Masters' Thesis in Mathematics

Pricing Some Path-dependent Discretely Sampled Options Using a Partial Differential Equation Approach

Daniel Eskilsson U.U.D.M. Project Report 2003:11

Examensarbete i matematik, 20 poäng Handledare och examinator: Johan Tysk September 2003

Department of Mathematics Uppsala University

Abstract Using a partial differential equation approach we in this thesis develop a method for the pricing of a certain class of path-dependent discretely sampled exotic options. By introducing auxiliary state variables the problems with the path-dependency are overcome. The resulting partial differential equation to be solved is the Black-Scholes equation with the pathdependent quantities treated as parameters and with jump conditions across sampling dates. This can be implemented into a numerical scheme. We also investigate the distribution of the path-dependent quantity for a particular option from the class of options dealt with.

2

“The greatest of all gifts is the power to estimate things at their true worth.” − La Rochefoucauld, Reflexions; ou sentences et maximes morales.

“Nowadays people know the price of everything and the value of nothing.” − Lord Henry Wotton, in Oscar Wilde’s, The Picture of Dorian Gray.

3

Acknowledgements I would like to thank Johan Tysk for his work as my advisor. I am also grateful to Jonas Persson for providing important comments on the numerical computations. I am greatly indebted to Lars Luthman for helping me turning the mathematics into Matlab-code. Special thanks to Daniel Pérez for encouraging discussions and for making the work during the summer more fun. Finally I would never have succeeded in finishing this thesis without the support and patience of Anna Ahlin.

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Table of Contents 1

INTRODUCTION ....................................................................................................................................... 7 1.1

2

3

ARBITRAGE THEORY IN CONTINUOUS TIME................................................................................ 9 2.1

BACKGROUND ...................................................................................................................................... 9

2.2

ASSUMPTIONS AND ASSET DYNAMICS .................................................................................................. 9

2.3

TRADING STRATEGIES ........................................................................................................................ 13

2.4

COMPLETENESS AND HEDGING ........................................................................................................... 15

GENERALISED MODEL FOR PRICING OF THE CONTINGENT CLAIMS................................ 18 3.1

SETUP ................................................................................................................................................. 18

3.2

SAMPLING STRUCTURE ....................................................................................................................... 19

3.2.1

4

5

DISPOSITION ......................................................................................................................................... 8

Jump Condition ............................................................................................................................. 20

3.3

DERIVATION OF THE PARTIAL DIFFERENTIAL EQUATION ................................................................... 21

3.4

PROPOSITION 3.1 (PRICING EQUATION 1)............................................................................................ 26

3.5

SIMILARITY REDUCTION ..................................................................................................................... 27

3.6

PROPOSITION 3.2 (PRICING EQUATION 2)............................................................................................ 29

NUMERICAL ANALYSIS....................................................................................................................... 30 4.1

OUTLINE ............................................................................................................................................. 30

4.2

THE GREEKS ....................................................................................................................................... 31

4.3

DIFFERENTIATION USING A GRID ....................................................................................................... 32

4.4

APPROXIMATING THE DERIVATIVES ................................................................................................... 33

4.5

FINAL CONDITIONS AND PAYOFFS ...................................................................................................... 34

4.6

BOUNDARY CONDITIONS .................................................................................................................... 34

4.7

JUMP CONDITIONS .............................................................................................................................. 35

4.8

THE EXPLICIT FINITE-DIFFERENCE METHOD ...................................................................................... 36

ANALYSIS OF THE DISTRIBUTION OF A PATH-DEPENDENT QUANTITY ............................ 38 5.1

A PRACTICAL EXAMPLE ..................................................................................................................... 38

5.2

MONTE CARLO SIMULATION .............................................................................................................. 39

5.3

RESULTS FROM THE SIMULATIONS ..................................................................................................... 40

5

5.4

ESTIMATION OF THE PROBABILITY DENSITY FUNCTION ..................................................................... 44

5.4.1

Estimated Probability Density Functions from Monte Carlo Simulation 1................................... 45

5.4.2

Estimated Probability Density Functions from Monte Carlo Simulation 2................................... 51

6

COMPARISON OF PRICES ................................................................................................................... 52

7

CONCLUDING REMARKS .................................................................................................................... 56

8

APPENDIX A ............................................................................................................................................ 58 8.1

9

APPENDIX B............................................................................................................................................. 63 9.1

10

HISTOGRAMS FROM SIMULATION 1 .................................................................................................... 58

MATLAB CODE ................................................................................................................................... 63

REFERENCES .......................................................................................................................................... 64

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1 Introduction This thesis is written to fulfil the requirements of the Master of Science degree in Mathematics at Uppsala University. When I for the first time came across Relax Sverige 1* a real world example from the class of European path-dependent discretely sampled exotic options, which this thesis mainly will focus on I became very puzzled, because I found the payoff structure very complex to grasp with my “financial common sense”. My first thought was that this is very confusing and I started to ponder about how one should price such a contract. According to Taleb [TAL] bad investment banks use to trick customers by taking advantage of the customers’ statistical misperception and he also claims that: “It is easier to fool someone with a distributional confusion. […] Until customers gained in sophistication, covered writes were the best game in town. As customers started understanding a little more statistics, the game moved to the fancier exotic options payoffs. The distributional confusion moved to the notion of path dependency.” [TAL, p. 351] Therefore it is important to investigate how the valuation of such a contract can be done. If one really understands the underlying logic and the tools used, one has a powerful knowledge applicable to many different situations, even outside the world of finance. A path-dependent option has a payoff at maturity that depends on the history of the underlying asset price as well as its price at maturity [WDH, p. 148]. In this thesis I will develop a mathematical model for pricing of some discretely sampled path-dependent European exotic options. In order to accomplish this I will apply a partial differential equation approach, based on the original Black and Scholes analysis [BSC]. The way I deal with the problem contains no approximations other than inherent in the final numerical solution. I will also investigate the probability density function of the path-dependent quantity in the case of Relax Sverige 1, and also compare the “fair” price I derive for this particular contract with the

*

For a precise description of the contract see pp. 38−39.

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issue price and the price on the market. The path-dependent quantity refers to the extra state variable carrying past information, which is necessary in order to keep track of the potential payoff at expiry.

1.1 Disposition Chapter two presents in a brief manner arbitrage pricing theory and its mathematical background. With this as a foundation, we develop in Chapter three a generalised mathematical model for pricing of some European path-dependent discretely exotic option in the standard Black-Scholes setting. In Chapter four we discuss the numerical solution of the problem using a finite difference approach. Since this is a thesis in mathematics we will use an intuitively understandable numerical method and not the most advanced and efficient one. Establishing the fact that the problem is solvable and that many techniques are available will satisfy us. We will, indeed, find a value numerically, but we will use a basic method with a lot more to wish, but it is enough for our purposes and has hopefully the advantage of transparency. Chapter five contains an analysis of the probability distribution function of the path-dependent quantity in the case of Relax Sverige 1. In Chapter six we compare our derived price with the actual price of the contract Relax Sverige 1. Finally we sum up in Chapter seven with some concluding remarks.

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2 Arbitrage Theory in Continuous Time 2.1 Background There are two main approaches to the valuation of derivative instruments. One of them is the “martingale method”, initiated by Cox and Ross (1976) and Harrison and Kreps (1979). This idea is based on writing the value of the claim as the expected value under a risk-adjusted measure of the discounted payoff. Using probabilistic methods the expectation can be calculated. The other approach is the “partial differential equation method”, due to Black and Scholes (1973) and Merton (1973). This technique consists of constructing for the price of the derivative instrument a partial differential equation together with appropriate boundary conditions. The solution to the partial differential equation can then be found using analytical or numerical methods [DKR]. Dewynne and Wilmott [DEW] advocate approaching the pricing of exotic options from a partial differential equation point of view. They claim this technique is more sophisticated and they encourage the mathematical modelling aspects and the numerical solving of such problems. In valuation of options they prefer this method to the search for explicit solutions because that inevitably leads to compromises in the underlying modelling [DEW]. Moreover, when the problem has been formulated in the form of a partial differential equation form we may consider ourselves to be on well-known territory, because we then have more than a century’s worth of theory which to rely on [WDH, p. iii]. Inspired by these ideas, we have chosen to utilise a partial differential approach when we try to price Relax Sverige 1 and other similar contracts. Before we start with the pricing we introduce some relevant theory in the next section.

2.2 Assumptions and Asset Dynamics All the results henceforth will be derived in the following frameworkunless otherwise statedwhich is an arbitrage-free market model with continuous trading. The foundations of the theory come from stochastic calculus [MUR, p. 109]. Below we will present a brief survey 9

of the important fundamental building blocks, which mainly is based upon [MUR] and [BJÖ], compare also [∅KS, pp. 247]. As in Black and Scholes seminal paper (1973), we, throughout this, thesis make the following assumptions regarding the conditions in the market for the underlying asset as well as for the derivative instrument: •

It is possible to borrow and lend money at the short-term interest rate, which is known and constant through time;

•

The underlying asset pays no dividends;

•

The derivative instrument can only be exercised at maturity;

•

There are no costs associated with buying or selling the derivative instrument or the underlying asset;

•

All securities are perfectly liquid and divisible. However, in contrast to Black and Scholes we do not assume that the market for the derivative instrument is liquid and exist a priori;

•

There are no restrictions on short selling or borrowing.

