and no explicit optimal hedging strategy has to be considered. For the general development of the theory of FBSDE see [6], [7], [18], [19], [20], [21], [22], [25].

2.6.4

Available Information

We consider a market with two agents A0 and Ain .The latter has more information about the evolution of the asset than the former. By making use of results by [10], [12] and [23] we will consider the prices under these different information structures (Ft0 ) and (Ftin ), respectively. Let α1 (t), α2 (t) be two nonsingular matrix valued deterministic, measurable functions of t with α1 α1∗ + α2 α2∗ = I

(2.6.36)

w(t) = α1 w1 (t) + α2 w2 (t)

(2.6.37)

for all t. Then

120


is again an Ft -Brownian motion if both w1 and w2 are independent Ft -Brownian motions. Let in the above model for our market w be decomposed in this way. Let Ft0 = Ft = Ftw and Ftin = σ(w1 (T )) ∨ Ft .

(2.6.38)

Both Ft0 and Ftin are again assumed to satisfy the usual conditions. In this case the results of [10], [12], and [23] allow us to describe the enlargement of filtration by Girsanov’s theorem: Theorem 2.6.2. Let △u (x) = α1 (T I − α1 α1∗ t)−1 (x − α1∗ w(t))

(2.6.39)

be the cross variation of the Ft -conditional density of (w1 (T )) and (wt ) divided by the density, then ∫

t

B(t) = w(t) −

△u (w1 (T ))du

(2.6.40)

s

( ) is a d-dimensional Ω, Ftin , P -standard Brownian motion. Exercise 2.6.3. Let αi , i = 1, 2 be deterministic • Compute the conditional density qs (x) of P (w1 (T )|Ft ) Ft = σ(ws , s ≤ t) to be the density of a normal distribution with mean α1 wt and variance T − α12 t. Use Ito’s formula to derive quadratic variation < q. , w. >s and show that ∂qs = qs △s (x). ∂x • Prove E[wt − ws |Ftin ] = (t − s)△s (x). Derive the representation result for is case. Remark 2.6.4. Note here that (Bt ) is an Ftin -Brownian motion for which in general the natural filtration (FtB ) will be different from (Ftin ). Below we consider an FBSDE with driving Brownian motion (Bt ). Following the arguments in the first section we will derive a solution to this FBSDE.


121

For the informed agent Ain the asset appears as

dPt0 = r(t)Pt0 dt dPtin = Ptin (µi (t, Ptin )dt + σij △jt (w1 (T ))dt + σij (t, Ptin )dBtj ), t ∈ [s, T ] , 1 ≤ i, j ≤ d. The risk premium processes for the two agents are respectively θ0 = σ −1 (µ − r1d )

(2.6.41)

θin = σ −1 (µ + σ△t − r1d ).

(2.6.42)

and

We now formally apply the arguments of section 1 to compute the prices yt0 and ytin for A0 and Ain : 0 be a solution of Let zst ] [ 0 0 dzst = −zst rdt + θ0∗ dwt 0 zss

(2.6.43)

= 1

in be a solution of and zst

] [ in in rdt + θin∗ dwt dzst = −zst in zss

(2.6.44)

= 1,

then the prices for the FTin = FT0 = FT claim is given by ∫ yt0

T

=ξ−

[

yu0 r

+

Zu0∗ θ0

]

∫

T

dt −

Zu0∗ dwu

(2.6.45)

Zuin∗ dBu

(2.6.46)

t

t

and ∫ ytin

=ξ− t

T

[ in ] yu r + Zuin∗ θi dt −

∫

T

t

) ) ( ( on Ω, F, Ft0 , P and Ω, F, Ftin , P respectively.

Theorem 2.6.5. The prices (yt0 ) and (ytin ) are equal, if the price (yt0 ) exists. Moreover under the above assumptions it is a standard result that (yt0 , Zt0 ) exists. Proof. Let (yt0 , Zt0 ) be a solution then (yt0 ) and (Zt0 ) are also Ftin −measurable. Then

122


∫ yt0

T

= ξ− t

∫

t

∫

T

= ξ−

∫

T

Zu0∗ dwu ∫ T ] [ 0 0∗ 0 0∗ yu r + Zu △u + Zu θ dt − Zu0∗ dBu t ∫ T ] [ in in∗ in yu r + Zu θ dt − Zuin∗ dBu

(2.6.47)

t

T

= ξ−

[ 0 ] yu r + Zu0∗ θ0 dt −

t

t

= ytin so that (yt0 , Zt0 ) solves for (ytin , Ztin ). Corollary 2.6.6. The optimal hedging strategies are equal for both agents, if (yt0 , Zt0 ) exists. Proof. From section 1 the optimal hedging strategies are given by πt0 = σ −1 Zt0 and πtin = σ −1 Ztin .

(2.6.48)

As Zt0 = Ztin by the uniqueness of solutions the result follows. Note, however,that the risk premium processes are different. Instead of studying the forward-backward equations we could also look at the corresponding pricing systems: ∫

T

0

u (t, x) = g(x) + ∫ −

t

[

] 1 0 0 0 0 0 0 0 σu σ + (µ − σθ )ux − ru + σkx − k θ ds 2 xx

T

k 0 (s, x)dws

(2.6.49)

t

and ∫ in

u (t, x) = g(x) + ∫ −

t

T

[

] 1 in in in in in in in σu σ + (µ − σθ )ux − ru + σkx − k θ ds 2 xx

T

k in (s, x)dBs

(2.6.50)

t

in an abbreviated ) ( ( 0 0 ) notation. in , if it exists, and hence Again ut , k solves the equation for uin t ,k yt0 = u0 (t, Pt , ω) = uin (t, Pt , ω) = ytin .

(2.6.51)

Remark 2.6.7. If in the above spde all coefficients are deterministic, then a solution is given by ( 0 0) ( 0 ) ( ) ( 0 ) in ut , k = ut , 0 and uin = ut , 0 . (2.6.52) t ,k Furthermore ys0 = ysin are deterministic as a consequence of the Markov nature of the processes.


2.6.5

123

The F¨ ollmer-Schweizer uninformed agent

Here we assume that ( )∗ Ptp = Pt1 , ..., Ptm

(2.6.53)

are the primary securities which are actually traded, m < d . Split up the matrix σ into p1 = (σij )i≤m,j≤d

(2.6.54)

p2 = (σij )m 0

(2.7.7)

(2.7.8)

j=1

0 0, t ∈ [0, T ]

(2.7.38)

Γ(t) = E (x∗t − pt )2

(2.7.39)

. ( ) Γ − 2 r − θ2 Γ − θ2 Γ = 0

(2.7.40)

Γ (0) = (x − p (0))2

(2.7.41)

satisfies

(iii) The smallest initial endowment to reach the contingent claim ξ is x = p(0)

(2.7.42)

and in this case the wealth process (xx,π∗ ) = (x∗t ) is equal to (pt ). So the t fair price of the contingent claim (- and so the solution of the BSDE (pt ) -) is the wealth process for the mean variance hedging problem with starting point x = p (0), and (qt ) is the corresponding portfolio multiplied by σ.

2.7. NEYMAN-PEARSON HEDGING

133

Note that the above result may be interpreted in the following way: If the initial capital is equal to the fair price of the contingent claim at time t = 0 then the wealth process is just the price of the contingent claim. Let us assume that the agent has a higher (respectively, a lower) capital then the ’rest’ is invested according to a Merton type strategy.

2.7.3

Superhedging and the upper price in an incomplete market

When we consider incomplete markets the target ξ might not be reachable, the liability ξ cannot be hedged in the sense that there might not be a starting point x and a portfolio π such that xx,π = ξ. This situation typically arises T when the number of tradeable securities is smaller than the number of random sources in the market. A most successful approach to solving this problem is the Föllmer et al. hedging or the mean variance hedging approach [14]. The research in this field gives a deep insight into the structure of hedging problems, though the main drawback of this theory is the fact that mean variance hedging makes no difference between a shortfall from the unreachable obligation and an overshooting: The influence of these two failures of completing a liability are ruled out by just considering the variance. A completely different approach was taken by Karatzas, Shreve, Cvitanic, El Karoui and Quenez [3], [4] ,[8] ,[11]. The idea consists in defining a selffinancing superstrategy (π, c) where π is a portfolio process and the increasing right continuous process (ct ) with c0 = 0 may be interpreted as a consumption process. The upper price of a contingent claim is then given by the generalized BSDE dpt = (rpt + θqt ) dt − dct + qt dwt

(2.7.43)

pT = ξ.

(2.7.44)

It is straightforward that the idea of introducing the above generalized BSDE has its counterpart in the martingale approach. Karatzas and Shreve [11] make use of the classical Doob Meyer decomposition to derive the existence of the cumulative consumption process. A similar approach was taken by El Karoui and Quenez [8]. Generalizations are found in Kramkow [10], and Föllmer-Kramkow [19], where a ”Doob-Meyer decomposition” with an optional process is derived. Yet another approach is possible by considering g − martingales [27]. At this point it makes sense to formalize the BSDE techniques for future use. The definitions and properties are taken from [24]. Many results below may be cited in a more general way, we restrict the assumptions to our needs. Given (Ω, F, P ) and an Rn -valued Brownian motion so we consider (i) (Ft )t∈[0,T ] the augmented filtration generated by (wt )t∈[0,T ] and it is assumed that the filtration satisfies the usual properties. P denotes the predictable σ-field.

134

CHAPTER 2. CLASSICAL THEORY ( ) (ii) L2 T ; Rd , the space of all FT -measurable random variables ( ) x: Ω −→ Rd with E |x|2 < ∞. ( ) (iii) H 2 T ; Rd the space of predictable processes ϕ : Ω × [0, T ] −→ Rd

(2.7.45)

with T

E ∫ (ϕt )2 dt < ∞.

(2.7.46)

0

( ) (iv) H 1 T ; Rd the space of predictable processes ϕ : Ω × [0, T ] −→ Rd with

(2.7.47)

√ T

E

∫ |ϕt |2 dt < ∞

(2.7.48)

0

( ) ( ) (v) Hβ2 T, Rd , β > 0 denotes the space H 2 T ; Rd endowed with the norm |.| defined by |.|2 = E ∫0T eβt ϕ2t dt. Definition 2.7.4. Consider the BSDE dpt = −f (t, pt, qt ) dt + qt dwt ,

(2.7.49)

pT = ξ

(2.7.50)

as an analogue of T

T

t

t

pt = ξ + ∫ f (t, pt, qt ) dt − ∫ qt dwt

(2.7.51)

where (i) ξ is an FT -measurable random variable (ii) the ”generator” f , f : Ω × R∗ × Rd × Rn×d −→ Rd

(2.7.52)

is P ⊗ B d ⊗ B n×d -measurable. We call (pt, qt ), such that (i) (pt ) is an adapted continuous Rd -valued process ∫T (ii) (qt ) is an Rn×d valued predictable process with 0 |qs |2 ds < ∞, P − a.s. a solution of the above BSDE. The next theorem gives a fairly general existence result [19]


135

( ) ( ) Theorem 2.7.5. Let ξ ∈ L2 T ; Rd , f (., 0, 0) ∈ H 2 T ; Rd such that there exists C > 0 with |f (ω, t, p1 , q1 ) − f (ω, t, p2 , q2 )| ≤ C(|p1 − p2 | + |q1 − q2 |), P − a.s. for all p1 , p2 , q1 , q2 . (In [8] (f, ξ) is then called a pair of standard parameters for the BSDE). ( ) ( ) Then there exists a unique pair (p, q) ∈ H 2 T ; Rd × H 2 T ; Rn×d which solves the above BSDE. We shall make use of a comparison theorem which is stated in ( 1 1) ( 2 2) Theorem ( 1 1 ) 2.7.6. ( 2Let2 ) f , ξ , f , ξ be standard parameters of BSDEs and let p , q and p , q be the associated square integrable solutions. Let (i) ξ 1 ≥ ξ 2 , P − a.s. (ii) f 1 (t, p1t , qt1 ) − f 2 (t, p2t , qt2 ) ≥ 0, dP ⊗ dt − a.s. Then for any t p1t ≥ p2t almost surely. If p1t = p2t on A ∈ Ft then (i) p1t = p2t on [t, T ] × A, a.s. (ii) ξ 1 =( ξ 2 , a.s.)on A ( ) (iii) f 1 s, p1s , qs1 = f 2 s, p2s , qs2 on A × [t, T ] , dP ⊗ ds − a.s. The comparison theorem implies Corollary 2.7.7. If ξ ≥ 0 a.s. and f (t, 0, 0) ≥ 0 dP ⊗dt− a.s. then pt ≥ 0 P − a.s. If pt = 0 on a set A ∈ Ft ,then ps = 0, f (s, 0, 0) = 0 on [t, T ]×A dP ⊗ds−a.s. and ξ = 0 a.s. on A. Remark 2.7.8. We shall mainly use the above theorems for the linear case, that is for the case −f (t, p, q) = rt pt + θt qt + ϕt n (whered )rt , θt are bounded R, resp. R -valued predictable processes, ϕ ∈ 2 H T, R . It is straightforward from the above theorem that in this case the BSDE has a unique solution and (pt ) is given by [ ∫ −1 pt = E z0t z0T ξ +

T

t

−1 z0t z0s ϕs ds|Ft

] P − a.s.

(2.7.53)

Definition 2.7.9. A supersolution of a BSDE with standard parameters (f, ξ) is a triple of processes (p, q, C) with dpt = −f (t, pt , qt ) dt + qt∗ dwt − dCt , pT = ξ

(2.7.54)

where • ξ is FT -measurable • (pt ) is a cadlag process. • (qt )is a predictable process with

∫T 0

qs2 ds < ∞ P − a.s.

• (Ct ) is an increasing, adapted, right continuous process with C0 = 0

136


• pt ≥ 0 P − a.s. ( ) Theorem 2.7.10. (i) Let (p1 , q 1 ) and p2 , q 2 be two solutions of BSDEs with standard parameters satisfying the( assumption ) of the comparison theorem. Then 1 , q 1 , C is a supersolution of the BSDE with there exists(a process C such that p ) parameter (f 2 , ξ 2 . ) ( ) (ii) Let p1 , q 1 , C 1 and p2 , q 2 , C 2 be two continuous supersolutions of BSDEs with parameters ( 1 1) ( 2 2) ∗ ∗ f , ξ and ( ∗ )f , ξ Then there exists a pair (q , C ) such that 1 2 ∗ ∗ p = p ∧ p , q , C is a supersolution of the BSDE with terminal condition ξ ∗ = ξ1 ∧ ξ2 and generator f t (t, p, q) = 1{p1t ≤p2t } f 1 (t, p, q) + 1{p2t xt }

(2.7.100)

In the very special case under consideration in this section we can compute the success ratio explicitly in terms of the density (z0T ) . This is a direct consequence of the Neyman-Pearson lemma: Theorem 2.7.18. The optimal randomized test φ∗ ∈ R is given by φ e = 1{z −1 >a∗ } + γ1{z −1 =a∗ }

(2.7.101)

{ ∫ } a∗ = inf a| p∗T dP = x0 −1 {z0T >a}

(2.7.102)

0T

0T

where

and ( γ=

) ∫ x∗ − p∗T dP / p∗T dP −1 −1 ∗ ∗ z >a z =a { 0T } { 0T } ∫

(2.7.103)

Proof. The optimization problem E (p∗T − xx,π t ) = minπ∈A(x) ! +

(2.7.104)

xx,π 0 ≤ x0

(2.7.105)

under the constraint

142

CHAPTER 2. CLASSICAL THEORY is equivalent to E (φ (T ) p∗ (T )) = max!

(2.7.106)

e (φ (T ) p∗ (T )) ≤ x0 E

(2.7.107)

under

Let us introduce the FT -densities e c |FT ) dQ p∗ (T ) E(ξ ξc ( )= = = ∗ dP E (p (T )) E(ξc ) e (ξc |FT ) E E e dQ p∗ (T ) p∗ (T ) = = e (p∗ (T )) e (ξc ) dPe E E

(2.7.108)

(2.7.109)

then the problem is equivalent to ∫ φ (T ) dQ = max!

(2.7.110)

under ∫

e≤x= φ (T ) dQ

x0 x0 = ∗ e p (0) E (ξc )

(2.7.111)

The Neyman Pearson lemma then implies that the optimal test of the hye is given by φ. pothesis Q against the alternative Q e Exercise 2.7.19. Derive the result from in detail by making use of the results from statistics. Remark 2.7.20. Just some few words for the non-statisticians: Testing two hypotheses H0 and H1 (the second hypothesis is usually called the anti-hypothesis or alternative) on the basis of an observation is a decision to accept one of the hypotheses as true. In statistics one treats this problem in an unsymmetric way: There are two kinds of mistakes we make when making the decision. First, the observation leads us to accepting H1 although H0 is true. This is called an error of first kind. Second a decision for H0 though H1 is true is called an error of second kind. Statisticians then find the optimal test as the one which keeps the first kind error below a pre-given boundary and minimizes the error of second kind. The reason for this unsymmetric way of approaching the problem lies in the structure of the problem itself. The probability of a possibly fatal error should be known in advance -in order to keep it small within acceptable bounds- and among all decisions which respect this one looks for the test which minimizes the probability of the second kind error. In our problem the initial investment gives the ”error bound” and we minimize the ”error” of a shortfall. The details are found in [38].


143

Remark 2.7.21. (i) Cvitanic and Karatzas [3] consider an additional requirement on the agent’s behaviour, namely that an observer gives a contingent claim which may be thought of as a minimal obligation to be fulfilled certainly ξa < ξ c

(2.7.112)

with price less than or equal to x0 , the present price of which must not be undergone by xx,π T . In the present case this problem is trivial as automatically ∗

xxT0 ,π ≥ pa (T )

(2.7.113)

by the well known comparison theorem and the martingale property of π ∗ . (ii) It is a simple generalization to consider the problem + E(l((ξc − xx,π T ) ))

(2.7.114)

where l is a concave or convex function. The respective optimal tests are easily computed by making use of the Legendre-Fenchel duals of l.

2.7.6

Uncertain real world

We stay in the market considered in the last sections but now we allow for additional uncertainty in the following sense: The real world measure P is assumed only to belong to a family P. The problem now consists in finding [ ]+ V (x) = sup inf EQ ξc − xx,π T

(2.7.115)

[ ]+ sup EQ ξc − xx,π T

(2.7.116)

−

Q∈P π∈A(x)

and −

V (x) =

inf

π∈A(x) Q∈P

−

We would now like to define a value of this game by V (x) = V (x) = V (x) . −

We shall only shortly describe one way to solve the problem without going into the details. The idea is to consider ( )+ kν ξc − xx,π T

(2.7.117)

where kν is the density of Q with respect to a basic measure P and P = {kν |ν ∈ D}

(

(2.7.118) ) + ξc − xx,π is a standard T

by abuse of notation. Let us now assume that kν driver for all (ν, π) and let ( ( ) ( )+ x,π )+ kνe ξc − xx,π = ess inf sup k ξ − x . ν c T T π

ν

Then a standard result [8] states that Isaacs equation holds:

(2.7.119)

144


(− − ) Theorem 2.7.22. Let p t , q t be the solution of the BSDE with standard driver ( )+ kνe ξc − xx,π . T Then ( )+ pt = ess inf sup E(kν ξc − xx,π |Ft ) T

(2.7.120)

( )+ = sup ess inf E(kν ξc − xx,π |Ft ) T

(2.7.121)

π

ν

ν

π

that is we have −

V (x) = V (x) −

(2.7.122)

and so a price of the contingent claim is well defined in this market.

2.7.7

Testing and Optimization

We reduced the measuring risk problem to solving the Neyman Pearson testing problem EQ (φ) = max!

(2.7.123)

EQe (φ) ≤ α

(2.7.124)

φ

under the constraint

and in section (2.7.6) we suggested a way to solve the problem in an uncertain world. Obviously, this problem may be translated into a testing problem of the following kind: Let P be a family of probability measures and let us assume that all Q ∈ P are absolutely continuous with respect to a measure µ. For each Q we find (a e and we may assume that all family of equivalent martingale measures Pe ∈ P e are also absolutely continuous with respect to µ. So we identify P with Pe ∈ P e with a set D e of densities fτ . Our special assumption a set D of densities fτ , P here is that we are given a parameter set Θ, disjoint union of Θ0 and Θ1 such that D = {fτ /τ ∈ Θ1 }

(2.7.125)

e = {fτ /τ ∈ Θ0 } . D

(2.7.126)

This assumption is not as strong as it might seem at first glance. Let us look at a certain world, certain about the physical measure Q, so that Θ1 contains a single point. Θ0 then may be seen as a parameter set for the risk premium processes in an incomplete market. If there is only one equivalent martingale measure then also Θ0 reduces to a set with one single point.


145

The problem of the last section then takes the form of finding a solution of }

{ max

inf Eτ (φ) | Eτ (φ) ≤ α, τ ∈ Θ0 , 0 ≤ φ ≤ 1 .

(2.7.127)

τ ∈Θ1

A solution of this problem is called a max-min test and standard results from statistics assure the existence for the case when µ is finite. As now we are following a Bayesian approach, we finally assume that Θ0 and Θ1 are measurable spaces. The following outline is more or less taken from a script on ”Testtheorie” for a lecture given by Professor Vogel at the University of Bonn in 1974. Now let ρ and ν be two finite measures on the measurable space (Θ, F ) and define the Lagrangian of the above problem by: ∫

∫

L(λ, φ, ν, ρ, h) =

∫

α(τ )dν+ Θ0

∫ ∫ dρ)+ (

hdµ+λ(1− Ω

Θ1

Ω

∫ fτ dρ−

Θ1

fτ dν−h)dµ

Θ0

(2.7.128) ℜ+ , 0

for 0 ≤ λ ∈ ≤ φ ≤ 1, h ∈ L1 (Ω, µ). From this we find the dual problem for 2.7.127 as {∫

∫ ∫ α(τ )dν + (

min Θ0

Ω

∫ fτ dρ −

Θ1

}

feτ dν) dµ | ρ(Θ0 ) = 1, ρ, ν ≥ 0 . +

Θ0

(2.7.129) We may now consider the following four equivalent problems: (S) the Lagrangian has a saddle point (λ0 , φ0 , ν0 , ρ0 , h0 ) (Kuhn Tucker KT) the Kuhn Tucker conditions hold: There is a five-tuple (λ0 , φ0 ,∫ν0 , ρ0 , h0 ) such that (i) Θ1 (Eτ (φ0 ) − λ)dρ0 = 0; Eτ (φ0 ) ≥ λ0 , τ ∈ Θ1 ∫ (ii) Θ0 (α(τ ) − Eτ (φ0 ))dν0 = 0; Eτ (φ0 ) ≤ α(τ ), τ ∈ Θ1 ∫ (iii) ∫Ω (1 ∫ − φ0 )h0 dµ∫= 0, 0 ≤ φ0 ≤ 1 ∫ ∫ (iv) Ω ( Θ1 fτ dρ0 − Θ0 fτ dν0 − h0 )dµ = 0, Θ1 fτ dρ0 − Θ0 fτ dν0 ≤ h0 (v) ρ0 (Θ1 ) = 1 (vi) φ0 , ν0 , ρ0 , h0 ≥ 0. (D) a duality formulation holds

max {λ/Eτ (φ) ≥ λ, τ ∈ Θ1 ; Eτ (φ) ≤ α(τ ), τ ∈ Θ0 ; 0 ≤ φ ≤ 1} {∫ = min

∫

Θ0

∫ hdµ |

α(τ )dν + Ω

∫ fτ dρ −

Θ1

(2.7.130)

} fτ dν ≤ h, ρ(Θ1 ) = 1; ν, ρ, h ≥ 0

Θ0

(2.7.131) (NP) a test has Neyman Pearson structure if [ φ0 =

∫ ∫ ] 1 f or Θ1 fτ dρ0 > Θ0 fτ dν0 ∫ ∫ 0 f or Θ1 fτ dρ0 < Θ0 fτ dν0

(2.7.132)

146

CHAPTER 2. CLASSICAL THEORY for ∫ (α(τ ) − Eτ (φ0 ))dν0 = 0, Eτ (φ0 ) ≤ α(τ ), τ ∈ Θ0

(2.7.133)

(Eτ (φ0 ) − λ0 )dρ0 = 0; Eτ (φ0 ) ≥ λ0 , τ ∈ Θ1

(2.7.134)

Θ1

∫ Θ0

ρ0 (Θ1 ) = 1, ν0 , ρ0 ≥ 0.

(2.7.135)

With this we have several equivalent formulations of the problem of measuring risk. The reader familiar with [11], [12] and [3] will recognize the different approaches to the problem as equivalent from the above results and the fact that a least favourable distribution (ν0 , ρ0 ) exists if one of the above conditions (S), (KT), (D), (NP) holds. Here (ν0 , ρ0 ) is a least favourable distribution if Eρ0 (φρ0 ,ν0 ) ≤ Eρ (φρ,ν ) for all ν, ρ where Eρ is the expectation with respect to ∫ fτ dρ.

(2.7.136)

(2.7.137)

Θ1

On the other hand (S), (KT), (D), (NP) hold if a least favourable a priori distribution exists with Eτ (φρ0 ,ν0 ) ≤ α, τ ∈ Θ1

(2.7.138)

and ∫ Eτ (φρ0 ,ν0 ) ≥

Eτ (φρ0 ,ν0 )dρ0 , τ ∈ Θ1 .

(2.7.139)

Θ1

To illustrate the application of these equivalent formulations let us consider the case #Θi = 1, i = 0, 1.

(2.7.140)

So we have the familiar testing problem max {Eτ1 (φ)/Eτ0 (φ) ≤ α, 0 ≤ φ ≤ 1}

(2.7.141)

which is solved by the standard Neyman Pearson lemma, namely the optimal test is given by [ ] 1 f or fτ1 > cfτ0 φ0 = (2.7.142) 0 f or fτ1 < cfτ0 for some c which has to be determined in the usual way. Equivalently we could solve { } ∫ + min αc + fτ1 − cfτ0 ) dµ/c ≥ 0 . (2.7.143)


147

The first formulation is the one chosen in [11], the latter was treated in [3]. This duality approach makes it easy to solve the slightly more general problem of measuring risk by [ ( ) ] ξ − xx,π + inf E l (2.7.144) P0 (T ) π∈A(x) which was shown to be equivalent to [ ( )] inf E l ((1 − φ)ξc )+

(2.7.145)

e E(φξ c ) ≤ α,

(2.7.146)

φ

under

where α was appropriately chosen. We consider two cases: (i) l is convex and its derivative exists with a well defined inverse I and l(0) = 0 (ii) l is concave and bounded, l(0) = 0. In the first case the results on Legendre-Fenchel duals and the transformation of densities show that the optimal Neyman Pearson test is given by φ∗ = 1 − (I(const

dPe )/ξc ∧ 1) dP

(2.7.147)

where again we assume that ξc > 0. The second case can be reduced to considering the Neyman Pearson lemma of section (2.7.4) after replacing ξc by l(ξc ). This for instance allows to compute the optimal tests φ∗p for l(x) = xp for both cases p < 1 and p > 1 where for p → 1 we refind the case treated in the previous sections, for p → 0 the problem becomes the quantile hedging problem p

( P

+ (ξc − xx,π T ) P0 (T )

) = min!

(2.7.148)

and for p → ∞ the optimal test is given by (ξc − γ)+ ξc

(2.7.149)

c − pγξc (0) = x0

(2.7.150)

φ∗∞ = where γ is the constant with

for the case 0 < x0 < c = pξc (0). The reader is referred to [13] which exhibits the results in much more detail.

148


2.7.8

Incomplete case

We now go back to section (2.7.5), here, however, we do no longer assume that the market is complete. So we have to be a bit more careful in stating the analogue of 2.7.82. Theorem 2.7.23. There exists a test function φ∗ ∈ R which solves the problem E ((1 − φ) ξc ) = min!

(2.7.151)

pφξc (0) ≤ x0

(2.7.152)

( ) dpφξc = rpφξc + θq φξc + q φξc dw − dC φξc ,

(2.7.153)

c pφξ T = φξc

(2.7.154)

where

for (Cs ) an increasing continuous process with C0 = 0 is the agent’s upper price of the claim. Proof. The result follows directly from 2.7.82 and section (2.7.3). Now we can follow the outline in the complete market case, and we can state the result in the following form: ( ) −1 φξ φξ c c Theorem 2.7.24. Let x0 , σ q , C be the strategy determined in 2.7.23. That is ( ) −1 x0 , σ q φξc , C φξc

(2.7.155)

( ) −1 is the upper hedging strategy for the contingent claim φξc . Then x0 , σ q φξc , C φξc solves the problem E(ξc − xx,σ T

−1 q φξc

)+ = min π∈A(x)

x0 ≥ pφξc (0) .

(

ξc − xx,π T

)+

(2.7.156)

(2.7.157)

Proof. The proof follows the same lines as in the complete market case, and so we find the following Corollary 2.7.25. The optimal test here is the optimal test in testing a compound hypothesis against a simple alternative. Without further assumptions - of course - we cannot give a more explicit form of the optimal (randomized) test.


2.7.9

149

The American contingent claim

Recall that by definition an American contingent claim has a price which is constrained to be greater than the payoff St of the claim at all times t ∈ [0, T ]. Even in a perfect market such claims cannot be perfectly hedged by a portfolio. The price process corresponds to the minimal superhedging strategy for the claim. In our notation we have to find a superhedging (self-financing, of course) strategy (π, C) such that xx,π,C ≥ St t

(2.7.158)

xx,π,C = ST T

(2.7.159)

for all t

and the infimum of all initial conditions x such that a superhedging strategy exists with the above properties is called the price of the American claim: inf {x|∃(π,C) xπ,C ≥ St , xπ,C = ST }. t T

(2.7.160)

Let us consider the European claims defined in the following way: We define the set of Ft -stopping times greater or equal to t by It,T = {σ/σ

(2.7.161)

is an Ft -stopping time, t ≤ σ ≤ T }. For all σ ∈ It,T we define the European price of the claim Sσ by p (t, σ) . So p(t, σ) satisfies dp(t, σ) = (rp (t, σ) + θq (t, σ))dt + q(t, σ)dw

p (σ, σ) = Sσ

(2.7.162)

(2.7.163)

Conditions which assure a unique solution of this random duration BSDE are given in [1]. The case under consideration here is simpler than the one in [1] as the stopping times under consideration are bounded and so the techniques from the deterministic terminal time case may be adopted. It is well known [11] that the price of the American claim is now given by the esssup of {p (t, σ) /σ ∈ ItT }. Furthermore [8],[17] the American price is described by a reflected backward stochastic differential equation: Definition 2.7.26. Let (i) ξ ∈ L2 (ii) a standard generator f and (iii) an obstacle (St , 0 ≤ t ≤ T ) which is a continuous Ft -adapted process bounded in L2 , ST ≤ ξ be given. By abuse of notation we denote (f, S, ξ) as a standard generator.

150


We now consider a triple (pt , qt, kt )0≤t≤T of Ft -progressively measurable processes taking values in R, Rn and R+ , respectively, and satisfying (i) k ∈ H 2 , p ∈ H 2 , kT ∈ L2 ∫T ∫T (ii)pt = ξ + t f (s, ps , qs ) ds + kT − kt − t qs dws (iii) pt ≥ St , 0 ≤ t ≤ T ∫ (iv) (kt ) is continuous and increasing, k0 = 0, and (pt − St ) dkt = 0. Note that because of (iv) (kt ) pushes the process upwards to stay above St . It is only increasing when kt = St . Conditions for well posedness are given in [8], which rely on comparison results similar to the results above. A special difficulty is the construction of the process (kt ). It may be defined by the limit of a sequence of penalized problems[8], [27]. The price of the American claim (St , ξ) where without loss of generality we assume ST = ξ, is now given by dpt = (rpt + θqt ) dt + qt dwt − dkt

(2.7.164)

pt ≥ St , pT = ST .

(2.7.165)

Now let us consider the situation that an agent is faced the obligation to be able to pay St at any time t ∈ [0, T ] . He will be able to do so if his initial capital is greater than or equal to p0 . If he is unwilling or unable to invest this amount at time t = 0, say he only has the capital x0 < p0 , then we are in a similar situation as in section (2.7.4). So we consider the European claim Sν for a stopping time ν ∈ I0T . Then from chapter 3 we know that there is a test function such that +

min E (Sν − xx,π ν ) = E ((1 − φ (ν)) Sν ) π

where −1 φ,ν x = pφ,ν 0 ,π = σ q

(2.7.166)

dpφ,ν = (rpφ,ν + θqtφ,ν ) dt + qtφ,ν dwt t t

(2.7.167)

pφ,ν = φ (ν) Sv . v

(2.7.168)

is given by

From the above consideration it is clear that the following proposition holds Theorem 2.7.27. Let x0 < p0. The optimal strategy (x0 , π φ , k φ ) is given as the solution of the rBSDE dpφ = (rpφ + θpφ ) dt + q φ dw − dk φ

(2.7.169)

∫ pφ t

≥ φ (t) St ,

pφ T

= φ (T ) ST ,

φ (pφ t − φ (t) St ) dkt = 0

(2.7.170)


2.7.10

151

Conclusion

We might consider a completely different approach to solve the problems considered in this paper. Let us assume that again we are in a complete market. As we have seen above∗ the claim hedgeable from the starting point x may be assumed to be xx,π under the assumptions of section (2.7.4). Let again T x < pξc (0) for a contingent claim ξc . Then the solution of dpξc = (rpξc + θq ξc )dt + q ξc dw

(2.7.171)

pξTc = ξc

(2.7.172)

may be considered as a supersolution of xx,π

dpt T

∗

= dp∗t = (rp∗t + θqt∗ )dt + qt∗ dwt p∗T = xxπ T

(2.7.173)

∗

(2.7.174) ) Then there is a right continuous process (Ct ), such that pξt c , qtξc , Ct solves dp∗t = (rp∗t + θqt∗ )dt + q ∗ dwt − dCt ∗

p∗T = xx,π . T

(

(2.7.175) (2.7.176)

Obviously, CT is just the amount of shortfall, namely CT = pξTc − pξTc ∧ p∗T

(2.7.177)

As the infimum of the two solutions is a solution

of the problem

pex,π t

(2.7.178)

( ) ∗ f, ξc ∧ xx,π T

(2.7.179)

CT = pξTc − pex,π T .

(2.7.180)

so that In this framework the quantile hedging problem may be formulated in the following form: Consider the simply coupled FBSDE dxx,π = (rt xx,π + θt σπt ) dt + σπt dwt , xx,π t t 0 =x

(2.7.181)

dpt = (rpt + θqt )dt + qt dwt − dCt

(2.7.182)

pT = xxπ T ∧ ξc .

(2.7.183)

152


e x,π ) Solve this problem for any π, compute the smallest supersolution (e px,π etx,π , C t ,q and find a pair (x∗ , π ∗ ) such that E(e px,π t ) = max

(2.7.184)

pex,π (0) ≤ x.

(2.7.185)

π

with respect to

Then the shortfall is simply given by e c) − x + C e x∗ ,π∗ (T ). E(ξ

(2.7.186)

Chapter 3

Mean Variance Hedging A short version of the mean variance approach to hedging in incomplete markets is available. This work Kohlmann, M. und Zhou, X.Y., Backward Stochastic Differential Equations and Stochastic Controls: A New Perspective, SICON 38 (2000), 1392-1407, is available. This might be better for a first reading as many technicalities of the following presentation are avoided in this basic paper.

3.1

The Toolbox

Let (Ω, F, P, {Ft }t≥0 ) be a fixed complete probability space on which is defined a standard d-dimensional Ft -adapted Brownian motion w(t) ≡ (w1 (t), · · · , wd (t))∗ . Assume that Ft is the completion, by the totality N of all null sets of F, of the natural filtration {Ftw } generated by w. Denote by {Ft2 , 0 ≤ t ≤ T } the P -augmented natural filtration generated by the (d − d0 )-dimensional Brownian motion (wd0 +1 , . . . , wd ). Assume that all the coefficients A, B, Ci , Di are Ft -progressively measurable bounded matrix-valued processes, defined on Ω × [0, T ], of dimensions n × n, n × m, n × n, n × m respectively. Also assume that M is an FT -measurable, nonnegative, and bounded n × n random matrix. Assume that Q and N are Ft -progressively measurable, bounded, nonnegative and uniformly positive n × n matrix processes, respectively. Consider the following backward stochastic Riccati differential equation (BSRDE in short):  d d  ∑ ∑  ∗ ∗   dK = −[A K + KA + Ci KCi + Q + (Ci∗ Li + Li Ci )     i=1 i=1   d d d  ∑ ∑ ∑   −(KB + Ci∗ KDi + Li Di )(N + Di∗ KDi )−1 (3.1.1) i=1 i=1 i=1    d d d  ∑ ∑ ∑   ∗ ∗  ×(KB + C KD + L D ) ] dt + Li dwi , 0 ≤ t < T, i i i  i    i=1 i=1 i=1   K(T ) = M. When the coefficients A, B, Ci , Di , Q, N, M are all deterministic, then L1 = · · · = Ld = 0 and the BSRDE (3.1.1) reduces to the following ordinary nonlinear 153

154

CHAPTER 3. MEAN VARIANCE HEDGING

matrix differential equation:  d d ∑ ∑   ∗ ∗  dK = −[A K + KA + Ci KCi + Q − (KB + Ci∗ KDi )     i=1 i=1   d d d ∑ ∑ ∑ ∗ −1 ∗ ×(N + D KD ) (KB + C KD + Li Di )∗ ] dt,(3.1.2) i i i i     i=1 i=1 i=1    0 ≤ t < T,   K(T ) = M, which was completely solved by Wonham [38] by applying Bellman’s quasilinear principle and a monotone convergence approach. Bismut [2, 3] initially studied the case of random coefficients, but he solved only some special simple cases. He always assumed that the randomness of the coefficients only comes from a smaller filtration {Ft2 }, which leads to L1 = · · · = Ld0 = 0. He further assumed in his paper [2] that Cd0 +1 = · · · = Cd = 0,

Dd0 +1 = · · · = Dd = 0,

(3.1.3)

under which the BSRDE (3.1.1) becomes the following one:  d0 ∑   ∗   dK = −[A K + KA + Ci∗ KCi + Q     i=1   d0 d0 d0  ∑ ∑ ∑   −(KB + Ci∗ KDi )(N + Di∗ KDi )−1 (KB + Ci∗ KDi )∗ ] dt (3.1.4) i=1 i=1 i=1    d  ∑    Li dwi , 0 ≤ t < T, +     i=d +1 0   K(T ) = M, and the generator does not involve L at all. In his work [3] he assumed that Dd0 +1 = · · · = Dd = 0,

(3.1.5)

under which the BSRDE (3.1.1) becomes the following one  d d ∑ ∑   ∗ ∗  Ci KCi + Q + (Ci∗ Li + Li Ci )  dK = −[A K + KA +    i=1 i=d0 +1    d0 d0 d0  ∑ ∑ ∑   −(KB + Ci∗ KDi )(N + Di∗ KDi )−1 (KB + Ci∗ KDi )∗ ] dt (3.1.6) i=1 i=1 i=1    d  ∑    + Li dwi , 0 ≤ t < T,     i=d +1 0   K(T ) = M, and the generator depends on the second unknown variable (Ld0 +1 , . . . , Ld )∗ in a linear way. Moreover his method was rather complicated. Later, Peng [29] gave a nice treatment on the proof of existence and uniqueness for the BSRDE (3.1.6),

3.1. THE TOOLBOX

155

by using Bellman’s quasilinear principle and a method of monotone convergence— a generalization of Wonham’s approach to the random situation. As early as in 1978, Bismut [3] commented on page 220 that:”Nous ne pourrons pas démontrer l’existence de solution pour l’équation (2.49) dans le cas général.” (We could not prove the existence of solution for equation (2.49) for the general case.) On page 238, he pointed out that the essential difficulty for solution of the general BSRDE (3.1.1) lies in the integrand of the martingale term which appears in the generator in a quadratic way. Two decades later in 1998, Peng [32] included the above problem in his list of open problems on BSDEs. In this paper, we prove the global existence and uniqueness result for the one-dimensional case of BSRDE (3.1.1), that is     

dK = −[aK +

d ∑

ci Li + Q + F (t, K, L)] dt +

i=1

K(T ) = M,

K∈

L∞ F (0, T ; R+ )

∩

d ∑

Li dwi ,

i=1 L∞ (Ω, FT , P ; C([0, T ]; R+ ))

(3.1.7)

with F (t, K, L) := −[B(t)K +

d ∑

Ci Di (t)K +

i=1

×[B(t)K + a(t) := 2A(t) +

d ∑

d ∑

d ∑

Di (t)Li ][N (t) + |K|

i=1

Ci Di (t)K +

i=1

Ci2 (t),

d ∑

d ∑

Di∗ Di (t)]−1

i=1

Di (t)Li ]∗ ,

0 ≤ t ≤ T;

(3.1.8)

i=1

0 ≤ t ≤ T;

i=1

ci (t) := 2Ci (t),

0 ≤ t ≤ T, i = 1, . . . , d.

The arguments given here are based on the following new observation that F (t, K, L) ≤ 0,

∀K ∈ R, ∀L ∈ Rd , 0 ≤ t ≤ T.

(3.1.9)

We make full use of this special structure for BSRDE (3.1.7). We apply an approximation technique, which is inspired by the works of Kobylanski [17] and Lepeltier and San Martin [22, 23]. Consider then the case where the control weight matrix N reduces to zero. Kohlmann and Zhou [13] discussed such a case. However, their context is rather restricted, as they make the following assumptions: (a) all the coefficients involved are deterministic; (b) C1 = · · · = Cd = 0, D1 = · · · = Dd = Im×m , and M = I;(c) A + A∗ ≥ BB ∗ . Their arguments are based on a result of Chen, Li and Zhou [2]. Kohlmann and Tang [18] considered a general framework along those analogues of Bismut [3] and Peng [29], which has the following features: (a) the coefficients A, B, C, D, N, Q, M are allowed to be random, but are only Ft2 -progressively measurable processes or FT2 -measurable random variable; (b) the assumptions in Kohlmann and Zhou [13] are dispensed with or generalised; (c) the condition (3.1.5) is assumed to be satisfied. Kohlmann and Tang [18] obtained a general result and generalised Bismut’s previous result on existence

156


and uniqueness of a solution of BSRDE (3.1.6) to the singular case under the following additional two assumptions: d ∑

M ≥ εI,

Di∗ Di (t) ≥ εI.

(3.1.10)

i=1

In this paper the existence and uniqueness result is also obtained for the singular case N = 0 under the assumption (3.1.10), but for a more general framework of the following features: the coefficients A, B, C, D, N, Q, M are allowed to be Ft -progressively measurable processes or FT -measurable random variable, and the coefficient D is not necessarily zero. The BSRDE (3.1.1) arises from solution of the optimal control problem inf

u(·)∈L2F (0,T ;Rm )

J(u; 0, x)

(3.1.11)

where for t ∈ [0, T ] and x ∈ Rn , ∫ J(u; t, x) := E Ft [

T

[(N u, u) + (QX t,x;u , X t,x;u )] ds + (M X t,x;u (T ), X t,x;u (T(3.1.12) ))]

t

and X t,x;u (·) solves the following stochastic differential equation  d ∑   dX = (AX + Bu) ds + (Ci X + Di u) dwi , t ≤ s ≤ T, (3.1.13) i=1   X(t) = x. The following connection is well known: if the BSRDE (3.1.1) has a solution (K, L), the solution for the above linear-quadratic optimal control problem (LQ problem in short) has the following closed form (also called the feedback form): u(t) := −(N +

d ∑

Di∗ KDi )−1 [B ∗ K

i=1

+

d ∑

Di∗ KCi

i=1

+

d ∑

Di∗ Li ]X(t) (3.1.14)

i=1

and the associated value function V is the following quadratic form V (t, x) :=

inf

u∈L2F (t,T ;Rm )

J(u; t, x) = (K(t)x, x),

0 ≤ t ≤ T, x ∈ Rn .(3.1.15)

In this way, on the one hand, the solution of the above LQ problem is reduced to solving the BSRDE (3.1.1). On the other hand, the formula (3.1.15) actually provides a representation—of Feynman-Kac type— for the solution of BSRDE (3.1.1). The reader will see that the proofs given here for Theorems 3.2.1 and 3.2.2 depend heavily on this kind of representation. As an application of the above results, the mean-variance hedging problem with random market conditions is considered. The mean-variance hedging problem was initially introduced by Föllmer and Sondermann [6], and later widely studied by Duffie and Richardson [4], Föllmer and Schweizer [7], Schweizer [26,

3.2. MAIN RESULTS

157

27, 28], Hipp [14], Monat and Stricker [25], Pham, Rheinländer and Schweizer [24], Gourieroux, Laurent and Pham [8], and Laurent and Pham [20]. All of these works are based on a projection argument. Recently, Kohlmann and Zhou [13] used a natural LQ theory approach to solve the case of deterministic market conditions. Kohlmann and Tang (3.1.10) used a natural LQ theory approach to solve the case of stochastic market conditions, but the market conditions are only allowed to involve a smaller filtration {Ft2 }. In this paper, the case of random market conditions is completely solved by using the above results, and the optimal hedging portfolio and the variance-optimal martingale measure are characterized by the solution of the associated BSRDE. The rest of the paper is organised as follows. Section 2 contains a list of notations and the statement of the main results which consist of Theorems 3.2.1 and 3.2.2. In Sections 3 and 4 the proofs of the main Theorems are given respectively. Section 5 provides a straightforward application of the main results to the regular and singular stochastic LQ problems. Section 6 presents an application to solution of the mean-variance hedging problem in finance.

3.2

Notation and The Main Results: Global Existence and Uniqueness

Notation. Throughout this paper, the following additional notation will be used: M∗

:

|M |

:

(M1 , M2 ) Rn R+ C([0, T ]; H)

: : : :

the√transpose of any vector or matrix M ; ∑ 2 = ij mij for any vector or matrix M = (mij );

the inner product of the two vectors M1 and M2 ; the n-dimensional Euclidean space; the set of all nonnegative real numbers; the Banach space of H-valued continuous functions on [0, T ], endowed with the maximum norm for a given Hilbert space H; L2F (0, T ; H) : the Banach space of H-valued Ft -adapted square-integrable stochastic processes f on [0, T ], endowed with the norm ∫T (E 0 |f (t)|2 dt)1/2 for a given Euclidean space H; L∞ : the Banach space of H-valued, Ft -adapted, essentially F (0, T ; H) bounded stochastic processes f on [0, T ], endowed with the norm ess supt,ω |f (t)| for a given Euclidean space H; L2 (Ω, F, P ; H) : the Banach space of H-valued norm-square-integrable random variables on the probability space (Ω, F, P ) for a given Banach space H; and L∞ (Ω, F, P ; C([0, T ]; Rn )) is the Banach space of C([0, T ]; Rn )-valued, essentially maximum-norm-bounded random variables f on the probability space (Ω, F, P ), endowed with the norm ess supω∈Ω max0≤t≤T |f (t, ω)|. The main results of this paper are stated by the following two theorems. Theorem 3.2.1. Assume that M ≥ 0, Q(t) ≥ 0 and N (t) ≥ εIm×m for some

158


positive constant ε > 0. Then, the BSRDE (3.1.7) has a unique Ft -adapted global solution (K, L) with ∞ K ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )),

L ∈ L2F (0, T ; Rd ).

Theorem 3.2.2. Assume that N (t) ≥ 0 and Q(t) ≥ 0. Also assume that M ≥ε

(3.2.1)

and d ∑

Di∗ Di (t) ≥ εIm×m

(3.2.2)

i=1

for some positive constant ε > 0. Then, the BSRDE (3.1.7) has a unique Ft -adapted global solution (K, L) with ∞ K ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )),

L ∈ L2F (0, T ; Rd ),

and K(t, ω) being uniformly positive w.r.t. (t, ω).

3.2.1

The Proof of Theorem 3.2.1

This section gives the proof of Theorem 3.2.1.

3.2.2

Construction of a Sequence of Decreasing Uniformly Lipschitz Drifts

Define for j = 0, 1, . . . , [ Fj (t, K, L) :=

sup d ˜ ˜ K∈R, L∈R

] ˜ L) ˜ − j|K − K| ˜ − j|L − L| ˜ , F (t, K,

d ∀K ∈ R, L ∈ R(3.2.3) .

Then, we have the following assertions. (i) The quadratic growth in (K, L): there is a deterministic positive constant ε0 which is independent of j, such that for each j = 0, 1, . . ., |Fj (t, K, L)| ≤ ε0 (1 + |K|2 + |L|2 ), ∀(t, K, L) ∈ [0, T ] × R × Rd . (ii)Monotonicity in j: {Fj ; j = 0, 1, . . .} is decreasingly convergent to F , that is F0 ≥ F1 ≥ · · · ≥ Fj ≥ Fj+1 ≥ · · · ≥ F,

Fj ↓ F.

(3.2.4)

(iii) The uniform Lipschitz property: for each j = 0, 1, . . ., Fj is uniformly Lipschitz in (K, L). (iv) The strong convergence: if limj→∞ K j = K and limj→∞ Lj = L, then limj→∞ Fj (t, K j , Lj ) = F (t, K, L). The proof of these four assertions is an easy adaptation to that of Lepeltier and San Martin [22]. Note that F0 (t, K, L) ≡ 0.

(3.2.5)

3.2. MAIN RESULTS

159

Then consider the following approximating backward stochastic differential equation (BSDE in short)  d d ∑ ∑   dK = −[aK + ci Li + Q + Fj (t, K, L)] dt + Li dwi , (3.2.6) i=1 i=1   1 K(T ) = M + j+1 . The generator of the BSDE (3.2.6) is given by Gj (t, K, L) := a(t)K +

d ∑

ci (t)Li + Q(t) + Fj (t, K, L),

K ∈ R, L ∈ Rd(3.2.7) .

i=1

In the following, we state Pardoux and Peng’s fundamental result on the existence and uniqueness of a nonlinear BSDE under the assumption of uniform Lipschitz on the generator. The reader is referred to Pardoux and Peng [26] for details of the proof. Lemma 3.2.3. (Pardoux and Peng (1990)) Assume that ξ ∈ L2 (Ω, FT , P ) and the real valued function f defined on Ω × [0, T ] × R × Rd satisfies the following conditions: (1) The stochastic process f (·, y, z) is Ft -adapted for each fixed pair (y, z); (2) f (t, ·, ·) is uniformly Lipschitz, i.e. there is a constant λ > 0 such that |f (t, y1 , z1 ) − f (t, y2 , z2 )| ≤ λ (|y1 − y2 | + |z1 − z2 |) ,

∀(yi , zi ) ∈ Rd+1 , i = 1, 2;

and (3) f (·, 0, 0) ∈ L2F (0, T ). Then, the following BSDE  d ∑   dy = −f (t, y, z) dt + zi dwi , i=1   y(T ) = ξ

(3.2.8)

has a unique solution (y, z) with y ∈ L2F (0, T ) ∩ L2 (Ω, F, P ; C[0, T ]) and z ∈ L2F (0, T ; Rd ). The next lemma states a comparison result due to Peng [31]. Lemma 3.2.4. (Peng (1992)) Suppose that (f i , ξ i ), i = 1, 2 satisfy the assumptions made in Lemma 3.1 for (f, ξ). Assume that f 1 (t, y, z) ≥ f 2 (t, y, z), ∀(y, z) ∈ R × Rd ;

ξ1 ≥ ξ2.

Let (y i , z i ), i = 1, 2 denote the solutions of BSDE (3.2.8) with (f, ξ) being replaced with (f i , ξ i ), i = 1, 2, respectively. Then, the following holds: y 1 (t) ≥ y 2 (t),

a.s.a.e.

By applying Lemma 3.1, we see that for each j = 0, 1, . . . , the BSDE (3.2.6) has a unique Ft -adapted global solution, denoted by (K j , Lj ). In view of the comparison result Lemma 3.2, we obtain K0 ≥ K1 ≥ · · · ≥ Kj ≥ Kj+1 ≥ · · · ,

a.s.a.e.

(3.2.9)

160


3.2.3

The Positivity of K j

Proposition 3.2.5. For each j = 0, 1, . . . , we have K j (t) > 0

a.s.a.e.

Proof of Proposition 3.2. Define τj := sup {t ∈ [0, T ] : K j (t) ≤ 0}. Since K j (T ) = M +

1 j+1

(3.2.10)

> 0 a.s., we have τj < T,

a.s.

(3.2.11)

We assert that τj = −∞,

i.e.

a.s.∀t ∈ [0, T ].

K j (t) > 0,

(3.2.12)

For this purpose, define ∫

t

σjl := T ∧ inf {t ∈ [0, T ] :

|Lj |2 ds ≥ l}.

(3.2.13)

0

Since Lj ∈ L2F (0, T ; Rd ), we see that ∫

T

|Lj |2 ds < ∞,

a.s.;

0

lim σjl = T,

l→∞

a.s.

Define the following feedback control uj := −(N + K j

d ∑ i=1

Di∗ Di )−1 (BK j +

d ∑

Ci Di K j +

i=1

d ∑

Di Lji )∗ X. (3.2.14)

i=1

Applying the existence and uniqueness result of Gal’Chuk [11], the stochastic equation has a unique solution X t,x;uj corresponding to the above feedback control starting from arbitrary initial data (t, x). It is easily seen that X t∨τj ,1;uj is well defined on the stochastic time interval [t ∨ τj , σjl ] for l = 1, 2, . . .. Using Itô’s formula, we can check out that K j (0 ∨[τj )

∫

σjl

]

Q|X 0∨τj ,1;uj |2 ds = E F0∨τj K(σjl )|X 0∨τj ,1;uj (σjl )|2 + j ∫ 0∨τ ∫ σjl σjl F0∨τj F0∨τj 2 (Fj − F )(s, K j , Lj ) ds N |uj | ds + E +E 0∨τj ] (3.2.15) [ 0∨τj ∫ ≥ E F0∨τj K j (σjl )|X 0∨τj ,1;uj (σjl )|2 + ∫ σjl N |uj |2 ds. +E F0∨τj 0∨τj

σjl

0∨τj

Q|X 0∨τj ,1;uj |2 ds

3.2. MAIN RESULTS

161

Letting l → ∞ and passing to the limit, we get E F0∨τj

∫

T

N |uj |2 ds < ∞,

E F0∨τj

∫

0∨τj

E F0∨τj (M + [

T

Q|X 0∨τj ,1;uj |2 ds < ∞,

0∨τj

< ∞, (3.2.16) ] ∫ T K j (T )|X 0∨τj ,1;uj (T )|2 + Q|X 0∨τj ,1;uj |2 ds > 0.

K j (0 ∨ τj ) ≥ E F0∨τj

1 0∨τj ,1;uj (T )|2 j+1 )|X

0∨τj

The last inequality implies that τj < 0, a.s., i.e. τj = −∞.

3.2.4

The Uniform Boundedness of (K j , Lj )

First we prove the following fact. Proposition 3.2.6. K 0 has the following Feynman-Kac representation: ∫ K (t) = E [ 0

Ft

T

Q|X t,1;0 |2 ds + (M + 1)|X t,1;0 (T )|2 ],

0 ≤ t ≤ T.(3.2.17)

t

It is uniformly bounded. Proof of 3.2.6 The first assertion results from computing |K 0 X t,1;0 |2 (s) with Itô’s formula. The second assertion is obtained by applying Theorem 2.1 of Peng [29]. The uniform boundedness of (K j , Lj ) is stated by Proposition 3.2.7. The sequence {(K j , Lj ); j = 0, 1, . . .} is uniformly bounded 2 d in the Banach space L∞ F (0, T ) × LF (0, T ; R ). That is ∫ j

ess sup K (t) + E (t,ω)

T

|Lj |2 ds ≤ β0

(3.2.18)

0

where β0 is a positive constant and is independent of j. Proof of 3.2.7 The uniform boundedness of K j is obvious from the following inequality K 0 (t) ≥ K j (t) ≥ 0, 0 ≤ t ≤ T and Proposition 3.2. We show the uniform boundedness for Lj in the following. In view of the BSDE (3.2.6), using Itô’s formula to compute |K j |2 (t), we get  j j j 2 Lj ) + Q + Fj (t, K j , Lj )] dt   d|K | (t) = −2Kj 2[aK + (c, +|L | dt + 2K j (Lj , dw), 0 ≤ t ≤ T, ( )2   (K j )2 (T ) = 1 M + j+1 .

(3.2.19)

162


Taking expectation on both sides, we have ∫ T j 2 E|K | (t) + E |Lj |2 ds t ∫ T (3.2.20) )2 ( j j j j j 1 K [aK + (c, L ) + Q + Fj (s, K , L )] ds. = E M + j+1 + 2E t

Our new observation is that 2Kj Fj (s, K j , Lj ) ≤ 0,

(3.2.21)

(since K j ≥ 0 and Fj ≤ 0) and so the following straightforward calculations hold: ∫ T j 2 E|K | (t) + E |Lj |2 ds t∫ T 2 (3.2.22) ≤ E(M + 1) + 2E K j [aK j + (c, Lj ) + Q] ds ∫ Tt 1 1 1 [2a|K j |2 + 2|c|2 |K j |2 + |Lj |2 + |K j |2 + Q2 ] ds. ≤ E(M + 1)2 + E 2 2 2 t Since the coefficients a(s), ci (s), Q(s) are uniformly bounded, there is a positive constant λ which is independent of j such that ∫ T ∫ T 1 j 2 j 2 E(K ) (t) + E |L | ds ≤ λ + λE |K j |2 ds. (3.2.23) 2 t t Using Gronwall’s inequality, we get 1 sup E|K | (t) + E 2 0≤t≤T

∫

j 2

3.2.5

T

|Lj |2 ds ≤ λ exp (λT ).

(3.2.24)

0

The Strong Convergence Result and The Existence

Proposition 3.2.8. We have the following convergence result: ∫ T lim E |K l − K r |2 ds = 0. l,r→∞

(3.2.25)

0

Proof of 3.2.8 Since the sequence {K j ; j = 0, 1, . . .} is decreasing and uniformly bounded, we have by the dominated convergence theorem of Lebesgue: ∫ T lim E |K l − K r |2 ds = 0. (3.2.26) l,r→∞

0

Since Lj is bounded in L2F (0, T ; Rd ), assume without loss of generality that as j → ∞, Lj → L weakly in L2F (0, T ; Rd ) for some L ∈ L2F (0, T ; Rd ). We also assume that l < r.

3.2. MAIN RESULTS

163

Set K lr := K l − K r ,

Llr := Ll − Lr ,

K l∞ := K l − K,

Ll∞ := Ll − L.

We have { dK lr = −[aK lr + (c, Llr ) + Fl (t, K l , Ll ) − Fr (t, K r , Lr )] dt + (Llr , dw), (3.2.27) 1 1 lr − 1+r . K (T ) = 1+l We now use a technique developed by Kobylanski [17] (see also Lepeltier and San Martin [23] in pages 236-237). Applying Itô’s formula with the following function (with the positive constant λ being specified later) Ψ(x) := λ−1 1 [exp(λ1 x) − 1] − x,

(3.2.28)

we have ( ) 1 ∫ T EΨ K lr (t) + E Ψ′′ (K lr )|Llr |2 ds 2 t ∫ ) ( T 1 1 Ψ′ (K lr )[aK lr + (c, Llr ) + Fl (s, K l , Ll ) − Fr (s, K r , Lr )] ds. = Ψ 1+l − 1+r + 2E t

Noting the following facts: K lr ≥ 0,

Ψ′ (K lr ) = exp(λ1 K lr ) − 1 ≥ 0,

Fl ≤ 0,

Fr ≥ F,

(3.2.29)

we obtain ( ) ( ) 1 ∫ T Ψ′′ K lr |Llr |2 ds EΨ K lr (t) + E 2 t ∫ (3.2.30) ( ) T 1 1 ≤ Ψ 1+l − 1+r + 2E Ψ′ (K lr )[aK lr + (c, Llr ) − F (s, K r , Lr )] ds. t

Note the following estimation −2F (s, K r , Lr ) ≤ 2ε−1 |BK r +

d ∑ i=1

Ci Di K r +

d ∑ i=1

Di Lri |2

≤ λ + λ|L | ≤ λ + 3λ(|L | + |L r 2

lr 2

(3.2.31)

| + |L| ).

l∞ 2

2

where λ is a positive constant and depends on ε and the bounds of K 0 (s), B(s), C(s), D(s) only (in view of 3.2.7), but independent of the integer r. Then we have ∫ T ( ) 1 EΨ K lr (t) + E ( Ψ′′ − 3λΨ′ )(K lr )|Llr |2 ds t ∫2 ( ) T 1 1 (3.2.32) ≤ Ψ 1+l − 1+r + 2E Ψ′ (K lr )[aK lr + (c, Llr )] ds t ∫ T +λE Ψ′ (K lr )(1 + 3|Ll∞ |2 + 3|L|2 ) ds. t

Take λ1 = 12λ. Since 1 ′′ Ψ (x) − 3λΨ′ (x) = 3λ exp (12λx) + 3λ, 2

164


we have that the term

√

converges strongly to

1 ′′ lr Ψ (K ) − 3λΨ′ (K lr ) 2

√

1 ′′ l∞ Ψ (K ) − 3λΨ′ (K l∞ ) 2 as r → ∞, and it is uniformly bounded in view of Proposition 3.3. Therefore, √ 1 ′′ lr Ψ (K ) − 3λΨ′ (K lr ) Llr 2 converges weakly to

√

1 ′′ lr Ψ (K ) − 3λΨ′ (K lr ) Ll∞ . 2

From the last weak convergence, we get ∫ T 1 ( Ψ′′ − 3λΨ′ )(K l∞ )|Ll∞ |2 ds E 2 ∫ t T 1 ≤ limr→∞ E ( Ψ′′ − 3λΨ′ )(K lr )|Llr |2 ds 2 ( ) t ∫ T 1 ≤ Ψ 1+l + 2E Ψ′ (K l∞ )[aK l∞ + (c, Ll∞ )] ds t ∫ T ′ l∞ +λE Ψ (K )(1 + 3|Ll∞ |2 + 3|L|2 ) ds.

(3.2.33)

t

Hence we have

∫

T

1 ( Ψ′′ − 6λΨ′ )(K l∞ )|Ll∞ |2 ds 2 ∫ T (t ) 1 Ψ′ (K l∞ )[aK l∞ + (c, Ll∞ )] ds ≤ Ψ 1+l + 2E t ∫ T +λE Ψ′ (K l∞ )(1 + 3|L|2 ) ds. E

(3.2.34)

t

Since

1 ( Ψ′′ − 6λΨ′ )(K l∞ ) = 6λ, 2 we have by passing to the limit l → ∞ and applying the dominated convergence theorem of Lebesgue the following ∫ T lim E |Ll∞ |2 ds = 0. (3.2.35) l→∞

0

At this stage, we can show that almost surely K j converges to K uniformly in t. The proof is standard, and the reader is referred to Lepeltier and San Martin [22] for details. With the uniform convergence in the time variable t of K j and the strong convergence of Lj , we can pass to the limit by letting j → ∞ in the BSDE (3.2.6), and conclude that the limit (K, L) is a solution.

3.2. MAIN RESULTS

3.2.6

165

A Feynman-Kac Representation Result and The Uniqueness

Consider the optimal control problem Problem P0

inf

u(·)∈L2F (0,T ;Rm )

J(u; 0, x)

(3.2.36)

where for t ∈ [0, T ] and x ∈ R, ∫ J(u; t, x) := E [

T

Ft

(N |u|2 + Q|X t,x;u |2 ) ds + M |X t,x;u (T )|2 ]

(3.2.37)

t

and X t,x;u (·) solves the following stochastic differential equation  d ∑   dX = (AX + Bu) ds + (Ci X + Di u) dwi , t ≤ s ≤ T, (3.2.38) i=1   X(t) = x. The associated value function is defined as V (t, x) :=

inf

u∈L2F (t,T ;Rm )

0 ≤ t ≤ T, x ∈ R.

J(u; t, x),

(3.2.39)

The following connection is straightforward. Proposition 3.2.9. Let (K, L) be an Ft -adapted solution of the BSRDE (3.1.7) ∞ 2 d with K ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )) and L ∈ LF (0, T ; R ). Then, the solution for the LQ problem P0 has the following closed form (also called the feedback form): u b = −(N +

d ∑

Di∗ KDi )−1 [B ∗ K

i=1

+

d ∑

Di∗ KCi

+

i=1

d ∑

b Di∗ Li ]X

(3.2.40)

i=1

and the associated value function V is the following quadratic form V (t, x) = K(t)x2 .

