The backward and forward dynamic utilities and the ... - UT Math

The backward and forward dynamic utilities and the associated pricing systems: Case study of the binomial model ∗ Marek Musiela and Thaleia Zariphopoulou BNP-Paribas, London and The University of Texas at Austin First version: July 2003 This version: December 2005 To appear in Volume on Indifference Pricing (ed. R. Carmona)

Abstract In this paper we introduce the concepts of backward and forward dynamic utilities. Moreover, we derive and analyze the associated with them indifference pricing systems for derivative securities.

1

Introduction

Derivatives pricing and investment management seem to have little in common. Even at the organizational level, they belong to two quite separate parts of financial markets. The so-called sell side, represented mainly by the investment banks, among other things offers derivatives products to their customers. Some of them are wealth managers, belonging to the so-called buy side of financial markets. ∗ This work was presented at the 2nd World Congress of the Bachelier Finance Society (2002), the AMS-IMS-SIAM Joint Summer Research Conference in Mathematical Finance (2003), Workshop on Financial Mathematics, Carnegie Mellon (2005), Workshop on Stochastic Modelling, CMR, University of Montreal (2005), SIAM Conference on Control and Optimization, New Orleans (2005), Workshop on PDEs and Mathematical Finance, KTH, Stockholm (2005), Meeting on Stochastic processes and their Applications, Ascona (2005) and in seminars at Princeton University (2003), Columbia University (2003), MIT (2004), Imperial College (2004), University of Southern California (2004), Cornell University (2004) and Brown University (2005). The authors would like to thank the participants of these conferences and seminars for fruitful suggestions and useful comments. The authors would also like to thank Rene Carmona for giving them the opportunity to present their work in this Volume. They would also like to thank Michael Anthropelos for thoroughly reading the manuscript and correcting typos and errors. The second author aknowledges support from the National Science Foundation (NSF grants: DMS-0102909, DMS-VIGRE-0091946, DMS-FRG-0456118)

1

So far, the only universally accepted method of derivative pricing is based upon the idea of risk replication. Models have been developed which allow for perfect replication of option payoffs via implementation of a replicating and self financing strategy. We call such models complete. The option price is calculated as the cost of this replication. Adjustments to the price are later made to cover for risks due to the unrealistic representation of reality. More accurate description of the market is given by the so-called incomplete models in which not all risk in a derivative product can be eliminated by dynamic hedging. However, this potential model advantage is hampered by another difficulty. Namely, the concept of price for a derivative contract is not uniquely defined. Many approaches have been proposed and extensively studied, however, up until now no clear consensus has emerged. On the other side of the spectrum of financial markets there are wealth managers. They have developed their own methodology for implementation of their investment decisions. They may use derivative products to improve their performance; however, their focus is on investment strategy with view to optimize returns rather than on risk replication. Therefore, it should not come as a surprise that the models they use are very different from the models used in derivatives pricing. The main aim of this chapter is to work towards convergence of the methodologies used in these apparently quite distant areas. The idea is to associate the concept of price for a derivative contract with a rather natural to a wealth manager constraint, that is, maximization of utility of wealth. We choose to work with exponential utility and a very simple model structure in order to eliminate all technical difficulties and to concentrate exclusively on the most important links between the two areas. The classical one-period binomial model and its multi-period generalization the Cox-Ross-Rubinstein model are the simplest examples of complete arbitrage free pricing models. Many textbooks and university courses on mathematical finance use these models to present the fundamental ideas without getting into difficulties of the general martingale theory necessary for the more realistic asset price dynamics. Throughout this paper we adopt the same method of presentation. Our incomplete models are very simple and hence mathematics behind them mostly trivial. The emphasis is on explaining the fundamental ideas and compare them with the classical arbitrage free theory. The chapter is organized as follows. In the next section we introduce and analyze in detail a single period binomial model. In particular, we derive an intuitively appealing formula for the indifference price of a general claim. Then, we study the various properties of the indifference price and exhibit the connection with convex risk measures. The link with the classical methodology of pricing by replication is analyzed next. It turns out that the analogue of the socalled delta retains its natural interpretation as the sensitivity of the price with respect to the movement of the instrument used for hedging. Moreover, it also appears that the components of risk that are left unhedged, in our incomplete model set up, have zero value from the perspective of valuation by indifference. Another important observation about the nature of pricing by the indiffer2

ence is exposed in the subsection dedicated to the relative pricing. Namely, this type of pricing scheme is relative to the agent’s portfolio in contrast to the arbitrage free pricing scheme which is relative to the market portfolio. When interpreted this way, the indifference valuation can be viewed as linear, while, of course, when seen as a functional over a set of random variables it is not. Going deeper into the comparisons with the pricing by arbitrage, we then investigate the issue of unit choice and the necessary consistency with the static no arbitrage constraint. We show, in particular, that in order to eliminate static arbitrage one needs to relate the risk aversion parameter with the unit of wealth. To our knowledge, this is the first time that modelling issues pertinent to consistency across units have been identified and addressed. Preparing the grounds for the introduction of the dynamic preference structure in a multi-period setup, where the risk aversion may be modified according to our local in time views about anticipated performance of the traded securities, we then study the case when the risk aversion parameter depends on the future value of the traded stock. It turns out that the pricing rule retains its intuitive form. Moreover, the associated value function exhibits an interesting relationship between the risk tolerance (the reciprocal of risk aversion) at the end and at the beginning of a time period. Namely, the end of a period risk tolerance can be viewed as an option payoff, and the consistent risk tolerance for the beginning of the period is its arbitrage free price. Motivated by the general need for absence of static arbitrage, and by the above observation in particular, we conclude the section on the one period case introducing the notions of the utility normalization and the concepts of the backward and forward utilities. In the second part of this chapter, we study the multi-period model. The main emphasis there is on the new concepts of backward and forward dynamic utilities and on the associated indifference pricing algorithms. We propose utilities that are stochastic processes depending on three arguments, namely, the wealth, the time and the normalization point. The latter is a novel concept and serves as a reference point in time with respect to which the risk preferences are specified. It plays a central role in the construction of the term structure of dynamic utilities and, ultimately, in building universal pricing systems. Given a normalization point for the utility of wealth measurement, the associated dynamic utility makes an investor indifferent to the choice of the investment horizon. For example, if the dynamic utility is normalized at the 5 year point, then the investments optimized over all shorter than 5 year time horizons represent the same value in terms of utility of wealth they generate. The dynamic utility which is normalized at some future date is called the backward utility associated with this date. If the normalization point is today, the associated dynamic utility is called the forward utility. The forward utility allocates the same value, in terms of utility of wealth, to the optimal investment over any investment horizon. Moreover, the forward utility is defined for any positive investment horizon, as opposed to the backward utility, which is defined only for the investment horizons between now and the corresponding normalization point. The indifference to the length of the investment horizon is a dynamic 3

property of utility that is necessary for the alignment of the indifference pricing methodology with the absence of static arbitrage. Indeed, any pricing methodology must provide the ability to forward transport known cash flows consistently with the shape of the yield curve. Going beyond the known cash flows case, any derivative pricing system must also satisfy the semigroup property. Otherwise, when reduced to the case of pricing by perfect replication, it will not be consistent with the arbitrage free valuation. Consequently, the pricing system based on the comparison of utility of wealth and referring to the claims with payoffs associated with different maturities, must be based on the dynamic utility concept that makes the investor indifferent to the choice of the investment horizon. The backward and forward dynamic utility functions lead to the associated concepts of the backward and forward indifference prices. It turns out, rather surprisingly, that the two concepts of value do not in general coincide. However, they do coincide under additional, and rather restrictive, assumptions about the model (cf. Musiela and Zariphopoulou (2004b)). The indifference price associated with the dynamic backward utility is very closely related with the traditional indifference price, as derived by Rouge and El Karoui (2000) (see, also, Delbaen et al. (2002)). The concept of indifference price associated with the forward utility as well as the corresponding explicit valuation algorithm are, to the best of our knowledge, entirely new. The backward and forward dynamic utilities expose, in our view, a very important and, in many aspects, novel modelling issue. Namely, in our dynamic utility set up one does not formulate preferences in isolation to the characteristics of the market in which one intends to invest. The dynamics of the markets can be observed but are exogenously given to the investor, modulo estimation of the relevant parameters by the equilibrium considerations or other procedures. The model, based on which the investor makes the investment decisions, represents the market dynamics in an equilibrium state. The investor formulates his preferences relatively to that equilibrium state. If she agrees with it, then she becomes indifferent to any action she can take in such a market. In our model set up this results to her indifference to the choice of an investment horizon. Consequently, formulation of the dynamic preferences, relatively to the exogenously given market dynamics, forces the investor to embed into his dynamic utility structure information describing the equilibrium market dynamics. This seems to be in a striking contrast with the majority of traditional formulations of the consumption and investment problems in which the market dynamics and the investor utility are specified independently of one another. We conclude the Introduction by stressing that addressing the issue of invariance across trading horizons, and understanding the emerging endogenous contraints on the dynamic utilities, is not sufficient for the development of a unified theory that bridges investments, fund management and derivatives. Indeed, an additional indispensable ingredient of such a theory is the consistency of prices and policies across units. As it was earlier mentioned, we study this issue herein, but only for the static case (see Section 2.5). For the dynamic case, we refer the reader to the work in progress by the authors and Zitkovic 4

(see Musiela and Zariphopoulou (2005a, 2005b) and Musiela, Zariphopoulou and Zitkovic (2005)).

2

One-period binomial model

In this section we introduce a simple one-period binomial model with one riskless and two risky assets, of which only one is traded. By construction, the model is incomplete and our aim is to develop a coherent approach for investment management and derive from it a pricing methodology for derivative contracts. Optimal investment management is based on maximization of expected utility of wealth. There is a number of constraints we want to impose on our investment decision process and on the derivatives valuation method. To mention just two, we want our investment decisions not to depend on units in which the wealth is expressed. This is mainly because we also need to ensure that our pricing method is consistent with the absence of arbitrage and that it is also numeraire independent. We also want our pricing concept to have a clear intuitive meaning, so effort is made to interpret the results and, whenever possible, to draw analogies with the classical arbitrage free theory of complete markets.

2.1

Indifference Price Representation

Consider a single period model in a market environment with one riskless and two risky assets. The riskless asset is assumed to offer zero interest rate. Only one of the traded assets can be traded, taken to be a stock. The current values of the traded and non-traded risky assets are denoted, respectively, by S0 and Y0 . At the end of the period T , the value of the traded asset is ST with ST = S0 ξ, ξ = ξ d , ξ u and 0 < ξ d < 1 < ξ u . Similarly, the value of the non-traded asset YT satisfies YT = Y0 η, η = η d , η u , with η d < η u , (Y0 , YT 6= 0) . We introduce randomness into our single period model by means of the probability space (Ω, FT , P), where Ω = {ω1 , ω2 , ω3 , ω4 } and P is a probability measure on the σ−algebra FT = 2Ω of all subsets of Ω. For each i = 1, .., 4, we assume that pi = P {ωi } > 0 and we model the upwards and the downwards movement of the two risky assets ST and YT by setting their values as follows ST (ω1 ) = S0 ξ u , YT (ω1 ) = Y0 η u ST (ω2 ) = S0 ξ u , YT (ω2 ) = Y0 η d

ST (ω3 ) = S0 ξ d , YT (ω3 ) = Y0 η u ST (ω4 ) = S0 ξ d , YT (ω4 ) = Y0 η d .

The measure P represents the so-called historical measure. (S,Y ) Observe that the σ-algebra FT coincides with the σ-algebra FT generated by the random variables ST and YT . In what follows we will also need the σalgebra FTS generated exclusively by the random variable ST . Consider a portfolio consisting of α shares of the traded asset and the amount β invested in the riskless one. Its current value X0 = x is equal to β + αS0 = x while its wealth XT , at the end of the period [0, T ], is given by XT = β + αST = x + α (ST − S0 ) . 5

(1)

Now introduce a claim, settling at time T and yielding payoff CT . In pricing of CT , we need to specify our risk preferences. We choose to work with the exponential utility U (x) = −e−γx, x ∈ R

and γ > 0.

(2)

Optimality of investments, which will ultimately yield the indifference price of the claim is examined via the value function   (3) V CT (x) = sup EP −e−γ(XT −CT ) α

  = e−γx sup EP −e−γα(ST −S0 )+γCT . α

Below, we recall the definition of indifference prices.

Definition 1 The indifference price of the claim CT = c(ST , YT ) is defined as the amount ν(CT ) for which the two value functions V CT and V 0 , defined in (3) and corresponding, respectively, to the claims CT and 0 coincide. Namely, ν(CT ) is the amount which satisfies V 0 (x) = V CT (x + ν(CT ))

(4)

for all initial wealth levels x ∈ R. Looking at the classical arbitrage free pricing theory, we recall that derivative valuation has two fundamental components which do not depend on specific model assumptions. Namely, the price is obtained as a linear functional of the (discounted) payoff representable via the (unique) risk neutral equivalent martingale measure. Our goal is to understand how these two components, namely, the linear valuation operator and the risk neutral pricing measure change when markets become incomplete. In the context of pricing by indifference, we will look for a valuation functional and a naturally related pricing measure under which the price is given as ν(CT ) = EQ (CT ). (5) Before we determine the fundamental features that E and Q should have, let us look at some representative cases. Examples: i) First, we consider a claim of the form CT = c (ST ) . Intuitively, the indifference price should coincide with the arbitrage free price, for there is no risk that cannot be hedged. Indeed, one can construct a nested complete one-period binomial model and show that ν(c(ST )) = EQ∗ (c(ST )),

(6)

with Q∗ being the relevant risk neutral measure. The indifference price mechanism reduces to the arbitrage free one and any effect on preferences dissipates.

6

ii) Next, we look at a claim of the form CT = c (YT ) and assume for simplicity that the random variables ST and YT are independent under the measure P. In this case, intuitively, the presence of the traded asset should not affect the price. Indeed, working directly with the value function (3) and Definition 1, it is straightforward to deduce that ν(c(YT )) =

1 log EP (eγc(YT ) ). γ

(7)

The indifference price coincides with the classical actuarial valuation principle, the so-called certainty equivalent value which is nonlinear in the payoff and uses as pricing measure the historical one. iii) Finally, we examine a claim of the form CT = c1 (ST ) + c2 (YT ) . One could be, wrongly, tempted to price CT by first pricing c1 (ST ) by arbitrage, next pricing c2 (YT ) by certainty equivalent, and adding the results. Intuitively, this should work when ST and YT are independent. However, this cannot possibly work under strong dependence between the two variables, for example, when YT is a function of ST . In general, ν(c1 (ST ) + c2 (YT )) 6= EQ∗ (c1 (ST )) +

1 log EP (eγc2 (YT ) ). γ

The above illustrative examples indicate certain fundamental characteristics E and Q should have. First of all, we observe that a nonlinear valuation functional must be sought. Clearly, any effort to represent indifference prices as expected payoffs under an appropriately chosen universal measure should be abandoned. Indeed, no linear pricing mechanism can be compatible with the concept of indifference based valuation as defined in (4). Note that this fundamental observation comes in contrast to the central direction of existing approaches in incomplete models that yield prices as expected payoffs under an optimally chosen measure. We also see that risk preferences may affect the valuation device given their inherent role in price specification. However, intuitively speaking, we would prefer to specify the pricing measure independently on the risk preferences. Finally, the pricing measure and the valuation device should ideally be the same for all claims to be priced. The next Proposition yields the indifference price in the desired form (5). Proposition 2 Let Q be a measure under which the traded asset is a martingale and, at the same time, the conditional distribution of the non-traded asset, given the traded one, is preserved with respect to the historical measure P, i.e. Q(YT |ST ) = P(YT |ST ).

(8)

Let CT = c(ST , YT ) be the claim to be priced under exponential preferences with risk aversion coefficient γ. Then, the indifference price of CT is given by  
1 . (9) log EQ eγCT |ST ν(CT ) = EQ (CT ) = EQ γ 7

Proof. We prove the above result by constructing the indifference price via its definition (4). We start with the specification of the value functions V 0 and V CT . We represent the payoff CT as a random variable defined on Ω with values CT (ωi ) = ci ∈ R, for i = 1, .., 4. Elementary arguments lead to  u V CT (x) = e−γx sup −e−γαS0 (ξ −1) (eγc1 p1 + eγc2 p2 ) α

−e−γαS0 (ξ

d

−1)

 (eγc3 p3 + eγc4 p4 ) .

Maximizing over α leads to the optimal number of shares αCT ,∗ , given by αCT ,∗ = +

1 (ξ u − 1) (p1 + p2 ) log u d γS0 (ξ − ξ ) (1 − ξ d ) (p3 + p4 )

(10)

1 (eγc1 p1 + eγc2 p2 ) (p3 + p4 ) . log γS0 (ξ u − ξ d ) (eγc3 p3 + eγc4 p4 ) (p1 + p2 )

Further straightforward, albeit tedious, calculations yield V CT (x) = −e−γx

1 qq

q

1−q

(1 − q)

1−q

(eγc1 p1 + eγc2 p2 ) (eγc3 p3 + eγc4 p4 )

where q=

1 − ξd . ξu − ξd

For CT = 0, the value function takes the form q  1−q  p3 + p4 p1 + p2 . V 0 (x) = −e−γx q 1−q

(11)

(12)

(13)

From the definition of the indifference price (4) and the representations (11), (13) of the relevant value functions, it follows that ν(CT ) = q

1 1 eγc1 p1 + eγc2 p2 eγc3 p3 + eγc4 p4 + (1 − q) log . log γ p1 + p2 γ p3 + p4

(14)

Next, we show that the above price admits the probabilistic representation (9). We first consider the terms involving the historical probabilities in (14) and we note that they can be actually written in terms of the conditional historical expectations, namely,

and


eγc1 p1 + eγc2 p2 = EP eγCT |A p1 + p2
eγc3 p3 + eγc4 p4 = EP eγCT |Ac , p3 + p4

where A = {ω1 , ω2 } = {ω : ST (ω) = S0 ξ u }. It is important to observe that conditioning is taken with respect to the terminal values of the traded asset. 8

We continue with the specification of the pricing measure defined in (12). For this, we denote (with a slight abuse of notation) by q1 , q2 , q3 , q4 the elementary probabilities of the sought measure Q. Straightforward calculations yield that q1 + q2 = q

(15)

with q as in (12). To compute q1 , we look at the conditional historical probability of {YT = Y0 η u } , given {ST = S0 ξ u } , and we impose (8), yielding p1 q1 = . p1 + p2 q The probabilities q2 , q3 and q4 , computed in a similar manner, are written below in a concise form pi pi qi = q , i = 1, 2 and qi = (1 − q) , i = 3, 4. (16) p1 + p2 p3 + p4 It follows easily that the nonlinear terms in (14) can be compiled as     1 eγc1 p1 + eγc2 p2 eγc3 p3 + eγc4 p4 1 log log IA + IAc γ p1 + p2 γ p3 + p4

1 1 log EP eγCT |A IA + log EP eγCT |Ac IAc γ γ
1 = log EQ eγCT |ST . γ Therefore, taking the expectation with respect to Q yields   1 γCT |ST ) log EQ (e EQ γ       eγc1 p1 + eγc2 p2 eγc3 p3 + eγc4 p4 1 1 log log IA + IAc = EQ γ p1 + p2 γ p3 + p4 =

=q

1 1 eγc1 p1 + eγc2 p2 eγc3 p3 + eγc4 p4 + (1 − q) log = ν(CT ), log γ p1 + p2 γ p3 + p4

where we used (14) to conclude. We next discuss the key ingredients and highlight the intuitively natural features of the probabilistic pricing formula (9). Interpretation of the Indifference Price: Valuation is done via a twostep nonlinear procedure and under a single pricing measure. i) Valuation procedure: In the first step, risk preferences are injected into the valuation process. The original derivative payoff is being distorted to the preference adjusted payoff, to be called thereof, the conditional certainty equivalent
1 C˜T = log EQ eγCT |ST . (17) γ 9

This new payoff has actuarial type characteristics and reflects the weight that risk aversion carries in the utility-based methodology. However, the certainty equivalent rule is not applied in a naive way. Indeed, we do not consider any classical actuarial type functional, since
1 C˜T 6= log EP eγCT γ


1 and C˜T 6= log EQ eγCT . γ

In the second step, the pricing procedure picks up arbitrage free pricing characteristics. It prices the preference adjusted payoff C˜T , dependent only on ST , through an arbitrage free method. The same pricing measure is being used in both steps. The price is then given by ν(CT ) = EQ (CT ) = EQ (C˜T ). It is important to observe that the two steps are not interchangeable and of entirely different nature. The first step prices in a nonlinear way as opposed to the second step that uses linear, arbitrage-free, valuation principles. In a sense, this is entirely justifiable: the unhedgeable risks are identified, isolated and priced in the first step and, thus, the remaining risks become hedgeable. One should then use a nonlinear valuation device for the unhedgeable risks and, linear pricing for the hedgeable ones. A natural consequence of this is that risk preferences enter exclusively in the conditional certainty equivalent term, the only term related to unhedgeable risks. Both steps are generic and valid for any payoff. ii) Pricing measure: One pricing measure is used throughout. Its essential role is not to alter the conditional distribution of risks, given the ones we can trade, from their respective historical values. Naturally, there is no dependence on the payoff. The most interesting part however is its independence on risk preferences. This universality is expected and quite pleasing. It follows from the way we identified the pricing measure, via (8), a selection criterion that is obviously independent of any risk attitude. Finally, the distorted payoff C˜T can be computed under both the historical and the pricing measure, indeed, we have
1
1 C˜T = log EQ eγCT |ST = log EP eγCT |ST . γ γ The remainder of this section is dedicated to a comparison of our representation of the indifference prices and of the associated value functions with the well known representations obtained by Rouge and El Karoui (2000) and Delbaen et al. (2002) (see also Kabanov and Stricker (2002)). The technical arguments are not difficult and therefore the discussion is provided in a casual fashion. The conclusions are presented in Proposition 4. In the aforementioned works, it has been established that the indifference price solves a stochastic optimization problem. The objective therein is to maximize, over all martingale measures, the expected payoff of the claim, reduced 10

by a (relative) entropic penalty term (see (20) below). This representation is a direct result of the choice of exponential preferences and of the duality approach used on the primary expected utility problem. Details on the duality approach will be presented in the next Chapter by Elliott and Van der Hock. A martingale measure that naturally arises in this analysis is the so-called ˜ It is defined minimal relative entropy measure, denoted, for the moment, by Q. as the minimizer of the relative entropy, namely,   ˜ |P = min H (Q |P ) H Q (18) Q∈Qe

where H (Q |P ) = EP



dQ dQ ln dP dP



.