Before we go on we need to know what a standard Brownian motion is. Definition 2.1

A stochastic process W defined on a probability space (Ω, F, P ) , is called

standard Brownian motion if the following conditions hold. 1. W(0) = 0. 2. The process W has independent increments, i.e. if r < s < t < u then W(u) − W(t) and W(s) − W(r) are independent stochastic variables. 3. For s < t the stochastic variable W(t) − W(s) has the Gaussian distribution with mean 0 and variance t − s. 4. W has continuous trajectories. [BJÖ, p. 27]

In this model the a priori existing market consists of two assets. The first asset in this model we refer to as the underlying risky asset, St, which is assumed to follow a geometric Brownian 10

motion [BSC]. This means that the dynamics of the asset price process is the following linear stochastic differential equation (SDE) of the form dS t = αS t + σS t dWt .

(2.1)

Equation (2.1) should be interpreted as the shorthand version of the following integral equation, where S 0 ∈ R + is the initial asset price,

t

t

0

0

S t = S 0 + α ∫ S u du + σ ∫ S u dWu , ∀t ∈ [0, T ].

The drift and the volatility of S are denoted by α ∈ R and σ > 0 , respectively. For t ∈ [0, T ] , Wt is defined to be a standard Brownian motion defined on a probability space (Ω, F, P). Here P denotes the objective probability measure and F = {Ft }t ≥0 denotes the filtration, where Ft can be said to encode all the information generated on the time interval [0, t ] . If a stochastic process Y is such that we have Y (t ) = Ft for all t ≥ 0 then we claim that Y is adapted to the filtration F. The first integral above is an ordinary Riemann integral; the second one is understood in the Itô sense. To fully understand (2.1) we need Itô’s formula. Theorem 2.1 (Itô’s formula) Let the process Xt have a stochastic differential given by dX t = αdt + σdWt . Let g (t , x ) ∈ C 2 ([0, ∞ ) × R ). Then Yt = g (t , X t ) has a stochastic differential given by

dYt =

2 ∂g (t , X t )dt + ∂g (t , X t )dX t + 1 ∂ g2 (t , X t ) ⋅ (dX t )2 , ∂t ∂x 2 ∂x

where (dX t ) = (dX t )(dX t ) is computed according to the rules 2

dt ⋅ dt = dt ⋅ dWt = dWt ⋅ dt = 0, dWt ⋅ dWt = dt. 11

Proof. A sketch of the proof can be found in [∅KS]. The process Xt is called an Itô process. This class of processes are continuous semimartingales and 't

t

0

0

X t = X 0 + ∫ αdu + ∫ β dWu , ∀t ∈ [0, T ],

gives what is called its canonical decomposition, which we will need later. As Björk puts it, every stochastic integral is a martingale, modulo an integrability condition [BJÖ, p. 24]. Starting from S0 at time 0, the solution to (2.1) is easily found using Itô’s formula to ln (S t ) ,

d (ln (S t )) =

1 1 1 1 2 dS t − 2 (dS t ) = (αS t dt + σS t dWt ) − 2 σS t2 dt St St 2S t 2S t

1 = αdt + σdWt − σ 2 dt. 2 Integration from 0 to t of both sides yields  1    ln (S t ) = ln (S 0 ) +  σWt + α − σ 2 t , ∀t ∈ [0, T ] , 2     which leads to  1    S t = S 0 exp σWt +  µ − σ 2 t , ∀t ∈ [0, T ] . 2    

(2.2)

The second asset the a priori market consists of is risk free and Bt denotes its price process. The dynamics is given by dBt = rBt dt .

(2.3) 12

Let us emphasise that the fact that it has no dW-term is a defining property of a risk free asset. By convention B0 = 1, thus, Bt = ert, for t ∈ [0, T ] , is a solution of (2.3). In this particular case we can interpret Bt as the price of a bond [BJÖ, p. 76].

2.3 Trading Strategies

(

)

Definition 2.2 A trading strategy φ = φ 1 , φ 2 which is an F-adapted stochastic process over the time interval [0, T ] is self-financing if its wealth process V (φ ) , which is set to equal

Vt (φ ) = φ t1 S t + φ t2 Bt , ∀t ∈ [0, T ] , satisfies the following condition t

t

Vt (φ ) = V0 (φ ) + ∫ φ dS u + ∫ φ u2 dBu , ∀t ∈ [0, T ] 1 u

0

0

The class of all self-financing trading strategies is denoted by Φ.

The discounted wealth process is denoted by Vt* (φ ) =

S Vt (φ ) = φ t1 S t* + φ t2 , where S t* = t Bt Bt

refers to the discounted underlying asset price. If φ is a self-financing strategy t

Vt* (φ ) = V0* (φ ) + ∫ φ u1 dS u* , ∀t ∈ [0, T ], 0

is satisfied. We need to exclude arbitrage opportunities from Φ. An arbitrage possibility is equivalent to the possibility of making a positive amount of money out of nothing with probability 1, a socalled free lunch on the financial market. We now define this central concept.

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Definition 2.3 An arbitrage possibility in a financial market is a trading strategy φ such that

V0 (φ ) = 0, P{VT (φ ) ≥ 0} = 1 and P{VT (φ ) > 0} > 0 . We say that the market is arbitrage free if there are no arbitrage possibilities [BJÖ, p. 80, and MUR, p. 232]. In order to restrict trading strategies to absence of arbitrage opportunities we need to define the notion of a P*-admissible trading strategy. Before that we must introduce what a martingale measure respectively a spot martingale measure is. Definition 2.4

A probability measure Q on (Ω, F ) , equivalent to P, is called a martingale

measure for the process S* if S* is a local martingale under Q. Similarly, a probability

measure P* is said to be a spot martingale measure if the discounted wealth of any selffinancing trading strategy follows a local martingale under P* [MUR, p. 113]. Lemma 2.1

A probability measure is a spot martingale measure if and only if it is a

martingale measure for the discounted stock price S*. Proof. For a proof, please see [MUR, pp. 113−114]. In the Black-Scholes framework, the martingale measure for the discounted stock price process is unique, and is explicitly known, as the following result shows. Lemma 2.2 The unique measure Q for the discounted stock price process S* is given by the

Radon-Nikodým derivative  r −α dQ 1 (r − α ) 2  = exp WT − T  , P−a.s. dP 2 σ2  σ 

Under the martingale measure Q, the discounted stock price S* satisfies dS t* = σS t* dWt * , 14

and the process W* which equals

Wt* = Wt −

r −α

σ

t , ∀t ∈ [0, T ] ,

follows a standard Brownian motion on the probability space (Ω, F, Q ) . Proof. See [MUR, p. 114] for an idea of the proof. We are now ready to make the following definitions. Definition 2.5

A trading strategy φ ∈ Φ is called P*-admissible if the discounted wealth

process Vt* (φ ) = Bt−1Vt (φ ), ∀t ∈ [0, T ] ,

( )

follows a martingale under P*. We write Φ P * to denote the class of all P*-admissible trading strategies.

2.4 Completeness and Hedging Definition 2.6 We say that a claim χ can be replicated, alternatively that it is reachable or hedgeable, if there exist a P* -admissible self-financing trading strategy φ such that VT (φ ) = χ , P − a.s.

In this case we say that φ is a hedge against χ. Alternatively, φ is called a replicating or hedging P*-admissible self-financing trading strategy. If every contingent claim is reachable

we say that the market is complete.

15

If a claim is hedgeable we thus have a natural price process ∏(t ; χ ) = Vt (φ ) where φ is consistent with Definition 2.6. By just accepting P*-admissible strategies we have excluded all arbitrage opportunities, consequently we can uniquely define a contingent claims arbitrage free price. Proposition 2.1 Suppose that the claim χ can be hedged using the P* -admissible self-

financing trading strategy φ. Then the only price process ∏(t ; χ ) that is consistent with no arbitrage is given by ∏(t ; χ ) = Vt (φ ) . Proof. If at some time t we have ∏(t ; χ ) > Vt (φ ) then we can make an arbitrage by buying the portfolio and selling the claim short, and vice versa if ∏(t ; χ ) < Vt (φ ) . Later on in this thesis we will need this extremely important theorem. Theorem 2.2 The Black-Scholes model given by

dBt = rBt dt , dS t = αS t dt + σS t dWt , where we assume that r, α and σ are deterministic constants and σ > 0 , is complete. Proof. The proof requires fairly deep results from probability theory and is thus outside the scope of this thesis.

We will refer to Μ = (S , B, Φ(P*)) as the arbitrage-free Black-Scholes model of a financial market.

Theorem 2.3 (Formula of Risk Neutral Valuation)

Let χ be a contingent claim that is attainable in M and matures at time T. The arbitrage-free price, ∏(t ; χ ) = Vt (φ ) , at time t ∈ [0, T ] , is given by

16

 χ ∏(t ; χ ) = Bt E P*  −1  BT χ Proof. Bt E P*   BT

 Ft , ∀t ∈ [0, T ]  V (φ )  Ft  = Bt E P*  T  BT 

 V (φ ) Ft  = Bt t = Vt (φ ) = ∏(t ; χ ) . Bt 

17

3 Generalised Model for Pricing of the Contingent Claims 3.1 Setup The payoff at the date of maturity for the contingent claim is defined by N   S − S i −1     , χ = max λ,β + ∑ min 0, i  S i =1 i −1   

where λ , β are positive real constants. Our main problem is to determine the arbitrage free

price of the claim. We will use the standard notation ∏(t ; χ ) for the price process of the claim χ. In contrast to ordinary vanilla options the function of the value will not just depend on t

and S(t). In this particular case we have to introduce two new variables Z(t) and S (t ) . Thus ∏(t ; χ ) = F (S (t ), t , S (t ), Z (t ) )

where S (t ) = the value of S at the previous sampling = Si.

and i   S j − S j −1    = Zi Z (t ) = max λ , β + ∑ min 0,   S j −1   j =1  

Here the index i refers to the sampling just prior to time t. Exactly what all this means will hopefully be clear in the next section.