(3.2.41)

Remark 3.2.10. Although the proof of Proposition 3.2.9 is straightforward (use Itô’s formula to do some calculations), we need to be careful about the solution of the optimal closed system: the coefficients of the closed system corresponding to the feedback control (3.2.40) involve the quantity L and might not be bounded. The reader is referred to Gal’chuk [11] for a rigorous argument on this respect. Using Proposition 3.2.9, we get the representation of K (as the first part of solution of BSRDE (3.1.7)) as K(t) = V (t, 1) =

inf

u∈L2F (t,T ;Rm )

Ft

E [M |X

∫ t,1;u

2

(T )| + t

T

(N |u|2 + Q|X t,1;u |2 ) ds], (3.2.42) 0 ≤ t ≤ T.

166


The uniqueness is a consequence of the representation result. In fact, assume e L) e are two Ft -adapted solutions of the BSRDE (3.1.7) with that (K, L) and (K, ∞ e e ∈ L2 (0, T ; Rd ). K, K ∈ LF (0, T ; R+ ) ∩ L∞ (Ω, FT , P ; C([0, T ]; R+ )) and L, L F Then, we have     

d ∑

dδK = −[aδK +

ci δLi + δF ] dt +

i=1

d ∑

δLi dwi ,

i=1

(3.2.43)

δK(T ) = 0.

Here, we use the notation: e δK := K − K,

ei , δLi := Li − L

e L). e δF := F (·, K, L) − F (·, K,

Applying Itô’s formula, we have ∫

T

2

∫ |δL| ds = 2E 2

E|δK(t)| + E t

T

δK(aδK + t

d ∑

ci δLi + δF ) ds. (3.2.44)

i=1

e has the same representation (3.2.42), we have δK = 0. Noting that K and K Putting this equality into (3.2.44), we have ∫ E

T

|δL|2 ds = 0.

0

e This implies that L = L.

3.2.7

A Remark

Theorem 3.2.1 can also be proved by nontrivially employing the result of Kobylanski [17]. However, the proof given here avoids doing an exponential transformation of the unknown variable of the BSDE under discussion, instead it makes full use of the special structure of the stochastic Riccati equation. Therefore we preferred this approach.

3.2.8

The Proof of Theorem 3.2.2

This section gives the proof of Theorem 3.2.2. The regular approximation method proposed by Kohlmann and Tang [18] is adapted to the present case. We begin with the citation of an a priori estimate for X t,x;u , which was established by Kohlmann and Tang [18]. Lemma 3.2.11. (a priori estimate) Assume that the assumption (3.2.2) is satisfied. Let u ∈ L2F (t, T ; Rm ). Then, there is β > 0 which only depends on the bounds of the coefficients A, B, C, D and ε, such that ε Ft E 2

∫ t

T

|u|2 ds + |x|2 ≤ exp (β(T − t))E Ft |X t,x;u (T )|2 ,

0 ≤ t ≤ T.(3.2.45)

3.2. MAIN RESULTS

167

Proof of Lemma 3.2.11 Using Itô’s formula, we have from (3.2.38) E Ft |X(T )|2 Ft

= E |X(r)| + 2E 2

Ft

∫

T

(AX + Bu, X) ds + E

Ft

∫

r

= E Ft |X(r)|2 + 2E Ft +2E Ft

∫

∫

r T

((A + r

T

((B +

d ∑

r

i=1

T

d ∑

|Ci X + Di u|2 ds

i=1

Ci∗ Ci )X, X) ds

Ci∗ Di )u, X) ds + E Ft

i=1∫

ε ≥ E Ft |X(r)|2 + E Ft 2

d ∑

T

|u|2 ds − βE Ft

T

T

(3.2.46)

d ∑ u∗ ( Di∗ Di )u ds

r

∫

r

∫

i=1

|X|2 ds

r

for some positive constant β. Write ρr := E Ft |X(r)|2 ,

t ≤ r ≤ T.

(3.2.47)

Then, the above reads ε ρt + E F t 2

∫

T

∫ |u| ds ≤ ρT + β

T

2

ρs ds.

(3.2.48)

t

t

By Gronwall’s inequality, we have ρr ≤ exp(β(T − r))ρT , ∫ ε Ft T 2 ρt + E |u| ds ≤ exp(β(T − t))ρT . 2 t

(3.2.49) (3.2.50)

This concludes the proof. Consider the following regular approximation of the original control problem P0 Problem Pα

min

u∈L2F (t,T ;Rm )

Jα (u; t, x)

(3.2.51)

with Jα (u; t, x) = J(u; t, x) + αE

Ft

∫

T

|u|2 ds,

α > 0.

(3.2.52)

t

It is associated with the following BSRDE  d d  ∑ ∑  ∗ ∗   dK = −[A K + KA + C KC + Q + (Ci∗ Li + Li Ci ) i  i    i=1 i=1   d d d  ∑ ∑ ∑   ∗ −(KB + Ci KDi + Li Di )(αI + N + Di∗ KDi )−1 (3.2.53) i=1 i=1 i=1    d d d  ∑ ∑ ∑   ∗ ∗  ×(KB + C KD + L D ) ] dt + Li dwi , 0 ≤ t < T, i i i  i    i=1 i=1 i=1   K(T ) = M.

168


The value function of the problem Pα is denoted by Vα (t, x). Proposition allows us to express the value function Vα (t, x) = Kα (t)x2 .

(3.2.54)

Here, (Kα , Lα ) is the unique Ft -adapted solution of the BSRDE (3.2.53) with ∞ 2 d Kα ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )) and Lα ∈ LF (0, T ; R ).

From Lemma 3.2.11, we immediately have Lemma 3.2.12. Suppose that the assumptions of Theorem 2.2 hold. Then, we have Vα (t, x) ≥ V (t, x) ≥ ε exp (−β(T − t))x2 .

(3.2.55)

This implies that Kα (t) ≥ ε exp (−β(T − t)).

(3.2.56)

The relationship between the original problem P0 and the approximating problem Pα is given in the next lemma. Lemma 3.2.13. Assume that the conditions (3.2.1) and (3.2.2) are satisfied. Then, for fixed x ∈ R, as α → 0+, Vα (t, x) converges in a decreasing way to ∞ V (t, x) strongly both in L∞ F (0, T ; R) and in L (Ω, FT , P ; C([0, T ]; R)). Proof of Lemma 3.2.13 It is obvious that Vα (t, x) is decreasing in α. Denote by u b the optimal control of the original problem, i.e. V (t, x) = J(b u; t, x). Then, V (t, x) ≤ Vα (t, x) ≤ Jα (b u; t, x) ∫ T ∫ Ft 2 Ft = J(b u; t, x) + αE |b u| ds = V (t, x) + αE t

T

|b u|2 ds.

(3.2.57)

t

It is easy to show that there is a constant β1 > 0 such that J(0; t, x) ≤ |x|2 exp (β1 (T − t)).

(3.2.58)

Noting the positivity of M and Lemma 3.2.11, we have Ft

J(b u; t, x) ≥ εE |X

t,x;b u

ε2 (T )| ≥ exp (−β(T − t))E Ft 2

∫

2

T

|b u|2 ds. (3.2.59)

t

Since J(b u; t, x) = V (t, x) ≤ J(0; t, x), we have ε2 exp (−β(T − t))E Ft 2

∫ t

T

|b u|2 ds ≤ |x|2 exp (β1 (T − t)).

(3.2.60)

3.2. MAIN RESULTS

169

Concluding the above, we have V (t, x) ≤ Vα (t, x) ≤ V (t, x) + 2αε−2 |x|2 exp ((β1 + β)(T − t)). This completes the proof of this lemma. With Lemma 3.2.13, the following is obvious: Lemma 3.2.14. Suppose that the assumptions of Theorem 3.2.2 are satisfied. Then, the value function V is a quadratic form. More precisely, there is an Ft ∞ adapted stochastic process K(·) ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )) such that V (t, x) = (K(t)x, x),

∀(t, x) ∈ [0, T ] × R, P − a.s.

(3.2.61)

Moreover, Kα converges to K strongly in the two Banach spaces L∞ F (0, T ; R+ )

and

L∞ (Ω, FT , P ; C([0, T ]; R+ )),

and K(t) is uniformly positive: K(t) ≥ ε exp (−β(T − t)). Lemma 3.2.15. Suppose that the assumptions of Theorem 3.2.2 are satisfied. Then, {Lα } is a Cauchy sequence in L2F (0, T ; Rd ). Proof First, we show that {Lα } is bounded in L2F (0, T ; Rd ). The arguments are similar to those in Section 3. Use Itô’s formula to compute |Kα (t)|2 . Then since Kα F (·, Kα , Lα ) ≤ 0, it can be left out in our estimation. The remainder is standard to show that {Lα } is bounded in L2F (0, T ; Rd ). Now we return to show that {Lα } is a Cauchy sequence in L2F (0, T ; Rd ). For this purpose, use Itô’s formula to compute |Kα (t) − Kγ (t)|2 . We get the following ∫

T

E|Kα − Kγ | (t) + E |Lα − Lγ |2 ds t ∫ T = 2E (Kα − Kγ )[a(Kα − Kγ ) + (c, Lα − Lγ ) + F (s, Kα , Lα ) − F (s, Kγ , Lγ )] ds. 2

t

Since Kα is uniformly bounded and uniformly positive (in view of Lemma 4.2) and Lα is uniformly bounded, we have that the right hand side of the last equality is less than the term ∥Kα − Kγ ∥L∞ F (0,T ;R) times the integral ∫ 2E 0

T

[|a||Kα − Kγ | + |c||Lα − Lγ | + |F (s, Kα , Lα )| + |F (s, Kγ , Lγ )|] ds

170


which is bounded uniformly in (α, γ) (more precisely, it is less than a positive constant times the term (1 + ∥Kα ∥2L∞ + ∥Kγ ∥2L∞ + ∥Lα ∥2L2 + ∥Lγ ∥2L2 )). While F

F

F

F

lim ∥Kα − Kγ ∥L∞ = 0, F (0,T ;R)

α,γ→0+

we then have the desired result. Proof of Theorem 3.2.2 Let L be the strong limit in L2F (0, T ; Rd ) of the Cauchy sequence {Lα }. Lemma 4.4 shows that K α uniformly converges ∞ to K. Moreover, K ∈ L∞ F (0, T ; R+ ) ∩ L (Ω, FT , P ; C([0, T ]; R+ )) is uniformly positive. Therefore, it is meaningful to take the limit in the approximating BSRDEs (3.2.53) by letting α → 0. As a result, (K, L) is shown to be an Ft -adapted solution to the BSRDE (3.1.7). The proof of the uniqueness assertion is similar as in the proof of Theorem 3.2.1, and is omitted here.

3.3

Application To The Stochastic LQ Problem

Consider the one-dimensional non-homogeneous stochastic LQ problem. Assume that ξ ∈ L2 (Ω, FT , P ),

q, f, g ∈ L2F (0, T ; R).

(3.3.1)

Consider the optimal control problem (denoted by P0 ): min

u∈L2F (0,T ;Rm )

J(u; 0, x)

(3.3.2)

with ∫

Ft

J(u; t, x) = E [M |X

t,x;u

(T ) − ξ| +

T

2

(Q|X t,x;u − q|2 + N |u|2 ) ds] (3.3.3)

t

and X t,x;u solving the following linear stochastic system  d ∑   dX = (AX + Bu + f ) ds + (Ci X + Di u + gi ) dwi , i=1   X(t) = x, u ∈ L2F (t, T ; Rm ).

t < s ≤ T, (3.3.4)

The value function V is defined as V (t, x) :=

min

u∈L2F (t,T ;Rm )

J(u; t, x),

(t, x) ∈ [0, T ] × R.

(3.3.5)

Define Γ : [0, T ] × R+ × Rd → Rm by Γ(·, S, L) = −(N +

d ∑ i=1

Di∗ SDi )−1 (B ∗ S

+

d ∑ i=1

Di∗ SCi

+

d ∑ i=1

Di∗ Li ). (3.3.6)

3.3. LQ-PROBLEM

171

and b := A + BΓ(·, K, L), C bi := Ci + Di Γ(·, K, L), b b+ A a := 2A

d ∑

b2 , b bi . C i ci := 2C(3.3.7)

i=1

Let (ψ, ϕ) be the Ft -adapted solution of the following BSDE  d d d ∑ ∑ ∑   ∗ ∗ b b dψ(t) = −[A ψ + Ci (ϕi − Kgi ) − Kf − Li gi + Qq] dt + ϕi dwi , (3.3.8) i=1 i=1 i=1   ψ(T ) = M ξ where (K, L) is the unique Ft -adapted solution of the BSRDE (3.1.7). The following can be verified by a pure completion of squares. Theorem 3.3.1. Suppose that the assumptions of Theorem 3.2.1 or Theorem 3.2.2 are satisfied. Let (K, L) be the unique Ft -adapted solution of BSRDE (3.1.7). Then, the optimal control u b for the non-homogeneous stochastic LQ problem P0 exists uniquely and has the following feedback law u b = −(N +

d ∑

Di∗ KDi )−1 [(B ∗ K +

i=1 d ∑

−B ∗ ψ +

d ∑

Di∗ KCi +

d ∑

i=1

b Di∗ Li )X

i=1

(3.3.9)

Di∗ (Kgi − ϕi )].

i=1

The value function V (t, x), (t, x) ∈ [0, T ] × R has the following explicit formula V (t, x) = K(t)x2 − 2ψ(t)x + V 0 (t), with

(t, x) ∈ [0, T ] × R

(3.3.10)

∫ T ∫ T V 0 (t) := E Ft M |ξ|2 + E Ft Q|q|2 ds − 2E Ft ψf ds t t ∫ T∑ d Ft +E (K|gi |2 − 2ϕi gi ) ds t

−E Ft

∫

i=1 T

((N +

d ∑

t

(3.3.11)

Di∗ KDi )u0 , u0 ) ds.

i=1

and 0

u := (N +

d ∑

Di∗ KDi )−1 [B ∗ ψ

i=1

+

d ∑

Di∗ (ϕi − Kgi )],

t ≤ s ≤ T. (3.3.12)

i=1

Proof Set u e = u − Γ(·, K, L)X.

(3.3.13)

Then the system (3.3.4) reads {

∑ b + Be bi X + Di u dX = (AX u + f ) ds + di=1 (C e + gi ) dwi , 2 m X(t) = x, u ∈ LF (t, T ; R ).

t < s ≤ T, (3.3.14)

172


Applying Itô’s formula, we have the equation for X =: X 2 :

        

∑ bi X(Di u = [b aX + 2X(Be u + f )] ds + di=1 [2C e + gi ) + |Di u e + gi |2 ] ds d ∑ (3.3.15) + [b ci X + 2X(Di u e + gi )] dwi , t < s ≤ T,

dX

i=1

X (t) = x2 ,

u ∈ L2F (t, T ; Rm ).

Note that the BSRDE (3.1.7) can be rewritten as

    

−dK = (b aK +

d ∑

b ci Li + Q + Γ∗ N Γ) dt −

i=1

d ∑

Li dwi ,

i=1

(3.3.16)

K(T ) = M.

So, application of Itô’s formula gives

Ft

Ft

E M |X(T )| + E ∫ 2 Ft = K(t)X (t) + 2E +E Ft

2

∫

∫

T

Q|X| ds + E

t T

t

d ∑

K|Di u e + gi |2 ds + 2E Ft

Ft

∫

T

Γ∗ N Γ|X|2 ds t t ∫ T∑ d T Ft bi X ds KX(Be u + f ) ds + E 2K(Di u e + gi )C 2

∫ t

i=1

t T

d ∑

i=1

Li (Di u e + gi )X ds,

i=1

and

∫T QqX ds] = E Ft [ψ(T )X(T ) + t QqX ds] ∫ T ∫ T∑ d Ft Ft = (ψ(t), X(t)) + E ψ(Be u + f ) ds + E ϕi (Di u e + gi ) ds E Ft [M ξX(T ) +

+E Ft

∫ t

∫T t

t T

t

d d ∑ ∑ ∗ b ( Ci Kgi + Kf + Li gi )X ds. i=1

i=1

i=1

3.4. MV-HEDGING

173

Combining the last two equations, we get ∫ T ∫ T Ft 2 2 E [M |X(T ) − ξ| + Q|X − q| ds + (N u, u) ds] t ∫ T t ∫ T QX 2 ds + Γ∗ N ΓX 2 ds] = E Ft [M |X(T )|2 + t∫ t T Ft −2E [M ξX(T ) + QqX ds] ∫ T t ∫ T Ft 2 2 Ft +E [M |ξ| + Qq ds] + E [(N u e, u e) + 2(N ΓX, u e)] ds t t ∫ T = (KX(t), X(t)) − 2(ψ(t), X(t)) + E Ft [M ξ 2 + Qq 2 ds] t ∫ T∑ ∫ T d +E Ft K|Di u e + gi |2 ds − 2E Ft ψ(Be u + f ) ds t

−2E Ft

∫ t

i=1 d T ∑

ϕi (Di u e + gi ) ds + E Ft ∫

i=1 Ft

∫

t T

(N u e, u e) ds

t T

= K(t)x − 2xψ(t) + E [M ξ + Qq 2 ds] t ∫ T ∫ T∑ d −2E Ft ψf ds + E Ft [Kgi2 − 2ϕi gi ] ds 2

+E Ft

∫

t

t

T

((N + t

−E Ft

∫

2

d ∑

i=1

Di∗ KDi )(e u − u0 ), u e − u0 ) ds

i=1

T

((N + t

d ∑

Di∗ KDi )u0 , u0 ) ds.

i=1

This completes the proof.

3.4

Application To The Mean-Variance Hedging Problem

In this section, we consider the mean-variance hedging problem when asset prices follow Itô’s processes in an incomplete market framework. The market conditions are allowed to be random, but are assumed to be uniformly bounded which implies by Novikov’s condition that there is an equivalent martingale measure. It will be shown that the mean-variance hedging problem in finance of this context is a special case of the linear quadratic optimal stochastic control problem discussed in Section 5, and therefore can be solved completely, by using the above results.

3.4.1

The Financial Market Model

Consider the financial market in which there are m + 1 primitive assets: one nonrisky asset (the bond) of price process ∫ t S0 (t) = exp ( r(s) ds), 0 ≤ t ≤ T, (3.4.1) 0

174


and m risky assets (the stocks) dS(t) = diag(S(t))(µ(t) dt + σ(t) dW (t)),

0 ≤ t ≤ T.

(3.4.2)

Here W = (w1 , . . . , wd )∗ is a d-dimensional standard Brownian motion defined on a complete probability space (Ω, F, P ), and {Ft , 0 ≤ t ≤ T } is the P augmentation of the natural filtration generated by the d-dimensional Brownian motion W . Assume that the instantaneous interest rate r, the m-dimensional appreciation vector process µ and the volatility m × d matrix process σ are progressively measurable with respect to {Ft , 0 ≤ t ≤ T }. For simplicity of exposing the main ideas, assume that they are uniformly bounded and there exists a positive constant ε such that σσ ∗ (t) ≥ εIm×m ,

0 ≤ t ≤ T, a.s.

(3.4.3)

The risk premium process is given by λ(t) = σ ∗ (σσ ∗ )−1 µ e(t),

0≤t≤T

(3.4.4)

where em = (1, . . . , 1)∗ ∈ Rm , and µ e := µ − rem .

3.4.2

Formulation of the problem

For any x ∈ R and π ∈ L2F (0, T ; Rm ), define the self-financed wealth process X with initial capital x and with quantity π invested in the risky asset S by { dX = [rX + (e µ, π)] dt + π ∗ σ dW, 0 < t ≤ T, (3.4.5) X(0) = x, π ∈ L2F (0, T ; Rm ). Given a random variable ξ ∈ L2 (Ω, FT , P ), consider the quadratic optimal control problem: Problem P0,x (ξ)

min

π∈ L2F (0,T ;Rm )

E|X 0,x;π (T ) − ξ|2

(3.4.6)

where X 0,x;π is the solution to the wealth equation (3.4.5). The associated value function is denoted by V (t, x), (t, x) ∈ [0, T ] × R. The minimum point of V (t, x) over x ∈ R for given time t is defined to be the approximate price for the contingent claim ξ at time t. The problem P0,x (ξ) is the so-called mean-variance hedging problem in mathematical finance. It is a one-dimensional singular stochastic LQ problem P0 . In the next subsection, we will give a complete solution of the mean-variance hedging problem P0,x (ξ).

3.4.3

A General Case of Random Market Conditions: A Complete Solution

For the case of the mean-variance hedging problem, we have A(t) = r(t), B(t) = µ e∗ (t), Di (t) = σi∗ , i = 1, . . . , d, u(t) = π(t), M

= 1,

n = 1,

Ci (t) = 0, d ∑ i=1

Di∗ Di

=

d ∑ i=1

σi σi∗ = σσ ∗

3.4. MV-HEDGING

175

where σi is the i-th column of the volatility matrix σ. The associated Riccati equation is a non-linear singular BSDE: ∑ ∑ ∑ dK = −[2rK − (e µ∗ K + di=1 Li σi∗ )(Kσσ ∗ )−1 (K µ e + di=1 Li σi )] dt + di=1 Li dwi = −[(2r − |λ|2 )K − 2(λ, L) − K −1 L∗ σ ∗ (σσ ∗ )−1 σL] dt + (L, dW ), 0 ≤ t < T (3.4.7) K(T ) = 1. Let (ψ, ϕ) is the Ft -adapted solution of the following BSDE dψ = −{[r − |λ|2 − (λ, K −1 L)]ψ ∑ ∑ − di=1 [λi + K −1 σi∗ (σσ ∗ )−1 σL]ϕi } dt + di=1 ϕi dwi , (3.4.8) = −{[r − |λ|2 − (λ, K −1 L)]ψ − (λ + K −1 σ ∗ (σσ ∗ )−1 σL, ϕ)} dt + (ϕ, dW ), ψ(T ) = ξ An immediate application of Theorem 3.3.1 provides an explicit formula for the optimal hedging portfolio: d d d ∑ ∑ ∑ π = −( σi Kσi∗ )−1 [(e µK + σi Li )X − µ eψ − σi ϕi ] i=1

i=1

i=1

= −(Kσσ ∗ )−1 [(e µK + σL)X − µ eψ − σϕ] ∗ −1 −1 = −(σσ ) [(e µ + σK L)X − µ eK −1 ψ − σK −1 ϕ]

(3.4.9)

where (K, L) is the Ft -adapted solution to the Riccati equation (3.4.45). The value function V is also given by Ft 2

V (t, x) = K(t)x − 2ψ(t)x + E ξ − E 2

Ft

∫

T

(e µψ + σϕ)∗ (σKσ ∗ )(e µψ + σϕ)(3.4.10) ds

t

where ϕ := (ϕ1 , . . . , ϕn )∗ . So, the approximate price p(t) at time t for the contingent claim ξ is given by p(t) = K −1 (t)ψ(t).

(3.4.11)

The above solution need not introduce the additional concepts of the socalled hedging numeraire and variance-optimal martingale measure, and therefore is simpler than that of Gourieroux et al [8], and Laurent and Pham [20]. To be connected to the latter, the optimal hedging portfolio (3.4.49) is rewritten as e − σ ϕ]. e e π = −(σσ ∗ )−1 [(e µ + σ L)(X − ψ)

(3.4.12)

Here, e := LK −1 , L

ψe := ψK −1 ,

ϕe := ϕK −1 − LψK −2 .

e ϕ) e solves the following BSDE: and the pair (ψ, { e ϕ)} e dt + (ϕ, e dW ), dψe = {rψe + (λ, e ψ(T ) = ξ

0 ≤ t < T,

(3.4.13)

(3.4.14)

176


with e := λ − [I − σ ∗ (σσ ∗ )−1 σ]LK −1 . λ

(3.4.15)

The process ψe is just the approximate price process, and the BSDE (3.4.51) is the approximate pricing equation. Note that the optimal hedging portfolio (3.4.49) consists of the following two parts: e π 1 := −(σσ ∗ )−1 (e µ + σ L)X

(3.4.16)

e e ψe + σ ϕ], π 0 := (σσ ∗ )−1 [(e µ + σ L)

(3.4.17)

π = π1 + π0.

(3.4.18)

and

and satisfies

The first part π 1 is the optimal solution of the homogeneous mean-variance hedging problem P0,x (0) (that is the case of ξ = 0 for the problem P0,x (ξ)). 1 The corresponding optimal wealth process X 0,1;π is the solution to the following optimal closed system {

e dt − (λ + σ ∗ (σσ ∗ )−1 σL, dW )], dX = X[(r − |λ|2 − (λ, L)) X(0) = 1,

0 < t ≤ T, (3.4.19)

and is just the hedging numéraire. So, the hedging numéraire is just the state (wealth) transition process of the optimal closed system (3.4.19) from time 0, or it is just the fundamental solution of the optimal closed system (3.4.19). e consider the BSDE satisfied by (K, L) To understand the quantity λ, {

dK = {(2r − |λ|2 )K + 2(λ, L) + K−1 L∗ [I − σ ∗ (σσ ∗ )−1 σ]L} dt + (L, dW ), (3.4.20) K(T ) = 1

with K := K −1 and L := −LK −2 . It is the BSRDE for the following singular ∗ ): stochastic LQ problem (denoted by P0,x ∗ Problem P0,x

min

θ∈L2F (0,T ;Rd )

E|X 0,x;θ (T )|2

(3.4.21)

where X 0,x;θ is the solution to the following stochastic differential equation { dX = X [−r dt − (λ, dW )] + ([I − σ ∗ (σσ ∗ )−1 σ]θ, dW ), 0 ≤ t ≤ T, (3.4.22) X (0) = x, θ ∈ L2F (0, T ; Rd ). Its optimal control θb has the following feedback form θb = −K−1 LX = LK −1 X .

(3.4.23)

3.4. MV-HEDGING

177

∗ is just the so-called dual problem of the problem P (0) in The problem P0,1 0,1 [8, 20], and so the variance-optimal martingale measure is P∗ defined as { ∫ T } ∫ 1 T e2 e dP∗ := exp − (3.4.24) (λ, dW ) − |λ| dt dP. 2 0 0

P∗ is an equivalent martingale measure. Note that ψe has the following explicit formula: ∫ T Ft e ψ(t) = E∗ ξ exp (− r(s) ds), 0 ≤ t ≤ T.

(3.4.25)

t

Here, the notation E∗Ft stands for the expectation operator conditioning on the σ-algebra Ft with respect to the probability P∗ . The discounted ϕe is just the integrand of the stochastic-integral-representation of the P ∗ -martingale ∫ ∫ T e dt). {E∗Ft ξ exp (− 0 r(s) ds), 0 ≤ t ≤ T } (w.r.t. the P ∗ -martingale W + 0 λ As in Kohlmann and Zhou [13], again, the formula (3.4.55) has an interesting interpretation in terms of mathematical finance. The optimal hedging ˜ portfolio π in (3.4.55) consists of the two components: (a) (σσ ∗ )−1 σ ϕ—it may be interpreted as the perfect hedging portfolio for the contingent claim ξ with ˜ (that is, under the variance-optimal martingale the risk premium process λ ∗ −1 ˜ ψ˜ − X)—it is a generalized Merton-type portfomeasure), (b) (σσ ) (˜ µ + σ L)( lio for a terminal utility function c(x) = x2 (see Merton [24]), which invests the capital (ψ˜ − X) left over after fulfilling the obligation from the perfect hedge under the variance-optimal martingale measure.

3.4.4

The Case of Markovian Market Conditions

Assume the following Markovian structure for the randomness of the market conditions: r(t, ω) := r(t, Yt ),

µ(t, ω) := µ(t, Yt ),

σ(t, ω) := σ(t, Yt )

with {Yt , 0 ≤ t ≤ T } defined by the stochastic differential equation { dY = η(t, Y ) dt + γ(t, Y ) dW, 0 ≤ t ≤ T, Y0 = y ∈ Rd .

(3.4.26)

(3.4.27)

In this case, the risk premium process {λ(t, ω), 0 ≤ t ≤ T } reads λ(t, ω) = σ ∗ (σσ ∗ )−1 (t, Yt )[µ(t, Yt ) − r(t, Yt )em ],

0 ≤ t ≤ T.

(3.4.28)

This context includes the stochastic volatility models usually studied in the literature (Hull and White [10], Stein and Stein [29], Heston [9]). Under the above assumption, the Riccati equation (3.4.45) and the stochastic differential equation (3.4.27) constitute a forward-backward stochastic differential equation. Define the function h as the generator of BSDE (3.4.45), that is h(t, y, z, v) := z(2r − |λ|2 )(t, y) − 2v ∗ λ(t, y) − z −1 v ∗ σ ∗ (σσ ∗ )−1 σ(t, y)v, (3.4.29) ∀(t, y, v) ∈ [0, T ] × R × Rd and z ̸= 0.

178


Then, it is straightforward in the literature that the solution to the Riccati equation (3.4.45) can be characterized by the parabolic partial differential equation:   Zt + (η(t, y), Zy ) + 21 tr (γγ ∗ (t, y)Zyy ) + h(t, y, Z, Zy γ(t, y)) = 0, y ∈ Rd , 0 ≤ t < T, (3.4.30)  Z(T, y) = 1, y ∈ Rd through the relation K(t) = Z(t, Yt ),

L(t) = [Zy γ(t, Yt )]∗ .

(3.4.31)

The reader is referred to Peng [30], Pardoux and Peng [23], and Pardoux and Tang [28] for details.

3.4.5

On a Modified Model

Consider the optimal control problem: [∫ Problem MP 0,x (ξ)

min

π∈ L2F (0,T ;Rm )

T

E

|X

0,x;π

(s) − qs | ds + |X 2

0,x;π

] (T ) − ξ| (3.4.32) 2

0

where qs := E Fs ξ and X 0,x;π is the solution to the wealth equation (3.4.5). Identically as before, we use Theorem 3.3.1 to solve it. The associated Riccati equation is a non-linear singular BSDE: dK = −[(2r − |λ|2 )K + 1 − 2(λ, L) − K −1 L∗ σ ∗ (σσ ∗ )−1 σL] dt + (L, dW ), (3.4.33) K(T ) = 1. Let (ψ, ϕ) is the Ft -adapted solution of the following BSDE dψ = −{[r − |λ|2 − (λ, K −1 L)]ψ − (λ + K −1 σ ∗ (σσ ∗ )−1 σL, ϕ) + q} dt + (ϕ, dW ), (3.4.34) ψ(T ) = ξ An immediate application of Theorem 3.3.1 provides an explicit formula for the optimal hedging portfolio: π = −(σσ ∗ )−1 [(e µ + σK −1 L)X − µ eK −1 ψ − σK −1 ϕ]

(3.4.35)

where (K, L) is the Ft -adapted solution to the Riccati equation (3.4.33). The value function V is also given by [ ] ∫ T ∫ T 2 Ft 2 2 Ft V (t, x) = K(t)x −2ψ(t)x+E ξ + qs ds −E (e µψ+σϕ)∗ (σKσ ∗ )(e µψ+σϕ) ds t

t

where ϕ := (ϕ1 , . . . , ϕn )∗ . So, the approximate price p(t) at time t for the contingent claim ξ is given by p(t) = K −1 (t)ψ(t).

(3.4.36)

The optimal hedging portfolio (3.4.35) is rewritten as e − σ ϕ]. e e π = −(σσ ∗ )−1 [(e µ + σ L)(X − ψ)

(3.4.37)

3.4. MV-HEDGING

179

Here, e := LK −1 , L

ψe := ψK −1 ,

ϕe := ϕK −1 − LψK −2 .

e ϕ) e solves the following BSDE: and the pair (ψ, { e ϕ) e + K −1 (ψ˜ − q)} dt + (ϕ, e dW ), dψe = {rψe + (λ, e ) = ξ ψ(T

(3.4.38)

0 ≤ t < T, (3.4.39)

with e := λ − [I − σ ∗ (σσ ∗ )−1 σ]LK −1 . λ

(3.4.40)

The process ψe is just the approximate price process, and the BSDE (3.4.39) is the approximate pricing equation. Similarly as in Kohlmann and Zhou [13], the economic interpretation for the approximate pricing equation (3.4.39) can also be given. Exercise 3.4.1. The following exercise generalizes the above results following [15]. • The system (3.4.5) can be generalized to allow for a stochastic income process ∫ t∑ ∫ t d gi (s) dwi (s), 0 ≤ t ≤ T fs ds + It = x + 0 i=1

0

by replacing x by It . See Karatzas and Shreve [11]. The state Xt is the wealth of the investor at time t. The control ut is the wealth vector invested in m risky securities whose price dynamics are described by dS(t) = diag(S(t))[(Bt∗ + At 1) dt + D(t)∗ dw(t)],

0 ≤ t ≤ T. (3.4.41)

Here 1 = (1, . . . , 1)∗ ∈ Rm . At is the instantaneous interest rate of a non-risky security with the price dynamics: ∫ t S0 (t) = exp ( As ds), 0 ≤ t ≤ T. (3.4.42) 0

Bt∗

is the m-dimensional appreciation vector process minus the instantaneous interest rate in its each component, and D∗ := (D1∗ , · · · , Dd∗ ) is the m × d volatility matrix process of the m risky securities. ξ ∈ L2 (Ω, FT , P ) is the given terminal contingent claim, and q ∈ L2F (0, T ) is the given intertemporal contingent claim. The stochastic LQ problem ∫ T 0,x;u 2 min [E|XT − ξ| + E Qs |Xs0,x;u − qs |2 ds] (3.4.43) u∈ L2F (0,T ;Rn )

0

is a generalized mean-variance hedging problem in financial economics: in the case of f = 0, g = 0, Q = 0, and q = 0, it reduces to the classical mean-variance hedging problem in the financial literature (see for instance Schweizer [26, 27, 28], Gourieroux et al [8], Laurent and Pham [20]).

180


• Derive the following explicit characterizations for the optimal hedging portfolio and the approximate price. Theorem 3.4.2. Assume that 1×m ), D ∈ L∞ (0, T ; Rd×m ), A ∈ L∞ B ∈ L∞ F (0, T ), F (0, T ; R F (3.4.44) f, q ∈ L2F (0, T ), g ∈ L2F (0, T ; Rd ), Q ∈ L∞ F (0, T ; S+ ).

Moreover, assume that D∗ (t)D(t) ≥ δIm×m for some positive constant δ. Let (K, L) be an {Ft , 0 ≤ t ≤ T }-adapted solution to the Riccati equation dK = −[(2A − |λ|2 )K + Q − 2⟨λ, L⟩ − K −1 L∗ D(D∗ D)−1 D∗ L] dt (3.4.45) +⟨L, dw⟩, 0≤t 0 a.s.. Therefore, P∗ is equivalent to Since E 0 |λ| P . It can be verified directly that P∗ is a martingale measure. The proof is then complete. • In the following the reader is invited to develop the comparison with some existing results: In the proof of Theorem 3.4.3, we actually give a new interpretation for the pricing measure P∗ : it is optimal in the sense of (3.4.64). When Q ̸= 0,, the discounting rate which is used in the computation of the approximate price is not the bond rate, instead it is A + K −1 Q. This seems to be a new feature in the financial literature. When the intertemporal hedging is not considered, i.e., Q = 0, then our interpretation for the pricing measure P∗ coincides with that of Schweizer [26, 27, 28], Gourieroux et al [8], Laurent and Pham [20]. In fact, the latter states that the pricing measure P∗ is given by ∫ T ∫ 1 T e e s νbs |2 ds) (3.4.68) dP∗ = exp ( ⟨−λs − Ds νs , dw(s)⟩ − |λs + D 2 0 0 where νb is the solution of the problem: ∫T 0

min |ν|2

ds 2, then ∫ ∫ ˆ ˆ H = H0 + δu dFu + νu dNu where δˆ is a hedging strategy and ν satisfies E

[∫

(3.4.78)

] v 2 du < ∞

(ii) The intrinsic value process satisfies ∫ ∫ ˆ ˆ 0 + δu dFu + vu dNu = E ˆ (H | Ft ) Vˆt = H P Exercise 4: Let Vˆ = EPˆ (H | Ft ) and G∗0 = 0 ( ) t ϕ(x) = δˆt + vm Vˆt − x 2F

dG∗t = ϕ(G∗t ) dFt with

t

and δˆ from (3.4.78).

t

(3.4.79)

3.4. MV-HEDGING

195

(i) dG∗t = ϕ(G∗t ) dFt , G∗0 = 0 (3.4.79) has a unique solution. The hedging strategy δt∗ = ϕ(G∗t ) has gains process Gt (δ ∗ ) = G∗ (ii) Let δ be any hedging strategy and define [( ) ] H(t) = E Vˆt − G∗t Gt (θ) If we can show that H(t) ≡ 0 then H(T ) = 0 which implies that δ ∗ is optimal. Apply the product rule to (Vˆt − G∗t )Gt (δ) to see that ∫ t 2 mu H(t) = − H(u) du. 2 0 vu With H(0) = 0 we derive H(t) ≡ 0. With this the problem is completely solved. (: −), yes indeed it is, mk) Some special cases: Let µ, m, σ, v, ρ be deterministic and define γ= { ∫ Yt = St exp −

and

σmρ −µ v }

T

γu du

YT = ST

t

where dSt = µt St dt + σt St dwt , S0 > 0 Exercise 5: (i) (F, Y ) is a Markov process (ii) Let H = g(FT , ST ), then Vˆt = EPˆ [H | Ft ] = f (Ft , Yt , t) with

[ ( { } { })] 1 1 1 1 2 2 f (x, y, t) = E g x exp w − var(w ) , y exp w − var(w ) 2 2

where w1 , w2 are normally distributed with variances ∫T and covariance t vσρ du.

∫T t

vu2 du and

∫T t

σu2 du

We now want to derive the representation of H in terms of f . Note that dVˆt = fx dFt + fy dYt + terms of finite variation As from above dVˆt = δˆt dFt + νt dNt , derive σt ρ t θˆt = fx (Ft , Yt , t) + fy (Ft , Yt , t) Yt vt Ft

196

CHAPTER 3. MEAN VARIANCE HEDGING √ νt = fy (Ft , Yt , t) Yt σt

1 − ρ2t

Solve the differential equation for G∗ and compute δ ∗ as in the preceding exercises. Exercise 6: Let H = k ST , k > 0, then [ { }] 1 2 2 f (x, y, t) = E −k y exp −w − var(w ) = −k y 2 This implies fx = 0, fy = −k and so σ t ρt δˆt = −k Yt vt Ft { ∫ ˆ Vt = −k Yt = −k St exp −

and

}

T

γu du

t

Finally: 1 δˆ∗ = Ft

(

) ) σρ mt ( ˆ t t ˆ ∗ Vt Vt − Gt + vt vt2

where G∗ is the solution of 1 dG = Ft ∗

(

) ) σρ mt ( ˆ t t ˆ ∗ Vt − G t + Vt dFt vt vt2 G∗0 = 0

With this we have the optimal hedging strategy in explicit form.

Summarizing Conclusion: To solve the problem follow the following steps: • Find the optimal terminal value of the gains process GT (δ ∗ ), where only the existence of δ ∗ is known. • Try to find δ ∗ by representing the random variable GT (δ ∗ ) as a martingale under a new measure. • Explicitly solve for δ ∗ in the Markovian case by making use of some sort of modified Feynman-Kac procedure as seen in the lectures.

3.5

Markovitz and Sons

This section is taken from a preprint by Li and Zhou: Portfolio selection is to seek a best allocation of wealth among a basket of securities. The mean-variance approach by Markowitz [15, 16] provides a fundamental basis for portfolio construction in a single period. The most important contribution of this model is

3.5. MARKOVITZ AND SONS

197

that it quantifies the risk by using the variance, which enables investors to seek highest return after specifying their acceptable risk level. This approach becomes the foundation of modern finance theory and inspires literally hundreds of extensions and applications. In particular, in the case where the covariance matrix is positive definite and short-selling is allowed, an analytic solution was obtained by Merton [17]. Moreover, Perold [20] developed a more general technique to locate the efficient frontier when the covariance matrix is nonnegative definite. After Markowitz’s pioneering work, the mean-variance model was soon extended to multi-period portfolio selection, see for example, Mossin [18], Samuelson [21], Hakansson [12], Elton and Gruber [7], Francis [10], and Grauer and Hakansson [11]. The literature in multi-period portfolio selection has been dominated by the works of maximizing expected utility functions of the terminal wealth, namely maximizing EU (xT ) where U is a utility function. Specifically, the investment situations where U is of a power form, log form, exponential form, or quadratic form have been extensively investigated. The resulting portfolio policies, in these situations, are often shown to be myopic optimal policies. However, when using utility functions of the terminal wealth in multi-period portfolio selection, besides the difficulty in eliciting utility functions from the investors, trade-off information between the risk and the expected return is implicit that makes an investment decision much less intuitive. In this sense, Markowitz’s mean-variance approach has not been fully utilized in the dynamic, multi-period setting. If in a multi-period model one is to completely mimic Markowitz’s formulation, then one should minimize an objective function involving a term [ExT ]2 (due to the variance term) or, more generally, a term of the form U (ExT ) where U is a nonlinear utility function. The seemingly harmless difference between EU (xT ) and U (ExT ) actually causes a major difficulty for the latter in view of applying the dynamic programming method. More precisely, For the objective function of the form EU (xT ), the dynamic programming ) is applicable due to the so-called “smooth( ing property” E U (xT )|Fm ) Fn = E(U (xT )|Fn ), where {Fk , k = 1, 2, · · · } is the and n ≤ m. However, no analogous relation, such as ( underlying filtration ) E U (ExT |Fm ) Fn = U (ExT |Fn ), is available for the case when the objective function involves a nonlinear U (ExT ). The mean-variance model in a continuous-time framework has been developed a bit later, see Föllmer-Sondermann [9], Duffie and Jackson [4], and Duffie and Richardson [5]. Once again the general setting in these works is to maximize certain objective function of the form EU (xT ) rather than U (ExT ) due to the same difficulty as in the multi-period case mentioned above. In [5], the case U (ExT ) is tackled by introducing an additional equality constraint on the expected return ExT . However, it works only under the assumption that all the coefficients (interest rate, volatility rate, etc.) are time-invariant constants. The purpose of this paper is to seek the efficient frontier for a continuoustime mean-variance portfolio selection problem. We develop a new approach to handle the difficulty arising from the term [ExT ]2 for the problem with timevarying coefficients. The idea is first to embed the original (not readily solvable)

198


problem into a tractable auxiliary problem, following a similar embedding technique introduced by Li and Ng [14] for the multi-period model, then to show that this auxiliary problem actually is a stochastic optimal linear-quadratic (LQ) problem and can be solved explicitly by the general stochastic LQ theory developed recently [2, 3]. The optimal solution to the original problem can then be located via the solution to the auxiliary problem. This approach does not need to introduce any state constraint, and therefore is easy to handle. One of the promising features of this approach is that it bridges up portfolio selection problems and standard stochastic control models (the theory of which are now very rich; see Fleming and Soner [8] and Yong and Zhou [23]) and provides a general solution procedure that may be used to solve more complicated situations such as those with random coefficients.

3.5.1

Problem Formulation

Throughout this paper (Ω, F, P, {Ft }t≥0 ) is a fixed filtered complete probability space on which defined a standard {Ft }t≥0 -adapted m-dimensional Brownian motion W (t) ≡ (W 1 (t), · · · , W m (t))′ . We denote by L2F (0, T ; Rm ) the set of all Rm -valued, measurable stochastic processes f (t) adapted to {Ft }t≥0 , such that ∫T E 0 |f (t)|2 dt < +∞. Notation. We make the following additional notation: M′ Mj |M |

: :

the transpose of any vector or matrix M ; the entry of any vector M ; √j-th ∑ 2 : = i,j mij for any matrix or vector M = (mij );

Sn n S+ n Sˆ+ C([0, T ]; X)

: : : :

L2 (0, T ; X)

:

the space of all n × n symmetric matrices; the subspace of all nonnegative definite matrices of S n ; the subspace of all positive definite matrices of S n ; the Banach space of X-valued continuous functions on [0, T ] endowed with the maximum norm ∥ · ∥ for a given Hilbert space X; the Hilbert space of X-valued integrable functions on [0, T ] (∫ )1 T 2 dt 2 for a given Hilbert space X. endowed with the norm ∥ f (t) ∥ X 0

Suppose there is a market in which m + 1 assets (or securities) are traded continuously. One of the assets is the bond whose price process P0 (t) is subject to the following (deterministic) ordinary differential equation: {

dP0 (t) = r(t)P0 (t)dt, t ∈ [0, T ], P0 (0) = p0 > 0,

(3.5.1)

where r(t) > 0 is the interest rate (of the bond). The other m assets are stocks whose price processes P1 (t), · · · , Pm (t) satisfy the following stochastic


199

differential equation: { } { ∑ j dPi (t) = Pi (t) bi (t)dt + m j=1 σij (t)dW (t) , t ∈ [0, T ], Pi (0) = pi > 0,

(3.5.2)

where bi (t) > 0 is the appreciation rate, and σi (t) ≡ (σi1 (t), · · · , σim (t)) : [0, T ] → Rm is the volatility or the dispersion of the stocks. Define the covariance matrix



 σ1 (t)   σ(t) =  ...  ≡ (σij (t))m×m . σm (t)

(3.5.3)

The basic assumption throughout this paper is σ(t)σ(t)′ ≥ δI,

∀t ∈ [0, T ],

(3.5.4)

for some δ > 0. This is the so-called non-degeneracy condition. We also assume that all the functions are measurable and uniformly bounded in t. Consider an investor whose total wealth at time t ≥ 0 is denoted by x(t). Suppose he/she decides to hold Ni (t) shares of i-th asset (i = 0, 1, · · · , m) at time t. Then x(t) =

m ∑

Ni (t)Pi (t), t ≥ 0.

(3.5.5)

i=0

Assume that the trading of shares takes place continuously and transaction cost and consumptions are not considered. Then one has ∑m  dx(t) =  { i=0 Ni (t)dPi (t) ∑ }   m   = r(t)N (t)P (t) + b (t)N (t)P (t) dt 0 0 i i i  i=1   ∑m ∑m  +{ i=1 Ni (t)Pi (t) j=1 σij (t)dWj (t)} ] ∑ [  = r(t)x(t) + m bi (t) − r(t) ui (t) dt  i=1   ∑ ∑m  j  + m  j=1 i=1 σij (t)ui (t)dW (t),   x(0) = x0 > 0,

(3.5.6)

where ui (t) ≡ Ni (t)Pi (t), i = 0, 1, 2 · · · , m,

(3.5.7)

denotes the total market value of the investor’s wealth in the i-th bond/stock. If ui (t) < 0 (i = 1, 2, · · · , m), then the investor is short-selling i-th stock. If u0 (t) < 0, then the investor is borrowing the amount |u0 (t)| at rate r(t). It is clear that by changing ui (t), the investor changes the “allocation” of his/her wealth in these m + 1 assets. We call u(t) = (u1 (t), · · · , um (t))′ a portfolio of

200


the investor. Notice that we exclude the allocation to the bond, u0 (t), from the portfolio as it will be determined completely by the allocation to the stocks. The objective of the investor is to maximize the mean terminal wealth, Ex(T ), and at the same time to minimize the variance of the terminal wealth [ ]2 [ ]2 Var x(T ) ≡ E x(T ) − Ex(T ) = Ex(T )2 − Ex(T ) .

(3.5.8)

This is a multi-objective optimization problem with two criteria in conflict. Definition 3.5.1. A portfolio u(·) is said to be admissible if u(·) ∈ L2F (0, T ; Rm ). Definition 3.5.2. The mean-variance portfolio optimization problem is denoted as ( Minimize Subject to

) ( ) J1 (u(·)), J2 (u(·)) ≡ − Ex(T ), Var x(T ) , { u(·) ∈ L2F (0, T ; Rm ), (x(·), u(·)) satisfy equation (3.5.6)

(3.5.9)

Moreover, an admissible portfolio u ¯(·) is called an efficient portfolio of the problem if there exists no admissible portfolio u(·) such that J1 (u(·)) ≤ J1 (¯ u(·)), J2 (u(·)) ≤ J2 (¯ u(·)),

(3.5.10)

and at least one of the inequalities holds strictly. In this case, we call (J1 (¯ u(·)), J2 (¯ u(·))) ∈ R2 an efficient point. The set of all efficient points is called the efficient frontier. In other words, an efficient portfolio is one that there exists no other portfolio better than it with respect to both the mean and variance criteria. The problem then is to identify the efficient portfolios along with the efficient frontier. By standard multi-objective optimization theory, an efficient portfolio can be found by solving a single-objective optimization problem where the objective is a weighted average of the two original criteria under certain convexity conditions (see, e.g., Yu [24]), which are satisfied in the present case. The efficient frontier can then be generated by varying the weights. Therefore, the original problem can be solved via the following optimal control problem Minimize Subject to

J 2 (u(·)) ≡ −Ex(T ) + µVar x(T ), {1 (u(·)) + µJ u(·) ∈ L2F (0, T ; Rm ), (x(·), u(·)) satisfy equation (3.5.6)

(3.5.11)


201

where the parameter (representing the weight) µ > 0. Denote the above problem by P (µ). Define ΠP (µ) = {u(·)|u(·) is an optimal control of P (µ)} .

3.5.2

(3.5.12)

An Auxiliary Problem

Note that Problem P (µ) is not a standard stochastic optimal control problem and is hard to solve directly due to the term [EX(T )]2 in its cost function, which is non-separable in the sense of dynamic programming (see the discussion in Introduction). We now propose to embed the problem into a tractable auxiliary problem that turns out to be a stochastic linear-quadratic (LQ) problem. To do this, set Minimize Subject to

J(u(·); µ, λ) ≡ E{µx(T )2 − λx(T )}, { u(·) ∈ L2F (0, T ; Rm ), (x(·), u(·)) satisfy equation (3.5.6),

(3.5.13)

where the parameters µ > 0 and −∞ < λ < +∞. Let us call the above Problem A(µ, λ). Define ΠA(µ,λ) = {u(·)|u(·) is an optimal control of A(µ, λ)} .

(3.5.14)

The following result tells the relationship between the problems P (µ) and A(µ, λ). Theorem 3.5.3. For any µ > 0, one has ΠP (µ) ⊆

∪

ΠA(µ,λ) .

(3.5.15)

−∞)0, ∀t ∈ [0, T ]. We introduce the following assumption for the coefficients of the above problem. (A) The data appearing in the LQ problem satisfy A ∈ C([0, T ]; Rn ), B, Dj ∈ C([0, T ]; Rn×m ), f ∈ L2 (0, T ; Rn ), n ), Q ∈ C([0, T ]; S+ m R ∈ C([0, T ]; S ), n. H ∈ S+ Note that here we do not assume that R is positive definite (therefore, R could be zero, indefinite, or even negative definite) as opposed to almost all the relevant existing research. Namely, the problem under consideration may include singular situations. We introduce the following stochastic Riccati equation:  ˙ P (t)    

= −P (t)A(t) − A(t)′ P (t) − Q(t) )−1 ( ∑ ′ B(t)′ P (t), +P (t)B(t) R(t) + m j=1 Dj (t) P (t)Dj (t)

  P (T ) = H,  ∑  ′ K(t) ≡ R(t) + m j=1 Dj (t) P (t)Dj (t) > 0, ∀t ∈ [0, T ],

(3.5.22) along with an equation {

( )−1 ∑ ′ g(t) ˙ = −A(t)′ g(t) + P (t)B(t) R(t) + m B(t)′ g(t) − P (t)f (t), j=1 Dj (t) P (t)Dj (t) g(T ) = 0. (3.5.23)

204


Theorem 3.5.4. If the equations (3.5.22) and (3.5.23) admit solutions P ∈ n ) and g ∈ C([0, T ]; Rn ), respectively, then the stochastic LQ problem C([0, T ]; S+

(3.5.20)-(3.5.21) has an optimal feedback control ∗

u (t, x) = −(R(t) +

m ∑

Dj (t)′ P (t)Dj (t))−1 B(t)′ (P (t)x + g(t)).

(3.5.24)

j=1

Moreover, the optimal cost value is   ∫ T m ∑ ( ) 1 −1 2f (t)′ g(t) − g(t)B(t) R(t) + Dj (t)′ P (t)Dj (t) B(t)′ g(t) dt+ J∗ = 2 0 j=1

(3.5.25) 1 ′ x P (0)x0 + x0 g(0). 2 0 ∑ ′ Proof. Denote K(t) ≡ R(t)+ m j=1 Dj (t) P (t)Dj (t). Applying Ito’s formula, we get 1 ′ 2 d(x P x)

=

1 2

{∑

m ′ ′ j=1 u Dj P Dj u

+ x′ (−Q + P BK −1 B ′ P )x } +2u′ B ′ P x + 2x′ P f dt

(3.5.26)

+ 21 {...}dW (t), and { } d(x′ g) = u′ B ′ g + x′ P BK −1 B ′ g + f ′ g − x′ P f + {...}dW (t).

(3.5.27)

Integrating both (3.5.27) and (3.5.28) from 0 to T , taking expectations, adding them together and noting (3.5.21), one obtains } ∫T { J(u(·)) = 12 E 0 u′ Ku + 2u′ B ′ (P x + g) + x′ P BK −1 B ′ P x + 2x′ P BK −1 B ′ g + 2f ′ g dt =

+ 21 x′0 P (0)x0 + x0 g(0) } ∫T { 1 −1 B ′ (P x + g)]′ K[u + K −1 B ′ (P x + g)] + 2f ′ g − gBK −1 B ′ g dt E [u + K 2 0 + 21 x′0 P (0)x0 + x0 g(0).

(3.5.28) It follows immediately that the optimal feedback control is given by (3.5.24) and the optimal value is given by (3.5.25) provided that the the corresponding equation 20 under (3.5.24) has a solution. But under (3.5.24), the system (3.5.20) reduces to  −1 ′ (P (t)x(t) + g(t))]dt dx(t) = [A(t)x(t)  ∑m − B(t)K−1 (t)B(t) − j=1 Dj (t)K (t)B(t)′ (P (t)x(t) + g(t))dW j (t),  x(0) = x0 .

(3.5.29)


205

This is a nonhomogeneous linear stochastic differential equation. Since P ∈ n ), g ∈ C([0, T ]; Rn ) and K −1 ∈ C([0, T ]; S m ), the equation (3.5.30) C([0, T ]; S+ +

admits one and only one solution. This completes the proof. 2 LQ models constitute an extremely important class of optimal control problems and their optimal solutions can be obtained explicitly via the Riccati equations, due to the nice underlying structures (see [1, 13, 22]). Mean-variance portfolio selection problems have inherent linear-quadratic structure and therefore it is very natural to solve them through solving the stochastic LQ problems. The general stochastic Riccati equation is introduced in [2] as a backward stochastic differential equation of the Pardoux-Peng type ([19]) for the case where all the coefficients are random. It reduces to (3.5.22) for the present case. A rather surprising discovery in [2] is that a stochastic LQ problem may still be meaningful and solvable even when the running control weighting cost R is indefinite, as opposed to the conventional belief (due to the deterministic case) that the positive definiteness of R is absolutely necessary for the LQ problem to be sensible. This phenomenon has to do with the deep nature of uncertainty as well as the way of controlling the uncertainty. While a detailed discussion on this point can be found in [2], one may also look at the stochastic Riccati equation (3.5.22) to get some rough idea as to why R may be allowed to be indefinite (or zero in particular). Indeed, even when R ≤ 0, the presence of ∑ ′ the term m j=1 Dj P Dj may offer compensation if it is positive enough so that ∑ ′ R+ m j=1 Dj P Dj > 0. The mean-variance portfolio model exemplifies such situations as we will see in the subsequent sections.

3.5.4

Solution to The Auxiliary Problem

Now let us come back to solve Problem A(µ, λ) introduced in Section 3.5.2, which specializes the general model discussed in Section 3.5.3. Set γ=

λ and y(t) = x(t) − γ. 2µ

(3.5.30)

Then Problem A(µ, λ) is equivalent to minimizing 1 E[ µy(T )2 ], 2

(3.5.31)

206


subject to

{ }  + B(t)u(t) + f (t) dt  dy(t) = A(t)y(t) ∑m j + j=1 Dj (t)u(t)dW (t),  y(0) = x0 − γ,

(3.5.32)

where {

A(t) = r(t), B(t) = (b1 (t) − r(t), · · · , bm (t) − r(t)), f (t) = γr(t), Dj (t) = (σ1j (t), · · · , σmj (t)).

(3.5.33)

Thus, the problem 3.5.32-3.5.33 is a special case of the problem 3.5.20-3.5.21 with (Q(t), R(t)) = (0, 0), H = µ,

(3.5.34)

and A(t), B(t), f (t), Dj (t) given by (3.5.34). Note that R(t) = 0 in this problem. This is why the mean-variance model gives rise to a inherently singular stochastic LQ problem. Also, a special feature of the problem is that the state x(t) is one dimensional, so is the unknown P (t) of the corresponding stochastic Riccati equation (3.5.22). This makes it easier to solve (3.5.22) explicitly. Denote m ∑ ρ(t) = B(t)[ Dj (t)′ Dj (t)]−1 B(t)′ = B(t)[σ(t)σ(t)′ ]−1 B(t)′ .

(3.5.35)

j=1

Then (3.5.22) reduces to   P˙ (t) = (ρ(t) − 2r(t)) P (t), P (T ) = µ,  P (t)[σ(t)σ(t)′ ] > 0, t ∈ [0, T ].

(3.5.36)

Clearly, the solution of (3.5.37) is given by P (t) = µe−

∫T t

(ρ(s)−2r(s))ds

.

(3.5.37)

Note that the third constraint in (3.5.37) is satisfied automatically due to the assumption (3.5.4). Moreover, the equation (3.5.23) becomes { g(t) ˙ = (ρ(t) − r(t)) g(t) − γr(t)P (t), g(T ) = 0,

(3.5.38)

which evidently admits a unique solution g ∈ C([0, T ]; R1 ). The optimal feedback control (3.5.24) then gives u ¯(t, y) ≡ (¯ u1 (t, y), · · · , u ¯m (t, y)) = −[σ(t)σ(t)′ ]−1 B(t)′ (y +

g(t) ). P (t)

(3.5.39)

3.5. MARKOVITZ AND SONS Let h(t) =

g(t) P (t) .

207

Then noting (3.5.37) and (3.5.39), one has ˙ h(t) =

P (t)g(t)− ˙ P˙ (t)g(t) P (t)2 r(t)P (t)g(t)−γr(t)P (t)2 P (t)2

= = r(t)h(t) − γr(t). Since h(T ) = 0, we can solve h(·) to get

∫T g(t) = h(t) = γ(1 − e− t r(s)ds ). P (t)

(3.5.40)

Substituting (3.5.41) into (3.5.40), and noting (3.5.31), we arrive at u ¯(t, x) ≡ (¯ u1 (t, x), · · · , u ¯m (t, x)) ∫T ′ −1 = −[σ(t)σ(t) ] B(t)′ [x − γ + γ(1 − e− t r(s)ds )] ∫T = [σ(t)σ(t)′ ]−1 B(t)′ (γe− t r(s)ds − x).

3.5.5

(3.5.41)

Efficient Frontier

In this section we proceed to derive the efficient frontier for the original meanvariance problem (3.5.9). Under the optimal feedback control (3.5.42) (for the problem A(µ, λ)), the wealth equation (3.5.6) evolves as } {  ∫ − T r(s)ds  ρ(t) dt  dx(t) = (r(t) − ρ(t))x(t) + γe t  

+B(t)(σ(t)σ(t)′ )−1 σ(t)(γe− x(0) = x0 .

∫T t

r(s)ds

− x(t))dW (t)

(3.5.42)

Moreover, applying Ito’s formula to x(t)2 , we obtain { }  ∫T 2 = (2r(t) − ρ(t))x(t)2 + γ 2 e−2 t r(s)ds ρ(t) dt  dx(t)   

+2x(t)B(t)(σ(t)σ(t)′ )−1 σ(t)(γe− x(0)2 = x20 .

∫T t

r(s)ds

− x(t))dW (t),

(3.5.43)

Taking expectations on both sides of (3.5.43) and (3.5.44), we conclude that Ex(t) and Ex(t)2 satisfy the following two nonhomogeneous linear ordinary differential equation: { ∫T { } dEx(t) = (r(t) − ρ(t))Ex(t) + γe− t r(s)ds ρ(t) dt, Ex(0) = x0 ,

(3.5.44)

and {

∫T { } dEx(t)2 = (2r(t) − ρ(t))Ex(t)2 + γ 2 e−2 t r(s)ds ρ(t) dt, Ex(0)2 = x20 .

(3.5.45)

208


Solving (3.5.45) and (3.5.46), we can express Ex(T ) and Ex(T )2 as explicit functions of γ, Ex(T ) = αx0 + βγ, Ex(T )2 = δx20 + βγ 2 ,

(3.5.46)

where ∫T

α=e

0

(r(t)−ρ(t))dt

, β = 1 − e−

∫T 0

ρ(t)dt

∫T

, δ=e

0

(2r(t)−ρ(t))dt

.

(3.5.47)

By Theorem 3.5.3, an optimal solution of the problem P (µ), if it exists, can ¯ so that (noting (3.5.47) and 31) be found by selecting λ ¯ λ ¯ = 1 + 2µE x λ ¯(T ) = 1 + 2µ(αx0 + β ). 2µ This yields

∫T ∫T ¯ = 1 + 2µαx0 = e 0 ρ(t)dt + 2µx0 e 0 r(t)dt . λ 1−β

(3.5.48)

Hence the optimal control for the problem P (µ) is given by (3.5.42) with γ = γ¯ =

¯ λ 2µ

¯ given by (3.5.49). In this case the corresponding variance of the and λ

terminal wealth is Var x ¯(T ) = E x ¯(T )2 − [E x ¯(T )]2 2 = β(1 − β)¯ γ − 2αβx0 γ¯ + (δ − α2 )x20 β(δ−α2 ) 2 ¯ 2 ¯ 2 − 2 αβ 2 x0 γ = 1−β β [β γ 1−β + 1−β x0 ] =

1−β γ β [(β¯

2

x0 γ ¯ + αx0 )2 − 2 αβ1−β +

(3.5.49)

β(δ−α2 ) 2 1−β x0 ].

Substituting β¯ γ = Ex ¯(T ) − αx0 (due to (3.5.47)) in the above and noting (3.5.48), we obtain Var x ¯(T ) = = =

2 1−β α x ¯(T ))2 − 2 1−β x0 E x ¯(T ) + βδ+α x20 ] β [(E 1−β ( )2 ∫T 1−β Ex ¯(T ) − x0 e 0 r(t)dt β ∫T ( )2 ∫T e− 0∫ ρ(t)dt r(t)dt 0 E x ¯ (T ) − x e . 0 T 1−e− 0 ρ(t)dt

(3.5.50)

To summarize the above discussion, we have the following result. Theorem 3.5.5. The efficient frontier of the bi-criteria optimal portfolio selection problem (3.5.9), if it ever exists, must be given by (3.5.51).


209

The classical formulation

New formulation including the risk free rate 4

3

3

2

expectation

expectation

2

1 1

0

0

1

2

3

variance

4

5

0

0

1 2 3 4 standard deviation

5

210


2,25 0

2,0

1

1,75

1,5 2

1,25

3

1,0 1,0 0,75 0,0

0,1

0,2 x

0,3

0,4

0,1 0,6 1,1 y 1,6

1,25 1,5

y

4

1,75 2,0

0,0

0,1

0,2

0,3

0,4

x

The relation (3.5.51) reveals explicitly the trade-off between the mean (return) and variance (risk). For example, if one has set an expected return level, then (3.5.51) tells the risk he/she has to take; and vice versa. In particular, if one cannot take any risk, namely, Var (¯ x(T )) = 0, then E x ¯(T ) has to be ∫T

x0 e

0

r(t)dt

meaning that he/she can only put his/her money in the bond. An-

other interesting phenomenon is that the efficient frontier (3.5.51) involves a perfect square. This is due to the possible inclusion of the bond in a portfolio (see first figure). In the case when the riskless bond is excluded from consideration, the efficient frontier may no longer be a perfect square, which means one cannot have a risk-free portfolio (see second figure). The next figures show the VAR depending on r and θ, respectively.

3.6. THE PARTIALLY OBSERVED CASE

3.5.6

211

Concluding Remarks

This section investigates a dynamic continuous-time mean-variance portfolio selection problem in the spirit of the original Markowitz’s work. An efficient frontier is traced out in a closed form for the model where the interest/appreciation rates and volatility rate are non-constants. The basic idea of solving the problem is to embed the original problem into a stochastic linear-quadratic (LQ) control problem. This represents a new approach different from the existing literature, which is however totally natural in view of the inherent LQ structure of the mean-variance model. More importantly, this approach opens up possible ways of solving more general models, such as the one with random interest/appreciation rates and volatility rate (since the general stochastic LQ technique can handle such a case) and the one with objective function of the form U (Ex(T )) (under certain convexity condition of the utility function U ). These are the problems currently under investigation. The mean-variance portfolio problem studied in this paper also nicely exemplifies the general stochastic LQ models where the control running cost R is indefinite. To be specific, observe that there is no direct running cost (over the time period [0, T ]) associated with a control (portfolio) in the cost function (3.1) or (5.2), namely, R ≡ 0 in this case. However, the portfolio does influence the diffusion term of the system dynamics (see (3.5.6)) which gives rise ∑ ′ ′ to a “hidden” running cost m j=1 Dj (t) P (t)Dj (t) ≡ P (t)σ(t)σ(t) (with P (·) given by (3.5.37)) that must be taken into consideration and may be regarded as a “cost equivalence” of the risk. To balance between this uncertainty/risk cost and the potential return is exactly the goal of the mean-variance portfolio selection problem.