(19)

For an extensive study of this measure, we refer to Frittelli (2000a, 2000b). Under general model assumptions, the following result was established by Rouge and El Karoui (2000) and by Delbaen et al. (2002). Proposition 3 The indifference price ν(CT ) is given by    1 ˜ |P H ( Q| P) − H Q . ν (CT ) = sup EQ (CT ) − γ Q∈Qe

(20)

where Qe is the set of martingale measures equivalent to P. The above formula has several attractive features. It is valid for general models and arbitrary payoffs. The entropic penalty directly quantifies the effects of incompleteness on the prices. The formula also exposes the limiting behavior of the price as the investor becomes risk neutral, namely, as γ converges to zero. Finally, it highlights, in an intuitively pleasing way, the monotonicity of the price in terms of risk preferences and its convergence to the arbitrage-free price as the market becomes complete. This representation has, however, some shortcomings. It provides the price via a new optimization problem, a fact that does not allow for a universal analogue to its arbitrage-free counterpart. It also yields a pricing measure that has the undesirable feature of depending on the specific payoff. Moreover, the price formula (20) considerably obstructs the analysis and study of certain important aspects of indifference valuation, as for example, its numeraire independence and its generalization when risk preferences become stochastic. It also provides limited intuition for the construction of the relevant risk monitoring strategies and the associated payoff decomposition formulae. We start our comparative analysis by exploring the relation between the ˜ appearing pricing measure Q used in (9), and of the minimal entropy measure Q in (20). We can readily see that the two measures coincide.

11

In fact, consider the relative entropy (19) and look at its minimizers. For ˆ is an arbitrary martingale measure defined by the simple model at hand, if Q the elementary probabilities qˆi , i = 1, ..., 4, then 4  X  qˆi ˆ |P = qˆi log . H Q p i i=1

Simple calculations yield that the minimizing elementary probabilities, say q˜i i = 1, ..., 4 are given by q˜i = q

pi pi , i = 1, 2 and q˜i = (1 − q) , i = 3, 4 p1 + p2 p3 + p4

and, thus, they are equal to the qi , i = 1, ..., 4 of Q (see (16)). Therefore, 4   X qi ˜ |P = qi log . H (Q |P ) = H Q p i i=1

(21)

Next, we recall that the pricing measure Q was shown to satisfy the intu˜ therefore, itively pleasing property Q (YT |ST ) = P (YT |ST ). The measure Q, is the unique martingale measure under which the conditional distribution of unhedgeable risks, given the hedgeable ones, remains the same as the one taken under the historical measure. The minimal entropy measure is not a mere technical element arising in the dual analysis; rather, it is a natural choice for a pricing measure as the one that allocates the same conditional weights to unhedgeable risks as the historical measure P. We also observe that the minimal relative entropy measure is not a maximizer in the pricing formula (20). Indeed, if this were the case, the indifference ˜ an obviously price would have been the expected value of the payoff under Q, incorrect conclusion. This can happen only if the market is complete in which case, the minimal relative entropy measure coincides with the unique risk neutral one and the indifference price reduces to the arbitrage free price. We can use the above observations to deduce alternative formulae for the involved value functions (3). These representations, first produced by Delbaen et al. (2002), are interesting on their own right. As we will see in subsequent sections, they offer valuable insights for the specification of the dynamic risk preferences and are instrumental in the construction of indifference prices in more complex model environments. To this end, we first observe that − log



p1 + p2 q

q 

p3 + p4 1−q

1−q

=

4 X i=1

qi log

qi , pi

which, in view of (21), implies that the left hand side represents the minimal relative entropy. Combining this with (13) yields the representation (22). This structural result is intuitively pleasing. It reflects how risk preferences are dynamically adjusted via the optimal investments. In fact, the value function V 0 12

is directly obtained from the terminal utility U by a mere translation of the wealth argument. In a sense, the entropy H(Q |P ) represents the wealth value adjustment due to the magnitude of the investment opportunities or, using the continuous time models language, the size of the corresponding Sharpe ratio. Also, if that ratio is equal to zero then the value and utility functions coincide. This is clearly the case when the historical and risk neutral distributions of the stock are the same. A similar representation can be derived for the value function V CT , see (23) below. It follows directly from the definitions of the indifference price and the value functions (see, respectively, (3) and (4)). Formula (23) shows that V CT can be obtained from the terminal utility through two wealth adjustments, one that is related to the indifference price and the other, already appearing in the absence of the claim, reflecting the magnitude of investment opportunities and market incompleteness. We summarize the above results below. Proposition 4 Let ν(CT ) be the indifference price of the claim, Q the pricing measure introduced in (8) and H (Q |P ) its associated relative entropy. ˜ satisfies i) The minimal relative entropy measure Q ˜ Q≡Q and therefore,

˜ (YT |ST ) = P (YT |ST ) . Q

ii) The value functions V 0 and V CT are represented, respectively, by   1 0 −γx−H(Q|P ) V (x) = −e = U x + H (Q |P ) γ

(22)

and V CT (x) = −e−γx−H(Q|P )+γν(CT )   1 = U x + H (Q |P ) − ν(CT ) γ with U as in (2). iii) The indifference price satisfies   1 ν (CT ) = sup EQ (CT ) − (H ( Q| P) − H (Q |P )) = EQ (CT ) , γ Q∈Qe where the nonlinear pricing functional EQ is given by   1 γCT |ST ) . log EQ (e EQ (CT ) = EQ γ

13

(23)

2.2

Properties of the Indifference Prices

The previous analysis produced the nonlinear price representation   ν (CT ) = EQ (CT ) = EQ C˜T ,

where the preference adjusted payoff C˜T is the conditional certainty equivalent (cf. (17)),
1 C˜T = log EQ eγCT |ST , γ

and the pricing measure Q is the minimal relative entropy martingale measure. This pricing formula yields a direct constitutive analogue to the linear pricing rule of the complete models. Our next task is to explore the structural properties of the indifference prices, their behavior with respect to various inputs as well as their differences and similarities to the arbitrage-free prices. Throughout we occasionally adopt the notation ν(CT ; γ). This is done for clarity and it is omitted whenever there is no ambiguity. Moreover, to ease the analysis and the presentation, it is also assumed that CT ≥ 0 P-a.s. This assumption is easily removed at the expense of more tedious arguments. i) Behavior with respect to the risk aversion coefficient. While risk preferences are not affecting the arbitrage free prices due to perfect risk replication, they represent an indispensable element of indifference prices. Indeed, the risk aversion coefficient γ appears in the construction of C˜T . It is through this conditional preference adjusted payoff that the indifference valuation mechanism extracts and valuates the underlying unhedgeable risks. Proposition 5 The function γ → ν (CT ; γ) = EQ




1 log EQ eγCT |ST γ

from R+ into R is increasing and continuous. Moreover, if for all claims CT we have ν (CT ; γ) = ν (CT ; 1) (24) then γ = 1. Proof. Continuity follows directly from the formula and the properties of conditional expectation. To establish monotonicity, let us assume that 0 < γ1 < γ2 . Holder’s inequality then yields EQ eγ1 CT |ST




≤ EQ eγ2 CT |ST

14



γ1 /γ2

and, in turn,

1 1 log EQ eγ1 CT |ST ≤ log EQ eγ2 CT |ST . γ1 γ2

Taking expectation, with respect to the pricing measure Q, we deduce the first statement. To establish the second assertion, we assume, without loss of generality, that p2 6= 0. Then, we consider claims of the form CT (ω1 ) = c1 , CT (ωi ) = 0, i = 2, 3, 4, and we observe that (24) leads to 1 ec1 p1 + p2 eγc1 p1 + p2 = log log γ p1 + p2 p1 + p2 for all c1 . To conclude it suffices to differentiate both sides with respect to c1 and rearrange terms. Proposition 6 The following limiting relations hold lim ν (CT ; γ) = EQ (CT ) ,

(25)

γ→0+

lim ν (CT ; γ) = EQ kCT kL∞

γ→∞

Q{·|ST

}

,

(26)

Proof. To show (25), we pass to the limit, as γ → 0, in the price formula (14),    γc1  γc3 1 1 e p1 + eγc2 p2 e p3 + eγc4 p4 ν(CT ; γ) = q log + (1 − q) log (27) γ p1 + p2 γ p3 + p4 to obtain lim ν(CT ; γ) = q

γ→0



p 2 c2 p 1 c1 + p1 + p2 p1 + p2



+ (1 − q)



p 4 c4 p 3 c3 + p3 + p4 p3 + p4



.

On the other hand, by the properties of the pricing measure, we have q

pi pi = qi , i = 1, 2 and (1 − q) = qi , i = 3, 4 p1 + p2 p3 + p4

and, in turn, lim ν(CT ; γ) =

γ→0

4 X

qi ci .

i=1

To establish (26), we pass to the limit as γ → ∞ in (27). We readily get that lim ν(CT ) = q max (c1 , c2 ) + (1 − q) max (c3 , c4 )

γ→∞

and the statement follows.

15

Proposition 7 The indifference price satisfies lim

γ→0

∂ν (CT ; γ) 1 = EQ (V arQ (CT |ST )) , ∂γ 2

(28)

and thus, 1 ν (CT ; γ) = EQ (CT ) + γEQ (V arQ (CT |ST )) + o (γ) . 2

(29)

Proof. We only show (28), since (29) is an easy consequence. We first differentiate ν (CT ; γ) with respect to γ, obtaining   1 1 EQ (CT eγCT |ST ) ∂ν (CT ; γ) γCT |ST ) + = EQ − 2 log EQ (e ∂γ γ γ EQ (eγCT |ST )     1 EQ (CT eγCT |ST ) = − ν(C ; γ) . EQ T γ EQ (eγCT |ST ) Therefore, 1 ∂ν (CT ; γ) = lim γ→0 γ γ→0 ∂γ lim

= lim EQ γ→0

    EQ (CT eγCT |ST ) − ν(C ; γ) EQ T EQ (eγCT |ST )


2 ! EQ (CT2 eγCT |ST )EQ (eγCT |ST ) − EQ (CT eγCT |ST ) 2

(EQ (eγCT |ST ))

− lim

γ→0

∂ν (CT ; γ) , ∂γ

and, thus,   ∂ν (CT ; γ) 1 2 = EQ EQ (CT2 |ST ) − (EQ (CT |ST )) . γ→0 ∂γ 2 lim

Proposition 8 The indifference price is consistent with the no arbitrage principle, namely, for all γ > 0, inf EQ (CT ) ≤ ν(CT ; γ) ≤ sup EQ (CT ) ,

Q∈Qe

(30)

Q∈Qe

where Qe is the set of martingale measures that are equivalent to P. Proof. We assume, without loss of generality that c1 < c2 and that c3 < c4 . The monotonicity of the price with respect to risk aversion implies lim ν(CT ; γ) ≤ ν(CT ; γ) ≤ lim ν(CT ; γ) γ→∞

γ→0

16

and, in turn, that EQ (CT ) ≤ ν(CT ; γ) ≤ EQ kCT kL∞

Q{·|ST

}

.

Taking the infimum over all martingale measures, yields inf EQ (CT ) ≤ EQ (CT ) ≤ ν(CT ; γ)

Q∈Q

and the left hand side of (30) follows. We next observe that EQ kCT kL∞

Q{·|ST

}

= EQ¯ (CT )

¯ has elementary probabilities where the martingale measure Q ¯ {ω1 } = 0, Q ¯ {ω2 } = q, Q ¯ {ω3 } = 0, Q ¯ {ω4 } = 1 − q. Q Observing that ν(CT ; γ) ≤ EQ¯ (CT ) ≤ sup EQ (CT ) Q∈Q

we conclude. ii) Behavior with respect to payoffs We first explore the monotonicity, convexity and scaling behavior of the indifference prices. We note that in the next two Propositions, all inequalities among payoffs and their prices hold both under the historical and the pricing measures P and Q. Since these two measures are equivalent, we skip any measure-specific notation for the ease of the presentation. Proposition 9 The indifference price is a non decreasing and convex function of the claim’s payoff, namely,

if CT1 ≤ CT2 then ν CT1 ≤ ν CT2 (31) and, for α ∈ (0, 1) ,




ν αCT1 + (1 − α)CT2 ≤ αν CT1 + (1 − α)ν CT2 .

(32)

Proof. Inequality (31) follows directly from formula (9). To establish (32), we apply Holder’s inequality to obtain    1 2 1 EQ log EQ eγ (αCT +(1−α)CT ) |ST γ    1−α  α   2 1 1 γCT γCT |ST ≤ EQ EQ e |ST log EQ e γ       1 2 1 1 + (1 − α) EQ log EQ eγCT |ST log EQ eγCT |ST = αEQ γ γ

and the result follows.

17

Proposition 10 The indifference price satisfies ν (αCT ) ≤ αν (CT ) for α ∈ (0, 1)

(33)

ν (αCT ) ≥ αν (CT ) for α ≥ 1.

(34)

and Proof. To show (33) we observe

where, for α ∈ (0, 1),




1 log EQ eγαCT |ST γ  
1 γ ¯ CT = αEQ |ST log EQ e γ¯

ν (αCT ) = EQ

γ¯ = αγ < γ. Using the monotonicity of the price with respect to risk aversion, we conclude. Inequality (34) follows by the same argument. The following result highlights an important property of the indifference price operator. We see that any hedgeable risk is automatically scaled out from the nonlinear part of the pricing rule, and it is priced directly by arbitrage. Hedgeable risks do not differ from their conditional certainty equivalent payoffs. In this sense, we say that the pricing operator has the property of additive 1 invariance with respect to hedgeable risks Note that this property is stronger, and more intuitive, than requiring mere translation invariance with respect to constant risks. Proposition 11 The indifference pricing operator is additively invariant with respect to hedgeable risks, namely, if CT = CT1 + CT2 , with CT1 = C 1 (ST ) and CT2 = C 2 (ST , YT ), then ν(CT ) = EQ (C 1 (ST ) + C 2 (ST , YT ))
= EQ C 1 (ST ) + ν(C 2 (ST , YT )).

(35)

Proof. The price formula (9), together with the measurability properties of C 1 (ST ) , yields    1 2 1 log EQ eγ (C (ST )+C (ST ,YT )) |ST ν(CT ) = EQ γ   
1 γC 2 (ST ,YT ) 1 |ST log EQ e = EQ C (ST ) + EQ γ
= EQ CT1 + ν(CT2 ). 1 The

authors would like to thank Patrick Cheridito for suggesting this terminology.

18

The above property yields the following conclusions for two extreme cases. Special Cases: i) Let CT = C 1 (ST ) + C 2 (ST , YT ) with YT depending functionally on ST . The payoff CT2 is then FTS −measurable and, therefore, C 2 (ST , YT ) = C˜ 2 (ST , YT ). Combining the above with (35) implies
ν(CT ) = EQ C 1 (ST ) + ν(C 2 (ST , YT ))
= EQ C 1 (ST ) + EQ (C 2 (ST , YT )).

The indifference price simplifies to the arbitrage-free one and the minimal relative entropy measure coincides with the unique risk neutral measure. ii) Let CT = C 1 (ST ) + C 2 (YT ) with YT and ST independent under P. Then,  1    2 2 1 C˜ 2 (YT ) = log EQ eγC (YT ) |ST = log EP eγC (YT ) . γ γ

The indifference price of CT consists of the arbitrage-free price of the first claim plus the traditional actuarial certainty equivalent price of the second, ν(CT ) = EQ (C 1 (ST )) +

  2 1 log EP eγC (YT ) . γ

We finish this section presenting the link between the indifference pricing functional ν and the so called convex risk measures (see, among others, Artzner et al. (1999), Follmer and Schied (2002a) and (2002b)). Definition 12 The mapping ρ : FT → R is called a convex risk measure if it 1 2 satisfies the following conditions for all
CT , CT 1∈
FT .
1 2 • Convexity: ρ αCT + (1 − α) CT ≤ αρ
CT + (1
− α) ρ CT2 , 0 ≤ α ≤ 1. • Monotonicity: If CT1 ≤ CT2 , then ρ CT1 ≥ ρ CT2 .

• Translation invariance: If m ∈ R, then ρ CT1 + m = ρ CT1 − m. For any CT ∈ FT define

ρ (CT ) = ν (−CT ) = EQ




1 . log EQ e−γCT |ST γ

(36)

Proposition 13 The mapping ρ given in (36) defines a convex risk measure. Proof. All conditions follow trivially from the properties of the indifference price discussed earlier. Note that the number ν (CT ) represents the indifference value of the payoff CT , while the number ρ (CT ) = ν (−CT ) is usually interpreted as a capital requirement imposed by a supervising body for accepting the position CT . It is interesting to observe that the concept of indifference value, deduced from the desire to behave optimally as an investor, is in the above sense consistent with a method that may be used to determine the capital amount, for a position to be acceptable to a supervising body. 19

2.3

Risk Monitoring Strategies

We now turn our attention to the important issue of managing risk generated by the derivative contract. In complete markets, the payoff is reproduced by the associated self-financing and replicating portfolio. Consequently, any risk associated with the claim is eliminated. For the model at hand, any FTS -measurable claim CT is replicable and the familiar representation formula CT = ν(CT ) +

∂ν(CT ) (ST − S0 ) ∂S0

(37)

holds, with ν(CT ) = EQ (CT )

and

∂EQ (CT ) ∂ν(CT ) = . ∂S0 ∂S0

(38)

The indifference price coincides with the arbitrage free price and its spatial derivative yields the so-called delta. When the market is incomplete however, perfect replication is not viable and a payoff representation similar to the above cannot be obtained. However, one may still seek a constitutive analogue to (37). We recall that the indifference price was produced via comparison of the optimal investment decisions with and without the claim in consideration. We should therefore base our study on the analysis of the relevant optimal portfolios, and the relation between the indifference price and the optimal wealth levels they generate. We start with an auxiliary structural result for the optimal policies of the underlying maximal expected utility problems (3). Proposition 14 Let ν(CT ) be the indifference price of the claim CT . The optimal number of shares αCT ,∗ in the optimal investment problem (3) is given by ∂ν(CT ) , (39) αCT ,∗ = α0,∗ + ∂S0 where α0,∗ = −

1 ∂H (Q |P ) γ ∂S0

(40)

represents the number of shares held optimally in the absence of the claim. Both optimal controls αCT ,∗ and a0,∗ are wealth independent. Proof. We first recall that αCT ,∗ was provided in (10), rewritten below for convenience, (ξ u − 1) (p1 + p2 ) 1 log αCT ,∗ = u d γS0 (ξ − ξ ) (1 − ξ d ) (p3 + p4 ) +

(eγc1 p1 + eγc2 p2 ) (p3 + p4 ) 1 . log γS0 (ξ u − ξ d ) (eγc3 p3 + eγc4 p4 ) (p1 + p2 )

20

When the claim is not taken into account, one can easily deduce, by setting ci = 0, i = 1, .., 4 above, that the corresponding optimal policy α0,∗ equals α0,∗ = =

1 γS0 (ξ u − ξ d )

log

(ξ u − 1) (p1 + p2 ) (1 − ξ d ) (p3 + p4 )

(41)

1 (1 − q) (p1 + p2 ) log . γ (S u − S d ) q (p3 + p4 )

Using that ∂ ∂q = ∂S0 ∂S0



S0 − S d Su − Sd



=

Su

1 − Sd

and differentiating the entropy expression (21) gives (1 − q) (p1 + p2 ) ∂q ∂H (Q |P ) = − log . ∂S0 q (p3 + p4 ) ∂S0 Differentiating, in turn, the price formula (9) gives 1 (eγc1 p1 + eγc2 p2 ) (p3 + p4 ) ∂ν(CT ) = log γc3 u d ∂S0 γS0 (ξ − ξ ) (e p3 + eγc4 p4 ) (p1 + p2 ) which, combined with the expressions for αCT ,∗ and α0,∗ yields (39). Next, we consider the optimal wealth variables X CT ,∗ and X 0,∗ representing, respectively, the agent’s optimal wealth with and without the claim. In the first case, the agent starts with initial wealth x + ν(CT ) and buys αCT ,∗ shares of stock. If the claim is not taken into account, the investor starts with x and follows the strategy α0,∗ . In other words,

and

XTCT ,∗ = x + ν(CT ) + αCT ,∗ (ST − S0 )

(42)

XT0,∗ = x + α0,∗ (ST − S0 ).