18

3.2 Sampling Structure The term to maturity is the fixed time interval [0, TN]. This interval has N number of deterministic equidistantly distributed time points, the sampling dates, denoted by Ti, where i = 1, …, N and

O < T1 < … < TN − 1 < TN. On the sampling dates the path-dependent quantity Z(t) is measured, and takes the constant value Zi for Ti ≤ t < Ti+1. Z is updated on the sampling date Ti according to the following rule   S − S    . Z i = max λ , Z i −1 + min 0, S   

(3.1)

On the time interval [0, T1) Z0 takes the value β. We define Ti to mean the time immediately after a sampling has occurred. When regarding the model on an infinitesimal time scale we have the interpretation that the sampling happens, not at the time point Ti, but rather at Ti −, which is equivalent to Ti − dt.

S SS3i

TT3i

t

Figure 3.1. Illustration of sampling structure.

19

3.2.1 Jump Condition From (3.1) it is obvious that Z might be discontinuous at Ti. A very important observation, though, is that the value of the option immediately before the sampling time and directly after the sampling takes place must be the same. This can easily be understood from an absence of arbitrage point of view; if the option behaved discontinuously across a known sampling date it would provide an arbitrage opportunity. Hence, across a sampling date the value of the contract is continuous. Expressed more compact in mathematical form as lim F (S , t , S , Z i −1 ) = F (S , Ti , S , Z i ) . t →Ti

Thus, in order to avoid arbitrage possibilities the following jump condition must hold F(S , Ti −, S , Z i −1 ) = F(S , Ti , S , Z i ) .

(3.2)

Using (3.1) we find that this can be written as    S − S      . F(S , Ti −, S , Z i −1 ) = F S , Ti , S , max λ , Z i −1 + min 0,  S      

(3.3)

Since Zi-1 does not change from Ti − to Ti we can, without loss of clarity, get rid of its subscript i - 1 and finally get the jump condition    S−S F(S , Ti −, S , Z ) = F S , Ti , S , max λ , Z + min 0,  S   

     .   

(3.4)

We must not forget that the final condition is

  S − S   . F(S, TN , S , Z ) = max λ , Z + min 0, S    

(3.5)

20

3.3 Derivation of the Partial Differential Equation By and large we in this section follow the line of argument Musiela and Rutkowski use in the derivation of the Black-Scholes option valuation formula [MUR, pp. 115−119]. Our task is to find the arbitrage free price process Π (t ; χ ) for the claim χ. First we note that according to our definitions the term to maturity which is the time interval [0, TN ] can be divided into intra-sampling intervals, that is time intervals where no sampling occur, namely, [0, T1 ) ,

[Ti , Ti +1 ) for i = 1, … , N − 1, and finally the interval [TN −1 , TN ) . On each of these intervals separately we can argue as follows. From Theorem 2.2 and Definition 2.6 we know that in Black-Scholes framework the value of the contingent claim can be replicated by the holding of a continuously rebalanced position in risk-free bonds and the underlying asset. The approach we use below will also give us formulae for the replicating strategy. As a starting point we have the assumption that the option price, Π (t ; χ ) , satisfies the equality Π (t ) = F (S (t ), t , S (t ), Z (t ) ) for some function F : R + × [0, T ]× R + × [λ , β ] → R . We may thus

assume that the replicating strategy φ we are looking for has the following form

φ (t ) = (φ t1 , φ t2 ) = (g (S t , t , S t , Z t ), h(S t , t , S t , Z t )),

(3.6)

For t ∈ [0, T ] , where g , h : R + × [0, T ]× R + × [λ , β ] → R are unknown functions. Since φ is assumed to be self-financing, the wealth process V (φ ) , which equals Vt (φ ) = g (S t , t , S t , Z t )S t + h(S t , t , S t , Z t )Bt = F (S t , t , S t , Z t ) ,

(3.7)

needs to satisfy, according to Definition 2.2, the following: dVt (φ ) = g (S t , t , S t , Z t )dS t + h(S t , t , S t , Z t )dB t .

(3.8)

21

(

)

( (

)

(

) )

From (3.7) we obtain, φ t2 = h S t , t , S t , Z t = B t− 1 F S t , t , S t , Z t − g S t , t , S t , Z t S t

substituting this together with the assumptions (2.1) and (2.2) into (3.8), leads to the following equation dV t (φ ) = g (S t , t , S t , Z t )(αS t dt + σS t dWt ) + Bt−1 (F (S t , t , S t , Z t ) − g (S t , t , S t , Z t )S t )rBt dt

= αg (S t , t , S t , Z t )S t dt + σS t g (S t , t , S t , Z t )dWt − rg (S t , t , S t , Z t )S t dt + rF (S t , t , S t , Z t )dt.

This can be re-written as dV t (φ ) = (α − r ) S t g (S t , t , S t , Z t )dt + σS t g (S t , t , S t , Z t )dWt + rF ( S t , t , S t , Z t )dt .(3.9)

We are going to look for the wealth function F in the class of smooth functions on the open domain D = (0,+∞ ) × (0, T ) × (0,+∞ ) × (λ , β ) ; more precisely, we assume that F ∈ C 2,1 ( D ) . Before we use Itô’s formula we note that the stochastic differentials for Z and S are degenerate dS t = 0 , and dZ t = 0 . The reason for this is that these variables can only change value at the discrete set of sampling dates Ti. Applying Itô’s formula to F yields1 1   dF (St , t , St , Z t ) =  Ft (St , t , St , Z t ) + αSt Fs (St , t , St , Z t ) + σ 2 S 2 Fss (St , t , St , Z t )dt + σSt Fs (St , t , St , Z t )dWt . 2   Now we form Yt = F (S t , t , S t , Z t ) − Vt (φ ) . We want to find the Itô differential of the process Y. This is easily accomplished by combining (3.9) and the expression above, which gives

1

Subscripts on F denote partial derivatives with respect to the corresponding variables.

22

1   dYt =  Ft (S t , t , S t , Z t ) + αS t FS (S t , t , S t , Z t ) + σ 2 S t2 FSS (S t , t , S t , Z t )dt + σS t FS (S t , t , S t , Z t )dWt 2   + (r − α ) S t g (S t , t , S t , Z t )dt − σS t g (S t , t , S t , Z t )dWt − rF (S t , t , S t , Z t )dt. (3.10) On the other hand, in view of (3.7), Y vanishes identically, thus dYt = 0 . By virtue of the uniqueness of canonical decomposition of continuous semi-martingales, which we mentioned in Chapter 3, the diffusion term in the above decomposition of Y vanishes. In this case, implying that for every t ∈ [0, T ] we will get

t

∫ σS (F (S , u, S , Z ) − g (S , u, S , Z ))dW u

S

u

u

u

u

ú

u

u

= 0,

0

or equivalently t

∫ S (F (S , u, S , Z ) − g (S , u, S , Z )) du = 0 . 2

2 u

S

u

u

u

u

u

u

(3.11)

0

Expression (3.11) holds if and only if the function g satisfies g (s, t , s , Z ) = Fs (s, t , s , Z ), ∀(s, t , s , Z ) ∈ R + × [0, T ]× R + × [λ , β ] .

(3.12)

From now on we assume that (3.12) holds. Then when plugging (3.12) into (3.10), we get still another representation for Y, namely t

1   Yt = ∫  Ft (S u , u , S u , Z ) + σ 2 S u2 FSS (S u , u , S u , Z ) + rS u Fs (S u , u , S u , Z ) − rF (S u , u , S u , Z )du. 2  0

Whenever F satisfies the following partial differential equation it is thus apparent that Y will disappear.