3.6

The Partially Observed Case

Pricing a financial instrument is an important issue in financial economics. When the market is complete, the replicating method can be used to price every contingent claim. This is the case for the classical Black-Scholes model.

212


Incomplete information typically makes the market incomplete, and the replicating method fails to be used universally for every contingent claim. The mean-variance hedging is a powerful approach to pricing contingent claims in an incomplete market with both tractability and intuition. In this paper, we study the mean-variance hedging approach to pricing contingent claims in a financial market where the stock appreciation rates are modeled by a random vector which has known distribution and is independent of the driving Brownian motion. The mean-variance criterion was initially introduced by Föllmer and Sondermann [6]. The mean-variance hedging problem was widely studied by Duffie and Richardson [4], Schweizer [26, 27, 28], Pham, Rheinländer and Schweizer [24], Gourieroux, Laurent and Pham [8], and Laurent and Pham [20]. All of these works use a projection argument. Recently, Kohlmann and Zhou [13], and the authors [14, 15, 16] developed a natural LQ theory approach. The mentioned works all assume that the market conditions are exactly known, that is, they all consider the case of complete information. The consequences of incomplete information in financial markets have been considered in the utility maximization problem. See Browne and Whitt [1], Detemple [2], Dothan and Feldman [3], Gennotte [5], Karatzas and Xu [12], Lakner [18, 19]. This paper explores the consequences of incomplete information in hedging contingent claims. Our problem is a partially observed optimal control problem. In the literature, the partially observed optimal control problem is solved by suitably formulating and then solving the separated problem, which is completely observed. It is found difficult to use the partially observed maximum principle to solve a partially observed optimal control problem in a completely natural way. However, in this paper, we develop such a new approach. First, we show the existence and uniqueness result by a functional argument. Second, we establish in the spirit of Tang [30, 31] the optimality conditions—so called the partially observed maximum principle—which should help to solve the optimal hedging portfolio. Note that in our case the maximum condition is an expected


213

equality conditioned on the history of the stock prices, due to the nature of incomplete information. Third, we use the Bayesian formula and filter the adjoint processes and the Kallianpur-Striebel functional. In this way, we derive a new set of conditions, which turns out to be the stochastic maximum principle for a mean-varaince hedging problem with complete information—which is actually the separated problem. This shows the separation principle for our problem. Fourth, the solution is given by using existing results for the case of complete information. Finally, we would like to comment that our present arguments heavily depend on the special structure of our problem, and they are still found difficult in applying to the general LQG problem with partial observation. The rest of the paper is organized as follows. Section 2 contains the formulation of the problem and some notations. Section 3 gives an equivalent formulation of the problem. In Section 4 we show the existence and uniqueness result. In Section 5 we study the partially observed maximum principle. In Section 6, we give the explicit formula for the optimal hedging policy and the approximate pricing equation. Finally, in section 7, we state the separated problem.

3.6.1

Formulation of the Problem

Consider a fixed complete probability space (Ω, F, P ) on which is defined a standard d-dimensional Brownian motion W = (W1 , · · · , Wm )′ . Assume that {FtW , 0 ≤ t ≤ T } is the completion, by the totality N of all null sets of F, of the natural filtration {FtW } generated by W . Consider a financial market in which there are m + 1 assets. One of them is non-risky, and the price dynamics is given by (∫ t ) S0 (t) = exp r(τ ) dτ , 0 ≤ t ≤ T 0

where r(·) is a measurable, adapted, and essentially bounded scalar process. The other m assets are risky, and the price dynamics are governed by the equation

{

∑ dSi (t) = Si (t)[Bi dt + m j=1 σij (t) dWi (t)], Si (0) = si > 0, i = 1, . . . , m.

(3.6.1)

214


Here B = (B1 , · · · , Bm )′ is the vector of stock appreciation rates which is unobservable and is modeled as a Rm -valued essentially bounded random vector, independent of the Brownian motion W , with known prior distribution µ. Assume that only the stock price process S(·) is observable. Neither the stock appreciation rate vector B nor the Brownian motion W (·) are observable. However, the processes ∫ t∑ m tBi + σij (s) dWi (s),

i = 1, · · · , m, 0 ≤ s ≤ t

0 j=1

are observable as they are equal to ( ) ∫ m Si (t) 1 t∑ ln + |σij |2 ds, si 2 0

i = 1, · · · , m

j=1

where σ =: (σij ) is the volatility matrix. Let {FtS , 0 ≤ t ≤ T } be the natural filtration generated by the stock price processes S(·), that is FtS =: σ(S(τ ); 0 ≤ τ ≤ t),

0 ≤ t ≤ T,

(3.6.2)

and {Ft , 0 ≤ t ≤ T } be the P -augmentation of the filtration {FtS , 0 ≤ t ≤ T }. Denote by {Gt , 0 ≤ t ≤ T } the P -augmentation of the auxiliary, enlarged filtration GtB,W := σ(B, W (τ ); 0 ≤ τ ≤ t)

(3.6.3)

generated by both the driving Brownian motion W (·) and the random variable B. Consider a small hedger with initial capital x > 0 in finite time-horizon [0, T ]. Denote by πi (t) the amount of money invested in the ith stock at time t, and set π = (π1 , · · · , πm )′ . Thus, the wealth process of this hedger satisfies the linear stochastic differential equation  m m ∑ ∑    πi (t)]S0−1 (t)dS0 (t) + πi (t)Si−1 (t)dSi (t)  dX(t) = [X(t) −    

i=1

i=1

= [r(t)X(t) + ⟨B − r(t)1m , π(t)⟩] dt + π ′ (t)σ(t) dW (t), X(0) = x

where 1m =: (1, · · · , 1).

(3.6.4) 0 < t ≤ T,


215

Assume that σ is non-singular and the inverse is uniformly bounded. Define the risk premium process λ(t) = σ −1 (B − r1m )(t),

0 ≤ t ≤ T.

(3.6.5)

It is not observable. It is emphasized that a trading strategy π(·) is adapted to the filtration {Ft , 0 ≤ t ≤ T }, which is required by the assumption that only the stock price can be observed and neither the drift B nor the Brownian motion are. Denote by Uad (t, T ) the set of all Rm -valued, {Fs , 0 ≤ s ≤ T }-adapted stochastic processes π on [t, T ] such that ∫ T |π(s)|2 ds < ∞

P − a.s.,

t

and by L2F (t, T ; Rm ) the set of those elements π of Uad (t, T ) such that ∫ T E |π(s)|2 ds < ∞. t

Note that the wealth equation (3.6.4) can be written as  ∫ t  ′ λ(s) ds), dX(t) = [r(t)X(t) + π (t)σ(t) d(W (t) + 0  X(0) = x Since the Brownian motion with drift ∫ t W (t) + λ(s) ds,

0 < t ≤ T,

(3.6.6)

0≤t≤T

0

is observable, the wealth process X is also observable. A contingent claim ξ is an integrable random variable, describing the net payoff of a financial instrument. Assume here that ξ ∈ L2 (Ω, F, P ). The meanvariance hedging problem is the following partially observable stochastic control problem: min

J(π; 0, x)

(3.6.7)

π∈Uad (0,T )

where X 0,x;π is the wealth process of the hedger associated with the hedging strategy π and the initial data (0, x), and J(π; 0, x) := E|X 0,x;π (T ) − ξ|2 .

216

CHAPTER 3. MEAN VARIANCE HEDGING We conclude this section by introducing some notations. M ′ is the transpose

of any vector or matrix M , and |M | is the Euclidean length of the vector or matrix M . ⟨M1 , M2 ⟩ stands for the inner product of the two vectors M1 and M2 . Rn is the n-dimensional Euclidean space. C([0, T ]; H) denotes the Banach space of H-valued continuous functions on [0, T ], endowed with the maximum norm for a given Hilbert space H. L2F (0, T ; H) is the Banach space of Hvalued {Ft , 0 ≤ t ≤ T }-adapted square-integrable stochastic processes f on ∫T [0, T ], endowed with the norm (E 0 |f (t)|2 dt)1/2 for a given Euclidean space H. L∞ F (0, T ; H) is the Banach space of H-valued, {Ft , 0 ≤ t ≤ T }-adapted, essentially bounded stochastic processes f on [0, T ], endowed with the norm ess supt,ω |f (t)| for a given Euclidean space H. L2 (Ω, F, P ; H) is the Banach space of H-valued norm-square-integrable random variables on the probability space (Ω, F, P ) for a given Banach space H.

3.6.2

An Equivalent Formulation

Theorem 3.6.1. Assume that σ has an inverse. Then the following holds: min π∈Uad (0,T )

J(π; 0, x) =

min

π∈L2F (0,T )

J(π; 0, x).

(3.6.8)

First, we have the following obvious fact: Lemma 3.6.2. If π ∈ Uad (0, T ) is an optimal hedging policy, then X 0,x;π (T ) is square-integrable. Theorem 3.6.1 is a consequence of Lemma 3.6.2 and the following a priori estimate. Lemma 3.6.3. Assume that X and π ∈ Uad (0, T ) satisfy the wealth equation (3.6.4). If the terminal state X(T ) is square-integrable, then the following estimate holds: 1 E 2

∫

T

⟨σσ ′ π, π⟩ ds + E|X(t)|2 ≤ exp (β(T − t))E|X(T )|2 ,

0 ≤ t ≤ T.(3.6.9)

t

Proof We first prove that Lemma 3.6.3 is true if π ∈ L2F (t, T ; Rm ).


217

Using Itô’s formula, we have from (3.6.4) E|X(T )|2

∫ T = E|X(r)|2 + 2E ⟨−rX(s) + π ′ (B − r(s)1m ), X(s)⟩ ds r ∫ T ′ ⟨σσ π(s), π(s)⟩ ds +E (3.6.10) r ∫ T ∫ T = E|X(r)|2 + 2E ⟨−rX(s) + π ′ σλ(s), X(s)⟩ ds + E ⟨σσ ′ π(s), π(s)⟩ ds r r ∫ T ∫ T 1 2 ′ ≥ E|X(r)| + E ⟨σσ π(s), π(s)⟩ ds − E (r(s) + 8|λ(s)|2 )|X(s)|2 ds. 2 r r Write ρr := E|X(r)|2 ,

t ≤ r ≤ T.

Then, the above reads ∫ T ∫ T 1 ⟨σσ ′ π(s), π(s)⟩ ds ≤ ρT + β ρs ds. ρt + E 2 t t

(3.6.11)

(3.6.12)

Here, β is the essential upper bound of r + 8|λ|2 . By Gronwall’s inequality, we have

1 ρt + E 2

∫

ρr ≤ exp(β(T − r))ρT , T

⟨σσ ′ π(s), π(s)⟩ ds ≤ exp(β(T − t))ρT .

(3.6.13) (3.6.14)

t

Then consider the general case π ∈ Uad (t, T ). For any integer N > 0, define the stopping time TN as follows:

∫

TN =: inf{τ ≤ T :

τ

⟨σσ ′ π(s), π(s)⟩ ds ≥ N }

(3.6.15)

t

with the convention inf ∅ = +∞. Clearly, lim TN = T,

N →∞

P − a.s.

and πN (τ ) =: π(τ )χ[t,TN ] (τ ),

t ≤ τ ≤ T, N = 1, 2, . . .

belong to L2F (t, T ; Rm ). Then, the estimate (3.6.9) is true for π = πN . Note that the constant β is independent of the integer N . Then passing to the limit by letting N → ∞, in view of Fatou’s lemma, we conclude the proof. Remark 3.6.4. We refer the reader to Pardoux and Peng [26] for the general theory of BSDEs. An a priori estimate for a BSDE can be found in Pardoux and Peng [23], and the proof given above is adapted from theirs.

218


3.6.3

Existence and Uniqueness of an Optimal Hedging Policy

Theorem 3.6.5. (existence and uniqueness) Assume that σ is non-singular. Then there is a unique π b ∈ L2F (0, T ; Rm ) such that J(b π ; 0, x) =

min

J(π; 0, x).

π∈Uad (0,T )

Proof We first show the uniqueness assertion. In fact, assume that π1 and π2 are two optimal controls, that is J(π1 ; 0, x) = J(π2 ; t, x) =

min

J(π; 0, x).

π∈Uad (0,T )

Then, we have by applying Lemma 3.6.3 that π1 + π2 0 ≥ J(π1 ; 0, x) + J(π2 ; 0, x) − 2J( ; 0, x) 2 1 = E|X 0,x;π1 (T ) − X 0,x;π2 (T )|2 2 ∫ T 1 |σ ′ (π1 − π2 )|2 ds ≥ exp (−βT )E 4 0 which gives π1 (s) = π2 (s), 0 ≤ s ≤ T . Then we show the existence. We can choose a sequence of admissible controls {πk }∞ k=1 such that lim J(πk ; 0, x) =

k→∞

inf

J(π; 0, x).

(3.6.16)

π∈Uad (0,T )

2 m We assert that {πk }∞ k=1 is a Cauchy sequence in LF (0, T ; R ). In fact, from

Lemma , we have πk − πl πk + πl J(πk ; 0, x) + J(πl ; 0, x) − 2J( ; 0, x) = 2J( ; 0, x) 2 2 ∫ T (3.6.17) 1 ≥ exp (−βT )E |σ ′ (πk − πl )|2 ds 4 0 which implies ∫ lim E

k,l→∞

T

|σ ′ (πk − πl )|2 ds = 0.

(3.6.18)

0

Then, we conclude that there is π b ∈ L2F (0, T ; Rm ) such that πk converges to π b strongly in L2F (0, T ; Rm ). As a consequence, X 0,x;πk (T ) converges to X 0,x;bπ (T ) strongly in L2 (Ω, FT , P ). Therefore, J(b π ; 0, x) = lim J(πk ; 0, x) = k→∞

and therefore π b is optimal.

inf π∈Uad (0,T )

J(π; 0, x),

(3.6.19)


3.6.4

219

The Partially Observed Maximum Principle for Optimal Hedging Policies

In the spirit of Tang [30, 31], we can prove the following partially observed maximum principle. Theorem 3.6.6. Let π b ∈ L2F ,σσ′ (0, T ; Rm ) be an optimal portfolio process. Denote by (p, q) the unique {Gt , 0 ≤ t ≤ T }-adapted solution of the following adjoint equation {

dp(t) = −rp(t) dt + ⟨q(t), dW (t)⟩, p(T ) = X 0,x;bπ (T ) − ξ.

0 ≤ t < T,

(3.6.20)

a.e.a.s..

(3.6.21)

Then, we have the following optimality condition: E Ft [p(t)(B − r(t)1m ) + σ(t)q(t)] = 0,

Proof Consider ∀π ∈ L2F (0, T ; Rm ). From the optimality of π b, we have d J(b π + επ; 0, x) ≥ 0. dε ε=0 A direct calculation yields d J(b π + επ; 0, x) = E[(X 0,x;bπ (T ) − ξ)X 0,0;π (T )]. dε ε=0 In terms of the adjoint processes (p, q), we have E[(X 0,x;bπ (T ) − ξ)X 0,0;π (T )] = E[p(T )X 0,0;π (T )] ∫ T = E ⟨π(t), p(t)(B − r(t)1m ) + σ(t)q(t)⟩ dt. 0

Therefore, we have ∫ E

T

⟨π(t), p(t)(B − r(t)1m ) + σ(t)q(t)⟩ dt ≥ 0,

0

∀π ∈ L2F (0, T ; Rm ).

This immediately gives the desired equality. From the above conditions, the Bayesian formula and filtering theory will be used to derive an explicit formula for the optimal hedging policy in the next section.

220


3.6.5

Optimal Hedging Policy and Approximate Pricing Equation

Introduce the exponential process ( ∫ t ) ∫ 1 t 2 Λ(t) =: exp − ⟨λ(s), dW (s)⟩ − |λ(s)| ds , 2 0 0

0 ≤ t ≤ T. (3.6.22)

It can be proved that W (·) is a (Gt , P )-Brownian motion, and Λ(·) is a (Gt , P )martingale; the reader is referred to Karatzas and Zhao [13] for the details of the proof. Therefore, we can define PeT (A) =: E[Λ(T )1A ],

∀A ∈ GT

(3.6.23)

a probability measure equivalent to P on GT . In the following, we denote by eT the expectation operator with respect to PeT . E Define

∫ Y (t) := W (t) +

t

λ(s) ds,

0≤t≤T

0

and the process ) ∫ 1 t 2 Z(t) := Λ (t) = exp ⟨λ(s), dW (s)⟩ + |λ(s)| ds 2 0) 0 (∫ t ∫ t (3.6.24) 1 = exp ⟨λ(s), dY (s)⟩ − |λ(s)|2 ds , 0 ≤ t ≤ T. 2 0 0 −1

(∫

t

On the one hand, we have from the stocks prices equation and the wealth equation {

dS(t) = diag(S(t))σ(t) dY (t), S(0) = s

(3.6.25)

and {

dX(t) = r(t)X(t) dt + π ′ (t)σ(t) dY (t), X(0) = x

0 < t ≤ T,

(3.6.26)

The two equations show that F S = F Y , and X is {Ft , 0 ≤ t ≤ T }-adapted, that is, X is observable. On the other hand, we can show that Y is a (Gt , Pe)Brownian motion, and Y is independent of B under the reference probability Pe. Note that Z is an exponential martingale under Pe. Note that Z has the following stochastic dynamics: { dZ(t) = Z(t)⟨λ(t), dY (t)⟩, 0 ≤ t < T, Z(0) = 1.

(3.6.27)


221

Ft b := E e Ft Z(t). Therefore, in view of E e Ft [Z(t)λ(t)] = Z(t)E b Write Z(t) T T T λ(t) (by

the Bayesian rule) and Theorem 8.1 of Liptser and Shiryayev [21] (Stochastics I-II, Satz 5.3.1 or by a direct argument), we have { b b b dZ(t) = Z(t)⟨ λ(t), dY (t)⟩, 0 ≤ t < T, b Z(0) = 1 b = E Ft λ(t). with λ(t) { b−1 (t) = dZ Zb−1 (0) =

(3.6.28)

Using Itô’s formula, we get b 2 dt − ⟨λ(t), b b−1 (t)[|λ(t)| Z dY (t)⟩], 1

0 ≤ t < T,

(3.6.29)

In the following, we compute the dynamics for the estimate of the adjoint process p, that is {E Ft p(t), 0 ≤ t ≤ T }. We have from the Bayesian rule the following E Ft p(t) =

e Ft [Z(t)p(t)] e Ft [Z(T )p(t)] E E T T b−1 (t)E e Ft [Z(t)p(t)] = =Z T Ft Ft e e ET Z(T ) ET Z(t)

The adjoint equation (3.6.20) can be rewritten as { dp(t) = [−rp(t) − ⟨λ(t), q(t)⟩] dt + ⟨q(t), dY (t)⟩, p(T ) = X(T ) − ξ.

0 ≤ t < T,

Using Itô’s formula, we have { d[Z(t)p(t)] = −rZ(t)p(t) dt + ⟨Z(t)p(t)λ(t) + Z(t)q(t), dY (t)⟩, Z(T )p(T ) = Z(T )X(T ) − Z(T )ξ.

(3.6.30)

0 ≤ t < T, (3.6.31)

Write P (t) =: Z(t)p(t),

Q(t) =: Z(t)p(t)λ(t) + Z(t)q(t),

Then, (P, Q) satisfies the following BSDE: { dP (t) = −rP (t) dt + ⟨Q(t), dY (t)⟩, P (T ) = Z(T )X(T ) − Z(T )ξ.

0 ≤ t ≤ T.

0 ≤ t < T,

(3.6.32)

Write e Ft P (t), Pb(t) =: E T

b =: E e Ft Q(t). Q(t) T

In view of Theorem 8.1 of Liptser and Shiryayev [21] (Stochastics I-II, Satz 5.3.1 or by a direct argument), we have from (3.6.32), { b dPb(t) = −rPb(t) dt + ⟨Q(t), dY (t)⟩, 0 ≤ t < T, FT b b e b )X(T ) − Z(T b )E FT ξ. (3.6.33) P (T ) = Z(T )X(T ) − ET [Z(T )ξ] = Z(T

222


b−1 (t)E e Ft [Z(t)p(t)]. Using Itô’s formula, we get Write pe(t) := E Ft p(t) = Z T { b de p(t) = [−r(t)e p(t) − ⟨λ(t), qe(t)⟩] dt + ⟨e q (t), dY (t)⟩, 0 ≤ t < T, (3.6.34) F T pe(T ) = X(T ) − E ξ where b p(t). b−1 (t)Q(t) b − λ(t)e qe(t) := Z

(3.6.35)

Applying the Bayesian rule, we derive from (3.6.21) 0 = E Ft [p(t)(B − r(t)1m ) + σ(t)q(t)] e Ft Z(T )[p(t)(B − r(t)1m ) + σ(t)q(t)] E T = e Ft Z(T ) E T e Ft Z −1 (t)E e Ft {Z(t)[p(t)(B − r(t)1m ) + σ(t)q(t)]}. = E

(3.6.36)

e Ft {Z(t)[p(t)(B − r(t)1m ) + σ(t)q(t)]} = 0. E T

(3.6.37)

T

Hence

From the definition of the risk premium process λ we have e Ft [Q(t)] = E e Ft [σ(t)Q(t)] = 0. σ(t)E T T

(3.6.38)

b = 0. σ(t)Q(t)

(3.6.39)

b p(t)] = 0. σ(t)[e q (t) + λ(t)e

(3.6.40)

Then, we have

In view of (3.6.35), we have

At this stage, in view of (3.6.40), (3.6.34) and (3.6.26), a direct calculation leads to the following assertions (see Kohlmann and Tang [15] for details): Theorem 3.6.7. Let (K, L) solve the Riccati equation: { b 2 ]K(t) − 2⟨λ(t), b dK(t) = −{[2r(t) − |λ(t)| L(t)⟩ − K −1 (t)|L(t)|2 } dt + ⟨L(t), dN (t)⟩, (3.6.41) K(T ) = 1 with K being uniformly positive and uniformly bounded and L being square ∫t b ds, 0 ≤ t ≤ T. Then, (e integrable. Let N (t) := Y (t) − 0 λ(s) p, qe) solves the following BSDE: {

de p(t) = −r(t)e p(t) dt + ⟨e q (t), dN (t)⟩, pe(T ) = X(T ) − E FT ξ.

(3.6.42)


223

e ϕ) e is characterized as Let ψe := KX − pe and ϕe := Kσ ′ π + LX − qe. Then, (ψ, the unique adapted solution of the following BSDE:  e b b + K −1 (t)L(t)⟩]ψ(t) e + ⟨λ(t) b + K −1 (t)L(t), ϕ(t)⟩} e  = {[−r(t) + ⟨λ(t), λ(t) dt  dψ(t) e (3.6.43) +⟨ϕ(t), dN (t)⟩,   ψ(T e ) = E FT ξ. The optimal portfolio process is given by b + K −1 L)X − K −1 (ϕe + λ bψ)]. e π = −(σ ′ )−1 [(λ Remark 3.6.8. Note that the process N defined in the above theorem is the so-called innovation process. Theorem 3.6.9. Let (K, L) be defined as in Theorem 6.1. Let ψ := K −1 ψe and ϕ := K −1 ϕe − K −1 Lψ. Then, (ψ, ϕ) is characterized as the unique adapted solution of the following BSDE: { b dψ(t) = {r(t)ψ(t) + ⟨λ(t), ϕ(t)⟩} dt + ⟨ϕ(t), dN (t)⟩, FT ψ(T ) = E ξ.

(3.6.44)

The optimal portfolio process is given by b + K −1 L)(X − ψ) − ϕ]. π = −(σ ′ )−1 [(λ The value function is given by V (t, x) = E Ft |x − ψ(t)|2 + E|ξ|2 − E|E FT ξ|2 , and the approximation price at time t for the contingent claim ξ is ψ(t). So, (3.6.44) is the approximate pricing equation.

3.6.6

The Separated Problem

The relations (3.6.40), (3.6.34) and (3.6.26) are the optimality conditions for the following completely observable optimal control: min

π∈L2F (0,T ;Rm )

e 0, x) J(π;

where X 0,x;π solves the following SDE: { dX(t) = r(t)X(t) dt + π ′ (t)σ(t) dY (t), X(0) = x

0 < t ≤ T,

(3.6.45)

224


and e 0, x) := |X 0,x;π (T ) − E FT ξ|2 . J(π; This assertion is clearer if we write (3.6.34) and (3.6.26) as follows: { de p(t) = −r(t)e p(t) dt + ⟨e q (t), dN (t)⟩, (3.6.46) FT e ψ(T ) = E ξ and { b dX(t) = [r(t)X(t) + ⟨σ(t)λ(t), π(t)⟩] dt + π ′ (t)σ(t) dN (t), X(0) = x

0 < t ≤ T, (3.6.47)

The above completely observable control problem is the so-called separated problem. Its equivalence to the original problem is also seen from the following e 0, x) = J(π; 0, x)+E|E FT ξ|2 −E|ξ|2 for ∀π ∈ L2 (0, T ; Rm ), three facts: (1) J(π; F (2) equation (3.6.47) is a filtered version of the original system equation (in view of Theorem 8.1 of Liptser and Shiryayev [21]), and (3) Y and N generates the same filtration.

Chapter 4

Bonds and Interest Rate Derivatives 4.1

Zero Coupon Bonds

We make the same assumptions as in section 2.5.2: We rewrite the model here for the reader’s convenience: Let Wt = (Wt1 , . . . , Wtd )∗ be a d-dimensional standard Brownian motion and let Ft be the augmentation of the filtration generated by the Brownian motion. This filtration satisfies the usual conditions. The trading time is here given by the deterministic interval [0, τ ]. Let us first model the bond by a nonnegative Ft -progressively measurable process r, the risk free interest rate, though it might be a random variable!. We assume that

∫

τ

|rt |dt < ∞; P -a.s.

(4.1.1)

0

Then the bond is given on [0, τ ] by

[∫

Bt := exp

]

t

rs ds .

(4.1.2)

0

It satisfies the differential equation dBt = rt Bt dt, B0 = 1.

(4.1.3)

The n risky investments S i (i = 1, . . . , n) are given by   ∫ t d d ∫ t ∑ ∑ 1 Sti := si0 exp  µis − (σsi,j )2 ds + σsi,j dWsj  . 2 0 0 j=1

j=1

225

(4.1.4)

226

CHAPTER 4. BONDS AND INTEREST RATE DERIVATIVES The processes µi , σ i,j are Ft -progressively measurable,and in order to make

all integrals well-defined we assume: n ∑ d ∫ ∑ i=1 j=1

τ

0

|µit | + |σti,j |2 dt < ∞; P -a.s.

(4.1.5)

By applying Itô’s formula we get: dSti

=

µit Sti dt

=

si0 .

+

d ∑

σti,j Sti dWtj ,

j=1

S0i

(4.1.6)

We make the following assumptions an the market data: (M0) The non-negative interest rate r, as well as the coefficients σ i,j , µi are progressively measurable and satisfy the conditions (4.1.1) and (4.1.5). (M1) d ≥ n. i,j (M2) The diffusion matrix σt∧τ := (σt∧τ )i=1,...,n, j=1,...,d has P ⊗λ-a.s. maximal

rank. With (M0) we guarantee the existence of the respective integrals used to model the market. The other conditions guarantee that at any time the number of independent random sources is greater or equal to the number of stocks. As we have seen in the last section, this of course again makes sure that the market is arbitrage free. For notational simplicity assume now that n = d = 1. Let us now consider a very special contingent claim, namely ξ = 1. Of course the reader will have noticed that this claim is closely connected to a notion defined earlier, namely the bond. So we define: Definition 4.1.1. A contingent claim which pays at time τ = T the amount 1 is called a zero coupon bond with maturing time τ . Its price is given from theorem 2.5.15 by dpξ (t, τ ) =

[

] rt pξ (t, τ ) + Zt θt dt + Zt dWt ,

pξ (τ, τ ) = ξ for ξ = 1,

4.2. FORWARD AND FUTURE PRICE

227

why we write pξ = p1 . Explicitly we get [ { ∫ p1t∧τ,τ = E exp −

τ

1 rs + |θs |2 ds − 2 t∧τ

∫

τ

t∧τ

} ] θs∗ dWs 1 Ft

(4.1.7)

in an obvious notation. Exercise 4.1.2.

• Show that under the risk neutral measure P ∗

p1t∧τ,τ

=E

∗

[

∫

]

τ

exp(−

rs ds)|Ft . t∧τ

• Derive the hedging strategy for the zero coupon bond for the initial wealth p1t∧τ,τ =: p(t, τ ) • What is the hedging strategy if r is deterministic?

4.2

Forward and Future Price

Definition 4.2.1. The T-forward price F (t, T ) of the risky asset St is a price fixed at time t ≤ T payable for ST at time T . To make this mathematically more precise, the T -forward price is defined by the property that the claim ST − F (t, T ) has discounted value zero under P ∗ . Exercise 4.2.2.

• The T-forward price F (t, T ) of the risky asset satisfies:

F (t, T )p(t, T ) = St .

(4.2.1)

where p(t, T ) is taken from 4.1.2. • Express the T-forward price of an arbitrary claim ξ F (ξ, t, T ) in terms of p(t, T ) as F (ξ, t, T ) =

pξt,T p(t, T )

.

228

CHAPTER 4. BONDS AND INTEREST RATE DERIVATIVES

This may be taken as a more practical definition!) Give an intuitive interpretation of this equation. Hint: We had considered pξt,T as the price of ξ at time t (this price is sometimes called the natural price) and p(t, T ) is the price at time t of the zero bond. Note also that for deterministic r the forward price is just the non-discounted price of ξ under the risk neutral measure. • The price at time 0 of the claim ξ − F (ξ, 0, T ) is zero. • The price at time t of ξ − F (ξ, 0, T ) is pξ (t) − F (ξ, 0, T )p(t, T ). The exercises show that the forward contract has • value zero at time t = 0 • nonzero value at time 0 < t < T • value ξ − F (ξ, 0, T ) = ξ −

pξ (0) p(0,T ) .

In this way the value of the forward contract moves away from zero during the time interval. One of the involved parties might now become concerned about a possible default of the other party. The worried agent might wish the other one deposit money into an escrew account. Any such additional agreement as part of the contract would of course change the reasoning which led to the forward price. This leads to the concept of the future price G(ξ, t, T ) for an asset with market value ξ at time T . This futures price process is constructed such that at all times t ∈ [0, T ) the futures contract has value zero. Suppose that an agent sells for the price zero the contract to another agent at time t. At time t + dt the futures price has moved by an amount G(ξ, t+dt, T )−G(ξ, t, T ). If this amount is positive -according to the provisions of the contract- the party holding the short position must transfer this amount to the party holding the long position. If the amount is negative the procedure works in the opposite direction. In


229

this way the futures contract is continuously resettled and the value of the contract is always zero. The futures price for an asset must agree with the market price on the delivery date. This motivates the definition: Definition 4.2.3. The T-future G(ξ, t, T ) price of a claim ξ is the solution of the BSDE: dG(ξ, t, T ) = [Zt θt ] dt + Zt dWt , G(ξ, T, T ) = ξ. Exercise 4.2.4.

• Compute the value of the futures contract ∫ T 1 ∗ Bt E [ dG(ξ, u, T )|Ft ] = 0. t Bu

(Note here that under P ∗ the solution of the BSDE for G is a martingale. • Show that G(ξ, t, T ) = E ∗ (ξ|Ft ). We can then make the assertion in the last exercise clearer: Corollary 4.2.5. If BT−1 and ξ are (conditionally) uncorrelated under the risk neutral measure P ∗ then G(ξ, t, T ) = F (ξ, t, T ). If they are positively correlated conditional on (Ft , P ∗ ) then G ≤ F. Proof. Under P ∗ we have F (ξ, t, T ) =

E ∗ [BT−1 ξ|Ft ] E ∗ [BT−1 |Ft ]E ∗ [ξ|Ft ] = = E ∗ [ξ|Ft ] = G(ξ, t; T ) E ∗ [BT |Ft ] E ∗ [BT |Ft ] (4.2.2)

. The second part is left to the reader. This motivates the following definition: Definition 4.2.6. The T-forward measure is defined as BT−1 dQT . = dP ∗ |FT E ∗ [BT−1 ]

230

CHAPTER 4. BONDS AND INTEREST RATE DERIVATIVES It is a nice exercise to derive the results in exercise 4.2.2 and in the last

corollary without making use of the equivalent martingale measure by only using the BSDE representations.

Exercise 4.2.7. Prove the following: What is a hedge for the claim ξ −F (ξ, 0, T )? Hint: Short F (ξ, 0, T ) zeroes with maturity T . This gives an amount large enough to hedge ξ at time zero, . This hedge requires no initial investment and show that it is worth ξ − E QT (ξ) at time T , so this forward contract has nonzero value. ∫T • Bt E ∗ [ t Bu−1 dG(ξ, u, T )|Ft ] = 0 and G(ξ, t, T ) is a (P ∗ , Ft )-martingale.

4.2.1

Forwards and Futures, an interpretation

A forward contract as considered above is an agreement to buy or sell a specified asset ST at a certain future time T for a delivery price K specified today t = 0. The purpose of such a contract is to share risk in the sense that the buyer takes a long position by agreeing to buy and the seller to sell the asset for the price K at time T . At the beginning neither party incurs any costs and the forward price of the contract at time t is the delivery price that would give the contract value zero. So at t = 0 the forward price is K which might change as time goes on. The payoff for the buyer is ST − K, for the seller it is K − ST . With a T-forward contract the only payment thus is F (ξ, 0, T ) at time T . As we have seen a forward contract has a non-zero value over the time interval. This is eliminated with so-called future contracts. Here the value is kept zero over the whole time interval: To this end the buyer receives a positive or negative cash flow over the whole time interval so that between time 0 and T he will have received ∫

T

dG(ξ, u, T ) = ξ − G(ξ, 0, T ),

0

so that if the buyer holds the contract over the whole time interval the buyer will have paid G(ξ, 0, T ) for the claim ξ with value ξ at time T .


4.2.2

231

Change of Numeraire, revisited

In the last sections we had considered the process Bt as a numeraire and had defined a risk neutral P ∗ measure as a measure equivalent to the original physical measure such that the discounted asset price S 1 (t)Bt−1 is an (Ft , P ∗ )(local)martingale. We had mentioned earlier already that any strictly positive process can take the role of a numeraire and other assets can be expressed in terms of the numeraire. We here state the following result: Theorem 4.2.8. Let Z be a numeraire, so a strictly positive process. Define a new probability measure by −1

∫

PZ (A) = Z(0)

Z(T )B −1 (T )dP ∗ .

A

Then PZ is equivalent to P ∗ and S 1 (t)Z −1 (t) is an (Ft , P ∗ ) martingale. Exercise 4.2.9.

• Prove the above theorem.

• Compute the forward price of of a zero coupon bond of maturity T ∗ ≥ T to be F (p(T, T ∗ ), t, T ) =

p(t, T ∗ ) . p(t, T )

• Take the T-maturity bond price as a numeraire and prove that the price of the risky asset in terms of this numeraire is given by the forward price. • Take the bond price p(t, T ) as a numeraire and compute the risk neutral measure for the bond. (Note that the measure change is only defined on FT .) The price of S 1 is given by the forward price F (t, T ) which then must be a martingale w.r.t. (Ft , Pp(t,T ) =: Pp ). So there is a martingale representation dF (t, T ) = σF (t, T )F (t, T )dwp (t), where wp is a standard Brownian motion under Pp . • Take S 1 as a numeraire. In terms of units of S 1 the value of a T-maturity bond turns out to be

1 F (t,T

=: F −1 (t, T ). Show that in the above martingale

representation σF = σF −1

232

CHAPTER 4. BONDS AND INTEREST RATE DERIVATIVES and express the Brownian motion in this representation wS by wp .

• Derive the price of a European Call option by making use of the forward price.

4.3

Term Structure

For a random interest rate r with the properties above we consider the associated zero coupon bond with time t price p(t, T ) which under the risk neutral measure (we assume that it exists!) is given by p(t, T ) = Bt E ∗ [BT−1 |Ft ]. Let us assume that zero coupon bonds are traded in the market and we will see in this section how these zeroes can be used to calibrate the market. • A term structure model is a mathematical model for

Definition 4.3.1.

the prices p(t, T ) for all 0 ≤ t ≤ T ≤ T ∗ . ) • R(t, T ) = − log(Tp(t,T −t) is called the yield. For fixed t the yield curve is given

as the graph of R(t, ·) against T . Note that

p(t,T ) Bt

are P ∗ -martingales for all T . p(t, T ) is a positive process

so that it must be of the form p(t, T ) = µ(t, T )p(t, T )dt + σ(t, T )p(t, T )dwt and the discounted version becomes: ( d

p(t, T Bt

) = (µ(t, T ) − r(t))

p(t, T ) p(t, T ) dt + σ(t, T ) dwt . Bt Bt

As now p(t, T )Bt−1 is a P ∗ -martingale we must have: µ(t, T ) ≡ r(t). In this way proposing a term structure means proposing an interest rate. We just give some names often used in economics literature:

4.3. TERM STRUCTURE

233

• The statement

Definition 4.3.2.

∫

∗

T

p(t, T ) = E [exp(−

r(u)du)|Ft ] t

is called the local expectation hypothesis. • If there is a probability measure P r such that ∫ T −1 r (p(t, T )) = E [exp( r(u)du)|Ft ] t

then we speak of the return to maturity expectations hypothesis. • The yield to maturity expectations hypothesis states that there is a P r such that ∫ p(t, T ) = exp(−E r [(

T

r(u)du)|Ft ])

t

Exercise 4.3.3. Give an intuitive interpretation for the above notions. In the following sections we shortly present some familiar models for the interest rate and so for the term structure:

4.3.1

The Vasicek Model

The Vasicek model proposes a mean reverting Ornstein-Uhlenbeck-like process given under P ∗ by drt = a(b − rt )dt + σdwt , 0 < r0 , a, b, σ. −2at

The solution rt is a N (e−at (r0 + b(eat − 1)), σ 2 ( 1−e2a

))-random variable and

so a zero coupon bond in this model is p(t, T ) = e−b(T −t) E ∗ [e−

∫T t

X(u)du

|Ft ]

where X(u) = r(u) − b. So X(u) is the solution of the classical OrnsteinUhlenbeck equation dX = −aXdt + σdw, X(0) = r(0) − b. Now let

∫t

Φ(t, x) = E ∗ [e

0

X(u,x)du

]

(4.3.1)

234


where X(u, x) is the solution of 4.3.1 with initial X(0, x) = x: ∫ t −au X(u, x) = e (x + σeas dws ). 0

This is a continuous Gaussian process. After a lengthy derivation we then arrive at the representation p(t, T ) = e−b(T −t) Φ(T − t, r(t) − b) = exp(−(T − t)R(T − t, r(t))). So we have expressed the zeroes by means of a sort of artificial interest rate R(T − t, r(t)) between t and T . R∞ = limt→∞ R(t, r) = b −

σ2 2a2

can be seen as the long term interest rate.

Note that R∞ does not depend on the instantaneous rate rt what practitioners consider the weakness of this model.

4.3.2

The Hull-White Model

Here the short rate process is given by drt = (α(t) − β(t)rt )dt + σ(t)dwt , r0 > 0 where α, β, σ are deterministic functions of t. Let us have a little exercise in probability theory to recall some properties of normal random variables and Gaussian processes: Exercise 4.3.4. •

∫T 0

• r is a Gaussian process. Compute mean and covariance.

rs ds is a normal random variable. Compute mean and variance.

∫T ∫T • Use this to derive: p(0, T ) = exp(−E[ 0 rs ds] + 21 var[ 0 rs ds]). Express this as exp(−r0 C(0, T ) − A(0, T )) • p(t, T ) = exp(−rt C(t, T ) − A(t, T )) • dp(t, T ) = rt p(t, T )dt − p(t, T )σ(t)C(t, T )dwt Bond Options Consider times 0 ≤ t ≤ T1 ≤ T2 and remember that in the framework of the last section r(T1 ) is Gaussian with -say- mean m1 and variance σ12 . Furthermore

4.3. TERM STRUCTURE

235

∫ T1

rs ds is Gaussian with mean -say- m2 and variance σ22 . The covariance of ∫T r(T1 ) and 0 1 rs ds is given by -say- ρσ1 σ2 . 0

With this we have the density of the distribution of ∫ T1 rs ds), (r(T1 ), 0

say f (x, y). Let us now consider the European option on p(T1 , T2 ) with expiration time T1 and strike K the price of which at time 0 is given E ∗ [e−

p(0) = =

∫ T1 0

r(u)du (p(T , T ) 1 2

∫∞ ∫∞

−y −∞ −∞ e (exp(−xC(T1 , T2 )

− K)+ ]

(4.3.2)

− A(T1 , T2 )) − K)+ f (x, y)dxdy. (4.3.3)

To determine the price at time t compute the density of ∫ T1 (r(T1 ), rs ds), t

say f (t, x, y) to find out that this density now depends on rt . This shows that the t-price is stochastic: p(t) = E ∗ [e− = E ∗ [e−

∫ T1 t

∫ T1 t

r(u)du (p(T , T ) 1 2

− K)+ |Ft ]

(4.3.4)

r(u)du (p(T

− K)+ |rt ].

(4.3.5)

1 , T2 )

This quantity can now be expressed as an integration w.r.t. f (t, x, y), what is a lengthy, boring thing to do ...: a nice exercise. The Hull-White model can be used to give a closed form of the price of the bond option. Also the parameters of the model can be estimated so the initial yield curve is matched exactly. The weaknesses of this model lie in the fact that all bond prices for all T are perfectly correlated. Furthermore the short rate will take negative values with positive probability and the bond price can exceed 1. So this includes the undesirable year-zero joke!

4.3.3

The Cox-Ingersoll-Ross Model

The C-I-R model proposes the following equation for r: drt = ( for r(0) > 0. We have:

√ nσ 2 − αrt )dt + σ rt dwt 4

(4.3.6)

236


• For n = 1 P (rt > 0) = 1 but P (there are infinitely many times t > 0 for which r(t) = 0). • For n ≥ 2 P (there is at least one time t > 0 for which r(t) = 0) = 0. Exercise 4.3.5.

• Start with n standard Brownian motions wi and define

n Ornstein-Uhlenbeck processes X i (t, x) as in 4.3.1 with initial X i (0). Then

∑

r(t) :=

(X i )2 (t)

i=1,··· ,n

is the process 4.3.6 for the Brownian motion ∑ ∫

w(t) :=

t

i=1,··· ,n 0

X i (u)dwi (u) √ . r(u)

• The C-I-R process is a Bessel process. So we define: Definition 4.3.6. A Cox-Ingersoll-Ross process is a solution of a SDE √ drt = (a − brt )dt + σ rt dwt , r(0) = r0 . From the above considerations we conclude that this CIR process is useful only if a ≥

σ2 2

so that n ≥ 2.

Write r0,t (x) for the solution of the CIR process with initial x. By computing again the density of ∫ (r0,t (x),

t

ru,t (x)du) 0

we can prove that the price of the zero coupon bond at time zero is ∫ p(0, T ) = E[exp(−

T

ru (x)du)] = exp(−aϕ0,1 (0, T ) − r0 (x)ψ0,1 (0, T )),

0

where T (γ+b) 2

2γe ϕ0,1 (0, T ) = − σ22 log( γ−b+e γT (γ+b) )

ψ0,1 (0, T ) = γ=

2(eγT −1) γ−b+eγT (γ+b)

√

b2 + 2σ 2 .

(4.3.7) (4.3.8) (4.3.9)

4.3. TERM STRUCTURE

237

The price of the zero coupon bond at time t then is: p(t, T ) = exp(−aϕ0,1 (T − t) − r0 (x)ψ0,1 (T − t)). The price of the European Call with expiration time T on the zero coupon bond p(t, T ∗ ) at time 0 is given by: p(0) = p(0, T ∗ )F 4a ,ξ1 ( σ2

r∗ r∗ ) − Kp(0, T )F 4a ,ξ2 ( ), σ2 K1 K2

where F solves ∂F ∂F 1 ∂2F = (a − by) − µyF + σ 2 y 2 . ∂t ∂y 2 ∂y ξi , Ki are constants.

4.3.4

The Heath-Jarrow-Morton Model

Let 0 ≤ t ≤ T ≤ T + ϵ ≤ T ∗ where t is today. We consider the following contract: We want to borrow 1 Euro at the future time T and repay it with interest at time t + ϵ. The interest rate to be paid must be agreed today, so it must be Ft -measurable. This transaction can be approximated by buying today a T -maturity zero for p(t, T ) and shorting an amount

p(t,T ) p(t,T +ϵ)

of (T +ϵ)-maturity

zeroes. The cost of this portfolio at time t is p(t, T ) −

p(t, T ) p(t, T + ϵ) = 0. p(t, T + ϵ)

With this we receive 1 Euro at time T for the T -maturity zero and at time T + ϵ we must pay p(t, T ) p(t, T + ϵ) for the (T + ϵ)-maturity zeroes. So the interest rate we pay is 1 R(t, T, T + ϵ) = − [log p(t, T + ϵ) − log p(t, T )], ϵ and so we define: Definition 4.3.7. The instantaneous interest rate for money borrowed at time T agreed upon at t ≤ T is the forward rate f (t, T ).

238

CHAPTER 4. BONDS AND INTEREST RATE DERIVATIVES In fact we have f (t, T ) = limt↓0 R(t, T, T + ϵ) ∂ − ∂T log p(t, T )

= So:

∫

(4.3.10) (4.3.11)

T

p(t, T ) = E[exp(−

f (t, u)du)] t

(note that p(t, t) = 1). We see that here the forward rate has taken the place of the short rate r and so the agreement is to pay the interest rate f (t, u) at time u ∈ [t, T ]. Exercise 4.3.8. Prove that f (t, t) = r(t). The HJM model now proposes a SDE for the dynamics of f (t, T ): df (t, T ) = α(t, T )dt + σ(t, T )dwt . Note that the coefficients must be adapted so that f (t, u), u ∈ (t, T ] is Ft ∫T measurable. Consider the processes X(t) := − t f (t, u)du and p(t, T ) = eX(t) . As the discounted p(t, T )-process is a martingale we finally find: ∫ T ∫ 1 T α(t, u)du = ( σ(t, u))2 2 t t or equivalently

∫ α(t, T ) = σ(t, T )

T

σ(t, u)du. t

If now the underlying probability measure is not risk neutral we have the following Theorem 4.3.9. (Heath-Jarrow-Morton): For each T ∈ (0, T ∗ ] let α(u, T ) and σ(u, T ) be adapted processes with σ > 0 and let f (0, T ) be a deterministic function. The instantaneous forward rate is defined by ∫ T ∫ T f (t, T ) = f (0, T ) + α(u, T )du + σ(u, T )dwu . 0

0

Then the term structure model determined by f (t, T ) does not allow arbitrage iff there is an adapted process θ(t) such that ∫ T α(t, T ) = σ(t, T ) σ(u, T )du + σ(t, T )θ(t), 0 ≤ t ≤ T ≤ T ∗ , t

4.3. TERM STRUCTURE and the process

239 ∫

T

exp(−

θ(u)dwu −

0

1 2

∫

T

θ2 (u)du) 0

is an (Ft , P )-martingale. Exercise 4.3.10. Note that the requirement of the theorem is that if θ exists it must be independent of T . Compute the rate of return of the bond under the physical measure and compare it to the interest rate. The rate of return above r(t) is

∫ −θ(t)

T

σ(t, u)du. t

4.3.5

Remarks

There are many more attempts to model the term structure. An interesting approach is described by Elliott and Kopp [9] basing on a Markov chain which models the short rate. Further approaches are described by Biagini.

240


Chapter 5

Continuous Time CAPM 5.1

Conditional Markets

In this chapter we are going to consider conditional properties of the market M = (Ω, (S, 1)), which reflect its dynamic properties. The main purpose of this (p)

chapter is to provide a link between the general model where S ∈ Aloc and the (p)

semimartingale market model where we assume S ∈ Sloc , which we will consider in the next chapter. We will then be mainly concerned with the convergence of simple self-financing hedging strategies. Many of the tools needed for this are developed in the present chapter. One could do this just for a semimartingale market. However, it is little extra work to start with the general market model, to see how far we get and to add new restricting assumptions along the way only at need. In the end we will have added strong enough assumptions, implying (p) the existence of a modification S˜ ∈ Sloc of S. We know of two results concerning

general markets: [24], Theorem 3.1, and [21], Theorem 7.2. The key idea of this chapter goes back to [60] and [16]: Assume Dq ̸= ∅ and set T1 := inf{t ≥ 0|Ztopt = 0}, where Z opt :=

Z min E[Z min |F0 ]

and we assume

E[Z min |F0 ] ̸= 0 for the moment. By optimality of Z min , Z opt will remain zero on [T1 , ∞). This makes it difficult, to study the convergence of a simple self-financing hedging strategy on [T1 , ∞). To overcome this problem, we will study conditional properties of the market M by considering the market

T1M.

The first step is now to find a condition such that ZTopt ̸= 0. This will allow 1− to express Z opt as a stochastic exponential. A counter example will show that 241

242

CHAPTER 5. CONTINUOUS TIME CAPM

this is not always possible. The next step is to consider the convergence in the market

T1M.

There we encounter the same problem, but recursively defining

Tn+1 := inf{t ≥ Tn |TnZtopt = 0}, where

TnZ opt

:=

TnZ min E[TnZ min |FTn ]

Dq (TnM) is the Lq -optimal martingale measure for the market

and

TnM,

TnZ min

∈

we get an

increasing sequence of stopping times. E[TnZ min |FTn ] ̸= 0 will be ensured by a conditional version of the SLPp assumption. The convergence behaviour of Tn will turn out to be crucial. We will derive necessary and sufficient conditions ensuring Tn → ∞ and show by an example what can typically go wrong. This corrects [60], Theorem 4.20. For p = 2, these conditions can be expressed in terms of a conditional version of the price for risk derived in the previous chapter. If all TnZ opt can be expressed as a stochastic exponential and if Tn converges to ∞, then there exists an exponential measure, a notion going also back to [60] and [16]. The idea is that the family {τZ opt | τ stopping time }, where τZ opt

:=

τZ min E[τZ min |Fτ ]

and τZ min denotes the Lq -optimal local martingale measure

in the market τ M, satisfies a certain consistency property, which will allow to calculate all τZ opt from one single object, namely τZ opt = E(τ N opt ) for a local martingale N opt . As we will see in the next section, optimality of Z min implies Z min Z0min

for Z0min ̸= 0 to satisfy a conditional optimization criterion with respect to

F0 . Similarly,

Z min Zτmin

for Zτmin ̸= 0 satisfies a conditional optimization criterion

with respect to Fτ , reflecting the consistency property.

5.2

Conditional Utility Maximization

Let c ∈ C p , 1 < p < ∞, set U := −c and denote by cˇ the conjugate Young function of c. Assume Dq ̸= ∅. Consider the conditional version of the expected terminal utility maximization problem (1.6.12): E[U (V sup )|F0 ] = ess supV ∈1+V¯ p E[U (V )|F0 ],

(5.2.1)

for a V sup ∈ 1 + V¯ p . Let V max denote the solution to (1.6.12) with respect to the utility function U and initial value x = 1. Proposition 5.2.1. V sup equals V max .

5.2. CONDITIONAL UTILITY MAXIMIZATION

243

Proof. The proof is a conditional version of the proof of Proposition 1.6.11: Let V ∈ V¯ p . We have U (V max + V ) ≥ U (V max ) − c′ (V max )V , hence E[U (V max + V )|F0 ] ≥ E[U (V max )|F0 ] − E[c′ (V max )V |F0 ] = E[U (V max )|F0 ], by Lemma 1.4.6 and since by Proposition 1.6.11

c′ (V max ) E[c′ (V max )]

∈ Dq . The assertion

follows now from the strict concavity of U . Define Z min := E[c′ (V max )|F· ]. By Proposition 1.6.11, Z min ∈ y˜Dq , where y˜ := E[c′ (V max )] > 0 is a solution to the dual problem (1.6.13). Consider the corresponding conditional dual problem E[ˇ c(Z inf )|F0 ] = ess inf Z∈˜yDq ∩Fy E[ˇ c(Z)|F0 ],

(5.2.2)

for a Z inf ∈ y˜Dq ∩ Fy , where y ∈ Lq is an F0 -measurable random variable such that E[y] = y˜ and Fy := {f ∈ Lq |E[f |F0 ] = y}. Proposition 5.2.2. Z min solves (5.2.2) for y := E[c′ (V max )|F0 ] = Z0min ∈ Lq and y˜ := E[y]. min ) + c min )(Z − Z min ), Proof. Let Z ∈ y˜Dq ∩ Fy . We have cˇ(Z∞ ) ≥ cˇ(Z∞ ˇ′ (Z∞ ∞ ∞

hence min min E[ˇ c(Z∞ )|F0 ] ≥ E[ˇ c(Z∞ )|F0 ] + E[V max (Z∞ − Z∞ )|F0 ] min = E[ˇ c(Z∞ )|F0 ] + 1(Z0 − Z0min ) min = E[ˇ c(Z∞ )|F0 ].

Proposition 5.2.3. Set A := {y = 0}. We have 1A ∈ V¯ p and for B ∈ F0 we have 1B ∈ V¯ p iff 1B ≤ 1A almost surely. Proof. We have by Lemma 1.6.9, 7.: min 0 ≤ E[c(V max )|F0 ] + E[ˇ c(Z∞ )|F0 ] = 1Z0min = y,

(5.2.3)

and we find V max = 0 on A. By Lemma 1.4.8, this implies 1A ∈ V¯ p . Let 1B ∈ V¯ p for B ∈ F0 . Since V := 1B c V max + 1B (1 − 1B ) ∈ 1 + V¯ p and

244


∥V ∥Lp ≤ ∥V max ∥Lp , we have by optimality 1B ≤ 1A almost surely. Conversely, again by Lemma 1.4.8, 1B ≤ 1A a.s. implies 1B ∈ V¯ p . This result motivates the following definition: Definition 5.2.4. For 1 ≤ p ≤ ∞, we say that in M the strong law of one price for Lp -integrable strategies, holds everywhere in F0 , (SLPp (F0 )), if the SLPp holds in the restricted market M|A for all A ∈ F0 with P (A) > 0. See [60], Definition 4.3, for the equivalent F0 -quasi-no-arbitrage condition. Corollary 5.2.5. Let 1 0 almost surely. Proof. This follows immediately from Proposition 5.2.3. Lemma 5.2.6. For 1 ≤ p ≤ ∞, the SLPp (F0 ) holds iff there exists a Z ∈ Dq with Z0 > 0 almost surely. Proof. We prove this by contradiction. Assume the SLPp (F0 ) not to hold. Then there exists an A ∈ F0 with P (A) > 0 and 1A ∈ V¯ p . Hence for any Z ∈ Dq , Z0 > 0 does not hold a.s. everywhere on A since E[Z∞ 1A ] = E[Z0 1A ] = 0. Note that E[(sgn(Z0 )1A + 1Ac )Z∞ ] ≥ 1. Hence for any Z ∈ Dq , Z + := Since E[Z + 1A |F0 ] =

(sgn(Z0 )1A + 1Ac )Z∞ ∈ Dq . E[(sgn(Z0 )1A + 1Ac )Z∞ ]

|Z0 |1A E[(sgn(Z0 )1A +1Ac )Z∞ ] ,

E[Z + 1A ] = 0 implies Z0 = 0 on A

for all Z ∈ Dq . Conversely, if for all Z ∈ Dq , Z0 > 0 does not hold almost surely, then for all Z ∈ Dq with Z0 ≥ 0, P ({Z0 = 0}) > 0 holds. Consider the family of random variables O := {1{Z0 >0} |Z ∈ Dq , Z0 ≥ 0}. 1A , 1B ∈ O implies 1A∪B ∈ O, hence by [50], Theorem A.3, the essential supremum of O exists and moreover, there exists a non-decreasing sequence 0 ≤ Z n ∈ Dq such that 1{Z0n >0} ∈ O converges to ess sup O. For an appropriately chosen sequence ∑∞ n =: Z ˜ ∈ Dq and λn of strictly positive constants we have 0 ≤ n=0 λn Z n 1∪∞ = 1{Z˜0 >0} ∈ O. Hence for all Z ∈ D we have 1{Z + >0} ≤ 1{Z˜0 >0} n=0 {Z0 >0} 0

a.s., implying Z0 = 0 a.s. on {Z˜0 = 0} for all Z ∈ Dq and since P ({Z˜0 = 0}) > 0 by assumption, we have 1{Z˜0 =0} ∈ (Dq )◦ = V¯ p . The assertion follows now from Proposition 1.4.10.


245

We now specialize these results to the case of the isoelastic utility functions Up , 1 < p < ∞. Corollary 5.2.7. Assume the SLPp (F0 ) to hold. Then the conditional optimization problems (5.2.1), resp. (5.2.2) have unique solutions V max ∈ 1 + V¯ p , min = sgn(V max )|V max |p−1 ∈ y min |F ] ∈ Lq resp. Z∞ ˜Dq ∩ Fy , where y := E[Z∞ 0

and y˜ = E[y]. Furthermore, we have y = E[|V max |p |F0 ] > 0. min |F ] = Z min = y. The other Proof. Note that E[|V max |p |F0 ] = E[V max Z∞ 0 0

assertions follow directly from the Propositions 5.2.1, 5.2.2 and 5.2.5.

As already mentioned in the introduction to this chapter, we want to show a certain consistency property for the family {τZ opt | τ stopping time }. To do this, we will first characterize Z opt as the unique solution of a modified version of problem (5.2.2). Formally, we can state problem (5.2.2) for y = 1, but since in general

Z min , E[Z min |F0 ]

which we expect to be the solution, is not Lq -integrable, there

might not exist a solution in Dq ∩F1 . To resolve this difficulty, we will introduce in the next subsection extended conditional expectations. In Subsection 5.3 we will then consider a modified version of problem (5.2.2), with

Z min E[Z min |F0 ]

as its

unique solution if the SLPp (F0 ) holds.

5.2.1

Extended Conditional Expectations

We will need a slightly generalized conditional expectation operator. This extended conditional expectation coincides with the generalized conditional expectation on its domain of definition, but has the advantage, that many useful properties of the ordinary condition expectation, like convergence properties and important inequalities, carry over without difficulties.

Definition 5.2.8. Let 1 ≤ p ≤ ∞ and let X be an F-measurable integrable random variable and τ a stopping time. We call X Lpτ -integrable and write X ∈ Lpτ if there exists an Fτ -measurable bounded random variable Y with Y > 0 almost surely and such that XY ∈ Lp .

246

CHAPTER 5. CONTINUOUS TIME CAPM Note that X ∈ L∞ τ iff there exists an almost surely finite Fτ -measurable

random variable Y such that |X| < Y almost surely. For X ∈ L1τ choose bounded Fτ -measurable random variables Y, Y˜ > 0 almost surely and such that XY, X Y˜ ∈ L1 . Then XY Y˜ ∈ L1 and [ [ [ [ ] ] ] ] E X Y˜ Fτ E XY Fτ E X Y˜ Y Fτ − E X Y˜ Y Fτ − = = 0, Y Y˜ Y Y˜ Hence we can extend the conditional expectation operator: Definition 5.2.9. For a stopping time τ and X ∈ L1τ , we define the extended conditional expectation of X with respect to Fτ as ] [ E XY Fτ Eτ [X] := , Y

(5.2.4)

where Y is a bounded and Fτ -measurable random variable with Y > 0 almost surely and such that XY ∈ L1 . Note that Eτ [X] < ∞ almost surely for X ∈ L1τ and Eτ [·] coincides with E[·|Fτ ] on L1 . For Y such that E[Y ] = 1, Y defines a probability measure Q equivalent to P with Fτ -measurable Radon-Nikodym derivative Y =

dQ dP

and

Eτ [·] is just the conditional expectation with respect to Q, see [51], Lemma 3.5.3. Proposition 5.2.10. Let 1 ≤ p ≤ ∞. X ∈ Lpτ iff for all ϵ > 0 there exists an A ∈ Fτ with P (A) > 1 − ϵ such that X1A ∈ Lp . Proof. Let X ∈ Lpτ and choose an Fτ -measurable Y > 0 such that XY ∈ Lp . Set An := {Y > δn } for a sequence of constants strictly decreasing to 0. Then there exists for 0 < ϵ < 1 an n ≥ 1 such that P (An ) > 1 − ϵ and we have |X|1An ≤

|XY | δn

∈ Lp . Conversely, assume that we can find for each member

of a sequence ϵn of constants, strictly decreasing to 0, a set An ∈ Fτ with c P (An ) > 1 − ϵn such that X1An ∈ Lp . Set Bn := ∪ni=1 Ai and Cn := An ∩ Bn−1 ∑n p for n ≥ 1. We then have P (Bn ) = i=1 P (Ci ) → 1 and X1Cn ∈ L for ∑∞ −i 1Cn all n ≥ 1. For 1 ≤ p < ∞ set Y := i=1 2 1∨E[|X|p 1Cn |Fτ ] , resp. for p = ∑∞ 1Cn ∞ set Y := i=1 1∨Yn , for Fτ -measurable Yn > |X|1Cn . Y is a bounded,


247

Fτ -measurable and almost surely strictly positive random variable such that XY ∈ Lp . In analogy the notion of σ-finiteness for measures, one could call X ∈ L1τ , resp. X ∈ Lpτ , σ-integrable, resp. σ-Lp -integrable, with respect to Fτ . Let X be an F-measurable random variable. For an arbitrary stopping time τ , there exists a generalized conditional expectation operator, also denoted as Eτ [·], such that Eτ [X] can be defined as an Fτ -measurable, (−∞, ∞]-valued τ,p random variable. For 1 ≤ p ≤ ∞, denote by OX := OX the family of all

indicator functions 1A for A ∈ Fτ satisfying X1A ∈ Lpτ . Lemma 5.2.11. There exists a set AX := Aτ,p X ∈ Fτ such that 1AX ∈ OX and for all 1B ∈ OX we have 1B ≤ 1AX almost surely. 1AX is unique as an equivalence class of almost surely identical random variables. Proof. Note that OX ̸= ∅ and 1A , 1B ∈ OX implies 1A∪B ∈ OX . By [50], Theorem A.3, there exists a non-decreasing sequence of sets An ∈ Fτ such that 1An ∈ OX converges to ess sup OX . Set AX := ∪∞ n=0 An ∈ Fτ . If P (AX ) = 0, then 1AX ∈ OX . Assume P (AX ) > 0. Since X1An ∈ Lpτ , there exists by Proposition 5.2.10 an increasing sequence Bin , i ≥ 1 of Fτ -measurable subsets of An such that P (An \ Bin ) < 2−n−i and X1Bin ∈ Lp . For Bi := ∪in=1 Bin we then have P (Bi ) → P (AX ) for i → ∞ since ∪in=0 An ↗ AX , and X1Bi ∈ Lp (Ω|AX ), hence again by Proposition 5.2.10, X|AX ∈ Lpτ (Ω|AX ) or equivalently X1AX ∈ Lpτ and we find 1AX ∈ OX . By definition of the essential supremum we have 1B ≤ 1AX almost surely for all 1B ∈ OX , implying the uniqueness of the equivalence class 1AX . Definition 5.2.12. For a stopping time τ and an F-measurable random variable X, we define the generalized conditional expectation of X with respect to Fτ as

[ ] Eτ [X] := Eτ 1Aτ,1 X + ∞1(Aτ,1 )c . X

Definition 5.2.13.

(5.2.5)

X

1. A process X ∈ A such that X∞ exists, Xt ∈ Lp0 , 1 ≤

q ≤ ∞ for all t ∈ [0, ∞] and Es [Xt ] = Xs for all 0 ≤ s ≤ t ≤ ∞, is

248

CHAPTER 5. CONTINUOUS TIME CAPM called an Lp0 -integrable martingale, resp. an extended uniformly integrable martingale for p = 1.

2. For X ∈ Lpτ , 1 ≤ p < ∞ we define the conditional Lp -norm (in τ Ω) by 1

∥X∥Lpτ := ∥X∥Lpτ (Ω) := (Eτ [|X|p ]) p = ∥X∥Lp0 (τ Ω) ∈ Lpτ .