(43)

We now introduce two important quantities that will help us produce a meaningful decomposition of the claim’s payoff. They are the residual optimal wealth and the residual risk, denoted respectively by L and R. These notions were first introduced by the authors in Musiela and Zariphopoulou (2001a) and (2001b); see also Musiela and Zariphopoulou (2004a). Definition 15 The residual optimal wealth process is defined as Lt = XtCT ,∗ − Xt0,∗

for t = 0, T.

(44)

In a complete model environment, the residual optimal wealth coincides with the value of the perfectly replicating portfolio. It is therefore a martingale under the unique risk neutral measure, and it generates the claim’s payoff at expiration. 21

When the market is incomplete, however, several interesting observations can be made. The residual terminal optimal wealth LT reproduces the claim only partially. In addition, it is an FTS −measurable variable and retains its martingale property under all martingale measures. Its most important property, however, is that it coincides with the conditional certainty equivalent. This fact will play an instrumental role in two directions, namely, in the identification of the replicable part of the claim and in the specification of the risk monitoring policy. Proposition 16 The residual optimal wealth process satisfies: L0 = ν(CT ) and LT = ν(CT ) +

(45)

∂ν(CT ) (ST − S0 ) . ∂S0

(46)

Moreover, LT coincides with the conditional certainty equivalent, LT = C˜T .

(47)

Finally, the process Lt is a martingale under all equivalent martingale measures, EQ (LT ) = L0 = ν(CT )

for Q ∈ Qe .

(48)

Proof. Assertions (45) and (46) follow easily from Definition 15, the optimal wealth representations (42) and (43) and the relation (39) between the optimal policies. To show (47), we first recall that ν(CT ) = EQ (C˜T ),

(49)

which, in view of (46) yields, ∂EQ (C˜T ) (ST − S0 ) . LT = EQ (C˜T ) + ∂S0 The claim C˜T however is FTS −measurable and, thus, replicable. Its arbitrage free decomposition is ∂EQ (C˜T ) (ST − S0 ) C˜T = EQ (C˜T ) + ∂S0 and the identity (47) follows. The martingale property (48) is an easy consequence of (46). Definition 17 The indifference price process νt (CT ), t = 0, T is defined as ν0 (CT ) = ν (CT )

and 22

νT (CT ) = CT .

(50)

Definition 18 The residual risk process Rt , t = 0, T , is defined as the difference between the indifference price and the residual optimal wealth, namely, Rt = νt (CT ) − Lt , i.e., R0 = ν (CT ) − L0

and

RT = CT − LT .

(51)

If perfect replication is viable, the residual risk is zero throughout and its notion degenerates. In general, it represents the component of the claim that is not replicable, given that risks that can be hedged have been already extracted optimally according to our utility criteria. As such, the residual risk should not generate any additional conditional certainty equivalent part nor it should, in consequence, acquire any additional indifference value. Proposition 19 The residual risk process has the following properties: i) It satisfies R0 = 0 and

RT = CT − C˜T .

(52) (53)

ii) Its conditional certainty equivalent is zero, ˜ T = 0. R

(54)

ν(RT ) = 0.

(55)

iii) Its indifference price is zero,

iv) It is a supermartingale under the pricing measure Q , EQ (RT ) ≤ R0 = 0.

(56)

v) Its expected, under the historical measure P, certainty equivalent is zero, 1 log EP (eγRT ) = 0. γ

(57)

Proof. Part (i) follows readily from the definition of the residual risk and the properties of Lt , t = 0, T. To show (ii), we apply directly the definition of the conditional certainty equivalent. This, together with the measurability of C˜T , yields   ˜ T = 1 log EQ eγ (CT −C˜T ) |ST R γ =


1 log EQ eγCT |ST − C˜T γ = C˜T − C˜T = 0. 23

Parts (iii) and (iv) are immediate consequences of (49), (53) and (54). To establish (57), we recall that ˜ T = 1 log EQ (eγRT |ST ) = 1 log EP (eγRT |ST ) R γ γ where we used (8). Using (54) and taking the expectation under P yields the result. Being a supermartingale, the residual risk can be decomposed according to the Doob-Meyer decomposition. The related components can be easily retrieved and are presented below. Proposition 20 The supermartingale Rt , for t = 0, T , admits the Doob-Meyer decomposition Rt = Rtm + Rtd where R0m = 0

and

RTm = RT − EQ (RT ),

(58)

RTd = EQ (RT ).

(59)

and R0d = 0 The and

and

(S,Y ) -martingale component Rtm is an FT (S,Y ) adapted to the trivial filtration F0 .

under Q,, while Rtd is decreasing

We are now ready to provide the payoff decomposition result. This result is central in the study of risks associated with the indifference valuation method since it provides in a direct manner the constitutive analogue of the arbitragefree payoff decomposition (37). Theorem 21 Let C˜T and RT be, respectively, the conditional certainty equivalent and the residual risk associated with the claim CT . Let also Rtm and Rtd be the elements of the Doob-Meyer decomposition (58) and (59). ˜ Define the process MtC , for t = 0, T , by ˜

˜

M0C = ν (CT ) and MTC = ν (CT ) +

∂ν (CT ) (ST − S0 ). ∂S0

(60)

i) The claim CT admits the unique, under Q, payoff decomposition CT = C˜T + RT = ν(CT ) +

∂ν(CT ) (ST − S0 ) + RT ∂S0

(61)

˜

= MTC + RTm + RTd . (S,Y )

ii) The indifference price process νt , defined in (50), is an FT martingale under Q. It admits the unique decomposition νt (CT ) = Mt + Rtd 24

− super(62)

where

˜

Mt = MtC + Rtm . The components Mt and Rtd represent, respectively, the associated martingale and the non-increasing parts of the price process νt . From the application point of view, one may think of RT and its moments as natural variables for the quantification of errors associated with the risk monitoring policy. As the Proposition below shows, the expected error obtains a rather intuitive form. It is proportional to the risk aversion and to the expected conditional variance of the non-traded risks. Naturally, both the expectation and the conditional variance need to be considered under the pricing measure Q. Proposition 22 The expected residual risk satisfies 1 EQ (RT ) = − γEQ (V arQ (CT |ST )) + o (γ) 2 and

1 EQ (RT ) = − γEQ (V arQ (RT |ST )) + o (γ) . 2

Proof. The proof follows from (53), yielding EQ (RT ) = EQ (CT ) − EQ (C˜T ), and the approximation formula (29). The second equality is obvious.

2.4

Relative Indifference Prices

From the previous analysis, we can deduce that the indifference price is not a linear function of the claim’s payoff, namely, for α 6= 0, 1, ν(αCT ) 6= αν(CT ).

(63)

Indeed, as it was established in Proposition 10, if α > 1 (resp. α < 1), the indifference price is a super-homogeneous (resp. sub-homogeneous) function of CT . Following simple arguments, we easily conclude that if two payoffs, say CT1 and CT2 , are considered, the indifference price functional is non-additive, namely, ν(CT1 + CT2 ) 6= ν(CT1 ) + ν(CT2 ).

(64)

Extending these arguments to the case of multiple payoffs, we obtain that for N , with N > 1, payoffs N N X X ν(CTi ). CTi ) 6= ν( i=1

i=1

25

(65)

The non-additive behavior of indifference prices is a direct consequence of the nonlinear character of the indifference valuation mechanism. Naturally, this nonlinear characteristic is inherited to the associated risk monitoring strategies. The nonadditivity property is perhaps the one that most differentiates the indifference prices and the relevant risk monitoring strategies from their complete market counterparts. This might then look as a serious deficiency of the indifference valuation approach both for the theoretical as well as the practical point of view. However, it should be noted that the aggregate valuation of the above claims was considered as if the individual risks were priced in isolation. In practice, risks and projects need to be valuated and hedged relative to already undertaken risks. In complete markets, perfect risk elimination makes this relative risk positioning redundant. But, when risks cannot be eliminated, one should develop a methodology that would quantify and price the incoming incremental risks, while taking into account the existing unhedgeable risk exposure. These considerations lead us to the relative indifference valuation concept. The notion of relative indifference price was first introduced by the authors in Musiela and Zariphopoulou (2001b) and was recently further developed in a stochastic volatility model by Stoikov (2005). Definition 23 Let C 1T = C 1 (ST , YT ) and C 2T = C 2 (ST , YT ) be two claims 1 2 1 2 that have indifference prices ν(CT1 ) and ν(CT2 ). Let V CT , V CT and V CT +CT be 1 2 the value functions (3) corresponding to claims C
T1 + CT2 .
CT , CT and 2 1 1 2 The relative indifference prices ν CT /CT and ν CT /CT are defined, respectively, as the amounts satisfying,

and


1 2 1 V CT (x) = V CT +CT (x + ν CT2 /CT1 )


2 1 2 V CT (x) = V CT +CT (x + ν CT1 /CT2 )

(66) (67)

for all wealth levels x ∈ R.

As the following result yields, when a new claim is being priced relatively to an already incorporated risk exposure, the associated indifference prices become linear. Proposition 24 Assume that the claims CT1 = C 1 (ST , YT ) and CT2 = C 2 (ST , YT )
1 2 1 2 have indifference
prices ν(CT ) and ν(CT ) and relative indifference prices ν CT /CT 2 1 and ν CT /CT . Then, the indifference price of the claim with payoff CT = CT1 + CT2 satisfies

and



ν (CT ) = ν CT1 + ν CT2 /CT1



ν (CT ) = ν CT2 + ν CT1 /CT2 . 26

(68)

Proof. We only show the first statement since the second follows by analogous arguments. For this, we recall the representation formula (23) which yields, respectively, 1 1 V C (x) = −e−γx−H(Q|P )+γν (CT ) and VC

1

+C 2

(x) = −e−γx−H(Q|P )+γν (CT +CT ) . 1

2

Moreover, the same formula together with the definition of the relative indiffer
ence price ν CT2 /CT1 implies that V C (x) = −e−γx−H(Q|P )+γν (CT ) 1

1

= −e−γ (x+ν (CT /CT ))−H(Q|P )+γν (CT +CT )
1 2 = V C +C (x + ν CT2 /CT1 ) 2

1

1

2

for all wealth levels. Equating the exponents yields (68). The following results are immediate consequences of the above.

Corollary 25 The indifference prices
ν CT1 and ν CT2 , and their relative
counterparts ν CT1 /CT2 and ν CT1 /CT2 , satisfy



ν CT1 − ν CT2 = ν CT1 /CT2 − ν CT2 /CT1 . Corollary 26 The indifference price of the claim CT = CT1 + CT2 is given by

1


1

ν CT1 + CT2 = ν CT1 + ν CT2 + ν CT1 /CT2 + ν CT2 /CT1 . 2 2

Moreover,

=





ν CT1 + CT2 − ν CT1 + ν CT2





1

1 ν CT1 /CT2 + ν CT2 /CT1 − ν CT1 + ν CT2 . 2 2

The latter formula yields the error emerging from the non-additive character of the indifference price. This error may vanish in certain cases, as the examples below demonstrate. These examples were discussed in detail in Section 2.2 in the context of the translation invariance property of indifference prices. They refer to the special cases of a complete and a fully incomplete market setting. Special Cases Revisited: i) Let CT = CT1 + CT2 with CT1 = C 1 (ST , YT ) and CT2 = C 2 (ST ). The translation invariance property (35) implies that the price is additive. In fact,

with ν(CT2 ) = EQ

ν(CT ) = ν(CT1 ) + ν(CT2 )
C 2 (ST ) . Proposition 24 then yields that

ν CT2 /CT1 = ν CT2 . 27

If additionally, YT depends functionally on ST , then, we easily deduce that
ν(CT ) = EQ CT1 + EQ (CT2 ) and, in turn, that

ν(CT1 /CT2 ) = ν(CT1 ) and ν(CT2 /CT1 ) = ν(CT2 ). ii) Let CT = C 1 (ST ) + C 2 (YT ) with YT and ST be independent under P. Then, it was shown that the price behaves additively, namely, ν(CT ) = ν(CT1 ) + ν(CT2 ) with ν(CT1 ) = EQ (CT1 )

and ν(CT2 ) =

Proposition 24 in turn implies

  2 1 log EP eγC (YT ) . γ

ν(CT1 /CT2 ) = ν(CT1 ) and ν(CT2 /CT1 ) = ν(CT2 ). The above examples demonstrate that the relative indifference prices reduce to the classical ones if the relevant risks are either fully replicable or independent from the traded ones.

2.5

Wealths, Preferences and Numeraires

The results of the previous sections were derived under the assumptions of zero interest rate and constant risk aversion. In this case, the wealths at the beginning and the end of a time period are expressed in a comparable unit (spot or forward) and, thus, the possible dependence of the underlying optimization problems on the unit choice is not apparent. Below we analyze this question by looking first at the relationship between the spot and forward units. Then, we consider a state dependent risk aversion coefficient in order to cover other cases of numeraires. In particular, we consider the stock itself as a numeraire and show that the indifference prices can be made numeraire independent and consistent with the static no arbitrage constraint if the appropriate dependence across units is built into the risk preference structure. i ) Indifference prices in spot and forward units Consider the one period model, introduced in Section 2.1, of a market with a riskless bond and two risky assets, of which only one is traded. The dynamics of the risky assets remain unchanged but we now allow for non-zero riskless rate. The price of the riskless asset, therefore, satisfies, B0 = 1 and BT = 1 + r with ξd ≤ 1 + r ≤ ξu. Because of the non-zero riskless rate, the price formula (9) cannot be directly applied. In order to produce meaningful prices, one needs to be consistent 28

with the units in which the quantities that are used in price specification are expressed. For the case at hand, we will consider the valuation problem in spot and in forward units and will force the price to become independent of the unit choice. We start with the formulation of the indifference price problem in spot units. Consider a portfolio consisting of α shares of stock and the amount β invested in the riskless asset. Its current value is given by β+αS0 = x, where x represents the agent’s initial wealth, X0 = x. Expressed in spot units, that is discounted to time 0, its spot terminal wealth XTs satisfies   ST XTs = x + α (69) − S0 . 1+r The investor’s utility is taken to be exponential with constant absolute risk aversion coefficient γ s . It is important to note that for the utility to be well defined this coefficient needs to be expressed in the reciprocal of the spot unit. Optimality of investments will be carried out through the relevant value function, in spot units, given by  “ ” CT s −γ s XT − 1+r s,CT . (70) (x) = sup EP −e V α

Note that the option payoff CT is also discounted from time T to time 0. The following definition is a natural extension of Definition 1. Definition 27 The indifference price, in spot units, of the claim CT is defined as the amount ν s (CT ) for which the two spot value functions V s,CT and V s,0 , defined in (70) and corresponding to claims CT and 0, coincide. Namely, it is the amount ν s (CT ) satisfying V s,0 (x) = V s,CT (x + ν s (CT ))

(71)

for any initial wealth levels x ∈ R. Proposition 28 Let Qs be the measure such that   ST = S0 , EQs 1+r and Qs (YT |ST ) = P (YT |ST ) .

(72)

Moreover, let CT = c(ST , YT ) be the claim to be priced, in spot units, with spot risk aversion coefficient γ s . Then, the spot indifference price is given by     s CT  1 CT γ 1+r s e = EQs log E ν s (CT ) = EQs . (73) |S Q T 1+r γs

29

Proof. Working along similar arguments to the ones used in the proof of Proposition 2, we first establish that the spot value functions, V s,0 and V s,CT , are given by V s,0 (x) = −e−γ

s

1

x qs (q s )

(1 −

qs

1−qs

(p1 + p2 ) (p3 + p4 )

1−qs qs)

and V s,CT (x) = −e−γ

s

1

x qs

(q s ) (1 − q s )

1−qs



eγ

s c1 1+r

p1 + e γ

s c2 1+r

p2

1−qs  s c3 s c4 , × eγ 1+r p3 + eγ 1+r p4

where

qs =

q s

(1 + r) − ξ d . ξu − ξd

(74)

Applying Definition 27 gives ν s (CT ) = q s

+ (1 − q s )

eγ 1 log γs 1 eγ log γs

s c1 1+r

s c3 1+r

p1 + eγ p1 + p2

p3 + eγ p3 + p4

s c2 1+r

s c4 1+r

p2

p4

!

!

(75)

.

Straightforward calculations yield that the spot pricing measure Qs has elementary probabilities, denoted by qis , i = 1, .., 4, qis = q s

pi pi , i = 1, 2 and qis = (1 − q s ) , i = 3, 4. p1 + p2 p3 + p4

(76)

We next introduce the conditional certainty equivalent in spot units  s CT  1 C˜Ts = s log EQs eγ 1+r |ST . γ Equation (75) then yields ν s (CT ) = EQs (C˜Ts ) and (73) follows. We next analyze the indifference valuation of CT assuming that all relevant prices, risk preferences and value functions are expressed in the reciprocal of the forward unit. To this end, we consider the forward terminal wealth XTf = XTs (1 + r) = x (1 + r) + α (ST − S0 (1 + r)) 30

= f + α (FT − F0 ) ,

(77)

where f = x (1 + r) is the forward value of the current wealth. Moreover, F0 = S0 (1 + r) and FT = ST is the forward stock price process. Implicitly, we assume existence of the forward market for the risky traded asset S, and hence of the quoted prices F0 and FT , for it can be replicated by trading in the forward market. The corresponding forward value function is   f f (78) V f,CT (f ) = sup EP −e−γ (XT −CT ) . α

The risk aversion coefficient γ f is naturally expressed in forward units. Definition 29 The indifference price, in forward units, of the claim CT is defined as the amount ν f (CT ) for which the two forward value functions V f,CT and V f,0 , defined in (78) and corresponding to claims CT and 0, coincide. Namely, it is the amount ν f (CT ) satisfying V f,0 (f ) = V f,CT (f + ν f (CT ))

(79)

for any initial wealth levels f ∈ R. Proposition 30 Let Qf be a measure under which EQf (FT ) = F0 and Qf (YT |FT ) = P (YT |FT ) .

(80)

Qf = Qs .

