Introduction to Stochastic Calculus Applied to Finance

~. r

Introduction to

I.

Stochastic Calculus Applied to Finance

I

Damien Loolberton L'Universite ffSfarne la Vallee France

and Bernard Lapeyre L'Ecole Nationale des Ponts et Chaussees France

Translated by Nicolas Rabeau Centre for Quantitative Finance Imperial College, London . and Merrill Lynch Int. Ltd., London

and

'

Francois Mantion Centrefor Quantitative Finance Imperial College London

CHAPMAN & HALUCRC Boca Raton London New York Washington, D.C.

L-G ~jj(5~3

L3(;/3

'9-96

Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying.

Introduction Options Arbitrage and put/call parity Black-Scholes model and its extensions Contents of the book Acknowledgements 1

2

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to .infringe,

Visit the CRC Press Web site at www.crcpress.com

?

First edition 1996 First CRC reprint 2000 © 1996 by Chapman & Hall No claim to original U.S. Government works International Standard Book Number 0-412-71800-6 Printed in the United States of America 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

3

Discrete-time models 1.1 Discrete-time formalism 1.2 Martingales and arbitrage opportunities 1.3 Complete markets and option pricing 1.4 Problem: Cox, Ross and Rubinstein model Optimal stopping problem and American options 2.1 Stopping time 2.2 The Snell envelope 2.3 Decomposition of supermartingales 2.4 Snell envelope and Markov chains 2.5 Application io American options 2.6 Exercises Brownian motion and stochastic differential equations 3.1 General comments on continuous-time processes 3.2 Brownian motion 3.3 Continuous-time martingales 3.4 Stochastic integral and Ito calculus 3.5 Stochastic differential equations 3.6 Exercises

vii Vll Vlll

IX X X

1 1 4 8 12

17 17 18 21 22 23

25 29 29 31 32 35

49 56

Contents

vi

4

5

The Black-Scholes model

63

4.1 4.2 4.3 4.4 4.5

63 65 67 72 77

Option pricing and partial differential equations 5.1 5.2 5.3 5.4

6

ModeIling principles Some classical models Exercises

Asset models with jumps 7.1 7.2 7.3 7.4

8

European option pricing and diffusions Solving parabolic equations numerically American options Exercises

Interest rate models 6.1 6.2 6.3

7

Description of the model Change of probability. Representation of martingales Pricing and hedging options in the Black-Scholes model American options in the Black-Scholes model Exercises

Poisson process Dynamics of the risky asset Pricing and hedging options Exercises

Simulation and algorithms for financial models 8.1 8.2 8.3

Simulation and financial models Some useful algorithms Exercises

Appendix Al Normal random variables A2 Conditional expectation A3 Separation of convex sets

Introduction

95 95 103 110 118 121 121 127 136

161 161 , 168 170

The objective of this book is to give an introduction to the probabilistic techniques required to understand the most widely used financial models. In the last few years, financial quantitative analysts have used more sophisticated mathematical concepts, such as martingales or stochastic integration, in order to describe the behaviour of markets or to derive computing methods. In fact, the appearance of probability theory in financial modeIling is not recent. At the beginning of this century, Bachelier (1900), in trying to build up a "Theory of Speculation' , discovered what is now called Brownian motion. From 1973, the publications by Black and Scholes (1973) and Merton (1973) on option pricing and hedging gave a new dimension to the use of probability theory in finance. Since then, as the option markets have evolved, Black-Scholes and Merton results have developed to become clearer, more general and mathematicaIly more rigorous. The theory seems to be advanced enough to attempt to make it accessible to students.

173'

Options

141 141 143 150 159

173, 174 178

References

179

Index

183

Our presentation concentrates on options, because they have been the main motivation in the construction of the theory and stilI are the most spectacular example of the relevance of applying stochastic calculus to finance. An option gives its holder the right, but not the obligation, to buy or seIl a certain amount of a financial asset, by a certain date, for a certain strike price. The writer of the option needs to specify: • the type of option: the option to buy is caIled a call while the option to seIl is a

put; • the underlying asset: typicaIly, it can be a stock, a bond, a currency and so on.