(5.2.6)

For arbitrary F-measurable X, ∥X∥Lpτ is defined using the generalized expectation operator: ∥X∥Lpτ

( )1 p := ∥X∥Lpτ (Ω) := Eτ [|X|p 1Aτ,p ] + ∞1(Aτ,p )c (5.2.7) X

=

X

∥X1Aτ,p ∥Lp0 (τ Ω) + ∞1(Aτ,p )c . X

X

∞ 3. For X ∈ L∞ τ we define the conditional L -norm (in τ Ω) by

∥X∥L∞ τ

{ } := ∥X∥L∞ := ess inf Y ≥ |X| Y Fτ -measurable τ (Ω) ∥X∥L∞ ∈ L∞ τ . 0 (τ Ω)

=

(5.2.8)

For arbitrary F-measurable X, we define ∥X∥L∞ := ∥1Aτ,∞ X∥L∞ + ∞1(Aτ,∞ )c . τ τ X

X

Let 1 ≤ p ≤ ∞. Note that X ∈ Lpτ˜ implies X ∈ Lpτ for all stopping times τ˜ ≤ τ . For X ∈ Lpτ and Y ∈ Lqτ we have XY ∈ L1τ . Furthermore, the extended expectation operator is linear in the following sense: for X, Y ∈ L1τ and Fτ measurable random variables f, g, we have Eτ [f X+gY ] = f Eτ [X]+gEτ [Y ]. We also have the extended Jensen inequality |Eτ [X]| ≤ Eτ [|X|] and the extended Hölder inequality holds: ∥XY ∥L1τ ≤ ∥X∥Lpτ ∥Y ∥Lqτ ,

(5.2.9)

for X ∈ Lpτ , Y ∈ Lqτ and 1 ≤ p ≤ ∞. For 1 < p < ∞, this follows from the conditional Hölder inequality, for p = 1 and by symmetry for p = ∞, observe,

that |XY | ≤ |X|∥Y ∥L∞ implies ∥XY ∥ 1 ≤ X∥Y ∥L∞ 1 = ∥X∥L1 ∥Y ∥L∞ . L τ τ τ L τ τ τ

For a sequence Xn ∈

Lpτ

˜ ∈ Lpτ , an extended version of the such that |Xn | ≤ X

dominated convergence theorem for conditional expectations holds: Eτ [Xn ] → Eτ [X] almost surely if Xn → X almost surely. We derive two auxiliary results:

5.3. LQ 0 -INTEGRABLE MARTINGALE MEASURES

249

Lemma 5.2.14. X is an Lp0 -integrable martingale iff for arbitrary 0 < ϵ < 1 there exists an A ∈ F0 with P (A) > 1 − ϵ, such that X|A ∈ U˜p (Ω|A ). If in ˜ addition Xτ ∈ L1 for some stopping time τ , then X ∈ U(Ω). Proof. Let X be an Lp0 -integrable martingale. Since X∞ ∈ Lp0 , we can find an F0 -measurable bounded Y > 0 such that Y X∞ ∈ Lp and E[Y X∞ |Fs ] = Y Xs , hence Y X ∈ U˜p . This implies for ϵn > 0 with P ({Y > ϵn }) > 0 that X ∈ U˜p (Ω|{Y >ϵn } ). If ϵn ↘ 0 as n → ∞, then we have P ({Y > ϵn }) → 1. Conversely, observe that X∞ exists almost surely. The equivalence follows now from Proposition 5.2.10. If Xτ ∈ L1 , then E[Eτ [X∞ ]] = E[Xτ ] < ∞, implying ˜ X∞ ∈ L1 , hence X ∈ U(Ω). This result suggests to call an extended uniformly integrable, resp. Lp0 integrable, martingale a σ-uniformly integrable, resp. a σ-uniformly Lp -integrable, martingale with respect to F0 . This result can also be interpreted in the following way: The property of a process to be an extended uniformly integrable, resp. Lp0 -integrable, martingale is invariant under equivalent change of probability measure from P to Q for F0 -measurable

dQ dP .

Lemma 5.2.15. Let X ∈ Lp0 , then the process ∥X∥Lp· : t 7→ ∥X∥Lpt , 0 ≤ t < ∞ is a semimartingale satisfying ∥X∥Lp∞ = X. Proof. For 1 ≤ p < ∞, this follows from the corresponding property of the conditional expectation, since our filtration satisfies the usual assumptions. For p = ∞, note that ∥X∥L∞ is decreasing and obviously ∥X∥L∞ = X holds. · ∞ Since ∥X∥Lpt+ := limϵ↘0 ∥X∥Lpt+ϵ is Ft -measurable by the right-continuity of the filtration and since ∥X∥Lpt+ ≤ ∥X∥Lpt , we have equality, implying ∥X∥L∞ · to be RCLL, hence the assertion follows.

5.3

Lq0 -integrable Martingale Measures

Definition 5.3.1. For 1 ≤ q ≤ ∞, the space of signed Lq0 -integrable local martingale measures is defined as { } Z q q q ˜ ˜ D := D (M) := Z ∈ D (M), E[Z|F0 ] ̸= 0 a.s. ⊆ Lq0 . E[Z|F0 ]

(5.3.1)

250


˜q = Confer [60], Definition 4.5. Note that by Lemma 5.2.6, D ̸ ∅ if and only if the SLPp (F0 ) holds for M. Let 1 < q < ∞ and consider the extended conditional version of the dual problem (5.2.2) for the isoelastic utility functions Up =: −cp : cp (Z)], E0 [ˇ cp (Z inf )] = ess inf Z∈D˜ q E0 [ˇ

(5.3.2)

˜ q . Let V max ∈ 1 + V¯ p and Z min = sgn(V max )|V max |p−1 ∈ for a Z inf ∈ D ∞ min |F ] ∈ Lq and y y˜Dq ∩ Fy , where y := E[Z∞ ˜ = E[y], be as in the previous 0

subsection. Assume the SLPp (F0 ) to hold, hence y > 0 almost surely. We now prove a conditional version of Proposition 1.6.11 for the special case of the isoelastic utility functions: ˜ q admits a representation Proposition 5.3.2. If Z opt ∈ D Z opt =

c′p (V opt ) , E[c′p (V opt )|F0 ]

(5.3.3)

for some V opt ∈ 1 + V¯ p , then Z opt solves problem (5.3.2) uniquely and V opt is the unique solution to the problems (5.2.1), resp. (1.6.12). Furthermore, we have ∥V opt ∥Lp0 ∥Z opt ∥Lq0 = 1,

(5.3.4)

and ∥V opt ∥Lp0 ≤ 1, resp. ∥Z opt ∥Lp0 ≥ 1. ˜ q , we have cˇp (Z) ≥ cˇp (Z opt ) + cˇ′p (Z opt )(Z − Z opt ) = cˇp (Z opt ) + Proof. For Z ∈ D V opt cˇ′p (E[c′p (V opt )|F0 ]) (Z

− Z opt ). Taking extended conditional expectations we find [ ] V opt opt opt E0 [ˇ cp (Z)] ≥ E0 [ˇ cp (Z )] + E0 ′ (Z − Z ) cˇp (E[c′p (V opt )|F0 ]) = E0 [ˇ cp (Z opt )],

since E0 [Z] = E0 [Z opt ] = 1. Uniqueness follows from the strict convexity of cˇp . V opt is the unique solution to the problems (5.2.1), resp. (1.6.12) by Proposition 1.6.11 and Proposition 5.2.1. Similar as in Proposition 1.6.12, we have for y := E[c′p (V opt )|F0 ], 1 = E0 [V opt Z opt ] =

E0 [|V opt |p ] y

which implies (5.3.4). 1 ∈ 1 + V¯ p implies ∥V opt ∥Lp0 ≤ 1.

= y q−1 E0 [|Z opt |q ],

5.3. LQ 0 -INTEGRABLE MARTINGALE MEASURES

251

Confer this result with [60], Proposition 4.7. Our result is a little stronger insofar the representation (5.3.3) implies optimality of Z opt and V opt , a property we will need in the next subsection. Furthermore, the duality technique we have used in the above proof can be extended to the case 0 ̸= p < 1, see Section 5.11.1.

˜ q , then we call Definition 5.3.3. If problem (5.3.2) admits a solution Z opt ∈ D the pair (V opt , Z opt ) ∈ Lp × Lq0 the (p, q)-optimal pair for the market M. For fixed p, we will call (V opt , Z opt ) just the optimal pair for M. In general (V opt , Z opt ) depends on p. We derive some properties of the optimal pair: Theorem 5.3.4. For the market M the SLPp (F0 ) holds iff the optimal pair (V opt , Z opt ) for M exists. Then for all A ∈ F0 with P (A) > 0 the optimal pair for the market M|A exists and equals (V opt , Z opt )|A . Furthermore, for the optimal pair the relation (5.3.3) holds. Proof. Note that optimal pairs are unique by the strict convexity of cp and cˇp . Assume the SLPp (F0 ) to hold. Then the optimal pair (V opt , Z opt ) for M exists by Corollary 5.2.7 and we have a representation (5.3.3). By Lemma 1.4.8, this representation can be restricted to A, for all A ∈ F0 with P (A) > 0, hence the optimal pair for M|A exists and equals (V opt , Z opt )|A . Conversely, if the SLPp (F0 ) does not hold, then there exists by Proposition 1.4.10 an A ∈ F0 with P (A) > 0 such that Dq (M|A ) = ∅. By Lemma 1.4.8 we have 1A ∈ V¯ p , ˜ q = ∅ and the optimal pair for hence Z0 = 0 a.s. on A for all Z ∈ Dq , hence D M can not exists. ¯ = ∪i∈N Ai , with Ai ∈ F0 and optimal Corollary 5.3.5. Given a partition Ω pairs (Viopt , Ziopt ) for the markets M|Ai , for all i ∈ N such that P (Ai ) > 0, ∑ then i∈N 1Ai (Viopt , Ziopt ) is the optimal pair for M. If for all 0 < ϵ < 1 there exists an A ∈ F0 with P (A) > 1 − ϵ and such that the optimal pair exists for M|A , then the optimal pair exists for M.

252


Proof. Assume that the optimal pair (V opt , Z opt ) for M does not exist. Then the SLPp (F0 ) does not hold by the previous corollary. Then there exists an A ∈ F0 with P (A) > 0 such that the SLPp does not hold in M|A and there exists an i such that P (Ai ∩ A) > 0, implying the SLPp not to hold in M|Ai ∩A . Hence the SLPp (F0 ∩ Ai ) does not in M|Ai , which is a contradiction. Again by Theorem 5.3.4, (Viopt , Ziopt ) equals (V opt , Z opt )|Ai for all i ∈ N such that P (Ai ) > 0, hence the first assertion follows. Choosing a strictly decreasing sequence 1 > ϵn ↘ 0 and Bn ∈ F0 with P (Bn ) > 1 − ϵn and such that the ¯ optimal pair exists for M|Bn , we can construct a F0 -measurable partition of Ω satisfying the conditions of the first part of this corollary. Remark 5.3.6. The previous result basically means that we can construct optimal pairs by restricting and patching them together. Note that S τ ∈ A(p) for a (p)

(p)

stopping time τ , implies τ S ∈ Aloc (τ Ω) if S ∈ Aloc . Then the results derived for M so far, can be applied to τ M. If τ is an arbitrary stopping time and τn a local(p)

(p)

izing sequence for S ∈ Aloc , we still have τ S|{τn ≤τ 0, where pi is a sequence of strictly positive constants such ∑ that ∞ i=1 pi = 1. Define a process by St = 1 on [0, 1) and St = 1 ± K on (N × {±1}) × [1, ∞) for a constant 0 < K < 1. S is a strictly positive bounded ¯ F, (Ft )0≤t , P ) and M := (Ω, (S, 1)) is a market semimartingale on Ω := (Ω, model. By Corollary 5.3.5, we can find the optimal pair (V opt , Z opt ) for M by patching together the optimal pairs for the markets Mi := M|{i}×{−1,1} for k

k

opt = 1+i i ∈ N. Considering Mi , we easily find Z opt = 1+i k on {(i, 1)}, resp. Z 2 ( k2i )q−1 (1+ik )q−1 1 1+i opt q on {(i, −1)}, implying E[|Z | |F0 ] = 2q + on {i} × {−1, 1} 2q ik ∑ C 1 for 0 < ϵ < q − 1 and C −1 = ∞ for 1 < q < ∞. For pi := iq−ϵ n=1 iq−ϵ we find ( ) ∑∞ (1+ik )q−1 ∑ q−ϵ opt |q |F ] (i, 1) ≥ C = ∞ for k > q−1 . E[|Z opt |q ] = ∞ 0 i=1 i=1 pi E[|Z 2q iq−ϵ

5.3.2

Consistency of Lp -optimal Martingale Measures

Assume for M the SLPp (F0 ) to hold. Let V opt be the solution to problem n → V opt ∈ 1 + V ¯ p in Lp . (5.2.1) and let V n ∈ 1 + V p (M) be such that V∞

Denote by Z˜ opt the solution to problem (1.6.13), such that E[Z˜ opt ] = 1 and Z opt =

˜ opt Z . ˜ Z0opt

Set τ := inf{t|Ztopt = 0}. So far V opt is only a random variable.

opt We will now define a process V·opt on [0, τ ) such that V∞ = V opt on {τ = ∞}

and such that Vtn → Vtopt on [0, τ ) in a sense to be explained later. Define on [0, τ ) Vtopt :=

E[V opt Z˜ opt |Ft ] . Z˜ opt

(5.3.5)

t

opt By Lemma 1.4.6, Vτñ Zτõpt → E[V opt Z∞ |Fτ˜ ] in L1 for any stopping time τ˜.

We want to construct τ˜Z opt from Z(opt . The idea ) is as follows: Assume opt opt Vτõpt , Zτõpt > 0. We then expect the pair V opt , τ˜ Zopt to equal the optimal pair Vτ˜

Zτ˜

(τ˜ V opt ,τ˜ Z opt ) for τ M, since otherwise it should be possible to concatenate Vτñ and V˜ n → τ˜ V opt ∈ V¯ p (τ M), resp. Zτõpt and τ˜Z opt in such a way, that the limiting pair (V, Z) exists and ∥V ∥Lp < ∥V opt ∥Lp , resp. ∥Z∥Lq < ∥Z opt ∥Lq , contradicting the optimality of (V opt , Z opt ). However, beforehand it is not clear why

V opt Vτõpt

254


should be Lp -integrable. If we show

V opt Vτõpt

∈ V¯ p (τ M), then the representation

(5.3.3) for Z opt leads to a corresponding representation relating

V opt Vτõpt

and

opt τ ˜Z opt Zτ˜

,

and by Proposition 5.3.2 we find the desired result. We need some auxiliary results: Lemma 5.3.8. Let τ˜ be a stopping time. We have |Vτõpt | > 0 almost surely on τ < τ }. {|Zτõpt | > 0}. In particular, we have |Vτõpt | > 0 almost surely on {˜ {

Proof. Note that

} (ω, τ˜(ω))| ω ∈ {|Zτõpt | > 0} ⊆ [0, τ ) almost surely, hence

Vτõpt is defined on {|Zτõpt | > 0} and we have 0 < E[V

opt opt Z ˜∞ |Fτ˜ ]

=

Vτõpt Z˜τõpt

on

{|Zτõpt |

˜ opt |q |Z τ ˜ ˜ ∥Z opt ∥Lq

≤

opt q ˜∞ E[|Z | |Fτ˜ ] ˜ ∥Z opt ∥Lq

=

> 0}.

Lemma 5.3.9. Let τ˜ be a stopping time and set A := {|Zτõpt | > 0}. Assume P (A) > 0. Then for P (A) > ϵ > 0 there exists a B ∈ Fτ˜ , B ⊆ A with n

P (B) > P (A) − ϵ and a subsequence V nj , j ∈ N, such that Vτ˜ j ̸= 0 on B and n

the restriction of

V∞j n Vτ˜ j

to B converges to the restriction of

V opt Vτõpt

to B in Lp (τ˜ Ω|B ).

opt Proof. Since Vτñ Z˜τõpt → E[V opt Z˜∞ |Fτ˜ ] in L1 we find a subsequence V nj , j ∈ opt N, such that Vτñ Zτõpt → E[V opt Z∞ |Fτ˜ ] pointwise almost surely. Then on A, n

n

Vτ˜ j → Vτõpt pointwise almost surely. This implies limn→∞ P ({supj≥n |Vτ˜ j − Vτõpt | > K}∩A) = 0 for all K > 0. We can therefore assume nj to be chosen such n

that P ({supj≥0 |Vτ˜ j − Vτõpt | > K} ∩ A)
0 satisfying P ({|Vτõpt | > n

K} ∩ A) > P (A) − 2ϵ . Set B := {|Vτõpt | > K} ∩ A ∩ {supj≥0 |Vτ˜ j − Vτõpt | ≤ K}. n

We then have Vτ˜ j ̸= 0 on B and P (B) > P (A) − ϵ. On B,

1 n Vτ˜ j

is bounded,

hence the assertion follows. (p)

Let τ˜k be a localizing sequence for S ∈ Aloc . Lemma 5.3.10. With B and τ˜ as in the previous lemma and(Ck := τk ≤ ) B ∩ {˜ opt τ˜ < τ˜k+1 }, we have for all k ≥ 1 such that P (Ck ) > 0, V opt ∈ 1+ Vτ˜

V¯ p (τ˜ M|Ck ). Proof. On Ck we have

V nj n Vτ˜ j

|Ck

n

= 1+

V nj −Vτ˜ j n Vτ˜ j

∈ 1 + V p (τ˜ M|Ck ), where nj is the

subsequence constructed in the previous lemma, from which now the assertion follows.

5.3. LQ 0 -INTEGRABLE MARTINGALE MEASURES ( Lemma 5.3.11. With B, Ck ,˜ τ and k as above,

255

V opt τ˜ Z opt , Vτõpt Zτõpt

) is the optimal |Ck

pair for the market τ˜ M|Ck . In particular for τ˜ M|Ck the SLPp (Fτ˜ ∩ Ck ) holds. ( Proof. Since

opt τ ˜Z Zτõpt

) |Ck

˜ q (τ˜ M|C ) admits a representation (5.3.3), the as∈D k

sertion follows from Proposition 5.3.2. Set Ak := A ∩ {˜ τk ≤ τ˜ < τ˜k+1 } for k ≥ 0 ( Lemma 5.3.12. For k ≥ 0 with P (Ak ) > 0,

V opt τ˜ Z opt , Vτõpt Zτõpt

pair for the market τ˜ M|Ak .

) is the optimal |Ak

Proof. Since ϵ in Lemma 5.3.9 can be chosen arbitrarily small, the assertion follows from Corollary 5.3.5 and the previous lemma. Proposition 5.3.13. Let τ˜ be a stopping time and set A := {|Zτõpt | > 0}. If P (A) > 0, ( then the SLP ) p (Fτ˜ ∩ A) holds and the optimal pair for the market V opt τ˜ Z opt . opt , opt τ˜ M|A is Vτ˜

Zτ˜

|A

Proof. Note that for P ({˜ τ = ∞}) = 1 the assertion holds. Assume now P ({˜ τ= ∞}) < 1. If for τ˜ M|A the SLPp (Fτ˜ ∩ A) does not hold, then there exists a D ∈ Fτ˜ ∩A and a k with P (D∩Ak ) > 0, such that for τ˜ M|Ak the SLPp (Fτ˜ ∩Ak ) does not hold, a contradiction to the previous lemma. Therefore there exists an optimal pair for τ˜ M|A . The assertion follows now from Theorem 5.3.4 and Corollary 5.3.5.

Remark 5.3.14. This result allows to show that if the SLPp (F) holds for M, then the family {τZ opt |τ stopping time } is a multiplicative family, i.e. τZτõpt = 1 on {˜ τ ≤ τ < ∞} and τZτ¯opt = τZτõpt τ˜Zτ¯opt for τ ≤ τ˜ ≤ τ¯ on {¯ τ < ∞}, see [60], Section 3 and the first part of Theorem 4.20. We have also proved this property for the family {τ V opt | τ stopping time }. As we will see in the sequel, the SLPp (F) assumption is not strong enough to ensure good convergence properties of self-financing hedging strategies in the market M.

256


Remark 5.3.15. Even so the consistency result of this subsection forms the technical core on which in some sense the whole approach is based, this property of the optimal pairs has an important direct consequence for numerical appli¯ < ∞ and trading takes place cations: In a finite, discrete time model, i.e. |Ω| only at a finite number of instances, the Lq -optimal martingale measure can be calculated by a backward iterative algorithm, which is much easier than solving the global optimization problems we have been considering. See [90] for explicit formulas for the case p = 2.

5.4

E-Martingale Measures

Given a sequence V n ∈ V p , such that V n → V¯ ∈ V¯ p in Lp , our aim is to determine a limiting process V such that V∞ = V¯ and Vtn → Vt is some sense. As we have seen in the previous section, the only candidate for V , defined in (5.3.5), is only given on [0, τ ). Working with signed measures, P ({τ < ∞}) might be greater than zero and we do not know much about V on [τ, ∞). The remedy to this problem was found by Choulli, Krawczyk and Stricker. The idea is to consider the sequence τV n in the market τ M. Formalizing this idea leads to the notion of an E-martingale, (read exponential martingale), which we introduce in this section following closely [60] and [16]. We derive some of their properties, introduce the set of E-martingale measures for a market M and study existence and optimality problems for E-martingale measures. For a semimartingale X ∈ S, it is known that the stochastic differential equation Y = 1 + Y− · X has a unique solution E(X) ∈ S, which is given by the Doléans-Dade exponential formula E(X)t = eXt −X0 − 2 t

∏

(1 + ∆Xs )e−∆Xs ,

(5.4.1)

0≤s≤t

where X c denotes the continuous martingale part of X. E(X) is called the stochastic exponential of X, see e.g. [48], I.4f. (p)

Let S ∈ Aloc , 1 ≤ p ≤ ∞, in this section.

5.4. E-MARTINGALE MEASURES

257

Definition 5.4.1. For a local martingale N ∈ L, the family E[N ] := {E(τ N )| τ stopping time}

(5.4.2)

is called a local E-martingale. The sequence of stopping times Tn , defined recursively by T0 = 0 and Tn+1 := inf{t > Tn |∆Nt = −1} is called its sequence of zeros.

If E(τ N ) is a uniformly integrable martingale for all stopping times

τ , then E[N ] is called an E-martingale. If for an E-martingale E[N ], E(τ N ) ∈ Lq , resp. E(τ N ) ∈ Lq0 (τ Ω), for all stopping times τ , then E[N ] is called an Lq -integrable, resp. Lq0 -integrable, Emartingale. Remark 5.4.2. Note that if E(τ N ) is an extended martingale for all stopping times τ , then, by Lemma 5.2.14, E(τ N ) is already a uniformly integrable martingale, since E(τ N )0 = 1, and E[N ] is an E-martingale. Definition 5.4.3. An increasing sequence of stopping times τn converges stationarily to ∞ if P (∪∞ n=0 {τn = ∞}) = 1. Lemma 5.4.4. For an E-martingale E[N ], the corresponding sequence of zeros Tn converges to ∞. For a finite stopping time τ , the corresponding sequence of zeros Tnτ for E[N τ ] converges stationarily to ∞. Proof. Note that Tn < Tn+1 on {Tn < ∞}, hence Tn → ∞, since N possesses left limits. We have Tnτ = Tn 1{Tn ≤τ } +∞1{τ 0, ˜ q (τ M|{τ ≤τ 0, such that Y E(TnN )∞ ∈ Lq . Define the RCLL extended uniformly integrable martingale W :=

E[|Y E(TnN )∞ |q | Tn F· ] . |Y |q

W is independent of Y

and Wt∗ := sup0≤s≤t Ws is almost surely finite. Since E(TnN ) ̸= 0 on [Tn , Tn+1 ) and E(TnN )Tn+1 − ̸= 0 on {Tn+1 < ∞} by the properties of the stochastic ex1

ponential, the first assertion follows from

Wq |E(TnN )|

=

TnX

on [Tn , Tn+1 ). Set

Aˆ := {Tn+1 = ∞} ∩ {E(TnN )∞− = 0} ∈ F∞− = F. By extended Jensen inequality and extended Hölder inequality we find, see (5.2.9), |E(TnN )t | ≤ ∥E(TnN )∞ ∥L1t = ∥E(TnN )∞ 1Aˆc ∥L1t ≤ ∥E(TnN )∞ ∥Lqt ∥1Aˆc ∥Lpt , hence on [Tn , Tn+1 ) 1≤

∥E(TnN )∞ ∥Lqt |E(TnN )t |

∥1Aˆc ∥Lpt .

Since ∥1Aˆc ∥Lpt → 0 on Aˆ for t → ∞, (∥1Aˆc ∥L∞ decreases to 1Aˆc ), we must have t TnX

t

→ ∞ on Aˆ as t → ∞. On the set {Tn+1 = ∞} ∩ {E(TnN )∞− ̸= 0},

converges to 1.

TnX

260

CHAPTER 5. CONTINUOUS TIME CAPM Now define an adapted increasing [0, ∞]-valued process by E q∗ [N ]t := sup

∞ ∑

0≤s≤t n=0

Xs∗ ,

Tn

(5.4.6)

for 0 ≤ t < ∞ and set E q∗ [N ]∞ := E q∗ [N ]∞− . Obviously E q∗ [N ] is an adapted increasing process. Lemma 5.4.7.

Tn E

q∗ [N ]

restricted to {E q∗ [N ]Tn < ∞} is an increasing RCLL

process on [Tn , Tn+1 ) . Proof. This follows from the properties of the conditional expectation, see Lemma 5.2.15, and the previous lemma. Proposition 5.4.8. E[N ] is Lq0 -integrable if and only if E q∗ [N ] is almost surely finite on [0, ∞). Proof. By the previous lemma and since Tn → ∞, the sum in (5.4.6) is finite on {Tn ≤ t < Tn+1 } for all t < ∞ and all n ∈ N. If E[N ] is Lq0 ( ) integrable, we have E q∗ [N ]t = max max0≤k≤n−1 TkXT∗k+1 − , TnXt∗ < ∞ on {Tn ≤ t < Tn+1 }. Conversely, if E q∗ [N ] < ∞ almost surely, then ∥E(TnN )∞ ∥Lq (

τ TnΩ)

|E(TnN )τ |

TnX

τ

=

1[Tn ,Tn+1 ) (τ ) = ∥E(τ N )∞ ∥Lqτ (Ω) < ∞ on {Tn ≤ τ < Tn+1 }.

The process E q∗ [N ] is closely related to the reverse Hölder condition. We extend the notion to the case 1 ≤ q ≤ ∞ and also define some weaker conditions: Definition 5.4.9. Let 1 ≤ q ≤ ∞ and E[N ] be an E-martingale. We say that E[N ] satisfies the reverse H¨ older condition (Rq ), resp. the local reverse H¨ older q condition (Rloc ), if E q∗ [N ] is bounded, resp. locally bounded. If E q∗ [N ]∞ < ∞

almost surely, we say that E[N ] satisfies the pathwise reverse H¨ older condition (Rωq ). If E q∗ [N ]∞ is in Lq , or equivalently if E q∗ [N ] is bounded in Lq , we say q E[N ] satisfies the integrable reverse H¨ older condition (Rint ).

With [16], Proposition 3.3, equivalence of our definition of Rq and the usual definition, see e.g. [16], Definition 3.2, follows. Note that an E-martingale, q which satisfies Rq , is already an Lq -integrable E-martingale. Rint implies Rωq

and Lq -integrability.


261

Corollary 5.4.10. If Ω = Ωτ for some finite stopping time, then E[N ] is an Lq0 -integrable E-martingale iff E[N ] satisfies Rωq . Proof. This follows directly from Proposition 5.4.8, since E q∗ [N ] = (E q∗ [N ])τ .

Definition 5.4.11. Define for all stopping times τ { } Tτ := τ˜ ≤ τ |˜ τ stopping time .

(5.4.7)

Lemma 5.4.12. For all stopping times τ we have ess supτˆ∈Tτ ∥E(τˆ N )∞ ∥Lq0 (τˆ Ω) ≤ E q∗ [N ]τ . Proof. Let τˆ ∈ Tτ . Set Ak := {Tk ≤ τˆ < Tk+1 } and Bk := {Tk ≤ τ < Tk+1 }. We have ∥E(τˆ N )∞ ∥Lq0 (τˆ Ω) = 1{ˆτ =∞} + = 1{ˆτ =∞} +

∞ ∑ k=0 ∞ ∑ k=0

≤ 1{ˆτ =∞} + ≤ 1{ˆτ =∞} +

∞ ∑ k=0 ∞ ∑

1Ak ∥E(τˆ N )∞ ∥Lq0 (τˆ Ω|A

k

)

E(TkN )∞

1Ak E(TkN )τˆ Lq (τˆ Ω|A 0

1Ak TkXτˆ∗ = 1{ˆτ =∞} +

k

)

∞ ∑

1Bl

l=0

1Bl

l=0

l ∑

1Ak TkXτ∗ ≤ 1{ˆτ =∞} +

k=0

∞ ∑ k=0 ∞ ∑ l=0

1Ak TkXτˆ∗ 1Bl max

0≤k≤l

Xτ∗

Tk

≤ E q∗ [N ]τ .

Let E[N ] be an Lq0 -integrable E-martingale, with corresponding sequence Tn of zeros. Define A := ∩∞ n=0 {Tn < ∞}, An := {TnXTn+1 − = 0} ⊆ {Tn+1 = ∞}, and set A∞ := ∪∞ n=0 An ∪ A

262


Proposition 5.4.13. For an Lq0 -integrable E-martingale E[N ], {E q∗ [N ]∞ = ∞} equals A∞ . In particular, E[N ] satisfies Rωq if and only if its corresponding sequence of zeros Tn converges stationarily to ∞ and E(Tn N )Tn+1 − ̸= 0 for all n ≥ 1. Furthermore, E[N ] satisfies Rωq if and only if it satisfies Rω1 . q∗ Proof. By Lemma 5.4.6, E q∗ [N ]∞ = ∞ on ∪∞ n=0 An and E [N ]∞ < ∞ on

Ac ∩ Ac∞ . We only have to show E q∗ [N ]∞ = ∞ almost surely on A if P (A) > 0. By the extended Jensen and Hölder inequality we find for P (A) > 0 on {Tn < ∞} 1 ≤ ∥E(TnN )∞ ∥L1 = ∥E(TnN )∞ 1Ac ∥L1 Tn

Tn

≤ ∥E(TnN )∞ ∥Lq ∥1Ac ∥Lp . Tn

Since ∥1Ac ∥Lp

Tn

Tn

→ 0 on A, we must have ∥E(TnN )∞ ∥Lq

Tn

→ ∞ on A, hence

E q∗ [N ]t → ∞ on A for t → ∞. For the last assertion note that A∞ does not depend on q Remark 5.4.14. This result explains the limiting behaviour of E[N ] at infinity. As it will turn out, the process E q∗ [N opt ] and the reverse Hölder conditions are the right tool to study the convergence of self-financing hedging strategies. We will see in Subsection 5.4.5, that Rωq is a minimum requirement excluding a number of undesirable phenomena. We close this subsection with an auxiliary result which will allow us later to use localization arguments. Lemma 5.4.15. Let E[N ] be an E-martingale and let τ be a stopping time. We then have E q∗ [N τ ] ≤ E q∗ [N ]. Proof. The sequence of zeros for E[N τ ] is Tnτ = Tn 1{Tn ≤τ } + ∞1{τ T0 |T0Ztopt = 0}. By Lemma 5.2.6, EDq ̸= ∅ implies the SLPp (F) to hold, hence in T1 M|{τn ≤T1 0 the SLPp (FT1 ∩{τn ≤ T1 < τn+1 }) holds and we find again by Proposition 5.4.17 T1Z opt

to be a stochastic exponential. Proceeding recursively like this, we find

the stochastic exponential

TnZ opt

and define Tn+1 := inf{t > Tn |TnZtopt = 0} for

all n ≥ 1. Using the representation (5.4.10), we can define semimartingales N n :=

(

Tn opt Z−

)−1

1[0,Tn+1 ] ·

Tn opt

Z

,

= E(N n ). By Lemma 5.2.14, TnZ opt is a local martingale, im)−1 ( opt 1[0,Tn+1 ] is locally bounded. plying N n to be a local martingale since TnZ− such that

TnZ opt

We also define for k ≥ 0 the local martingales Nk :=

k ∑

N n 1[Tn ,∞) .

n=0

Note that Nk = Nl on [0, Tl+1 ] for l ≤ k. If we show Tn → ∞, then we can define the local martingale N opt :=

∞ ∑

N n 1[Tn ,∞) .

(5.4.11)

n=0

Note that {∆N opt = −1} = ∪n [Tn ], since NTnn = 0 almost surely. Set τ¯ := limn→∞ Tn and A˜ := {¯ τ < ∞}

(5.4.12)

266


˜ we have ∥TnZ opt ∥ q Exactly as in Lemma 5.4.16 it is shown that on A, L0 (TnΩ) → ∞ ˜ = 0, hence and by the same argument as in Proposition 5.4.17 be conclude P (A) Tn → ∞. Now, let τ¯ be a stopping time and set Ak,l := {τk ≤ τ¯ < τk+1 } ∩ {Tl ≤ τ¯ < opt Tl+1 }. By Corollary 5.3.5, it suffices to show that E(τ¯ N )∞ = τ¯Z∞ on Ak,l for all

k, l ≥ 0 such that P (Ak,l ) > 0. On Ak,l we have E(τ¯ N )∞ =

E(Tl N )∞ E(Tl N )τ¯

=

TlZ opt ∞ TlZ opt τ ¯

,

opt which equals τ¯Z∞ on Ak,l for all k, l such that P (Ak,l ) > 0 by Proposition

5.3.13. The following result characterizes E[N opt ]: Proposition 5.4.19. Assume EDq ̸= ∅. Then almost surely E q∗ [N opt ] ≤ E q∗ [N ], for all E[N ] ∈ EDq . In particular, if Deq ̸= ∅, then the Lq -optimal E-martingale measure exists and satisfies Rωq . Proof. By optimality we have ∥E( τ N opt )∞ ∥Lqτ ≤ ∥E( τ N )∞ ∥Lqτ for all stopping times τ , hence the first assertion follows from the definition of E q∗ [·]. Z ∈ Deq −1 defines an element E[N ] ∈ EDq for N := Z− · Z ∈ L. By Proposition 5.4.13,

E[N ] satisfies Rωq , thus E q∗ [N opt ] as well, by the first part. Our aim is to derive a necessary and sufficient condition for the existence of the Lq -optimal E-martingale measure. As we will see in the next chapter, it is of fundamental importance for the pricing and hedging of non-attainable claims.

5.4.4

Existence of the Lq -Optimal E-Martingale Measure

Definition 5.4.20. For a stopping time τ , define the following [1, ∞]-valued random variables Z q∗ (τ ) := ess supτˆ∈Tτ ∥τˆZ opt ∥Lq0 (τˆ Ω) ,

(5.4.13)

V p∗ (τ ) := ess inf τˆ∈Tτ ∥τˆ V opt ∥Lp0 (τˆ Ω) .

(5.4.14)


267

Theorem 5.4.21. We have equivalence between 1. The Lq -optimal E-martingale measure exists, 2. EDq (M) ̸= ∅, 3. The SLPp (F) holds and Z q∗ (τ ) < ∞ almost surely, for all finite stopping times τ , 4. The SLPp (F) holds and V p∗ (τ ) > 0 almost surely, for all finite stopping times τ . Proof. Equivalence of 2. and 1. was shown in Proposition 5.4.18, equivalence of 3. and 4. follows immediately from (5.3.4). If the Lq -optimal E-martingale measure E[N opt ] exists, then the SLPp (F) holds by Lemma 5.2.6, and 3. follows from Proposition 5.4.8 and Lemma 5.4.12. If Z q∗ (τ ) is finite for all finite stopping times τ , then, for A defined in (5.4.9) and τ˜ defined in (5.4.8), P (A ∩ {˜ τ < ∞}) = 0 holds and by Proposition 5.4.17, N opt can be defined as ˜ = 0 holds again by 3., hence the in (5.4.11). For A˜ defined in (5.4.12), P (A) Lq -optimal E-martingale measure exists.

In order to understand the meaning of the two conditions 3. and 4. we show in the next subsection two typical cases where 3. fails to hold, and in Section 5.5 we give an interpretation in terms of a conditional version of the price for risk.

5.4.5

Counterexamples

In this subsection we construct two simple market models, showing that in general EDq might be empty, even if the SLPp (F) holds.

5.4.6

Example 1

First we construct a market such that there exists a unique local martingale opt measure Z opt and such that Z opt approaches zero, i.e. inf 0≤s≤T Zs− = 0 for

a T < ∞, on a set with probability greater than 0. Then Z opt can not be a stochastic exponential and we have EDq = ∅.

268


Choose a sequence of constants 0 < kn < 1, n ≥ 1, strictly increasing to 1, ∏ but doing so slowly enough, such that limn→∞ nj=1 kj = 0. Let K > 1 be a constant and set pn :=

K− K−

∏n−1 ∏j=1 n

kj

j=1 kj

.

We have 0 < pn < 1 and pn will play the role of a branching probability. ∏ K−1 ∏N Note, that we have PN := N for all N ≥ 1 and PN strictly n=1 pn = K−

decreases to

j=1

kj

K−1 K .

Now set k¯n :=

1−pn kn 1−pn

and let Zn , n ≥ 1 be a sequence of independent

¯ F, P ) such that P (Zn = kn ) = pn random variables on a probability space (Ω, and P (Zn = k¯n ) = 1 − pn . It follows E[Zn ] = 1 and by a simple calculation ∏ we have k¯n n−1 j=1 kj = K. Define sets An := {Zi = ki , i = 1, . . . , n − 1}, ¯ A := ∩∞ n=1 An . Note that we have A1 = Ω, P (AN +1 ) = PN for N ≥ 1 and P (AN +1 ) ↘ P (A) =

K−1 K

as N → ∞.

Set t0 := 0 and choose a strictly increasing sequence of constant tn , n ≥ 1 with t1 = 1 and limn→∞ tn = T ≤ ∞. We now define Z opt by Z opt := 1Ac ×[T,∞) K + 1[0,T )

∞ ( ∏

) 1[0,tj ) + Zj 1[tj ,∞) .

j=1

Note that for 0 ≤ t < T , in the infinite product only a finite number of factors ¯ F, P ) for all 0 ≤ are different from 1, hence Ztopt is a random variable on (Ω, t < ∞. Denote by (Ft )0≤t the augmented filtration generated by Z opt . Ω := ¯ F, (Ft )t≥0 , P ) is then a probability space, satisfying the usual conditions. (Ω, Note that Zn is Ftn -measurable and An ∈ Ftn−1 = Ftn − . ∏ For n ≥ 0 and t ∈ [tn , tn+1 ), we have Ztopt = nj=1 kj on An+1 and Ztopt = K on Acn+1 . For t ≥ T we have Ztopt = 0 on A and Ztopt = K on Ac . Since for t ∈ [tn , tn+1 ), E[Ztopt |Ftn−1 ] = Ztopt E[Zn |Ftn−1 ] = Ztopt , we find (Z opt )τ to be n−1 n−1 a uniformly bounded, strictly positive martingale for all stopping times τ < T . ∏ opt Since limt→T Ztopt = limn→∞ nj=1 kj = 0 on A, we find ZTopt which − = ZT implies Z opt to be a non-negative, uniformly bounded martingale. Now we are going to construct a bounded, strictly positive price process S, such that Z opt is the unique local martingale measure for the market (Ω, (S, 1)). Since 0 < pn kn < 1, we can choose a sequence of constants pn kn < ln
0 as N → ∞. Set xn := ln − 1. Then ( ) −1 ∏N ∏N pn kn increases to C1 as N → ∞. Set x ¯n := −xn 1−p n=1 (1 + xn ) = n=1 ln n kn and define the random variable Xn by Xn = xn on An+1 , Xn = x ¯n on An ∩Acn+1 and Xn = 0 on Acn . Define the adapted process S˜ := 1[0,T )

∞ ∏

(1 + Xn 1[tn ,∞) ).

n=1

S˜t strictly increases to

1 C

on A and becomes eventually constant on Ac as

t ↗ T . Furthermore, by a simple calculation we see that pn kn < ln < 1 implies −1 < x ¯n < 0. We therefore can define a bounded, strictly positive semimartingale on Ω by S := 1[0,T ) S˜ + 1[T,∞) S˜T − . Since xn q + x ¯n (1−q) = 0 iff q =

x ¯n x ¯n −xn

(5.4.15)

= pn kn , we have E[Xn Zn |Ftn−1 ] = 0 and

xn pn w1 + x ¯n (1−pn )w2 = 0 for constants w1 , w2 such that w1 pn +w2 (1−pn ) = 1 opt iff w1 = kn and w2 = k¯n . Together with ST − = ST and ZTopt − = ZT , this implies

that Z opt is a local martingale measure for M and since FT − = FT , implying ZT = E[ZT |FT − ] = ZT − for all local martingales, we find Z opt to be the unique local martingale measure for M. Since limt→T Ztopt = 0 on A and P (A) > 0, we find EDq (M) = ∅ for all 1 ≤ q ≤ ∞, if T < ∞. Since A ̸∈ Ft for all t < T and τZ opt = 1 for all stopping times τ ≥ T , the SLPp (F) holds. Remark 5.4.22. For an arbitrage opportunity V ∈ V p , we must have {V∞ > opt 0} ⊆ A. Since Vt Ztopt = E[V∞ Z∞ |Ft ] = 0 there can not exist an arbitrage in opt V p . For a predictable, non-simple H such that (H·S)t Ztopt = E[(H·S)∞ Z∞ |Ft ],

the same argument shows, that there does not exist an arbitrage in the space of such portfolios, defined using the stochastic integral (we will study these spaces in detail in the next chapter). On the other hand, we know that Z q∗ (t) → ∞ for t ↗ T and a simple calculation shows that E|tZ opt |q |Ft ] → ∞ on A. This means, e.g. for p = 2, that an unbounded price for risk does in general not cause arbitrage. Remark 5.4.23. The case T = ∞ will be of interest later. We make the following observations: Dq = {Z opt }, EDq ̸= ∅ since Z opt > 0 almost surely on [0, ∞)

270


opt and since Z∞ = K1Ac , we have 1A ∈ V¯ p for all 1 ≤ p ≤ ∞, hence there n → 1 . Since Z opt 1 = 0, we find exists a sequence V n ∈ V p such that V∞ ∞ A A

with Lemma 1.4.6 Vtn → 0 in probability. The limiting process is therefore 0 on [0, ∞), hence we can not define an adapted process V such that V∞ = opt 1A ∈ V¯ p and Vt Ztopt = E[V∞ Z∞ |Ft ]. In particular, V opt = 1Ac exists only

as a random variable and it is not possible to define the hedging numéraire as a process. This also means that, even so there exists a unique absolutely continuous local martingale measure, the market is not complete and we can not introduce an adapted price process X with X∞ = 1A to the market M without causing Dq (MX ) = ∅ (which is equivalent to 1 ∈ V¯ p (MX )). For T < ∞, we have X = 1A×[T,∞) if and only if Dq (MX ) ̸= ∅, hence any pricing rule, ˜ e.g. utility indifference pricing, would lead to introducing an arbitrage to the market. Instead of defining S as in (5.4.15), we can even define S on A × [T, ∞) to equal an arbitrary deterministic function, e.g. set St :=

1 C

+ (t − T ) for t ≥ T

on A. We then encounter immediate arbitrage a concept introduced in [22]. However, in T M the SLPp (T F0 ) does not hold in this case. Confer also [21], Example 7.7, where a continuous market with similar properties is constructed.

5.4.7

Example 2

The second example shows that, in general, the sequence of stopping times Tn , defined recursively by T0 = 0 and Tn+1 := inf{t > Tn | TnZtopt = 0}, does not converge to ∞. Let Y n , n ≥ 1 be a sequence of independent random variables on a probabil¯ G, Q), such that P (Y n = −1) = qn and P (Y n = qn ) = 1−qn and ity space (Ω, 1−qn ∏∞ i ∞ j=1 qj = C > 0. Set Bn := {Y = −1, i = 1, · · · , n − 1} and B := ∩n=1 Bn . ∏n−1 Note that P (Bn ) = j=1 qj and P (B) > 0. Define a process N opt by N opt := 1[0,T )

∞ ∑

Y n 1Bn ×[tn ,∞) ,

n=1

where tn and T are chosen as in the previous example. Denote by (Gt )0≤t the ¯ G∞ , (Gt )t≥0 , Q) is then a probaugmented filtration generated by N opt . Ω := (Ω,


271

ability space, satisfying the usual conditions. Note that Y n is Gtn -measurable and Bn ∈ Gtn−1 = Gtn − . We have N opt ↘ −∞ on B as t ↗ T , hence N opt is not a semimartingale if T < ∞. However, it is a semimartingale on [0, T ). Nonetheless we can define for all stopping times τ the process τ opt Zt

:= E(τ N opt )t :=

∏

(1 + ∆Nsopt ) =

tnZ opt

(1 + Zn 1Bn ) .

τ 0.

272


5.5

Conditional Mean-Variance Efficiency (2)

We proceed very similar as in Section 1.7. We now assume S ∈ Aloc . Assume also the SLP2 (F0 ) to hold for M and let (V opt , Z opt ) denote the (2, 2)-optimal pair for M. We consider the conditional constraint minimization problem: F (e) := E[(V e )2 |F0 ] = ess inf V ∈1+V¯ 2 E[V 2 |F0 ],

(5.5.1)

with the constraint E[V |F0 ] = e, for an F0 -measurable random variable e and V e ∈ 1 + V¯ 2 , with E[V e |F0 ] = e. There are two cases, where we can immediately solve the above problem: For e = E[V opt |F0 ] =: eˆ, V opt trivially solves (5.5.1). Note also, that (5.3.4) implies eˆ ≤ 1. Define the conditional variance by Var0 (V ) := E[V 2 |F0 ] − E[V |F0 ]2 for V ∈ L2 . We have E[V 2 |F0 ] ≥ E[V |F0 ]2 and equality holds where V is F0 -measurable. For V ∈ 1 + V¯ 2 with E[V |F0 ] = 1, we have E[V 2 |F0 ] ≥ E[V |F0 ]2 = 1 and equality holds iff V = 1 ∈ 1 + V¯ 2 . Again, problem (5.5.1) might not have a solution for every F0 -measurable e. On A := {ˆ e = 1}, we have E[(V opt )2 |F0 ] = 1, hence by Proposition 5.3.2 E0 [|Z opt |2 ] = 1, implying Z opt = 1 on A. This means that if P (A) > 0, then S|A is a local martingale with respect to P|F ∩A and E[V |F0 ] = eˆ = 1,

(5.5.2)

for all V ∈ 1 + V¯ 2 (M|A ). On Ac , given an F0 -measurable e, we can always find an F0 -measurable λ such that e = (1 − λ)ˆ e + λ1 and we expect V e = (1 − λ)V opt +λ1 on Ac in analogy to the non-conditional case if (1−λ)V opt +λ1 ∈ L2 . Define the closed set V¯e := {V ∈ 1 + V¯ 2 |E[V |F0 ] = e} ⊆ L2 . Lemma 5.5.1. If V¯e ̸= ∅, then there exists a unique solution to problem (5.5.1). Proof. Uniqueness follows from strict convexity. Let V1 , V2 ∈ V¯e and set B := {E[V12 |F0 ] ≤ E[V22 |F0 ]} ∈ F0 and W := 1B V1 + 1B c V2 . Then W ∈ V¯e by Lemma 1.4.7, and E[W 2 |F0 ] = E[V12 |F0 ] ∧ E[V22 |F0 ]. This implies that the element with minimal norm in V¯e is the solution to problem (5.5.1).

5.5. CONDITIONAL MEAN-VARIANCE EFFICIENCY

273

Proposition 5.5.2. Let e, e¯ be F0 -measurable random variables such that we have V¯e , V¯e¯ ̸= ∅ and P ({ˆ e = e¯}∩Ac ) = 0. Define an F0 -measurable λ by λ|A = 0 and e = (1 − λ)ˆ e + λ¯ e on Ac . Then V e = (1 − λ)V opt + λV e¯. Furthermore, there exists an F0 -measurable random variable γ with γ|A = 0 and γ|Ac > 0, such that for all e with V¯e ̸= ∅, F (e) = F (ˆ e) + γ(e − eˆ)2 , holds. Proof. This is proved similar as Proposition 1.7.1, replacing E[·] with E0 [·]: On A we have e = 1 = eˆ, we set γ = 0 and there is nothing to prove. Set Vλ := (1 − λ)V opt + λV e¯. For all 0 < ϵ P (Ac ) − ϵ, such that λ is uniformly bounded on B. Then (Vλ )|A∪B ∈ L2 . It suffices to prove the assertion for the market M|A∪B . Since 0 < ϵ < P (Ac ) was arbitrary, the result holds for the market M too. We now work on Ω|A∪B . We have E0 [Vλ ] = e and ∥Vλ ∥2L2 0

= E0

[(

)2 ] (1 − λ)V opt + λV e¯

[ ] = (1 − λ)2 ∥V opt ∥2L2 + λ2 ∥V e¯∥2L2 + 2λ(1 − λ)E0 V opt V e¯ 0 0 ( ) opt 2 2 2 e¯ 2 = 1 − λ ∥V ∥L2 + λ ∥V ∥L2 + 2λ(1 − λ)E0 [V opt (V e¯ − V opt )] 0 0 ( ) opt 2 2 2 e¯ 2 = 1 − λ ∥V ∥L2 + λ ∥V ∥L2 , 0

since 1 − V opt

0

[ ] ∈ V¯ 2 implies E0 V opt (V e¯ − V opt ) = 0. For constants 0 < k
0 the assertion follows.

Lemma 5.5.3. On Ac , we have γ > 1. Proof. We have Var0 (V ) ≥ 0 for all V ∈ L2 . This implies Var0 (V e )|Ac = (F (e) − e2 )|Ac ≥ 0. Since (F (e) − e2 )|Ac is a polynomial of at most second order in e, we must have γ|Ac ≥ 1. γ|A˜ = 1 for an F0 -measurable set A˜ ⊆ Ac would imply eˆ|A˜ = 0, in contradiction to Z0opt ̸= 0, see Corollary 5.2.7. Now consider the following conditional variance optimization problem: G(e) := Var0 (V e ) =

inf

¯2 V ∈1+V

Var0 (V ),

(5.5.4)

under the constraint E0 [V ] = e, for an F0 -measurable e with e|A = 1 and V e ∈ 1 + V¯ 2 . This problem is again solved uniquely by the solutions of (5.5.1): Lemma 5.5.4. Let e be F0 -measurable with e|A = 1 and V e ∈ 1 + V¯ 2 . On Ac , we have G(e) = (γ − 1) (e − 1)2 , Furthermore, γ|Ac =

F (ˆ e) F (ˆ e)−ˆ e2

=

1 1−ˆ e.

(5.5.5)

5.5. CONDITIONAL MEAN-VARIANCE EFFICIENCY

275

Proof. Since F (e) − e2 = F (ˆ e) + γ(e − eˆ)2 − e2 = G(e) ≤ G(1) = 0, we find on Ac the pointwise minimum of G(e) in

γ ˆ γ−1 e

= 1, hence γ =

γ 0 = G(1) = G( γ−1 eˆ), which implies F (ˆ e) = eˆ and γ|Ac =

F (ˆ e) F (ˆ e)−ˆ e2

1 1−ˆ e

=

on Ac and 1 1−ˆ e.

Definition 5.5.5. Define the conditional Sharpe-ratio with respect to F0 for M as β˜0 := β˜0 (M) :=

Lemma 5.5.6. We have β˜0 =

√

√

Var0 (Z opt ).

(F (ˆ e))−1 − 1 and on Ac , β˜0 =

(5.5.6)

√

1 γ−1 ,

resp.

on A, β˜0 = 0, holds. Proof. Since

V opt E[V opt |F0 ]

= Z opt by Proposition 5.3.2 and the assumption that

SLP2 (F0 ) holds for M, we have E[V opt (V opt − 1)|F0 ] = 0, hence F (ˆ e) = eˆ. By (5.3.4) and Lemma 5.5.5, the assertion follows. Proposition 5.5.7. For all V ∈ 1 + V¯ 2 the following inequality holds: −β˜0

√

Var0 (V ) ≤ E0 [V ] − 1 ≤ β˜0

√

Var0 (V ).

(5.5.7)

Furthermore, on {Var0 (V ) ̸= 0}, |E0 [V ] − 1| √ ≤ β˜0 , Var0 (V )

(5.5.8)

and equality holds if and only if V = V e for some e with e|A = 1. Proof. On A, (5.5.7) holds since (β˜0 )|A = 0 and E[V |F0 ] = 1 for all V ∈ 1 + V¯ 2 . On Ac and if V = V e , (5.5.7) follows from (5.5.5). Since Var0 (V e ) ≤ Var0 (V ) for all V ∈ 1 + V¯ 2 with E0 [V ] = e, (5.5.7) holds for all V ∈ 1 + V¯ 2 .

276


5.5.1

The Conditional Intertemporal Price for Risk (2)

We still assume S ∈ Aloc , but drop the assumption that SLP2 (F0 ) holds for M. Definition 5.5.8. Define the conditional intertemporal price for risk in M as { β0 (M) := ess inf δ δ [0, ∞]-valued, F0 -measurable and on {δ < ∞}, } √ 2 . (5.5.9) |E[V |F0 ] − 1| ≤ δ Var0 (V ), ∀ V ∈ 1 + V∞

Note that we have β0 (M) = β˜0 (M) by Proposition 5.5.7, if the SLP2 (F0 ) holds for M. For B ∈ F0 with P (B) > 0, 1B ∈ V¯ 2 implies β0 (M) = ∞ on B. Note also that β0 (M) = ∞ iff D2 (M) = ∅. We immediately find the following conditional version of Theorem 1.7.9: Theorem 5.5.9. The following conditions are equivalent: 1. β0 (M) < ∞ almost surely. 2. The SLP2 (F0 ) holds in M. 3. For B ∈ F0 , P (B) > 0 implies 1B ̸∈ V¯ 2 (M). ˜ 2 (M) ̸= ∅. 4. D Furthermore,

( β0 (M) = ess inf Z∈D2 (M) Var0 (

where we set Var0

Z Z0

)

Z Z0

) ,

(5.5.10)

:= ∞ on {Z0 = 0}.

Proof. Equivalence of 2., 3. and 4 follows from Proposition 5.2.3 and Lemma 5.2.6. 1. follows then from Proposition 5.5.7. For B := {β0 (M) = ∞} the SLP2 (F0 ∩ B) does not hold in M|B if P (B) > 0, hence 1. implies 2. Corollary 5.5.10. The SLP2 (F) holds in M if and only if β0 (τ M) < ∞ almost surely for all stopping times τ .

5.6. THE MODIFIED MARKET

277

Definition 5.5.11. Define for an arbitrary stopping time τ the following [0, ∞]valued random variable: β ∗ (τ ) := ess supτˆ∈Tτ β0 (τˆ M).

(5.5.11)

Theorem 5.5.12. The existence of the L2 -optimal E-martingale measure is equivalent to β ∗ (τ ) < ∞ almost surely for all finite stopping times τ . Furthermore, β ∗ (∞) < ∞ almost surely iff the L2 -optimal E-martingale measure exists and satisfies Rω2 . Proof. If the SLP2 (F) does not hold in M, then the L2 -optimal E-martingale measure can not exist and β ∗ (τ ) < ∞ almost surely can not hold for all finite stopping times τ . If the SLP2 (F) holds in M, then the optimal pair exists √ for τ M for all stopping times τ . Note that β0 (τˆ M) = ∥τˆZ opt ∥2L2 ( M) − 1, ˆ 0 τ

hence

β ∗ (τ )

< ∞ if and only if

Z q∗ (τ )

< ∞ almost surely. The first assertion

follows now from Theorem 5.4.21. The second assertion follows from Lemma 5.4.12.

5.6

The Modified Market

This section will provide the link to the next chapter, where we are going to consider semimartingale models. In fact, up to a modification of the price process, we are already working in such a model: (p)

Proposition 5.6.1. Let S ∈ Aloc and assume EDq ̸= ∅. Then there exists a (p) modification S˜ ∈ Sloc of S. (p)

Proof. Choose a localizing sequence τn for S ∈ Aloc with τ0 = 0 and let (Tm ) be the sequence of zeros for E[N ] ∈ EDq . For P ({Tm ∨ τn < Tm+1 ∧ τn+1 }) > 0, we know that on {Tm ∨ τn < Tm+1 ∧ τn+1 } X m,n :=

Tm ∨τn(SE(Tm N ))

Tm+1 ∧τn+1

is a uniformly integrable martingale on Ω|{Tm ∨τn 0 almost surely, hence EDq ̸= ∅. In this situation we can use a result by Yor about the closedness of certain spaces of stochastic integrals with respect to a local martingale. This idea goes back to [23]. ˆ = (H, h) ∈ SF v such Proposition 5.8.1. There exists an Rd × R-valued H ˆ H that V∞ = V¯

Proof. Denote by H n ∈ Hp , the processes generating V n , i.e. V n = vn + H n · S. n −v → V ¯ − v in With respect to Q, V n ∈ U, by Proposition 1.4.4, and V∞ n

L1 (Q). By Yor’s result, see [98], Corollaire 2.5.2, resp. [46], Theoreme 4.60 for the multi-dimensional case, there exists an H ∈ L(S) with respect to Q such that H · S = E Q [V¯ − v|F· ]. By [46], Proposition 7.26, we have H ∈ L(S) with respect to P and the two stochastic integrals coincide. Now choose h := ˆ := (H, h), we then have H ˆ Sˆ = HS + h = v + (H · S)− − HS− ∈ P1 . With H ˆ

v+(H ·S)− +HS−HS− = v+(H ·S)− +H∆S = v+(H ·S)− +∆(H ·S) = v+V H , ˆ ∈ SF vZ (MZ˜ ) for all ˆ ∈ SF v . By Lemma 5.7.3, we also have H hence H 0 ˆ ˆ · SZ. ˆ ˆ Z˜ ∈ Dq , i.e. HSZ = V H Z = vZ0 + H

ˆ

In this situation, we could study the convergence of Vtn → VtH directly, however, in order to generalize the result to the case of signed local martingale measures, we introduce the notion of an E[N ]-martingale, which also goes back to [60] and [16]. It will turn out, that E[N ]-martingales share many properties with uniformly integrable martingales. Definition 5.8.2. Let E[N ] be a local E-martingale and Tn its sequence of zeros. A semimartingale X such that τ XE(τ N ) is a uniformly integrable martingale, resp. an extended uniformly integrable martingale, (local martingale), in τ Ω for any stopping time τ , is called an E[N ]-martingale, resp. an extended, (local), E[N ]-martingale in Ω. We say X is generated by an F-measurable random ¯ if variable X, τ Xt E(τ N )t

¯ τ N )∞ ], = Et [XE(

(5.8.1)

5.8. E[N ]-MARTINGALES

285

holds in τ Ω for all finite stopping times τ . X is called closed, if X∞ exists. Note that if X is an extended E[N ]-martingale, then Xτ ∈ L1 for all stopping times τ implies X to be a E[N ]-martingale.

5.8.1

Properties of E[N ]-Martingales

The following results generalize and extend part of [16], Proposition 3.12. For a uniformly integrable martingale X it is well known that X∞ exists almost surely and Xt → X∞ in L1 for t → ∞. The situation for E[N ]-martingales is a little more complicated: Lemma 5.8.3. Let X be an extended E[N ]-martingale with sequence of zeros Tn . X∞ exists on the set ( ) ( ∞ ) {E 1∗ [N ]∞ < ∞} = ∪∞ n=0 {E(TnN )Tn+1 − = 0} ∪ ∩n=0 {Tn < ∞} . (5.8.2) In particular, X is closed if E 1∗ [N ]∞ < ∞ almost surely. Proof. The equality (5.8.2) follows from Proposition 5.4.13. Since

Tn XE(Tn N )

is an extended uniformly integrable martingale in Tn Ω, it converges to a random ¯ n by Lemma 5.2.14. On Bn := {Tn < ∞}∩{Tn+1 = ∞}∩{E 1∗ [N ]∞ < variable X ∞} we have E(Tn N )∞ ̸= 0 almost surely by Proposition 5.4.13, hence X∞ exists there almost surely. ¯ be an F-measurable random variable, such that for all Lemma 5.8.4. Let X ¯ Tn N )∞ ∈ L1 . Then the process X defined on [Tn , Tn+1 ) by n ≥ 0, XE( 0 Xt :=

¯ Tn N )∞ ] Et [XE( , E(Tn N )t

where Et [·] is the extended conditional expectation operator in

(5.8.3) Tn Ω,

is an ex-

tended E[N ]-martingale. Proof. Note that X has left limits at Tn , hence it is a semimartingale. Let τ be a stopping time. On the set {Tn ≤ τ < Tn+1 }, we have τ Xt E(τ N )t = E(Tn N )τ ∨t TnXτ ∨t E(T N )τ n

=

¯ T N )∞ ] Eτ ∨t [XE( n E(Tn N )τ

in τ Ω, which is an extended uniformly inte-

grable martingale, hence X is an extended E[N ]-martingale.

286


Proposition 5.8.5. Let X be an extended E[N ]-martingale. The random vari¯ := X∞ 1{E 1∗ [N ] 0}, the assertion follows form Lemma 5.8.9.

5.8.2

Convergence of E[N ]-Martingales

Uniformly integrable martingales satisfy nice convergence properties. As it will turn out, the situation is similar for E[N ]-martingales. Proposition 5.8.13. Let V¯ m → V¯ ∈ Lp , let E[N ] be an Lq0 -integrable Emartingale and denote by V N , resp. V N,m , the extended E[N ]-martingales generated by V¯ , resp. V¯ m . Then V¯τN,m → V¯τN in probability for all finite stopping

288


N =V ¯ and V¯τN,m → V¯τN in probability for times τ . If E[N ] satisfies Rωq , then V∞

all stopping times τ . Proof. If P (Tn < ∞) > 0 then both V N , V N,m are extended uniformly integrable martingales on [Tn , Tn+1 ). By Lemma 5.2.14 , there exists for 0 < ϵ P (Tn < ∞) − ϵ and E(Tn N )V N 1A E(Tn N )V N,m 1A ∈ U(Tn Ω). Since on TN , Tn+1 ∩ A × [0, ∞) , Eτ [V¯ m E(Tn N )∞ ] → Eτ [V¯ E(Tn N )∞ ] ∈ L1 (Tn Ω|A ) and E(Tn N ) > 0 there, the assertion follows from the definition of E[N ]-martingales generated by V¯ , resp. V¯ m , see 5.8.4 and Corollary 5.8.7 . n →V ¯ ∈ v+V¯ p , with F0 -measurable vn , v ∈ Lp Let V n ∈ vn +V p such that V∞ (p)

and let τn be a localizing sequence for S ∈ Sloc . Set An (τ ) := {τn < τ < τ n + 1} for a stopping time τ . Lemma 5.8.14. Vn is an E[N ]-martingale for all E[N ] ∈ EDq . n Proof. We have τ V|A ∈ (Vτn )|An (τ ) + V p (τ M|An (τ ) ) for all stopping times τ n (τ )

and n ∈ N with P (An (τ )) > 0. The assertion follows now from Proposition 1.4.4 and since Vτn ∈ Lp for all stopping times τ . Corollary 5.8.15. For a sequence V n as above, there exists a unique semimartingale V , such that V is an extended E[N ]-martingale for all E[N ] ∈ EDq and Vτn → Vτ in probability for all finite stopping times τ . If there exists an E[N ] ∈ EDq , satisfying Rωq , then V∞ = V¯ and Vτn → Vτ in probability for all stopping times τ . Proof. By the previous lemma, we can apply Proposition 5.8.13 to the sequence V n and find an extended E[N ]-martingale V N for all E[N ] ∈ EDq such that ¯

¯ ] ∈ EDq . The last Vτn → VτN in probability. This implies V N = V N for all E[N assertion holds by Corollary 5.8.7. If a process V is an extended E[N ]-martingale, then V is in some sense not too far away from being an extended uniformly integrable martingale. For example, if E[N ] ∈ U q with E[N ]∞ with E[N ]∞ > 0 and if V is generated by V¯ ∈ Lp , then V ∈ U p with respect to the probability measure Q defined by

dQ dP

=


289

E[N ]∞ . The Doob maximal inequalities imply nice convergence properties for uniformly integrable martingales. It is natural to ask whether these inequaliies still hold after a change of measure or more general for E[N ]-martingales. This is the topic of the next section.