(81)

Then Let CT = c(ST , YT ) be the claim to be priced under exponential preferences with forward risk aversion coefficient γ f . Then, the indifference price in forward units of CT is given by ν f (CT ) = EQf (CT ) (82)     f 1 . = EQf log EQf eγ CT |FT f γ Proof. Given the deterministic interest rate assumption, the fact that the measures Qf and Qs coincide is obvious. We next observe that the forward value function V f,CT can be written as   f f V f,CT (f ) = sup EP −e−γ (XT −CT ) α



= sup EP −e−γ α



f

(x(1+r)+α(ST −S0 (1+r))−CT )

= sup EP −e α

“ “ ” ” ST CT −˜ γ x+α 1+r −S0 − 1+r

31



with γ˜ = γ f (1 + r) . Therefore, V f,CT , and in turn V 0,CT , can be directly retrieved from their forward counterparts. The rest of the proof follows easily and it is therefore omitted. For the rest of the analysis, we denote by Q the common spot and forward pricing measure. We are now ready to investigate when the spot and forward indifference prices are consistent with the static no arbitrage condition and independent of the units, spot or forward, chosen in the supporting investment optimization problem. The result below gives the necessary and sufficient conditions on the spot and forward risk aversion coefficients. Proposition 31 The indifference prices, expressed in spot and forward units are consistent with the no arbitrage condition, that is, ν f (CT ) = (1 + r) ν s (CT ) ,

(83)

if and only if the spot and forward risk aversion coefficients satisfy γf =

γs . 1+r

(84)

Proof. We first show that if (84) holds then (83) follows. Recalling (73) and (82), we deduce that, if (84) holds, then ν f (CT ) can be written as    s CT 1 γ 1+r f ν (CT ) = (1 + r) EQ |ST log EQ e γs

and one direction of the statement follows. We remind the reader that Q = Qs = Qf . We next show that for (83) to hold for all CT we must have (84). Indeed, if the consistency relationship (83) holds, then, for all claims CT ,      f  s CT 1 1 1 γ 1+r γ CT |S = E |S e e log E log E EQ T Q T Q Q 1+r γf γs and, in turn,      f  γs 1 1+r γ CT 1+r CT |S . EQ = E |S e e log E log E T Q T Q Q γf γs The statement then follows from straightforward rescaling arguments and (24). We continue with a representation result for the spot and forward value functions. We recall that the spot and forward pricing measures reduce to the same measure Q which, therefore, has elementary probabilities qi , i = 1, .., 4 given in (76). Working along similar arguments to the ones used in Section 2.1 we can easily establish that, among all equivalent martingale measures, Q has 32

the minimal, relative to the historical measure P, entropy. The minimal relative entropy is given by   X pi qi log . H (Q |P ) = qi i=1,..,4 Proposition 32 The value functions V s,CT and V f,CT are given by   1 s −γ s (x−ν s (CT ))−H(Q|P ) s s,CT (x) = −e = U x − ν (CT ) + s H (Q |P ) , V γ   f f 1 V f,CT (f ) = −e−γ (f −ν (CT ))−H(Q|P ) = U f f − ν f (CT ) + f H (Q |P ) , γ where U s (x) = −e−γ utility functions.

s

x

and U f (f ) = −e−γ

f

f

represent the spot and forward

Recall that the argument x in U s (x) and in V s,CT (x) is expressed in the spot units, while the same argument f in U f (f ) and V f,CT (f ) is expressed in the forward units. Therefore, the utility and the value functions represent the same utility and value, independently on the units in which the relevant optimization problems are solved, if and only if (84) holds. More generally, the indifference based valuation as well as the associated optimal investment problems can be formulated and solved in a numeraire independent fashion provided the appropriate relations are built into the preference structure. In fact, these problems can be analyzed without making any reference to a unit by optimizing over unitless quantities like γ s XTs or γ f XTf . ii) Indifference prices and state dependent preferences Before we proceed with the specification of the price and the conditions for numeraire independence, we extend our previous set up to the case of random risk aversion coefficient. Specifically, we assume that it is a function of the states of the traded asset. We may then conveniently represent the risk aversion at time T as the FTS -measurable random variable γT = γ (ST ) taking the values
γ u = γ (S0 ξ u ) and γ d = γ S0 ξ d ,   when events {ω : ST (ω) = S0 ξ u } = ω 1 , ω 2 and ω : ST (ω) = S0 ξ d =  3 the ω , ω 4 occur. Clearly, the risk aversion γT is expressed in the unit which is the reciprocal of the wealth XT unit. Alternatively, one may think of the risk tolerance δT =

1 γT

which is obviously expressed in the units of wealth at time T . 33

(85)

Here we assume the same model as in the previous section and choose to work with spot units, so XT = XTs as in (69). We have mentioned already that representations of the indifference prices under minimal model assumptions have been derived via duality arguments. These results can be extended even to the cases when the risk aversion coefficient is random. The related pricing formulae, however, take a form that reveals limited insights about the numeraire and units effects. For this, we seek an alternative price representation, provided below, which, to the best of our knowledge, is new. We also adopt the notation ν (CT ; γT ) , V CT (x; γT ) and V 0 (x; γT ) , for the price and the relevant value functions, so that the random nature of risk preferences is conveniently highlighted. Proposition 33 Assume that the risk aversion coefficient γT is of the form γT = γ (ST ). Let Q be the measure defined in (72) and let CT = c(YT , ST ) be a claim to be priced under exponential utility with risk aversion coefficient γT . Then, the indifference price of CT is given by    CT 1 . ν (CT ; γT ) = EQ log EQ eγT 1+r |ST γT Proof. In order to construct the indifference price, we need to compute the value functions V CT (x; γT ) and V 0 (x; γT ). We recall that  “ “ ” ” ST CT −γT x+α 1+r −S0 − 1+r CT (x; γT ) = sup EP −e V (86) α

and we introduce the notation

u

β =γ

u



ξu −1 1+r



  ξd . and β = γ 1 − 1+r d

d

(87)

Further calculations yield V CT (x; γT ) = sup φ (α) α

where

  c1 c2 u u u φ (α) = −e−αS0 β e−γ (x− 1+r ) p1 + e−γ (x− 1+r ) p2   c3 c4 d d −αS0 β d e−γ (x− 1+r ) p3 + e−γ (x− 1+r ) p4 −e

with β u and β d given in (87). Differentiating with respect to α yields that the maximum occurs at  u c1  c2 γ 1+r γ u 1+r u −γ u x e β e p + e p 1 2 1 .  d c3 log αCT ,∗ = d c4 S0 (β u + β d ) β d e−γ dx eγ 1+r p + eγ 1+r p 3

34

4

(88)

CT ,∗ S0 β u

CT ,∗ S0 β d

Calculating the terms −e−α

e

−αCT ,∗ S0 β u

=



=



u

β βd

−

βu β u +β d

and −e−α

(

β u γ u −γ d

e

β u +β d

)x

e

c

1 γ u 1+r

e

c

3 γ d 1+r

gives

p1 + e p3 + e

c

2 γ u 1+r c

4 γ d 1+r

p2 p4

!−

βu β u +β d

and e

−αCT ,∗ S0 β d

βu βd

−

βd β u +β d

(

β d γ u −γ d

e

β u +β d

)x

eγ

u c1 1+r

p1 + eγ

u c2 1+r

eγ

d c3 1+r

p3 + eγ

d c4 1+r

p2 p4

!−

βd β u +β d

.

It then follows, after tedious but routine calculations that
V CT (x; γT ) = φ αCT ,∗    u − uβu d  u  uβd d β u γ d +β d γ u β +β β +β β β  e− βu +βd x + = − d β βd

 uβd d  d c3  uβu d  u c1 c2 c4 β +β β +β γ u 1+r γ d 1+r γ 1+r γ 1+r p1 + e p2 p3 + e p4 × e e .

(89)

Substituting CT = 0 in turn implies    u − uβu d  u  uβd d β u γ d +β d γ u β +β β +β β β  e− βu +βd x + V 0 (x; γT ) = −  d β βd βd

βu

× (p1 + p2 ) βu +βd (p3 + p4 ) βu +βd .

(90)

Using (89), (90) and (4) (cf. Definition 1) we get ν(CT ; γT ) =

×

βd eγ log βu + βd

u c1 1+r

p1 + eγ p1 + p2

u c2 1+r



βuγd + βdγu βu + βd

p2

−1

βu eγ + u log β + βd

d c3 1+r

p3 + eγ p3 + p4

d c4 1+r

p4

!

.

(91) We next observe that γ uβd (1 + r) − ξ d = βuγd + βd γu ξu − ξd = and

S0 (1 + r) − S d = q, Su − Sd

γ dβu ξ u − (1 + r) = βuγ d + βdγ u ξu − ξd 35

(92)

=

S u − S0 (1 + r) = 1 − q, Su − Sd

where S u = S0 ξ u and S d = S0 ξ d . The above equalities combined with (91) then yield ν (CT ; γT ) = q

1 eγ log u γ

1 eγ + (1 − q) d log γ

u c1 1+r

d c3 1+r

p1 + e γ p1 + p2

p3 + eγ p3 + p4

u c2 1+r

d c4 1+r

p4

p2

(93)

.

Following the arguments developed in the proof of Proposition 2 we see that eγ and

eγ

u c1 1+r

d c3 1+r

p1 + e γ p1 + p2

p3 + e γ p3 + p4

u c2 1+r

d c4 1+r

p2

  CT = EQ eγT 1+r |ST = S u

p4

  CT = EQ eγT 1+r ST = S d .

(94)

(95)

Finally, using the equalities (94) and (95), and the expression in (93) we easily conclude. Proposition 34 Assume that the risk aversion coefficient γT is of the form γT = γ (ST ) and let δT be the risk tolerance coefficient introduced in (85). Let Q be the measure defined in (72) and let CT = c(YT , ST ) be a claim to be priced under exponential utility with risk aversion coefficient γT . Then, the value functions V 0 (x; γT ) and V CT (x; γT ) , defined in (86), admit the following representations   x 0 ∗ V (x; γT ) = − exp − − H (Q |P ) (96) EQ (δT )   x − ν (CT ; γT ) V CT (x; γT ) = − exp − − H (Q∗ |P ) (97) EQ (δT ) where ν (CT ; γT ) = EQ and



  CT 1 log EQ eγT 1+r |ST γT

δT (ω) dQ∗ (ω) = . dQ EQ (δT )

(98)

(99)

Proof. To show (96) and (97), we work with formulae (89) and (90) interpreting appropriately the involved quantities. We first observe that −1  βuγd + βdγu βu + βd = βu + βd βuγ d + βdγ u 36

= and, that



1 1 (1 − q) + u q d γ γ

−1

=

1 EQ (δT )


−1 γd (1 − q) βu = , u d β +β EQ (δT )

βu γ u (1 − q) = , βd γdq

−1

(γ u ) q βd = . βu + βd EQ (δT )

Combining the above we obtain 0

V (x; γT ) = −e

−E



x Q (δT )

= −e

−E

p1 + p2 q∗

x Q (δT )

q∗ 

−H(Q∗ |P )

p3 + p4 1 − q∗

1−q∗

.

∗

Herein, the measure Q is defined by qi∗ = q ∗

pi , p1 + p2

i = 1, 2 and qi∗ = (1 − q ∗ )

with

pi , p3 + p4

i = 3, 4,

−1

q∗ =

(γ u ) q EQ (δT )

(100)

where q given in (92). Note that the emerging measure Q∗ satisfies EQ∗ (γT (ST − S0 (1 + r))) = 0

(101)

and gives the same conditional distribution of YT given ST as the measure P. Therefore, Q∗ is the measure which has the minimal, relative to P, entropy and satisfies (101). Alternatively one can define Q∗ by its Radon-Nikodym density with respect to the minimal relative entropy martingale measure Q. Namely, −1

q∗ (γ u ) dQ∗ (ωi ) = = , dQ q EQ (δT ) and

i = 1, 2


−1 γd 1 − q∗ dQ∗ (ωi ) = = , dQ 1−q EQ (δT )

i = 3, 4.

To complete the proof, it remains to show (97). For this, it suffices to observe that the terms appearing in (89), containing the payoff, can be written as e

c

1 γ u 1+r

c

2 γ u 1+r

p1 + e p1 + p2

p2

!

βd β u +β d

= exp



e

c

3 γ d 1+r

ν (CT ; γT ) EQ (δT ) 37

c

4 γ d 1+r

p3 + e p3 + p4 

p4

!

βu β u +β d

where we used (98). Formula (97) then follows from the latter equality, and assertions (89) and (96). We stress that both measures, Q and Q∗ , have minimal, relative to P, entropy. However, they have different martingales properties, namely, under Q, EQ (ST − S0 (1 + r)) = 0, while, under Q∗ , EQ∗ (γT (ST − S0 (1 + r))) = 0. We finish this section with an important decomposition result for the writer’s optimal policy. Proposition 35 The writer’s optimal investment policy αCT ,∗ (cf. (88)) admits the following decomposition αCT ,∗ = α0,∗ + α1,∗ + α2,∗ , where α0,∗ = −

∂H (Q∗ |P ) EQ (δT ) , ∂S0

and α2,∗ = EQ (δT )

∂ ∂S0

α1,∗ =



(102)

∂ log EQ (δT ) x ∂S0

ν (CT ; γT ) EQ (δT )



.

Proof. We first recall that

αCT ,∗

 p 2 1 .  d c3 log = d c4 S0 (β u + β d ) β d e−γ dx eγ 1+r p3 + eγ 1+r p4 β u e−γ

u

x



eγ

u c1 1+r

p1 + eγ

u c2 1+r

We then write αCT ,∗ as

αCT ,∗ = α0,∗ + α1,∗ + α2,∗ where α0,∗ =

1 S0 (β u + β d )

log

β u (p1 + p2 ) , β d (p3 + p4 ) u

1,∗

α and α2,∗

1 e−γ x = , log S0 (β u + β d ) e−γ d x

 u c1  u c2 eγ 1+r p1 + eγ 1+r p2 (p3 + p4 )  log  d c3 . = d c4 S0 (β u + β d ) eγ 1+r p3 + eγ 1+r p4 (p1 + p2 ) 1

38

We will next represent the various quantities in terms of the measures Q and Q∗ . Recall that q= and

S0 (1 + r) − S d , Su − Sd

∂q 1+r = u , ∂S0 S − Sd

(γ u )−1 q βd = = q∗ , βu + βd EQ (δT )
−1 γd (1 − q) βu = = 1 − q∗ . βu + βd EQ (δT )

It follows, trivially, that log

(1 − q ∗ ) (p1 + p2 ) β u (p1 + p2 ) = log . d β (p3 + p4 ) q ∗ (p3 + p4 )

Moreover, the relative entropy H (Q∗ |P ) is given by H (Q∗ |P ) = q ∗ log and, hence,

∂H (Q∗ |P ) =− ∂S0

q∗ 1 − q∗ + (1 − q ∗ ) log p1 + p2 p3 + p4



∂q ∗ ∂S0



log

(1 − q ∗ ) (p1 + p2 ) . q ∗ (p3 + p4 )

The sensitivity of q ∗ to S0 can be easily calculated. Indeed, we get
−1 −1 γd ∂q ∗ ∂q (γ u ) . = ∂S0 ∂S0 (EQ (δT ))2
Also, because the coefficient S0 β u + β d can be written as
−1 −1 γd ∂q (γ u ) = , S0 (β u + β d ) ∂S0 EQ (δT ) 1

we get

1 ∂q ∗ EQ (δT ). = S0 (β u + β d ) ∂S0

The above formulae imply that α0,∗ = −

∂H (Q∗ |P ) EQ (δT ). ∂S0

Moving to the second term, α∗,1 , we notice that α∗,1 = −

1 S0

(β u

+

βd)

39


γ u − γ d x.

Moreover, because, ∂ log EQ (δT ) ∂ log EQ (δT ) ∂q = ∂S0 ∂q ∂S0 =

we easily obtain that


−1  ∂q 1 −1 (γ u ) − γ d EQ (δT ) ∂S0
1 =− γu − γd S0 (β u + β d ) α∗,1 =

∂ log EQ (δT ) x. ∂S0

Obviously, the last term, α∗,2 , has to do with the indifference price sensitivity. Indeed, after rather tedious calculations, we obtain    CT ∂q ∗ ∂  γT 1+r ∗ ∗ e log E E |S Q Q T ∂S0 ∂q ∗    CT ∂  EQ∗ log EQ∗ eγT 1+r |ST = EQ (δT ) ∂S0   ∂ ν (CT ; γT ) = EQ (δT ) ∂S0 EQ (δT )

α∗,2 = (EQ (δT ))

and the proof is complete. The above Propositions demonstrate the effects of the random nature of risk aversion on the form of the value function, the indifference price and the optimal policy. The appearance of the expected value of the risk tolerance, the reciprocal of risk aversion, seems to indicate that this quantity, rather then risk aversion, is more natural from the structural and interpretation points of view. Moreover, two interesting new features appear. First of all, the optimal policy αCT ,∗ is no longer independent of the initial wealth. However, the hedging demand α2,∗ , coming from the presence of a derivative contract, remains independent on the initial wealth. So does the shares amount α0,∗ , which is aiming to benefit from the opportunities created by the differences in the probabilities allocated to the outcomes by the historical measure P and by the minimal relative entropy martingale measure Q∗ . The number of shares α1,∗ depends linEQ (δT ) early on the initial wealth x with the slope ∂ log∂S representing the relative 0 sensitivity of the current risk tolerance to the changes in the stock price. The ∂ ν(CT ;γT ) of the option second interesting and new feature is the sensitivity ∂S 0 EQ (δT ) price expressed in a unitless fashion, i.e., relatively to the current risk tolerance. iii) Indifference prices and general numeraires Recall that the wealth XT at time T , given in (69), is expressed in the spot units. Observe that if the stock price is taken as the numeraire, the wealth will be expressed in the number of shares of stock and not in a dollar amount. 40

Specifically, the terminal wealth XTS is given by   S0 x 1 , +α − XTS = ST 1 + r ST and the current wealth, equal to the number of shares at time 0, by X0S =

x = xS . S0

Note that XT is discounted to time 0, and hence XTS is the time 0 equivalent of the number of shares held in the portfolio at time T . The related value function is given by  “ ” CT S
−γ S (ST ) XT − S (1+r) S S,CT T (103) x = sup EP −e V α

where γ S (ST ) represents the risk aversion associated with this unit.

The following Definition is a direct extension of Definition 1. It is important to notice that in the current unit framework, the indifference price is expressed in number of shares and not as a dollar amount. Definition 36 The indifference price of the claim CT is defined as the number of shares ν S (CT ) for which the two value functions V S,CT and V S,0 , defined in (103) and corresponding to claims CT and 0, coincide. Namely, it is the number of stock shares ν S (CT ) satisfying V S,0 (xS ) = V S,CT (xS + ν S (CT ))

(104)

for any initial number of shares xS ∈ R. Proposition 37 Let QS be a measure under which the discounted, by the traded asset, riskless bond, Bt /St , t = 0, T is a martingale and, at the same time, the conditional distribution of the nontraded asset, given the traded one, is preserved with respect to the historical measure P, i.e. QS (YT |ST ) = P(YT |ST ).

(105)

Let CT = c(ST , YT ) be the claim to be priced under exponential preferences with state dependent risk aversion coefficient γ S (ST ). Then, the indifference price of CT , quoted in the number of shares of stock, is    CT 1 γ S (ST ) S (1+r) T . (106) log E S e |S ν S (CT ) = EQS T Q γ S (ST ) Proof. We start with the specification of the measure QS . We recall that, given t the choice of numeraire, the martingale in consideration is BtS = B St , where Bt and St stand, respectively, for the original bond and stock process. We denote 41

by qiS , i = 1, .., 4 the elementary probabilities of QS . Simple calculations yield that qiS = q S

pi pi , i = 1, 2 and qiS = (1 − q S ) , i = 3, 4, p1 + p2 p3 + p4

where qS =



1 1 − ξd 1+r



ξu ξd . − ξd

ξu

Alternatively, the measure QS can be defined by its Radon-Nikodym density with respect to the measure Q, namely dQS ST . = dQ (1 + r) S0 Indeed, we have EQS



1+r ST



= EQ



ST 1+r (1 + r) S0 ST



=

1 . S0

We next observe that the value function V S,CT (cf. (103)) can be written as  “ “ ” ” ST CT
−λ x+α 1+r −S0 − 1+r , V S,CT xS = sup EP −e T α

where

λT =

γ S (ST ) . ST

Working as in the proof of Proposition 35 we get
V S,CT xS = V CT (x; λT )

where

!   x − ν (CT ; λT ) e |P
−H Q = − exp − EQ λ−1 T e dQ λ−1 T
. = dQ EQ λ−1 T

The wealth argument x in V CT (x; λT ) as well as the price    CT 1 λT 1+r |ST ν (CT ; λT ) = EQ log EQ e λT are expressed in the spot units and, hence, for all claims CT we have ν (CT ; γT ) = ν (CT ; λT ) .

42

(107)

Considering as in Proposition 5 claims of the form CT1 (ω1 ) = c1 , CT1 (ωi ) = 0, i = 2, 3, 4 and CT2 (ω3 ) = c3 , CT2 (ωi ) = 0, i = 1, 2, 4 we get that γT = λT . Consequently, the risk aversion γ S (ST ) associated with the numeraire S must satisfy γ S (ST ) = γT ST . We recall, however, that for any payoff, say G, dependent only on ST     G (ST ) G (ST ) EQ = S0 EQS . 1+r ST Therefore,



  CT 1 log EQ eγT 1+r |ST γT    CT 1 γT 1+r log EQ e = (1 + r) S0 EQS |ST γT ST    CT 1 γ S (ST ) S (1+r) T . log E S e |S = (1 + r) S0 EQS T Q γ S (ST ) ν (CT ; γT ) = EQ

The statement then follows because the quantity (cf. (98))    CT ν (CT ; γT ) 1 γ S (ST ) S (1+r) T = EQS |ST log EQS e (1 + r) S0 γ S (ST ) is the indifference price quoted in the equivalent number of shares.