viii

Introduction

• the amount of an underlying asset to be purchased or sold; • the expiration date: if the option can be exercised at any time before maturity, it is called an American option but, if it can only be exercised at maturity, it is called a European option; • the exercise price which is the price at which the transaction is done if the option is exercised. The price of the option is the premium. When the option is traded on an organised market, the premium is quoted by the market. Otherwise, the problem is to price the option. Also, even if the option is traded on an-organised market, it can be interesting to detect some possible abnormalities in the market. Let us examine the case of a European call option on a stock, whose price at time t is denoted by St. Let us call T the expiration date and K the exercise price. Obviously, if K is greater than ST, the holder of the option has no interest whatsoever in exercising the option. But, if ST > K, the holder makes a profit of ST - K by exercising the option, i.e. buying the stock for K and selling it back on the market at ST. Therefore, the value of the call at maturity is given by

(ST - K)+

= max (ST

- K,O).

If the option is exercised, the writer must be able to deliver a stock at price K. It means that he or she must generate an amount (ST - K)+ at maturity. At the time of writing the option, which will be considered as the origin of time, Sr is unknown and therefore two questions have to be asked: . 1. How much should the buyer pay for the option? In other words, how should we price at time t = 0 an asset worth (ST - K)+ at time T? That is the problem, of pricing the option. 2. How should the writer, who earns the premium initially, generate an amount (ST - K)+ at time T? That is the problem of hedging the option.

Arbitrage and put/call parity We can only answer the two previous questions if we make a few necessary assumptions. The basic one, which is commonly accepted in every model, is the absence of arbitrage opportunity in liquid financial markets, i.e. there is no riskless profit available in the market. We will translate thatinto mathematical.terms in the first chapter. At this point, we will only show how we can derive formulae relating European put and call prices. Both the put and the call which have maturity T and exercise price K are contingent on the same underlying asset which is worth St at time t. We shall assume that it is possible to borrow or invest money at a constant rate r. Let us denote by Ct and P; respectively the prices of the call and the put at time t. Because of the absence of arbitrage opportunity, the following equation called

Introduction

ix

put/call parity is true for all t < T C t - Pt

= St -

K e-r(T-t).

To understand the notion of arbitrage, let us show how we could make a riskless profit if, for instance,

c, .; Pt > S,

- K e-r(T-t).

At time t, we purchase a share of stock and a put, and sell a call. The net value of the operation is

Ct -

Pt -

St.

If this amount is positive, we invest it at rate r until time T, whereas if it is negative we borrow it at the same rate. At time T, two outcomes are possible: • ST > K: the call is exercised, we deliver the stock, receive 'the amount K and clear the cash account to end up with a wealth K + er(T -t) (Ct - P, - St) .> O. • ST ::; K: we exercise the put and clear our bank account as before to finish with the wealth K + er(T-t)(ct - Pt - St) > O. In both cases, we locked in a positive profit without making any initial endowment: this is an example of an arbitrage strategy. There are many similar examples in the book by Cox and Rubinstein (1985). We will not review all these formulae, but we shall characterise mathematically the notion of a financial market without arbitrage opportunity.

Black-Scholes model and its extensions Even though no-arbitrage arguments lead to many interesting equations, they are not sufficient in themselves for deriving pricing formulae. To achieve this, we need to model stock prices more precisely. Black and Scholes were the first to suggest a model whereby we can derive an explicit price for a European call on a' stock that pays no dividend. According to their model, the writer of the option can hedge himself perfectly, and actually the call premium is the amount of money needed at time 0 to replicate exactly the payoff (ST - K)+ by following their dynamic hedging strategy until maturity. Moreover, the formula depends on only one non-directly observable parameter, the so-called volatility. It is by expressing the profit and loss resulting from a certain trading strategy as a stochastic integral that we can use stochastic calculus and, particularly, Ito formula, to obtain closed form results. In the last few years, many extensions of the Black-Scholes methods have been considered. From a thorough study of the Black-Scholesmodel, we will attempt to give to the reader the means to understand those extensions. r