5.8.3

Doob Maximal Inequalities under a Change of Measure

In this subsection we derive a kind of Doob inequality for E[N ]-martingales, for Lq0 -integrable E[N ]. The results go back to [29], [49] and were generalized by [60] and [16] to the case of E[N ]-martingales for Lq -integrable E[N ], see [16], Theorem 4.1.ii, resp. [60], Theorem 6.6.ii. We extend the results to the case 1 ≤ q ≤ ∞ and prove a local version for extended E[N ]-martingales.

We first cite a lemma, which can be found in [16], resp. in [60] for the infinite time horizon case. Lemma 5.8.16. Let A, B ∈ A1 be positive RCLL processes, A increasing, and let U be a random variable, such that for all t ∈ [0, ∞], E[sup Bs |Ft ] ≤ E[U |Ft ], t≤s

then E[sup As Bs ] ≤ E[A∞ U ]. 0≤s

¯ := Let X be an E[N ]-martingale. By Lemma 5.8.3, X is generated by X X∞ 1{E 1∗ [N ]∞ 0} > ϵ) ≤ Kn

¯ L1 ∥X∥ . ϵ

(5.8.5)


291

∗p 2. If E[N ] satisfies Rωq , then X ∈ Sstat− and the sequence τn in 1. can be

chosen to be stationarily converging to ∞. 3. If E[N ] satisfies Rq , then X ∈ S ∗p and τ1 in 1. can be chosen to equal ∞. q ∗p 4. If E[N ] satisfies Rloc , then X ∈ Sloc and in (5.8.4) and (5.8.5), τn − can

be replaced by τn . Furthermore, X is a special semimartingale. q 5. If E[N ] satisfies Rint for 1 ≤ q < ∞, then X ∈ S ∗1 , in particular X is a

special semimartingale. Proof. Up to localization, we copy part of the proof of [16], Theorem 4.1. Define stopping times τ := inf{t ≥ 0|E q∗ [N ]t ≥ n} for n ≥ 1. Since E[N ] is Lq0 integrable, τn increases to ∞, and doing so stationarily, if E[N ] satisfies Rωq . If E[N ] satisfies Rq , τn = ∞ for some n ≥ 1. We first prove the case 1 0 and such that Z := Z n,m := E(TnN )|An,m ∈ Lq (TnΩ|An,m ) n,m and g := g n,m := ∥Z∞ ∥Lq ≤ G for all m, n and a constant G. We work for

the moment on A := Am,n , all appearing random variables are restricted to A. Define a probability measure Q on

TnΩ|A

by

dQ dP|F ∩A

=

|Z∞ |q ∥Z∞ ∥Lq

with density

process dqt :=

E[|Z∞ |q |Ft ] g

and set ¯ := W ¯ n,m := X ¯ Z∞ 1{T 0. Then for all constants x, y > 0 there exists a unique process V opt,x such that V opt,x = V H for some H ∈ SF + x and opt,x V∞ solves problem (5.11.5), resp. a unique Z opt,y ∈ yD1+ solving problem

(5.11.6). Furthermore, u is continuously differentiable and if y = u′ (x), then VTopt,x = I(ZTopt,y ), resp. ZTopt,y = U ′ (VTopt,x ), and V opt,x Z opt,y is a uniformly integrable martingale. (The assumption that F0 is trivial enters into the proof of equality (4.2) in [59].) We shall see in the following subsection, that a modification of the above theorem still holds for U = Up , p < 1 and general F0 . Set V opt := V H

opt,1

and Z opt := Z opt,1 .

306


Remark 5.11.7. Note that under the assumptions of Theorem 5.11.6, U ′ (x) = ∞ iff x = 0 and ZTopt < ∞ a.s., imply VTopt > 0 a.s., resp.

U ′ (x) = 0 iff

x = ∞ and VTopt < ∞ a.s., imply ZTopt > 0 a.s., and therefore we have opt Z opt , Z− , V opt , V−opt > 0, since Z opt is a non-negative supermartingale with

ZTopt > 0 almost surely, resp. since V opt Z opt is a non-negative uniformly integrable martingale with VTopt ZTopt > 0 almost surely, see [48], Lemma III.3.6.

5.11.3

Optimal Pairs

Let U ∈ C + and denote the conjugate function by V . Again, we consider conditional optimization problems:

u(M) := E0 [U (V opt )] = ess supV ∈V + (M) E0 [U (V )], 1

(5.11.7)

where V opt ∈ V1+ (M) and the corresponding dual problem ˆ (Z opt )] = ess inf ˆ u ˆ(M) := E0 [U ∞ Z∈D + (M) E0 [U (Z∞ )], 1

(5.11.8)

where Z opt ∈ D1+ (M). For U = Up , p < 1, set up (M) := u(M), resp. u ˆp (M) := u ˆ(M). Definition 5.11.8. For 0 ̸= p < 1 and |X|p ∈ L10 define 1

∥X∥Lp0 := (E0 [|X|p ]) p . Let p < 1 and set q :=

p p−1 ,

so that

1 p

+

1 q

(5.11.9)

= 1 for all p ̸= 0, 1.

Proposition 5.11.9. Assume the existence of a V ∈ V + and a Z ∈ D+ such that Z∞ = cUp′ (V ),

(5.11.10)

holds almost surely for an F0 -measurable strictly positive c. If E[V Z∞ |F0 ] = 1 almost surely, then V, Z solve the optimization problems (5.11.7) and (5.11.8). q Furthermore, for 0 ̸= p < 1 we have V p , Z∞ ∈ L10 and

∥V ∥Lp0 ∥Z∞ ∥Lq0 = 1. ∥V ∥Lp0 ≥ 1 holds for 0 < p < 1, resp. ∥V ∥Lp0 ≤ 1 holds for p < 0.

(5.11.11)

5.11. CONTINUOUS SEMIMARTINGALE MARKETS

307

Proof. Note that (5.11.10) together with E[V Z∞ |F0 ] = 1 implies E0 [U (V )] < ˆ (Z∞ )] < ∞ almost surely. Let V˜ ∈ V + . Since Up is concave, ∞ and E0 [U we have Up (V˜ ) ≤ Up (V ) + Up′ (V )(V˜ − V ) and taking extended conditional expectations we find [

] Z ∞ E0 [Up (V˜ )] ≤ E0 [Up (V )] + E0 (V˜ − V ) c E[Z∞ V˜ |F0 ] − 1 ≤ E0 [Up (V )] + c ≤ E0 [Up (V )]. ˆp = −Uq , resp. U ˆ0 = −U0 − 1. For Z˜ ∈ D+ , we have Note that for 0 ̸= p < 1, U Uq (Z˜∞ ) ≤ Uq (Z∞ ) + Uq′ (Z∞ )(Z˜∞ − Z∞ ), hence taking extended conditional expectations we find [ 1 ] E0 [Uq (Z˜∞ )] ≤ E0 [Uq (Z∞ )] + E0 c p−1 V (Z˜∞ − Z∞ ) 1

≤ E0 [Uq (Z∞ )] + c p−1 (E[Z˜∞ V |F0 ] − 1) ≤ E0 [Uq (Z∞ )]. ¿From V p = V V p−1 = V

Z∞ c

− p1

and E[V Z∞ |F0 ] = 1 it follows ∥V ∥Lp0 = c p q

1− 1q

q Since Z∞ = (cV p−1 )q = cq V p , we have ∥Z∞ ∥Lq0 = c∥V ∥Lp = c

.

1 p

= c ,

0

hence (5.11.11) holds. The last assertion follows from optimality of V since 1 ∈ V +. Definition 5.11.10. A pair of processes (V opt , Z opt ), such that V opt = V H opt opt for some H ∈ SF + , Z opt ∈ D+ , E[V∞ Z∞ |F0 ] = 1 and (5.11.10) holds for opt opt V∞ , Z∞ and a p < 1, is called a (p, q)-optimal pair, or just optimal pair, for

M. Corollary 5.11.11. If there exists a (p, q)-optimal pair for M, then it is unique. opt opt Proof. The uniqueness of V∞ , Z∞ follows by the previous proposition imme˜ opt + H = VH ˜ diately from the concavity of Up . Assume V∞ ∞ = V∞ for H, H ∈ SF , ˜

˜

opt Z∞ = Z∞ for a Z ∈ D+ and V H Z, V H Z opt ∈ U. Set A := {VtH > VtH } for

¯ := H1[0,t] + H1Ac ×(t,∞) + a 0 < t < ∞. Then H

VtH ˜ ˜ H1A×(t,∞) VtH

∈ SF + , since

308


˜

¯

¯

H ≥ V opt and V H > V opt on VtH > 0 almost surely by [48], Lemma III.3.6, V∞ ∞ ∞ ∞ ˜

A, hence P (A) = 0 by optimality. This implies V H = V H and Z = Z opt .

The optimal pair for M satisfies again a consistency property: Proposition 5.11.12. If (V opt , Z opt ) is the optimal pair for M, then for any stopping time τ , the optimal pair (τ V opt , τZ opt ) for the market τ M is given by ( opt opt ) Vτ ∨· Zτ ∨· opt , opt . Vτ

Zτ

Proof. ¿From (5.11.10) and since V opt Z opt ∈ U, it follows V opt , Z opt > 0 almost surely, hence

Zτopt ∨· Zτopt

∈ D+ (τ M),

opt V∞ Vτopt

∈ V + (τ M) and we find a corresponding

representation (5.11.10) with Fτ -measurable c. Now the assertion follows from Proposition 5.11.9.

We are now going to extend Theorem 5.11.6 for U = Up , p < 1 to the case of a non-trivial F0 . Proposition 5.11.13. Assume M = MT for a constant T < ∞, De1 ̸= 0 and up (M) < ∞ almost surely for a p < 1. Then the (p, q)-optimal pair (V opt , Z opt ) opt opt for M exists and the relation (5.11.10) holds for V∞ and Z∞ .

Proof. To prove the existence of V opt , Z opt , we construct a new market with trivial initial σ-algebra, such that we can apply Theorem 5.11.6. Choose a ∑ Z ∈ De1 and define B := { di=1 |Z0 S0i | + |S0i | < K} ∩ {up (M) < K} for a ( ) ˜ := B, F ∩ B, (F˜t )0≤t , P|F ∩B with constant K > 0 such that P (B) > 0. Set Ω P (B) F˜t := Ft−1 ∩ B for t ≥ 1 and F˜t = {P −1 ({0, 1}) ∩ B} for 0 ≤ t < 1. Note ˜ is a filtered probability space satisfying the usual assumptions and F˜0 that Ω is stochastically trivial. Define the process S˜ on B by S˜t =

EB [S0 Z0 ] EB [Z0 ] 1[0,1) (t)

+

St−1 1[1,∞) (t), where EB [·] denotes the expectation operator on Ω|B . S˜ is then ˜ Define a strictly positive Z˜ ∈ U(Ω) ˜ by Z˜t := 1[0,1) (t) + a semimartingale on Ω. (Zt−1 )|B EB [Z0 ] 1[1,∞) (t)

˜ ˜ := (Ω, ˜ (S, ˜ 1)), we and note that S˜Z˜ ∈ L(Ω). Hence for M

˜ Note that up (M) < K on B implies u(1, M|B ) < ∞. Since have Z˜ ∈ De1 (M). ∑d ˜i ˜ i=1 |∆S1 | is uniformly bounded, u(1, M|B ) < ∞ implies u(1, M) < ∞ and ˜ and find we are now in the position to apply Theorem 5.11.6 to the market M


309

˜ By Proposition 5.11.12 and a time shift the optimal pair (V˜ opt , Z˜ opt ) for M. ( opt opt ) ˜ V˜ Z 1+· argument the optimal pair for M|B is ˜1+· . Since K can be chosen opt , ˜ opt V1

Z1

arbitrarily large, the assertion follows now by patching together optimal pairs on F0 -measurable sets. By Theorem 5.11.6, we find a representation (5.11.10) opt opt for V˜∞ and Z˜∞ . By the consistency property, Proposition 5.11.12, we find a opt opt representation (5.11.10) for V∞ and Z∞ .

It is shown in [59], Example 5.1’, that in general Z opt is not a local martingale. However, for continuous price process S, we shall see in the next subsection, that Z opt is at least a local martingale.

5.11.4

The Optimal Local Martingale

We assume in this subsection the price process S to be a continuous semimartingale with canonical decomposition S = S0 + M + A, where M ∈ L and A ∈ N ∩ P with M0 = A0 = 0. M, A are known to be continuous too, see [48], I.4.24.

We first slightly generalize [5], Théorème 4: Proposition 5.11.14. There exists a non-negative Z ∈ L with Z > 0 almost surely and such that SZ ∈ L if and only if there exists a λ ∈ L2loc (M ) such that A = −λ · [M, M ] = −λ· < M, M >, and then Z = Z0 E(λ · M + L), with L ∈ L orthogonal to M and [M, N ] = 0 holds.

Proof. Assume SZ ∈ L. Zt > 0 for all 0 ≤ t < ∞ implies Z− > 0 almost surely since Z is a non-negative supermartingale. Hence the stochastic logarithm of Z exists and Z = Z0 E(N ) for N := Z is one and

1 Z−

1 Z−

· Z, which is a local martingale since

is locally bounded in the sense, that for all ϵ > 0 there exists

an A ∈ F0 such that P (A) > 1 − ϵ and the restriction of

1 Z−

to A is locally

bounded in the usual sense. By [48], III.4.11, there exists a λ ∈ L2loc (M ) and an L ∈ L orthogonal to all components of M and with [L, M ] = 0, such that

310


N = λ · M + L. We calculate, using Z = Z0 + Z− · N , SZ − S0 Z0 = S− · Z + Z− · S + [S, Z] = S− · Z + Z− · M + Z− · A + [M, Z− · N ] = S− · Z + Z− · M + Z− · A + Z− · [M, λ · M + L] = Z− · (A + λ · [M, M ]) + local martingale. Since SZ ∈ L, we have Z− · (A + λ · [M, M ]) ∈ L. Since Z− > 0 almost surely, we find A + λ · [M, M ] to be a predictable local martingale of finite variation, hence A + λ · [M, M ] = 0 up to indistinguishability, see [48], I.3.16. Conversely, A = −λ · [M, M ] for a λ ∈ L2loc (M ), implies SZ ∈ L for Z := E(λ · M ). Since M is continuous, Z ∈ L and Z > 0 almost surely.

Assume S = S0 + M − λ · [M, M ] for λ ∈ L2loc (M ) and continuous M ∈ L with M0 = 0. λ is often called the risk premium process. We will see later, that another related process deserves this name. Proposition 5.11.15. If for an H ∈ SF + and a Z ∈ D+ , V H Z is a uniHZ formly integrable martingale with V∞ ∞ > 0 almost surely, then Z is a local

martingale.

Proof. Since Z is a non-negative supermartingale, we have by [48], Lemma III.3.6, that Z, Z− > 0 almost surely. Hence Z = E(Y ) for Y :=

1 Z−

· Z. Since

Z is a supermartingale, Z is a special semimartingale. This implies Y to be special as well, since

1 Z−

is locally bounded and thus sup0≤s≤t |( Z1− ·Z)s | is locally

integrable, see [48], I.4.23. Therefore Y admits a decomposition Y = L + B, where B ∈ N ∩ P, L ∈ L and L0 = B0 = 0. There exists a K ∈ L2loc (M ) and an N ∈ L orthogonal to all components of M and with [N, M ] = 0, such that H > 0 and again by L = K · M + N . Then Z = E(B + K · M + N ) holds. Since V∞

[48], Lemma III.3.6, we find V H Z, (V H Z)− > 0 almost surely, hence V H , V−H > ˜ ∈ L1 (S) = L2 (S), such 0 almost surely, and by Lemma 5.7.5, we find an H loc loc


311

( ) ¯ := H, ˜ 1 − HS ˜ that H generates V H , see Definition 5.7.6. We have ¯ ˜ · S)E(B + K · M + N ) V H Z = V (H) Z = E(H ( ) ˜ · S + B + K · M + N + [H ˜ · S, B + K · M + N ] = E H ( ) ˜ · (λ · [M, M ]) + (H ˜ + K) · M + N + [H ˜ · M, K · M ] = E B−H ( ) ˜ · ((K − λ) · [M, M ]) + (H ˜ + K) · M + N = E B+H

˜ · ((K − λ) · [M, M ])) = V H Z − The Doléon-Dade SDE implies (V H Z)− · (B + H ˜ + K) · M + N ) − 1 to be a predictable local martingale of finite (V H Z)− · ((H variation, since (V H Z)− is locally bounded. Since (V H Z)− > 0 almost surely, we find ˜ · ((K − λ) · [M, M ]) = 0, B+H

(5.11.12)

¯ := (H ′ , 1 − H ′ S) and up to indistinguishability. Let H ′ ∈ L2loc (S), set H ¯ ¯ ¯ We have V (H) consider the value process V (H) generated by H. = E(H ′ · S) > ¯

0 almost surely since S is continuous and V (H) Z = V H Z is a non-negative supermartingale, since Z ∈ D+ . Using (5.11.12), we calculate V (H) Z = E(H ′ · S)E(B + K · M + N ) ( ) = E H ′ · S + B + K · M + N + [H ′ · S, B + K · M + N ] ( ) = E H ′ · S + B + K · M + N + [H ′ · M, K · M ] ( ) = E H ′ · ((K − λ) · [M, M ]) + B + (H ′ + K) · M + N ( ) ˜ · M, (K − λ) · M ] + (H ′ + K) · M + N . = E [(H ′ − H) ¯

˜ + K − λ ∈ L2 (S) we find, again by the Doléon-Dade SDE, For H ′ := H loc ¯

(V (H) Z)− · [(K − λ) · M, (K − λ) · M ] = ( ) ¯ ¯ ˜ + 2K − λ) · M + N − 1, V (H) Z − (V (H) Z)− · (H ¯

to be a non-decreasing local supermartingale. Since (V (H) Z)− > 0 almost surely, this implies [(K − λ) · M, (K − λ) · M ] = 0 up to indistinguishability, hence B = 0 up to indistinguishability, by (5.11.12), and Z = E(λ · M + N ) ∈ L since λ · M = K · M up to indistinguishability.

312


5.12

Locally Efficient Portfolios

We consider in this section the market model M := (Ω, (S, 1)), where S is assumed to be an Rd -valued continuous semimartingale. Assume S = S0 + M − λ · [M, M ] for λ = (λi )1≤i≤d ∈ L2loc (M ) and continuous M = (M i )1≤i≤d ∈ L with M0 = 0. We can find an adapted increasing, continuous, thus predictable and locally integrable, process α with α0 = 0, such that [M, M ] = C · α for a symmetric non-negative d × d-matrix-valued predictable process C = C(α) = ∑ (C ij )1≤i,j≤d , see [48], II.2. Set ||C|| := 1≤i,j≤d |C ij |. Definition 5.12.1. We call α a clock for the market M. See [40]. Remark 5.12.2. One can take α ¯ :=

∑d

i=1 [M

i, M i]

as a clock for M. For C¯

¯ > 0 holds P ⊗ d¯ defined by C¯ · α ¯ = [M, M ], then ||C|| α-almost surely. Note that we have dα ¯ ≪ dα for any clock α for M and d¯ α ∼ dα for all α such that ||C(α)|| > 0 holds P ⊗ dα-almost surely. ˜ := (H, 1 − HS), the value process For H = (H i )1≤i≤d ∈ L2loc (M ) and H ˜

˜

˜ is given as V (H) = E(H ·S) = E(−HCλ·α+H ·M ), where V (H) generated by H ∑ HCλ := di,j=1 H i C ij λj is predictable and locally integrable with respect to α. √ Define the predictable process β := λCλ. √ √ √ √ We have |HCλ| ≤ HCH λCλ = β HCH and HCλ = β HCH holds P ⊗dα-almost surely iff P ⊗dα-almost surely H = kλ+N , where k ∈ L2loc (β 2 ·α) is a non-negative predictable R-valued process and N is a predictable process such that CN = 0, or equivalently N · M = 0, up to indistinguishability, √ resp. HCλ = −β HCH holds P ⊗ dα-almost surely iff P ⊗ dα-almost surely H = −kλ + N , with k, N as before. ˜

˜ := (H, 1− Definition 5.12.3. We call a portfolio process V (H) generated by H HS) for H = ±kλ P ⊗ dα-almost surely with non-negative k ∈ L2loc (β 2 · α), a locally efficient portfolio (with respect to α). Locally efficient strategies are also called myopic. This notion goes back to [73]. See also [65], [36] and [1].

5.12. LOCALLY EFFICIENT PORTFOLIOS

313

Proposition 5.12.4. Local efficiency of a portfolio does not depend on the chosen clock α for M. Proof. The assertion follows from Remark 5.12.2.

In the following subsection we give an interpretation of the process β.

5.12.1

The Instantaneous Price for Risk

We consider the same market M as in the previous subsection. Let α be a clock for M and C, β as before. ˜ = (H, 1 − HS) with H ∈ L2 (S) be a generator for a self-financing Let H loc ˜

portfolio V (H) . We then have [

] ( ) ˜ ˜ ˜ 2 V (H) , V (H) = V (H) HCH · α.

( ) ˜ 2 The expression V (H) HCH is a measure for the instantaneous or local vari√ ˜ ation of the portfolio V (H) with respect to α, hence HCH can be interpreted ˜

as the relative instantaneous risk of the portfolio V (H) measured with respect to α. We also have ˜

˜

˜

V (H) = E(−HCλ · α + H · M ) = 1 − V (H) HCλ · α + V (H) H · M. ˜

The expression −V (H) HCλ is a measure for the instantaneous or locally ex˜

pected drift of the portfolio V (H) with respect to α, hence −HCλ can be in˜

terpreted as the expected relative instantaneous return of the portfolio V (H) measured with respect to α. √ For HCH ̸= 0, the expression

−HCλ √ HCH

can therefore be interpreted as an ˜

instantaneous Sharpe-ratio for the portfolio V (H) measured with respect to the clock α. We have seen in the previous subsection that −β ≤

−HCλ √ HCH

≤ β and

˜

that equality on the left or right side holds iff V (H) is a locally efficient portfolio √ with HCH ̸= 0. Definition 5.12.5. The process β is called the instantaneous price for risk process for M with respect to the clock α.

314

CHAPTER 5. CONTINUOUS TIME CAPM We summarize: ˜

Proposition 5.12.6. The portfolio V (H) with

√

HCH ̸= 0 is locally efficient

iff the absolute value of its instantaneous Sharpe-ratio process

|HCλ| √ HCH

equals the

instantaneous price for risk process β. Remark 5.12.7. The importance of locally efficient portfolios stems from the fact, that they depend only on local properties of the market M. The globally optimal portfolios studied earlier depend on the whole law of the price process S. If one wants to determine globally optimal portfolios in a real market, then one has to know the future law of the price process. The advantage of locally efficient portfolios is, that they can be calculated from adapted, i.e. observable data only, which can in principle be estimated without knowing the whole law of the involved price processes.

In the next subsection we will show that for a certain class of market models, the globally optimal portfolios with maximal expected terminal utility for Up , p ̸= 1, are locally efficient portfolios.

5.12.2

Totally Unhedgeable Instantaneous Price for Risk

In this subsection we introduce the notion of a totally unhedgeable instantaneous price for risk. See [50], Example 6.7.4, for the similar notion of a market with totally unhedgeable coefficients. This notion describes a market model where the uncertainty in the coefficients defining the model is, in a certain sense, orthogonal to the uncertainty of the local martingale, given by the canonical decomposition of the price process, such that we can not hedge against this risk. We continue to work with the market M such that S = S0 − λ · [M, M ] + M for λ ∈ L2loc (M ). Let p ̸= 1 and q :=

p p−1 .

Note that p−1 + q −1 = 1 for all

p ̸= 0, 1. Set H p :=

λ p−1 .

˜ p := (H p , 1 − H p S) generates a locally efficient Then H

portfolio. We seek a condition allowing to find a representation (5.11.10). We


315

calculate ) ( ( )p−1 ˜ p p−1 V (H ) = E(−H p · (λ · [M, M ]) + H p · M ) ( q ) = E − [λ · M, λ · M ] + λ · M 2 ( q ) = E(λ · M ) exp − [λ · M, λ · M ] . 2

(5.12.1)

Since E(λ · M ) is a non-negative supermartingale, E(λ · M )∞ ≥ 0 exists. We assume E(λ · M )∞ > 0 almost surely. This holds e.g. if M = MT for some constant T < ∞, since λ ∈ L2loc (M ). By [81], Proposition V.1.8, we have ( ) exp − 2q [λ · M, λ · M ]∞ > 0 almost surely iff E(λ · M )∞ > 0 almost surely. Assume now the existence of an N ∈ L, with N0 = 0, orthogonal to M , or equivalently [M, N ] = 0, and a strictly positive F0 -measurable c, such that ( q ) c exp − [λ · M, λ · M ]∞ = E(N )∞ . 2 ˜ (H)

Then V∞

(5.12.2)

∈ V + , E(λ · M + N ) ∈ D+ and a representation (5.11.10) holds. For

p = 2, 2q [λ · M, λ · M ] is known as the mean-variance tradeoff process, see e.g. [76]. Definition 5.12.8. If there exists a representation (5.12.2), such that E(N ) is a uniformly integrable martingale and E(N )∞ > 0 almost surely, then we call the instantaneous price for risk in M to be totally p-unhedgeable. See [96] for an equivalent definition in the case p = 2 and an extension to a price process with jumps. Remark 5.12.9. For p = 0 we have q = 0 and we find a representation (5.12.2) with c = 1, N = 0, hence the instantaneous price for risk in M is always totally 0-unhedgeable. For p = 0, the corresponding expected terminal utility optimization problem for U0 is also known as maximizing the Kelly-criterion, see [54], [14] and [50]. For general results see [1] and [40]. The following proposition generalizes a result in [82], Chapter 5.2. Proposition 5.12.10. Assume the instantaneous price for risk in M to be totally p-unhedgeable and a representation (5.12.2) to hold. For p > 1 assume

316


E(λ · M ) ∈ Dq or E(qλ · M ) ∈ U, for 0 < p < 1 assume E(qλ · M + N ) ∈ U and for p < 0 assume E(qλ · M ) ∈ U. Then the (p, q)-optimal pair for M is given by

(

) ) ( p V opt , Z opt = V (H ) , E(λ · M + N ) .

(5.12.3)

Furthermore, for p ̸= 0, )] [ ( q opt p ∥V∞ ∥Lp = E0 exp − [λ · M, λ · M ]∞ , 0 2

(5.12.4)

opt opt and ∥V∞ ∥Lp0 ≥ 1 for 0 < p < 1, resp. ∥V∞ ∥Lp0 ≤ 1 for p ̸∈ [0, 1] holds.

Proof. By (5.12.1) and (5.12.2) we find a representation (5.11.10). Hence for p

p < 1, we can apply Proposition 5.11.9, if we show that V (H ) E(λ · M + N ) ∈ U. We have for p ̸= 1 (

V

(H p )

λ [λ · M, λ · M ] + ·M E(λ · M + N ) = E − p−1 p−1 = E(qλ · M + N ).

) E(λ · M + N )

For 0 ≤ p < 1, E(qλ · M + N ) ∈ U by assumption. For p > 1 or p < 0 we have q > 0, hence c−1 E(N ) is a uniformly bounded martingale, since ) ( exp − 2q [λ · M, λ · M ]∞ ≤ 1. For p < 0 this implies E(qλ · M + N ) ∈ U. ( ) For p > 1, E(λ · M ) ∈ U q implies E(λ · M )q = E q(q − 1)[λ · M, λ · M ] + qλ · M ) to be uniformly integrable, hence E(qλ · M ) ∈ U and E(qλ · M + N ) ∈ U. It ˜p

remains to show that V (H ) Z ∈ U for all Z ∈ Dq . By Proposition 5.9.4 and Theorem 5.11.1 it suffices to show this for all Z ∈ Deq . By Proposition 5.11.14, ˜ ) for a local martingale N ˜ orthogonal to for Z ∈ Deq , we have Z = E(λ · M + N ˜ ] = 0, hence V (H˜ p ) Z ∈ L. Observe that M with [M, N ( )p ( ) [λ · M, λ · M ] λ ˜ p) p (H V = E − + ·M p−1 p−1 ) ( q = E − [λ · M, λ · M ] + qλ · M = c−1 E(qλ · M + N ) (5.12.5) 2 ≤ E(qλ · M ), { ˜p } (H ) hence Vτ | τ stopping time is uniformly bounded in Lp . By Lemma 1.4.3 we find V (H ) Z ∈ U. This proves (5.12.3). By (5.12.5) we find ∥V (H ) ∥pLp = c−1 . ˜p

˜p

0

With (5.12.2) and since E(N ) ∈ U, the equation (5.12.4), follows.


317

Remark 5.12.11. Confer [62], Section 6, for a related result in a markovian setting.

5.12.3

Deterministic Instantaneous Price for Risk

√ Assume αt = t to be a clock for M and that β := λCλ is F0 -measurable, or ∫T weaker 0 βs2 ds is F0 -measurable, where C is defined by [M, M ] = C · α. In this case the instantaneous price for risk in MT for a constant T < ∞ turns out to be totally p-unhedgeable for all p ̸= 1, since we trivially find a representation ( ∫ ) T (5.12.2) with N = 0 and c = exp 2q 0 βs2 ds . We immediately find for p ̸= 0, 1 ( ∥V opt,T ∥Lp0 = exp − (

where V opt,T , Z

) opt,T

(

= (V

(H p )

1 2(p − 1)

∫

)

T

βs2 ds

=

0

1 , ∥Z opt,T ∥Lq0

) )T , E(λ · M )T denotes the (p, q)-optimal pair

for the market MT . This example generalizes results by Merton for the case of a price process given by a Brownian geometric motion, see [71]. See also [62], Section 5.1. Remark 5.12.12. For constant β, the quantity

d ln(∥VTopt,T ∥Lp ) 0

dT

2

β can be = − 2(p−1)

interpreted as an implied forward growth rate of the maximal expected terminal utility of wealth. See [10] for an approach based on risk-sensitive stochastic control, where related quantities are considered.

5.12.4

Totally Hedgeable Instantaneous Price for Risk

There is a situation, where the minimal local martingale measure is easily seen to be the Lq -optimal local martingale measure. The idea goes back to [62], Section 5.2. ˜ = (H, ¯ 1 − HS) ¯ ¯ = Let H with H

λ p−1

+ H ∈ L2loc (S) be given. We calculate

( ) ( ) ˜ p−1 ¯ ¯ · M ) p−1 V (H) = E(−HCλ ·α+H ( ( ) ( ))p−1 β2 λ = E −qHCλ · α − ·α+H ·M E ·M p−1 p−1 ( ( ))p−1 p 2 = E(λ · M ) E −qHCλ · α + H · M − β ·α , 2(p − 1)2 ( )p−1 p−2 λ 2 where we have used that E(λ · M ) = E p−1 · M − 2(p−1) . 2β · α

318


Definition 5.12.13. We say that the price for risk in M is p-totally hedgeable, ¯ = if there exists an H

λ p−1 +H

∈ L2loc (S) and an F0 -measurable strictly positive

c such that ( E(−qHCλ · α + H · M ) = c exp

) p 2 β ·α . 2(p − 1)2

(5.12.6)

Remark 5.12.14. With h := E(−qHCλ · α + H · M )H equation (5.12.6) can be transformed into (

(

)

h · M − qCλ · α = c exp

) p 2 β ·α . 2(p − 1)2

(5.12.7)

By Girsanov’s theorem M − qCλ · α is a local martingale with respect to the measure Q defined by

dQ dP

= E(qλ · M + N ), where we assume E(qλ · M + N ) ∈ U

and N to be a local martingale orthogonal to M . This means that (5.12.6) is ( ) p 2 · α , see [62], basically a martingale representation condition for exp 2(p−1) β 2 Section 5.2. We immediately find Proposition 5.12.15. Assume the instantaneous price for risk in M to be totally p-hedgeable and a representation (5.12.6) to hold. For p > 1 assume ˜

˜ ∈ SF p and E(λ · M ) ∈ U p , for 0 ̸= p < 1 assume E((qλ + H) · M ) ∈ U. V (H) H Then the (p, q)-optimal pair for M is given by (

) ) ( ˜ V opt , Z opt = V (H) , E(λ · M ) ,

˜ := (H, ¯ 1 − HS) ¯ ¯ := where H with H

5.13

λ p−1

(5.12.8)

+ H.

Globally Efficient Portfolios

In this section we describe an approach to the problem of finding optimal pairs based on transformations of martingale representations. We will work with a ( ) market model M := Ω, (S, 1) , where S = S0 + M + λ · [M, M ] for a continuous local martingale M and λ ∈ L2loc (M ). Fix p ̸= 0, 1 and set q :=

p p−1 .

We assume

Deq ̸= ∅ for p > 1, resp. De1 ̸= ∅ for 0 ̸= p < 1. (The case p = 0 was already treated in the previous subsection.)

5.13. GLOBALLY EFFICIENT PORTFOLIOS

319

For p > 1, Deq ̸= ∅ implies by Theorem 5.11.1 the existence of the Lq optimal E-martingale measure E[N opt ] satisfying Rωq and E(τ N opt )∞ > 0 almost surely for all stopping times τ . We can assume N0opt = 0. By Proposition 5.10.2 there exists an H opt ∈ L(S), depending on p, such that the optimal ( ) ( ) pair τ V opt , τZ opt for the market τ M equals E(τ (H opt · S)), E(τ N opt ) for all stopping times τ and E(τ (H opt · S)), E(τ (H opt · S))∞ > 0 almost surely. By Proposition 5.11.14, we have N opt = λ · M + L, where L ∈ L, with L0 = 0 is orthogonal to M and thus [M, L] = 0 For 0 ̸= p < 1, we assume M = MT for some constant T < ∞ and up (M) < ∞ almost surely. Then by Proposition 5.11.13, the (p, q)-optimal pair exists for τ M for all stopping times τ and admits the same representation ( ) E(τ (H opt · S)), E(τ N opt ) as for the case p > 1. Now assume in addition the existence of a continuous local martingale N orthogonal to M such that (M, N ) has the predictable representation property. This implies the existence of a ν opt ∈ L2loc (N ), depending on p, such that L = ν opt · N and N opt = λ · M + ν opt · N . Furthermore, we choose a clock α such that [M, M ] = C · α and [N, N ] = C˜ · α. Remark 5.13.1. The following considerations carry through with appropriate changes even if we do not assume the existence of a continuous local martingale N orthogonal to M such that (M, N ) has the predictable representation property. Making this assumption ensures all martingales to be continuous, which considerably simplifies the formulas appearing in the sequel. This is due to the fact that there exists a simple formula allowing to express the p-th power of a continuous stochastic exponential again as a stochastic exponential. Summarizing, under the above assumptions there exists for p ̸= 0, 1 an H opt ∈ L2loc (M ) and a ν opt ∈ L2loc (N ), both depending on p, such that the (p, q)( ) optimal pair τ V opt , τZ opt for the market τ M equals up to indistinguishability ( ) E(τ (H opt · S)), E(τ N opt ) for all stopping times τ . For p > 1, E(τ N opt ) ∈ U is Lq0 -integrable. For 0 ̸= p < 1, we have E(τ N opt ) ∈ L. For all p ̸= 0, 1, E(τ N opt ), E(τ N opt )− , E(τ N opt )∞ > 0 almost surely. For p > 1, we have

320


E(τ (H opt · S))∞ ∈ Lp and for all p ̸= 0, 1, we have E(τ (H opt · S)), E(τ (H opt · S))− , E(τ (H opt · S))∞ > 0 almost surely. Furthermore, we have E(τ (H opt · ( ( )) S))E(τ N opt ) = E τ (H opt + λ) · M + ν opt · N ∈ U for all p ̸= 0, 1. By Theorem 5.3.4 for p > 1, resp. by Proposition 5.11.13 for 0 ̸= p < 1, there exists a unique strictly positive Fτ -measurable random variable cτ such that

opt opt p−1 |Up′ ( τ V∞ )| | τ V∞ | opt = = τZ∞ , cτ cτ

(5.13.1)

or equivalently ( ) opt 1−p τ opt ln cτ | τ V∞ | Z∞ = 0,

(5.13.2)

opt p opt p−1 τ opt opt τ opt and since | τ V∞ | = | τ V∞ | V∞ = cτ τ V∞ Z∞ , we find opt p cτ = ∥ τ V∞ ∥Lp (

0 τ Ω)

.

By the consistency property of optimal pairs and the uniqueness of cτ , we find for an arbitrary stopping time τ ≤ τ˜ cτ˜ =

cτ τZτõpt | τ Vτõpt |p−1

.

(5.13.3)

In the next subsection we shall transform equation (5.13.2), into a backward stochastic differential equation (BSDE). See e.g. [97] for an introduction to BSDE theory.

5.13.1

The BSDE Approach

Under the above assumptions, set (V opt , Z opt ) := (0 V opt , 0Z opt ) and define the following semimartingale on Ω: ( ) Y := ln c0 | V opt |1−p Z opt . We know by (5.13.2), that Y∞ = 0 and we have Y0 = ln(c0 ). By (5.13) and (5.13.3), we find for all stopping times τ (

opt p Yτ = ln (cτ ) = ln ∥ τ V∞ ∥Lp (

0 τ Ω)

)

[ ] opt p E |V∞ | Fτ . = ln  | Vτopt |p 


321

We have τ opt 1−p τ opt V Z = E(H opt · S)1−p τE(λ · M + ν opt · N ) ( ) = τE (p − 1)[H opt · M, λ · M ] + (1 − p)H opt · M ( ) p(p − 1) opt τ opt E [H · M, H · M ] τE(λ · M + ν opt · N ) 2 ) ( p(p − 1) opt opt opt opt τ [H · M, H · M ] + ((1 − p)H + λ) · M + ν · N , = E 2 τ

=

hence we find, using (5.4.1): ( ) ( ) − Yτ = ln c0 | τV opt |1−p τ Z opt − ln c0 | Vτopt |1−p Zτopt ) ( ( ) τV opt 1−p τ Z opt τ V opt 1−p τZ opt = ln = ln | Vτopt |1−p Zτopt ) ( ˜ opt (p − 1)H opt CH opt − λCλ − ν opt Cν opt + (p − 1)H Cλ · τ α = 2 ( ) + (1 − p)H opt + λ · τ M + ν opt · τ N. (5.13.4)

τY

This last equality can be interpreted as a BSDE. Equation (8.2.6) is a backward type of problem, since we have Y∞ = 0. In our situation the solution to the BSDE (8.2.6) is a tuple Y = (Y, (H, ν)) of adapted processes such that Y is a ( ) semimartingale on Ω with Y∞ = 0 almost surely and (H, ν) ∈ L2loc (M, N ) . We have shown under the above assumptions, that there exists a solution Y(M) to the non-linear BSDE (8.2.6), given by   [ opt ]   E |V∞ |p F·  , (H opt , ν opt ) . Y(M) = ln  | V opt |p Remark 5.13.2. For a twice continuously differentiable utility function, using Itô’s formula, the above approach would lead to a quite complicated coupled forward-backward stochastic differential equation system (FBSDE). The homogeneity of Up causes the optimal value process for initial capital x > 0 to be xV opt and the invested proportions are independent of the actual wealth. This is the reason, why for the isoelastic utility functions Up , the FBSDE decouples and we end up with the much simpler BSDE (8.2.6). By Lemma 5.7.5, a selffinancing hedging strategy with corresponding value process V opt is given by

322


V opt (H opt , 1 − H opt )S, which is in a feedback form (through the dependence on V opt ), as often found in control theory. However, the invested proportions (Htopt , 1 − Htopt St ) at time t depend only on the market t M. A second feature of the BSDE (8.2.6), important for numerical applications, is that the drift and diffusion terms do not depend on the semimartingale part of the solution, (Y ). This facilitates the numerical approximation of the solution considerably, since (8.2.6) can be solved by a backward iterative algorithm. If drift and diffusion terms of a BSDE depend on the semimartingale part of the solution, then in order to approximate the solution, a second iteration becomes necessary, see [97], Chapter 7. Conversely, given a solution Y(M) to (8.2.6), we are now going to show, under certain additional integrability assumptions, that we can derive the opti( ) ˜ := (H, 1 − HS) mal pair for the market M from Y(M) = Y, (H, ν) . Define H and ˜

V := V (H) = E(H · S),

(5.13.5)

Z := E(λ · M + ν · N ).

(5.13.6)

and ( ) Theorem 5.13.3. Let Y(M) = Y, (H, ν) be a solution to (8.2.6) and define ( (V, Z) by (5.13.5) resp. (5.13.6). Assume for 0 ̸= p < 1, that E (H + λ) · M + ( ˜ q and E (H + λ) · ν · N ) ∈ U, and for p > 1, assume in addition, that Z ∈ D ˜ q . Then (V, Z) M + ν˜ · N ) ∈ U for all ν˜ ∈ L2loc such that E(λ · M + ν˜ · N ) ∈ D is the optimal pair for M. Proof. Transforming equation (8.2.6) back, we find (5.13.1) to hold for V, Z with c0 := eY0 . In particular V, V− , Z, Z− > 0 holds up to an evanescent set and V∞ , Z∞ > 0 holds almost surely. For 0 ̸= p < 1 the assertion follows now ˜ q together immediately from Proposition 5.11.9. For p > 1, note that Z ∈ D with (5.13.1) implies that for all 0 < ϵ < 1, there exists an A ∈ F0 with P (A) > 1 − ϵ > 0, such that (V∞ )|A ∈ Lp (Ω|A ), hence by Proposition 5.9.4, we ( ) have (V∞ )|A ∈ V¯ p (M|A ) and (V∞ )|A , Z|A is the optimal pair for M|A . By a patching argument using Corollary 5.3.5, the assertion follows.


5.13.2

323

Markovian Market Models

As an example, we will now apply the result of the previous subsection to the case of a markovian market model. For d, d˜ ≥ 1, let W = (W 1 , W 2 ) ˜ with W0 = 0 almost surely, be a (d + d)-dimensional Brownian Motion on ( ) W , (F W ) W Ω := Ω, F∞ t 0≤t , P , where (Ft )0≤t is the completion of the filtration generated by W . It is known that W has the predictable martingale representation property, see [51]. We now define a market model M on Ω: For simplicity, let µ ˆ = (µ, µ ˜) : ( ) ˜ ˜ ˜ Rd+d → Rd+d , (µ = (µi )1≤i≤d , µ ˜ = (˜ µi )1≤i≤d˜), and σ ˆ = σσ˜ : Rd+d → ˜

˜

R(d+d)×(d+d) be smooth uniformly bounded functions with uniformly bounded ˜

derivatives of all orders. Assume in addition that for all x = (x1 , x2 ) ∈ Rd+d ˜

˜

with x1 = (xi1 )1≤i≤d and x2 = (xi2 )1≤i≤d˜, σ ˆσ ˆ ∗ (x) : Rd+d → Rd+d is invertible with an inverse uniformly bounded in x. Furthermore, assume σ ˆσ ˆ ∗ to factorize ˜ ˜ ˜ such that σ ˆσ ˆ ∗ (x) = C(x) × C(x) : Rd × Rd → Rd × Rd . In this situation there ˜ ˜ strong uniquely exists a Rd+d -valued continuous Markov process Sˆ = (S, S)

solving the SDE dSˆt = µ ˆ(Sˆt )dt + σ ˆ (Sˆt )dWt , Sˆ0 = sˆ0 = (s0 , s˜0 ), ˜

for sˆ0 ∈ Rd+d , see e.g. [51]. ( ) Define M := Ω, (S, 1) . S˜ can be interpreted as a non-traded state variable. We can choose αt := t as a clock for M. We have the canonical semimartingale ˆ · α + σ(S) ˆ · W , resp. S˜ = S˜0 + µ ˆ ·α+σ ˆ ·W. decompositions S = S0 + µ(S) ˜(S) ˜ (S) ˆ · W and N := σ ˆ · W , we have [M, M ] = C(S) ˆ · α, With M := σ(S) ˜ (S) ˜ S) ˆ · α and [M, N ] = 0. Since C and C˜ are assumed to be in[N, N ] = C( vertible with uniformly bounded inverse, (M, N ) inherits the predictable rep˜ ˆ ˜ resentation property from W . Define for x ∈ Rd+d , λ(x) := (λ(x), λ(x)) :=

˜ = (λ ˜i) (ˆ σσ ˆ ∗ (x))−1 µ ˆ(x), such that µ(x) = C(x)λ(x), (λ = (λi )1≤i≤d , λ 1≤i≤d˜). √ ˆ is the instantaneous price for risk Set β(x) := λ(x)C(x)λ(x). then β(S) process for M with respect to the clock α and we have the canonical semiˆ S) ˆ · α + M with λ(S) ˆ ∈ L2 (M ). martingale decomposition S = S0 + C(S)λ( loc

324


Furthermore, we have for Z ∈ D+ (MT ) ∩ L(ΩT ) with Z∞ > 0 almost surely ˆ · M + ν · N )T and where T < ∞ is a constant, the representation Z = E(λ(S) for some ν ∈ L2loc (N T ). The idea is now to derive in this markovian setting a (deterministic) PDE, such that H opt and ν opt for the market MT , (T < ∞ constant, p ̸= 0, 1), see Section 5.13, can be calculated from the solution. We shall achieve this by the following ansatz: In the situation considered here, the BSDE (8.2.6) for the market MT and p ̸= 0, 1 reads as ∫ T (p − 1)Hsopt Cs Hsopt − λs Cs λs − νsopt C˜s νsopt Yt = − ds 2 t ∫ T − (p − 1)Hsopt Cs λs ds t ∫ T ∫ T ) ( opt νsopt dNs , (1 − p)Hs + λs dMs − −

(5.13.7)

t

t

ˆ ˜ ˜ ˆ ˆ for all 0 ≤ t ≤(T , [where Cs ]:= ) C(Ss ), Cs := C(Ss ), λs = λ(Ss ) for 0 ≤ s. We opt p E |V∞ | F· expect Y = ln to be function of Sˆ and the remaining time to | V opt |p maturity T − t, and therefore we make the ansatz Yt = Y p (St , S˜t , T − t) for a ( ) ˜ deterministic function Y p ∈ C 2,2,1 Rd × Rd × [0, ∞) with boundary condition ˜

Y p (s, s˜, 0) = 0 for all (s, s˜) ∈ Rd+d . Applying Itô’s formula to Yt = Y p (St , S˜t , T − t) we find ) ∫ T( ∂Y p 0 = YT = Yt + − + L1 Y p + L2 Y p (Ss , S˜s , T − s)ds (5.13.8) ∂t t ∫ T ∫ T + ∇x1 Y p (Ss , S˜s , T − s)dMs + ∇x2 Y p (Ss , S˜s , T − s)dNs , t

t

where the differential operators L1 , L2 are defined as L

1

L

2

:=

:=

d ∑

i

d 1 ∑ ij ∂ 2 ∂ + µ C ∂xi1 2 i,j=1 ∂xi1 ∂xj1 i=1

d˜ ∑

i

d˜ 1 ∑ ˜ ij ∂ 2 ∂ µ . + ˜ C ∂xi2 2 i,j=1 ∂xi2 ∂xj2 i=1

We are now going to derive a PDE for Y p by comparing the terms of (5.13.7) and (5.13.8):


325

Comparing the last line of (5.13.7) with the second line of (5.13.8), we find Hsopt =

λs − ∇x1 Y p (Ss , S˜s , T − s) , p−1

(5.13.9)

and νsopt = ∇x2 Y p (Ss , S˜s , T − s).

(5.13.10)

Plugging these two expressions into (5.13.7) and comparing the resulting first two lines of (5.13.7) with the first line of (5.13.8), we find Y p to be a solution to the following non-linear PDE: −

∂Y β2 + L1 Y + L2 Y = L(p) Y + L3 Y + q ∂t 2

(5.13.11)

with boundary condition Y (·, ·, 0) = 0, where the differential operators L3 , L(p) are defined by L(p) f

L f 3

:=

d d ∑ ∑ ∂f ij ∂f 1 ∂f ij j C − q C λ i j i 2(p − 1) ∂x ∂x ∂x 1 1 1 i,j=1 i,j=1

d˜ 1 ∑ ∂f ˜ ij ∂f := − C 2 ∂xi2 ∂xj2 i,j=1

( ) ˜ for f ∈ C 1,1,0 Rd × Rd × [0, ∞) . Remark 5.13.4. In general it is difficult to give conditions ensuring the existence of a classical solution to the PDE (8.2.39) with boundary condition. In addition, we need certain integrability conditions on the derivatives of the solution in order to be able to express the optimal pair (V opt , Z opt ) for MT in terms of H opt , ν opt as defined in (5.13.9) and (5.13.10). ( ) ˜ Assume Y p ∈ C 2,2,1 Rd × Rd × [0, ∞) to be a solution to the PDE (8.2.39), satisfying the boundary condition Y p (·, ·, 0) = 0. Define H opt , ν opt by (5.13.9) ˜ and (5.13.10). Since Sˆ is continuous, C(x), C(x), λ(x) are uniformly bounded ˜

in x, and ∇x1 Y p , ∇x2 Y p are continuous on Rd × Rd × [0, ∞), we have H opt ∈ ( ) ˜ L2loc (M T ) and ν opt ∈ L2loc (N T ), (if only Y p ∈ C 2,2,1 Rd × Rd × (0, ∞) holds, ˜ := (H opt , 1 − H opt S) and define we have to assume this additionally). Set H ˜

V opt := V (H) = E(−H opt Cλ · α + H opt · M )T ,

326


and Z opt := E(λ · M + ν opt · N )T . Since H opt ∈ L2loc (M T ) and ν opt ∈ L2loc (N T ), we have V opt , Z opt > 0 up to an evanescent set. ( Theorem 5.13.5. Assume for 0 ̸= p < 1, that E (H opt + λ) · M + ν opt · N ) ∈ U, ( ˜ q and E (H + λ) · M + ν˜ · N ) ∈ U and for p > 1, assume in addition, that Z ∈ D ˜ q . Then (V opt , Z opt ) is the optimal for all ν˜ ∈ L2loc such that E(λ · M + ν˜ · N ) ∈ D pair for M. Proof. By construction Y(MT ) :=

( ) Y p (St , S˜t , T − t), (H opt , ν opt ) solves the

BSDE (8.2.6) and the assertion follows immediately form Theorem 5.13.3. Remark 5.13.6. For p < 1 the portfolio V opt is optimal for an investor maximizing terminal expected utility with respect to the increasing utility function Up . For all p ̸= 0, 1, Z opt can be used for utility indifference pricing. We have also seen, that for p = 2, the mean-variance efficient portfolios can be constructed from the hedging numéraire V opt . To have an explicit formula for the hedging numéraire is especially important, since the mean-variance hedging problem can be solved explicitly by considering the market discounted with respect to the hedging numéraire, see [41]. So far an explicit formula for the hedging numéraire was only known for the totally hedgeable, resp. totally unhedgeable, case was known, see [76], [62]. Remark 5.13.7. In [56], the hedging numéraire and the variance optimal local martingale measure are derived from the solutions of a system of BSDEs. This approach provides an alternative method to solve the mean-variance hedging problem explicitly. See also [57].

5.13.3

Example

For a constant instantaneous price of risk β, resp. in a time-inhomogeneous model for bounded deterministic β = β(t), one easily finds ∫ Y (s, s˜, t) := − p

t

q 0

β(u)2 du, 2

5.14. NON-DISCOUNTED MARKETS

327

to be a solution for (8.2.39). This situation is just the case of a totally unhedgeable price for risk, described in Subsection 5.12.1, and we find H opt =

ˆ λ(S) p−1

resp.

ν opt = 0. Remark 5.13.8. Starting with suitable, e.g. smooth, uniformly bounded with ˜ and Y p ∈ ˜ λ, λ uniformly bounded derivatives of all orders, functions C, C, ( ) ˜ C 2,2,1 Rd × Rd × [0, ∞) with Y p (·, ·, 0) = 0, we can try to solve (8.2.39), ˜. With this method we can which is linear in µ ˜ through L2 if ∇x2 Y p ̸= 0, for µ generated a wide class of market models, where we have explicit formulae for the optimal pair.

5.14

Non-discounted Markets

In this section we show how to deal with a non-discounted market by a change of probability measure technique, see [91] and [82]. Fix p ̸= 0, 1 and let M = (Ω, (S, B)), be a semimartingale market, where B > 0 is a numéraire such that |B∞ |p ∈ L1 . We can then define a probability ¯ F) by measure P B on (Ω,

dP B dP

=

|B∞ |p E[|B∞ |p ]

and consider the filtered probability )) ( ( ¯ F, (Fs )s≥0 , P B ), resp. the market BM := ΩB , S , 1 , which space ΩB := (Ω, B ( ( S )) , 1 . For a V ∈ v + is a usual discounted market. Recall MB := Ω, B

V¯ p (BM) we find that the law of |V |p under P B equals the law of |V |p

dP B

dP ¯ H B∞ V∞

|B∞ V |p E[|B∞ |p ]

=

¯ H ¯ = (H, h) ∈ SF(BM) such that V∞ exists, we have under P . For H

S ¯ ∈ SF(M), (see the proof of = B∞ (v +H · B )∞ = (vB0 +H ·S)∞ and H

Lemma 5.7.3). Therefore the optimization problem with respect to the isoelastic utility function Up for the non-discounted market M is transformed into an optimization problem for the discounted market BM. Note that in general for 1 < p < ∞ the two optimization problems are not equivalent, since SF p (MB ) not necessarily equals SF p (BM). Equality holds e.g. if B is uniformly bounded and uniformly bounded away from 0 and then

dP B q B dP D ( M)

= Dq (MB ).

With this observation, for B uniformly bounded and uniformly bounded away from 0, we find the existence of corresponding optimal pairs for the nondiscounted markets. The results of this chapter carry over with appropriate

328


changes in a straight forward way, see [63] and [64] for details. Assuming B = E(r · α) for a predictable process r representing the short rate with respect to the clock α, the BSDE (8.2.6) reads now as − Yτ ( ) ˜ opt (p − 1)H opt CH opt − λCλ − ν opt Cν opt = + (p − 1)H Cλ − pr · τ α 2 ( ) (5.14.1) + (1 − p)H opt + λ · τ M + ν opt · τ N,

τY

whereas the PDE (8.2.39) is modified to −

∂Y β2 + L1 Y + L2 Y = L(p) Y + L3 Y + q − pr, ∂t 2

(5.14.2)

where now the short rate is defined as rt := r(St , S˜t ) for a smooth bounded ˜

function r : Rd+d → R and β 2 := λCλ for λ := C −1 (µ − rS). Because of its importance for economical applications, we present in the next subsection a result concerning the intertemporal price for risk for a nondiscounted market.

5.14.1

The Price for Risk in a Non-discounted Market

Let M = (Ω, (S, B)), with stochastically trivial F0 , be a semimartingale market, where B ̸= 0 is a numéraire such that B∞ ∈ L2 . In a continuous market, one often works with a continuous strictly positive numéraire of finite variation, B = E(r · α), for αt = t for all t ≥ 0, and a process r ∈ P, representing the instantaneous interest rate or short rate, and B represents a bank account with an initial capital of one currency unit. For constant T < ∞, define P B B TMT

T

and

as above. Assuming r to be uniformly bounded on finite intervals, we T

have D2 (B MT ) =

E[BT2 ] 2 D (MTB T ), BT2

T

SF 2 (MTB T ) = SF 2 (B MT ). Consider the

non-discounted optimization problem E[|VTH

opt

|2 ] =

sup T H∈SF 2 (B MT ) H0 S0 =1

E[|VTH |2 ],

(5.14.3)

T

for H opt ∈ SF 2 (B MT ) with H0opt S0 = 1. Similar as in Subsection 1.6.5, the duality principle implies

BT VTH

opt

E[BT VTH

opt

]

to be the solution to the corresponding

5.14. NON-DISCOUNTED MARKETS

329

discounted dual optimization problem   Z opt 2 E  T  = sup BT Z∈D2 (MT

BT

[ ] ZT 2 , E BT )

(5.14.4)

for Z opt ∈ D2 (MTB T ). Assume the zero bond B (T ) > 0 with maturity time T to be attainable in the market M and consider the optimization problem (5.14.3) under the constraint E[VTH ] = e, (5.14.5) ( ) opt (T ) −1 for e ∈ R. Assuming eˆ := E[VTH ] ̸= B0 , we find, similar as in Section 1.7, that the non-discounted mean-variance efficient portfolio V e with E[V e ] = e is given by V e = λV H

opt

(T )

+ (1 − λ) B (T ) , where λ ∈ R is defined by e = λˆ e+ B0

Furthermore, we have the following inequality √ ( opt ) Z Var BTT √ ( H) 1 Var VT , E[V H ] − (T ) ≤ (T ) B B 0

1−λ (T ) . B0

(5.14.6)

0

T

for all H ∈ SF 2 (B MT ) with H0 S0 = 1, and equality holds iff V H = V e for some e ∈ R. The quantity √

( Var

β¯ :=

ZTopt BT

(T )

) ,

(5.14.7)

B0

can be interpreted as an intertemporal price for non-discount risk. ) ( (T ) opt (T ) −1 BT For eˆ = B0 it follows that V H = B (T ) , Z opt = E[B , β¯ = 0, T] B0 ( ) T (T ) −1 (T ) E[V H ] = B0 for all H ∈ SF 2 (B MT ) with H0 S0 = 1 and Bt = [ 1 E

BT Bt

|Ft

],

i.e. the so-called Return-to-Maturity Expectation Hypothesis holds.

See [63] and [64] for details.

330


5.15

Concluding Remarks

( ) Let 1 < p < ∞ be fixed and consider a market model M = Ω, (S, 1) with S ∈ A. In order to find a workable market model, we have imposed increasingly strong conditions on M and studied the implied properties for such a market. We collect these conditions as well as equivalent formulations and the most important implications in the following list. p 1. The LPp holds for M ⇐⇒ 1 ̸∈ V∞ . (see Sections 1.3 and 1.4)

2. Dq (M) ̸= ∅ ⇐⇒ the SLPp holds for M ⇐⇒ 1 ̸∈ V¯ p (M). =⇒ (a) Optimization problems (1.6.8) and (1.6.9) can be solved. (see Section 1.6) (b) The Lq -norm optimal signed local martingale measure Z min exists. (c) The Lp -norm optimal value process V opt exists in an approximative sense. (d) The duality relation ∥V opt ∥Lp ∥Z min ∥Lq = 1 holds. (e) For p = 2 there exists a finite intertemporal risk premium β(M). (see Section 1.7) ˜ q (M) ̸= ∅ ⇐⇒ the SLPp (F0 ) holds for M ⇐⇒ for all A ∈ F0 with 3. D P (A) > 0, 1A ̸∈ V¯ p (M) holds. (see Sections 5.2 and 5.3) =⇒ (a) The conditional optimization problems (5.2.1) and (5.2.2) can be solved. (b) The optimal pair (V opt , Z opt ) for the market M exists. (c) The conditional duality relation ∥V opt ∥Lp0 ∥Z min ∥Lq0 = 1 holds. (d) A consistency property for optimal pairs holds. (see Subsection 5.3.2)

5.15. CONCLUDING REMARKS

331

(e) For p = 2 there exists an almost surely finite conditional intertemporal risk premium β0 (M). (see Section 5.5) ˜ q (τ M) ̸= ∅ for all stopping times τ ⇐⇒ the SLPp (F) holds for M 4. D =⇒ The optimal pairs (τV opt , τZ opt ) for the markets τ M form a multiplicative group. (see Section 5.3, Remark 5.3.14) 5. EDq (M) ̸= ∅ ⇐⇒ the Lq -optimal E-martingale measure E[N opt ] exists ⇐⇒ τZ opt is a stochastic exponential for all stopping times τ . (see Section 5.4) =⇒ (a) E q∗ [N opt ] is an increasing finite valued process. (b) For p = 2 the supremum of the conditional risk premiums up to time τ , β ∗ (τ ), is almost surely finite for all finite stopping times τ . (see Section 5.5) (c) There exists a semimartingale modification for the price process S. (see Section 5.6) 6. E[N opt ] satisfies Rωq ⇐⇒ E q∗ [N opt ]∞ < ∞ almost surely. (see Section 5.4) =⇒ (a) Limits of self-financing hedging strategies exists. (see Section 5.9) (b) The exponential hedging numéraire exists as a stochastic integral. (see Section 5.10) (c) Generalized Föllmer-Schweizer decompositions exist. (d) For p = 2 the supremum of the conditional risk premiums, β ∗ (∞), is almost surely finite for all finite stopping times τ . (see Section 5.5)

332

CHAPTER 5. CONTINUOUS TIME CAPM (e) For continuous price process S, Deq ̸= ∅. (see Section 5.11.1)

7. E[N opt ] satisfies Rq ⇐⇒ E q∗ [N opt ]∞ is uniformly bounded. =⇒ (a) Lp -Approximation of contingent claims implies convergence in the semimartingale norm ||| · |||p . (b) The Föllmer-Schweizer decomposition exists. From the counterexamples in Section 5.3 and in Subsection 5.4.5 it is clear that in the above list, each condition is strictly stronger than its predecessors.

Chapter 6

Markets Basing on Fractional Brownian Motions 6.1

Introduction

Since the seminal paper by Pardoux and Peng [26] backward stochastic differential equations (BSDEs) are of increasing importance in stochastic control and mathematical finance. These equations are of the form ∫ T ∫ T Yt = ξ − f (t, Yt , Zt )dt − Zt dBt t

t

Here B is a classical Brownian motion and the data ξ and f are given. ξ is an FT -measurable random variable where Ft denotes the filtration generated by the Brownian motion B. f (t, y, z) is an Ft -adapted process for every pair (y, z) ∈ R2 . A pair (Y, Z) is a solution of the BSDE, if (Y, Z) fulfill the equation P -a.s., are Ft -adapted and satisfy a suitable integrability condition. In this paper we consider linear fractional BSDEs, i.e. the generator f is linear and the driving Brownian motion is replaced by a fractional Brownian motion B H with arbitrary Hurst parameter 0 < H < 1. Under suitable assumptions we construct explicit solutions of such linear fractional BSDEs by adopting the Ma/Protter/Yong four step scheme [21] to the case of a fractional Brownian motion. To this end we solve a backward parabolic partial differential equation, which is time degenerate for H > 1/2 respectively singular in the case H < 1/2 (section 5.2). The solution can then be obtained by applying a fractional Itô formula that we prove in section 3. Note that there are at least two different extensions of the classical Itô 333

334

CHAPTER 6. FRBM-MARKETS

integral to a fractional Brownian motion. As the pathwise integral (see Lin [20]) has not zero expectation in general (Duncan/Hu/Pasik-Duncan [8]), it cannot be used as a model for a random perturbation. Hence, we interpret the integral with respect to a fractional Brownian motion as a fractional Wick-Itô integral. There are two essentially equivalent definitions for this integral. It can either be defined as a Pettis type integral on the space of Hida distributions using the Wick product (see Hu/Øksendal [16], Bender [2] or Duncan/Hu/Pasik-Duncan [8] for a more basic but less general definition) or as divergence operator using ¨ unel [6] and Alos et al. [1]). In the Malliavin calculus (see Decreusefond/Ust¨ section 2 we give a rather simple definition of the fractional Wick-Itô integral in terms of the S-transform and the usual Lebesgue integral. The advantage of this definition is that we neither need the complicate constructions of the white noise calculus nor the Malliavin calculus throughout this paper. In section 6 we prove a generalization of the famous Black and Scholes pricing formula for derivatives of the stock price in a fractional market model by solving a fractional BSDE. Here the stock price is given by a geometric fractional Brownian motion. Notice that there are several definitions of geometric fractional Brownian motions. Ours is motivated by a fractional analogue of the Doléans-Dade identity, that we prove in section 4. Moreover, it coincides with the definition of Hu/Øksendal [16] and Elliott/van der Hoek [10] in the case of constant coefficients. The related problem to prove a no arbitrage property for the fractional market model is going to be treated in a forthcoming paper [3]. Finally, we should mention that both problems, the solvability of linear fractional BSDEs and the pricing of contingent claims in a fractional Black Scholes market have been treated by Hu/Øksendal/Sulem [17], Hu/Øksendal [16] and Elliott/van der Hoek [10] by change of measure. However in neither of these papers the integral with respect to the fractional Brownian motion ˜ H under the new measure is defined and it is assumed that the not defined B integral has the same properties than the integral with respect to B H . Hence, the argumentation in these papers is rather informal. Moreover, our pricing formula holds for time dependent parameters in the case H > 1/2, whereas the

6.2. PRELIMINARIES

335

formulae in [16] and [10] are obtained in the case of constant parameters only.