2.6

Functions of value and utility

The analysis on numeraires in the previous section exposed the important fact that the arguments of the functions of value and utility can be arbitrary as long as we are careful with the units in which the relevant quantities are expressed. Specifically, we saw that in order to refer to the same value and utility, independently on the arguments, and, also, in order to eliminate static arbitrage opportunities from our model, one has to make sure that the risk aversion multiplied by wealth represents the same quantity independently on the wealth units. This produced necessary and sufficient conditions on risk preferences in spot and forward units, and for the case of general numeraires. In other words, natural properties of indifference prices imposed a certain structure on preferences and, in turn, on the involved utility and value functions. As one moves to the multi-period set up and to more complex dynamic payoffs, additional price considerations will impose further structural properties on the supporting utilities and value functions. We will then see that the concept of utility, as a static expression of risk attitude, needs to be put in the correct dynamic context in order to ultimately generate consistent prices across times, units etc. 43

Even though the single period framework impedes us from exposing all such issues, we, nevertheless, provide below some motivational observations. For convenience, we fix the benchmark risk aversion parameter γT = γ (ST ) , as representing our aversion to risk associated with wealth expressed in the spot units, i.e., discounted to the current time. Whenever convenient, we also use the corresponding risk tolerance δT = γT−1 (cf. (85)). From the previous analysis, the utility and value function in the single period Merton problem can be written as U (x; δT ) = −e and V 0 (x; δT ) = −e where

−E

− δx

x Q (δT )

T

−H(Q∗ |P )

,

(108)

δT dQ∗ = , dQ EQ (δT )

and Q is a martingale measure with the minimal, relatively to P, entropy. We recall that indifference prices are built via the values of the relevant investment opportunities, and these values are generated by the preference structure and the assumptions on the market. In the next section, we will see that the cornerstone for the construction of a consistent pricing system is the correct specification of dynamic preferences and the understanding of the interplay between utilities and value functions. To gain some intuition, let us simply look at how zero wealth is being valued at the beginning and at the end of the trading horizon. Setting x = 0 in (108) yields ∗ V 0 (0; δT ) = −e−H(Q |P ) . This is the value of zero wealth at the beginning of the period, as measured by the value function. Hence, it depends on the entropy term H (Q∗ |P ) and thus on the model. On the other hand, the utility of zero wealth at the end of the time period T , as measured by the utility function, equals UT (0; δT ) = −1. We then say that the utility value of zero wealth is −1. An investor may prefer to associate 0, rather than −1, utility value with zero wealth. This requirement is easily met by adding the normalizing constant 1 to UT (x; δT ). We choose not to do it because this does not change anything in our analysis and lengthens many expressions. Alternatively, the investor may also want to associate −1 to the value of zero wealth at the beginning of the period, thus making it independent on the model. This can be easily achieved by normalizing the utility function (108) accordingly. Indeed, let x

∗

¯ (x; δT ) = −e− δT +H(Q U 44

|P )

(109)

represent the utility of wealth x at time T. Obviously, at the beginning of the period, the associated value function V¯ 0 (x; δT ) satisfies x

∗ − V¯ 0 (x; δT ) = eH(Q |P ) V 0 (x; δT ) = −e EQ (δT ) .

(110)

The corresponding values of zero wealth then become ¯ (0; δT ) = −eH(Q∗ |P ) U and

∗ V¯ 0 (0; δT ) = eH(Q |P ) V00 (0; δT ) = −1.

The above considerations might look pedantic in the single period setting. However, in the multi-period case, one needs to reconcile single period and multi period concepts of the value and utility functions. Indeed, when dealing concurrently with investment problems over multiple investments horizons, one needs to identify the utility function at a given horizon with the value function obtained by solving the optimal investment problem over the next time period. Together, the functions of utility and value lead to the natural concept of a dynamic utility or of a term structure of utilities. Such a utility is a function of wealth, of the investment horizon and of the market model. To prepare the reader for the upcoming new notions of backward and forward dynamic utilities, we revert to the notation we will be using in the dynamic setting. In the above simple case, the dynamic utility for the beginning and the end of period, as given by the functions U F (x, 0) = −e

−

x EQ (δT )

,

U F (x, T ) = −e

− δx +H(Q∗ |P ) T

(111)

is normalized at the beginning of the period. We call it the forward utility. On the other hand, the dynamic utility, given by U B (x, 0, T ) = −e

−E

x Q (δT )

−H(Q∗ |P )

,

U B (x, T, T ) = −e

− δx

T

,

(112)

is normalized at the end of the time period. We call it the backward utility.

3

Multi-period model

We consider the multi-period extension of the single period binomial incomplete market model that was analyzed in Section 2.1. As before, there are two securities available for trading, a riskless bond and a risky stock. A third asset is also present but is not traded. We are interested in pricing European type contingent claims that are written on both, traded and non-traded assets. The first aim is to show that all fundamental pricing features of the single period case have their natural counterparts in this multi-period framework. In essence, we will see that they are local in time and can be, thus, analyzed by arguments used in the one period case. The second aim is to expose the dynamic over time valuation features that are absent in the static case. They refer, among 45

others, to the dynamic nature of risk preferences and the semigroup property of the proposed pricing schemes. The claim in consideration is taken to yield payoff CT at expiration T . We aim at representing its indifference price, at time t < T , in a natural valuation format, namely, νt (CT ) = E (t,T ) (CT ) (113) where E (t,T ) stands for the appropriate dynamic pricing operator. A fundamental property that the operator E (t,T ) should have is the semigroup property, νt (CT ) = E (t,s) (E (s,T ) (CT )) = E (t,s) (νs (CT ))

for t ≤ s ≤ T.

(114)

This states that the price νt (CT ) can be computed in two steps in reference to an intermediate time t ≤ s ≤ T . Namely, one may first determine the indifference price at time s, νs (CT ) = E (s,T ) (CT ), by applying (113), and, then, use νs (CT ) as a new claim to be priced at initial time t. Clearly, this property states that the current price can serve as a new payoff which is, in turn, priced, moving backwards in time, via the time-adjusted price operator. If property (114) fails, it is easy to see that arbitrage opportunities may arise. We recall that, in the single period case (r = 0), the indifference price was represented in terms of the nonlinear valuation functional 1 ν(CT ) = EQ (CT ) = EQ ( log EQ (eγCT |ST )) γ

(115)

(cf. (9)). The indifference pricing measure Q turned out to be the unique martingale measure that satisfies Q(YT |ST ) = P(YT |ST ).

(116)

Given the inherent non-linearities of the indifference pricing mechanism, we stress that, a priori, it is not at all obvious what are the multi-period analogues of (115) and (116), and why the semigroup price property (114) must follow. We will see that there is a delicate interplay among optimality of policies, semigroup properties of the involved value functions, the nature of the pricing measure and the structure of the pricing schemes that ultimately yields the desired pricing ingredients. An important issue that emerges once the market environment becomes dynamic is the proper specification of preferences across times and investment opportunities. This leads us to the development of dynamic utilities. Two kinds of such utilities are considered, namely, the backward and the forward ones. With reference to these dynamic risk preferences, we introduce the backward and forward indifference prices and construct their associated pricing schemes. While the backward utility and forward price are, in many aspects, similar to the traditional value function and classical indifference price, the concepts of forward utility and forward indifference price are, to the best of our knowledge, entirely new. 46

3.1

Preliminaries

We start with the probabilistic set-up of the multi-period model. We denote by St , t = 0, 1, ... the values of the traded risky asset. It is assumed that St > 0, and we define the variables ξt as ξt+1 =

St+1 d u , ξt+1 = ξt+1 , ξt+1 St

d u with 0 < ξt+1 < 1 < ξt+1 .

The second traded asset is riskless and yields zero interest rate. We also denote by Yt , t = 0, 1, .., the values of the non-traded asset and we assume that, for all t = 0, 1, ..T, Yt 6= 0. We introduce the variables ηt satisfying ηt+1 =

Yt+1 d u , ηt+1 = ηt+1 , ηt+1 Yt

d u with ηt+1 < ηt+1 .

We then view {St , Yt : t = 0, 1, ...} as a two-dimensional stochastic process defined on the probability space (Ω, F , (Ft ) , P). The filtration Ft is generated by the random variables Ss and Ys , or, equivalently, by ξs and ηs , for s = 0, 1, ..., t. We also consider the filtrations FtS an FtY generated, respectively, by Ss and Ys for s = 0, 1, ..., t. The real (historical) probability measure on Ω and F is denoted by P. Using notation compatible with the single period case, we take Xt , t = 0, 1, ... to represent the wealth process associated with a multi-period self-financing portfolio. We denote by αt , t = 0, 1, ... the number of shares of the traded asset held in this portfolio over the time period [t − 1, t). Then, △Xt = Xt − Xt−1 = αt △ St and, hence, for t = 0, 1, .., T − 1, XT = Xt +

T X

αs △ Ss .

s=t+1

Consider the optimal investment problem over the trading horizon [t, T ] and assume, for the moment, that the investor has exponential risk preferences of constant absolute risk aversion γ > 0. This assumption will be removed when the dynamic, backward and forward, risk preferences are introduced. The investor aims at maximizing the expected utility of terminal wealth XT . For t = 0, 1, .., T − 1, the associated value function, corresponding to wealth x at time t, is then defined by
(117) V (x, t) = sup EP −e−γXT |Ft αt+1 ,...,αT

=

sup αt+1 ,...,αT

  PT EP −e−γ (x+ s=t+1 αs △Ss ) |Ft .

47

This is the classical value function, studied, among others, by Rouge and El Karoui (2000), Delbaen et al. (2002) (see, also, Kabanov and Stricker (2002)) who established that, for t = 0, 1, .., T − 1, V (x, t) = −e−γx−HT (Q(·|Ft )| P(·|Ft ) ) . Herein, Q is a martingale measure, defined on FT , that has the minimal, relative to P (restricted to FT ) entropy. The quantity HT (Q (· |Ft ) | P (· |Ft ) ) denotes the aggregate relative entropy of the conditional on Ft measures P (· |Ft ) and Q (· |Ft ), defined on FT , namely,   Q (· |Ft ) HT (Q (· |Ft ) | P (· |Ft ) ) = EQ log |Ft . P (· |Ft ) The relative entropy will play an important role in the pricing and specification of dynamic risk preferences. The following result provides an analytic expression for it. Proposition 38 Let Q be an equivalent to P martingale measure, defined on FT and having the minimal, relative to P, entropy. For all t = 0, 1, .., T − 1, let Q (· |Ft ) and P (· |Ft ) be defined on FT and standing for the conditional, on Ft , measures. Then, their relative entropy, HT (Q (· |Ft ) | P (· |Ft ) ), is given by ! T X (118) HT (Q (· |Ft ) | P (· |Ft ) ) = EQ hs |Ft , s=t+1

where, for s = t + 1, ..., T, hs = Hs (Q (· |Fs−1 ) |P (· |Fs−1 ) ) = qs log

1 − qs qs + (1 − qs ) log ≥ 0, P (As |Fs−1 ) 1 − P (As |Fs−1 )

(119)

As = {ξs = ξsu } and qs = Q (As |Fs−1 ) =

1 − ξsd . ξsu − ξsd

Proof. Let Q be an equivalent to P martingale measure on FT . Recall that, for any Q, the conditional on Ft measure Q (· |Ft ) is defined by Q (A |Ft ) = EQ (IA |Ft ) ,

A ∈ FT .

Recall also that the, relative to P( · |Ft ), entropy of Q(· |Ft ) is given by HT (Q (· |Ft ) |P (· |Ft ) ) 48

log

= EQ

=

T −1 X s=t

where we used that

QT −1

s=t Q (ξs+1 , ηs+1 |Fs ) QT −1 s=t P (ξs+1 , ηs+1 |Fs )

|Ft

!

  Q (ξs+1 , ηs+1 |Fs ) EQ log |Ft , P (ξs+1 , ηs+1 |Fs )

P(ξt+1 , ..., ξT , ηt+1 , ..., ηT |Ft ) =

TY −1

P (ξs+1 , ηs+1 |Fs )

s=t

and Q(ξt+1 , ..., ξT , ηt+1 , ..., ηT |Ft ) =

TY −1

Q (ξs+1 , ηs+1 |Fs ) .

s=t

To calculate the minimal relative entropy, it suffices to minimize the individual terms of the above sum. To this end, for s = t + 1, .., T , we look at the elementary events As = {ξs = ξsu } ,

Bs = {ηs = ηsu }

(120)

and we observe that,   Q (ξs , ηs |Fs−1 ) |Ft EQ log P (ξs , ηs |Fs−1 )

  Q (As Bs |Fs−1 ) Q (As Bsc |Fs−1 ) = EQ log IA B + log IA B c |Ft P (As Bs |Fs−1 ) s s P (As Bsc |Fs−1 ) s s   Q (Acs Bsc |Fs−1 ) Q (Acs Bs |Fs−1 ) c c c . I I + log |F +EQ log A B A B t P (Acs Bs |Fs−1 ) s s P (Acs Bsc |Fs−1 ) s s

In turn,    Q (ξs , ηs |Fs−1 ) Q (As Bs |Fs−1 ) EQ log |Ft = EQ Q (As Bs |Fs−1 ) log P (ξs , ηs |Fs−1 ) P (As Bs |Fs−1 ) + (Q (As |Fs−1 ) − Q (As Bs |Fs−1 )) log

Q (As |Fs−1 ) − Q (As Bs |Fs−1 ) |Ft P (As |Fs−1 ) − P (As Bs |Fs−1 )



 Q (Acs |Fs−1 ) − Q (Acs Bsc |Fs−1 ) +EQ (Q (Acs |Fs−1 ) − Q (Acs Bsc |Fs−1 )) log P (Acs |Fs−1 ) − P (Acs Bsc |Fs−1 )  Q (Acs Bsc |Fs−1 ) , |F +Q (Acs Bsc |Fs−1 ) log t P (Acs Bsc |Fs−1 )

where Acs and Bsc stand for the complements of the events As and Bs . Recall that, for any martingale measure, Q (As |Fs−1 ) = qs = 49

1 − ξsd . ξsu − ξsd

Therefore, one needs to minimize the above two conditional expectations only with respect to the probabilities Q (As Bs |Fs−1 ) and Q (Acs Bsc |Fs−1 ). Note however, that this minimization problem is analogous to the one already solved in the single period case. Straightforward, albeit tedious, arguments then yield that the martingale measure Q at which the minimum occurs, is unique and needs to satisfy, for s = t + 1, ..., T, Q (As Bs |Fs−1 ) P (As Bs |Fs−1 ) = Q (As |Fs−1 ) P (As |Fs−1 )

(121)

and

P (Acs Bsc |Fs−1 ) Q (Acs Bsc |Fs−1 ) = . (122) Q (Acs |Fs−1 ) P (Acs |Fs−1 ) Consequently, for any martingale measure Q,     Q (ξs , ηs |Fs−1 ) Q (ξs , ηs |Fs−1 ) EQ log |Ft ≥ EQ log |Ft P (ξs , ηs |Fs−1 ) P (ξs , ηs |Fs−1 )  P (As Bs |Fs−1 ) qs = EQ qs log P (As |Fs−1 ) P (As |Fs−1 )    qs P (As Bs |Fs−1 ) |Ft log +qs 1 − P (As |Fs−1 ) P (As |Fs−1 )    P (Acs Bsc |Fs−1 ) 1 − qs log +EQ (1 − qs ) 1 − c P (As |Fs−1 ) 1 − P (As |Fs−1 )  c c 1 − qs P (As Bs |Fs−1 ) log |F + (1 − qs ) t P (Acs |Fs−1 ) 1 − P (As |Fs−1 )   qs 1 − qs = EQ qs log + (1 − qs ) log |Ft . P (As |Fs−1 ) 1 − P (As |Fs−1 ) Finally, observe that the function h (q) = q log

satisfies h′ (q) = log

q 1−q + (1 − q) log , 0 < q < 1, p 1−p

q 1−q − log p 1−p

and h′′ (q) =

1 1 + . q 1−q

Therefore h (q) ≥ h (p) = 0 and the result follows. In the sequel, we extend the martingale measure Q to the σ-algebra F . Consequently, both measures, P and Q are defined on F , and the relevant entropy terms of their restrictions to the σ-algebra FT are given by (118) and (119). Note that, with a slight abuse of notation, we use P and Q to denote the measures of F as well as their restrictions to FT . We observe that (121) and (122) can be written in the more compact form (123), appearing below. This, together with the above Proposition, leads us to the following result. 50

Proposition 39 Let Q be a martingale measure that satisfies

S S , = P ηt+1 Ft ∨ Ft+1 Q ηt+1 Ft ∨ Ft+1

(123)

for t = 0, 1.., T − 1. Then Q is unique and it has the minimal, relative to P , entropy.

3.2

Backward and forward dynamic utilities

The reader is referred back to the Introduction for a general motivation behind the concept of dynamic risk preferences and to Section 2.6 for a preliminary discussion on backward and forward2 dynamic utilities. Before we move to the multi-period setting, we illustrate the main ideas using a simple two-period example. We then give general definitions of the dynamic backward and forward utilities and analyze their properties to the specific multi-period setting. Let us consider an investor with an initial capital x who chooses to optimize his wealth over the investment horizon T = 2, using the exponential utility U (x; 2) = −e−γx , x ∈ R and γ > 0. The earlier results show that the associated value function, given in (117) and now denoted by V (x, t; 2), satisfies, for t = 0, 1, V (x, 0; 2) = −e−γx−EQ( h1 +h2 |F0 ) and V (x, 1; 2) = −e−γx−EQ( h2 |F1 ) . Herein hi , i = 1, 2, is given in (119), and F1 and F0 represent, respectively, the σ-algebras generated by both risks, at t = 1 and t = 0. We carry the notation with regards to the trivial σ-algebra F0 for consistency. Alternatively, the investor can also choose to optimize his wealth only over the first time period, that is to choose as investment horizon T = 1. If he chooses again the exponential utility at this new horizon, i.e. if U (x; 1) = −e−γx, the associated value function becomes V (x, 0; 1) = −e−γx−EQ( h1 |F0 ) . Clearly, for all wealth levels x, we have that V (x, 0; 2) ≥ V (x, 0; 1) and that V (x, 0; 2) > V (x, 0; 1) , if and only if P (A2 |F1 ) 6= q2 . Consequently, the investor gets more value (in terms of the value function) from the longer term investment, unless the historical and the risk neutral measures coincide. This strongly suggests that he should never choose to optimize his wealth solely during the first period. Note though that, in general, the investor would face the problem of choosing his investment horizon. Moreover, once she has made her choice, she will stay with it regardless of any additional information about the market revealed in upcoming times. Alternatively, the investor may want to become indifferent to 2 With a slight abuse, the ‘forward’ terminology is being used both for the corresponding dynamic utilities and the relevant wealth units. These two concepts are entirely different since the former corresponds to the evolution of the utility level across time while the latter in the unit choice.

51

the investment horizon. To achieve this goal, she may be prepared to modify her utility functions for T = 1, or for T = 2, or, in general, for both investment horizons. In what follows, we will see that such an approach is in accordance with the desire to formulate individual preferences in relation to the market dynamics, rather than in complete isolation from them. To gain some intuition, let us see if, indeed, for an appropriate choice of preferences, the investor’s value function becomes invariant with regards to the investment horizon. For this, let us fix the future terminal time T = 2 and consider the backward dynamic utility, U B (x, t; 2), t = 0, 1, 2, defined by  t=0  −e−γx−EQ( h1 +h2 |F0 ) B U (x, t; 2) = −e−γx−EQ( h2 |F1 ) t=1  −e−γx t = 2.

We adopt the terminology that U B (x, t; 2) is normalized at T = 2, in that U (x, 2; 2) = −e−γx. We next introduce the associated backward dynamic value function V B (x, 0, t; 2) by
for t = 1, 2, V B (x, 0, t; 2) = sup EP U B (Xt , t; 2) |F0 B

α1 ,αt

Pt with X0 = x and Xt = x + s=1 αs △ Ss . We are going to calculate the values V B (x, 0, t; 2) for t = 0, 1, 2, corresponding to the different choices of trading horizon [0, t]. For t = 0, it is, naturally, taken that V B (x, 0, 0; 2) = −e−γx−EQ( h1 +h2 |F0 ) . For the trading horizon [0, 1], we have V B (x, 0, 1; 2) = sup EP U B (X1 , 1; 2) |F0 α1



= sup EP −e−γ(x+α1 ∆S1 )−EQ ( h2 |F1 ) |F0 α1



= sup EP e α1

−γ(x+α1 ∆S1 )

sup EP −e

−γα2 ∆S2

α2

Above we used that









|F1 |F0


−e−EQ ( h2 |F1 ) = sup EP −e−γα2 ∆S2 |F1 ,



.

α2

which follows from a direct adaptation of the arguments used in the single period case. Therefore, the value of V B (x, 0, 1; 2) turns out to be   V B (x, 0, 1; 2) = sup EP −e−γ(x+α1 ∆S1 +α2 ∆S2 ) |F0 α1 ,α2

= −e−γx−EQ ( h1 +h2 |F0 ) . 52

For the trading horizon [0, 2] we, in turn, obtain V B (x, 0, 2; 2) = sup EP U B (X2 , 2; 2) |F0 α1 ,α2




 
= sup EP e−γ(x+α1 ∆S1 ) sup E −eγα2 ∆S2 |F1 |F0 α2

α1

= −e−γx−EQ (h1 +h2 |F0 ) .