23

1 Ft (s, t , s , Z ) + σ 2 sFSS (s, t , s , Z ) + rsFS (s, t , s , Z ) − rF (s, t , s , Z ) = 0 . 2

(3.13)

This is the famous Black-Scholes partial differential equation [BSC], but its important to note that F here is a function of four variables, later we will treat the path-dependent quantities as parameters. Now it remains to verify that the replication strategy φ, which equals

φt1 = g (St , t , St , Z t ) = FS (St , t , St , Z t ),

φt2 = h(St , t , St , Z t ) = Bt−1 (F (St , t , St , Z t ) − g (St , t , St , Z t )St ), is P*-admissible. Let us start by investigating if φ is self-financing. Although this property is here almost trivial by the construction of φ, it is nevertheless important to check directly the self-financing property of a given strategy. According to Definition 2.2 we must verify that dVt (φ ) = φ t1 dS t + φ t2 dBt . Since we know that Vt (φ ) = φ t1 S t + φ t2 Bt = F (S t , t , S t , Z ), an application of Itô’s formula gives 1 dVt (φ ) = FS (St , t , St , Z )dSt + σ 2 St2 Fss (St , t , St , Z )dt + Ft (St , t , St , Z )dt , 2

(3.14)

When viewing (3.13) again, we see that 1 Ft (s, t , s , Z ) + σ 2 sFSS (s, t , s , Z ) = rF (s, t , s , Z ) − rsFS (s, t , s , Z ) . 2 Substituting this expression into (3.14) yields dVt (φ ) = FS (s, t , s , Z )dSt + rF (s, t , s , Z )dt − rSt FS (s, t , s , Z )dt , and thus 24

F (S t , t , S t , Z ) − φ t1 S t dVt (φ ) = φ dS t + rBt dt = φ t1 dS t + φ t2 dBt . Bt 1 t

Hence the self-financing property is verified. According to Definition 2.5 of admissibility of trading strategies, we need to check that the discounted wealth process V * (φ ) , which satisfies

t

Vt* (φ ) = V0* (φ ) + ∫ FS (S u , u , S u , Z )dS u* ,

(3.15)

0

follows a martingale under the martingale measure P*. Using dSu* = σSu*dWu* we get

t

Vt* (φ ) = V0* (φ ) + ∫ σS u* FS (S u , u , S u , Z )dWu* 0

From the general properties of the Itô stochastic integral, it is thus clear that the discounted wealth process V * (φ ) follows a local martingale under P*. To conclude that Vt * (φ ) follows a martingale we need to show that the integrand satisfies some integrability conditions. Sufficient integrability follows from the fact that Fs is bounded. For every intra-sampling interval separately we need also to impose boundary value conditions. They are in line with what we found earlier, namely    s − s    F (s, Ti −, s , Z ) = F  s, Ti , s, max λ , Z + min 0,    , s    

and the final condition is   s − s  F(s, TN , s , Z ) = max λ , Z + min 0,  . s     In conclusion, and to stress the procedure’s recursive nature, we formulate this result as follows in Proposition 3.1. 25

3.4 Proposition 3.1 (Pricing equation 1) On the time interval [TN −1 , TN ] we have F (s, t , s , Z ) = F N (s, t , s , Z ) , where FN solves the boundary value problem  ∂F N (s, t , s , Z ) ∂F N (s, t , s , Z ) 1 2 2 ∂ 2 F N (s, t , s , Z ) + rs + s σ − rF N (s, t , s , Z ) = 0,  2 ∂t 2 ∂s ∂s     s − s   F N (s, TN , s , Z ) = max λ , Z + min 0,  .  s     On each half open time interval [Ti −1 , Ti ) we have F (s, t , s , Z ) = F i (s, t , s , Z ) , for i= 2, 3 ,…, N − 1 where Fi, over the closed interval [Ti −1 , Ti ] , solves the boundary value problem  ∂F i (s, t , s , Z ) ∂F i (s, t , s , Z ) 1 2 2 ∂ 2 F i (s, t , s , Z ) + rs + sσ − rF i (s, t , s , Z ) = 0,  2 2 ∂t ∂s ∂s      s − s     F i (s, Ti , s , Z ) = F i +1  s, Ti , s, max λ , Z + min 0,  .  s      

Finally, on the time interval [0, T1 ) we have F (s, t , s , Z ) = F 1 (s, t , s , Z ) , where F1 over the closed interval [0, T1 ] , solves the boundary value problem  ∂F 1 (s, t , s , Z ) ∂F 1 (s, t , s , Z ) 1 2 2 ∂ 2 F 1 (s, t , s , Z ) rs + + s σ − rF 1 (s, t , s , Z ) = 0,  2 ∂t ∂s 2 ∂s      s − s     F1 (s, T1 , s , Z ) = F 2  s, T1 , s, max λ , Z + min 0,   .  s    

26

3.5 Similarity Reduction Now it is time to take a look at the payoff at expiry again; N   S − S i −1   . χ = max λ,β + ∑ min 0, i S i = 1 i −1   

The particular mathematical structure of this claim enables a reduction in the dimensionality, called similarity reduction. The use of a similarity variable will reduce the problem to three dimensions. With the substitution x =

S the payoff can be written as S

N   χ = max λ,β + ∑ min (0, xi − 1) , i =1  

where

xi =

Si . S i −1

According to our previous way of argumentation in Chapter 3.1−3.4 it seems reasonable that the contract value is of the following form F (S t , t , S t , Z t ) = H ( x t , t , Z t ) . As before the path-dependent quantity Z(t) is measured on the sampling dates, and takes the value Zi for Ti ≤ t < Ti+1. Z is updated on the sampling date Ti according to the following rule Z i = max(λ , Z i −1 + min(0, x − 1)) . On the time interval [0, T1) Z0 takes the value β.

27

In this case H satisfies the governing equation ∂H ( x, t , Z ) ∂H ( x, t , Z ) 1 2 2 ∂ 2 H ( x, t , Z ) + rx + x σ − rH ( x, t , Z ) = 0 . ∂t ∂x 2 ∂x 2 The terminal condition is H ( x, TN , Z ) = max(λ , Z + min (0, x − 1)) , and the jump conditions this time become H ( x, Ti −, Z ) = H (1, Ti , max(λ , Z + min (0, x − 1)))

Figure 3.2. Plot of the payoff at maturity.

28

3.6 Proposition 3.2 (Pricing equation 2) On the time interval [TN −1 , TN ] we have H ( x, t , Z ) = H N ( x, t , Z ) , where HN solves the boundary value problem  ∂H N ( x, t , Z ) ∂H N ( x, t , Z ) 1 2 2 ∂ 2 H N ( x, t , Z ) + rx + x σ − rH N ( x, t , Z ) = 0,  2 ∂t ∂x 2 ∂x  N  H ( x, T N , Z ) = max(λ , Z + min (0, x − 1)). 

On each half open time interval

[Ti −1 , Ti )

we have H ( x, t , Z ) = H i ( x, t , Z ) , for

i = 2, 3 ,…, N − 1 where Hi, over the closed interval [Ti −1 , Ti ] , solves the boundary value problem  ∂H i ( x, t , Z ) ∂H i ( x, t , Z ) 1 2 2 ∂ 2 H i ( x, t , Z ) + rx + x σ − rH i ( x, t , Z ) = 0,  2 ∂ ∂ t x 2 ∂ x   H i ( x, Ti , Z ) = H i +1 (1, Ti , max(λ , Z + min (0, x − 1))). 

Finally, on the time interval [0, T1 ) we have H ( x, t , Z ) = H 1 ( x, t , Z ) , where H1 over the closed interval [0, T1 ] , solves the boundary value problem  ∂H 1 ( x, t , Z ) ∂H 1 (x, t , Z ) 1 2 2 ∂ 2 H 1 ( x, t , Z ) + + x σ − rH 1 (x, t , Z ) = 0, rx  2 ∂ ∂ t x 2 ∂x  1 2  H ( x, T1 , Z ) = H (1, T1 , max(λ , Z + min (0, x − 1))). 

To solve this partial differential equation we need to use numerical methods, in our case we will implement a basic explicit finite-difference method.

29

4 Numerical Analysis For the exotic options we deal with in this thesis it is not possible to find a closed-form solution for the value. In order to solve the problem numerically we will use a basic explicit finite difference method with obvious limitations, but as we knowbrevity is the soul of wit, and this approach is enough for our purposes and has hopefully the advantages of being intuitively understandable and transparent. It is also beneficial that the finite difference approach easily permits the model to be extended to for example to allow for stochastic volatility etc. In this chapter we will start by an overall planning of how we are going to tackle the problem. Then the notions of the Greeks are introduced and we define our notation and set up the grid. After that the Greeks are approximated by our grid of values and the initial condition, and boundary conditions are presented. In addition, due to the discretely sampled path-dependent quantity we also need to implement jump conditions. Finally, a comprehensive description of the explicit finite difference method, which we will use, can be found.

4.1 Outline The partial differential equation for the option value between sampling dates is just the basic Black-Scholes equation with Z treated as a parameter. Thus the strategy for valuing the option is as follows: •

Solve ∂H ( x, t , Z ) ∂H ( x, t , Z ) 1 2 2 ∂ 2 H ( x, t , Z ) + rx + x σ − rH ( x, t , Z ) = 0 ∂t ∂x 2 ∂x 2 Between sampling dates, using the value of the option immediately before the next sampling date as final data. This gives the value of the option until immediately after the present sampling date.

30

•

Then apply the appropriate jump condition across the current sampling date to deduce the option value immediately before the present sampling date.

•

Repeat this process as necessary to arrive at the current value of the option.

This outline is identical to Wilmott et al suggestion on how to handle discretely sampled Asian options [WDH, p. 176].

4.2 The Greeks In Chapter 3.5 we found that the function H of the value of the option satisfies the governing partial differential equation ∂H ( x, t , Z ) ∂H ( x, t , Z ) 1 2 2 ∂ 2 H ( x, t , Z ) + rx + x σ − rH ( x, t , Z ) = 0 . ∂t ∂x 2 ∂x 2 Before we start with the numerical solution its time to introduce the notion of what usually is referred to as the Greeks. The Greeks are closely related to hedging. We will use the standard notation Definition 4.1 ∂H ∂x ∂2H Γ= 2 ∂ x ∂H Θ= ∂t ∆=

The delta, ∆, of a contingent claim is the sensitivity of the option to the underlying asset. The second derivative of the value function is called gamma, Γ. This means that gamma is the

31

sensitivity of the delta to the underlying. Theta, Θ, indicates the rate of change of the option price with time. Now we can rewrite the partial differential equation in its so-called Greek form

Θ + rx∆ +

1 2 2 x σ Γ − rH = 0 2

(4.1)

In the sequel we will refer to the Greek form of the partial differential equation.