x

Introduction

Introduction

Contents of the book

The first two chapters are devoted to the study of discrete time models. The link between the mathematical concept of martingale and the economic notion of arbitrage is brought to light. Also, the definition of complete markets and the pricing of options in these markets are given. We have decided to adopt the formalism of Harrison and Pliska (1981) and most of their results are stated in the first chapter, taking the Cox, Ross and Rubinstein model as an example. The second chapter deals with American options. Thanks to the theory of. optimal stopping in a discrete time set-up, which uses quite elementary methods, we introduce the reader to all the ideas that will be developed in continuous time in subsequent chapters. Chapter 3 is an introduction to the main results in stochastic calculus that we will use in Chapter 4 to study the Black-Scholes model. As far as European options are concerned, this model leads to explicit formulae. But, in order to analyse American options or to perform computations within more sophisticated models, we need numerical methods based on the connection between option pricing and partial differential equations. These questions are addressed in Chapter 5. Chapter 6 is a relatively quick introduction to the main interest rate models and Chapter 7 looks at the problems of option pricing and hedging when the price of the underlying asset follows a simple jump process. In these latter cases,' perfect hedging is no longer possible and we must define a criterion to achieve optimal hedging. These models are rather less optimistic than the Black-Scholes model and seem to be closer to reality. However, their mathematical treatment is still a matter of research, in the framework of so-called incomplete markets. Finally, in order to help the student to gain a practical understanding, we have included a chapter dealing with the simulation of financial models and the use of computers in the pricing and hedging of options. Also, a few exercises and longer questions are listed at the end of each chapter. This book is only an introduction -to a field that has already benefited from considerable research. Bibliographical notes are given in some chapters to help the reader to find complementary information. We would also like to warn the reader that some important questions in financial mathematics are not tackled. Amongst them are the problems of optimisation and the questions of equilibrium for which the reader might like to consult the book by D. Duffie (1988).· A good level in probability theory is assumed to read this book: The reader is referred to Dudley (1989) and Williams (1991) for prerequisites. Ho~ever, some basic results are also proved in the Appendix. Acknowledgements

This book is based on the lecture notes taught at l'Ecole Nationale des Ponts et Chaussees since 1988. The-organisation of this lecture series would not have

Xl

been possible without the encouragement ofN. Bouleau. Thanks to his dynamism, CERMA (Applied Mathematics.Institute of ENPC) started working on financial modelling as early as 1987, sponsored by Banque Indosuez and subsequently by Banque Intemationale de Placement. Since then, we have benefited from many stimulating discussions with G. Pages and other academics at CERMA, particularly O. Chateau and G. Caplain. A few people kindly jead the earlier draft of our book and helped us with their remarks. Amongst them are S. Cohen, O. Faure, C. Philoche, M. Picque and X. Zhang. Finally, we thank our colleagues at the university and at INRIA for their advice and their motivating comments: N. El Karoui, T. Jeulin, J.E Le Gall and D. Talay.

,

"

i

t

1

Discrete-time models

The objective of this chapter is to present the main ideas related to option theory within the very simple mathematical framework of discrete-time models. Essentially, we are exposing the first part of the paper by Harrison and Pliska (1981). Cox, Ross and Rubinstein's model is detailed at the end of the chapter in the form of a problem with its solution. 1.1 Discrete-time formalism 1.1.1 Assets

A discrete-time financial model is built on a finite probability space (0, F, P) equipped with a filtration, i.e. an increasing sequence of o-algebras included in F: F o, F 1 , •.. , F N. F n can be seen as the information available at time nand is sometimes called the a-algebra of events up to time n. The horizon N will often correspond to the maturity of the options. From now on, we will assume that F o = {0,O}, FN = F = P(n) and Vw E 0, P ({w}) > O. The market consists in (d + 1) fiflanci;l assets, whose prices at time n are given by the non- negative random variables S~, S~, ... ,S~, measurable with respect to_:fn (investors know past arid present prices but obviously not the future ones). nie vector Sn = (S~, S~, .... , S~) is the vector of prices at time n. The asset indexed by 0 is the riskless asset and we have sg = 1. If the return of the riskless asset over one period is constant and equal to r, we will obtain S~ ~ (1 + rt.'The coefficient /3n = 1/ S~ is interpreted as the discount factor (from time n to time 0): if an amountAn is invested.in.the riskless a~et at time 0, then one dollarwill called risky assets. be available at time n. The assets indexed by i = 1 ... d are ----. . 1.1.2 Strategies Atrading strategy is defined as a stochastic Rrocess (i.e. a§e.q~e in the discrete .(:(. 0 1 d)) . d+1 where rPni denotes the number of c!!.~e)rP = rPn' ~n"'" ~n O~n~N In lR

Discrete-time models

2

shares of asset i held in the portfolio at time n. if> is predictable, i.e.