The above calculations demonstrate that under the dynamic utility U B , the associated dynamic value function (maximal expected dynamic utility) V B remains invariant with respect to the investment horizon, in that V B (x, 0, 0; 2) = V B (x, 0, 1; 2) = V B (x, 0, 2; 2)

(124)

= −e−γx−EQ(h1 +h2 |F0 ) for x ∈ R. Note that the term V B (x, 0, 1; 2) represents, under the above risk preferences, the value the investor gets from investing optimally over the first period (T = 1), while the term V B (x, 0, 2; 2) yields the value he gets from investing optimally over two periods (T = 2). The above concept of utility, normalized at some future time, resembles in spirit the concept of a forward rate. Indeed, the dynamic backward utility converges at the normalization point to the classical exponential utility, in much the same way as the forward rate converges to the spot rate. Additionally, in analogy to the notions of the savings account and of the associated spot rate, we also introduce the concept of dynamic utility which is normalized at initial time, taken for simplicity to be 0, and which makes the investor indifferent to the choice of any future investment horizon. To provide some motivation for the upcoming notion of forward dynamic utility, we look at the following example. We fix the current time at 0, and we define the utility process U F (x, t; 0) , for t = 0, 1, 2, by  −e−γx t=0  F −e−γx+h1 t=1 U (x, t; 0) = 2  t = 2, −e−γx+Σi=1 hi with hi , i = 1, 2 as in (119). We observe that U F (x, t; 0) is normalized at t = 0, since U F (x, 0; 0) = −e−γx for x ∈ R. We next define the associated forward dynamic value function by
for t = 1, 2, V F (x, 0, t; 0) = sup EP U F (Xt , t; 0) |F0 α1 ,αt

P with X0 = x and Xt = x + ti=1 αi △ Si . We are going to calculate the values V F (x, 0, t; 0) for t = 0, 1, 2, corresponding to the different choices of trading horizon [0, t]. For t = 0, we naturally have V F (x, 0, 0; 0) = −e−γx 53

for x ∈ R, which yields that V F is itself normalized at t = 0. For the trading horizon [0, 1], we obtain V F (x, 0, 1; 0) = sup EP U F (X1 , 1; 0) |F0 α1

= sup EP −e−γX1 +h1 |F0 α1







  = sup EP −e−γ(x+α1 △S1 )+h1 |F0 α1


= e−γx sup EP −e−α1 △S1 +h1 |F0 = −e−γx. α1

Finally, for the trading horizon [0, 2], we have

V F (x, 0, 2; 0) = sup EP U F (X2 , 2; 0) |F0 α1 ,α2

= sup EP α1, α2






   EP −e−γX2 +(h1 +h2 ) |F1 |F0

    −γX2 +(h1 +h2 ) |F1 |F0 = sup EP sup EP −e α1

α2

 
−γα2 △S2 +h2 −γX1 +h1 |F1 |F0 sup EP −e = sup EP e α2

α1

= sup EP −e−γX1 +h1 |F0 α1




= sup EP U F (X1 , 1; 0) |F0 α1

= V F (x, 0, 1; 0) = −e−γx .




Therefore, the above choice of forward dynamic preferences yields, for t = 0, 1, 2 and x ∈ R, V F (x, 0, 0; 0) = V F (x, 0, 1; 0) (125) = V F (x, 0, 2; 0) = −e−γx , which, in turn, implies that the forward dynamic value function is invariant with regards to the investment horizon. We are now ready to introduce the above concepts in the general multi-period setting. We start with an auxiliary definition. For convenience we recall the local entropy terms, (cf. (119)), hi = qi log

qi 1 − qi + (1 − qi ) log P (Ai |Fi−1 ) 1 − P (Ai |Fi−1 )

with Ai = {ξi = ξiu }

and qi = Q (Ai |Fi−1 ) = 54

1 − ξid ξiu − ξid

for i = 0, 1, .., T. We also recall the form of the aggregate entropy, (cf. (118)), ! T X hi |Ft HT (Q (· |Ft ) | P (· |Ft ) ) = EQ i=t+1

and we remind the reader that hi = Hi (Q (· |Fi−1 ) |P (· |Fi−1 ) ) . We continue with the introduction of the backward and forward dynamic utility functions.  Definition 40 For T = 1, 2.., the process U B (x, t; T ) : t = 0, 1, ..., T defined, for x ∈ R, by  −e−γx if t=T  B U (x, t; T ) = (126)  −e−γx−HT (Q(·|Ft )| P(·|Ft ) ) if 0 ≤ t ≤ T − 1,

is called the backward, normalized at time T , dynamic exponential utility. The aggregate entropy term, HT (Q (· |Ft ) | P (· |Ft ) ) , is given above.  Definition 41 For s = 0, 1, .., the process U F (x, t; s) : t = s, s + 1, ... defined, for x ∈ R, by  −e−γx if t=s  F U (x, t; s) = (127) Pt  −e−γx+ i=s+1 hi if t ≥ s + 1

is called the forward, normalized at time s, dynamic exponential utility. The local entropy terms, hi , are given above.

The following measurability results follow directly from the above Definitions and the properties of the local entropy terms hi , i ≥ 0. Proposition 42 Let U B be the backward dynamic exponential utility process with normalization point T . Then, for 0 ≤ t ≤ T , and x ∈ R, U B (x, t; T ) ∈ Ft . Proposition 43 Let U F be the forward dynamic exponential utility process with normalization point s ≥ 0. Then, for t ≥ s, and x ∈ R, U F (x, t; s) ∈ Ft−1 . It is important to notice that the investor incorporates into the above dynamic utilities information about both the historical and the risk neutral representations of the model. However, the historical assessment and the risk neutral view are assimilated in the backward and forward preferences in a completely different manner.

55

In the backward utility U B (x, t; T ) this information is given by the aggregate entropy term HT (Q (· |Ft ) | P (· |Ft ) ) which represents the distance between Q (· |Ft ) and P (· |Ft ) , restricted to FT . However, the forward utility U F (x, t; s) captures information about the model along the path and incrementally, through the local in time entropy terms hi = Hi (Q (· |Fi−1 ) |P (· |Fi−1 ) ) , i = s + 1, ..., T. Indeed, over a single period (i − 1, i) the restrictions to Fi of the measures Q (· |Fi−1 ) and P (· |Fi−1 ) correspond to the conditional on Fi−1 distributions of (ξi , ηi ) under Q and P. In fact, it follows from (119), that hi is the relative entropy of the conditional on Fi−1 distributions of ξi under the measures Q and P. Therefore, only information about the transition densities of the risky traded asset is used. It is important to notice that the transition densities are computed with respect to the filtration F and not with respect to F S . The next two results provide a relation between the dynamic backward and forward utilities for different normalization points. Their proofs follow directly from Definitions 40 and 41 and are, therefore, omitted. Corollary 44 Let T1 < T2 and 0 ≤ t ≤ T1 . Consider the backward dynamic exponential utilities U B (x, t; T1 ) and U B (x, t; T2 ), normalized, respectively, at T1 and T2 . Then, for α ∈ R, U B (x, t; T2 ) = −U B ((1 − α) x, t; T1 ) U B (αx, T1 ; T2 ) , for all wealth levels x ∈ R. Corollary 45 Let 0 ≤ s1 < s2 and s2 ≤ t. Consider the forward dynamic exponential utilities U F (x, t; s1 ) and U F (x, t; s2 ) normalized, respectively, at s1 and s2 . Then, for α ∈ R, U F (x, t; s1 ) = −U F ((1 − α) x, t; s2 ) U F (αx, s2 ; s1 ) for all wealth levels x ∈ R. Next, we introduce the associated backward and forward dynamic value functions. For simplicity we skip the ”exponential” nomenclature. Definition 46 Let U B be the backward dynamic utility process defined in (126), normalized at T . The associated backward dynamic value function V B is defined, for x ∈ R, s = 0, 1, .. and s ≤ t ≤ T , as
(128) V B (x, s, t; T ) = sup EP U B (Xt , t; T ) |Fs αs+1 ,..,αt

with Xt = Xs +

Pt

i=s+1

αi △ Si and Xs = x.

Note that four arguments enter in the representation of the backward dynamic value function, namely, the normalization point T , the time arguments s and t ≤ T , denoting the trading horizon [s, t], and the initial wealth Xs = x. If t = T , the above definition coincides with the one yielding the classical solution of the Merton’s optimal investment problem in the absence of intermediate consumption (Merton (1969)). 56

Definition 47 Let U F be the forward dynamic utility process defined in (127), normalized at s, for s = 0, 1, ... The associated forward dynamic value function V F is defined, for x ∈ R, T = s + 1, ... and s ≤ t ≤ T as
(129) V F (x, t, T ; s) = sup EP U F (XT , T ; s) |Ft αt+1 ,..,αT

with XT = Xt +

PT

i=t+1

αi △ Si and Xt = x.

In resemblance to its backward counterpart, the forward dynamic value function has four arguments, namely, the level of initial wealth x, the time arguments t and T , associated with the length of the trading horizon [t, T ], and the forward normalization point s. Note that the end of the horizon point T is arbitrary while the forward normalization point s is fixed. The backward and forward dynamic value functions have the following fundamental properties. Proposition 48 The backward dynamic value function V B , associated with the normalized at T backward dynamic utility U B is itself normalized at T , i.e., V B (x, T, T ; T ) = −e−γx

for x ∈ R.

(130)

It is also invariant with respect to the investment horizon, in that V B (x, s, t1 ; T ) = V B (x, s, t2 ; T )

for t1 6= t2

(131)

and s ≤ min (t1 , t2 ) and t1 , t2 ≤ T . Moreover, for s ≤ t ≤ T and x ∈ R, V B (x, s, t; T ) is given by V B (x, s, t; T ) = V B (x, s, s; T ) = −e−γx−EQ(Σi=s+1 hi |Fs ) T

(132)

for x ∈ R,

with the entropic terms hi as in (119). Proof. The fact that V B (x, T, T ; T ) = −e−γx is a direct consequence of Definition 46 and the normalization properties of U B . To establish (131), we consider the solution of the traditional Merton’s optimal investment problem (see Merton (1971)), considered in (117) and rewritten below for convenience,
V (x, s) = sup EP −e−γXT |Fs for 0 ≤ s ≤ T, αs+1 ,...,αT

PT with XT = Xs + i=s+1 αi △ Si and Xs = x. The Dynamic Programming Principle will then yield, for t1 , t2 ≥ s,

V (x, s) = sup EP V Xtj , tj |Fs for j = 1, 2. αs+1 ,...,αtj

57

We recall (see Delbaen et. al. (2002)) that


V Xtj , tj = −e

“ −γXtj −EQ ΣT i=t

j +1

hi |Ftj

”

for

j = 1, 2.

(133)

Note, however, that Definition 40 implies

V Xtj , tj = U B Xtj , tj ; T .

Therefore,

V B (x, s, t1 ; T ) = =

sup αs+1 ,...,αt1

sup αs+1 ,...,αt1

EP U B (Xt1 , t1 ; T ) |Fs




EP (V (Xt1 , t1 ) |Fs ) = V (x, s).

On the other hand V (x, s) =

sup αs+1 ,...,αt2

=

sup αs+1 ,...,αt2

and the equality

EP (V (Xt2 , t2 ) |Fs )


EP U B (Xt2 , t2 ; T ) |Fs = V B (x, s, t2 ; T ) ,

V B (x, s, t1 ; T ) = V B (x, s, t2 ; T ) follows. To show assertion (132), we apply (131) at the above points t1 = s and t2 = t, obtaining V B (x, s, s; T ) = V B (x, s, t; T ) = V (x, s) = −e−γx−EQ(Σi=s+1 hi |Fs ) . T

Proposition 49 The forward dynamic value function V F , associated with the, normalized at s, forward dynamic utility U F is itself normalized at s, i.e., V F (x, s, s; s) = −e−γx

for x ∈ R.

(134)

It is invariant with respect to the investment horizon, namely, V F (x, t, T1 ; s) = V F (x, t, T2 ; s)

for T1 6= T2

(135)

and s ≤ t ≤min(T1 , T2 ). Moreover, for s ≤ t ≤ T , t

V F (x, t, T ; s) = V F (x, t, t; s) = −e−γx+Σi=s+1hi

for x ∈ R.

(136)

Proof. The fact that V F is normalized at time s is a direct consequence of Definition 40 and the normalization properties of U F . To show (135), it suffices to show V F (x, t, T ; s) = V F (x, t, T + 1; s) 58

since the rest of the proof follows by direct induction arguments. To this end, we recall
EP U F (XT +1 , T + 1; s) |Ft V F (x, t, T + 1; s) = sup αt+1 ,..,αT +1

=

sup αt+1 ,..,αT +1

=

sup αt+1 ,..,αT +1

=

sup αt+1 ,..,αT

  T +1 EP −e−γXT +1 +Σi=s+1 hi |Ft

 
T EP e−γXT +Σi=s+1 hi EP −e−γαT +1 ∆ST +1 +hT +1 |FT |Ft

 
T EP e−γXT +Σi=s+1 hi sup EP −e−γαT +1 ∆ST +1 +hT +1 |FT |Ft . αT +1

Using the measurability properties of hT +1 and that
sup EP −e−γαT +1 ∆ST +1 |FT = −e−hT +1 αT +1

yields

V F (x, t, T + 1; s) =

sup αt+1 ,..,αT

=


EP U (XT , T ; s) |Fs = V F (x, t, T ; s) F

sup αt+1 ,..,αT

  T EP e−γXT +Σi=s+1 hi |Ft

and the claim follows. To show (136), it suffices to establish it for T = t + 1, namely, t

V F (x, t, t + 1; s) = −e−γx+Σi=s+1hi . This follows easily since   t+1 V F (x, t, t + 1; s) = sup EP e−γ(x+αt+1 ∆St+1 )+Σi=s+1 hi |Ft αt+1

t

= e−γx+Σi=s+1 hi sup EP −e−γαt+1 ∆St+1 +ht+1 |Ft αt+1

t

= e−γx+Σi=s+1hi





where we used that supαt+1 EP −e−γαt+1 ∆St+1 |Ft = −e−ht+1 .

In the following result, we compare the notions of the backward utility and of the classical value function. Indeed, the normalized at T backward dynamic utility corresponds, in a direct manner, to the value function of the Merton problem, associated with the investment horizon T. On the other hand, the normalized at s forward dynamic utility is constructed in such a way that the associated with it optimal value of wealth at time s is given by the classical exponential utility function. One can say that in our dynamic utility context, the backward dynamic utility is in spirit close to the classical value function, while the forward dynamic utility turns the traditional value function into the associated with it classical utility. 59

Theorem 50 The backward dynamic value function V B (x, s, t; T ), associated with the normalized at T backward dynamic utility U B (x, s; T ), is Fs -measurable and satisfies, for x ∈ R and s ≤ t ≤ T , V B (x, s, t; T ) = U B (x, s; T )

(137)

= −e−γx−HT (Q(·|Fs )| P(·|Fs ) ) PT = −e−γx−EQ( i=t+1 hi |Fs ) where s = 0, 1, .., T . Theorem 51 The forward dynamic value function V F (x, t, T ; s) , associated with the normalized at s spot dynamic utility U F (x, t; s) is Ft−1 -measurable and satisfies, for x ∈ R, 0 ≤ s ≤ t and all T ≥ t, V F (x, t, T ; s) = U F (x, t; s) = −e−γx+

Pt

i=s+1

(138)

Hi (Q(·|Fi−1 )| P(·|Fi−1 ) )

= −e−γx+

Pt

i=s+1

hi

.

The following results are direct consequences of the above Theorems and Propositions 42 and 43. Corollary 52 Let T1 < T2 and s, t such that 0 ≤ s ≤ t ≤ T1 . Consider the backward dynamic utilities U B (x, t; T1 ) and U B (x, t; T2 ) normalized, respectively, at T1 and T2 , and the associated backward dynamic value functions V B (x, s, t; T1 ) and V B (x, s, t; T2 ). Then, for α ∈ R and x ∈ R, V B (x, s, t; T2 ) = −U B ((1 − α) x, s; T1 ) U B (αx, T1 ; T2 ) = −V B ((1 − α) x, s, t; T1 ) U B (αx, T1 ; T2 ) . Corollary 53 Let s1 < s2 and t, T such that s2 ≤ t ≤ T . Consider the forward dynamic utilities U F (x, t; s1 ) and U F (x, t; s2 ), normalized, respectively at s1 and s2 , and the associated forward dynamic value functions V F (x, t, T ; s1 ) and V F (x, t, T ; s2 ). Then, for α ∈ R and x ∈ R, V F (x, t, T ; s1 ) = −U F ((1 − α) x, t; s2 ) U F (αx, s2 ; s1 ) = −V F ((1 − α) x, t, T ; s2 ) U F (αx, s2 ; s1 ) .

60

3.3

Backward and forward indifference prices

The classical notion of indifference price was introduced and analyzed in the context of a single contingent claim with a given maturity. The backward and forward dynamic utility concepts give rise to two different concepts of the indifference price. However, more importantly, they allow for the development of the derivatives pricing system which is consistent with the concept of dynamic risk preferences as well as with the static no arbitrage requirement. In what follows we develop and analyze these ideas. It turns out that our pricing systems, based on the backward and forward utility frameworks, satisfy all of the fundamental properties. However, the prices obtained in the backward and forward frameworks do not, in general, coincide. Moreover, it seems that the prices associated with the forward framework retain more of an intuitive appeal. This is primarily due to the fact that the forward utility is normalized at the current time, as opposed to the backward, which is normalized at a future date. Consequently, information about the distance between the pricing measure and the equilibrium state of the market enters the valuation procedure in drastically different ways. Clearly, there is a ”price to pay” for the desire to control deterministically the utility of wealth in the future. We start with the definition of the backward indifference price. For this, we introduce a backward dynamic preference structure with normalization point T (cf. Definition 40). Next, we consider a claim, written, at time t0 ≥ 0, on both risky assets yielding payoff CT¯ by time T¯ ≤ T . The claim’s payoff is represented as an FT¯ -measurable random variable and we are interested in specifying its forward indifference price at an arbitrary time t, with t0 ≤ t ≤ T¯. Note that, in general, the derivative may yield a series of cash flows at intermediate times. In order to simplify the presentation and, more importantly, to concentrate on the new ideas and concepts we assume, throughout, that there is a single cash flow. The indifference valuation of additional cash flows may be developed by similar arguments but we choose not to present it herein. The interested reader might look at Musiela and Zariphopoulou (2005b). As in the single period case, we start by examining the writer’s optimal behavior in [t, T¯], under the backward dynamic exponential utility and in the presence of the liability CT¯ . The writer’s backward dynamic value function is then defined as


(139) sup EP U B XT¯ − CT¯ , T¯; T |Ft V B,CT¯ x, t, T¯; T = αt+1 ,...,αT¯

¯

with XT¯ = Xt + ΣTi=t+1 αi ∆Si and Xt = x. For convenience, we skip any w-notation, referring to writer-related quantities.
¯; T Clearly, in the absence of the claim the writer’s function V B,0 x, t, T
coincides with the backward dynamic value function V B x, t, T¯; T , introduced in Definition 46. An interesting observation is that, due to (137), one can interpret the backward dynamic utility as the writer’s backward dynamic value of zero liability. 61

In analogy to the single period price Definition (cf. (4)), the backward indifference price of CT¯ , denoted by ν B (CT¯ , t; T ), is constructed from the comparison of the value functions V B,CT¯ and V B,0 , evaluated at the appropriate wealth arguments. Definition 54 Consider the backward dynamic utility U B , normalized at time T (cf. (126)). Let T¯ ≤ T and consider a claim, written at time t0 ≥ 0, yielding payoff CT¯ at time T¯, with CT¯ ∈ FT¯ . Let, also, V B,CT¯ and V B,0 be, respectively, the backward dynamic value functions with and without the claim (cf. (139) and (126)). For t0 ≤ t ≤ T¯, the backward indifference price, associated with the backward, normalized at time T, utility U B , is defined as the amount ν B (CT¯ , t; T ) for which

V B,0 x, t, T¯; T = V B,CT¯ x + ν B (CT¯ , t; T ), t, T¯; T (140)

for all initial wealth levels x ∈ R.

If we take T¯ = T , the above definition yields the classical indifference price in a binomial setting (see, Musiela and Zariphopoulou (2004b) and the next Chapter by Elliott and Van der Hoek). Note that the indifference price ν B (CT¯ , t; T ) is a function of three arguments, namely, the payoff CT¯ , the time t, at which the price is calculated, and the time T at which the backward dynamic risk preferences are normalized. Recall that, in general, t does not coincide with the claim’s inscription time. Let us now examine how the indifference price is constructed if the normalization point changes. For this, let us assume that the backward dynamic utility is now normalized at a different point, say T ′ , with T¯ ≤ T ′ < T with T¯ as before. In order to construct the relevant backward price ν B (CT¯ , t; T ′ ) we proceed as follows. We consider the backward dynamic utility U B (x, t; T ′ ) , given, for x ∈ R, by  −e−γx if t = T′   U B (x, t; T ′ ) = “P ′ ” T  −γx−EQ  s=t+1 hs |Ft if 0 ≤ t ≤ T ′ − 1, −e and the associated backward dynamic value functions with and without the claim, namely,


sup EP U B XT¯ , T¯; T ′ |Ft V B,0 x, t, T¯; T ′ = αt+1 ,...,αT¯

and


V B,CT¯ x, t, T¯; T ′ =

sup αt+1 ,...,αT¯



EP U B XT¯ − CT¯ , T¯; T ′ |Ft .

The backward indifference price ν B (CT¯ , t; T ′ ) is, then, the amount for which

V B,0 x, t, T¯; T ′ = V B,CT¯ x + ν B (CT¯ , t; T ′ ) , t, T¯; T ′ 62

for all initial wealth levels x ∈ R. Note that, in general,

V B,CT¯ x, t, T¯; T ′ 6= V B,CT¯ x, t, T¯; T

and V B,0 x, t, T¯; T ′ 6= V B,0 x, t, T¯; T . Therefore, backward indifference prices, corresponding to different normalization points, do not in general coincide, ν B (CT¯ , t; T ′ ) 6= ν B (CT¯ , t; T )

for

T ′ 6= T.