4.3 Differentiation Using a Grid Let us first define our notation. The underlying asset step will be δx , the time step δt , and δZ is the step size for the path-dependent quantity Z. Thus we have a three dimensional grid made up of the points at underlying asset values x = lδ x ,

times t = T − kδt ,

and path-dependent quantity values Z = j δZ where 0 ≤ l ≤ L , 0 ≤ k ≤ K and 0 ≤ j ≤ J . We will write the option value at each grid point as H lk, j = H (lδx, T − kδt , jδZ ) , thus the time variable is the superscript and the subscript the

asset variable and the path-dependent quantity respectively. Observe that we have changed the direction of time; as k increases real time decreases.

32

4.4 Approximating the Derivatives From the definition of the first time-derivative of H we have H ( x, t + g , Z ) − H ( x, t , Z ) ∂H = lim . g → 0 ∂t g

Consequently we can approximate the options Θ, the time derivative, from our grid of values using H lk, j − H lk, +j 1 ∂H ( x, t , Z ) ≈ . δt ∂t

Reasoning in the same way we choose to approximate the options ∆, the first x derivative, using the central difference H lk+1, j − H lk−1, j ∂H ( x, t , Z ) ≈ . 2δx ∂x

The second derivative of the option with respect to the underlying, is called the Γ. The natural approximation for this is H lk+1, j − 2 H lk, j + H lk−1, j ∂2H ( x, t , Z ) ≈ . ∂x 2 δx 2 The delta and gamma are going to appear in the algorithm in these forms but the theta will we get from the governing partial differential equation because we can rewrite (4.1) as

Θ = rH − rx∆ −

1 2 2 x σ Γ 2

Recall that theta indicates the rate of change of the option price with time. In the algorithm theta will therefore be used in the following way

33

H lk, +j 1 = H lk, j − δt ⋅ Θ lk, j .

4.5 Final Conditions and Payoffs We know for certain that the value of the option at maturity just is the payoff function. Hence we do not have to solve anything at time T. At maturity we have H ( x, T , Z ) = Payoff ( x, Z )

or, in our finite difference notation, H l0, j = Payoff (lδx, jδZ ) .

We know the function on the right-hand side. This final condition will start our finitedifference scheme. In our case we have changed the direction of time and thus the originally final condition has become an initial condition instead of a final condition. H l0, j = max(λ , jδZ + min (0, lδx − 1))

4.6 Boundary Conditions At the stage when we solve the partial differential equation numerically we need to specify the option value at the extremes of the region. More precisely, we must prescribe the option value at x = 0 and at x = Lδx. Since the payoff is guaranteed at x = 0, we set H 0k,+j1 = H 0k, j (1 − r ⋅ δt ) .

After a look at Figure 3.2 again we see that the payoff at maturity indicates that an appropriate upper boundary condition is 34

∂H (x, t , Z ) → 0 as x → ∞ . ∂x The finite difference representation of this is H Ik, j = H Ik−1, j .

4.7 Jump Conditions As we already have seen we have jump conditions due to the discretely sampled quantity. So as well as having final conditions (initial conditions), boundary conditions, and the partial differential equation to satisfy we also have jump conditions. Recall from Chapter 3.5 that the derived jump conditions mathematically were written as H ( x, Ti −, Z ) = H (1, Ti , max(λ , Z + min (0, x − 1))) .

(4.2)

When we are solving numerically we will find the option value just after the discretely sampling date, which is the right-hand side of equation (4.2), having “timestepped” from maturity. This means that we know H ( x, t , Z ) at a finite number of values of x and Z at t = Ti and we want to find H ( x, t , Z ) at t = Ti − . It is not likely that the value of max(λ , Z + min (0, x − 1)) will correspond to one of the finite number of values of Z for which

the option value is known. Hence, we must try to find an approximate value by interpolation. Observe that the value of the option after the jump is always in terms of its value before the jump but for smaller or equal values of Z. Before continuing with the “timestepping" backward in time we want to implement (4.2). When setting up the grid we ensure that the sampling dates coincide with grid points. The implementation of the jump condition (4.2) is carried out by the use of linear interpolation. This method is of the same accuracy as the other methods described above. To implement the jump condition we must find the grid points between which lies the point Z = max(λ , Z + min (0, x − 1)) . Then we interpolate between these two grid points to find an

accurate value for the option value before the path-dependent quantity is updated. 35

For further discussion of the treatment of jump conditions see [WIL, pp. 907−910] and [WDH, p. 342]

4.8 The Explicit Finite-Difference Method In this section we will introduce the ideas behind the finite-difference method in a very brief manner. This presentation follows Wilmott’s exposition in [WIL, pp. 878−879], for further reading see for example [MOM]. As before the governing equation is ∂H ( x, t , Z ) ∂H ( x, t , Z ) 1 2 2 ∂ 2 H ( x, t , Z ) + rx + x σ − rH ( x, t , Z ) = 0 . ∂t ∂x 2 ∂x 2 We now write this as ∂H ∂H ∂2H + a ( x, t , Z ) + b( x, t , Z ) 2 + c( x, t , Z )H = 0 ∂t ∂x ∂x

(4.3)

If we are solving a backward equation the only constraint we need to impose on the coefficients is that a > 0. At this stage we put the approximations of the derivatives into (4.3) and get H lk, j − H lk, j

δt

 H lk, j − 2 H lk, j + H lk, j +a   δx 2  k l, j

 H k − H lk−1, j   + blk, j  l , j   2δx 2  

  + c lk, j H lk, j  

The trick we now perform is to rearrange this difference equation to put all of the k + 1 terms on the left-hand side:

(

)

H lk, +j 1 = Alk, j H lk−1, j + 1 + Blk, j H lk, j + C lk, j H lk, j

(4.4)

36

where 1 v 2 blk, j 2 k = −2v1 a l , j + δtc lk, j

Alk, j = v1 a lk, j − Blk, j and

C lk, j = v1 a lk, j +

1 v 2 blk, j 2

where

v1 =

δt δt and v 2 = . 2 δx δx

Equation (4.4) only holds for l = 1, ... , L − 1 , i.e. for interior points, since H −k1, j and H Lk +1, j are not defined. Thus we have L + 1 unknowns and L – 1 equations. The remaining two equations come from the two boundary conditions on l = 0 and l = L. One can easily realise that if we know H lk, j for all l, then equation (4.4) gives us H lk, +j 1 . The point is now that we know the payoff at maturity, i.e. H l0, j , thanks to this we can easily calculate the option value one timestep before maturity, H l1, j . In other words, we can work backwards from maturity step by step to find the option value today. Because the relationship between the option values at timestep k + 1 is a simple function of the option values at timestep k this method is called the explicit finite difference method.

37

5 Analysis of the Distribution of a Path-dependent Quantity As suggested in the introduction it might be easy to be fooled by the subtle properties of the path-dependent structure in this particular kind of exotic option. In this chapter we will investigate the distribution of the path-dependent quantity for an existing exotic option. The particular recursive properties of Z (see Chapter 3) make it difficult to derive the distribution theoretically. Therefore we have chosen an empirical approach in the analysis of the distribution of the path-dependent quantity. In this chapter we start by describing the real world example contract Relax Sverige 1. It is this claim we will investigate in this chapter. Then we set up the Monte Carlo simulation. After that, two simulations are performed, simulation 1 and simulation 2, and the results from the simulations are presented. The chapter is concluded by the estimation of the probability density function for the path-dependent quantity in the case of Relax Sverige 1. This is done using a graphical device, which teaches us about the shape of the probability density function of the path-dependent quantity at all sampling dates.

5.1 A Practical Example The payoff of the contract Relax Sverige 1 is based on the monthly performance of the Swedish stock index OMX. The payoff starts at 70 percent but is reduced by the sum of the negative monthly returns of OMX during the term to maturity. The term to maturity is three years. The investor is, however, guaranteed at least a payoff of 3 percent. Each monthly period ends on the eighth every month except for the last period that ends on January the fifth 2006. Relax Sverige 1 is defined by the terms below: Start date:

8/01/2003

Maturity date:

24/01/2006 38

Underlying asset:

OMX Index, S

Index levels:

Si = Closing level of OMX Index on start date + i months

Contract buyer receives at maturity: 36   S − S i −1     Notional Amount * max1.03,1.7 + ∑ min 0, i  S 1 i = i −1   

5.2 Monte Carlo Simulation Choosing the drift rate when trying to find the distribution of the path-dependent quantity might be confusing. In practice we have a choice between using the risk-free rate or the real world drift rate. Well, in this thesis we have assumed the underlying asset to follow a geometric Brownian motion and we have also shown that the assumed market model is complete. This implies that the martingale measure is unique and therefore we should use the risk-free rate as our drift in the simulations. Consistently with the argument above the risk-neutral dynamics of the asset price process is the following linear stochastic differential equation (SDE) of the form dS t = rS t + σS t dWt .

(5.1)

As we know from Chapter 2.2 we were able to find a simple and exact time-stepping algorithm for the geometric Brownian motion. We obtained (2.2) which we now express differently, over a timestep δt,  1    S (t + δt ) = S (t ) + δS = S (t ) exp σ δtφ +  r − σ 2 δt  , 2     where φ is a standardised normal random variable [BOR, p. 114]. Since this expression is exact, δt does not need to be small. Due to this it is the best time-stepping algorithm to use. For further reading about Monte Carlo simulations see [BEN] and [WIL].