3

(iii) For any n E {l, ... , N},

~

if>b is Fo-measurable ViE{O,I, ... ,d}

Discrete-time formalism

n

Vn(if»

{ and, for n ~ 1:

= Vo(if» + L

if>~ is F n_ 1-measurable.

This assumption means that the positions in the portfolio at time n (if>~, if>~, ... , if>~) ,are decided with respect to the information available at time (n -1) and kept until time n when new quotations are available.

if>j . !::J.Sj,

j=1 where !::J.Sj is the vector Sj - Sj-l = {JjSj - {Jj- 1Sj-l. Proof. The equivalence between (i) and (ii) results from Remark 1.1.1. The equivalence between (i) and (iii) follows from the fact that if>n,Sn = if>n+l,Sn if and only if = if>n+l.Sn. 0

«s;

The value ofthe portfolio at time n is the scalar product d

Vn(~) =

»;s: =

Lif>~S~. ;=0

Its discounted value is

Vn(if»

= {In (if>n,Sn) = «:s:

s:

with'{Jn = 1/ S~ and = (1, (JnS;, ... , (JnS~) is the v~tor of disco~nted prices. A strategy is called self-financing if the following equation is satisfied for all nE {O,I, ... ,N-I}

if>n,Sn

= if>n+l' S;".

This proposition shows that, if an investor follows a self-financing strategy, the discounted value of his portfolio, hence its value, is completely defined by the ~itial wealth and the strategy (if>~, ... , if>~) O:::;n:::;N (this is only justified because

!::J.SJ

= 0). More precisely, we can prove the following proposition.

Proposition 1.1.3 For any predictable process (( if>~, . . . , if>~))O
any Fo-measurable variable Yo, there exists a unique predictable pr~ce~s (if>~) On+l.Sn+l - if>n,Sn,

Vn(if»

if>~+if>~S~+"'+if>~S~~

(1 -1 + .... + if>j!::J.S d -d)j .

Vo + ~ L.J if>j!::J.Sj j=1

which defines if>~. We just have to check that if>0 is predictable, but this is obvious ITw~"~Ifeequation

or to

Vn+l(if» - Vn(if»

= if>n+dSn+l

-Sn). At time n +'1, the portfolio is worth if>n+l,Sn+l a~d ,Sn+l - if>n+l,Sn is the net gain caused by the price changes between times nand n + I-:--Hence;-tI1e

«:

o

profit or loss realised by following a self-financing strategy is only due to the price moves. The following proposition makes this clear in tenns of discounted prices.

Proposition 1.1.2 The following are equivalent (i)' The strategy if> is self-financing. (ii): For any n E {l, ... , N},

1.1.3 Admissible strategies and arbitrage

+L j=1

where 6.Sj is the vector Sj - Sj-l.

We did not make any assumption on the sign of the quantities if>~. If if>~ -: 0, we have borrowed the amount 1if>~1 in the riskless asset. If if>~ < for i ~ 1, we say that we are short a number if>~ of asset i. Short-selling and borrowing is allowed but the value of our portfolio must be' positive at all times.

°

n

Vn(if» = Vo(if»

II' '

if>j . !::J.Sj,

Definition 1.1.4 A'~trategy if> is admissible if it is self-financing and !jVn( if» ~ foranyn E {O,I, ... ,N}.

°

4

Discrete-time models

The investor must be able to pay back his debts (in riskless or risky asset) at any time.. The notion of arbitrage (possibility of riskless profit) can be formalised as follows: Definition 1.1.5 An arbitrage strategy is an admissible strategy with zero initial value and non-zero final value.

Martingales and arbitrage opportunities

5

Definition 1.2.2 An adapted sequence (Hn)05,n5,N of random variables is predictable if, for all n ~ 1, n; is Fn~1 measurable. Proposition 1.2;3 Let (Mn)05,n5,N be a martingale and (Hn)O
Most models exclude any arbitrage opportunity and the objective of the next section is to characterise these models with the notion of martingale.