(141)

Next, we revert our attention to the notion of indifference prices with regards to forward, rather than backward, dynamic risk preferences. In analogy to the setting considered in the backward case, we assume that the claim in consideration is written, at time t0 , on both traded and non-traded assets and that it offers a single cash flow at t¯ > t0 . Its payoff can be then represented by a random variable Ct¯ ∈ Ft¯. It is assumed that the forward dynamic utility U F , introduced in Definition 41, is normalized at the point s, with s ≤ t0 < t¯. For simplicity, we take s = t0 . The writer’s forward dynamic value function, representing the writer’s optimal investment behavior in [t, t¯], and taking into account the liability Ct¯, is defined, for s ≤ t ≤ t¯, by
(142) V F,Ct¯ (x, t, t¯; s) = sup EP U F (Xt¯ − Ct¯, t¯; s) |Ft αt+1 ,..,αt¯

¯ where Xt¯ = Xt + Σti=t+1 αi ∆Si and Xt = x. We are now ready to introduce the new concept of forward indifference price.

Definition 55 Consider the forward dynamic utility U F normalized at time s ≥ 0. Let s ≤ t¯ and consider a claim yielding, at time t¯, payoff Ct¯ ∈ Ft¯. For s ≤ t ≤ t¯, let V F,Ct¯ and V F,0 be, respectively, the forward dynamic value functions with and without the claim (cf. (142) and (129)). For s ≤ t ≤ t¯, the forward indifference price, associated with the forward, normalized at time s, utility U F , is defined as the amount ν F (Ct¯, t; s) for which
V F,0 (x, t, t¯; s) = V F,Ct¯ x + ν F (Ct¯, t; s), t, t¯; s (143) for all initial wealth levels x ∈ R.

In principle, as in the backward case, the forward indifference price ν F (Ct¯, t; s) is a function of three arguments, namely, the payoff Ct¯, the time t, at which the price is calculated, and the time s ≤ t with respect to which the forward dynamic risk preferences are normalized. However, because Pt
V F,Ct¯ (x, t, t¯; s) = e i=s+1 hi sup EP U F (Xt¯ − Ct¯, t¯; t) |Ft αt+1 ,..,αt¯

=e

Pt

i=s+1

hi

V F,Ct¯ (x, t, t¯; t)

it readily follows that for all normalization points s ≤ t ν F (Ct¯, t; s) = ν F (Ct¯, t; t) 63

and, hence, the forward indifference price depends only on the payoff and the time at which it is calculated. The above new kinds of prices, directly derived from a dynamic, rather than the traditional static, choice of risk preferences generate several interesting issues and questions. The first issue is related to the existence, uniqueness and measurability properties of the backward and forward indifference price. We address these points in the next two sections, where we explicitly construct these prices via two iterative pricing schemes. The second question has to do with the relation between the backward and forward prices of the same claim. Recall that so far, we have chosen the riskless interest rate to be zero and one could potentially expect that the backward and forward prices coincide, in analogy to similar behavior of various quantities (expressed in spot and forward units) in complete markets. Note, however, that, in incomplete markets, this may not be the case due to the very different way the backward and forward prices assimilate unhedgeable risks. Indeed, in general, the two prices do not coincide, i.e. if s and T are respectively the backward and forward normalization points, then ν B (CT¯ , t; T ) 6= ν F (CT¯ , t; s) .

(144)

Herein, s ≤ t0 ≤ t ≤ T¯ ≤ T where t0 and T¯ are, respectively, the time the claim is introduced and the one at which its cash flow occurs. In a later section (see Section 3.5), we consider a reduced binomial model in which the transition probability of the traded asset does not depend on the non-traded one. We will then see that, under zero interest rate, the two prices, backward and forward, coincide at all times. The third issue is related with the semigroup property of indifference prices. This property is an indispensable ingredient of any valuation approach and has been discussed in detail in the reduced binomial setting analyzed by the authors in Musiela and Zariphopoulou (2004b). With regards to the new concept of backward price introduced herein, the semigroup property would hold if, for 0 ≤ t′ < t ≤ T¯ ≤ T and ν B (CT¯ , t; T ) ∈ Ft ,
ν B (CT¯ , t′ ; T ) = ν B ν B (CT¯ , t; T ) , t′ ; T where T¯, is the time at which the claim’s cash flow occurs and T is the normalization point of the backward preferences. Respectively, for s ≤ t0 ≤ t′ < t ≤ t¯, the forward indifference price operator
will have the semigroup property, if, for for s ≤ t0 ≤ t′ < t ≤ t¯ and ν F Ct¯, , t; T ∈ Ft ,

ν F (Ct¯, t′ ; s) = ν F ν F Ct¯, , t; s , t′ ; s

where t¯, is the time the claim’s cash flow occurs and s is the normalization point of the forward preferences. The semigroup property of backward and forward prices will be addressed through the properties that the prices inherit from the algorithms developed for their construction. 64

3.4

The indifference valuation algorithm under backward dynamic utility

We are going to construct an iterative pricing scheme for backward indifference prices. In spirit, the backward indifference algorithm resembles the algorithm developed by the authors (see Musiela and Zariphopoulou (2004b)). However, the results herein considerably generalize the ones in the latter work in two directions. Firstly, they incorporate the dynamic notion of backward risk preferences allowing for arbitrary claim structures and consistent valuation across times and investment opportunities. Secondly, they are applicable to general binomial models without any restrictive assumptions on the distribution of the returns of the traded asset. We recall that in the single period case, the fundamental quantities for the representation of the indifference price were the pricing measure Q (cf. (8)) and the single step nonlinear pricing functional EQ (cf. (9)). As it would have been expected, their multi-period analogues, in the context of backward dynamic risk preferences, are determined in a similar manner. In order to facilitate the exposition, we proceed differently than in the one period setting. We do not first construct the price and then recover their form. Rather, we first introduce the multi-period counterparts of the pricing measure and the backward price functional and we subsequently justify their role in the price representation. The backward normalization point is taken to be T and it is assumed to dominate the occurrence time of the cash flow. The backward price functional is denoted by EQB (· ; T ). In its definition, the backward dynamic utility U B (x, t; T )
−1 and its inverse U B (x, t; T ) appear. The inverse backward dynamic utility
B −1 − U : R × [0, T ] is defined with respect to the wealth argument and is given by ! T X
1 1 B −1 (145) hi |Ft . U (x, t; T ) = − log (−x) − EQ γ γ i=t+1 To simplify the presentation, we use the compressed notation B U B (x, t; T ) = Ut,T (x)

and

UB

whenever appropriate.


−1

(x, t; T ) = U B


−1 t,T

(x)

(146)

Definition 56 Let Z be a random variable on (Ω, F , P) and Ft and FtS be the previously introduced filtrations. Let T be the backward normalization point and
−1 U B and U B be, respectively, the backward dynamic utility and its inverse. We define: i) the pricing measure Q to be a martingale measure with the minimal relative to P entropy, that is, satisfying, for t = 0, 1, ...

S S , (147) = P ηt+1 Ft ∨ Ft+1 Q ηt+1 Ft ∨ Ft+1 65

ii) the single step backward price functional  


−1 B,(t,t+1) B S |Ft EQ (Z; T ) = EQ − U B t+1,T EQ Ut+1,T (−Z) Ft ∨ Ft+1 (148) and iii) the iterative backward price functional B,(t,t′ )

B,(t,t+1)

B,(t+1,t+2)

B,(t′ −1,t′ )

(Z; T )); T ); T ),

(149)

We caution the reader that, in general, for t′ 6= t + 1,  


−1 B,(t,t′ ) (Z; T ) 6= EQ − U B t′ ,T EQ UtB′ ,T (−Z) Ft ∨ FtS′ |Ft . EQ

(150)

EQ

(Z; T ) = EQ

(EQ

(...(EQ

across the time interval (t, t′ ).

We are now ready to present a multi-period iterative valuation algorithm for the calculation of the indifference prices under the backward dynamic utility U B (x, t; T ). The results below are, to the best of our knowledge, entirely new. As assumed earlier, the claim in consideration is introduced at t0 with 0 ≤ t0 < T and offers a single cash flow CT ∈ FT . We are interested in its backward indifference price at intermediate times t, with t0 ≤ t ≤ T. Note that the expiration time is taken to be equal to the time that the backward risk preferences are normalized. This is done only for simplicity and the arguments below can be extended to address the general case, i.e. when the cash flow occurs before the normalization point. However, the analysis is more involved and we choose not to present it herein. Rather, we refer the interested reader to Musiela and Zariphopoulou (2005b). Theorem 57 Let CT ∈ FT be the claim, introduced at t0 , and to be priced under the backward dynamic utility U B (x, t; T ). The following statements are true: i) The backward indifference price ν B (CT , t; T ), defined in (140), is given, for t0 ≤ t ≤ T , by the algorithm B,(t,t+1)

ν B (CT , t; T ) = EQ

(ν B (CT , t + 1; T ); T ),

ν B (CT , T ; T ) = CT ,

(151)

B,(t,t+1)

with EQ and Q defined respectively by (148) and (147). ii) The backward indifference price process ν B (CT , t; T ) ∈ Ft and satisfies B,(t,T )

ν B (CT , t; T ) = EQ

(CT ; T ),

t = 0, 1, ...T,

(152)

B,(t,T )

with the iterative price functional EQ defined in (149). iii) The backward pricing algorithm is consistent across time in that, for 0 ≤ t0 ≤ t ≤ t′ ≤ T , the semigroup property B,(t,t′ )

ν B (CT , t; T ) = EQ

B,(t′ ,T )

(EQ

66

(CT ; T ) ; T )

(153)

B,(t,t′ )

= EQ

′

B,(t ,T )

((ν B (CT , t′ ; T ); T ) = ν B (EQ

(CT ; T ), t; T )

holds. iv) For 0 ≤ t ≤ t′ ≤ T , the writer’s backward dynamic value function B V (x, t, t′ ; T ), defined in (139), can be written in the following form V B,CT (x, t, t′ ; T ) = −e−γ (x−ν

B

(CT ,t;T ))−EQ (ΣT i=t+1 hi |Ft )

(154)


= U B x − ν B (CT , t; T ) , T ; T . To ease the notation and simplify the presentation, we skip for the moment the T argument in the above price functional, using hereafter the notation B,(t,t+1) B,(t,t′ ) EQ (Z) and EQ (Z). The reference to the backward normalization point will be reinstated whenever appropriate. Proof. In analogy to the single period case, we define, for t = 1, 2, ..., T, the conditional probabilities
p1,t = P (ξt = ξtu , ηt = ηtu |Ft−1
) p3,t = P ξt = ξtd , ηt = ηtu |Ft−1
. (155) p2,t = P ξt = ξtu , ηt = ηtd |Ft−1 p4,t = P ξt = ξtd , ηt = ηtd |Ft−1 We are going to establish (152) for t = T − 1 and (153) for t = T − 2, t′ = T − 1. The rest of the proof follows by routine, albeit tedious, induction arguments and is not presented. We start by showing that B,(T −1, T )

ν B (CT , T − 1; T ) = EQ 
−1 = EQ − U B T,T EQ

(CT )



B UT,T (−CT ) FT −1 ∨ FTS |FT −1 .

Using the normalization properties of U B and above equality is equivalent to  1 B ν (CT , T − 1; T ) = EQ log EQ eγCT γ

UB


−1

(156)

(cf. (126) and (145)) the


FT −1 ∨ FTS |FT −1



(157)

with Q as in (147). To show the above, we first consider the value function V B,CT at T − 1, given by   V B,CT (x, T − 1, T ; T ) = sup EP −e−γ(x+αT △ST −CT ) |FT −1 αT

(cf. (139)). Imitating the calculations for the single period case (cf. (10)), appropriately modified to accommodate the conditioning on FT −1 , we deduce that the optimal policy is given by αTCT ,∗ =

γST −1

1 (ξ u − 1) (p + p2,T )
log T d
1,T u d ξT − ξT 1 − ξT (p3,T + p4,T ) 67

+

γST −1

  u u u d eγCT (ξT ,ηT ) p1,T + eγCT (ξT ,ηT ) p2,T (p3,T + p4,T ) 1
log   d u d d ξTu − ξTd eγCT (ξT ,ηT ) p3,T + eγCT (ξT ,ηT ) p4,T (p1,T + p2,T )

with the probabilities pi,T , i = 1, .., 4 defined above. Rearranging terms yields V B,CT (x, T − 1, T ; T ) = −e−γx−hT +γλT −1 (CT )

(158)

with hT given in (119) and λT −1 (CT ) = EQ We easily deduce that




1 log EQ eγCT FT −1 ∨ FTS |FT −1 γ



.

(159)

V B,CT (x + ν B (CT , T − 1; T ) , T − 1, T ; T ) = −e−γ (x+ν

B

(CT ,T −1;T )−λT −1 (CT ))−hT


= U B x + ν B (CT , T − 1; T ) − λT −1 (CT ), T − 1; T .

Therefore, the backward indifference pricing condition,

V B,CT (x + ν B (CT , T − 1; T ) , T − 1, T ; T ) = V B,0 (x, T − 1, T ; T ) or, equivalently, (cf. (137)), the equality V B,CT (x + ν B (CT , T − 1; T ) , T − 1, T ; T ) = U B (x, T − 1; T ) would hold if and only if ν B (CT , T − 1; T ) = λT −1 (CT ) . This proves the validity of (156). Next, we establish the semigroup property (153) for the two time steps T − 1 and T − 2, i.e. B,(T −2,T −1)

ν B (CT , T − 2; T ) = EQ

(ν B (CT , T − 1; T )),

(160)

with ν B (CT , T − 1; T ) given by (156). We first construct ν B (CT , T − 2; T ) directly via the pricing formula (140). To this end, we compute V B,CT (x, T − 2, T ; T ) defined as
V B,CT (x, T − 2, T ; T ) = sup EP U B (XT , T ; T ) |FT −2 αT −1 ,αT

=

sup αT −1 ,αT

  EP −e−γ(x+αT −1 △ST −1 +αT △ST −CT ) |FT −2

    = sup EP e−γ(x+αT −1 △ST −1 ) sup EP −e−γ(αT △ST −CT ) |FT −1 |FT −2 . αT −1

αT

68

Using, from the previous single step arguments, that   B sup EP −e−γ(αT △ST −CT ) |FT −1 = −eγν (CT ,T −1;T )−hT , αT

we, in turn, deduce that V B,CT (x, T − 2, T ; T )   B 1 = sup EP −e−γ (x+αT −1 △ST −1 −ν (CT ,T −1;T )+ γ hT ) |FT −2 αT −1

= −e−γx−hT −1 +γµT −2

where hT −1 is as in (119) and     B 1 S γν (CT ,T −1;T )−hT µT −2 = EQ FT −2 ∨ FT −1 |FT −2 . log EQ e γ

(161)

(162)

We next observe that      B 1 µT −2 = EQ |FT −2 log −EQ −eγν (CT ,T −1;T )−hT FT −2 ∨ FTS−1 γ      1 γν B (CT ,T −1;T )−EQ (hT |FT −1 ) S log −EQ −e FT −2 ∨ FT −1 |FT −2 = EQ γ  


1 = EQ log −EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2 , γ

where we used the measurability properties of hT and the form of UTB−1;T (x). We now define

Z = EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 , (163)

and recall that, for x ∈ R− , UB


−1

Therefore,

(x, T − 1; T ) = U B


−1

T −1,T

1 1 (x) = − log (−x) − EQ (hT |FT −1 ) . γ γ

  1 µT −2 = −EQ − log (−Z) |FT −2 γ  1 
−1 = −EQ U B T −1,T (Z) |FT −2 − EQ (hT |FT −2 ) . γ

Equality (161) then yields that V B,CT (x, T − 2, T ; T ) = −e

“ “ −1 −γ x−EQ (U B )

T −1,T

(Z)|FT −2

”” −EQ (hT −1 +hT |FT −2 )

with Z as above. We also recall that V B,0 (x, T − 2, T ; T ) = U B (x, T − 2; T ) 69

= −e−γx−EQ(hT −1 +hT |FT −2 ) . Combining the above yields that V B,CT (x + ν B (CT , T − 2; T ) , T − 2, T ; T ) = V B,0 (x, T − 2, T ; T ), if only if

 
−1 ν B (CT , T − 2; T ) = EQ − U B T −1;T (Z) |FT −2

or, by using (163), if and only if

B,(T −2,T −1)

ν B (CT , T − 2; T ) = EQ

ν B (CT , T − 1; T ) ; T




 



−1 = EQ − U B T −1,T EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2

and the assertion follows. The claimed measurability property of the backward price, ν B (CT , t; T ) ∈ Ft , and the representation (154) of the backward dynamic value function follow easily. Note that even though hT ∈ FT −1 , it may fail to be FTS−1 -measurable. As a consequence, for the auxiliary quantity µT −2 , used above to establish the semigroup property, we have     B 1 S γν (CT ,T −1;T )−hT FT −2 ∨ FT −1 |FT −2 log EQ e µT −2 = EQ γ      B 1 1 γν (CT ,T −1;T ) S |FT −2 . 6 EQ − hT + log EQ e = FT −2 ∨ FT −1 γ γ

On the other hand, we showed that

−EQ



  1 µT −2 = EQ − hT |FT −2 γ

UB


−1

T −1;T




EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2 .

Comparing the above we see that, in general,  



−1 −EQ U B T −1,T EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2 6= EQ



  B 1 log EQ eγν (CT ,T −1;T ) FT −2 ∨ FTS−1 |FT −2 γ



.

(164)

While at this point, this might look like a technical nuisance, it will turn out that (164) has very important consequences on the nature of the backward and 70

forward indifference prices as well as on the emerging differences between them. These issues are analyzed in detail in Section 3.5. We continue with a discussion on the features of the backward indifference pricing algorithm. We first note that valuation is done via an iterative pricing scheme applied backwards in time. At each time step, say (t, t + 1), t = t0 , t0 + 1, ..., T −1, the price ν B (CT , t; T ) is computed via the single step price functional B,(t,t+1) EQ (· ; T ) applied to the end of the period payoff ν B (CT , t + 1; T ). The B,(t,t+1)

valuation role of EQ (· ; T ) is in many aspects identical to the one of its single period counterpart, analyzed in detail in Section 2.1. B,(t,t+1) Highlighting the main properties of EQ (· ; T ), we first observe that it is nonlinear and time dependent. Nonlinearity arises due to the way unhedgeable risks are priced. Risk preferences are injected, in a dynamic way, via the current backward dynamic value function and its inverse, to the end of the period payoff ν B (CT , t + 1; T ). A new payoff (cf. (146) for the compressed notation), namely,


S B −ν B (CT , t + 1; T ) Ft ∨ Ft+1 EQ Ut+1,T (165) emerges, to be called the backward conditional certainty equivalent . It is the dynamic analogue of the static conditional certainty equivalent C˜T , introduced in (17). This step amounts to the specification, isolation and valuation of the unhedgeable risks and it has naturally a nonlinear character. Once this sub-step is executed, the remaining risks are hedgeable and are, thus, priced by arbitragefree arguments, yielding the price as ν˜B (CT , t + 1; T ) = − U B


−1

t+1,T

ν B (CT , t; T ) = EQ (˜ ν B (CT , t + 1; T ) |Ft ). B,(t,t+1)

As in the single period case, it is worth noticing that EQ (· ; T ) is affected by the backward dynamic risk preferences (and its inverse) only at its first step, the one that considers the unhedgeable risks. It is also interesting to notice its independence on the specific payoff and its universality with respect to the pricing measure Q. Finally, observe that, at all valuation steps, the generic local entropy term ht = Ht (Q (· |Ft−1 ) |P (· |Ft−1 ) ) enters the indifference pricing algorithm of the claim. This is a direct consequence of the fact that the forward risk preference structure is normalized at the future time T .

3.5

The indifference valuation algorithm under forward dynamic utility

We now revert our attention to the construction of an iterative valuation algorithm for the forward indifference price. In analogy to the presentation for the forward pricing algorithm, we first introduce the pricing measure and the associated forward price functionals, namely, the single step functional and the iterative one. 71

The forward normalization point is taken to be s and is assumed to be smaller than the occurrence time of the cash flow. In the definition of the forward price functionals, the forward dynamic utility U F (x, t; s) and its inverse
−1
−1 UF (x, t; s) appear. The inverse forward dynamic utility U F : R− × [0, T ] is defined with respect to the wealth argument and is given, for t ≥ s, by UF


−1

t 1 1 X (x, t; s) = − log (−x) − hi . γ γ i=s+1

(166)

To simplify the presentation we will occasionally use the compressed notation
−1
−1 F U F (x, t; s) = Us,t (x) and UF (x, t; s) = U F s,t (x) for t ≥ s, (167) whenever appropriate. Note that in contrast to the compressed backward notation (146), the point s, at which the forward dynamic utility is normalized, precedes the valuation time t. Definition 58 Let Z be a random variable on (Ω, F , P) and Ft and FtS be the previously introduced filtrations. Let s be the forward normalization point
−1 F and Us,t (x) and U F s,t (x) be, respectively, the forward dynamic utility and its inverse. We define: i) the pricing measure Q to be a martingale measure with the minimal relative to P entropy, that is, satisfying, for t = 0, 1, ...