39

5.3 Results From the Simulations We made two simulations of the risk-neutral random walk as described above, starting at the initial value of the underlying asset, S0, for the lifetime of the exotic option. We simulated 60 000 realisations of the underlying price path for two different choices of the volatility. For these realisations we calculated the path-dependent quantity Zi, for i = 1, …, 36, according to (3.1). Throughout the simulations the interest rate was set to three percent per year and the timestep to 1/12 year. In simulation 1 the volatility was set to 30% per year and in what we call simulation 2 the volatility was set to 20% per year. 250

200

S

150

100

50

34

36

32

30

28

26

24

22

20

18

16

14

12

8

10

6

4

2

0

0

t

Figure 5.1 Examples of simulated realisations of the price path for the underlying asset.

The tables below summarise the data from the simulations in a compact form. We have also constructed a confidence interval for the expected value of the path-dependent quantity at all sampling points. This was done in the following fashion. First we calculated the sample arithmetical average, z i , and the sample standard deviation si for Zi, for i = 1, …, 36, from the aggregate of simulated realisations of the underlying asset path. Then we used the fact that any average of many independent identically distributed random variables are approximately normally distributed, a confidence interval for the expected value can then be constructed according to the following formula  s s  E [Z i ] =  z i − tα / 2 (ni − 1) ⋅ i , z i + tα / 2 (ni − 1) ⋅ i  , for i = 1, …, 36. ni ni   40

We chose to construct at 99% confidence interval for the expected value for Zi [BLH, p. 97 and GUT, p. 173]. Which can be found in Table 5.1 and Table 5.2. Table 5.1. Descriptive statistics from Monte Carlo simulation 1. Variable

Mean

Median

St.Dev.

Minimum

99% Conf. Int. for the Maximum Expected Value

Z0

1,7000

1,7000

0,0000

1,7000

1,7000

1,700

1,700

Z1

1,6666

1,6998

0,0475

1,4061

1,7000

1,666

1,667

Z2

1,6334

1,6482

0,0668

1,2389

1,7000

1,633

1,634

Z3

1,6001

1,6125

0,0816

1,0306

1,7000

1,599

1,601

Z4

1,5663

1,5785

0,0943

1,0300

1,7000

1,565

1,567

Z5

1,5332

1,5452

0,1052

1,0300

1,7000

1,532

1,534

Z6

1,4996

1,5121

0,1150

1,0300

1,7000

1,498

1,501

Z7

1,4658

1,4778

0,1243

1,0300

1,7000

1,464

1,467

Z8

1,4330

1,4446

0,1325

1,0300

1,7000

1,432

1,434

Z9

1,4004

1,4116

0,1392

1,0300

1,7000

1,399

1,402

Z10

1,3679

1,3778

0,1449

1,0300

1,7000

1,366

1,369

Z11

1,3359

1,3451

0,1493

1,0300

1,7000

1,334

1,337

Z12

1,3050

1,3120

0,1518

1,0300

1,7000

1,303

1,307

Z13

1,2750

1,2790

0,1527

1,0300

1,7000

1,273

1,277

Z14

1,2462

1,2452

0,1515

1,0300

1,7000

1,245

1,248

Z15

1,2196

1,2118

0,1487

1,0300

1,7000

1,218

1,221

Z16

1,1946

1,1780

0,1445

1,0300

1,7000

1,193

1,196

Z17

1,1717

1,1455

0,1386

1,0300

1,7000

1,170

1,173

Z18

1,1510

1,1109

0,1321

1,0300

1,6741

1,150

1,152

Z19

1,1323

1,0774

0,1245

1,0300

1,6741

1,131

1,134

Z20

1,1159

1,0435

0,1163

1,0300

1,6674

1,115

1,117

Z21

1,1016

1,0300

0,1077

1,0300

1,6674

1,100

1,103

Z22

1,0891

1,0300

0,0987

1,0300

1,6263

1,088

1,090

Z23

1,0783

1,0300

0,0899

1,0300

1,6263

1,077

1,079

Z24

1,0692

1,0300

0,0812

1,0300

1,6261

1,068

1,070

Z25

1,0616

1,0300

0,0730

1,0300

1,6161

1,061

1,062

Z26

1,0553

1,0300

0,0652

1,0300

1,5865

1,055

1,056

Z27

1,0501

1,0300

0,0581

1,0300

1,5517

1,049

1,051

Z28

1,0458

1,0300

0,0512

1,0300

1,5253

1,045

1,046

Z29

1,0425

1,0300

0,0451

1,0300

1,5253

1,042

1,043

Z30

1,0398

1,0300

0,0396

1,0300

1,5253

1,039

1,040

Z31

1,0376

1,0300

0,0347

1,0300

1,5253

1,037

1,038

Z32

1,0358

1,0300

0,0299

1,0300

1,5253

1,035

1,036

Z33

1,0344

1,0300

0,0261

1,0300

1,5253

1,034

1,035

Z34

1,0334

1,0300

0,0225

1,0300

1,5253

1,033

1,034

Z35

1,0326

1,0300

0,0196

1,0300

1,5253

1,032

1,033

Z36

1,0319

1,0300

0,0167

1,0300

1,4330

1,032

1,032

Note: This Table is based on data from Monte Carlo simulation 1, when r = 3% per year, volatility = 30% per year, timestep = 1/12 year and the total number of simulations where 60 000.

41

To get an intuitive feeling of the probability distribution of the path-dependent quantity given simulation 1, see Appendix A for histograms from simulation 1. Table 5.2. Descriptive statistics from Monte Carlo simulation 2. Variable

Mean

Median

St.Dev.

Minimum

99% Conf. Int. for the Maximum Expected Value

1,7000 1,7000 0,0000 1,7000 1,7000 1,700 1,700 Z0 1,6782 1,7000 0,0317 1,4824 1,7000 1,678 1,679 Z1 1,6564 1,6673 0,0449 1,4065 1,7000 1,656 1,657 Z2 1,6346 1,6438 0,0550 1,3195 1,7000 1,634 1,635 Z3 1,6129 1,6216 0,0636 1,2748 1,7000 1,612 1,614 Z4 1,5909 1,5998 0,0712 1,2166 1,7000 1,590 1,592 Z5 1,5691 1,5779 0,0781 1,1637 1,7000 1,568 1,570 Z6 1,5472 1,5557 0,0845 1,1075 1,7000 1,546 1,548 Z7 1,5253 1,5335 0,0905 1,0602 1,7000 1,524 1,526 Z8 1,5037 1,5124 0,0959 1,0300 1,7000 1,503 1,505 Z9 1,4819 1,4908 0,1010 1,0300 1,7000 1,481 1,483 Z10 1,4600 1,4684 0,1058 1,0300 1,7000 1,459 1,461 Z11 1,4381 1,4463 0,1105 1,0300 1,7000 1,437 1,439 Z12 1,4163 1,4242 0,1147 1,0300 1,7000 1,415 1,418 Z13 1,3945 1,4024 0,1188 1,0300 1,7000 1,393 1,396 Z14 1,3731 1,3810 0,1223 1,0300 1,7000 1,372 1,374 Z15 1,3516 1,3593 0,1254 1,0300 1,7000 1,350 1,353 Z16 1,3304 1,3375 0,1280 1,0300 1,7000 1,329 1,332 Z17 1,3094 1,3155 0,1301 1,0300 1,6937 1,308 1,311 Z18 1,2886 1,2938 0,1316 1,0300 1,6937 1,287 1,290 Z19 1,2683 1,2719 0,1327 1,0300 1,6937 1,267 1,270 Z20 1,2484 1,2505 0,1326 1,0300 1,6937 1,247 1,250 Z21 1,2291 1,2286 0,1319 1,0300 1,6727 1,228 1,230 Z22 1,2105 1,2069 0,1305 1,0300 1,6693 1,209 1,212 Z23 1,1928 1,1849 0,1282 1,0300 1,6591 1,191 1,194 Z24 1,1760 1,1627 0,1253 1,0300 1,6591 1,175 1,177 Z25 1,1604 1,1414 0,1214 1,0300 1,6426 1,159 1,162 Z26 1,1455 1,1197 0,1170 1,0300 1,6332 1,144 1,147 Z27 1,1319 1,0978 0,1123 1,0300 1,6289 1,131 1,133 Z28 1,1192 1,0755 0,1068 1,0300 1,6249 1,118 1,120 Z29 1,1078 1,0540 0,1013 1,0300 1,6249 1,107 1,109 Z30 1,0974 1,0322 0,0954 1,0300 1,6164 1,096 1,098 Z31 1,0881 1,0300 0,0895 1,0300 1,6164 1,087 1,089 Z32 1,0798 1,0300 0,0836 1,0300 1,5658 1,079 1,081 Z33 1,0724 1,0300 0,0776 1,0300 1,5599 1,072 1,073 Z34 1,0658 1,0300 0,0716 1,0300 1,5599 1,065 1,067 Z35 1,0602 1,0300 0,0658 1,0300 1,5599 1,060 1,061 Z36 Note: This Table is based on data from Monte Carlo simulation 2, when r = 3% per year, volatility = 20% per year, timestep = 1/12 year and the total number of simulations where 60 000.