Xn

HoMo

=

HoMo

+ H 1.6M1 + ... + H n6.Mn

for n ~ 1

is a martingale with respect to (Fn)05,n5,N'

1.2 Martingales and arbitrage opportunities In order to analyse the connections between martingales and arbitrage, we must first define a martingale on a fj.nite probability space. The conditional expectation plays a central role in this definition and the reader can refer to the Appendix for a quick review of its properties.

(Xn) is sometimes called the martingale transform of (Mn) by (Hn). A consequence of this proposition and Proposition 1.1.2 is that if the discounted prices of the assets are martingales, the expected value of the wealth generated by following a self-financing strategy is equal to the initial wealth. Proof. Clearly, (Xn ) is an adapted sequence. Moreover, for n > 0

E (Xn+l - XnlFn) , E (Hn+lUv/n+l - Mn)IFn) = Hn+lE (Mn+1 - MnlFn) since Hn+l is Fn-measurable .;.' = O.

1.2.1 Martingales and martingale transforms

In this section, we consider a finite probability space (D, F, P), with F = P(D) and Vw E D, P ({w}) > 0, equipped with a filtration (Fnh~n::;N (without necessarily assuming that F N = F, nor F o = {0, D}). A sequence '(Xn)O::;n::;N ofrandom variables is adapted to the filtration!f for any n, X!, is Fn-measurable. Definition 1.2.1 An adapted sequence

(Mn)O::;n~N

of real random variables is:

:S N - 1;

o

a martingale ifE (Mn+1IFn) = Mnfor all n

o

asupermartingale ifE (Mn+lIFn)

o

asubmartingale ifE (Mn+lIFn) ~ Mnforalln:S N-1.

:S Mnforallri:S

N -1;

1. (M n)O::;n5,N is a martingale if and only if

u;

Vj ~ 0

2. If (Mnk:~o is a martingale, thus for any n: E (Mn) = E (MQ) 3. The sum of two martingales is a martingale.

'A E (Xn+1IFn) = E (XnIFn) That shows that (Xn ) is martingale ..

a

= X n. o

The following proposition is a very useful characterisation of martingales.

These definitions can be extended to the multidimensional case: for instance, a sequence (Mn)O
E (Mn+jIFn) ==

Hence

Proposition 1.2.4 An ~dapted sequence ofreal random variables (M~) tingale ifand only iffor any predictable sequence (Hn), we have

E

.-,

6. u;) = O.

Proof. If (Mn) is a ~artingale, the sequence (~n) defined by X o = 0 ~~d, for n ;::: 1, X n = bn=1 H n6.Mn for any predictable process (Hn) is also a martingale, by Proposition 1.2.3. Hence, E(XN) = E(Xo) = O. Conversely, we notice that if j E {I, ... , N}, we can associate the sequence (H n ) defined by H n = 0 for n # j + 1 and Hj+l = lA, for any Frmeasurable A. Clearly, (H n ) is predictable and E

(2::=1 H n6.Mn) = 0 becomes

.

4. Obviously, similar properties can be shown for supennartingales and submartingales.

(t. n;

i~ a ~ar-

E(iA (Mj+l - M j))

Therefore E (!'v!j+ 11 F j

)

=

u;

~O. 0

Martingales and arbitrage opportunities

Discrete-time models

6

If.the.strategy is

1.2.2 Viable financial markets

ti.!!J1!l{I·

that

n ~ N - 1,

(¢) <

k

P (Gn(¢)

< 0) > 0 and 'v'm > n

G m(¢)

(¢)) = 0,

That is the cumulative discounted gain realised by following the self-financing strategy¢;', ... , ¢~. According to Proposition 1.1.3, there exists a (unique) process (¢~) su~h that the strategy (( ¢~, ¢;" ... , ¢~)) is self-financing with zero initial (.¢) is the discounted value of this strategy at time n and because the value. market IS vIabl~, th~ fact tha! this value is positive at any time, i.e Gn (¢ ) 2: 0 for n = 1, ... , N, implies ~hat G N (¢) = O. The following lemma shows that even if we do not assume that G n(¢) are no~-negative, we still have GN(¢) ¢ r. ' (b2) The set V of random varia~es G N(¢), with ¢ predictable process in IRd ; is clearly a vector subspace of IR(where IRo is the set of real random variables defined on n). According to Lemma 1.2.6, the subspace V does not intersect r

2: O.