6.2 6.2.1

Preliminaries Construction of the Fractional Brownian Motion

We assume the underlying probability space (Ω, F, P ) to be the white noise space. So Ω is S ′ (R), the space of tempered distributions, F is the σ-field generated by the open sets in S ′ (R) with respect to the weak∗ topology of S ′ (R). Finally, by the Bochner-Minlos theorem the probability measure P is uniquely determined by the property that for all rapidly decreasing functions f ∈ S(R)

{ } 1 2 exp{i⟨ω, f ⟩}dP (ω) = exp − |f |0 2 S ′ (R)

∫

(6.2.1)

Here ⟨ω, f ⟩ denotes the dual action and | · |0 is the usual L2 (R)-norm. The corresponding inner product is denoted by (·, ·)0 . From (6.2.1) we may conclude that for all finite families (f1 , . . . , fn ) ⊂ S(R) the random vector (⟨·, f1 ⟩, . . . , ⟨·, fn ⟩) is a centered Gaussian random vector with covariance matrix ((fi , fj )0 )i,j . The details of the above construction can be found in Hida, [12]. Because of the isometry E[⟨·, f ⟩2 ] = |f |20 ; f ∈ S(R) we can extend ⟨·, g⟩ to g ∈ L2 (R). Hence, we have for f, g ∈ L2 (R): E[⟨·, f ⟩⟨·, g⟩] = (f, g)0

(6.2.2)

We subsequently use the notation (L2 ) := L2 (Ω, G, P ), where the σ-field G is generated by ⟨·, f ⟩, f ∈ L2 (R). The (L2 )-norm is denoted by ∥Φ∥20 := E[Φ2 ]. For a, b ∈ R we define the indicator function:   1, if a ≤ t < b −1, if b ≤ t < a 1(a, b)(t) =  0, otherwise

(6.2.3)

From (6.2.2) and (6.2.3) immediately follows that a continuous version of ⟨·, 1(0, t)⟩ 1/2

is a classical Brownian motion Bt . Consequently, approximating with step

336


functions yields P -a.s. the identity ∫ 1/2 ⟨·, f ⟩ = f (t)dBt ; f ∈ L2 (R).

(6.2.4)

R

Here the integral on the right hand side is the Wiener integral with respect to the above constructed Brownian motion. We now recall a construction of fractional Brownian motions with arbitrary Hurst parameter 0 < H < 1: Definition 6.2.1. A continuous stochastic process (BtH )t∈R is called a fractional Brownian motion with Hurst parameter H, if the family (BtH )t∈R is centered Gaussian with E[BtH BsH ] =

) 1 ( 2H |t| + |s|2H − |t − s|2H ; t, s ∈ R 2

(6.2.5)

Let (∫

∞(

KH := Γ(H + 1/2)

H−1/2

(1 + s)

−s

H−1/2

0

)

1 ds + 2H

)−1/2 ,

and define the operator  −(H−1/2)  f,  KH D± f, M±H f :=   K I H−1/2 f, H ±

0 < H < 1/2 H = 1/2 1/2 < H < 1

(6.2.6)

H−1/2

Here I±

is the fractional integral of Weyl’s type defined by ∫ ∞ 1 H−1/2 (I− f )(x) := f (t)(t − x)H−3/2 dt Γ(H − 1/2) x ∫ x 1 H−1/2 (I+ f )(x) := f (t)(x − t)H−3/2 dt Γ(H − 1/2) −∞ −(H−1/2)

if the integrals exist for all x ∈ R. D±

is the fractional derivative of

Marchaud’s type given by (ϵ > 0) (

and

−(H−1/2) D±,ϵ f

)

−(H − 1/2) (x) := Γ(H + 1/2)

∫ ϵ

∞

f (x) − f (x ∓ t) dt t3/2−H

( ) ( ) −(H−1/2) −(H−1/2) D± f := lim D±,ϵ f ϵ→0+

if the limit exists almost surely. With these definitions we have:

6.2. PRELIMINARIES

337

Theorem 6.2.2. For 0 < H < 1 let the operators M±H be defined by (6.2.6). Then a fractional Brownian motion is given by a continuous version of ⟨·, M−H 1(0, t)⟩ Proof. Using elementary integration and (6.2.4) one can easily show that ⟨·, M−H 1(0, t)⟩ coincides with the well known Mandelbrot-Van Ness representation, see [22]. Note that the existence of a continuous version follows from the KolmogorovCentsov theorem, see Karatzas/Shreve [18]. A more detailed proof can be found in Bender [2].

6.2.2

Integration

In this section we give rather simple definitions of integrability in terms of the S-transform and the usual Lebesgue integral. We begin with the definition and some properties of the S-transform: Definition 6.2.3. For Φ ∈ (L)2 the S-transform is defined by its P -convolvution: ∫ SΦ(η) =

Φ(η + ω)dP (ω); η ∈ §.

Using the translation property of the Gaussian measure one can show that the S-transform can be equivalently defined by (see Kuo [19], p.36): [ ] SΦ(η) := E Φ· : e⟨·,η⟩ : .

(6.2.7)

Here the Wick exponential of ⟨·, η⟩ is given by : e⟨·,η⟩ : = e⟨·,η⟩− 2 |η|0 . 1

2

The following theorem ensures that the below notions of integrability are well defined: Theorem 6.2.4. The S-transform is injective, i.e. SΦ(η) = SΨ(η) for all η ∈ § implies Φ = Ψ. Proof. In view of (6.2.7) the assertion is a straightforward consequence of the { } fact, that the linear span of the set : e⟨·,η⟩ : ; η ∈ § is dense in (L2 ), see Hida et al. [15], p. 7. One can also characterize (L2 )-convergence in terms of the S-transform:

338


Theorem 6.2.5. Let Φn be a sequence in (L2 ) and Φ ∈ (L2 ). Then the following assertions are equivalent: (i) Φn (strongly) converges to Φ in (L2 ). (ii) ∥Φn ∥0 → ∥Φ∥0 and for all η ∈ § S(Φn )(η) → (SΦ)(η). Proof. (i) ⇒ (ii): obvious. (ii) ⇒ (i): By the preceding theorem the linear span of {: eI(η) :; η ∈ §} is dense in (L2 ). Hence, by theorem 3, p.121, in Yosida [30], Φn weakly converges to Φ in (L2 ). Then ∥Φn ∥0 → ∥Φ∥0 implies convergence even in the strong topology. The following lemma can be proved easily: Lemma 6.2.6. Let f, g ∈ L2 (R). Then: E[: eI(f ) : · : eI(g) : ] = e(f,g)0 . In particular, (S : eI(f ) : )(η) = e(f,η)0 . Proof. If g = 0, we have: I(f )

E[: e

1 :]= √ 2π|f |0

∫

{ ( )} u2 2 exp u − 1/2 |f |0 + 2 du = 1. |f |0 R

The general case can be reduced to this by : eI(f ) : · : eI(g) : = e(f,g)0 : eI(f +g) : .

Lemma 6.2.6 and theorem 6.2.5 imply: Corollary 6.2.7. Let fn be a sequence that converges in L2 (R) to f . Then : eI(fn ) : converges to : eI(f ) : in (L2 ). From lemma 6.2.6 we know that E[: eI(f ) : ] = 1 for f ∈ L2 (R). Hence we can define a probability measure on G by dQf =: eI(f ) : dP.

(6.2.8)

6.2. PRELIMINARIES

339

One can easily check, that P and Qf are equivalent. With the measures Qη , η ∈ §, we can rewrite the S-transform as (SΦ)(η) = E Qη [Φ].

(6.2.9)

Under appropriate conditions the equation ∫ ∫ ( H ) ( H ) f (s) M− g (s)ds = M+ f (s)g(s)ds. R

(6.2.10)

R

holds. We refer to (6.2.10) as the fractional integration by parts rule. It is valid under the following conditions: Theorem 6.2.8. (i) Let 0 < H < 1/2. Then (6.2.10) holds, if M+H f ∈ Lp (R), M−H g ∈ Lr (R), f ∈ Ls (R), g ∈ Lt (R) and p > 1, r > 1, 1 s

=

1 p

+ H − 12 ,

1 t

=

1 r

+H −

1 p

+

1 r

=

3 2

− H,

1 2

(ii) Let 1/2 < H < 1. Then (6.2.10) holds, if f ∈ Lp (R), g ∈ Lr (R) and p > 1, r > 1,

1 p

+

1 r

=

1 2

+H

Proof. In view of the definition of M±H this is a simple reformulation of corollary 2 (p. 129) and formula (5.16) in Samko/Kilbas/Marichev [28]. From the proof of lemma 2.6 in Bender [2], we know that M+H η ∈ L1/(1−H) (R), if H < 1/2 and η ∈ §. Moreover, it is well known that § ⊂ L2 (R). Hence, we have: Corollary 6.2.9. Let f ∈ §. Then: (i) (6.2.10) holds, if 0 < H < 1/2, M−H g ∈ L2 (R) and g ∈ L1/H (R). (ii) (6.2.10) holds, if 1/2 < H < 1 and g ∈ L1/H (R). The Classical Case from an S-Transform Point of View Let 0 ≤ a ≤ b and X : [a, b] × Ω → R a progressively measurable process satisfying

[∫

b

E Then the classical Itô integral

] |Xt | dt < ∞. 2

∫b a

(6.2.11)

a

Xt dBt with respect to the Brownian motion

B exists. By the isometric property of the Itô integral it is an element of (L2 ). We are now going to calculate its S-transform:

340


Let Qη , η ∈ §, the measure defined by (6.2.8). Then by the Girsanov ∫ ˜t := Bt − t η(t)dt is a two sided Brownian motion under the measure theorem B 0 Qη . Moreover, by (6.2.11) and Hölder’s inequality √  ∫ b |Xt |2 dt < ∞. E Qη  a

Consequently,

∫s a

˜t , a ≤ s ≤ b, is a Qη -martingale with zero expectation. Xt dB

Using the above considerations, (6.2.9) and Fubini’s theorem we get: ] ) [∫ b (∫ b Qη Xt dBt S Xt dBt (η) = E a a [∫ b ] ∫ b Qη ˜ = E Xt dBt + Xt η(t)dt ∫

a

a

b

E Qη [Xt ]η(t)dt

= a

∫

b

(SXt )(η)η(t)dt

= a

If we replace X by a function f ∈ L2 (R) we can repeat the argumentation and get for the Wiener integral: ∫ S(I(f ))(η) =

f (t)η(t)dt. R

In particular,

∫

t

η(s)ds.

S(Bt )(η) = 0

Let us summarize: Theorem 6.2.10. (i) Let 0 ≤ a ≤ b and X : [a, b] × Ω → R a progressively ∫b measurable process satisfying (6.2.11). Then the Itˆ o integral a Xt dBt is the unique element in (L2 ) with S-transform given by ∫

b

(SXt )(η)η(t)dt. a

(ii) The Wiener integral I(f ), f ∈ L2 (R), is the unique element in (L2 ) with S-transform given by

∫ f (t)η(t)dt. R

6.2. PRELIMINARIES

341

Using this result we can define an extension of the Itô integral in terms of the S-transform: Definition 6.2.11. Let M ⊂ R, X : M → (L2 ). Then X is said to be HitsudaSkorohod integrable, if S(Xt )(η)η(t) ∈ L1 (M ) for any η ∈ § and there is a Φ ∈ (L2 ) such that for all η ∈ § ∫ SΦ(η) =

S(Xt )(η)η(t)dt M

In that case Φ is uniquely determined by theorem 6.2.4 and we denote it by ∫b 1/2 a Xt dBt . Remark 6.2.12. In this terminology, theorem 6.2.10, (i), states, that the HitsudaSkorohod integral is an extension of the Itô integral. This result is well known in a white noise setting. However, in this setting the proof is usually given by calculating the chaos decomposition, see e.g. Hida et al. [13]. Note, that our proof is rather elementary.

6.2.3

Fractional Integration - continued

We can now define: Definition 6.2.13. Let X : [a, b] → (L2 ). Then X is said to be integrable, if S(Xt )(η) ∈ L1 ([a, b]) for any η ∈ § and there is a Φ ∈ (L2 ) such that for all η∈§

∫ SΦ(η) =

b

S(Xt )(η)dt a

In that case Φ is uniquely determined by the above theorem and we denote it ∫b by a Xt dt. { } As the set : e⟨·,η⟩ : ; η ∈ § is a subset of (L2 ) integrability in the sense of definition 6.2.13 is implied by Pettis integrability in (L2 ). The reader not so familiar with Pettis integrability may think of a Banach space valued Riemann integral as long as X is continuous. Moreover, Pettis integrability is implied by Bochner integrability. Hence, we have the following criterion:

342


Theorem 6.2.14. Let X : [a, b] → (L2 ) such that S(Xt )(η) is measurable for all η ∈ § and ∥Xt ∥0 ∈ L1 ([a, b]). Then X is integrable and

∫ b ∫ b

∥Xt ∥0 dt X dt ≤ t

a

0

(6.2.12)

a

{ } Proof. As the linear span of the set : e⟨·,η⟩ : ; η ∈ § is dense in (L2 ), E[Xt F ] is measurable for any F ∈ (L2 ). Hence, Xt is weakly measurable. The separability of (L)2 implies that Xt is strongly measurable, see Hille/Phillips [14], p.73. Thus, Xt is Bochner integrable by theorem 3.7.4 in Hille/Phillips [14].

For more information concerning Pettis and Bochner integrals we refer to Hille/Phillips [14]. Let us now introduce the fractional Itô integral with respect to a fractional Brownian motion: Definition 6.2.15. Let X : [a, b] → (L2 ). Then X is said to be fractional Itˆ o integrable, if S(Xt )(η)(M+H η)(t) ∈ L1 ([a, b]) for any η ∈ § and there is a Φ ∈ (L2 ) such that for all η ∈ § ∫

b

S(Xt )(η)(M+H η)(t)dt

SΦ(η) = a

In that case Φ is uniquely determined and we denote it by

∫b a

Xt dBtH .

As the expectation coincides with the S-transform at η = 0 an immediate consequence of this definition is: Theorem 6.2.16. Let X : [a, b] → (L2 ) be fractional Itˆ o integrable. Then: [∫

]

b

Xt dBtH

E

= 0.

a

Example 6.2.17. Let us calculate

∫T 0

BtH dBtH : Again, Qη , η ∈ §, denotes the

˜ is the Brownian motion under Qη given by the classical measure in (6.2.8). B Girasanov theorem. Then we can apply integration by parts and the classical

6.2. PRELIMINARIES

343

Girsanov theorem to obtain: ∫ ∫ T H H 2 S(Bt )(η)(M+ η)(t)dt = 2 0

(∫

t

0

(M+H η)(s)ds

= 0

= E

∫

(M+H η)(s)ds(M+H η)(t)dt

0

)2

T

T

[( ∫

Qη R

∫ )2 ]

[( ∫ = E Qη

R

M−H 1(0, T )(s)dBs

)2 ]

T

−

˜s M−H 1(0, T )(s)dB

(M+H η)(s)ds 0

− |M−H 1(0, T )|20

( ) − T 2H = S (BTH )2 (η) − T 2H

by theorem 6.2.10 and (6.2.9). Hence, ∫ T ) 1 (( H )2 H H 2H Bt dBt = . BT − T 2 0 Remark 6.2.18. (M+H η)(t) is continuous in t for η ∈ §. Consequently, S(Xt )(η) ∈ L1 ([a, b]) implies S(Xt )(η)(M+H η)(t) ∈ L1 ([a, b]). We shall now motivate the above definition: Although a fractional Brownian motion B H : R → (L2 ) is not differentiable, it has a derivative, if we look at B H : R → (S)∗ , see Bender [2]. Here (S)∗ denotes a space of generalized random variables, the so called Hida distributions. For more information about Hida distributions see Hida et al. [13], Holden et al. [15] and Kuo [19]. Note that the S-transform can be extended to (S)∗ and that integrability can be defined as in definition 6.2.13. With this definition we have by Bender [2] ∫ b ∫ b d H Xt dBt = Xt ⋄ BtH dt, dt a a the diamond denoting the Wick product. In the case of a classical Brownian ∫b d 1/2 motion a Xt ⋄ dt Bt dt can be proved to be an extension of the Itô integral, see Holden et al. [15]. Thus, definition 6.2.15 is a sensible generalization of the Itô integral to the case of a fractional Brownian motion. Moreover, we should notice that for suitable integrands ∫ b ∫ Xt dBtH = M−H (1(a, b)X) (t)dBt , a

R

where the integral on the right hand side is a Hitsuda-Skorohod integral with 1/2

respect to the Brownian motion Bt . A sufficient condition for fractional Itô integrability can be given in terms of the operator M−H and the white noise derivative operator. The details can be found in Bender [3].

344


6.2.4

The Relationship between the Fractional Itˆ o Integral and the Hitsuda-Skorohod Integral

In this section we first extend the fractional integral (resp. derivative) operators H−1/2

I±

1/2−H

(resp. D±

) to stochastic processes X : R → (L2 ). Then we prove

that under appropriate conditions the fractional Itô integral coincides with a Hitsuda-Skorohod integral. To this end let us first recall the notion of Pettis integrability: Definition 6.2.19. Let X : M → (L2 ) (M ⊂ R). Then X is said to be Pettis integrable, if E[Xt Ψ] ∈ L1 (M ) for any Ψ ∈ (L2 ). In that case there is an unique Φ ∈ (L2 ) such that for all Ψ ∈ (L)2 such that ∫ E[ΦΨ] = E[Xt Ψ]dt M

Φ is called the Pettis integral of Xand is denoted by

∫ M

Xt dt.

Note that by this definition we have for Pettis integrable X: [∫ ] ∫ E[Xt Ψ]dt = E Xt Ψdt M

(6.2.13)

M

for all Ψ ∈ (L2 ). In particular, the Pettis integral interchanges with the Stransform. A proof for the existence and uniqueness of Φ can be found in Hille/Phillips [14]. As we look at stochastic processes as functions from M ⊂ R into (L2 ) the (L2 )-valued Pettis integral fits better than the ordinary pathwise integral. However, let X : [a, b] × Ω → R be measurable and pathwise integrable such that ∥Xt ∥0 ∈ L1 ([a, b]). Then by Fubini’s theorem: [∫ b ] ∫ b E Xt dt · Ψ = E [Xt Ψ] dt, a

a

where the integral on the left hand side is the ordinary pathwise integral. Hence, the Pettis integral defined in definition 6.2.19 coincides with the pathwise integral, whenever ∥Xt ∥0 ∈ L1 ([a, b]) holds. Let X : M → (L2 ), M ⊂ R. Then we say that X ∈ Lp (M, (Lq )), p, q ≥ 1, if

∫ M

p/q

E [|Xtq |]

dt < ∞

(6.2.14)

6.2. PRELIMINARIES

345

We can now define for X : R → (L2 ) and 1/2 < H < 1 the fractional integral of Weyl’s type by H−1/2 (I− X)x H−1/2

(I+

:=

X)x :=

∫ ∞ 1 Xt (t − x)H−3/2 dt Γ(H − 1/2) x ∫ x 1 Xt (x − t)H−3/2 dt Γ(H − 1/2) −∞

if the integrals exist for all x ∈ R as Pettis integrals. If 0 < H < 1/2 the fractional derivative of Marchaud’s type are given by (ϵ > 0)

∫ ( ) −(H − 1/2) ∞ Xx − Xx∓t −(H−1/2) D±,ϵ X := dt Γ(H + 1/2) ϵ x t3/2−H

and

( ) ( ) −(H−1/2) −(H−1/2) D± X := lim D±,ϵ X ϵ→0+

if the above integrals exist for all ϵ > 0 and x ∈ R as Pettis integrals and the limit exists in L2 (R, (L2 )) . The operators M±H can then be defined as in (6.2.6). With this notation we have: Lemma 6.2.20. Let M±H X exist for some X : R → (L2 ). Then we have for all Ψ ∈ (L2 ) :

[ ] E (M±H X)t Ψ = M±H (E[Xt Ψ]) .

In the case H < 1/2 the convergence of the fractional derivative on the right hand side is in the L2 (R) sense. In particular, the operators M±H interchange with the S-transform. Proof. The case H ≥ 1/2 is straightforward in view of (6.2.13). Let H < 1/2: Then by (6.2.13): ∫ [ 2 ] −(H−1/2) −(H−1/2) X)x Ψ − D±,ϵ (E[Xx Ψ]) dx E (D± ∫R [ ( )] 2 −(H−1/2) −(H−1/2) = E Ψ (D X) − (D X) dx x x ± ±,ϵ R [( ∫ )2 ] −(H−1/2) −(H−1/2) 2 ≤ E[Ψ ] · E (D± X)x − (D±,ϵ X)x dx R

→ 0 as ϵ → 0+ by the definition of the fractional derivative of a stochastic process. In view of the definition of M±H in the case H < 1/2 the proof is finished.

346

CHAPTER 6. FRBM-MARKETS We also need the following lemma:

Lemma 6.2.21. Let X : M → (L2 ), M ⊂ R such that X ∈ Lp (M, (Lp )), [ ] p > 1. Then E X : eI(f ) : ∈ Lp (M ) for all f ∈ L2 (R). Proof. By Hölder’s inequality: ∫ [ E Xt : eI(η) : M [ ]q/p ∫ I(η) q ≤ E :e :

] p dt |E [|Xt |p ] dt < ∞,

M

where 1/q = 1 − 1/p. We can now prove the main result of this section: Theorem 6.2.22. Let 0 < H < 1, a ≤ b ∈ R and X : [a, b] → R such that X ∈ L1/H ([a, b], (L1/H )). Moreover assume M−H (1(a, b)X) ∈ L2 (R, (L2 )) in the case H < 1/2. Then: ∫ b

∫ Xt dBtH

a

= R

1/2

M−H (1(a, b)X)(t)dBt

in the usual sense, i.e. if one of the integrals exists, then the other one does and both integrals coincide. Proof. From the previous lemma we know, that (SX)(η) ∈ L1/H ([a, b]) for all η ∈ §. Moreover, S(M−H (1(a, b)X))(η) ∈ L2 (R), if H < 1/2. Now we obtain from lemma 6.2.20 and the fractional integration by parts rule (corollary 6.2.9): ∫ ( ) S (M−H (1(a, b)X)t ) (η) · η(t)dt ∫R = M−H (S(1(a, b)(t)Xt )(η)) · η(t)dt R ∫ b = (SXt )(η) · (M+H η)(t)dt, a

which proves (in view of definition 6.2.15) the assertion.

6.3

An Itˆ o Formula

Let H ∈ (0, 1) and define the spaces { } L2H := f : R → R; M−H f ∈ L2 (R)

ˆ FORMULA 6.3. AN ITO

347

We endow L2H with the inner product (f, g)L2 := (M−H f, M−H g)0 . H

Notice that L2H is not complete, if H > 1/2, see Pipiras/Taqqu [27]. Let us now consider processes of the form Xt := ⟨·, M−H (1(0, t)σ)⟩ with a continuous function σ : R → R such that 1(0, t) · σ ∈ L2H . Recall that the special case σ(x) = 1 yields P -a.s. a fractional Brownian motion BtH . In general, we have: Theorem 6.3.1. Let σ : R → R continuous such that 1(0, t) · σ ∈ L2H . Then in (L2 ):

∫ Xt :=

⟨·, M−H (1(0, t)σ)⟩

t

σ(s)dBsH .

= 0

Proof. By the assumption 1(0, t) · σ ∈ L2H Xt is an element of (L2 ). Its Stransform is given by: ∫ ⟨η + ω, M−H (1(0, t)σ)⟩dP (ω) ∫ H = ⟨η, M− (1(0, t)σ)⟩ dP (ω) + E[Xt ]

S(Xt )(η) =

= ⟨η, M−H (1(0, t)σ)⟩ Applying the fractional integration by parts rule (which can be justified in the same way as in [2], lemma 2.6) yields the assertion. We are going to prove the following Itô formula: Theorem 6.3.2. Fix 0 < a ≤ b and let X as in the above theorem. Furthermore assume that (i) there is a constant K > 0 such that for all t ∈ [a, b] H M− (1(0, t)σ) 2 ≤ 1 ; 0 K (ii)

d dt

H M− (1(0, t)σ) 2 exists as an element of L1 ([a, b]); 0

(iii) F ∈ C 1,2 ([a, b] × R) and there a constants C ≥ 0 and λ < K/4 such that

348


for all (t, x) ∈ [a, b] × R 2 } { ∂ ∂ ∂ 2 max |F (t, x)|, F (t, x) , F (t, x) , 2 F (t, x) ≤ Ceλx . ∂t ∂x ∂x Then the following equality holds in (L2 ): ∫ b ∫ b ∂ ∂ H σ(t) F (t, Xt )dBt = F (b, Xb ) − F (a, Xa ) − F (t, Xt )dt(6.3.1) ∂x a a ∂t ∫ 2 ∂ 2 1 b d H − M− (1(0, t)σ) 0 2 F (t, Xt )dt 2 a dt ∂x Proof. By definition 6.2.15 we have to prove that the right hand side is an element of (L2 ) and has S-transform given by ( ) ∫ b ∂ H H (M+ η)(t)S σ(t) F (t, Bt ) (η)dt ∂x a ( ) ∫ b ∂ H H = (M+ η)(t)σ(t)S F (t, Bt ) (η)dt ∂x a

(6.3.2)

2

∂ ∂ ∂ Notice first that by the growth condition (iii) for G = F, ∂t F, ∂x F, ∂x 2 F and

t ∈ [a, b] 2 ∥G(t, Xt )∥20 ≤ C 2 (1 − 4λ M−H (1(0, t)σ) 0 )−1/2 ≤ const.

(6.3.3)

Consequently all terms on the right hand side are elements of (L2 ). For the last integral this can be proved in the following way:

∫ b

2 ∂ 2 d H

M− (1(0, t)σ) 0 2 F (t, x)dt

∂x a dt

0 ∫ b d H 2 ∂ 2

M− (1(0, t)σ)

dt ≤ F (t, x) dt

0 ∂x2 a 0 ∫ b 2 d H M− (1(0, t)σ) dt < ∞ ≤ const. 0 a dt using theorem 6.2.14, (ii) and (6.3.3). We shall now calculate the S-transform of the right hand side of (6.3.1): To { 2} 1 this end let g(t, x) := √2πt exp −x be the heat kernel. By the definition of 2t the S-transform we have:

∫

S (F (t, Xt )) (η) = F (t, Xt (η + ω))dP (ω) ∫ ( ) = F t, ⟨ω, M−H (1(0, t)σ)⟩ + ⟨η, M−H (1(0, t)σ)⟩ dP (ω) (

∫ =

F R

∫

)

t

(M+H η)(s)σ(s)ds

t, u + 0

g(|M−H (1(0, t)σ)|20 , u)du

ˆ FORMULA 6.3. AN ITO

349

Here the last equality follows from the fact that Xt is centered Gaussian with variance |M−H (1(0, t)σ)|20 . By the growth condition (iii) we may interchange integration and differentiation and obtain: d S(F (t, BtH ))(η) dt ( ) ∫ ∫ t ∂ F t, u + (M+H η)(s)σ(s)ds g(|M−H (1(0, t)σ)|20 , u)du R ∂t 0

=

(M+H η)(t)σ(t) ( ) ∫ t ∫ ∂ H (M+ η)(s)σ(s)ds g(|M−H (1(0, t)σ)|20 , u)du F t, u + × ∂x 0 R d H 2 + |M (1(0, t)σ)|0 dt − ( ) ∫ ∫ t ∂ H × F t, u + (M+ η)(s)σ(s)ds g(|M−H (1(0, t)σ)|20 , u)du ∂t R 0 =: (I) + (II) + (III). +

As the heat kernel fulfills

∂ ∂t g

=

1 ∂2 2 ∂x2 g

integration by parts yields:

d |M H (1(0, t)σ)|20 dt − ( ) ∫ ∫ t ∂2 H × F t, u + (M+ η)(s)σ(s)ds g(|M−H (1(0, t)σ)|20 , u)du 2 ∂x R 0 ( 2 ) d ∂ H 2 H = |M (1(0, t)σ)|0 · S F (t, Bt ) (η). dt − ∂x2

(III) =

Moreover,

( (I) = S

) ∂ H F (t, Bt ) (η) ∂t

and

( (II) =

(M+H η)(t)σ(t)S

) ∂ H F (t, Bt ) (η) ∂x

Consequently, the S-transform of the right hand side of (6.3.1) equals (6.3.2) and the proof is finished. An important corollary is: Corollary 6.3.3. Let H ∈ (0, 1), T > 0 and F ∈ C 1,2 ([0, T ] × R) such that there a constants C ≥ 0 and λ < (4T 2H )−1 such that for all (t, x) ∈ [a, b] × R 2 } ∂ ∂ ∂ 2 max |F (t, x)|, F (t, x) , F (t, x) , 2 F (t, x) ≤ Ceλx . ∂t ∂x ∂x {

350


Then the following equality holds in (L2 ): ∫ T ∫ T ∂ ∂ F (t, BtH )dBtH = F (T, BTH ) − F (0, 0) − F (t, BtH )dt ∂x ∂t 0 0 ∫ T 2 ∂ −H t2H−1 2 F (t, BtH )dt ∂x 0 Proof. Let 0 < ϵ ≤ T and recall that BtH = ⟨·, M−H 1(0, t)⟩ P -a.s. As |M−H 1(0, t)|20 = t2H we can apply theorem 6.3.2 with K = T −2H to obtain: ∫ T ∫ T ∂ ∂ F (t, BtH )dBtH = F (T, BTH ) − F (ϵ, BϵH ) − F (t, BtH )dt ∂x ∂t ϵ ϵ ∫ T 2 ∂ −H t2H−1 2 F (t, BtH )dt ∂x ϵ 2

∂ ∂ ∂ By the growth condition on G = F, ∂t F, ∂x F, ∂x 2 F one can check (see (6.3.3)

for a similar argument), that all terms on the right hand side converge in (L2 ) as ϵ tends to zero. Hence, the left hand side also converges and the proof is finished. Remark 6.3.4. Corollary 6.3.3 generalizes the Itô formula from Alos et. al [1] to the case H < 1/4. It also contains the versions of Itô’s formula in Bender [2] provided F is classically differentiable. Moreover, in corollary 6.3.3 some kind of exponential growth is allowed. However, note that the Itô formula in [2] holds in the framework of tempered distributions. The next lemma shows that the assumptions (i) and (ii) in theorem 6.3.2 are superfluous in the case H > 1/2: Lemma 6.3.5. Let σ : R → R be continuous and H > 1/2. Then: ∫ t∫ τ H M− (1(0, t)σ) 2 = 2H(2H − 1) σ(s)σ(τ )|s − τ |2H−2 dsdτ 0 0

0

In particular: (i) For all t ≥ 0

H M− (1(0, t)σ) 2 ≤ max |σ(s)|2 t2H 0 s∈[0,t]

(ii)

is differentiable in t and for all t ≥ 0 ∫ t 2 d H σ(s)σ(t)|s − t|2H−2 ds M− (1(0, t)σ) 0 = 2H(2H − 1) dt 0 ≤ 2H max |σ(s)|2 t2H−1

|M−H (1(0, t)σ)|20

s∈[0,t]

(6.3.4)

6.4. GFRBM

351

Proof. By an identity of Gripenberg/Norros [11], we have (see Hu/Øksendal [16] for a detailed proof): H M− (1(0, t)σ) 2 ∫ ∫0 = H(2H − 1) 1(0, t)(s)σ(s)1(0, t)(τ )σ(τ )|s − τ |2H−2 dsdτ. R

R

(6.3.4) easily follows. The other assertions are direct implications of (6.3.4).

6.4

Geometric Fractional Brownian Motion

In this section we introduce geometric fractional Brownian motions with deterministic (but not necessarily constant) coefficients: Definition 6.4.1. Let H ∈ (0, 1), x0 > 0 and σ, r : [0, ∞) → R continuous. Furthermore assume: (i) For any T > 0 there is a constant KT > 0 such that for all t ∈ [0, T ] H M− (1(0, t)σ) 2 ≤ 1 0 KT and

2 lim M−H (1(0, t)σ) 0 = 0;

t→0

2 d H (ii) For any t > 0 dt M− (1(0, t)σ) 0 exists and

d dt

H M− (1(0, t)σ) 2 ∈ L1 ([0, T ]) 0

for any T > 0. Then we call {∫ Pt := x0 exp 0

t

} 2 1 H H r(s)ds − M− (1(0, t)σ) 0 + ⟨·, M− (1(0, t)σ)⟩ 2

a geometric fractional Brownian motion with coefficients H, x0 , σ, r. Remark 6.4.2. The conditions (i) and (ii) are fulfilled, if σ is constant, because then we have |M−H (1(0, t)σ)|20 = σ 2 t2H . By lemma 6.3.5 conditions (i) and (ii) are also satisfied whenever H > 1/2. The following Itô formula is a consequence of theorem 6.3.2. Theorem 6.4.3. Let P be a geometric fractional Brownian motion, T > 0 and F ∈ C 1,2 ([0, T ] × R) such that F,

∂ ∂ ∂2 ∂t F (t, x), ∂x F (t, x), ∂x2 F (t, x)

are of

352


polynomial growth. Then the following equality holds in (L2 ): ∫

T

0

∫ −

0

T

∫ T ∂ ∂ H σ(t)Pt F (t, Pt )dBt = F (T, PT ) − F (0, x0 ) − F (t, Pt )dt ∂x ∂t 0 ∫ 2 ∂ 1 T d H ∂2 r(t)Pt F (t, Pt )dt − M− (1(0, t)σ) 0 Pt2 2 F (t, Pt )dt ∂x 2 0 dt ∂x

Proof. Apply theorem 6.3.2 to b = T , a = ϵ < b and F (t, g(t, x)) with {∫

t

g(t, x) := x0 exp 0

} 2 1 H r(s)ds − M− (1(0, t)σ) 0 + x 2

and let ϵ → 0+ in (L2 ). The special case F (t, x) = x yields: Corollary 6.4.4. Let P be a geometric fractional Brownian motion. Then for all t ≥ 0:

∫

∫

t

σ(s)Ps dBsH

Pt = x0 +

t

r(s)Ps ds.

+ 0

0

Remark 6.4.5. The above corollary is a fractional analogue of the Doléans-Dade identity. It is a justification for the name geometric fractional Brownian motion.

6.5

Solvability of a Class of Linear FrBSDEs

A fractional BSDE is an equation of the form: ∫

T

Yt = ξ −

∫

T

f (t, Yt , Zt )dt −

t

Zt dBtH

(6.5.1)

t

Here the data ξ and f are given. ξ is an FT -measurable random variable where Ft denotes the filtration generated by the fractional Brownian motion B H . f (t, y, z) is an Ft -adapted process for every pair (y, z) ∈ R2 . Definition 6.5.1. A pair (Y, Z) is called a solution of the fractional backward stochastic differential equation (FrBSDE) (8.2.6), if the equation (8.2.6) holds in (L2 ) for any t ∈ [0, T ] and Y , Z are Ft -adapted processes such that ∫

T

E 0

[

Yt2

]

[( ∫ dt + E 0

T

)2 ] |Zt |dt

< ∞.

(6.5.2)

6.5. LINEAR FRBSDES

6.5.1

353

Reduction to a PDE

In this section we show how to reduce a special case of the FrBSDE (8.2.6) to the solvability of a partial differential equation (PDE). The procedure is similar to the Ma-Protter-Yong four step scheme [21] for BSDEs driven by a classical Brownian motion: Let Xt , t ∈ [0, T ], as in theorem 6.3.2 and assume that the PDE 2 1 d H M− (1(0, t)σ) 0 uxx (t, x) − f (t, x) (6.5.3) 2 dt = (B(t, x)σ(t) − b′ (t))ux (t, x) + A(t, x)u(t, x); (t, x) ∈ (0, T ) × R ut (t, x) +

u(T, x) = g(x); x ∈ R

(6.5.4)

has a classical solution that fulfills the growth condition of theorem 6.3.2 for 0 < a < b < T . Then theorem 6.3.2 yields for Φt = x0 + Xt + b(t): ∫ b σ(t)ut (t, Φt )dBtH u(a, Φa ) = u(b, Φb ) − a ∫ b − f (t, Φt ) + A(t, Φt )u(t, Φt ) + B(t, Φt )σ(t)ux (t, Φt )dt a

Let us now set Yt := u(t, Φt ) and Zt := σ(t)ux (t, Φt ). Provided u has nice properties we can go to the limits b → T and a → 0 and the pair (Y, Z) fulfills the integrability conditions (6.5.2). Then the pair (Y, Z) is a solution of the linear FrBSDE: ∫ Yt = g(ΦT ) −

T

∫ f (s, Φs ) + A(s, Φs )Ys + B(s, Φs )Zs ds −

t

T

Zs dBsH . (6.5.5) t

We shall carry out that procedure under suitable conditions on the coefficients in the next sections.

6.5.2

Solvability of the PDE

In this section we explicitly solve the PDE (8.2.39) in the case of only time dependent coefficients A, B and f . Throughout this section we impose the following set of conditions: (P1) S ∈ C 1 ((0, T ], R) ∩ C([0, T ], R) is strictly increasing with S(0) = 0. (P2) r, A ∈ C((0, T ], R) ∩ L1 ([0, T ], R) and f ∈ C((0, T ], R).

354


(P3) g ∈ C(R, R) and there are constants C and λ < (8S(T ))−1 such that for all (t, x) ∈ [0, T ] × R |g(t, x)| ≤ Ceλx

2

Theorem 6.5.2. Assume (P1)–(P3). Then the PDE 1 ut (t, x) + S ′ (t)uxx (t, x) 2 = r(t)ux (t, x) + A(t)u(t, x) + f (t); (t, x) ∈ (0, T ) × R u(T, x) = g(x); x ∈ R

(6.5.6)

(6.5.7)

has a classical solution given by ∫ u(t, x) := −

T

f (s)ds + √

t

∫ ×

g(y) exp R

Proof. Let 1 v(t, x) := √ 2πt

e−

∫T t

A(s)ds

2π(S(T ) − S(t))  ( )2  ∫T    − x − t r(s)ds − y   

 

2S(T ) − 2S(t) {

∫ g(y) exp R

− (x − y)2 2t

(6.5.8)

dy

} dy

It is well known that v is a solution of the heat equation 1 vt (t, x) − vxx (t, x) = 0; (t, x) ∈ (0, S(T )) × R 2 v(0, x) = g(x); x ∈ R Noting ∫ u(t, x) = −

T

−

f (s)ds + e

∫T t

( A(s)ds

∫

v S(T ) − S(t), x −

t

)

T

r(s)ds t

the assertion can be obtained by differentiating. Lemma 6.5.3. Assume (P1)–(P3) and let u be given by (6.5.8). Then there are constants C and κ < (4S(T ))−1 independent of t and x such that: |u(t, x)| ≤ C exp{κx2 } 1 |ux (t, x)| ≤ C √ exp{κx2 } S(T ) − S(t) 1 |uxx (t, x)| ≤ C exp{κx2 } (S(T ) − S(t))3/2

6.5. LINEAR FRBSDES

355

Proof. We prove the estimate for ux , the other estimates are similar. Notice first, that ∫T

e− t A(s)ds ux (t, x) = − √ 2π(S(T ) − S(t))  ( )2  ∫T ∫T   ∫   − x − r(s)ds − y (x − t r(s)ds − y) t × g(y) √ exp dy   2S(T ) − 2S(t) S(T ) − S(t) R   Consequently, for any ϵ > 0 there is a constant Kϵ such that |ux (t, x)| ≤ Kϵ (S(T ) − S(t))

∫

−1

(ϵ−1/2)

|g(y)|e

R

Using (P3) and substituting y for x −

∫T

(x−

∫T 2 t r(s)ds−y S(T )−S(t)

)

dy

r(s)ds − y we have:

t

|ux (t, x)| ≤ Kϵ (S(T ) − S(t))−1 ∫ ∫T 2 y2 y2 (−1/4) S(T )−S(t) (ϵ−1/4) S(T )−S(t) × e e eλ(x− t r(s)ds−y) dy R

By Young’s inequality ∫ 2 y2 (ϵ−1/4) S(T )−S(t) λ(x− tT r(s)ds−y )

e

e

∫ y2 −(1/4−ϵ−2λS(T )) S(T )−S(t) 2λ(x− tT r(s)ds)2

≤ e

e

≤ e2λ(x−

∫T t

r(s)ds)2

if ϵ is sufficiently small by (P3). Choosing ϵ sufficiently small, there is a constant K ′ such that ′

−1

|ux (t, x)| ≤ K (S(T ) − S(t))

∫

2

y − 4S(T )−4S(t)

e R

dy · e2λ(x−

∫T t

r(s)ds)2

Hence, the assertion is true for any 2λ < κ < (4S(T ))−1 by Young’s inequality.

A straightforward corollary is: Corollary 6.5.4. Assume (P1)–(P3), let u be given by (6.5.8) and fix an ϵ > 0. Then there are constants Cϵ and κ < (4S(T ))−1 such that for all (t, x) ∈ [ϵ, T − ϵ] × R: |ut (t, x)| ≤ Cϵ exp{κx2 }

356


6.5.3

Explicit Solutions of a Class of Linear FrBSDEs

We now apply the results from the previous sections to solve linear FrBSDEs. Let us first consider the case H > 1/2. We assume: (B1) T > 0, 1/2 < H < 1 and σ : [0, T ] → R+ continuous is bounded away from zero by a constant k > 0. (B2) x0 ∈ R, b ∈ C 1 ((0, T ], R) ∩ C([0, T ], R). (B3) (P2) holds with r(t) = σ(t)B(t) − b′ (t) and B : [0, T ] → R. (B4) g ∈ C(R, R) and there are constants λ < (8T 2H max0≤t≤T |σ(t)|2 )−1 and C such that for all (t, x) ∈ [0, T ] × R |g(t, x)| ≤ Ceλx

2

Theorem 6.5.5. Assume (B1)–(B4) and let Φt = x0 +b(t)+

∫t 0

σ(s)dBsH . Then

the frBSDE ∫ Yt = g(ΦT ) −

T

∫ f (s) + A(s)Ys + B(s)Zs ds −

T

Zs dBsH t

t

has a solution given by Yt := u(t, Φt ), Zt := σt ux (t, Φt ) and ∫ u(t, x) := −

T

f (s)ds + √

e−

∫T t

A(s)ds

2π(|M−H (1(0, T )σ)|20 − |M−H (1(0, t)σ)|20 )  ( )2  ∫T  ′ ∫  − x − t (σ(s)B(s) − b (s))ds − y   dy × g(y) exp H 2 H 2  R  2|M− (1(0, T )σ)|0 − 2|M− (1(0, t)σ)|0   t

Proof. By lemma 6.3.5 and (B1) S(t) := |M−H (1(0, t)σ)|20 satisfies (P1). The same lemma together with (B4) implies (P3). Finally, (P2) follows from (B2). Hence, by theorem 6.5.2 u is a classical solution of the PDE 1d ut (t, x) + |M H (1(0, t)σ)|20 uxx (t, x) 2 dt − [ ] = σ(t)B(t) − b′ (t) ux (t, x) + A(t)u(t, x) + f (t); (t, x) ∈ (0, T ) × R u(T, x) = g(x); x ∈ R Moreover, by lemma 6.5.3 and corollary 6.5.4 F (t, x) := u(t, x0 + b(t) + x) fulfills the growth condition (iii) of theorem 6.3.2. Applying theorem 6.3.2 for

6.5. LINEAR FRBSDES

357

0 < a < b < T we get in (L2 ) ∫

b

u(a, Φa ) = u(b, Φb ) − σ(t)ut (t, Φt )dBtH a ∫ b − f (t) + A(t)u(t, Φt ) + B(t)σ(t)ux (t, Φt )dt.

(6.5.9)

a

We now check the integrability condition (6.5.2) for (Y, Z): By theorem 6.2.14, lemma 6.5.3 and (B1) [( ∫

T

E ∫ ≤ C

)2 ]1/2 |Zt |dt

[ (∫ =E

0 T

√

0

T

)2 ]1/2 |σ(t)||ux (t, Φt )|dt

(6.5.10)

0

1 |M−H (1(0, T )σ)|20 − |M−H (1(0, t)σ)|20

exp{κΦ2t } dt 0

As in (6.3.3) we have exp{κΦ2t } 0 ≤ const. From lemma 6.3.5 and (B1) ∫

T

0

∫

T

= 0

≤ k

−1

= k −1

1

√

dt |M−H (1(0, T )σ)|20 − |M−H (1(0, t)σ)|20 ]−1/2 [ ∫ T∫ τ 2H−2 σ(s)σ(τ )|s − τ | dsdτ dt 2H(2H − 1) t

∫ ∫

T

[ ∫ 2H(2H − 1)

T

T

∫

τ

]−1/2 |s − τ |

2H−2

t

0

0

0

dsdτ

dt

0

1 [ 2H ]−1/2 Γ( 21 )Γ( 2H ) T − t2H dt = k −1 T 1−H 1 2HΓ( 2H + 12 )

Note, that the last identity holds for arbitrary 0 < H < 1. Thus, we obtain from (6.5.10)

[( ∫ E

T

)2 ] |Zt |dt 0, 0 < H < 1/2 and σ > 0 is constant. In the new situation we have: Theorem 6.5.6. Assume (B1’) and (B2)–(B4) and let Φt = x0 + b(t) + σBtH . Then the frBSDE ∫ Yt = g(ΦT ) −

T


t

T

Zs dBsH t

has a solution given by Yt := u(t, Φt ), Zt := σux (t, Φt ) and ∫ u(t, x) := −

T

f (s)ds + √

t

∫ ×

g(y) exp R

e−

∫T t

A(s)ds

2πσ 2 (T 2H − t2H )  ( )2  ∫T  ′  − x − t (σB(s) − b (s))ds − y    

2σ 2 (T 2H − t2H )

 

dy

Proof. The proof is the same as the proof of theorem 6.5.5 in the case of constant σ. In this case all applications of lemma 6.3.5 can be replaced by the identity: |M−H (1(0, t)σ)|20 = σ 2 t2H

Remark 6.5.7. If we look at the special cases f = A = B = 0 and let t = 0, then theorem 6.5.5 and theorem 6.5.6 yield Clark-Ocone type representations of the random variables g(ΦT ) in terms of the fractional Itô integral.

6.6. PRICING OF DERIVATIVES IN FRBSM Remark 6.5.8. Considering the case b(t) =

∫t 0

359

2 r(s)ds− 21 M−H (1(0, t)σ) 0 , g(x) =

G(ex ) and G of polynomial growth, theorem 6.5.5 and 6.5.6 give existence results for frBSDEs ∫ Yt = g(PT ) −

T


t

T

Zs dBsH t

where P is a geometrical fractional Brownian motion (with σ constant in the case 0 < H < 1/2). We shall use that special case for the pricing of derivatives in the next section.

6.6

The Pricing of Derivatives in a Fractional Black Scholes Market

In this section we apply the obtained results to finance. We first model a fractional Black Scholes market. Then we solve the pricing problem for some kind of options. The question of arbitrage is going to be answered in a forthcoming paper [3].

6.6.1

The Model

We model a market consisting of one bond and one stock on the time interval [0, T ]. The bond is given by

{∫

t

} r(s)ds ,

At := exp

(6.6.1)

0

where the nonnegative continuous function r is the interest rate. The stock price is given by a geometric fractional Brownian motion {∫ t } 2 1 H H Pt := p0 exp µ(s)ds − M− (1(0, t)σ) 0 + ⟨·, M− (1(0, t)σ)⟩ . (6.6.2) 2 0 Here x0 > 0, the drift µ and the volatility σ are continuous, nonnegative and σ is bounded away from zero. In the case 0 < H < 1/2 σ is assumed to be constant. A portfolio πt is an Ft -adapted process, the amount of money invested in the stock at time t. The rest of the wealth is invested in the bond. The portfolio is called feasible for the initial wealth y0 , if [( ∫ )2 ] T E |σ(t)πt |dt 1/2 is more important, because statistical analysis of stock prices indicates Hurst parameters of about 0.7, see Shiryaev [29], p.378. (ii) In the case of constant coefficients our results coincide with the pricing formulae obtained in Hu/Øksendal [16] and Elliott/van der Hoek [10]. But their results rely on a rather informal argument of integration with respect to a frac˜ H under an equivalent measure. tional Brownian motion B (iii) The above argumentation does not justify to call Y0 the fair price of the contingent claim G(PT ). It is an upper bound for the price considering feasible portfolios.

(iv) To complement the above results and to see that Y0 indeed is the fair price of G(PT ) the following steps have to be done: (a) Define a subclass of the feasible portfolios, the admissible portfolios. (b) Show that the market is arbitrage free with this class of admissible portfolios. (c) Prove that the above constructed hedging portfolios are admissible and that there is no admissible hedging portfolio with smaller initial wealth. These steps are carried out in the following sections:

6.7. CHANGE OF MEASURE

363

6.7

The Fractional Itˆ o Integral under Change of Measure

6.7.1

Expectation of the Fractional Itˆ o Integral under Change of Measure

In this section we calculate the expectation of a fractional Itô integral under the measure Qf , f ∈ L2 (R), given by (6.2.8). We begin with the rather simple case Qη , η ∈ §: To this end let X : [a, b] → (L2 ) be H-fractional Itô integrable (0 < H < 1). In that case: [∫ E Qη

b

] (∫ b ) ∫ b Xt dBtH = S Xt dBtH (η) = E Qη [Xt ] (M+H η)(t)dt (6.7.1)

a

a

a

by definition 6.2.15 and (6.2.9). Let us now consider the general case f ∈ L2 (R). We choose a sequence (ηn )n∈N ⊂ § such that ηn converges to f in L2 (R). By corollary 6.2.7 the left [∫ ] b hand side of (6.7.1) (with η replaced by ηn ) converges to E Qf a Xt dBtH . So we have to impose conditions that ensure the convergence of the right hand side:

Theorem 6.7.1. Let f ∈ L2 (R) and Qf be given by (6.2.8). Moreover assume that X : [a, b] → (L2 ) is H-fractional Itˆ o integrable (0 < H < 1), X ∈ L1/H ([a, b], (L1/H )) and M−H (1(a, b)X) ∈ L2 (R, (L2 )). Additionally, suppose M+H f ∈ L1/(1−H) (R) in the case H < 1/2. Then: [∫ E

]

b

Qf

Xt dBtH a

∫

b

E Qf [Xt ] (M+H f )(t)dt.

= a

Proof. Let (ηn )n∈N ⊂ § be given such that ηn converges to f in L2 (R). By the considerations at the beginning of this section it remains to prove, that the right ] [∫ b hand side of (6.7.1) (with η replaced by ηn ) converges to E Qf a Xt (M+H η)(t)dt . By the supposed integrability conditions, lemma 6.2.20 and lemma 6.2.21 we may apply the fractional integration by parts rule in the form of corollary 6.2.9

364


to obtain: ∫ b [ ] [ ] I(η ) H I(f ) H n E Xt : e : (M+ )(ηn ) − E Xt : e : (M+ )(f )dt ∫a ( ) ≤ |ηn (t)|E M−H (1(a, b)X)t : eI(ηn ) : − : eI(f ) : dt R ∫ + |ηn (t) − f (t)|E M−H (1(a, b)X)t ) · : eI(f ) : dt R

= I1 + I2 By Hölder’s inequality (∫ I2

≤

|ηn − f |0 ·

R

(∫ ≤

|ηn − f |0 ·

R

2 )1/2 H I(f ) E M− (1(a, b)X)t ) · : e : dt )1/2 ( [ [ ])1/2 2 ] H E M− (1(a, b)X)t ) dt E : eI(f ) : 2

→ 0 as n → ∞. The same argument applied to I1 yields: (∫ ≤

I1

|ηn |0 ·

R

)1/2 [ 2 ] H E M− (1(a, b)X)t ) dt

( [ ])1/2 × E | : eI(ηn ) : − : eI(f ) : |2 → 0 as n → ∞ by corollary 6.2.7. Hence, ∫

∫

b

lim

E

n→∞ a

Qηn

[Xt ] (M+H ηn )(t)dt

b

E Qf [Xt ] (M+H f )(t)dt.

= a

which proves the assertion.

Remark 6.7.2. If

∫b a

Xt (M+H f )(t)dt exists as Pettis integral and the assumptions

of the preceding theorem are valid, then by (6.2.13): [∫ E

Qf

]

b

Xt dBtH a

[∫ =E

Qf

]

b

Xt (M+H f )(t)dt

.

(6.7.2)

a

Note that Xt (M+H f )(t) is Pettis integrable, if H < 1/2 and the assumptions of


365

the preceding theorem are valid. For ∫ b [ ] E Xt (M+H f )(t)Ψ dt a

(∫ ≤

b

)1−H (∫ |(M+H f )(t)|1/(1−H) dt

a

(∫ ≤

b

1 |(M+H f )(t)| 1−H dt

)1−H (∫

a b

b

)H 1/H

|E [Xt Ψ]|

[

E |Xt |

1 H

]

dt

)H 1 1−H dt · E[|Ψ| 1−H ] H

a

a

< ∞ for all Ψ ∈ (L2 ), as (1 − H)−1 < 2, if H < 1/2.

6.7.2

A Girsanov Theorem for the Fractional Itˆ o Integral

The classical Girsanov theorem states that a Brownian motion with drift becomes a Brownian motion under some change of measure. In the fractional Brownian motion case similar results have been obtained by Norros et al. [23], Hu/Øksendal [16] and Elliott/van der Hoek [10]. In this section we explore, how a fractional Itô integral behaves under change of measure. In that way we generalize the results mentioned above. So let again Qf , f ∈ L2 (R), be the measure defined by (6.2.8). The probability space (Ω, G, Qf ) carries a two sided Brownian motion given by ∫ ˜t := Bt − t f (t)dt by virtue of the classical Girsanov theorem. Hence, all B 0 constructions from the preliminary section carry over to this probability space. We shall use the notation SQf for the S-transform with respect to this new probability space:

[ B˜ ] (SQf X)(η) := E Qη : eI (η) : X .

S and I denote the S-transform with respect to the space (Ω, G, P ), resp. the Wiener integral with respect to B as before. Subsequently we shall need the following identity which can be verified directly: : eI

˜ B (f )

: · : eI(g) : =: eI(g+f ) :

(6.7.3)

Our Girsanov theorem has the following form: Theorem 6.7.3. Let the assumptions of theorem 6.7.1 hold. Moreover, assume that Xt (M+H f ) is Pettis integrable over [a, b] (all assumptions with respect to

366


the probability space (Ω, G, P )) and that [ ∫ 2 ] ∫ b b E Qf Xt dBtH − Xt (M+H f )(t) < ∞ a

Then the identity ∫

b

∫ ˜H = Xt dB t

a

holds in

L2 (Ω, G, Q

b

∫

b

Xt dBtH −

a f)

(6.7.4)

a

Xt (M+H f )(t)dt a

and consequently Qf -a.s. and P -a.s.

Remark 6.7.4. From the remark following theorem 6.7.1 we see that the additional assumption of Pettis integrability is superfluous in the case H < 1/2. Proof. We want to apply theorem 6.7.1 to f + η, η ∈ §. So recall that in the 1

case H < 1/2 M+H (η) ∈ L 1−H (R) for all η ∈ § by the proof of lemma 2.6 in 1

Bender [2]. Hence, M+H (f + η) ∈ L 1−H (R), if H < 1/2. By (6.7.3), theorem 6.7.1 and the assumed Pettis integrability we have: ) (∫ b ∫ b H H Xt (M+ f )(t)dt (η) SQf Xt dBt − a a [∫ b ] ∫ b Qf +η H H = E Xt dBt − Xt (M+ f )(t)dt ∫

a

a

∫

b

b

+ η)(t)dt − E Qf +η [Xt ] (M+H f )(t)dt a a ∫ b ∫ b ) ( SQf Xt (η)(M+H η)(t)dt E Qf +η [Xt ] (M+H η)(t)dt = = =

E

Qf +η

[Xt ] M+H (f

a

a

In view of definition 6.2.15 applied to the space (Ω, G, Qf ) the asserted identity is proven. Note, that it also holds P -a.s., as the measures P and Qf are equivalent.

6.7.3

Removal of a Drift

Usually (and in particular in no arbitrage arguments) the classical Girsanov ∫b theorem is applied to remove a given drift a Xt g(t)dt. The fractional version stated in theorem 6.7.3 allows to kill a drift of the form: ∫ b Xt (M+H f )(t)dt. a

Hence, we need to explore, whether the fractional equation (M+H f ) = g a.s.

(6.7.5)


367

has a solution f ∈ L2 (R) for given g. We start with the case H < 1/2: Note that in that case the theorems 6.7.1 and 6.7.3 require that (M+H f ) ∈ L1/(1−H) (R). So the best we can hope for is: Theorem 6.7.5. Let 0 < H < 1/2 and g ∈ L1/(1−H) (R). Then the equation (6.7.5) has a solution f ∈ L2 (R). −1 Proof. Let f (x) = KH (I+

1/2−H

g)(x), where KH is the constant given by (6.2.6).

Then by theorem 6.1 in Samko et al. [28] and (6.2.6): 1/2−H

M+H f = D+

(

1/2−H

(I+

) ) (g) = g

Now the Hardy-Littlewood theorem (theorem 5.3 in Samko et. al [28]) implies that f ∈ L2 (R). In the case H > 1/2 we have: H−1/2

Theorem 6.7.6. Let 1/2 < H < 1 and assume that g = I+

(ϕ) for a

ϕ ∈ L2 (R). Then equation (6.7.5) has a solution f ∈ L2 (R). The proof is given in Samko et al. [28], theorem 30.6. Moreover, the solution f is explicitly calculated. For our purpose it is sufficient to solve equation (6.7.5) on the interval [a, b]. Hence, the following corollary is useful, as its assumptions are easier to verify: Corollary 6.7.7. Let 1/2 < H < 1, g = 1(a, b)ψ such that ψ : [a, b] → R H¨ older continuous with exponent λ > H − 1/2. Then equation (6.7.5) has a solution f ∈ L2 (R). Proof. Let us first extend ψ to a λ-Hölder continuous function ψ˜ : R → R. Then: g = 1(a, b)ψ˜ and the assertion follows from Lemma 5.3 in Pipiras/Taqqu [27], theorem 6.7 in Samko et al. [28] and the previuos theorem. Thus, we have proven the following variant of theorem 6.7.1: Theorem 6.7.8. Let 0 < H < 1 and a < b ∈ R. Moreover assume that X : [a, b] → (L2 ) is H-fractional Itˆ o integrable, X ∈ L1/H ([a, b], (L1/H )) and M−H (1(a, b)X) ∈ L2 (R, (L2 )). Additionally, suppose that g ∈ L1/(1−H) (R), if

368


H < 1/2, resp. that g is λ-H¨ older continuous with λ > H − 1/2, if H > 1/2. Then there is a f ∈ L2 (R) such that: [∫ b ] ∫ b Qf H E Xt dBt = E Qf [Xt ] g(t)dt. a

a

Remark 6.7.9. We could also formulate a similar variant of the fractional Girsanov theorem 6.7.3. But notice, that this variant would require the integrability condition (6.7.4) to hold with respect to the measure Qf , where f is the solution of equation (6.7.5) on [a, b]. However, for the no arbitrage argument in the next section we are going to use theorem 6.7.8.

6.8

Absence of Arbitrage in a Fractional Black Scholes Market

In this section we first model a market, in which the stock price is given by a geometric fractional Brownian motion. In the case H > 1/2 we allow the coefficients to be non constant (but deterministic). Hence, the model generalizes the constant coefficient case in Hu/Øksendal [16] and Elliott/van der Hoek [10]. Then we prove that the market is arbitrage free considering an appropriate class of admissible portfolios. Note, that from an economical point of view the case H > 1/2 is the more important one, because statistical analysis of stock prices indicates Hurst parameters of about 0.7, see Shiryaev [29], p.378.

6.8.1

The Model

We recall the market 6.6.1, 6.6.2 consisting of one bond and one stock on the time interval [0, T ]. The bond is given by {∫ t } At := exp r(s)ds ,

(6.8.1)

0

where the nonnegative continuous function r is the interest rate. The stock price is given by a geometric fractional Brownian motion } {∫ t ∫ t 2 1 σ(s)dBsH . (6.8.2) Pt := p0 exp µ(s)ds − M−H (1(0, t)σ) 0 + 2 0 0 Here p0 > 0, the drift µ and the volatility σ are continuous, nonnegative and σ is bounded away from zero. In the case 0 < H < 1/2 σ is assumed to be

6.8. ABSENCE OF ARBITRAGE

369

constant. We use the notation θ(t) := (µ(t) − r(t))/σ(t), 0 ≤ t ≤ T for the risk premium and assume that θ is λ-Hölder continuous with λ > H − 1/2, if H > 1/2. A portfolio is a pair (u, v) of FtH -adapted processes, which are (L2 )-random variables for fixed t ∈ [0, T ]. Here FtH denotes the filtration generated by the fractional Brownian motion B H . u and v are the numbers of bond units, resp. stock units held by an investor. Hence, the corresponding value process is: Vt = ut At + vt Pt . Definition 6.8.1. We call a portfolio (u, v) self-financing, if the corresponding value process fulfills for all t ∈ [0, T ] ∫ t ∫ t ∫ t Vt = V 0 + r(s)As us ds + µ(s)Ps vs ds + σ(s)vs Ps dBsH 0

0

(6.8.3)

0

Note, that this definition requires, that the first two integrals exist as (L2 )valued Pettis integrals and the last one as fractional Itô integral. We shall give a short motivation of this definition. From the theorems 6.2.22 and 6.2.10 we know that ∫ t 0

( ) σ(s)dBsH = I M−H (1(0, t)σ) .

Thus, by corollary 4.3 in Bender [3] the stock price P is the solution of the integral equation ∫

∫

t

σ(s)Ps dBsH +

Pt = p0 + 0

t

µ(s)Ps ds.

(6.8.4)

0

Substituting this in (6.8.3) we get: ∫ t ∫ t Vt = V 0 + us dAs + vs dPs 0

0

The discrete time analogon is Vt+h = Vt + ut (At+h − At ) + vt (Pt+h − Pt )

(6.8.5)

which means that no withdrawals and/or inputs occur. Some further discussion on our definition of a self-financing portfolio can be found in section 6.8.3. We can now introduce the notion of admissibility:

370


Definition 6.8.2. A portfolio (u, v) is called admissible, if it is self-financing and the process Xt := σs vs Ps satisfies X ∈ L1/H ([0, T ], (L1/H )) and M−H (1(0, t)X) ∈ L2 (R, (L2 )) for all t ∈ [0, T ]. Remark 6.8.3. In the case of a classical Brownian motion H = 1/2 the integrability conditions in the definition of an admissible portfolio reduce to ∫ T E[|Xt |2 ]dt < ∞ 0 1/2

as M−

is the identity mapping. Note also, that in this case all Pettis integrals

can be replaced by pathwise integrals by Fubini’s theorem provided X : [0, T ] × Ω → R is measurable. Hence, in this case our definition of admissible portfolios coincides with an usual definition, which is given e.g. in El Karoui et. al. [9].

6.8.2

Absence of Arbitrage

We are now in the position to prove, that the market modeled in the previous section is arbitrage free in the following sense: Definition 6.8.4. (i) A portfolio (u, v) is said to be an arbitrage, if the corresponding value process V satisfies, V0 ≤ 0, VT ≥ 0 and P (VT > 0) ̸= 0. (ii) A market is called arbitrage free, if there is no arbitrage in the class of admissible portfolios. Theorem 6.8.5. The market modeled in the previous section is arbitrage free. Proof. Let (u, v) be an admissible portfolio. By letting X := σs vs Ps we can rewrite (6.8.3) as ∫

∫

t

Vt = V 0 +

∫

t

r(s)Vs ds + 0

t

Xs dBsH

θs Xs ds + 0

(6.8.6)

0

Note that the first two integrals exist as Pettis integrals by the assumed continuity of r and θ and the definition of a self-financing portfolio. We can now apply theorem 6.7.8 with g = θ to obtain a f ∈ L2 (R) such that for all t ∈ [0, T ]: [∫ t ] ∫ t Qf Qf E [Vt ] = V0 + E r(s)Vs ds = V0 + r(s)E Qf [Vs ] ds 0

0

Now assume, that V0 ≤ 0. Then for all t ∈ [0, T ]: ∫t

E Qf [Vt ] = V0 e

0

r(s)ds

≤0


371

If VT ≥ 0, then by the previous inequality VT = 0 Qf -a.s. and hence P -a.s., as the measures are equivalent. Consequently (u, v) is not an arbitrage.