S S Q ηt+1 Ft ∨ Ft+1 = P ηt+1 Ft ∨ Ft+1 , (168)

ii) the single step forward price functional  


−1 F,(t,t+1) F S |Ft , (−Z) Ft ∨ Ft+1 (Z; s) = EQ − U F s,t+1 EQ Us,t+1 EQ (169) and iii) the iterative forward price functional F,(t,t′ )

EQ

F,(t,t+1)

(Z; s) = EQ

F,(t+1,t+2)

(EQ

F,(t′ −1,t′ )

(...(EQ

(Z; s))); s)

(170)

across the time interval (t, t′ ). As in the backward case, we caution the reader that, in general, for t′ = 6 t+1,   ′


−1 F,(t,t ) F S (171) |Ft . EQ (Z; s) 6= EQ − U F s,t′ EQ Us,t ′ (−Z) Ft ∨ Ft′

We are now ready to present a multi-period iterative valuation algorithm for the calculation of the indifference prices under the fowrad dynamic utility U F (x, t; s). The claim in consideration is introduced at t0 and offers a single cash flow Ct¯ ∈ Ft¯ at time t¯. It is assumed that the forward normalization point s satisfies s ≤ t0 < t¯. We are interested in the development of a scheme for the forward indifference price calculation at intermediate times t, with t0 ≤ t ≤ t¯. 72

Theorem 59 Let Ct¯ ∈ Ft¯ be the claim, introduced at t0 and to be priced under the forward utility U F (x, t; s). The following statements are true: i) The forward indifference price ν F (Ct¯, t; s), defined in (143), is given, for t0 ≤ t ≤ t¯, by the algorithm F,(t,t+1)

ν F (Ct¯, t; s) = EQ

(ν F (Ct¯, t + 1; s); s),

ν F (Ct¯, t¯; s) = Ct¯,

(172)

F,(t,t+1)

with EQ (· ; s) and Q defined respectively by (169) and (168). ii) The forward indifference price process ν s (Ct¯, t; s) ∈ Ft and satisfies F,(t,t¯)

ν F (Ct¯, t; s) = EQ

(Ct¯; s),

t ≥ t0 ,

(173)

with the iterative forward price functional defined in (170). iii) The forward pricing algorithm is consistent across time in that, for s ≤ t ≤ t′ ≤ t¯, the semigroup property F,(t,t′ )

ν F (Ct¯, t; s) = EQ F,(t,t′ )

= EQ

′

F,(t ,t¯)

(EQ

(Ct¯; s) ; s)

(174)

′

F,(t ,t¯)

(ν F (Ct¯, t′ ; s); s) = ν F (EQ

(Ct¯; s), t; s)

holds. iv) For s ≤ t0 ≤ t ≤ t′ ≤ t¯, the writer’s forward dynamic value function F,Ct¯ V (x, t, t′ ; s), defined in (142), can be written in the following forms V F,Ct¯ (x, t, t′ ; s) = −e−γ (x−ν

F

(Ct¯,t;s))+Σti=s+1 hi

(175)




= U F x − ν F (Ct¯, t; s) , t; s . Before we present the proof of the above Theorem, we present the following Lemma. Lemma 60 Let Z be a random variable on the probability space (Ω, F , P). Let
−1 also s be the forward normalization point and U F and U F be, respectively, F,(t,t+1) the forward dynamic utility and its inverse. If EQ (; s) is the single step forward price functional, introduced in (169), then, for all t ≥ s,  

1 F,(t,t+1) S (176) |Ft . log EQ eγZ Ft ∨ Ft+1 EQ (Z ; s) = EQ γ

Proof. We recall that the local entropy terms hi ∈ Fi−1 for i ≥ 1. Therefore, t+1 X hi ∈ Ft . Calculating, at the generic argument Z, the single step forward

i=s+1

price functional yields F,(t,t+1)

EQ


−1 (Z; s) = EQ − U F s,t+1 EQ 73



F S |Ft (−Z) Ft ∨ Ft+1 Us,t+1

= EQ

1 log − EQ γ

t+1 1 X = hi + EQ γ i=s+1

=



F Us,t+1

t+1


1 X S hi |Ft + (−Z) Ft ∨ Ft+1 γ i=s+1

!

    1 S γZ−Σt+1 hi i=s+1 |Ft Ft ∨ Ft+1 log EQ e γ

 t+1 1 1 X hi + EQ log EQ γ i=s+1 γ

and the statement follows.



S |Ft eγZ Ft ∨ Ft+1



−

t+1 1 X hi γ i=s+1

We continue with the proof of the above Theorem. Since the technical arguments are similar to the ones used for the construction of the backward iterative algorithm (see proof of Theorem 57), we only present the main steps. Proof. We start by showing that F,(t¯−1,t¯)

ν F (Ct¯, t¯ − 1; s) = EQ 
−1 = EQ − U F s,t EQ

or, equivalently, that F

ν (Ct¯, t¯ − 1; s) = EQ



(Ct¯; s)



F Us,t (−Ct¯) Ft¯−1 ∨ Ft¯S |Ft¯−1


1 log EQ eγCt¯ Ft¯−1 ∨ Ft¯S |Ft¯−1 γ



(177)

with Q as in (168). To prove the above, we first consider the spot value function V F,Ct¯ at t¯ − 1, given by
V F,Ct¯(x, t¯ − 1, t¯; s) = sup EP U F (Xt¯ − Ct¯, t¯; s) |Ft¯−1 . αt¯

We easily deduce that

  ¯ t V F,Ct¯(x, t¯ − 1, t¯; s) = sup EP −e−γ(x+αt¯△St¯−Ct¯)+Σi=s+1 hi |Ft¯−1 αt¯

  ¯−1 t = −e−γx+Σi=s+1hi sup EP −e−γ(αt¯△St¯−Ct¯)+ht¯ |Ft¯−1 , αt¯

where we used the measurability properties of the local entropy terms. Imitating the calculations for the single period case (cf. (14)), appropriately modified to accommodate the conditioning on Ft¯−1 , we deduce that ¯−1 t

V F,Ct¯(x, t¯ − 1, ¯t; s) = −e−γx+Σi=s+1hi +γλt¯−1 (Ct¯) with λt¯−1 (Ct¯) = EQ




1 log EQ eγCt¯ Ft¯−1 ∨ Ft¯S |Ft¯−1 γ 74



.

It is then immediate that the pricing condition V F,Ct¯(x + ν F (Ct¯, t¯ − 1; s) , t¯ − 1, ¯t; s) = V F,0 (x, t¯ − 1, ¯t; s) or, equivalently, due to (cf. (138)), the equality −e−γ (x+ν

F

¯−1 (Ct¯,t¯−1;s))+Σti=s+1 hi +γλt¯−1 (Ct¯)

¯−1 t

= −e−γx+Σi=s+1hi

would hold if and only if ν F (Ct¯, t¯ − 1; s) = λt¯−1 (Ct¯) . This proves the validity of (177). Next, we establish the semigroup property (174) for the two time steps t¯− 1 and t¯ − 2, i.e. that F,(t¯−2,t¯−1)

ν F (Ct¯, t¯ − 2; s) = EQ

(ν F (Ct¯, t¯ − 1; s) ; s),

or, equivalently, ν F (Ct¯, t¯ − 2; s) = EQ



  F 1 ¯ log EQ eγν (Ct¯,t−1;s) Ft¯−2 ∨ Ft¯S−1 |Ft¯−2 γ



(178)

with ν F (Ct¯, t¯ − 1; s) as in (177). We first construct ν F (Ct¯, t¯ − 2; s) directly via the pricing formula (143). To this end, we start with the computation of V F,Ct¯(x, t¯ − 2, t¯; s). From (142) we have,
V F,Ct¯(x, t¯ − 2, t¯; s) = sup EP U F (Xt¯, t¯; s) |Ft¯−2 αt¯−1 ,αt¯



 ¯ t = sup EP −e−γ (x+αt¯−1 △St¯−1 +αt¯△St¯−Ct¯)+Σi=s+1 hi |Ft¯−2 . αt¯−1 ,αt¯

Further calculations, together with the fact that   F 1 ¯ sup EP −e−γ (αt¯△St¯−Ct¯− γ ht¯) |Ft¯−1 = −eγν (Ct¯,t−1;s) , αt¯

yield ¯−2 t

= e−γx+Σi=s+1hi

V F,Ct¯(x, t¯ − 2, ¯t; s)   F 1 ¯ sup EP −e−γ (αt¯−1 △St¯−1 −ν (Ct¯,t−1;s)− γ ht¯−1 ) |Ft¯−2

αt¯−1

with ν (Ct¯, t¯ − 1; s) as in (177). We then easily conclude that F

¯−2 t

V F,Ct¯(x, t¯ − 2, t¯; s) = −e−γx+Σi=s+1 hi +γλt¯−2 where λt¯−2 = EQ



  F 1 ¯ log EQ eγν (Ct¯,t−1;s) Ft¯−2 ∨ Ft¯S−1 |Ft¯−2 γ 75



.

From the forward pricing condition (143) and the above equalities, it is then immediate that the pricing condition V F,Ct¯(x + ν s (Ct¯, t¯ − 2; s) , t¯ − 2, ¯t; s) = V F,0 (x, t¯ − 2, t¯; s) will hold if and only if ν F (Ct¯, t¯ − 2; s) = λt¯−2 , and assertion (178) follows. The claimed measurability property of the forward price, ν F (Ct¯, t; s) ∈ Ft , and the representation (175) of the forward dynamic value function follow easily. As in the backward case, the forward indifference valuation is done via an iterative pricing scheme applied backwards in time. At each time step, say (t, t + 1), with t0 ≤ t ≤ t¯ − 1, the price ν F (Ct¯, t; s) is computed via the single F,(t,t+1) step forward price functional EQ applied to the end of the period payF,(t,t+1)

off ν F (Ct¯, t + 1; s). The valuation role of EQ is similar to its backward counterpart in that it nonlinear and time dependent. The forward dynamic risk preferences are injected via the current forward dynamic value function and its inverse to the end of the period payoff ν F (Ct¯, t+ 1; s). The new payoff (cf. (167) for the compressed notation)



−1 S F −ν F (Ct¯, t + 1; s) Ft ∨ Ft+1 ν˜F (Ct¯, t + 1; s) = − U F s,t+1 EQ Us,t+1 = EQ



  F   1 γν (Ct¯,t+1;s) S Ft ∨ Ft+1 |Ft log EQ e γ

(179)

emerges, to be called the forward conditional certainty equivalent. It is the dynamic analogue of the static conditional certainty equivalent C˜T (cf. (17)) and, also, the forward analogue of the backward conditional certainty equivalent ν˜B (CT , t + 1; T ) (cf. (165)). Naturally, it is of nonlinear character since it is at this valuation step that the unhedgeable risks are identified and priced. Once this sub-step is executed, the remaining risks are hedgeable and are, thus, priced by arbitrage-free arguments, yielding the forward price as ν F (Ct¯, t; s) = EQ (˜ ν F (Ct¯, t + 1; s) |Ft ). F,(t,t+1)

As in the single-period and forward cases, EQ (· ; s) is affected by the forward risk preferences (and its inverse) only at its first step, the one that considers the unhedgeable risks. It is independent on the specific payoff and it uses the same pricing measure Q through all time steps. Finally, it is important to observe that the same pricing measure Q appears in both backward and forward valuational functionals even though the associated risk preferences are of very different nature. Note, however, that there is a fundamental difference in the way market incompleteness appears in the backward and in the forward indifference valuation algorithms. 76

To elaborate, let us first recall that the single-step backward and forward functionals are given, respectively, by  



−1 B,(t,t+1) S B |Ft −Z B Ft ∨ Ft+1 EQ (Z B ; T ) = EQ − U B t+1,T EQ Ut+1,T

and

F,(t,t+1)

EQ


−1 (Z F ; s) = EQ − U F s,t+1 EQ




S F |Ft , −Z F Ft ∨ Ft+1 Us,t+1

B where T and s are the backward and forward normalization points, and Ut+1,T ,
−1
−1 F B F Us,t+1 and Ut+1,T the backward and forward dynamic utilities , Us,t+1 and their inverses. The variables Z B and Z F play the role of the end of the period backward and forward local payoffs. B Recalling the definition of Ut+1,T (and its inverse) we see that, in every step B,(t,t+1)

(t, t + 1), the local entropy ht+1 enters in EQ (· ; T ), via the backward dynamic preferences. It is important to notice that these entropic terms are independent of the specific claim and change dynamically in time. Moreover, the single step backward price functional depends on the normalization point, and hence so does the backward indifference price. On the other hand, as it is stated in the above Lemma, the single step forward F,(t,t+1) (· ; s) is essentially time invariant with regards to the price functional EQ local in time changes of the forward dynamic risk preferences. This property strongly indicates that the forward dynamic utility measures utility across times in a more intuitive fashion. Indeed, it is time invariant and model independent for the next time period. However, further it looks ahead in the future, the more information it incorporates about the degree of market incompleteness through the local entropic terms. These successive entropic terms give an idea of excess return locally in time and along the path. The forward dynamic utility does not provide such refined information since it only aggregates the entropies and incorporates their estimate. F,(t,t+1) Finally, as evidenced by (176), the functional EQ (· ; s) is independent on the normalization point s and hence so is the forward indifference price F,(t,t+1) ν F (Ct¯, t; s). As a consequence, from now on, we will use the notation EQ (· F ) and ν (Ct¯, t) for the single step forward price functional and the forward indifference price of Ct¯, calculated at time t.

3.6

A reduced binomial model and its backward and forward prices

The discussion at the end of the previous section sheds light to the different way the backward and forward dynamic preferences incorporate information on market incompleteness and how, in turn, the two algorithms process it. Due to the distinct nature of backward and forward preferences and valuation schemes, it is therefore natural to expect that the backward and forward indifference prices do not, in general, coincide. 77

Naturally, such differences should dissipate when the market becomes complete and the indifference prices converge to the arbitrage-free ones. If the market remains incomplete, however, an important question arises, related to when the backward and forward indifference prices coincide across times. This clearly points to a relationship between the model definition and the specification of the dynamic preference structure. To gain some intuition for the upcoming result, let us review the main steps in the construction of the backward price algorithm. Referring to the proof of Theorem 57, and specifically to the remark following it, we recall that, at each B,(t,t+1) valuation step, the single step backward price functional EQ (· ; T ) acts on two emerging terms, namely, the local entropy term, ht+1 , and the local payoff ν B (CT ; t + 1; T ). As it was mentioned earlier, while ht+1 is always Ft −measurable, it may fail to be FtS −measurable for arbitrary incomplete binomial models. Therefore, the B,(t,t+1) additive invariance property (cf. Proposition 11) of EQ (· ; T ) cannot be applied and ht+1 does not scale out. The above observation clearly indicates that discrepancies between prices could be remedied if additional conditions are imposed on the model resulting in the entropy terms ht+1 to be FtS −measurable, for all t ∈ [0, T ]. This is formulated in the upcoming condition (180). In this case, as the proof below shows, the backward and forward price functionals produce, at each time step, the same outcome which ultimately yields identical prices under the two kinds of preferences (cf. (181)). Lemma 61 Assume that for all t ∈ [0, T ],

u u S P ξt+1 = ξt+1 |Ft = P ξt+1 = ξt+1 Ft .

(180)

S Then the local entropy terms, defined in (119), satisfy ht ∈ Ft−1 , for 0 ≤ t ≤ T.

The proof of the above Lemma follows from the definition of ht and it is omitted. We next introduce a binomial model similar to the one introduced in Section 3.1 but with the additional assumption (180) on the transition probabilities of the traded asset. This is the same setting considered by the authors in Musiela and Zariphopoulou (2004b) as well as in Smith and Nau (1995), Becherer (2002), and Elliott and Van der Hoek herein. We will refer to this model as the reduced binomial model. Proposition 62 Consider the reduced binomial model introduced above. Assume dynamic risk references of backward and forward kind with respective normalization points s and T , 0 ≤ s < T . Consider a claim that yields a single cash flow CT ∈ FT . Let ν B and ν F be its backward and forward indifference prices. Then, for all CT and t ≤ T the two prices coincide, namely, ν B (CT , t; T ) = ν F (CT , t) . 78

(181)

Proof. We first observe, from (157) and (177), that ν B (CT , T − 1; T ) = ν F (CT , T − 1)  

1 = EQ log EQ eγCT FT −1 ∨ FTS |FT −1 . γ We are going to establish (181) for t = T − 2, namely, ν B (CT , T − 2; T ) = ν F (CT , T − 2) . Recall (cf. (178)) that ν F (CT , T − 2) = EQ



  F   1 |FT −2 . log EQ eγν (CT ,T −1) FT −2 ∨ FTS−1 γ

Combining the above, we deduce that it suffices to show    B   1 |FT −2 . log EQ eγν (CT ,T −1;T ) FT −2 ∨ FTS−1 ν B (CT , T − 2; T ) = EQ γ From Theorem 57, we obtain


B,(T −2,T −1) ν B (CT , T − 2; T ) = EQ ν B (CT , T − 1; T ) ; T  



−1 = EQ − U B T −1,T EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2 .

Note, however, that because hT ∈ FTS−1 ,



EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1   B = EQ −eγν (CT ,T −1;T )−EQ (hT |FT −1 ) FT −2 ∨ FTS−1   B = EQ −eγν (CT ,T −1;T )−hT FT −2 ∨ FTS−1   B = e−hT EQ −eγν (CT ,T −1;T ) FT −2 ∨ FTS−1 .

Therefore,  

−1

EQ − U B T −1,T EQ UTB−1,T −ν B (CT , T − 1; T ) FT −2 ∨ FTS−1 |FT −2     
−1 B = EQ − U B T −1,T e−hT EQ −eγν (CT ,T −1;T ) FT −2 ∨ FTS−1 |FT −2      1 1 −hT γν B (CT ,T −1;T ) S = EQ EQ −e log −e FT −2 ∨ FT −1 + hT |FT −2 γ γ     1 1 1 γν B (CT ,T −1;T ) S FT −2 ∨ FT −1 + hT |FT −2 = EQ − hT + log EQ e γ γ γ      1 γν B (CT ,T −1;T ) S |FT −2 FT −2 ∨ FT −1 log EQ e = EQ γ and the result follows. To conclude, we need to show that equality (181) holds for t = T −3, T −4, ..s. The involved arguments can be easily reproduced in an inductive manner and are not presented. 79

4

Bibliography

Artzner P., Delbaen F., Eber J.-M. and Heath D., Coherent measures of risk, Mathematical Finance 9 (3), 203–228 (1999). Artzner P., Delbaen F., Eber J.-M. and Heath D., Multi-period risk and coherent mult-iperiod risk measurement, preprint (2003). Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion, preprint (2002). Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties, Mathematical Finance, 12 (2), 99–123 (2002). Foellmer, H. and Schied, A.: Convex measures of risk and trading constraints, Finance and Stochastics, 6, 429–447 (2002a). Foellmer, H. and Schied, A.: Robust preferences and convex measures of risk, Advances in Finance and Stochastics, Springer, Berlin, (2002b). Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance, 10, 39–52 (2000a). Frittelli, M.: Introduction to a theory of value coherent with the no-arbitrage principle, Finance and Stochastics, 4, 275–297 (2000b). Kabanov, Y. and Stricker, C.: On the optimal portfolio of exponential utility maximization: Remarks to the six-author paper, Mathematical Finance, 12 (2), 125–134 (2002). Merton, R.: Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3, 373–413, (1971). Musiela, M. and Zariphopoulou, T.: Indifference prices and related measures, The University of Texas at Austin, Technical report, (2001a). Musiela, M. and Zariphopoulou, T.: Pricing and Risk Management of derivatives written on non-traded assets, The University of Texas at Austin, Technical report, (2001b). Musiela, M. and Zariphopoulou, T.: An example of indifference prices under exponential preferences, Finance and Stochastics, 8, 229–239, (2004a). Musiela, M. and Zariphopoulou, T.: A valuation algorithm for indifference prices in incomplete markets, Finance and Stochastics, 8, 399–414 (2004b). Musiela, M. and Zariphopoulou, T.: Forward indifference valuation in incomplete binomial models, preprint, (2005a). Musiela, M. and Zariphopoulou, T.: Backward indifference valuation in incomplete binomial models, in preparation, (2005b). Musiela, M., Zariphopoulou, T. and Zitkovic, G. : Numeraire consistency and indifference valuation, in preparation (2005). Rouge, R. and El Karoui, N.: Pricing via utility maximization and entropy, Mathematical Finance, 10, 259–276 (2000). Smith, J., and McCardle, K.F.: Valuing oil properties: integrating option pricing and decision analysis approaches, Operations Research, 46 (2), 198–217 (1998). Smith, J. and Nau, R.: Valuing risky projects: Option Pricing and Decisions Analysis, Management Science, 41 (5), 795–816, (1995). 80

Stoikov, S. F.: Option pricing from the point of view of a trader, submitted for publication, (2005).

[email protected] [email protected]

81