C!n

n

if j ~ if j > n

u

<

N

0) > o}. It follows from the definition ofn

We can now introduce a new process 'ljJ

where A is the event {Gn(¢)

(V

(bI) To any admissible process (¢;" ... , ¢~) we associate the process defined by

Proof. Let us assume thatG N(¢) E r. First, ifG n(¢) 2: 0 for all n E {O,... , N} the market is obviously not viable. Second, if the Gn (¢) are not all non-negative,

{kiP (G

its initial value is zero, then E*

with VN (¢) 2:0. Hence VN (¢) = 0 since P* ({w}) > 0, for all wEn. (b) The proof of the converse implication is more tricky. Let us call r the convex cone of strictly positive random variables. The market is viable if and only if for any admissible strategy ¢: Vo (¢) = 0 => VN (¢) ¢ r.

Let us get back to the discrete-time models introduced in the first section. Definition 1.2.5 The market is viable if there is no arbitrage opp0'!!!:Eity. Lemma'1.2.6 jf the market is viable, any ~ble process (¢i , ... , ¢d) satis-- -" fies

we define n = sup

admissi~le and

7

o}. Because ¢ is predictable and A is F n-

measurable, 'ljJ is also predictable. Moreover

Gj('ljJ)= { lA(G j(¢)-G1;l(¢))

~

n if j > n

if j

0,

thus, G ('ljJ) 2: 0 for all j E {O,... , N} and G N ('ljJ) > 0 on A. That contradicts j 0 the assumption of market viability and completes the proof of the lemma.

- -----_/ ~:_YYiY.q.le~~he:-diSt;ounteJ£pric.:s._(fl!S~e..~~_P* -

Theorem 1.2.7 .The market is viable if and onl)' if there exists-a-probabil.ity . . ...------.----"-

There~ore.it?oesnoti~tersecttheconvexcompacts·etK = {X E fI Ew X(w)::: I} WhICh IS included m r. As a result of the convex sets separation theorem (see

I,

the Appendix), there exists (oX (w)tEo such that: 1.

vx

E K,

L oX(w)X(w) > O. w

2. For any predictable ¢

martingales.

~ (a) Let us assume that there exists a probability P* equivalent to P under which discounted prices are martingales. Then, for any self-financing strategy

(¢n), (1.1.2) implies n

Vn(¢) =

Vo(¢) +

L ¢j.f:::.Sj. j=l

Thus by Proposition 1.2.3,

(Vn (¢))

is a P" - martingale. Therefore VN (¢) and

Vo (¢) have the same expectation under P*: , E*

t Recall

(VN (¢)) = E* (Vo (¢)).

that two probability measures P I and P2 are equivalent if and only if for any event A. PI (A) = ¢} P2 (A) = 0, Here, P" equivalent to P means that. for any wEn.

p·({w}»o,

°

w

From Property 1: we deduce that oX(w) !, P* defined by '. P* ({w})

=

> 0 for all wEn, so that the probability .

'

oX(w)

Ew' EO oX(w')

is equivalent to P. , Moreover, if we denote by E* the expectation under measure P*, Property 2. means that, for any predictable process (¢n) in IRd,

8

Discrete-time models

Complete markets and option pricing

It follows that for all i E {I, ... ,d} and any predictable sequence (¢~) in JR, we have E*

(t ¢;6.8;)

9

Theore.~ 1.3.4 A viable m~rket is complete if and only if there exists a unique probability measure P equivalent to P under which discounted prices are mar. tingales.

= O.

The probability P* will appear to be the computing tool whereby we can derive closed-form pricing formulae and hedging strategies. Proof. (a) Let us assume that the market is viable and complete. Then, any non-neg~tlve, F N-~~asurable random variable h can be written as h VN (¢), where ¢ IS an admissible strategy that replicates the contingent claim h. Since ¢ is self-financing, we know that

J=I

Therefore, according to Proposition 1.2.4, we conclude that the discounted prices (8~), ... , (8~) are P* martingales. 0

=

)

h SO

1.3 Complete markets and option pricing 1.3.1 Complete markets

N

We shall define a European option" of inaturity N by giving its payoff h 2: 0, FN-measurable. For instance, a call on the underlying SI with strike price K will be defined by setting: h (S}y - K) +. A put on the same underlying asset

=

=

=

+.