From Table 1, consisting of the data from simulation 1, we can see that the 99% confidence interval for the final payoff, Z36, is [1.032, 1.032] . Table 2 shows the results from simulation 42

2when the volatility was set to 20% instead of 30% as in simulation 1, everything else equaland we see that the confidence interval at 1% level of significance for the payoff at maturity is [1.060, 1.061] . In both cases one should note the steady decline of the value of the path-dependent quantity, Z. To further stress this property we plot graphs of the mean value of Z found in Table 1 and Table 2 as a function of time, see Figure 5.1 and Figure 5.2. As we know the Z is updated on the sampling dates. Between these dates the path-dependent quantity is constant.

1,7 1,6 1,5

N

1,4 1,3 1,2 1,1 1,0

0

10

20

30

40

t

Figure 5.1.

The mean value of Z in Table 1 plotted as a function of time.

1,7 1,6 1,5

N

1,4 1,3 1,2 1,1 1,0

0

10

20

30

40

t

Figure 5.2.

The mean value of Z in Table 2 plotted as a function of time.

43

5.4 Estimation of the Probability Density Function

( )

To find the unknown probability density function of Z i which we denote by f z i , for i = 1, ..., 36 , we use the results from the Monte Carlo simulation. Here i as a superscript denotes the same thing as i as a subscript did before. But we use the superscript instead because we now need the subscript to indicate the random sample of the variable. We i being a random sample of the random variable Z i . We divide the consider z1i , ... , z 60000

sample into k classes. A class consists of data of almost the same values. Let p1, … , pk be the

( )

unknown areas below the density function f z i . From the classification of the sample we plot a histogram and we let f i denote the absolute frequency and

f f1 , ... , k denote the n n

relative frequencies. These frequencies are unbiased estimates of p1, … , pk. This method does not give us a real estimate of the density function, but of the areas below it. But when we choose a fine classification, which we can do because we have a large sample, we get an illustrative picture of what the density function looks like [BLH, pp. 78−79]. In all graphs in Chapter 5.4.1 and Chapter 5.4.2 we have chosen a class width of 0.01. From simulation 2 we just show six graphs, the pattern should be clear from the graphs presented from simulation 1. These graphs are coherent with our findings in Chapter 5.3. Starting with all the probability mass in the point 1.7, after each sampling the mass moves left towards 1.03. The probability for a Z36 = 1.03 in simulation 1 is almost 1 and in simulation 2 the probability of Z36 = 1.03 is almost 0.75. In passing, note that for a lower volatility the convergence towards the lower limit 1.03 of the Z value takes longer time.

44

5.4.1 Estimated Probability Density Functions from Monte Carlo Simulation 1 Estimated Density Function of Z1 n = 60000

Estimated Density Function of Z2 n = 60000 0,3

0,5

Rel. Freq.

Rel. Freq.

0,4 0,3 0,2

0,2

0,1

0,1 0,0

0,0 1,4

1,5

1,6

1,7

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint

Estimated Density Function of Z3 n = 60000


0,15

0,07

0,10

0,05

Rel. Freq.

Rel. Freq.

0,06

0,05

0,04 0,03 0,02 0,01

0,00

0,00

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

Midpoint

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint



0,04

0,03

Rel. Freq.

Rel. Freq.

0,03

0,02

0,02

0,01

0,01

0,00

0,00 1,0

1,1

1,2

1,3

1,4

Midpoint

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

45



Rel. Freq.

Rel. Freq.

0,03

0,02

0,01

0,00

0,01

0,00

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint



0,03

0,03

0,02

0,02

Rel. Freq.

Rel. Freq.

0,02

0,01

0,00

0,01

0,00

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

Midpoint

1,3

1,4

1,5

1,6

1,7

Midpoint



0,04

0,06 0,05

Rel. Freq.

Rel. Freq.

0,03

0,02

0,01

0,04 0,03 0,02 0,01 0,00

0,00

1,0

1,1

1,2

1,3

1,4

Midpoint

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

46



Rel. Freq.

Rel. Freq.

0,10

0,05

0,00

0,10

0,05

0,00

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint



0,2

Rel. Freq.

Rel. Freq.

0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint



0,3

Rel. Freq.

Rel. Freq.

0,3

0,2

0,1

0,2

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

Midpoint

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

47



0,4 0,4

Rel. Freq.

Rel. Freq.

0,3

0,2

0,1

0,0

0,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint



0,6

0,6

0,5

0,5

0,4

0,4

Rel. Freq.

Rel. Freq.

0,2

0,1

1,0

0,3 0,2

0,3 0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

Midpoint

Midpoint



0,7

0,7

0,6

0,6

0,5

0,5

Rel. Freq.

Rel. Freq.

0,3

0,4 0,3 0,2

0,4 0,3 0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

Midpoint

1,5

1,6

1,0

1,1

1,2

1,3

1,4

1,5

1,6

Midpoint

48

0,8

0,8

0,7

0,7

0,6

0,6

0,5 0,4 0,3

0,4 0,3 0,2

0,1

0,1

0,0

0,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,1

1,2

1,3

1,4

1,5

1,6

Midpoint

Midpoint



0,9

0,9

0,8

0,8

0,7

0,7

0,6

0,6

Rel. Freq.

Rel. Freq.

0,5

0,2

1,0

0,5 0,4 0,3

0,5 0,4 0,3

0,2

0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,1

1,2

1,3

1,4

1,5

Midpoint

Midpoint



0,9

0,9

0,8

0,8

0,7

0,7

0,6

0,6

Rel. Freq.

Rel. Freq.


Rel. Freq.

Rel. Freq.


0,5 0,4 0,3

0,5 0,4 0,3

0,2

0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

Midpoint

1,4

1,5

1,0

1,1

1,2

1,3

1,4

1,5

Midpoint

49

1,0

1,0

0,9

0,9

0,8

0,8

0,7

0,7

0,6 0,5 0,4 0,3

0,5 0,4 0,3 0,2

0,1

0,1

0,0

0,0

1,1

1,2

1,3

1,4

1,5

1,0

1,1

1,2

1,3

1,4

1,5

Midpoint

Midpoint



1,0

1,0

0,9

0,9

0,8

0,8

0,7

0,7

Rel. Freq.

Rel. Freq.

0,6

0,2

1,0

0,6 0,5 0,4 0,3

0,6 0,5 0,4 0,3

0,2

0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

1,5

1,0

1,1

1,2

1,3

1,4

1,5

Midpoint

Midpoint



1,0

1,0

0,9

0,9

0,8

0,8

0,7

0,7

Rel. Freq.

Rel. Freq.


Rel. Freq.

Rel. Freq.


0,6 0,5 0,4 0,3

0,6 0,5 0,4 0,3

0,2

0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

Midpoint

1,4

1,5

1,0

1,1

1,2

1,3

1,4

Midpoint

50

5.4.2 Estimated Probability Density Functions from Monte Carlo Simulation 2

Estimated Probability Distribution Function of Z6 n = 60000

Estimated Probability Distribution Function of Z12 n = 60000 0,04

0,05

0,03

Rel. Freq.

Rel. Freq.

0,04 0,03 0,02

0,02

0,01 0,01

0,00

0,00

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint


Estimated Probability Distribution Function of Z24 n = 60000 0,2

0,02

Rel. Freq.

Rel. Freq.

0,03

0,01

0,00

0,1

0,0

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

Midpoint

Midpoint



0,5

0,8 0,7 0,6

Rel. Freq.

Rel. Freq.

0,4 0,3 0,2

0,5 0,4 0,3 0,2

0,1

0,1

0,0

0,0

1,0

1,1

1,2

1,3

1,4

Midpoint

1,5

1,6

1,0

1,1

1,2

1,3

1,4

1,5

1,6

Midpoint

51

6 Comparison of Prices The prices in this section are quoted in percent of the notional amount. When the volatility is set to 30 percent per year and the risk-free interest rate is set to 3 percent per year our Matlabprogram based on Chapter 4 (see Appendix B for the program code) suggests that the value of the exotic option Relax Sverige 1 (defined in Chapter 5.2) three years before maturity is 95 percent. In the case when everything is equal but the volatility instead is set to 20 percent per year, the suggested price is 97 percent. These two prices should be compared to the values we can derive from the Monte Carlo simulations. Using the formula of risk-neutral valuation (Theorem 2.3) and the values of the mean from Table 5.1 and Table 5.2 give us the suggested prices 94 percent and 97 percent respectively. Thus, these two different approaches indicate approximately the same values of the exotic option. The real issue price of Relax Sverige 1 was 100 percent. As we note, there are, indeed, differences that should not be neglected. But as we already clearly have stated, our finite difference implementation is very basic and simple with a lot more to be desired, which might explain the disparity between the suggested prices from the numerical methods. Furthermore, our model assumptions in Chapter 3 oversimplify the reality, and this should explain the discrepancies between our “theoretical fair price” and the observed real price. In Figure 6.1 to Figure 6.6 the interest rate is set to 3 percent per year. In Figures 6.1−6.3 the volatility is everywhere 20 percent per year and in Figures 6.4−6.6 the volatility is set to 30 percent per year.

Figure 6.1. The derived value of Relax Sverige1, 3 years before maturity, when the volatility is 20% per year.

52

Figure 6.2. The derived value of Relax Sverige1, 2 years before maturity, when the volatility is 20% per year.

Figure 6.3. The derived value of Relax Sverige1, 1 year before maturity, when the volatility is 20% per year.

Figure 6.4. The value of the exotic 3 year before maturity, when the volatility is 30% per year.