6.8.3

Remarks on Related Results

Finally, we discuss related results on arbitrage in fractional Brownian motion market models. First, we compare our proof for the absence of arbitrage with the arguments given by Hu/Øksendal [16] and Elliott/van der Hoek [10] in the case of constant coefficients. Then we have a look at some other models that lead to arbitrage. By discussing these models we try to give some arguments for ours. The No Arbitrage Argument by Hu/Øksendal and Elliott/ van der Hoek The models in Hu/Øksendal [16] and Elliott/van der Hoek [10] only slightly differ from our model. They use different constructions of fractional Brownian motions and their definitions explicitly make use of the white noise setting ([10]) resp. a fractional white noise setting ([16]). Moreover, they only consider the constant coefficient case. In both papers the fractional Itô integral is defined as a (fractional) Hida distribution valued Pettis integral. In the white noise setting these definitions are essentially equivalent to our definition of the fractional Itô integral. However, the (fractional) white noise measure is crucial in the definition of the space of (fractional) Hida distributions. So the notions of the fractional Itô integral are not defined under another measure than the (fractional) white noise measure in these papers. Consequently, the argument that we sketch below is rather a heuristic: (i) A Girsanov theorem for the fractional Brownian motion (not for the fractional Itô integral) is proven. (ii) It is assumed without proof that the Girsanov theorem carries over to the fractional Itô integral (although the fractional Itô integral under the new measure is not defined). (iii) Finally, the Girsanov theorem is applied to (6.8.6) and it is assumed that the not defined Itô integral under the new measure has zero expectation.

372


(iv) Then the argument for no arbitrage can be given in the lines of the classical Brownian motion case. In the present paper we provided all tools to make this argument rigorous. We gave a general definition of the fractional Itô integral independent of the white noise space and proved a fractional Girsanov theorem (theorem 6.7.3) for the fractional Itô integral. However, we slightly modified the argument by using theorem 6.7.8 instead of the fractional Itô integral. The reason is, that we would have had to additionally assume fractional Itô integrability of X under the new measure, if we had used the Girsanov theorem (compare with the assumptions of theorem 6.7.3).

The Arbitrage Result by Dasgupta/Kallianpur Dasgupta and Kallianpur [5] model stock price with the aid of the pathwise integral defined in Lin [20] for H > 1/2. It is given as the solution of the integral equation: ∫ Pt = p0 +

∫

t

t

σ(s)Ps δBsH .

µ(s)Ps ds + 0

0

The last integral is to be understood in the pathwise sense. As a matter of fact the pathwise integral has not zero expectation in general, see Duncan et al. [8] for a simple example. Hence, one cannot interpret the stock price as an exponential growth with a random perturbation in this model, as the stochastic integral is part of the drift, too. This is a serious argument against this model and the main reason, why one should rather use the fractional Itô integral for a random perturbation with long time memory. Moreover, Dasgupta and Kallianpur [5] provide an easy proof, that the model based on the pathwise integral gives opportunities for arbitrage. It is interesting to note, that this proof can also be applied to the Brownian motion case (H = 1/2), if one replaces the Itô integral by the Stratonovich integral, which was pointed out by Hu/Øksendal [16].


373

The Arbitrage Results by Rogers and Cheridito There is a serious argument against our model in the case H ̸= 1/2 based on the seminal paper by Delbaen/Schachermeyer [7]. As the stock price in our model is not a semi-martingale for H ̸= 1/2, there is a kind of approximative arbitrage with a combination of buy-and-hold portfolios. This result has been sharpened from approximative arbitrage to arbitrage by Rogers [25] and more general by Cheridito [4] for some fractional Brownian motion models. Moreover, the result of Cheridito contains our stock price model in the case of constant coefficients. Hence, we should have a closer look at his setting: Definition 6.8.6. (i) A portfolio (u, v) is called a buy-and-hold strategy, if there are stopping times 0 ≤ τ1 ≤ τ2 ≤ T , FτH1 -measurable random variables F1 , G1 and constants F0 , G0 such that u = 1[0,τ1 ] F0 + 1(τ1 ,τ2 ] F1 and v = 1[0,τ1 ] G0 + 1(τ1 ,τ2 ] G1 . (ii) A buy-and-hold strategy is called self-financing*, if the value process V satisfies for all t ∈ [0, T ]: Vt = V0 + F0 (Aτ1 ∧t − A0 ) + F1 (Aτ2 ∧t − Aτ1 ∧t ) +G0 (Pτ1 ∧t − P0 ) + G1 (Pτ2 ∧t − Pτ1 ∧t ) Note, that this definition of a self-financing* buy-and-hold strategy is in perfect analogy of (6.8.5). So it is usually taken as granted. Cheridito calls a portfolio almost simple predictable, if it is a countable linear combination of buy-and-hold strategies, which has only finitely many summands on almost every path. A notion of self-financing* can be defined analogously to the above definition. After excluding the usual doubling strategies Cheridito proves the existence of an arbitrage within the class of self-financing* almost simple predictable portfolios in the case of constant coefficients and H ̸= 1/2. The reason, why Cheredito’s result is not in conflict with ours, is that the notions of a self-financing* buy-and-hold strategy and a self-financing portfolio (in the sense of (6.8.3)) are not compatible. To see this, we consider a simple form of a buy-and-hold strategy (u, v) with F0 = G0 = 0 and deterministic

374


stopping times τ1 = t1 > 0 and τ2 = t2 . If it is self-financing*, then by (6.8.4): Vt = V0 + F1 (At2 ∧t − At1 ∧t ) + G1 (Pt2 ∧t − Pt1 ∧t ) ∫ t2 ∧t = V0 + F1 r(s)As ds t1 ∧t t2 ∧t

∫ +G1

t1 ∧t t

µ(s)Ps ds + G1

∫ = V0 +

∫

r(s)As us ds + ∫

+G1

0 t2 ∧t

t1 ∧t

∫

σ(s)Ps dBsH

t2 ∧t

t1 ∧t

σ(s)Ps dBsH

∫ t µ(s)Ps vs ds + σ(s)vs Ps dBsH 0 0 ∫ t2 ∧t − G1 ⋄ σ(s)Ps dBsH t

t1 ∧t

which does not coincide with (6.8.3) in general, if H ̸= 1/2. At first glance the notion of self-financing* seems to be more plausible than the definition of self-financing in (6.8.3). But a justification for (6.8.3) may be given in the following way: In reality the stock price evolution and the trading takes place in discrete time. So we look at the continuous time model as an approximation of a discrete model with a stock price satisfying (compare with (6.8.4)): Ptn = Ptn−1 + µ(tn )Ptn−1 (tn − tn−1 ) + σ(tn )Ptn−1 ⋄ (BtHn − BtHn−1 )

(6.8.7)

With a stock price evolution given by (6.8.7) we would like to rewrite the discrete time condition for a self-financing portfolio (6.8.5) as: Vtn

= Vtn−1 + utn−1 (Atn − Atn−1 ) + vtn−1 µ(tn )Ptn−1 (tn − tn−1 ) +vtn−1 σ(tn )Ptn−1 ⋄ (BtHn − BtHn−1 )

The continuous time analogue of this identity is the condition (6.8.3) for a self-financing portfolio. Hence, in sticking to (6.8.3) we would like to view the complete market model (including the notion of self-financing) as an approximation of the discrete time model with a stock evolution given by (6.8.7). Unfortunately this reasoning is not correct as we have to be careful to write down formula 6.8.8. The last term should read as · · · + vtn−1 (σ(tn )Ptn−1 ⋄ (BtHn − BtHn−1 )) what is not equal to · · · + (vtn−1 σ(tn )Ptn−1 ) ⋄ (BtHn − BtHn−1 ).


375

As we see from the correct formulae we would get an additional term. This looks very similar to the approximations of SDEs basing on Brownian motions where a smoothing and integrating of the approximating sequence leads to the Stratonovitch integral. So, at this point some work remains to be done to arrive at the correct interpretation.

376


Chapter 7

Conclusion

377

378

CHAPTER 7. CONCLUSION

Chapter 8

Appendix: A Short Note on The Classical Theory of BSDE’s The examination of backward stochastic differential equations with uniform Lipschitz conditions goes back to Pardoux and Peng [26]. In the first chapter we are going to introduce and prove some basic results under uniform Lipschitz conditions which have been achieved since then. The chapter is divided as follows: First we motivate by a simple example how to formulate the problem in a sensible way. The problem means the examination of stochastic differential equations with final condition. Then we prove by the Pardoux/Peng-theorem the first important result of existence and uniqueness and give a Picard-type approximation procedure. Furthermore we examine the structure of linear and Markov BSDE and the connection between BSDEs and partial BSDEs.

379

380

CHAPTER 8. APPENDIX BSDES Summary: As this chapter will try to make you familiar with some more

abstract mathematical techniques I here give a short overview of the relevant results:

• Existence and Uniqueness of Solutions (Y, Z) – simple BSDEs: Let ξ ∈ L2T (Rn ) and g ∈ HT2 (Rn ). Then the BSDE dYt = gt dt + Zt dWt , YT

= ξ

has exactly one solution and the following holds: ] [ ∫ T Yt = E ξ − gs ds Ft . t

– BSDEs with standard parameter dYt = −f (t, Yt , Zt )dt + Zt dWt , YT

= ξ

• Comparison of Solutions of BSDEs: What happens with the solutions if (f, ξ) grows? This will be relevant when we penalize the cost of a claim for failing to avoid constraints. So these results will be necessary to treat imperfect and incomplete markets. • Linear BSDEs, relevant to treat the classical case. [ ] d ∑ dYt = At Yt + Bti Zti + gt dt + Zt dWt , i=1

YT

= ξ

with explicit solution Yt =

Φ−1 t E

[

∫ ΦT ξ − t

T

] Φs gs ds Ft ,

where Φ is the strong solution of the following SDE: dΦt = −Φt At dt −

d ∑ i=1

Φ0 = E n .

Φt Bti dWti ,

381 • Markovian FBSDEs and Feynman Kac procedures. This will become relevant to derive Black Scholes equations in more general cases. dXt = b(t, Xt )dt + σ(t, Xt )dWt , Xs = x dYt = −f (t, Xts,x , Yt , Zt )dt + Zt dWt , YT and

= g(XTs,x ).

[ Yts,x

=E

∫ g(XTs,x )

+ t

T

f (r, Xrs,x )dr Ft

] = u(t, Xts,x ).

where ] 1 [ tr ∂x,x ui (t, x)(σσ ∗ )(t, x) + ⟨∂x ui (t, x), b(t, x)⟩ 2 = −f i (t, x, u(t, x), ∂x u(t, x)σ(t, x)) ,

∂t ui (t, x) +

x ∈ Rm .

u(T, x) = g(x);

• FBSDE with explicit dependence on ω and the Feynman Kac procedure (BPDE) This will be important when we extend the classical model to more realistic models. Again we consider a weakly coupled system consisting of a SDE and a BSDE: dXts,x = b(t, Xts,x )dt + σ(t, Xts,x )dWt , dYts,x = −f (t, Xts,x , Yts,x , Zts,x )dt + Zts,x dWt , Xss,x = x, YTs,x = g(XTs,x );

0 ≤ s ≤ t ≤ T.

where now all coefficients explicitly depend on Ω in a Ft -progressively measurable way. Then the pde to consider turns into: ∫ T u(t, x) = g(x) + {f (s, x, u(s, x), z(s, x, ∂x u(s, x), q(s, x))) t

1 + tr[∂x,x u(s, x)(σσ ∗ )(s, x)] + tr[σ ∗ (s, x)∂x q(s, x)] 2 ∫ T +⟨∂x u(s, x), b(s, x)⟩}ds − ⟨q(s, x), dWs ⟩. t

Here z(t, x, w, q) := [q +

σ ∗ (t, x)w]∗

382

CHAPTER 8. APPENDIX BSDES

8.1

General Notation

On the vector space Rn×d of real (n × d)-matrices a scalar product is defined by ⟨A, B⟩ := tr(AB ∗ ). At that tr denotes the trace of a matrix and B ∗ the transformed matrix. The Euclidean norm induced by the scalar product is denoted by | · |. Note the difference of the notation of scalar products of Rn×d -valued stochastic processes ⟨Yt , Xt ⟩ and the notation of the cross variation of these processes ⟨Y, X⟩t . We expect the reader to be familiar with the spaces C([0, T ], U ) of continuous functions from [0, T ] to U , C l (Rm , Rn ) of l-times continuously differentiable functions, Cbl (Rm , Rn ) the space of l-times continuously differentiable functions with all partial derivatives up to the first order bounded, Lp (U, µ) the space of the U -valued p-times µ-integrable functions and Wpl (Rm , R), the Sobolevspace of the functions with derivatives in distributional sense up to first order in Lp (R, λ), as well as the usual topologies on these space. Here λ denotes Lebesgue measure on R. Analogously λ[0,T ] is the Lebesgue measure on [0, T ]. For the partial derivatives of a Rn -valued function f (t, x1 , . . . , xm ) ∈ C k,l (Rn ), which so is k-times in t and once in xi continuously differentiable, we use the notation ∂t f resp. ∂xi f . Higher partial derivatives are denoted in a similar way, say e.g. ∂xi ,xj f for ∂xi ∂xj f . ∂x f denotes the Jacobi-matrix of first derivatives in directions xi , (i = 1, . . . , m), if n ≥ 2. For n = 1 ∂x f is the x-gradient, a row vector. In tis case ∂x,x f denotes the Hessian of second derivatives in x = (x1 , . . . , xm ).

8.1.1

Equivalence classes and appropriate spaces

There are a filtered probability space (Ω, F, Ft , P ) which carries a d-dimensional Brownian motion W and let [0, T ] be a time interval. We assume that the filtration Ft is the augmentation of the filtration given by the Brownian motion and thus satisfies the usual conditions (see Protter [56], p.3). Definition 8.1.1. Let X, Y : Ω × [0, T ] → Rn be two stochastic processes. They are

8.1. GENERAL NOTATION

383

(i) indistinguishable, if P (∀0≤t≤T Xt = Yt ) = 1, (ii) almost surely indistingusishable, if P (Xt = Yt a.e. t ∈ [0, T ]) = 1, (iii) modifications of each other, if ∀0≤t≤T P (Xt = Yt ) = 1. We denote the respective equivalence classes by [X]1 , [X]2 and [X]3 .

For FT -measurable random variables consider the interval [T, T ] and define the notion of modification analogously. Obviously two indistinguishable processes are also almost surely indistinguishable and identical up to modification. In addition one can show that for RCLL-processes, i.e. for right-continuous processes with left-side limes, all three classifications are identical. At measurable random variable one should consider the interval and define the term of modification analogously. Next we define some spaces which play an important role in the following work. Let:

[ ] L2T (Rn ) := {ξ; Rn -valued and FT -measurable with E |ξ|2 < ∞}, HT1 (Rn ) := {Y ; Rn -valued and progressively measurable with √ ∫ T E |Yt |2 dt < ∞}, 0

HT2 (Rn )

:= {Y ; Rn -valued and progressively measurable with ∫ T E |Yt |2 dt < ∞}, 0

ST2 (Rn ) := {Y ; Rn -valued and progressively measurable with E sup |Yt |2 < ∞}, 0≤t≤T

STc,2 (Rn )

:= {Y ; Y ∈ ST2 (Rn ) and continuous}.

To define a norm on these spaces we must pass over to the appropriate equivalence classes. We put:

384


L2T (Rn ) := {[ξ]3 ; ξ ∈ L2T (Rn )}, HT1 (Rn ) := {[Y ]2 ; Y ∈ HT1 (Rn )}, HT2 (Rn ) := {[Y ]2 ; Y ∈ HT2 (Rn )}, ST2 (Rn ) := {[Y ]1 ; Y ∈ ST2 (Rn )}, STc,2 (Rn ) := {[Y ]1 ; Y ∈ STc,2 (Rn )}. Now equivalent norms on the spaces L2T (Rn ), HT1 (Rn ), HT2 (Rn ) and ST2 (Rn ) depend on β ∈ R given by:

∥ξ∥β := ∥Y ∥1,β

√

E[e2βT |ξ|2 ], √ ∫ T e2βt |Yt |2 dt, := E 0

√ ∥Y ∥2,β := ∥Y ∥s,β :=

∫ E

√

T

e2βt |Yt |2 dt,

0

E sup [e2βt |Yt |2 ]. 0≤t≤T

On the left hand side we left out the notion of equivalence classes. We will continue to do so as long as the correct meaning is clear from the context. Obviously we have the following relation between the spaces: STc,2 (Rn ) ⊂ ST2 (Rn ) ⊂ HT2 (Rn ) ⊂ HT1 (Rn ).

(8.1.1)

and from Hölder’s inequality we have: Y ∈ ST2 (Rn ), Z ∈ HT2 (Rn×d ) ⇒ Z ∗ Y ∈ HT1 (Rd ).

(8.1.2)

When we try to carry these relations over to the equivalence classes we face a problem as ST2 (Rn ) and HT2 (Rn ) use different classifications. Let [Y ]1 ∈ ST2 (Rn ) and let Y ∈ ST2 (Rn ) represent the equivalence class. Then [Y ]2 is independent of the representing object as indistinguishability implies almost sure indistingusihability. Now [Y ]2 is in HT2 (Rn ). So expressions like ST2 (Rn ) ⊂ HT2 (Rn ) make sense and they should be understood in this way.

8.2. THE PROBLEM

385

By combination of the spaces introduced by now we also get:

MT,d (Rn ) := STc,2 (Rn ) × HT2 (Rn×d ), M T,d (Rn ) := HT2 (Rn ) × HT2 (Rn×d ), MT,d (Rn ) := STc,2 (Rn ) × HT2 (Rn×d ), MT,d (Rn ) := HT2 (Rn ) × HT2 (Rn×d ). So (Y, Z), (Y ′ , Z ′ ) ∈ MT,d (Rn ) are equivalent, if P (Yt = Yt′ a.e. t ∈ [0, T ] ∧ Zt = Zt′ a.e. t ∈ [0, T ]) = 1, and (Y, Z), (Y ′ , Z ′ ) ∈ MT,d (Rn ) are equivalent, if P (∀0≤t≤T Yt = Yt′ ∧ Zt = Zt′ a.e. t ∈ [0, T ]) = 1. As in the preliminary considerations MT,d (Rn ) can also be seen as a subspace of MT,d (Rn ). We give the spaces MT,d (Rn ) and MT,d (Rn ) the norms √ ∥Y ∥22,β + ∥Z∥22,β , √ := ∥Y ∥2s,β + ∥Z∥22,β .

∥(Y, Z)∥β := ∥(Y, Z)∥s,β

As (HT2 (Rn ), ∥ · ∥2,0 ) just is the usual L2 -space w.r.t. the measurable space (Ω×[0, T ], F ⊗B[0,T ] , P ⊗λ[0,T ] ), this space is a Hilbert space. From the definition of MT,d (Rn ) and the equivalence of the β-norms we get: Theorem 8.1.2. For all β ∈ R, (MT,d (Rn ), ∥ · ∥β ) is a Hilbert space.

8.2

The Problem

To find the correct framework to treat SDEs with terminal condition remember that a strong solution of an SDE is by definition adapted to the given filtration (see e.g. Karatzas/Shreve [18], p.285, for the exact definition). We will need this adaptedness also for BSDEs to fulfill the requirements of the financial

386


applications we have in mind: The agent should not have knowledge about the future. So when we define the notion of BSDE and its solution, a crucial requirement will be that the solution is adapted to a given filtration Ft . When we consider ordinary differential equations an easy way to treat a problem with terminal condition is just to invert time and then to treat a problem with initial condition. This way of treating the problem is not possible when we consider BSDEs as the following example shows: Example 8.2.1. Let (Ω, F, Ft , P ) be a filtered probability space with a onedimensional Brownian motion Wt . We assume that Ft is the augmentation of the filtration generated by the Brownian motion. Then consider the equation: dYt = 0, YT

= ξ,

(8.2.1)

where ξ is an FT -measurable, non-constant random variable. Obviously Yt ≡ ξ is the only process which solves equation (8.2.1). But Yt is not Ft -adapted as ξ is not constant. So for such a trivial equation like (8.2.1) we would find a solution only in very special cases. The example shows that the requirement to be adapted makes it impossible to adopt techniques from the theory of ordinary differential equations. In order to fulfill the requirement in the above example we look for a process Y˜t which is Ft -adapted and which is not too far away from the non-adapted solution Yt . So project Y by making use of the conditional expectation Y˜t := E[Yt |Ft ] = E[ξ|Ft ]. Now Y˜t no longer fulfills the equation (8.2.1). So we have to find next an appropriate reformulation of (8.2.1). To this end we need a martingale representation theorem like the one in Karatzas/Shreve [18], p.182,: Theorem 8.2.2 (martingale represenattion theorem). Let W be a ddimensional Brownian motion on (Ω, F, Ft , P ) and let Ft be the augmentation of the filtration generated by the Brownian motion. Then for every squareintegrable martingale M there is a progressively measurable process Z, so that

8.2. THE PROBLEM

387

for 0 ≤ T < ∞

∫

T

E

|Zt |2 dt < ∞

(8.2.2)

0

and for 0 ≤ t < ∞

∫

t

Zs dWs ; P -a.s.

Mt = M0 + 0

Remark 8.2.3. Karatzas/Shreve assume the martingale to be RCLL. As every martingale has an RCLL modification (Protter [56], p.8) we may omit this additional assumption. [ ] Let E |ξ|2 < ∞, so from Jensen’s inequality we find that Y˜t is a square integrable martingale: [ ] [ ] [ ] E |Y˜t |2 = E |E[ξ|Ft ]|2 ≤ E |ξ|2 .

(8.2.3)

˜ such that for So theorem 8.2.2 may be applied to find an adapted process Z, 0≤t≤T

∫ Y˜t = E[ξ] +

t

Z˜s dWs .

0

So: ∫ Y˜t = E[ξ] + ∫ ˜ = YT −

0 T

T

∫ Z˜s dWs − t

T

Z˜s dWs

∫

Z˜s dWs = ξ −

t

T

Z˜s dWs .

t

˜ so is an adapted pair of solutions of the SDE with terminal The pair (Y˜ , Z) condition: dYt = Zt dWt , YT

= ξ.

(8.2.4)

Next we have to find out whether this pair is unique. To construct further solutions we use a result by Dudley [14]: Theorem 8.2.4. Let W be a one-dimensional Brownian motion and Ft the augmentation of the filtration generated by the Brownian motion. Let ξ for 0 < T < ∞ be a P -a.s. finite FT -measurable random variable. Then there is a progressively measuarable process Z, such that ∫ T |Zt |2 dt < ∞; P -a.s. 0

388


and

∫

T

Zs dWs ; P -a.s.

ξ= 0

With this we have the following proposition: [ ] Proposition 8.2.5. Let ξ be an FT -measurable random variable with E |ξ|2 < ∞. Then for each y ∈ R there is a pair of adapted processes (Y y , Z y ) with Y0 = y which solves (8.2.4). Proof. Let y ∈ R. For ξ −y we have from Dudley’s theorem 8.2.4 an Ft -adapted process Z y with

∫

T

ξ−y =

Zsy dWs . 0

Define Y y by

∫ Yty

Then:

∫ Yty = y +

and

Y0y

Zsy dWs . 0

∫

T

t

:= y +

Zsy dWs −

0

T

∫ Zsy dWs = ξ −

t

T

Zsy dWs t

= y. So (Y y , Z y ) is the pair we are looking for.

Of course we would like to have uniquenss of our solutions. From the integrability condition (8.2.2) we have: ∫

t

Z˜s dWs is a martingale.

0

For the processes Z y the respective Itô-integrals are local martingales and not martingales. We however have: Proposition 8.2.6. Under the additional assumption ∫

t

Zs dWs is a martingale 0

the equation (8.2.4) has a unique solution in the following sense: For two solutions the Y -parts are modifications of each other and the Z-parts are almost surely indistinguishable.

8.2. THE PROBLEM

389

˜ is a solution. So we have to prove uniqueness: Consider two Proof. (Y˜ , Z) solutions and let (Y, Z) be the difference of these two solutions. Then: ∫ T Yt = − Zs dWs . t

So from the martingale property: Yt = E[Yt |Ft ] = 0. So the Y -parts are equal up to a modification. From the isometry of Itô’s integral we then have: [ ∫ 0 = E

T

0

2 ] [∫ Zs dWs = E

T

] |Zs |2 ds .

0

So the Z-parts are indistinguishable.

Let us summarize the above considerations: (i) A solution of an SDE with terminal condition is a pair of adapted processes. The second part of the solution makes it possible to have an adapted solution. (ii) To guarantee the uniqueness of the solution the Itô integral must be a martingale. Before we formulate the general problem let us look at some properties of ˜ again. (Y˜ , Z) ˜ satisfies the following integrability condition: (Y˜ , Z) ∫ T 2 ˜ |Z˜t |2 dt < ∞. E sup |Yt | + E 0≤t≤T

(8.2.5)

0

For Z˜ this follows from the martingale representation theorem, for Y˜ this follows from the square integrability of ξ by making use of Doob’s inequality. Obviously (8.2.5) implies the martingale property of the Itô-integrals, and we will make this stronger assumption. Moreover Y˜ has a continuous modification. So we require the Y -part of the solution to be continuous. As the Y -parts are equal up to a modification they are indistinguishable. Finally the processes Y˜ and Z˜ are not only adapted but even progressively measurable. Also this will be part of the formulation of the general problem:

390

CHAPTER 8. APPENDIX BSDES Let (Ω, F, Ft , P ) be a filtered probability space with a d-dimensional Brown-

ian motion Wt . Let Ft be the augmentation of the filtration generated by the Brownian motion. Then the problem of treating SDEs with terminal condition may be formulated in the following way: Definition 8.2.7. Let f : Ω × [0, T ] × Rn × Rn×d → Rn , so that for all (y, z) ∈ Rn × Rn×d f (·, ·, y, z) is Ft -progressively measurable. Let ξ be an FT measurable random variable. (i) An equation of the form dYt = −f (t, Yt , Zt )dt + Zt dWt , YT

= ξ

(8.2.6)

is called a backwards stochastic differential equation (BSDE). (ii) A pair of processes (Y, Z) ∈ MT,d (Rn ) is called a solution of the BSDE (2.5.11), if for all 0 ≤ t ≤ T ∫ ∫ T f (s, Ys , Zs )ds − Yt = ξ + t

T

Zs dWs ; P -a.s.

(8.2.7)

t

(iii) The BSDE (2.5.11) is called uniquely solvable, if for two solutions (Y 1 , Z 1 ) and (Y 2 , Z 2 ) ( ) P ∀0≤t≤T Yt1 = Yt2 ∧ Zt1 = Zt2 a.e. t ∈ [0, T ] = 1.

(8.2.8)

(iv) The function f is called the generating function or generator of the BSDE (2.5.11). Remark 8.2.8. Note that when speaking of solutions of the BSDE in principle we must make the difference between an equivalence class in MT,d (Rn ) and an element in it. The reader will keep this in mind. Remark 8.2.9. Let us explicitly point to the fact that Ft is the Brownian filtration. As the martingale representation theorem is used even in solving easiest problems this assumption is indeed crucial. Extensions to more general filtrations in the literature lead to basic changes in the notion of solution. So the solution in El Karoui/Huang [17] and El Karoui/Peng/Quenez [9], chapter 5, consist of three processes.

8.2. THE PROBLEM

391

To make the role of the process Z more transparent let us shortly look at an application in finance. Example 8.2.10. We consider the usual Samuelson-Black-Scholes-Mertonmarket with one risky and one non-risky asset. One portion of our initial capital (wealth) is invested in the non-risky asset, the rest is put into the risky one. Let πt be the amount invested in the risky asset. Then the wealth process Yt is descriebed by the following SDE: dYt = rt Yt dt + (µt − rt )πt dt + σt πt dWt , Y0 = y.

(8.2.9)

Here rt is the risk free interest rate, µt the drift and σt the (strictly positive) volatility of the risky asset. y is the initial wealth. The problem consists in reaching at time T a given FT -measurable goal ξ. In other words: We search for an initial wealth y and a portfolio πt , so that the corresponding wealth process Yt has at time T teh value ξ. In view of (8.2.9) this leads with Z := σπ to the following BSDE dYt = rt Yt dt + σt−1 (µt − rt )Zt dt + Zt dWt , YT

= ξ.

(8.2.10)

Let (Y, Z) be a solution of the BSDE (8.2.10). Then a portfolio is given by πt := σt−1 Zt , and Z may be identified with this portfolio. The corresponding initial wealth is Y0 .

8.2.1

Existence and Uniqueness

In this paragraph we prove that the problem defined by 8.2.7 is correctly posed under certain conditions. The essential condition will be that the generator is uniformly Lipschitz-constant. Under this condition Pardoux and Peng [26] showed existence and uniqueness. Furthermore the solutions continuously depend on the final condition. We commence to examine existence and uniqueness by a group of ”simple” BSDEs. Herewith ”simple” means that the generator is independent from Y

392


and Z. The treatment of these ”simple” BDSEs will be the basement for the general case. We get the following result: Proposition 8.2.11. Let ξ ∈ L2T (Rn ) and g ∈ HT2 (Rn ). Then the BSDE dYt = gt dt + Zt dWt , YT

= ξ

(8.2.11)

has exactly one solution and the following holds: [

∫

T

Yt = E ξ − t

] gs ds Ft .

(8.2.12)

Proof. The proof imitates the ideas of the previous paragraph. The proof of uniqueness can be overtaken from proposition 8.2.6 word by word. To prove the existence we proceed heuristically for the present. Let us assume that (Y, Z) is a solution of equation (8.2.11). Then the following holds: [

∫

Yt = E[Yt |Ft ] = E ξ −

T

t

] gs ds Ft .

So we define: [

∫

T

] gs ds Ft ,

Mt := E ξ − 0 ∫ t gs ds. Yt := Mt + 0

With this Mt is a square integrable martingal, as from Jensen’s, Young’s and H”older’s inequalities we have: [

2

E |Mt |

]

[ ∫ ≤ E ξ −

0

T

2 ] [ ∫ 2 gs ds ≤ 2E |ξ| + T

T

] |gs | ds < ∞. 2

0

So we may apply the martingale representation theorem 8.2.2, and we get a process Z ∈ HT2 (Rn×d ) with ∫ Mt = M0 +

t

Zs dWs . 0

8.2. THE PROBLEM

393

So: ∫

∫

t

Yt = M0 +

Zs dWs

0

∫ = M0 + ∫ = ξ−

t

gs ds + 0

∫

t

gs ds + 0 ∫ T gs ds −

t

T

∫ Zs dWs −

0 T

T

Zs dWs t

Zs dWs .

t

It remains to prove that Y ∈ STc,2 (Rn ). We have: [ ] [ E

sup |Yt |2 0≤t≤T

∫ t 2 ] ≤ 2E sup |Mt |2 + sup gs ds 0≤t≤T 0≤t≤T 0 [ ] ∫ T 2 2 ≤ 2E 4 |MT | + T |gs | ds < ∞. 0

Here we used the inequalities by Young, Hölder and Doob, and the square integrability of M . By choosing a continuous modification of Y the proof is complete.

We are now going to extend this result to a more general sizuation. To this end we introduce the so-called ”‘standard data”’ : Definition 8.2.12. We call a pair (f, ξ) standard data, if: (i) the generator fulfills the following uniform Lipschitz condition: there is a constant K, so that for all (y, z, y˜, z˜) ∈ Rn × Rn×d × Rn × Rn×d |f (t, y, z) − f (t, y˜, z˜)| ≤ K (|y − y˜| + |z − z˜|) ; P ⊗ λ[0,T ] -a.s.

(8.2.13)

(ii) f (t, 0, 0) ∈ HT2 (Rn ). (iii) ξ ∈ L2T (Rn ). Remark 8.2.13. A generator with (i) and (ii) is called a standard generator. The main result of this section is the following theorem by Pardoux und Peng [26]: Theorem 8.2.14 (Pardoux/Peng 1990). Let (f, ξ) be standard data. Then the BSDE dYt = −f (t, Yt , Zt )dt + Zt dWt , YT

= ξ

(8.2.14)

394


has a unique solution. In preparation of the proof we prove two lemmas. Lemma 8.2.15. Let f be a standard generator and (y, z) ∈ M T,d (Rn ), then f (t, yt , zt ) ∈ HT2 (Rn ). Proof. With Minkowskis and Young’s inqualities and from the properties of a standard generator we have: √

∫

T

E 0

√ ≤

|f (t, yt , zt )|2 dt

∫

T

E

√ 2

|f (t, yt , zt ) − f (t, 0, 0)| dt +

0

√

≤

√ 2K

∫

√

T

|yt |2 + |zt |2 dt +

E

E

T

|f (t, 0, 0)|2 dt

0

∫

T

|f (t, 0, 0)|2 dt < ∞.

E

0

∫

0

Lemma 8.2.16. Let X ∈ HT1 (Rn ). Then

∫t 0

Xs dWs on [0, T ] is a martingale.

Proof. X ∈ HT1 (Rn ) implies ∫

T

|Xs |2 ds < ∞; P -a.s.

0

So

∫t

Xs dWs is a local martingale. Now let τn be a sequence of stopping times ∫ t∧τ with τn → T P -a.s., so that 0 n Xs dWs is a martingale. From the Burkholder0

Davis-Gundy-inequality we find: ∫ E sup n [

t∧τn

0

√ ] ∫ Xs dWs ≤ KE

T

|Xs |2 ds.

0

Here K is independent of n. As X ∈ HT1 (Rn ), we may apply Lebesgue’s theorem, to find for 0 ≤ t ≤ u ≤ T : [∫ E 0

u

] [∫ Xs dWs Ft = lim E n→∞

0

u∧τn

] ∫ t Xs dWs . Xs dWs Ft = 0

8.2. THE PROBLEM

395

We are going to prove the Pardoux/Peng-theorem now. The proof will be done in two steps. First of all we will transform the problem into a fixed-point problem in space MT,d (Rn ) and then solve it with the help of Banach’s fixedpoint theorem. We call back to mind that italic letters denote spaces of the equivalence classes. proof of theorem 8.2.14: (i) Reformulation into a fixed point problem: We define a mapping Π : MT,d (Rn ) → MT,d (Rn ) ⊂ MT,d (Rn ) in the following way: Choose an element (y, z) from an equivalence class in MT,d (Rn ). Then the simple BSDE dYt = −f (t, yt , zt ) + Zt dWt , YT

= ξ

(8.2.15)

has a unique solution in the sense of definition 8.2.7 by Lemma 8.2.15 and Proposition 8.2.11. We now put Π(y, z) := (Y, Z), where we omit the notation of equivalence classes for simplicity. The reader easily sees that the above does not depend on the chosen member of an equivalence class. As for Π(y, z) we always find a member of an equivalence class in MT,d (Rn ), we may identify the fixed points of Π with the solutions of the BSDE (8.2.14). Note that two members of an equivalence class with continuous Y -part have indistinguishable Y -parts,so that each equivalence class in MT,d (Rn ) contains at most one solution of the BSDE in the sense of definition 8.2.7. (ii) Treating the fixed point problem: It remains to prove that the mapping Π has exactly one fixed point. As (MT,d (Rn ), ∥ · ∥β ) for β ∈ R is a Hilbert space, we may apply the Banach fixed point theorem. So it suffices to prove that Π for an appropriate β is a contraction. Let β > 12 . Then we know from Lemma 8.2.16, that ∫ E

T

e2βs ⟨Ys − Y˜s , (Zs − Z˜s )dWs ⟩ = 0,

0

˜ ∗ (Y − Y˜ ) is in H 1 (Rn ). So from Itô’s as from H”older’s inequality (Z − Z) T

396


formula for (y, z), (˜ y , z˜) ∈ MT,d (Rn ) we have: ∫ E|Y0 − Y˜0 |2 = −E

∫

T 2βt

e

|Zt − Z˜t |2 dt − 2βE

0

∫

T

e2βt |Yt − Y˜t |2 dt

0 T

e2βt ⟨Yt − Y˜t , f (t, yt , zt ) − f (t, y˜t , z˜t )⟩dt ∫ T e2βt |Zt − Z˜t |2 dt − 2βE e2βt |Yt − Y˜t |2 dt 0 0 ∫ T +2KE e2βt |Yt − Y˜t | (|yt − y˜t | + |zt − z˜t |) dt 0 ∫ T ∫ T 2βt 2 ˜ ≤ −E e |Zt − Zt | dt − 2βE e2βt |Yt − Y˜t |2 dt 0 0 ∫ T +(2β − 1)E e2βt |Yt − Y˜t |2 dt 0 ∫ T ( ) 2K 2 E e2βt |yt − y˜t |2 + |zt − z˜t |2 dt + 2β − 1 0 2K 2 = − ∥Π(y, z) − Π(˜ y , z˜)∥2β + ∥(y, z) − (˜ y , z˜)∥2β . 2β − 1 +2E ∫ ≤ −E

0 T

Note that we made use of the Cauchy-Schwarz-inequality, the Lipschitz condition of the standard generator and Young’s inequality. So Π for β > (1 + 2K 2 )/2 is contracting and the proof is complete.

Remark 8.2.17. The original proof of Pardoux and Peng essentially deviates from the proof given here. The solution has been constructed in a two-step iteration procedure there. In the more recent literature the Banach fixed-point appraoch is considered to be more elegant. Nevertheless we draw the readers’ attention to the fact that the contraction constant generally depends on T , see e.g. El Karoui/Peng/Quenez [9], Theorem 2.1, oder Yong/Zhou [62], Theorem 7.3.2. In our proof however, the contraction constant is independent from T . This allows for a generalization to unbounded stopping times as terminal time (Bender/Kohlmann [3]). We will carry this out in chapter 3. We find the following iteration procedure to determine the solution of the BSDE:

Corollary 8.2.18. Let (f, ξ) be standard data. Then the sequence (Y i , Z i ) is

8.2. THE PROBLEM

397

recursively defined by Y 0 ≡ 0 and Z 0 ≡ 0 and for i ≥ 1 by dYti = −f (t, Yti−1 , Zti−1 )dt + Zti dWt , YTi

= ξ.

Furthermore let (Y, Z) be the solution of the (8.2.14). Then for all β ∈ R: ∫ T lim E e2βt |Yti − Yt |2 + e2βt |Zti − Zt |2 dt = 0 i→∞

and

0

] [ lim Yt − Yti + Zt − Zti = 0; P ⊗ λ[0,T ] -a.s.

i→∞

Proof. Let β be sufficiently large. From the estimate in the proof of Pardoux/Peng’s theorem, as (Y i , Z i ) = Π(Y i−1 , Z i−1 ):

i i

(Y , Z ) − (Y i−1 , Z i−1 ) 2 ≤ 1 (Y i−1 , Z i−1 ) − (Y i−2 , Z i−2 ) 2 . 2,β 2,β 2 Iterating this we find:

i i

(Y , Z ) − (Y i−1 , Z i−1 ) 2 ≤ 2,β

( )i−1

1 1 2 1

(Y , Z ) . 2,β 2

Now let β be arbitrary. Then from the equivalence of the β-norms we find a constant C depending on β, but not on i so that: ( )i−1 1 C. 2

(8.2.16)

( )i ( )j 1 1 = 2C . 2 2

(8.2.17)

i i

(Y , Z ) − (Y i−1 , Z i−1 ) 2 ≤ 2,β So:

∞ ∑

(Y, Z) − (Y i , Z i ) 2 ≤ C 2,β j=i

This proves the first of the two proposed convergences. The second proposed convergence: For β = 0 we find from (8.2.17) with Tschebytschev’s inequality: ) ( [ ] 1 i i − 4i . P ⊗ λ[0,T ] Yt − Yt ≥ 2 ≤ 2C √ 2 The right hand side is the i-th term of a convergent series, so the Borel/Cantelli lemma gives:

[ ] i P ⊗ λ[0,T ] ∃j≥0 ∀i≥j Yt − Yti < 2− 4 = T.

This implies the almost sure convergence of Y i . By making use of a similar argument this gives the almost sure convergence of Z i .

398

CHAPTER 8. APPENDIX BSDES We give a simple example that in this framework the Lipschitz condition

must not be dispensed with: Example 8.2.19. Let n = d = 1. Then the BSDE −1 Yt dt + Zt dWt , exp {T − t} − 1 = 0

dYt = YT

(8.2.18)

has infinitely many solutions. We construct solutions of this BSDE by making use of proposition 8.2.11. {√ } Let y ∈ R and gt := exp 2Wt . We consider the following family of simple BSDEs dYty = y · gt dt + Zty dWt , YTy = 0. From proposition 8.2.11 the unique solution (Y y , Z y ) has the form [∫ Yty

= −yE t

T

] gs ds Ft .

So the family (Y y , Z y )y∈R solves the BSDE [∫ dYt = −gt E t

YT

T

]−1 gs ds Ft Yt dt + Zt dWt ,

= 0.

(8.2.19)

From standard techniques like developing the exponential series, using the normal moments and integrating we find from the form of gt : [∫

T

E t

gt−1 gs ds Ft

] = exp{T − t} − 1.

So the BSDEs (8.2.18) and (8.2.19) are equal, and the BSDE (8.2.18) has infinitely many solutions. Remark 8.2.20. The generator of BSDE (8.2.18) is not uniformly Lipschitz continuous, yet it fulfills a stochastic Lipschitz condition. BSDEs with such generators are considered in chapter 3.

8.2. THE PROBLEM

399

Finally we prove an estimate for the difference of solutions of BSDEs with different data. For the special case f 1 ≡ f 2 this proves the continuous dependence of the solutions on the terminal condition.

Theorem 8.2.21. Let (f 1 , ξ 1 ), (f 2 , ξ 2 ) be standard data and let (Y 1 , Z 1 ), and (Y 2 , Z 2 ) be the unique solutions of the corresponding BSDEs. Then for all β ∈ R there is a constant C, such that

1

(Y − Y 2 , Z 1 − Z 2 ) 2 s,β (

2 ) 2 ≤ C ξ 1 − ξ 2 β + f 1 (·, Y 2 , Z 2 ) − f 2 (·, Y 2 , Z 2 ) 2,β . Here the constant C depends on β, T and on the Lipschitz constants of f 1 , it does not depend on ξ 1 , ξ 2 and f 2 .

Proof. First, from the equivalence of the norms it suffices to prove the result for a sufficiently large β. To make the notation easier we set ξ := ξ 1 − ξ 2 , Y := Y 1 − Y 2 and Z := Z 1 −Z 2 . Furthermore let K be the Lipschitz constant of f 1 . C denotes possibly different constants. From Itô’ formula we get: ∫ e2βt |Yt |2 + t

T

∫

∫ e2βs |Zs |2 ds + 2

T

e2βs ⟨Ys , Zs dWs ⟩

t T

e2βs |Ys |2 ds = e2βT |ξ|2 − 2β t ∫ T +2 e2βs ⟨Ys , f 1 (t, Ys1 , Zs1 ) − f 2 (t, Ys2 , Zs2 )⟩ds t ∫ T 2βT 2 ≤ e |ξ| − 2β e2βs |Ys |2 ds t ∫ T ( ) +2 e2βs |Ys | K|Ys | + K|Zs | + |f 1 (s, Ys2 , Zs2 ) − f 2 (s, Ys2 , Zs2 )| ds t ∫ ∫ T 1 T 2βs 2 2βT 2 e |Zs | ds + (C − 2β) e2βs |Ys |2 ds ≤ e |ξ| + 2 t t ∫ T + e2βs |f 1 (s, Ys2 , Zs2 ) − f 2 (s, Ys2 , Zs2 )|2 ds. t

Note that here we made use of Young’s inequality twice.

400

CHAPTER 8. APPENDIX BSDES For sufficiently large β we have: ∫ T ∫ T e2βs ⟨Ys , Zs dWs ⟩ 2e2βt |Yt |2 + e2βs |Zs |2 ds ≤ −4 t t ) ( ∫ T 2βs 1 2 2 2 2 2 2 2βT 2 e |f (s, Ys , Zs ) − f (s, Ys , Zs )| ds .(8.2.20) +2 e |ξ| + t

By computing the expectation by making use of Lemma 8.2.16) we find: ∫

T

e2βs |Zs |2 ds 0 [ ] ∫ T 2βT 2 2βs 1 2 2 2 2 2 2 ≤ 2E e |ξ| + e |f (s, Ys , Zs ) − f (s, Ys , Zs )| ds .(8.2.21) E

0

Looking at the supremum in (8.2.20) only the first term presents some problems which we first treat separately. We make use of the Burkholder-Davis-Gundyinequality: ∫ T 2βs E sup e ⟨Ys , Zs dWs ⟩ 0≤t≤T t ∫ T 2βs ≤ E e ⟨Ys , Zs dWs ⟩ + E sup

0≤t≤T

0

√

∫

T

≤ CE

∫ t e2βs ⟨Ys , Zs dWs ⟩ 0

e4βs |Ys |2 |Zs |2 ds

0

v( )∫ u u ≤ CE t sup e2βt |Yt |2 0≤t≤T

T

e2βs |Zs |2 ds

0

∫ T [ ] 1 2βt 2 2 e2βs |Zs |2 ds ≤ E sup e |Yt | + C E 4 0≤t≤T 0 [ ] ∫ T 2 2βT 2 2βs 1 2 2 2 2 2 2 ≤ 2C E e |ξ| + e |f (s, Ys , Zs ) − f (s, Ys , Zs )| ds 0

[ ] 1 + E sup e2βt |Yt |2 . 4 0≤t≤T

(8.2.22)

Note that the last inequality makes use of (8.2.21. So by (8.2.20) and (8.2.22): ( E sup

e2βt |Yt |2

0≤t≤T

[

≤ CE e

2βT

)

∫ |ξ| + 2

]

T 2βs

e

|f

1

(s, Ys2 , Zs2 )

−f

2

(s, Ys2 , Zs2 )|2 ds

(. 8.2.23)

0

Combining (8.2.21) and (8.2.23) we get the assertion for sufficiently large β. From the remark in the beginning of the proof this gives the general case.

8.2. THE PROBLEM

401

With this result we can make the covergence result from corollary 8.2.18 better. Corollary 8.2.22. Let (f, ξ) be standard data, and let the sequence Y i be defined as in corollary 8.2.18. Furthermore let Y be the first part of the solution of the BSDE corresponding to the respective data. Then for all β ∈ R: lim E sup

i→∞

0≤t≤T

( ) e2βt |Yt − Yti |2 = 0

[

and

] sup |Yt −

lim

i→∞

0≤t≤T

Yti |

= 0; P -a.s.

Proof. From theorem 8.2.21 we have by making use of the Lipschitz condition of the generator: ( E sup 0≤t≤T ∫ T

e2βt |Yt − Yti |2

)

≤ CE

e2βt |f (t, Yti , Zti ) − f (t, Yti−1 , Zti−1 )|2 dt ∫ T ( ) 2 ≤ 2K CE e2βt |Yti − Yti−1 |2 + |Zti − Zti−1 )|2 dt. 0

0

Here C is independent of i, and K denotes the Lipschitz constant of f . Then we find from (8.2.16) in the proof of corollary 8.2.18 for a constant independent of i, which is denoted by C again: ( )i ( ) 1 2βt i 2 E sup e |Yt − Yt | ≤ C . 2 0≤t≤T This proves convergence in expectation. Almost sure convergence is then derived in a similar way as in the proof of corollary 8.2.18 by making use of Tschebytschev’s inequality and the Borel/Cantelli lemma.

8.2.2

Linear BSDEs

In this section we are going to study the structure of linear BSDEs. This type of BSDEs finds interesting applications in mathematical finance and in stochastic control where they are found under the notion of ”‘adjoint equations”’ in maximum principles. In this context BSDEs are first found in Bismut [5, 6]. A thorough study of control applications is presented in the Stochastik I

402


script. For more recent results the reader is referred to Yong/Zhou [62], and Kohlmann/Zhou [18], and to Kohlmann und Tang [14, 15, 16, 37] for recent results on backwards stochastic Riccati equations, which allow to treat LQcontol problems with srtochastic coefficients. Applications of the latter results are found in this book. We define: Definition 8.2.23. A BSDE is called linear, if the generator f has the following form:

d ∑

−f (t, y, z) = At y +

Bt z i + gt .

(8.2.24)

i=1

Here z := (z1 , . . . , zd ), and A, B1 , . . . , Bd : Ω × [0, T ] → Rn×n , g : Ω × [0, T ] → Rn are Ft -progressively measurable. The following theorem generalizes the representation of Y in simple BSDEs (Proposition 8.2.11) to linear BSDEs. Theorem 8.2.24. Let ξ ∈ L2T (Rn ) and g ∈ HT2 (Rn ), and let the coefficients A, B 1 , . . . , B d be essentially bounded. Then the BSDE ] [ d ∑ i i dYt = At Yt + Bt Zt + gt dt + Zt dWt , i=1

YT

= ξ

(8.2.25)

has a unique solution. We have: Yt =

Φ−1 t E

[

∫ ΦT ξ − t

T

] Φs gs ds Ft ,

(8.2.26)

where Φ is the strong solution of the following SDE: dΦt = −Φt At dt −

d ∑

Φt Bti dWti ,

i=1

Φ0 = En .

(8.2.27)

Remark 8.2.25. For n = 1 and A ≡ 0 Φ is just the Girsanov functional, as equation (8.2.27) is the Dolans-Dade-equation. If furthermore g ≡ 0, then Y is the conditional expectation of ξ w.r.t. the measure given by the Girsanov

8.2. THE PROBLEM

403

functional, and under this measure it is a martingale. Martingale theory can be generalized by making use of BSDEs. The theory of such g-martingales is found in Peng [53, 54]. Remark 8.2.26. A mathematical finance interpretation for Φ in the case g ≡ 0 and n = 1 is found in the theory of pricing contingent claims ξ. Then Φ is the pricing-kernel. The solution Y of the linear BSDE is interpreted as the expectation of the discounted claim w.r.t. the riskneutral measure. This is the classical approach to the pricing of options, see Karatzas [31], section 1.1.2. The following properties of Φ are summarized: Lemma 8.2.27. Let Ψ be the strong solution of the SDE [ dΨt =

At Ψt +

d ∑

] Bti Bti Ψt

dt +

d ∑

Bti Ψt dWti ,

i=1

i=1

Ψ0 = En .

(8.2.28)

Then the following holds: (i) Φ−1 = Ψ. (ii) Φ ∈ STc,2 (Rn ). Proof. (i) With (8.2.27) and (8.2.28) we find from Itô’s formula: ∫

t

[ Φs As Ψs +

Φt Ψt = En + 0

+

d ∫ t ∑ i=1

−

Bsi Bsi Ψs ds

∫

Φs Bsi Ψs dWsi −

t

Φs As Ψs ds 0

t

Φs Bsi Ψs dWsi 0

]

i=1

0

d ∫ ∑ i=1

d ∑

−

d ∫ ∑ i=1

t

Φs Bsi Bsi Ψs ds

0

= En . (ii) is a standard result from the thory of SDEs (see e.g. Yong/Zhou [62], S.49).

Proof of 8.2.24: Existence and uniqueness follow from the Pardoux/Pengtheorem, as a linear generator with essentially bounded coefficients which fulfill

404


a uniform Lipschitz condition. So we still have to prove the special form of Y . Apply Itô’s formula to Φt Yt : [ ] ∫ T d d ∫ ∑ ∑ i i Φt Yt = ΦT ξ − Φs As Ys + Bs Zs + gs ds − t

∫

i=1

T

+

Φs As Ys ds + t

d ∫ ∑ i=1

∫

T

= ΦT ξ −

Φs gs ds −

t

T

Φs Bsi Ys dWsi +

t

d ∫ T ∑

i=1

Φs Zsi dWsi

t

i=1

d ∫ ∑

T

T

Φs Bsi Zsi ds

t

Φs [Zsi − Bsi Ys ]dWsi .

t

i=1

From the boundedness of B i Zsi −Bsi Ys is in HT2 (Rn ). From the preceding lemma we have Φ ∈ STc,2 (Rn ) , the integrand of the Itô integral is in HT1 (Rn ), and from Lemma 8.2.16 the Itô-integral is a martingale. Now look at the conditional expectation to find:

[

∫

Φt Yt = E[Φt Yt |Ft ] = E ΦT ξ −

T

t

Finally multiply with

Φ−1 ,

] Φs gs ds Ft .

which exists by the preceding lemma, to arrive at

the assertion.

In the case n = 1 this theorem on linear BSDEs has an important consequence for BSDEs under standard data. Under appropriate assumptions it allows to compare the Y -parts of the solutions of BSDEs with different data. ˜ be standard data with correTheorem 8.2.28. Let n = 1, (f, ξ) and (f˜, ξ) ˜ Assume: sponding solutions (Y, Z) and (Y˜ , Z). ( ) P ξ ≥ ξ˜ = 1, ( ) P f (t, Y˜t , Z˜t ) ≥ f˜(t, Y˜t , Z˜t ) a.e. t ∈ [0, T ] = 1. Then we find: (i) P (∀t∈[0,T ] Yt ≥ Y˜t ) = 1. (ii) If Yt = Y˜t on a set F ∈ Ft , then Ys = Y˜s on [t, T ] × F P -a.s. ˜ Define with Z¯ 0 := Z Proof. Denote by Z resp. Z˜ i the components of Z and Z. and Z¯ i := (Z˜ 1 , . . . , Z˜ i , Z i+1 , . . . , Z d ) for 1 ≤ i ≤ d −f (t, Yt , Zt ) + f (t, Y˜t , Zt ) , Yt − Y˜t −f (t, Y˜t , Z¯ti−1 ) + f (t, Y˜t , Z¯ti ) := 1{Z i ̸=Z˜ i } . t t Zti − Z˜ti

At := 1{Yt ̸=Y˜t } Bti

8.2. THE PROBLEM

405

˜ solves the linear BSDE Now (Y − Y˜ , Z − Z) ( dyt =

At yt +

d ∑

) Bti zti − f (t, Y˜t , Z˜t ) + f˜(t, Y˜t , Z˜t ) dt + zt dWt ,

i=1

yT

˜ = ξ − ξ.

(8.2.29)

As f is a standard generator, the processes A, B 1 , . . . , B d are essentially bounded ˜ is unique from theorem by the Lipschitz constant. The solution (Y − Y˜ , Z − Z) 8.2.24 and has the following form: [ ∫ −1 ˜ ˜ Yt − Yt = Φt E ΦT (ξ − ξ) +

[

T

Φs

] ] ˜ ˜ ˜ ˜ ˜ f (s, Ys , Zs ) − f (s, Ys , Zs ) ds—Ft .

t

(8.2.30) As we consider the case n = 1, Φ can be explicitly described and is is positive (see e.g. Karatzas/Shreve [18], p.361). This proves (i). To derive (ii) we multiply (8.2.30) with the indicator function of F and get: ( ) P 1F ξ = 1F ξ˜ = 1, ( ) P 1F f (s, Y˜s , Z˜s ) = 1F f˜(s, Y˜s , Z˜s ) a.e. s ∈ [t, T ] = 1. With this we find for s ∈ [t, T ]: 1F (Ys − Y˜s ) = 0, that is (ii).

Apply this theorem to f˜ = 0 and ξ˜ = 0 to find the following criterion for non-negativity: Corollary 8.2.29. Let n = 1, (f, ξ) be standard data with ξ ≥ 0 P -a.s. and P (f (t, 0, 0) ≥ 0 a.e. t ∈ [0, T ]) = 1. Then the following holds for the solution of the corresponding BSDE: P (∀t∈[0,T ] Yt ≥ 0) = 1. Finally we give a further characterization of solutions of linear BSDEs which establishes a connection to an adjoint (forward-) SDE. We still assume the coefficients to be essentially bounded. For (y, z) ∈ M T,d (Rn ) let us consider the SDE dXt =

[−A∗t Xt

+ yt ]dt +

d ∑ ( i=1

X0 = 0.

) zti − (Bti )∗ Xt dWti ,

406


The unique (by the boundedness assumption) strong solution gives a linar functional F on M T,d (Rn ) by ∫ F (y, z) := −E

T

⟨Xs , gs ⟩ds + E⟨XT , ξ⟩.

0

Continuity of F follows from the continuity of the SDE in the data. We have the following connection between the functional F and the linear BSDE (8.2.25): Theorem 8.2.30. Let (Y, Z) ∈ MT,d (Rn ), and let the assumptions of theorem 8.2.24 hold. Then the following assertions are equivalent: (i) (Y, Z) is solution of the BSDE (8.2.25). (ii) For alle (y, z) ∈ M T,d (Rn ) we have: ∫ T E ⟨Ys , ys ⟩ + ⟨Zs , zs ⟩ds = F (y, z).

(8.2.31)

0

Proof. (i) ⇒ (ii): Let (Y, Z) be the solution of the BSDE (8.2.25). From die Itô’s formula: ∫

T

0 = E⟨X0 , Y0 ⟩ = E⟨XT , ξ⟩ − E

⟨Xs , As Ys +

0

∫ +E

T

⟨Ys , A∗s Xs − ys ⟩ds −

0

d ∑ i=1

Also:

∫ E

T

∫ E

d ∑

Bsi Zsi + gs ⟩ds

i=1 T

⟨Zsi , zsi − (Bsi )∗ Xs ⟩ds.

0

⟨Ys , ys ⟩ + ⟨Zs , zs ⟩ds = F (y, z).

0

(ii) ⇒ (i): We interpret F as a mapping in the Hilbert space MT,d (Rn ). Then the representation (8.2.31) is unique by the Rieszs representation theorem. As from above a solution of the linear BSDE (8.2.25) is a representation of F in the form (8.2.31) we have for each representation of this form (Y ′ , Z ′ ) ∈ MT,d (Rn ): P (∀0≤t≤T Yt = Yt′ ∧ Zt = Zt′ a.e. t ∈ [0, T ]) = 1.

(8.2.32)

Here, the indistinguishability of Y and Y ′ follows from continuity as both processes are in MT,d (Rn ) (the Riesz representation theorem only gives almost sure indistinguishability.). From (8.2.32) we find however that (Y ′ , Z ′ ) is solution of the linear BSDE.

8.2. THE PROBLEM

8.2.3

407

Markovian BSDEs

In this section we treat BSDEs with a very special structure. The coefficients are assumed to depend on ω only implicitly by the solution flow X of a forward SDE. This has certain implications for the form of the solutions: The solution processes (Y, Z) of the BSDE can be represented as deterministic functions of X. For Y we will prove this under rather general assumptions, for Z the result will be stated in an important special case, where the SDE and the BSDE are related by a parabolic partial differential equation (PDE). Consider for (s, x) ∈ [0, T ] × Rm and s ≤ t ≤ T the SDE: dXt = b(t, Xt )dt + σ(t, Xt )dWt , Xs = x

(8.2.33)

and assume that the coefficients fulfill the following assumptions: (F1) b : [0, T ] × Rm → Rm and σ : [0, T ] × Rm → Rm×d are measurable and lipschitz continuous in the second variable (uniformly in t). (F2) b and σ have at most linear growth in the second component, i.e. there is a constant C such that for all (t, x) |b(t, x)| + |σ(t, x)| ≤ C(1 + |x|). Under these assumptions we have the following classical result: Theorem 8.2.31. Under assumptions (F1),(F2) we have: (i) The SDE (8.2.33) has for all pairs (s, x) ∈ [0, T ] × Rm a unique strong solution denoted by Xts,x . (ii) For p ≥ 1/2 there is a constant K depending only on p and T with E sup s≤t≤T

[ s,x 2p ] |Xt | ≤ K(1 + |x|2p ). t,Xts,x

(iii) The following flow property holds for s ≤ t ≤ r: Xr

(8.2.34) = Xrs,x .

The following BSDE depends on the solution flow in the following way: dYt = −f (t, Xts,x , Yt , Zt )dt + Zt dWt , YT

= g(XTs,x ).

(8.2.35)

408


The coefficients are assumed to have the following properties: (B1) f : [0, T ] × Rm × Rn × Rn×d → Rn is measurable and lipschitz continuous in the third and fourth component (uniformly in the two first). (B2) g : Rm → Rn is measurable. (B3) There is a constant C and a p ≥ 1/2 such that for all (t, x, y, z) |f (t, x, y, z)| + |g(x)| ≤ C(1 + |x|p ). First we assume that (F1),(F2) and (B1)–(B3) hold. With this we get: Theorem 8.2.32. The (8.2.35) has -for each pair (s, x) ∈ [0, T ]×Rm - a unique solution in the time interval [s, T ] denoted by (Yts,x , Zts,x ). Remark 8.2.33. Note that the initial time now is s. So the definition of the spaces MT,d (Rn ), HT2 (Rn ), L2T (Rn ) have to be adapted in a straightforward way. Proof. We prove that for each (s, x) ∈ [0, T ] × Rm (f (·, X s,x , ·, ·), g(XTs,x )) are standard data. Then the assertion follows from the multiply cited theorem by Pardoux and Peng. The integrability conditions folow immediately from (B3) and (8.2.34), and (B1) is just the Lipschitz condition on the generator.

We are interested in the structure of solutions (Yts,x , Zts,x ). Let us first look at Yts,x . Under the assumptions of theorem 8.2.32 we will show that Yts,x may be represented as a deterministic function of Xts,x . Then the BSDE (8.2.35) is called markovian as in this case the Markov property of Xts,x is carried over to Yts,x . In preparation we prove that Ytt,x is deterministic. We introduce the follow˜ ts := Wt+s − Ws be the by s translated Brownian ing notation: For fixed s let W ˜ ts . motion and Gts be the augmentation of the filtration generated by W The following holds: s -adapted and Y t,x P -a.s. is deLemma 8.2.34. Xts,x , Yts,x and Zts,x are Gt−s t

terministic.

8.2. THE PROBLEM

409

Proof. By translation of time we get (s ≤ t ≤ T ): ∫ t ∫ t s,x s,x Xt = x+ b(r, Xr )dr + σ(r, Xrs,x )dWr s s ∫ t−s ∫ t−s s,x s,x ˜ rs . = x+ b(r + s, Xr+s )dr + σ(r + s, Xr+s )dW 0

So

s,x Xt+s

0

solves for 0 ≤ t ≤ T − s the SDE: ˜ ts , dXt = b(t + s, Xt )dt + σ(t + s, Xt )dW X0 = x.

(8.2.36)

s,x Because of (F1),(F2) the solution of this SDE is pathwise unique. So Xt+s s is indistinguishable from the Gts -adapted solution of (8.2.36). So Xts,x is Gt−s

adapted. ˜ be the (from theorem 8.2.32) unique G s -adapted solution of Now let (Y˜ , Z) t the BSDE s,x ˜ ts , dYt = −f (t + s, Xt+s , Yt , Zt )dt + Zt dW

YT −s = g(XTs,x ); 0 ≤ t ≤ T − s. By making use of the same time transformation as above we find that (Y˜t−s , Z˜t−s ) s -adapted solution of the BSDE (8.2.35) on [s, T ]. As W ˜ s is Ft is a Gt−s t−s

measurable so (Y˜t−s , Z˜t−s ) is Ft -adapted. Because of the uniqueness of the solution of (8.2.35) we then have: (Y˜t−s , Z˜t−s ) = (Yts,x , Zts,x ). So Yts,x and Zts,x s -adapted. Gt−s

Ytt,x is measurable w.r.t. the trivial σ-algebra G0t , so from the Blumenthal 0-1-law P -a.s. deterministic.

In the following we use the notation u(t, x) for the deterministic modification of Ytt,x . We can now prove the desired result for simple BSDEs: Proposition 8.2.35. Let f be independent of (y, z). Then: ] [ ∫ T s,x s,x s,x Yt = E g(XT ) + f (r, Xr )dr Ft = u(t, Xts,x ). t

Especially u(t, Xts,x ) is a modification of Yts,x .

410


Proof. The first equality is the usual representation for simple BSDEs (Proposition 8.2.11). Let us look at the second equation: Again from proposition 8.2.11 we have:

[ u(t, x) =

Ytt,x

=E

∫ g(XTt,x )

T

+ t

f (r, Xrt,x )dr Ft

] . t,Xts,x

So from the Markov property and the flow property for s ≤ t ≤ r ≤ T Xr

=

Xrs,x we get: [ u(t, Xts,x )

= E

t,X s,x g(XT t )

[ = E

∫ + ∫

g(XTs,x )

T

+ t

t T

t,X s,x f (r, Xr t )dr Ft

f (r, Xrs,x )dr Ft

]

]

.