(V

n

.

(¢)) O
=2

--

and, since h is arbitrary, PI equal to F.

Remark 1.3.2 In a viable financial market, we just need to find a self-financing strategy worth h at maturity to say that h is attainable. Indeed, if ¢ is a selffinancing strategy and if P* is a probability measure equivalent to P under which

= {0, O}. Therefore

= P 2 on the whole a-algebra F N

assumed to be

(b) Let us assume that the market is viable and incomplete. Then, there exists a random variable h ~ 0 which is not attainable. We call V the set of random variables of the form

(V (¢ )) is also a P*-martingale, being n

a martingale transform. Hence, for n E {O,... , N} Vn(¢)

= E*

N

(VN(¢)IFn).

Uo +

Clearly, if VN (¢) ~ 0 (in particular if VN (¢) = h), the strategy ¢ is admissible.

To assume that a financial market is complete is a rather restrictive assumption that does not have such a clear economic justification as the no-arbitrage assumption. The interest of complete markets is that it allows us to derive a simple theory of contingent claim pricing and hedging. The Cox-Ross-Rubinstein model, that we shall study in the next section, is a very simple example" of complete market modelling. The following theorem gives a precise characterisation of complete, viable financial markets.

L ¢n.6.8n,

(1.1)

n=I

":here Uo is Fo-measlJrable and dictable process,

, Definition 1.3.3 The market is complete if every contingent claim is attainable.

claim.

¢j.6.8j.

j=I

the last equality coming from the fact that ·Fo

Definition 1.3.1 The contingent claim defined by h is attainable if there exists an admissible strategy worthli at time N.

* Or more generally a contingent

N

Thus, if PI and P: are two probability measures under which discounted prices are martingales, that, for i 1 or i

with the same strike price K will be defined by h (K - S}y) In those two examples, which are actually the two most important in practice, h is a function of S N only. There are some options dependent on the whole path of the underlying asset, i.e. h is a function of SO, S1>' .. , SN. That is the case of the so-called Asian options where the strike price is equal to the average of the stock prices observed during a certain period of time before maturity..

discounted prices are martingales, then

_

= VN (¢) = Vo (¢) + L

':

((¢~'''''¢~))o
is an JRd-valued pre-

- -

It follows fr,?m Proposition 1.1.3 and Remark 1.3.2 that the variable hiSf}y does not belong to V. Hence, V is a strict subset of the set of all random variables on (0, F). Therefore, if P* is a probability equivalent to P under which discounted prices are riJ.~ngales and if we define the following scalar product on the set of random ~ariables (X, Y) t-+ E: (XY), we notice that there exists a non-zero random vanable X orthogonal to V. We also write P** ({w})

==

(1 + 211Xlloo X(w)) P*({w}) .

Discrete-time models

10

with II XII"" = sUPwEn IX(w)l. Because E* (X) = 0, that defines a new probability measure equivalent to P and different from P*. Moreover

E**

(t, ¢n.~Sn) =

for any predictable process (( ¢~, ... , ¢~)) O::;n::;N· It follows from Proposition 0

J.3.2 Pricing and hedging contingent claims in complete markets The market is assumed to be viable and complete and we denote by P* the unique probability measure under which the discounted ~rices of finan~ial assets are martingales. Let h be an .1'N -measurable, non-negative random variable and ¢ be an admissible strategy replicating the contingent claim hence defined, i.e.