53



As we can see in the figures above, the value of the option for small values of x and Z, are just the present value of the minimum guaranteed return. When comparing the Figures 6.1−6.3 with the Figures 6.4−6.6 we see that the value of the exotic option is always greater for lower volatility, everything else equal. This property is in line with our expectations.

At July the ninth 2003, a half-year into the contract’s life, the actual market price of Relax Sverige 1 was 101 percent, and the value of the path-dependent quantity was 1.581. Our Matlab-program generates at this particular time a price of 99 percent, given as before that the

54

interest rate is set to 3 percent per year and the volatility is set to 20 percent annually. The plot of the value at this point in time is shown in Figure 8.7.

Figure 6.7.

A plot of the value for Relax Sverige 1, 2.5 year before maturity, when the volatility is 20% per

year.

55

7 Concluding Remarks Using a partial differential equation approach this thesis has developed a mathematical model for pricing of some European path-dependent discretely sampled exotic options in the classical Black and Scholes setting. By introducing auxiliary state variables carrying information about the underlying asset price path, the problems with the path-dependency are overcome. The resulting partial differential equation is the Black-Scholes equation with the path-dependent quantity treated as a parameter, and also jump conditions need to be implemented across sampling dates. This can be solved numerically. In order to quantify the distributional effects for the final payoff when adding path-dependency aspects, the thesis also investigates the distribution of the path-dependent quantity for a particular exotic option. There are no reasons for recapitulating all the results. Instead we would like to discuss and comment some results and to stress obvious limitations and to suggest possible improvements. Some further research is also hinted. Our model for finding the arbitrage free price contains several simplifying assumptions. This means that the comparison of the prices our model suggests to real actual prices is far from straightforward. For instance, we assume that there are no costs associated to transactions, which of course in reality there are. Furthermore, we assume that the volatility is constant over the term to maturity, in real life we know that the future volatility is uncertain and that volatility tends to vary a lot. Moreover, in practice there might be problems with liquidity, although we assume that all the securities are perfectly liquid and divisible. Hence, this should mean that our model tends to underestimate the “real fair” price. As clearly stressed before the numerical methods implemented have obvious limitations. The results therefrom just indicate the prices the model suggests. Still it shows that the problem can be solved numerically and provides us with important insights about the approximate value of the option. However, too strong conclusions should not be drawn. When we study a real world example of an option from the class of exotic options our model price, we note that the prices the Monte Carlo approach indicates are approximately the same as the suggested prices the explicit finite difference method generates. Assuming a constant volatility of 30 percent per year and a constant risk-free rate of 3 percent per year, which 56

should not be too unrealistic, the hinted value for the option’s issue price is a bit below the actual issue price. This is in conformity with our expectations, because “real fair” prices should be higher due to our oversimplifying assumptions of the market conditions. We also find that when 5/6 of the term to maturity remains the deviation between our derived price and the actual market price is smaller than when comparing the issue price. This is sensible because in this particular case it is the emittent who is the market maker in the exotic option. When the emittent sets the issue price, the emittent wants to have a margin to allow for potential hedging costs and profit. Therefore the discrepancy between the price the model suggests and the actual price should be quite large. But as time goes by the emittent could be assumed to set the price more and more close to the real arbitrage free price, because after the issue date the emittent is the one who buys back the contract, due to the market making, and is thus of course not willing to pay a too high price. The most important desirable developments as we see it, are to improve the efficiency and accuracy of the numerical solving of the partial differential equation. It would also be a great advantage to be able to circumvent the assumption of constant volatility by treating the uncertain volatility aspects in a sophisticated way. All this would lead to a more reliable “fair” price. Further research could be to investigate the risk management and hedging of the class of exotic options dealt with in this thesis.

57

8 Appendix A 8.1 Histograms From Simulation 1 In this section the histograms from simulation 1 are presented. The simulations were done with the following parameters: r = 3% per year, volatility = 30% per year, timestep = 1/12 year, and the total number of simulations where 60 000. On the horizontal axis, we have divided the values of the random variable Zi into subintervals. In each class interval we have

30000

15000

20000

10000

Frequency

Frequency

erected rectangles equal in height to the numbers of observations in that class interval.

10000

5000

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

Z1

1,4

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

Z2

4000

8000 7000

3000

Frequency

Frequency

6000 5000 4000 3000 2000

2000

1000

1000

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

Z4

Z3

2000

Frequency

Frequency

1000

1000

500

0

0 1,0

1,1

1,2

1,3

1,4

Z5

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

Z6

58

1000

1000

900 800

Frequency

Frequency

700 500

600 500 400 300 200 100 0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

1400

1000 900

1200

800

1000

700 600

Frequency

Frequency

1,4

Z8

Z7

500 400 300 200

800 600 400 200

100

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

Z10

Z9

4000 2000

Frequency

Frequency

3000

1000

2000

1000

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

6000

1,4

8000 7000

5000

6000

4000

Frequency

Frequency

1,3

Z12

Z11

3000 2000 1000

5000 4000 3000 2000 1000 0

0 1,0

1,1

1,2

1,3

1,4

Z13

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

Z14

59

15000

Frequency

Frequency

10000

5000

10000

5000

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

Z16

Z15

20000

Frequency

Frequency

15000

10000

5000

10000

0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

Z18

Z17

25000

30000

15000

Frequency

Frequency

20000

10000 5000

20000

10000

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

0

1,7

Z19

1,0

1,1

1,2

1,3

1,4

Z20

40000 30000

Frequency

Frequency

30000 20000

10000

0

20000

10000

0 1,0

1,1

1,2

1,3

1,4

Z21

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

Z22

60

40000

40000

Frequency

Frequency

30000

20000

10000

30000 20000 10000 0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

50000

40000

40000

Frequency

Frequency

50000

30000 20000

0

0 1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,6

1,7

1,3

1,4

1,5

1,6

1,7

1,5

1,6

1,7

1,5

1,6

1,7

Z26

Z25

50000

50000

40000

40000

Frequency

Frequency

1,5

20000 10000

1,1

1,4

30000

10000

1,0

1,3

Z24

Z23

30000 20000 10000

30000 20000 10000 0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,3

1,4

Z28

Z27

60000 50000

50000

Frequency

Frequency

40000 30000 20000

40000 30000 20000

10000

10000

0

0 1,0

1,1

1,2

1,3

1,4

Z29

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

Z30

61

60000

50000

50000

40000

40000

Frequency

Frequency

60000

30000

20000

As 10000

30000 20000 10000 0

0 1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

60000

50000

50000

40000

40000

Frequency

Frequency

60000

30000

20000

0

0 1,2

1,3

1,4

1,5

1,6

1,0

1,7

1,1

1,2

1,6

1,7

1,3

1,4

1,5

1,6

1,7

1,5

1,6

1,7

Z34

Z33

60000

60000

50000

50000

40000

40000

Frequency

Frequency

1,5

20000 10000

1,1

1,4

30000

10000

1,0

1,3

Z32

Z31

30000 20000 10000

30000 20000 10000 0

0 1,0

1,1

1,2

1,3

1,4

Z35

1,5

1,6

1,7

1,0

1,1

1,2

1,3

1,4

Z36

62

9 Appendix B 9.1 Matlab Code function H = exotic(r, vol, lambda, beta, maturity, interval, xzSteps) % step sizes and limits xSteps = xzSteps; minX = 0; maxX = 2; xStep = (maxX - minX) / (xSteps - 1) zSteps = xzSteps; minZ = lambda; maxZ = beta; zStep = (maxZ - minZ) / (zSteps - 1); tStep = 1/(vol^2*xSteps^2) % 1/(20*vol^2*xSteps^2) tStepsPerInt = ceil(interval / tStep); tStep = interval / tStepsPerInt; % make it an integer fraction of interval pause % calculate initial values X = minX:xStep:maxX; Z = minZ:zStep:maxZ; for l = 1:xSteps for j = 1:zSteps H(l, 1, j) = max(minZ, Z(j) + min(0, X(l) - 1)); end end % calculate values for the next step T = 0:tStep:maturity; for k = 1:size(T, 2) disp(['Step number ',num2str(k),' out of ',num2str(size(T, 2))]) % use jump condition or normal updating? if (mod(k,tStepsPerInt)==1) & (k~=1) for j = 1:zSteps for l = 1:xSteps zAfter = max(minZ, Z(j) + min(0, X(l) - 1)); fz = find(zAfter == Z); if size(fz, 2) == 1 H(l, k+1, j) = H(ceil(xSteps/2), k, fz(1)); else fz = find(zAfter < Z); highZ = fz(1); lowZ = fz(1) - 1; H1 = H(ceil(xSteps/2), k, lowZ); H2 = H(ceil(xSteps/2), k, highZ); H(l, k+1, j) = H1 + (zAfter - Z(lowZ)) * (H2 - H1)/(Z(highZ) - Z(lowZ)); end end end else for j = 1:zSteps H(1, k+1, j) = H(1, k, j) * (1 - r * tStep); for l = 2:(xSteps-1) delta(l, k, j) = (H(l+1, k, j) - H(l-1, k, j)) / (2 * xStep); gamma(l, k, j) = (H(l+1, k, j) - 2*H(l, k, j) + H(l-1, k, j)) / xStep^2; theta(l, k, j) = r*H(l, k, j) - r*X(l)*delta(l, k, j) - X(l)^2*vol^2*gamma(l, k, j) / 2; H(l, k+1, j) = H(l, k, j) - tStep * theta(l, k, j); end H(xSteps, k + 1, j) = H(xSteps - 1, k + 1, j); end end end

63

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