Now make use of the approximation described in corollary 8.2.22 to carry over this result to the general case: Theorem 8.2.36. Under the standard assumptions (F1),(F2) and (B1)–(B3) of this section Ytt,x has a deterministic modification u(t, x), and we have: For all pairs (s, x) ∈ [0, T ] × Rm u(t, Xts,x ) is a modification of Yts,x . Proof. From lemma 8.2.34 u(t, x) is deterministic, and the assertion follows from proposition 8.2.35 for simple BSDEs. For the general case we make use of the iteration procedure of corollary 8.2.22. So let the sequence Yts,x,k be given by Yts,x,0 := 0, Zts,x,0 := 0 and dYts,x,k = −f (t, Xts,x , Yts,x,k−1 , Zts,x,k−1 )dt + Zts,x,k dWt , YTs,x,k = g(XTs,x ). Put uk (t, x) := Ytt,x,k . From lemma 8.2.34 also uk is a deterministic function and from the approximation in corollary 8.2.22 uk (t, x) converges pointwise to u(t, x). Furthermore, from corollary 8.2.22 for fixed (t, s, x) Yts,x,k also converges to Yts,x P -a.s. So finally we have from proposition 8.2.35: u(t, Xts,x ) = lim uk (t, Xts,x ) = lim Yts,x,k = Yts,x ; P -a.s. k→∞

k→∞

8.2. THE PROBLEM

411

Remark 8.2.37. Under the standard assumptions of this section also Zts,x may be represented as a deterministic function of Xts,x . This again is first proved for simple BSDEs where a result by C ¸ inlar et al. [8], Theorem 6.27, can be used. The result then follows from the iteration of corollary 8.2.18. For details the reader is referred to El Karoui/Peng/Quenez [9], chapter 4. Now we consider a special case where we assume that the coefficients are regular enough so that u(t, x) := Ytt,x ∈ C 1,2 (Rn ). Then Itô’s formula may be applied and as Yts,x = u(t, Xts,x ) we get for 1 ≤ i ≤ n: ∫ Yts,x,i

= g

i

(XTs,x ) ∫

T

−

−

∫

T

∂t u

i

(r, Xrs,x )dr

t

−

T

⟨∂x ui (r, Xrs,x ), b(r, Xrs,x )⟩dr

t

⟨∂x ui (r, Xrs,x ), σ(r, Xrs,x )dWr ⟩

t

1 − 2

∫

T

[ ] tr ∂x,x ui (r, Xrs,x )(σσ ∗ )(r, Xrs,x ) dr.

(8.2.37)

t

Compare (8.2.35) with (8.2.37), to see that Zts,x = ∂x u(t, Xts,x )σ(t, Xrs,x )

(8.2.38)

so that -by putting s = t- for (t, x) ∈ [0, T ] × Rm (1 ≤ i ≤ n): ] 1 [ tr ∂x,x ui (t, x)(σσ ∗ )(t, x) + ⟨∂x ui (t, x), b(t, x)⟩ 2 = −f i (t, x, u(t, x), ∂x u(t, x)σ(t, x)) ,

∂t ui (t, x) +

u(T, x) = g(x);

x ∈ Rm .

(8.2.39)

In this way we get both a representation for Zts,x and a characterization for Ytt,x : Theorem 8.2.38. If in addition to the assunptions (F1),(F2) and (B1)–(B3) the coefficients are so regular that u(t, x) := Ytt,x ∈ C 1,2 (Rn ), then: (i) Zts,x = ∂x u(t, Xts,x )σ(t, Xts,x ). (ii) u(t, x) is a classical solution of the system of quasilinear parabolic PDEs (8.2.39). (iii) Let Dt be the Malliavin derivative at time t. Then Dt Yts,x is a modification of Zts,x .

412


Proof. After the above preparations we only have to prove (iii). As Yts,x = u(t, Xts,x ), we get from the chain rule for Malliavin derivatives (see e.g. Nualart [24], Proposition 1.2.2) up to a modification: Dt Yts,x = ∂x u(t, Xts,x )Dt Xts,x = ∂x u(t, Xts,x )σ(t, Xts,x ).

Remark 8.2.39. A sufficient condition for Ytt,x ∈ C 1,2 (Rn ) is the assumption that all coefficients belong to the appropriate Cb3 -space. We refer to Pardoux/Peng [23] for the technically interesting proof which makes extensive use of Malliavin’s calculus. We are now going to prove a sort of reverse of theorem 8.2.38. If the quasilinear system of parabolic partial differential equations (8.2.39) has a classical solution then we get the following scheme for solving the BSDE (8.2.35). This is aspecial case of the 4-step-scheme by Ma/Protter/Yong [21]: Theorem 8.2.40. Let v be a classical solution of the system (8.2.39), and assumke that for all (t, x) ∈ [0, T ] × Rm there is a constant C and a p ≥ 1/2 with |v(t, x)| + |∂x v(t, x)σ(t, x)| ≤ C (1 + |x|p ) .

(8.2.40)

Then the unique solution of the BSDE (8.2.35) is given by: Yts,x := v(t, Xts,x ), Zts,x := ∂x v(t, Xts,x )σ(t, Xrs,x ). Proof. For the classical solution we have v ∈ C 1,2 (Rm ). So the assertion (except for (Y, Z) to be in MT,d (Rn )) folllows by applying Itô’s formula as above. The integrability requirements on (Y, Z) can easily be derived as in the proof of theoreem 8.2.32 from (8.2.40).

From the uniqueness of the solution of the BSDE (8.2.35) we have that the system (8.2.39) has at most one solution. We have the following Feynman-Kactype formula:

8.2. THE PROBLEM

413

Corollary 8.2.41. Let v be a classical solution of the system of PDEs (8.2.39) which together with its first derivative is bounded. Then: v(t, x) = Ytt,x . Proof. From the boundedness and (F2) we have (8.2.40) with p = 1. So theorem 8.2.40 gives: Ytt,x = v(t, Xtt,x ) = v(t, x).

Remark 8.2.42. In order to apply the scheme in theorem 8.2.40 it is important to know if the PDE (8.2.39) has a solution. For this problem the reader is referred to the bibliography in Ma/Protter/Yong [21], especially Ladyzenskaja et al. [41].

8.2.4

BSDEs and partial BSDEs

In the last section we studied the relation between solutions of BSDEs and solutions of PDEs in the Markovian case. Now we are going to consider the same problem but now we allow the coefficients of the BSDE to depend explicitly on ω. This leads to a stochastic backward PDE BPDE. In the context of this book we are not so much interested in existence and uniqueness results for BPDEs (we will cite the results below). We are mainly concerned with the question under what kind of assumptions we will have the equality: Yt = u(t, Xt ), where u is the solution of a BSPDE. To make the notation easier we restrict our considerations to the case n = 1. Again we consider a weakly coupled system consisting of a SDE and a BSDE: dXts,x = b(t, Xts,x )dt + σ(t, Xts,x )dWt , dYts,x = −f (t, Xts,x , Yts,x , Zts,x )dt + Zts,x dWt , Xss,x = x,

YTs,x = g(XTs,x );

0 ≤ s ≤ t ≤ T.

(8.2.41)

where now all coefficients explicitly depend on Ω in a Ft -progressively measurable way. Except for the simplification n = 1 all dimensions are the same as in the last section.

414

CHAPTER 8. APPENDIX BSDES The following BPDE will now play the role of the PDE (8.2.39): ∫ u(t, x) = g(x) +

T

{f (s, x, u(s, x), z(s, x, ∂x u(s, x), q(s, x)))

t

1 + tr[∂x,x u(s, x)(σσ ∗ )(s, x)] + tr[σ ∗ (s, x)∂x q(s, x)] 2 ∫ T +⟨∂x u(s, x), b(s, x)⟩}ds − ⟨q(s, x), dWs ⟩. (8.2.42) t

Here z(t, x, w, q) := [q + σ ∗ (t, x)w]∗ . As in the case of BSDEs the solution of a BPDE is a pair of processes. We define: Definition 8.2.43. A pair of processes (u, q) is called a classical solution of the BSPDE (8.2.42), if: (i) equation (8.2.42) holds for all t ∈ [0, T ] P -a.s. ( ( )) ( ) (ii) (u, q) ∈ C [0, T ], L2t C 2 (B R , R) × HT2 C 1 (B R , Rd ) for all R ∈ R. Here BR is the open ball with radius R with center zero in Rm . ( ) ) ( Remark 8.2.44. The spaces HT2 C 1 (B R , Rd ) and L2t C 2 (B R , R) are defined in analogy to the spaces HT2 (Rn ) and L2T (Rn ). The solution (u, q) is called classical as it is sufficiently differentiable in x in the classical sense. Besides classical solutions we may also find weak and strong solutions for BSPDEs and for details the reader is referred to Ma/Yong [46], chapter 5. Remark 8.2.45. If the coefficients do not depend on ω and if u is a classical solution of the PDE (8.2.39) then (u, 0) is a classical solution of the BSPDE (8.2.42) as in this case (8.2.42) is just the t-integrated form of (8.2.39). So the notion of BSPDE (8.2.42) is a natural generalization of the PDE (8.2.39) when the coefficients depend on Ω. We need the following generalization of Itô’s formula: Theorem 8.2.46 (Generalized Itˆ o’s formula). Let for all R ∈ R u(t, x) ∈ ( ( )) C [0, T ], L2t C 2 (B R , R) of the form: ∫ u(t, x) = u(0, x) +

∫

t

p(s, x)ds + 0

0

t

⟨q(s, x), dWs ⟩

8.2. THE PROBLEM

415

( ) ( ) with p ∈ HT2 C 1 (B R , R) and q ∈ HT2 C 2 (B R , Rd ) for all R ∈ R. Futhermore let X = (X 1 , . . . X m )∗ be a continuous semimartingale. Then: ∫

∫

t

u(t, Xt ) = u(0, X0 ) + +

+

m ∫ ∑

0

∂xi u(s, Xs )dXsi

⟨

i=1

⟨q(s, Xs ), dWs ⟩

0

t

i=1 0 m ∫ . ∑

t

p(s, Xs )ds +

m ∫ 1 ∑ t + ∂xi ,xj u(s, Xs )d⟨X i , X j ⟩s 2 0 i,j=1

⟨∂xi q(s, Xs ), dWs ⟩, X⟩t .

0

Proof. We show that this is a special case of the generalized Itô’s formula in Kunita [19], Theorem 3.3.1. For all notations we refer to this book. Because of the differentiability assumptions on p and q u is a C 1 -semimartingale (see Kunita [19], Theorem 3.1.2 and Exercise 3.1.5) and for 1 ≤ i ≤ m: ∫

∫

t

t

∂xi p(s, x)ds +

∂xi u(t, x) = ∂xi u(0, x) +

⟨∂xi q(s, x), dWs ⟩.

(8.2.43)

0

0

So (⟨q(t, x), q(t, y)⟩, p(t, x)) are te local characteristics of u. From the integrability of p and q in t it is easily seen that the local characteristics are of class B 1,0 )) ( ( (indeed weaker assumptions would suffice.). As u(t, x) ∈ C [0, T ], L2t C 2 (B R , R) , u is a C 2 -process. So all assumptions in theorem 3.3.1 in Kunita [19] are fulfilled. After having applied this theorem we only have to prove that m ∫ . m ∫ . ∑ ∑ ⟨ ⟨∂xi q(s, Xs ), dWs ⟩, X⟩t ⟨ ∂xi u(ds, Xs ), X⟩t = 0

i=1

and

∫

i=1

∫

t

u(ds, Xs ) = 0

∫

t

p(s, Xs )ds + 0

(8.2.44)

0

t

⟨q(s, Xs ), dWs ⟩.

(8.2.45)

0

Now (8.2.45) first only holds for simple integrands because of the definition of the Itô-integral with parameters (see Kunita [19] p.80 for the definition) from some technical computations. The general case then follows by approximation. Using the same argument because of (8.2.43) we have: ∫

∫

t

∂xi u(ds, Xs ) = 0

This implies (8.2.44).

∫

t

∂xi p(s, Xs )ds + 0

0

t

⟨∂xi q(s, Xs ), dWr ⟩.

416

CHAPTER 8. APPENDIX BSDES Let (u, q) be a classical solution of the BSPDE (8.2.42). Put: p(s, x) := f (s, x, u(s, x), z(s, x, ∂x u(s, x), q(s, x))) 1 + tr[∂x,x u(s, x)(σσ ∗ )(s, x)] + tr[σ ∗ (s, x)∂x q(s, x)] 2 +⟨∂x u(s, x), b(s, x)⟩. (8.2.46)

As p is just the integrand in the Lebesgue-integral in (8.2.42), we have: ∫ u(t, x) = g(x) +

T

∫ p(s, x)ds −

t

T

⟨q(s, x), dWs ⟩.

t

Let us assume that the coefficients and the solution (u, q) are regular enough to allow the generalized Itô’s formula to be applied. Let the coefficients satisfy assumptions such that the system (8.2.41) has a unique solution (Xts,x , Yts,x , Zts,x ). Apply the generalized Itô’s formula to find: ∫ T ∫ T ⟨q(r, Xrs,x ), dWr ⟩ p(r, Xrs,x )dr − u(t, Xts,x ) = u(T, XTs,x ) + t t ∫ T ⟨∂x u(r, Xrs,x ), b(r, Xrs,x )⟩dr − t ∫ T − ⟨∂x u(r, Xrs,x ), σ(r, Xrs,x )dWr ⟩ t ∫ 1 T − tr[∂x,x u(r, Xrs,x )(σσ ∗ )(r, Xrs,x ))dr 2 t ∫ T tr[σ ∗ (r, Xrs,x )∂x q(r, Xrs,x )]dr. (8.2.47) − t

Putting p and z in (8.2.47) gives as u(T, x) = g(x): ∫ u(t, Xts,x )

T

f (r, Xrs,x , u(r, Xrs,x ), z(r, Xrs,x , ∂x u(r, Xrs,x ), q(r, Xrs,x ))) dr

= t

∫

− t

T

z(r, Xrs,x , ∂u(r, Xrs,x ), q(r, Xrs,x ))dWr + g(XTs,x )

So the pair (u(t, Xts,x ), z(r, Xrs,x , ∂u(r, Xrs,x ), q(r, Xrs,x ))) satisfies the backward equation of system (8.2.41). If this pair is sufficiently integrable it must be equal with the unique solution (Yts,x , Zts,x ). So: Yts,x = u(t, Xts,x ) Zts,x = z (t, Xts,x , ∂x u(t, Xts,x ), q(r, Xrs,x )) .

8.2. THE PROBLEM

417

A set of assumptions on the coefficients which allows for the BSPDE (8.2.42) to have a regular pair of solutions so that all transformations above can be made. (P1) f ist von der Form f (t, x, y, z) = a1 (t, x)y + ⟨a2 (t, x), z⟩ + a0 (x). Here a1 : Ω × [0, T ] → Cbk (Rm , R), a2 : Ω × [0, T ] → Cbk (Rm , Rd ) are bounded and Ft -progressively measurable and ( ) a0 ∈ HT2 W2k (Rm , R) . Under these assumptions the BSDE in the system (8.2.41) is linear: dYts,x = − [a1 (t, Xts,x )Yts,x + ⟨a2 (t, Xts,x ), Zts,x ⟩ + a0 (t, Xts,x )] dt + Zts,x dWt , YTs,x = g(Xts,x ). (P2) σ : Ω × [0, T ] → Cbk+1 (Rm , Rm×d ), b : Ω × [0, T ] → Cbk (Rm , Rm ) are bounded and Ft -progressively measurable and ( ) g ∈ L2T W2k (Rm , R) . (P3) For 1 ≤ i ≤ m gilt: P ([σ(∂xi σ ∗ )]∗ = σ(∂xi σ ∗ ); a.e. t ∈ [0, T ]) = 1. Apply Theorem 5.2.1 in Ma/Yong [46] to arrive at the following existence result: Theorem 8.2.47. Let (P1)–(P3) hold for a k > l + m/2 and l ≥ 2. Then the BSPDE (8.2.42) has a unique classical solution (u, q) where: ∫ T 2 max E ∥u(t, ·)∥C l + E ∥q(t, ·)∥2C l−1 dt b 0≤t≤T b 0 [ ] ∫ T ≤ KE ∥g∥2W k + ∥a0 (t, ·)∥2W k dt . 2

0

2

(8.2.48)

418


Proof. The reduction to theorem 5.2.1 in Ma/Yong [46] is evident. Note that for the derivation of the estimate the Sobolev imbedding theorem is used, see Triebel [61], Satz 10.10.

We need some more information on the regularity of the coefficients: Lemma 8.2.48. Let (P1) and (P2) hold for k > l + m/2. ( ) ( ) Then a0 ∈ HT2 Cbl (Rm , R) and g ∈ L2T Cbl (Rm , R) . Proof. The result follows again from Sobolev’s imbedding theorem.

The main result of this section is the following theorem: Theorem 8.2.49. Let (P1)–(P3) hold for a k > 3 + m/2. Then the system (8.2.41) has a unique solution (Xts,x , Yts,x , Zts,x ). Let (u, q) the by theorem 8.2.47 unique solution of the BSPDE (8.2.42). Then: Yts,x = u(t, Xts,x ), Zts,x = [q(t, Xts,x ) + σ ∗ (t, Xts,x )∂x u(t, Xts,x )]∗ . Finally u has the following representation: [ u(t, x) = E

∫ t,x ϕt,x T g(XT )

+ t

T

t,x ϕt,x s a0 (s, Xs )ds Ft

]

with [∫ ϕt,x s

s

a1 (r, Xrt,x )

:= exp t

1 − |a2 (r, Xrt,x )|2 dr + 2

∫

s

] ⟨a2 (r, Xrt,x ), dWr ⟩

.

t

Proof. Note that under the above assumptions SDE dXts,x = b(t, Xts,x )dt + σ(t, Xts,x )dWt , Xss,x = x has a unique solution for all (s, x). (It is even a flw of C 2 -diffeomorphisms, buit thsi result is not used here).

8.2. THE PROBLEM

419

Remember that the BSDE under consideration is linear because of (P1) and now look at: dYts,x = − [a1 (t, Xts,x )Yts,x + ⟨a2 (t, Xts,x ), Zts,x ⟩ + a0 (t, Xts,x )] dt + Zts,x dWt , YTs,x = g(XTs,x ). From (P1) and lemma 8.2.48 it is seen that the coefficients satisfy the assumptions in the theorem on linear BSDEs 8.2.24. Then for each (s, x) we have a unique solution (Yts,x , Zts,x ) and we have a representation for Yts,x in the following form: [ Yts,x

=E

∫ ϕt,s,x g(XTs,x ) T

+ t

T

ϕt,s,x a0 (s, Xvs,x )dv Ft v

] ,

(8.2.49)

where ϕ solves the following linear SDE (s ≤ t ≤ v ≤ T ): dϕt,s,x = ϕt,s,x a1 (v, Xvs,x )dv + ⟨ϕt,s,x a2 (v, Xvs,x ), dWv , ⟩ v v v ϕt,s,x = 1. t Then finally (see Karatzas/Shreve [18], pp.360): [∫ v ] ∫ v 1 t,s,x s,x s,x 2 s,x ϕv = exp a1 (r, Xr ) − |a2 (r, Xr )| dr + ⟨a2 (r, Xr ), dWr ⟩ . 2 t t (8.2.50) Let us now turn to the BSPDE (8.2.42). By theorem 8.2.47 it has a unique solution (u, q) and from the estimate (8.2.48) we have: ( ) ( ( )) (u, q) ∈ C [0, T ], L2t Cb3 (Rm , R) × HT2 Cb2 (Rm , Rd ) .

(8.2.51)

Let p be as in (8.2.46). Then from the boundedness of the coefficients, lemma ) ( 8.2.48, and (8.2.51) we find that p ∈ HT2 Cb1 (Rm , Rd ) . With this the solution of the BSPDE (8.2.42) is regular enough to aplly the generalized Itô-formula. After the considerations above we only have to show for the representation of (Yts,x , Zts,x ) that for all (s, x): (u(t, Xts,x ), [q(t, Xts,x ) + σ ∗ (t, Xts,x )∂x u(t, Xts,x )]∗ ) ∈ MT,d (R). From the estimate (8.2.48) and the boundedness of σ we have (u(t, Xts,x ), [q(t, Xts,x ) + σ ∗ (t, Xts,x )∂x u(t, Xts,x )]∗ ) ∈ M T,d (R).

(8.2.52)

420


This proves (8.2.52) after making use of the lemma below.

Continuity of

u(t, Xts,x ) in t follows from the continuity of u and X. Finally we have to prove the representation of u. We already have: u(t, x) = u(t, Xtt,x ) = Ytt,x . With this the represenattion of u is just (8.2.49),(8.2.50) with s = t.

We made use of the following lemma: Lemma 8.2.50. Let (f, ξ) be standard data and for (Y, Z) ∈ M T,d (Rn ) let: Y be continuous and ∫ Yt =

T

∫ f (s, Ys , Zs )ds −

t

T

Zs dWs . t

So (Y, Z) ∈ MT,d (Rn ) and with this we have the unique solution of the BSDE with data (f, ξ). Proof. We only have to prove that Y ∈ ST2 (Rn ). From the same argument as in the theorem on ”‘simple”’ BSDEs (Proposition 8.2.11) we get: [ ] [ ] ∫ T 2 E sup |Yt | ≤ KE |ξ|2 + |f (s, Ys , Zs )|2 ds . 0≤t≤T

0

The assertion follows by making use of the Lipschitz condition of the generator and the integrability properties of (Y, Z). Remark 8.2.51. It is easily seen that the assertion of the lemma also holds when we replace the initial time 0 by an arbitrary s.

8.2.5

Remarks

8.2: The especially simple BSDE with f ≡ 0 is also used by Yong/Zhou [62], chapter 7, as the motivation for introducing the notion of solution as a pair of processes. Due to the special assumptions here the proof of uniqueness differs from their proof. The construction of further solutions which are not sufficiently integrable follows the ideas in El Karoui [16] where a consequence of Dudley’s theorem is used.

8.2. THE PROBLEM

421

The relevant literature on BSDEs gives different definitions for the solutions of BSDEs. Here with the assumption (Y, Z) ∈ MT,d (Rn ) we followed Yong/Zhou. Some authors however make weaker assumptions like (Y, Z) ∈ M T,d (Rn ), see Pardoux/Peng [26] or El Karoui [16], El Karoui/Peng/Quenez [9]. For standard data however (Y, Z) ∈ M T,d (Rn ) already implies (Y, Z) ∈ MT,d (Rn ) (see lemma 8.2.50). So in the end it is not relevant which definition is used. section 8.2.1: The theorem by Pardoux/Peng is the existence and uniqueness result for BSDEs. References on variants of the proof are found at the end of our proof. The proof is restrictive under variant aspects: (i) The filtration must be the Brownian filtration. (ii) The terminal time is deterministic and finite. (iii) The uniform Lipschitz condition is very restrictive. So there are many attempts to generalize the theorem: If we allow more general filtrations we must add another process to the notion of solution (El Karoui/Huang [17], El Karoui/Peng/Quenez [9], chapter 5). The other generalizations are possible under slight generalizations of the notion of solution. Peng [51] and Yong/Zhou [62] use a ”‘continuation”’-method to allow unbounded stopping times as terminal times. Basing on Bender/Kohlmann [3] we will give a further generalization which also weakens the Lipschitz condition. section 8.2.3: The study of linear BSDEs goes back to 1970, see e.g. Bismut [6]. The transformation making use of the matrix-valued process Φ is first found in Bensoussan [2]. We only studied the structure of linear BSDEs, existence and uniqueness is treated in Yong/Zhou [62] by making use of this transformation. The comparison theorem is found in many variants. One of the first results is found in Peng [52] which has been improved under several aspects (see Peng [53]). Our formulation is taken from El Karoui/Peng/Quenez [9]. The characterization of solutions of linear BSDEs with the Riesz representation of a continuous functional in M T,d (Rn ) is the workout of a remark in Yong/Zhou [62]. section 8.2.3: Markovian BSDEs are a special case of coupled systems of SDEs

422


and BSDEs. These general FBSDEs of the form dXt = b(t, Xt , Yt , Zt )dt + σ(t, Xt , Yt , Zt )dWt , dYt = f (t, Xt , Yt , Zt )dt + Zt dWt , X0 = x,

YT = g(XT )

were first studied by Antonelli [1]. A good survey on the state of art is given in the monograph by Ma/Yong [46]. The representation of Y as a deterministic function of X follows chapter 4 in El Karoui/Peng/Quenez [9]. Our arguments are different what allows to explicitly determine u as Ytt,x . The presented connections between BSDEs und PDEs base on results by Peng [51], Pardoux/Peng [26] and Ma/Protter/Yong [21]. The solution scheme for markovian BSDEs is a special case of the 4-step-scheme by Ma/Protter/Yong. The Feynman-Kac-formula was first proved by Peng [51]. Other versions of Feynman-Kac-type formulas are found in Yong/Zhou [62]. The representation of Z as a Malliavin derivative of Y holds under much more general assumptions. Results are found in Pardoux/Peng [23], El Karoui/Peng/Quenez [9], chapter 5, and Ma/Yong [23]. section 8.2.4: The connection between BSDEs and BSPDEs was first studied by Ma/Yong [23], and the basic ideas of our presentation are taken from there. We use a slightly more general existence and uniqueness result from Ma/Yong [46], chapter 5, and this leads to slightly more general results. Finally the representation of u is a generalization of the stochastic FeynmanKac-formula in Ma/Yong [23], where only the case a2 ≡ 0 is treated.

8.3

(a, β)-Theory

In this chapter we are going to prove several results from the previous chapter under weaker assumptions. So we aim at generalizing the results under the following two aspects: On the one hand we want to allow unbounded stopping times as terminal times. On the other hand we will weaken the assumptions to allow a stochastic process to take the role of the Lipschitz constant in the

8.3. APPENDIX (A, β)-THEORY

423

Lipschitz condition on the coefficients of the BSDE. Under these weaker assumptions we will prove existence in the classical sense (Definition 8.2.7) if the data are sufficiently integrable. It seems not to be possible to derive classical uniqueness in this case, we will show however that uniqueness holds in a smaller space. Finally we prove the continuity of the solutions in the data and derive a comparison theorem in this more general situation. The structure of solutions of linear BSDEs is then studied under these aspects. Some of the results are described in more detail in (Bender/Kohlmann [3]).

8.3.1

Formulation of The Main Result

In the proof of Pardoux/Peng’s theorem we made use of weighted norms and thus used the fact that for all β ∈ R ∫

T

∫ |Yt | dt < ∞ ⇔ E 2

E 0

T

eβt |Yt |2 dt < ∞.

0

Let us now consider a BSDE with infinite time horizon, so a BSDE on the (stochastich) interval [0, τ ] with a possibly unbounded stopping time τ . In this case the above cited equivalence does not hold in general. Of course we still have to use weighted norms so that we have to impose assumptions of integrability on the data and the coefficients which depend on β. As furthermore we are going to weaken the Lipschitz condition these assumptions will also depend on a process a. This leads to a slightly modified definition of a solution. For β = 0, a ≡ 1 this new notion of solution coincides of course with the notion in the classical approach. Let us first introduce soem new notations: We consider a probability space with the same properties as in the last section. Especially the assumption of a Brownian filtration still holds. Let a be a positive Ft -progressively measurable process a. With this process we define an increasing continuous process: ∫

t

a2s ds.

At := 0

424


Let β ≥ 0 and let τ be a stopping time w.r.t. the filtration Ft . Then define: L2τ,a,β (Rn ) := {ξ; Rn -valued and Fτ -measurable with [ ] ∥ξ∥2a,β := E eβAτ |ξ|2 < ∞}, 2 Hτ,a,β (Rn ) := {Y ; Rn -valued and progressively measurable with ∫ τ ∥Y ∥22,a,β := E eβAt |Yt |2 dt < ∞}, 0 a,2 2 Hτ,β (Rn ) := {Y ; aY ∈ Hτ,a,β (Rn )}, c,2 Sτ,a,β (Rn ) := {Y ; Rn -valued, progressively measurable and continuous with

∥Y ∥2s,a,β := E sup eβAt |Yt |2 < ∞}. 0≤t≤τ

Remark 8.3.1. Continuity of the process Y on [0, τ ] here means thatthe stopped process Yt∧τ is continuous on [0, ∞). Remark 8.3.2. Note that in this chapter we will make no notational difference between equivalence classes and their members in order to keep the notation easier. The reader will easily fill in the necessary technical parts of the proof from the preceding section. A further consequence of the unbounded time horizon is that even in the c,2 a,2 case a ≡ 1 we do not have Sτ,a,β (Rn ) ⊂ Hτ,β (Rn ). So we define: a,2 2 M τ,a,β,d (Rn ) := Hτ,β (Rn ) × Hτ,a,β (Rn×d ), ( ) a,2 c,2 2 Mτ,a,β,d (Rn ) := Hτ,β (Rn ) ∩ Sτ,a,β (Rn ) × Hτ,a,β (Rn×d ).

The spaces M τ,a,β,d (Rn ) and Mτ,a,β,d (Rn ) are endowed with the norms ∥(Y, Z)∥a,β := ∥(Y, Z)∥s,a,β :=

√ √

∥aY ∥22,a,β + ∥Z∥22,a,β , ∥Y ∥2s,a,β + ∥aY ∥22,a,β + ∥Z∥22,a,β .

Similar arguments as in chapter 1 show that M τ,a,β,d (Rn ) with the norm ∥(·, ·)∥a,β is a Hilbert space. Let us now give a new formulation of the problem. If no further restrictions are stated the terminal time τ is a Ft -stopping time which may take the value +∞. Let f : Ω×[0, ∞)×Rn × Rn×d → Rn be given such that for all (y, z) ∈ Rn × Rn×d f (·, ·, y, z) is Ft -progressively measurable and let ξ be an Fτ -measurable


425

random variable. Consider the BSDE dYt = −f (t, Yt , Zt )dt + Zt dWt , Yτ

= ξ.

(8.3.1)

Definition 8.3.3. Let β ≥ 0 and a be positive Ft -progressively measurable processes. (i) A pair of processes (Y, Z) ∈ Mτ,a,β,d (Rn ) is called an (a, β)-solution of the BSDE (8.3.1) if for all 0 ≤ t < ∞ ∫ τ ∫ Yt∧τ = ξ + f (s, Ys , Zs )ds − t∧τ

τ

Zs dWs ; P -a.s.

(8.3.2)

t∧τ

(ii) We call an (a, β)-solution classical if a ≡ 1 and β = 0. (iii) The BSDE (8.3.1) is (a, β)-uniquely solvable if for each two (a, β)-solutions (Y 1 , Z 1 ) and (Y 2 , Z 2 ) ( ) 1 2 1 2 P ∀0≤t 0. Then (Y, Z) is a classical solution. Proof. We have:

∫

τ

1 |Yt | dt ≤ 2 E ϵ

∫

2

E 0

τ

eβAt |at Yt |2 dt < ∞.

0

The other integrability conditions are evident.

Next we adapt the definition of standard data to the new situation: Definition 8.3.6. Let β ≥ 0 . A pair (f, ξ) is called a pair of (a, β)-standard data if: (i) There are d + 1 nonnegative Ft -progressively measurable processes rt and uit such that P ⊗ λ-a.s. for all (y, z, y˜, z˜) ∈ Rn × Rn×d × Rn × Rn×d we have: |f (t ∧ τ, y, z) − f (t ∧ τ, y˜, z˜)| ≤ rt∧τ |y − y˜| +

d ∑ i=1

uit∧τ |z i − z˜i |.

(8.3.4)

426


We call (8.3.4) a stochastic Lipschitz condition. √ ∑ (ii) at := rt + di=1 (uit )2 > 0. (iii) ξ ∈ L2τ,a,β (Rn ). 2 (iv) a−1 f (·, ·, 0, 0) ∈ Hτ,a,β (Rn ).

Remark 8.3.7. The condition a > 0 is easily forced to hold by replacing for an ϵ > 0 rt by rt + ϵ. In this way of course the conditions (iii) and (iv) become stronger. We first give two examples of situations where we have (a, β)-Standard data: Example 8.3.8. (i) Let τ = T be a deterministic and finite terminal time and let (f, ξ) be standard data in the sense of Definition 8.2.12. Then (f, ξ) are (C, β)-standard data for arbitrary β ≥ 0 and a sufficiently large constant C which depends on the Lipschitz constant. (ii) Let the generator f be linear, so that −f (t, y, z) = Ct y +

d ∑

Bti z i + gt .

i=1

Then the linear generator f fulfills the stochastic Lipschitz condition with rt := ∑ |Ct | and uit := |Bti |. Let a2t := rt + di=1 (uit )2 (+ϵ) - that means we add a positive constant -if necessary to guarantee that a > 0 - and let for a β > 0 2 ξ ∈ L2τ,a,β (Rn ) and a−1 g ∈ Hτ,a,β (Rn ). Then (f, ξ) are (a, β)-standard data.

The main result of this chapter is the following existence and uniqueness theorem which generalizes theorem 3 in Bender/Kohlmann [3]: Theorem 8.3.9. Let (f, ξ) be (a, β)-standard data for β > 2. Then the BSDE (8.3.1) has a unique (a, β)-solution. Furthermore (a, β)-uniqueness holds for (a, 2)-standard data. The proof will be found in the next two sections. Note that theorem 8.2.14 is a special case of theorem 8.3.9 from example 8.3.8 (i). With example 8.3.8 (ii) we get an existence and uniqueness result for linear BSDEs where boundedness of the coefficients plays no role:


427

Corollary 8.3.10. Let the generator f be linear, say −f (t, y, z) = Ct y +

d ∑

Bti z i + gt ,

i=1

and with rt := |Ct | and ut =: |Bti | let a2t := rt +

∑d

i 2 i=1 (ut ) (+ϵ).

We assume

2 that β > 2. Furthermore let ξ ∈ L2τ,a,β (Rn ) and a−1 g ∈ Hτ,a,β (Rn ). Then the

linearBSDE for the corresponding data has a unique (a, β)-solution. From Lemma 8.3.5 we have in adition: Corollary 8.3.11. Let (f, ξ) be (a, β)-standard data for β > 2 and a ≥ ϵ > 0. Then BSDE (8.3.1) has a classical solution. The following example shows that we cannot expect uniqueness in corollary 8.3.11: Example 8.3.12. We consider for n = d = 1 and for a finite deterministic terminal time the BSDE −1 Yt dt + Zt dWt , exp {T − t} − 1 = 0.

dYt = YT

For this in example 8.2.19 we had found infinitely many classical solutions. Here we have f (t, y, z) := (1−exp {T − t})−1 y. As f is linear we put uit := 0 and a2t := rt := (exp {T − t}−1)−1 ≥ (exp T −1)−1 . With this choice f satisfies the stochastic Lipschitz condition, and as f (t, 0, 0) = 0 and ξ = 0, so (f, ξ) are (a, β)-standard data for the just defined a and arbitrary β ≥ 0. With this all assumptions in corollary 8.3.11 are fulfilled and the BSDE above has infinitely many classical solutions. In the existence and uniqueness results above we always assume β > 2. We will prove that for β < 2 uniqueness does not hold in general: Example 8.3.13. Look at the BSDE 1 Yt dt + Zt dWt , t−1 = 0.

dYt = Y1

428


Just as in the preceding example one easily sees that the pair ((t − 1)−1 y, 0) is a pair of (a, β)-standard datea for a2t := (1 − t)−1 and arbitrary β ≥ 0. Furthermore (0, 0) is an (a, β)-solution. We prove now that the pair (Yt , Zt ) := (1 − t, 0) for β < 2 also is an (a, β)-solution. First for 0 ≤ t ≤ 1 we have: ∫ Yt = 1 − t =

1

∫ 1ds = −

t

t

1

1 Ys ds − s−1

∫

1

Zs dWs . t

2 As obviously 0 = Z ∈ Hτ,a,β (Rn×d ) (here with τ = 1) we only have to prove

sufficient integrability of Y . First we have (0 ≤ t < 1): ∫

∫

t

a2s ds

At = 0

So:

∫

t

=

(1 − s)−1 ds = − log[1 − t].

0

∫

1

eβAt a2t Yt2 dt

=

0

1

(1 − t)1−β dt =

0

1 < ∞. 2−β

(8.3.5)

As for β < 2, t < 1 we have d [ βAt 2 ] e Yt = −(2 − β)(1 − t)1−β < 0, dt so we get:

[ sup 0≤t≤1

] eβAt Yt2 = eβA0 Y02 = 1.

(8.3.6)

As Y is independent of ω, so (8.3.5)-(8.3.6) must be shown to hold. So (Y, Z) is a second (a, β)-solution for β < 2.

8.3.2

”‘Simple”’ BSDEs

This and the next section is concerned with the proof of the main result 8.3.9. The basic underlying idea of the proof is the same as in Pardoux/Peng: First we prove existence and uniqueness for ”‘simple”’ BSDEs and then apply Banach’s fixed point theorem. Due to the more general situation here however the proof is technically more difficult what we will already see when we treat ”‘simple BSDEs”’ in this section: The assumptions of the following prposition are assumed to hold through the whole section.


429

Proposition 8.3.14. Let β > 0 and a be a positive Ft -progressively measurable 2 process. If a−1 f ∈ Hτ,a,β (Rn ) and ξ ∈ L2τ,a,β (Rn ) then the BSDE

dYt = −ft dt + Zt dWt , Yτ

= ξ

(8.3.7)

has exactly one (a, β)-solution. Note that for β = 0, a ≡ 1 -what is excluded here- this is for finite time horizon the assertion of proposition 8.2.11. The proof there motivates the following definitions (0 ≤ t < ∞): [

∫

τ

Mt := E ξ + 0

∫ Yt := Mt∧τ −

] fs ds Ft ,

t∧τ

fs ds. 0

We first prove that

∫ E

τ

0

fs ds < ∞.

This guarantees existence of the conditional expectation in a classical sense. Here the assumption β > 0 plays a crucial role. We have: Lemma 8.3.15.

∫ E

0

τ

fs ds < ∞.

(8.3.8)

Furthermore: ∫ τ [ ] 2 |fs |2 2 2 E |Mt | ≤ 2E|ξ| + E eβAs 2 ds, (8.3.9) β as 0 [ ] [ ] 8 ∫ τ |fs |2 E sup eβAt |Yt |2 ≤ 8E |ξ|2 eβAτ + E eβAs 2 ds. (8.3.10) β as 0≤t≤τ 0 Especially Mt is a square integrable martingale. Proof. First we have for 0 ≤ t < ∞ from H”older’s inequality: ∫

τ

t∧τ

2 (∫ f (s)ds ≤ ≤

) (∫ τ 2 βAs |fs | e ds a2s t∧τ t∧τ ∫ τ 2 1 |f | s eβAs 2 ds. βeβAt∧τ 0 as τ

e−βAs a2s ds

)

(8.3.11)

430


For t = 0 we find (8.3.8) from the expectation (note that (Ω, F, P ) is finite). We also get (8.3.9) from (8.3.11) with the Jensen and Young inequalities. Now turn to the estimation for Y : [√ β/2At∧τ

e

] |Yt∧τ | ≤ E |ξ + fs ds|2 eβAt∧τ Ft∧τ t∧τ [√ ] ∫ τ √ ≤ 2E |ξ|2 eβAt∧τ + | fs ds|2 eβAt∧τ Ft∧τ t∧τ ] [√ ∫ τ 2 √ |f | 1 s ≤ eβAs 2 ds Ft∧τ . 2E |ξ|2 eβAτ + β 0 as ∫

τ

The last inequality makes use of (8.3.11). So we see that eβ/2At∧τ |Yt∧τ | is dominated by a martingale and we may apply Doob’s inequality. For fixed 0 ≤ T < ∞ we so get (make use of Jensen’inequality): [ ] [ ] 8 ∫ τ |fs |2 βAt 2 2 βAτ E sup e |Yt | ≤ 8E |ξ| e + E eβAs 2 ds. β as 0≤t≤T ∧τ 0 By taking the limit T → ∞ the estimate for Y follows from Fatou’s lemma. Remark 8.3.16. The assertions of the lemm do not hold in general for β = 0 if the stopping time τ is not bounded. The reason is that (Ω × [0, ∞), F ⊗ B[0,∞) , P ⊗ λ) is no longer finite so that an L2 -function stopped in τ might not be an L1 -function. The condition β > 0 gives the required integrability. From the above lemma Mt is a square integrable martingale an we may apply the martingale representation theorem (theorem 8.2.2). We get a process Z with the following properties: For 0 ≤ T < ∞ we have ∫ T E |Zt |2 dt < ∞

(8.3.12)

0

and for 0 ≤ t < ∞

∫

t

Mt = M0 +

Zs dWs ; P -a.s. 0

Use this in the definition of Yt (and choose a continuous modification what we tacitly assume to be done in the following), to get for 0 ≤ t ≤ T < ∞ (up t indistinguishability): ∫ Yt∧τ = YT ∧τ +

T ∧τ

t∧τ

∫ fs ds −

T ∧τ

Zs dWs . t∧τ

(8.3.13)


431

With this (Y, Z) seems to be a good candidate for being an appropriate pair of solutions. Next we prove the required integraqbility: Lemma 8.3.17. (Y, Z) ∈ Mτ,a,β,d (Rn ). More exactlyy: There is a constant Kβ , so that

∫ τ ∫ τ [ ] E sup eβAt |Yt |2 + E eβAs |as Ys |2 ds + E eβAs |Zs |2 ds 0≤t≤τ 0 0 [ ] ∫ τ 2 ≤ Kβ E eβAτ |ξ|2 + eβAs |a−1 s fs | ds . 0

Proof. We define the following localization by the sequence of stopping times (N ∈ N): σN := inf{t; eβAt ≥ N } ∧ N ∧ τ.

(8.3.14)

This sequence converges P -a.s. to σ := inf{t; eβAt = ∞} ∧ τ.

(8.3.15)

With this for N ∈ N the stopped process eβAt∧σN has bounded variation and with (8.3.13) Itô’s formula gives: ∫ σN ∫ σN 2 βAs 2 |Y0 | + e |Zs | ds + β eβAs |as Ys |2 ds 0 0 ∫ σN ∫ σN βAs 2 βAσN eβAs ⟨Ys , Zs dWs ⟩. e ⟨fs , Ys ⟩ds − 2 = e |YσN | + 2 0

0

Now recall the integrability of Y proved in the last lemma, the integrability of Z implied by the martingale representation theorem. The localization then implies again with Lemma 8.2.16 that on [0, N ] ∫ t 1{σN >s} eβAs ⟨Ys , Zs dWs ⟩ 0

is a martingale with expectation zero. So we get for the expectation (use the Cauchy-Schwarz’s and der Young’s inequality): ∫ σN ∫ σN β eβAs |as Ys |2 ds E eβAs |Zs |2 ds + E 2 0 0 [ ] 2 ∫ σN βAσN 2 2 ≤ E e |YσN | + E eβAs |a−1 s fs | ds. β 0 Now let N tend to infinity so that from (8.3.10) we finally get ∫ σ ∫ σ β βAs 2 E e |Zs | ds + E eβAs |as Ys |2 ds 2 0 0 [ ] 2 ∫ σ βAσ 2 2 ≤ E e |Yσ | + E eβAs |a−1 s fs | ds. β 0

(8.3.16)

432


Let us now look at the interval (σ, τ ). Due to the assumed and proven integrability properties the processes f and Y are zero on this interval. We also have 1{σ 2 the following holds for an appropriate choice of c for a constant Cβ1


437

depending only on β: ∫ σN ∫ σN E eβAs |as Ys |2 ds + E eβAs |Zs |2 ds 0 0 [ ] 1 ∫ σN 2 2 2 2 2 2 1 βAσN 2 βAs f (s, Ys , Zs ) − f (s, Ys , Zs ) ≤ Cβ E e |YσN | + e ds . as 0 (8.3.22) From (8.3.21) as in the proof of theorem 8.2.21 we get with the BurkholderDavis-Gundy-inequality the existence of another constant Cβ2 depending only on β, so that [ ] E sup eβAt |Yt |2 0≤t≤σN [ ∫ 2 2 βAσN ≤ Cβ E e |YσN | + 0

σN

] 1 2 , Z 2 ) − f 2 (s, Y 2 , Z 2 ) 2 f (s, Y s s s s eβAs ds . as (8.3.23)

Combining (8.3.22) and (8.3.23) let N tend to infinity so the assertion follows from Lebesgue’s theorem and the by now well known discussion of the interval (σ, τ ). Remark 8.3.22. From example 8.3.8 theorem 8.3.20 may also be applied for standard data (f, ξ) in the sense of Definition 8.2.12 in the case of a deterministic finite terminal time τ = T . At first sight theorem 8.3.20 might look like a better result than theorem 8.2.21 as the constant in theorem 8.2.21 also depends on parts of the data and the terminal time. In theorem 8.3.20 the norms depend on the Lipschitz condition so that an adaptation to theorem 8.2.21 leads to the same dependences. In the case f 1 ≡ f 2 the last term of the right hand side of (8.3.18) vanishes and we may put in the proof going on c = 0. With this we have the following corollary: Corollary 8.3.23. We consider (a, 2)-standard data (f, ξ 1 ), (f, ξ 2 ) and let (Y 1 , Z 1 ), resp. (Y 2 , Z 2 ) be (a, 2)-solutions of the corresponding BSDEs, if they exist. Then there is a constant C, such that

2

2

1

Y − Y 2 2 + Z 1 − Z 2 2,a,β ≤ C ξ 1 − ξ 2 a,β . s,a,β

438


Especially we have also for (a, 2)-standard data (a, β)-uniqueness. Next we prove a version of the comparison theorem 8.2.28 which fits into the framework here. As noted before the proof of theorem 8.2.28 cannot simply be imitated as in the most general case we have no representation for linear BSDEs. So here we try to estimate the positive part of the difference Y˜ − Y directly. ˜ be (a, β)-standard data Theorem 8.3.24. Let n = 1 and let (f, ξ) and (f˜, ξ) ˜ be (a, β)-solutions of the for β ≥ 2 and the same a. Let (Y, Z) and (Y˜ , Z) corresponding BSDEs (the existence of which is not proved in the case β = 2). The following holds true:

( ) P ξ ≥ ξ˜ = 1

and ( ) P f (t ∧ τ, Y˜t∧τ , Z˜t∧τ ) ≥ f˜(t ∧ τ, Y˜t∧τ , Z˜t∧τ ) a.e. t ∈ [0, ∞) = 1. Then: P (∀t∈[0,∞) Yt∧τ ≥ Y˜t∧τ ) = 1. Proof. To simplify the notation put: Y

:= Y˜ − Y,

Z := Z˜ − Z. + Our aim is to show that the positive part Yt∧τ vanishes for all 0 ≤ t < ∞. The

assertion then follows from the continuity of Y + . First note that because of the assumptions and the stochastic Lipschitz condition for 0 ≤ t < ∞ (up to indistingusihability) the following holds: [ ] 1{Yt∧σN >0} f˜(t ∧ σN , Y˜t∧σN , Z˜t∧σN ) − f (t ∧ σN , Yt∧σN , Zt∧σN ) ≤ 1{Yt∧σN >0} f (t ∧ σN , Y˜t∧σN , Z˜t∧σN ) − f (t ∧ σN , Yt∧σN , Zt∧σN ) ≤

+ rt∧σN Yt∧σ N

+ 1{Yt∧σN >0}

d ∑ i=1

i uit∧σN |Zt∧σ |. N

(8.3.24)


439

+ Apply Itô’s formula to eβAt∧σN (Yt∧σ )2 . Compute the expectation to get for N

0 ≤ t < ∞ by making use of (8.3.24) and Young’s inequality as β ≥ 2: [ ] + 2 E eβAt∧σN (Yt∧σ ) N ∫ σN [ ] [ ] βAσN + 2 ≤ E e (YσN ) − E eβAs β(as Ys+ )2 + 1{Ys >0} |Zs |2 ds ∫

σN

+2E

eβAs Ys+

t∧σN

[

t∧σN

(

rs Ys+ + ∫

]

≤ E eβAσN (Yσ+N )2 − E ∫

σN

+2E [

( βAs

]

t∧σN

∫

) 1{Ys >0} uis |Zsi | ds

i=1

σN

rs (Ys+ )2

e t∧σN

d ∑

[ ] eβAs β(as Ys+ )2 + 1{Ys >0} |Zs |2 ds

+

d ∑

) uis Ys+ 1{Ys >0} |Zsi |

ds

i=1

[ ] eβAs β(as Ys+ )2 + 1{Ys >0} |Zs |2 ds t∧σN ∫ σN ∫ σN 1 βAs + 2 1{Ys >0} eβAs |Zs |2 ds e (as Ys ) ds + E +2E 2 t∧σN t∧σN ] [ βAσN + 2 ≤ E e (YσN ) . ≤ E e

βAσN

(Yσ+N )2

−E

σN

Now we get by taking the limit N → ∞ with Lebesgue’s theorem: [ [ ] [ ] ( )2 ] + 2 βAt∧σ βAσ + 2 βAτ + ˜ E e (Yt∧σ ) ≤ E e (Yσ ) = E e [ξ − ξ] = 0. + This implies Yt∧τ = 0 for all 0 ≤ t < ∞ as Y = Y˜ = 0 on (σ, τ ).

Remark 8.3.25. Now put f˜ ≡ 0 and ξ˜ = 0 to again get a criterion for nonnegativity.

8.3.5

One-Dimensional Linear BSDEs

We conclude the generalizations of the ”‘classical theory”’ to (a, β)-data with a study of the structure of solutions of linear BSDEs. We have to impose further assumptions and will get a strict version of the above comparison theorem. Let us look at the linear BSDE: [ ] d d ∑ ∑ i i dYt = Ct Yt + Bt Zt + gt dt + Zti dWti , i=1

Yτ

i=1

= ξ.

(8.3.25)

We assume that the BSDE is on-dimensional and define: d ∑ 2 at := |Ct | + (Bti )2 (+ϵ) i=1

(8.3.26)

440


Remember that A is defined by (0 ≤ t ≤ τ ) ∫

t

a2s ds.

At := 0

In addition we assume that for a β > 2 ∫

τ

E 0

2 eβAt a−2 t gt dt < ∞, [ ] E eβAτ ξ 2 < ∞.

(8.3.27) (8.3.28)

From corollary 8.3.10 we know that the linear BSDE under these assumptions has a unique (a, β)-solution. Now we are going to study under what kind of assumptions this solution has a similar representation as was derived in theorem 8.2.24 for the case of bounded coefficients and deterministic finite terminal time. In the proof of theorem 8.2.24 we made use of the fact that the SDE dΦt = −Φt Ct dt −

d ∑

Φt Bti dWti ,

i=1

Φ0 = 1

(8.3.29)

has a square integrable solution. This result cannot be expected when we have unbounded coefficients. Yet we get the following result: Theorem 8.3.26. Let a be defined as in (8.3.26) and let Aτ < ∞ P -a.s. Then [ ∫ Φt := exp − 0

t∧τ

∑ 1∑ i 2 Cs + (Bs ) ds − 2 d

d

i=1

∫

i=1

]

t∧τ

Bsi dWsi

(8.3.30)

0

is a solution of the SDE (8.3.29) on the interval [0, τ ]. Furthermore, we have for β > 2:

[ E sup 0≤t≤τ

] e−βAt Φ2t < ∞.

Proof. All integrals in Φ are defined as Aτ < ∞. Φ solves the SDE on [0, τ ] as is easily seen from Itô’s formula. So the integrability of Φ remains to be proved. In a first step we prove that for β > 2 ∫

τ

E 0

e−βAt |at Φt |2 dt ≤

1 . β−2

(8.3.31)


441

From Itô’s formula we get (0 ≤ t < ∞): −βAt∧τ

e

∫ |Φt∧τ |

2

t∧τ

≤ 1−β

−βAs

e

|as Φs | ds + 2

0

∫

t∧τ

+2

e−βAs |Cs ||Φs |2 ds −

0

∫

≤ 1 − (β − 2) −

i=1 0 d ∫ t∧τ ∑ i=1

t∧τ

t∧τ

e−βAs |Bsi Φs |2 ds

e−βAs Bsi Φ2s dWsi

0

e−βAs |as Φs |2 ds

0

d ∫ ∑ i=1

d ∫ ∑

t∧τ

e−βAs Bsi Φ2s dWsi .

(8.3.32)

0

Define the sequence of stopping times (depending on T ) { ∫ T ∧u } −βAs i 2 2 |e Bs Φs | ds ≥ N f”ur ein i ∧ τ. τN,T := inf u; 0

Now for all 0 ≤ T < ∞ τN,T converges to τ P -a.s. (for N → ∞) and the stopped Itô-integrals are martingales. So for all (T, N ): ∫ T ∧τN,T 1 e−βAs |as Φs |2 ds ≤ E . β−2 0 Now let first (for fixed T ) N tend to infinity and then find from Fatou’s lemma that for all 0 ≤ T < ∞:

∫

T ∧τ

E

e−βAs |as Φs |2 ds ≤

0

1 . β−2

Then let T go to infinity and apply Fatou’s Lemma again to see that (8.3.31) follows. Now let for (N, T ) ∈ N × [0, ∞) { σN,T := inf

u;

sup

0≤t≤T ∧u

} |e−βAt Φ2s |2 ≥ N

∧τ

Again σN,T converges for fixed T and N → ∞ to τ P -a.s. Now it follows from (8.3.32) with the Burkholder-Davis-Gundy-inequality and der Young’s inequality that for all (N, T ) and a constant K not depending on N and T : √ ∫ T ∧σN,T [ ] −βAt 2 E sup e Φt |e−βAt at Φ2t |2 dt ≤ 1+K E 0≤t≤T ∧σN,T

0

[ ] 1 sup e−βAt Φ2t ≤ 1+ E 2 0≤t≤T ∧σN,T ∫ τ 2 +K E e−βAt |at Φ2t |dt. 0

442


So:

[ E

sup

0≤t≤T ∧σN,T

∫ ] e−βAt Φ2t ≤ 2 + 2K 2 E

τ

e−βAt |at Φ2t |dt.

0

Because of (8.3.31) the expression on the left hand side is uniformly bounded in N und T . Let N and then T tend to infinity to find the assertion from Fatou’s Lemma.

Now we can prove the following theorem: Theorem 8.3.27. Assume (8.3.26)-(8.3.28). Let Aτ < ∞ P -a.s. Then the uniquely existing (a, β)-solution (Y, Z) of the linear one-dimensional BSDE (8.3.25) has the following form: Yt∧τ =

Φ−1 t∧τ E

[

∫ Φτ ξ −

τ

t∧τ

] Φs gs ds Ft .

Here Φ is defined as in (8.3.30). Proof. As in the proof of theorem 8.2.24 one proves that ∫ Φt∧τ Yt∧τ = Φτ ξ −

τ

d ∫ ∑

Φs gs ds −

t∧τ

τ

Φs [Zsi − Bsi Ys ]dWsi .

t∧τ

i=1

From the just proved integrability of Φ we get: √∫ τ

E ( ≤

|Φs [Zsi − Bsi Ys ]|2 ds

0

[ E sup 0≤t≤τ

−βAt

e

Φ2t

])1/2 ( ∫ E

)1/2

τ βAt

e

[Zti

−

Bti Yt ]2 dt

< ∞.

0

So the Itô-integrals are martingales and the assertion follows from appplying the conditional expectation. Remark 8.3.28. The interpretation from martingale theory which is possible in the case of bounded coefficients here fails as Φ for C ≡ 0 in general is a strict local martingale. For an example illustrating this see Lipster/Shiryaev [23], S.224. Just as in the case of bounded coefficients from the above we get a strict comparison theorem:


443

Theorem 8.3.29. Assume the assumptions of theorem 8.3.24. In addition let Aτ < ∞ P -a.s. Then the comparison is strict in the following sense: If Yt∧τ = Y˜t∧τ holds on a set F ∈ Ft then Ys∧τ = Y˜s∧τ holds on [t, ∞) × F P -a.s. Proof. The proof follows exactly the proof of theorem 8.2.28, though the appropriate linear BSDE might have unbounded coefficients. Nevertheless the reasoning of theorem 8.2.28 holds as in the case under consideration the assumptions of the preceding theorem are fulfilled.

8.3.6

Remarks

section 8.3.1: The use of a stochastic Lipschitz condition goes back to El Karoui/Huang [17]. Further generalizations in the cited work however require an extension of the notion of solution to three processes. Our aim was to keep the notion of solution as near as possible to the definition 8.2.7. So the definition of a classical solution (Definition 8.3.3) is the natural extension of definition 8.2.7 for arbitrary terminal times. When an unbounded time horizon is considered it is common to use a notion of solution depending on weighted norms. So our definition of an (a, β)-solution for a ≡ 1 is a slight generalization of the notion of solution in Yong/Zhou [62], section 7.3.2. for unbounded time horizon. Theorem 8.3.9 is an extension of an earlier result (Theorem 3 in Bender/Kohlmann [3]). Some modifications in the proof allowed to get away rom the assumption that a is strictly bounded away from zero. With β = 2 we here could derive a sharp boundary for the theorem to hold. section 8.3.2: In the case a ≡ 1 the existence and uniqueness for simple BSDEs result was proved in Yong/Zhou [62], Lemma 7.3.7. The extension to general a required essential changes. section 8.3.3: In Bender/Kohlmann [3] we got the existence and uniqueness result on the basis of the Banach fixed point theorem from a priori estimates. In T this approach the estimates are not good enough to prove the result for arbitrary β > 2.

444


section 8.3.4: Theorem 8.3.20 is the natural generalization of theorem 8.2.21. It is stronger than theorem 4 in Bender/Kohlmann [3] where the same result was proved for sufficiently large β. The version of the comparison theorem is new. The basic idea of estimating the positive part is found in El Karoui et al. [19]. However, here we use weighted norms and a localization what is not necessary in [19]. section 8.3.5: All results of this section are new. Even if the basic idea of the proof is similar to the case of bounded coefficients we would like to stress the following: (i) The proof requires the integrability of Φ which is here proved for the first time. This then ensures the martingale property of the Itô-integrals. (ii) The results in this section are relevant for the mathfinance applications. The strict comparison theorem will be used to derive that the market is arbitrage free and then allows to treat the pricing problem for options in markets with unbounded risk premium.

Chapter 9

General References The references are put together for different subjects for the readers’ convenience:

• References on BSDEs • References on Neyman-Pearson-Hedging • References on Grossisement de Filtration • References on LQ-Control and MV-Hedging • References on The Markovitz Problem • References Partially Observed MV-Hedging will be found in the following separated bibliographies:

445

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Index XX · XXLpτ

β(M), 46

0 ̸= p < 1, 306

β ∗ (τ ), 277

p ≥ 1, 248

β0 (M), 276 βx , 44

Ad (Ω), 5

β˜0 , β˜0 (M), 275

XXX · XXXp , 294

Bear Spread, 71

(p) (p) A(p) , Ad , Ad (Ω), 6 ∗p A∗p , A∗p d , Ad (Ω), 7

binary market, 10 Black Scholes, 68

adjustment process, 302

Black Scholes Formel, 68

AE(U ), 305

Black-Scholes formula

American contingent claim, 74

classical case, 65

American Option, 65

generalized, 99

American Put, 73

bond, 56

annulator, 12

Brownian market, 56

approximate price, 176

BSDE, 87, 320

arbitrage, 10

BSDE formulation of the pricing prob-

G-, 10

lem, 89

-free, 10

BSPDE, 99

NFLVR, 13

Bull Spread, 71

arbitrage free, 59 C, 25

arbitrage opportunity, 59

C + , 304

example, 92 Asian Option, 70, 74

C p , 27

asymptotic elasticity, 305

C0 , 25 C, 312

¯ 329 β,

˜ 319 C,

β, 312

Cauchy probem, 70 486

INDEX

487

clock, 312

European Call, 71

conditional Lp -norm, 248

European Option, 55, 65

conditional expectation

discrete case, 55

extended, 245

European Put, 71

generalized, 245

exchange option, 85

constrained market, 101 consumption process, 57 contingent claim, 40, 63, 66 attainable, 63 non-attainable, 40 continuation region, 84 cumulative consumption process, 57

F, 5 Föllmer-Schweizer hedging, 111 (Fs )s≥0 , 5 τ˜ τ Fs , Fs , Fs

∩ A, 8

fair price of an option, 56 FBSDE, 321 forward price, 227

D+ ,

304

forwards, 230

Dv+ , Dv+ (M), Dq , Dq (M), Deq , Daq ,

304

13

13

fractional Brownian motion, 86 future price, 227 futures, 230

˜q, D ˜ q (M), 250 D d, 5

GH , 9

Delta, 70

gains process, 9

discounted, 9

Gamma, 70

Dudley’s representation, 85, 92

generator of E-martingale, 286

Eτ [·], 246 E-martingale, 257 local, 257 E(X),τ E(X), 8

of hedging strategy, 283 Girsanov functional, 61 Greeks, 70

E[N ], 257

H · X, 7

E q∗ [N ], 260

H opt , 319

early exercise price, 85

1 < p < ∞, 302

EDq , EDq (M), 257

Hp , Hdp (M), 12

efficient frontier, 207

Hp (X), 294

equivalent martingale measure, 60

hedging

488

INDEX discrete example, 53

hedging numéraire, 31

loc−, 290 local martingale measure, 13

hedging numeraire, 175

Lq -integrable, 13

hedging price

Lq -optimal, 31

lower, 65

Lq -optimal E-, 258

upper, 65

Lq0 -integrable, 250

hedging strategy, 9, 55, 67, 279

Lq0 -integrable E-, 257

Lp -integrable, 298

E-, 257

discrete example, 55

signed, 13

non-simple, 279

variance optimal, 31

self-financing, 9, 279

LPp , 15

simple, 9, 12 M , 312 incomplete market, 85

M, 8

indifference price

MB , 9

for information, 48 risk-premium-, 296 utility-, 41 Lp0 -integrable martingale, 248

µ, µ ˜, 323 B M,

327

Mτ˜ ,τ Mτ˜ , 8 market, 8

Lpτ , 245

complete, 63, 64

L, Ld , 7

discounted, 9

Ld (Ω), L˜d (Ω), 5

non-discounted, 327

λ, 310

semimartingale, 279

Lp (X), 294

Markovitz problem, 197

Lp , Lp (P ), Lp (Ω), 6

martingale

L1loc (M ), 7

E[N ]-, 285

Lvar (A), 7

closed, 285

L(X), 7

extended E[N ]-, 285

law of one price, 15

extended uniformly integrable,

strong, 15 everywhere in F, 252 everywhere in F0 , 244

248 generated, 285 mean variance hedging, 173

INDEX

489

mean-variance hedging

instantaneous, 314

approximate price, 176

intertemporal, 46

Markovian market, 177

non-discounted, 329

running risk, 178

totally p-unhedgeable, 315

measures of risk, 128

price process, 8

N , 319

q q Rq , Rloc , Rωq , Rint , 260

N opt , 265

reverse Hölder condition, 260

Nd (Ω), 6

Rho, 70

ν opt , 319

risk neutral martingale measure, 62

nds-admissible strategy, 58

Russian Option, 73

Neyman-Pearson hedging, 127

S, 8

no free lunch, 59

S, Sd , 7

Novikov condition, 61

S p , 294

numéraire, 9, 279

Sd (Ω), 5

Ω, 5

S p , Sdp , Sdp (Ω), 7

¯ 5 Ω,

S (p) , Sd , Sd (Ω), 7

(p)

optimal pair 1 < p < ∞, 251 p < 1, 307 option, 66

(p)

S ∗p , Sd∗p , Sd∗p (Ω), 7 self-financing, 9 semimartingale norm, 294 sequence of zeros, 257 SF, SF(M), SF v (M), 279

P, 5

SF p , SF p0 , 298

P, Pd , 7

SF pv , SF pv (M), 298

Pd (Ω), 6

SF s , SF s (M), SF sv (M), 9

Pdb (Ω), 6

SF + , SF + 1 , 304

partially informed agent, 115

+ SF + v , SF v (M), 304

portfolio process, 57

SF s,p v , 12

pre-local, 290

Sharpe-ratio, 46

price for risk

conditional, 275, 276

conditional, 276

generalized, 44

deterministic, 317

instantaneous, 313

490

INDEX

SLPp , 15

V¯ p , V¯ p (M), 13

SLPp (F), 252

V(G), 9

SLPp (F0 ), 244

V p , V p (M), V0p (M), 12

Snell envelope, 75

Vvp , Vvp (M), V(SF s,p v ), 12

stat−, 290

p p V∞ , V∞ (M), V∞ (SF s,p 0 ), 12

stationary convergence, 257

V + , 304

stochastic exponential, 256

Vv+ , Vv+ (M), 304

stock, 56

◦ Vp◦ , V(SF s,p 0 ) , 12

Straddle, 71

V H, 9

Strangle, 71

V e , 272

Strip, 71

V (H) , V (H,H0 ) , 283

superhedging, 100

V opt

˜

˜

1 < p < ∞, 251

Tτ , 261 tame strategy, 58 term structures, 225, 232 bond options, 234 Cox-Ingersoll-Ross model, 235 Heath-Jarrow-Morton model, 237 Hull-White model, 234 Vasicek model, 233

p < 1, 307 V p∗ (τ ), 266 τ V opt

1 < p < ∞, 252 p < 1, 308 value process, 9 variance-optimal martingale measure, 175

zero coupon bond, 226

Vega, 70

Theta, 70 Up , 27

W , 323

U, Ud , 7

Wick product, 10

Ud (Ω), U˜d (Ω), 5 Udp (Ω), U˜dp (Ω),

Tn X ,Tn t

Xt∗ , 259

6

utility function, 27

Y p , 324

isoelastic, 27

Y(M), 321

p-moderate, 28

yield equating martingale measure,

marginal utility, 20 paradox, 20

62 Young function, 25

INDEX conjugate, 28 Z opt 1 < q < ∞, 251 q < 1, 307 Z q∗ (τ ), 266 τ Z opt

1 < q < ∞, 252 p < 1, 308 zero coupon bond, 226 zeros of E[N ], 257

491