The sequence

(V

n)

O::;n::;N

model will show how we can compute the option price and the hedging strategy in practice.

is a P* -rnartingale, and consequently

Since an American option can be exercised at any time between 0 and N, we shall define it as a positive sequence (Zn) adapted to (.1'n), where Zn is the immediate profit made by exercising the option at time n. In the case of an American option on the stock SI with strike price K, Zn (S~ - K) +; in the case of the put,

=

Zn = (K - S~) -t-' In order to define the price of the option associated with (Zn)O::;n::;N, we shall think in terms of a backward induction starting at time N. Indeed, the value of the option at maturity is obviously equal to UN = Z N. At what price should we sell the option at time N ., I? If the holder exercises straight away he will earn Z N -1, or he might exercise at time N in which case the writer must be ready to pay the amount ZN. Therefore, at time N - 1, the writer has to earn the maximum between Z N -1' and the amount necessary at time N - 1 to generate ZN at time N. In other words, the writer wants the maximum between Z N -1 and the value at time N - 1 of an admissible strategy paying off Z N at time

N, i.e. S~_I E* (ZNI.1'N- I)' with ZN = ZN /S~. As we see, it makes sense to price the option at time N - 1 as

UN- I that is Vo(¢)

= E* (h/S~) and more generally ,

= S~E* ( ; .

_ Un- I =

l.1'n ) ,

n

= 0,1, ... ,N.

S~ =

At any time, the value of an admissible strategy replicating h is completely determined by h. It seems quite natura} to call Vn (¢) the price of the option: that is the wealth needed at time n to replicate h at time N by following the strategy ¢. If, at time 0, an investor sells the option for'

and

E.* (:~),

let

he can follow a replicating strategy ¢ in order to generate an amount h at time N. In other words, the investor is perfectly hedged. Remark 1.3.5 It is important notice that the computation of the option price only requires the knowledge of P* and not P. We could have just considered a measurable space (fl,.1') equipped with the filtration (.1'n). In other words, we would only define the set of all possible states and the evolution of the information over time. As soon as the probability space and the filtration are specified, we do not need to find the true probability of the possible events (say, by statistical means) in order to price the option. The analysis of the Cox-Ross-Rubinstein

~ax ( Z~-I' S~_I E* ( ~~ l.1'n-

1) )

.

If we assume that the interest rate over one period is constant and equal to r,

N

to

= max (ZN-I, S~_IE* ( ZNI ~N-l)) .

By induction, we defin~ the American option price for n :: 1, ... , N by

'-'

Vn (¢ )

11

J.3.3 Introduction to American options

0

1.2.4 that (Sn)O::;n::;N is a P" -m.artingale.

Complete markets and option pricing

Un- I

ii;

(1 + r)"

= max ( Zn-I, 1 ~ r E* (Un l.1'n-l )) ,

= Un / S~ be the discounted price of the American option.

Proposition 1.3.6 The "sequence (Un)

is a P* -supermartingale. It is the . O
We should note that, as opposed to the European case, the discounted price of the American option is generally not a martingale underP". Proof. From the equality .,

Un- I

= max ( Zn-I, E*

(Un l.1'n-l ) ) ,

it follows that (Un)O::;n::;N is a supermartingale dominating (Zn)O::;n::;N. Let us

12

Discrete-time models

now consider a supermartingale (Tn)O~n~N that dominates (Zn)O~n~N. Then TN 2: lJN and if i; 2: ii; we have

Problem: Cox, Ross and Rubinstein model

13 3. Give examples of arbitrage strategies if the no-arbitrage condition derived in Question (2.) is not satisfied. ' Assume for instance that r :S a. By borrowingan amount So at time 0, we can purchase one share of the risky asset. At time N, we pay the loan back and sell the risky asset. We realised a profit equal to SN - So(1 + r)N which is always positive, since SN 2: So(1 + a)N. Moreover, it is strictly positive with non-zero probability. There is arbitrage opportunity. If r 2: b w,e can make a riskless profit by short-selling the risky asset.

Tn- l 2: E* (Tn l.1'n-l) 2: E* (Un l.1'n-1 ) whence

Tn- l 2: max ( Zn-l, E* (Un l.1'n-1 ))

= Un-I' o

A backward induction proves the assertion that (Tn) dominates (Un).

4. From now o~, w.e assume that r E Ja, b[ and we write p (b - r)/(b _ a). ShowTthat (~n) IS a P-martingale if and only if the random variables T I, T , 2 . : ., N are mdependent, identically distributed (lID) and their distribution is grven by: P(TI 1 + a) p 1 - P(TI 1 + b). Conclude that the market is arbitrage-free and complete.

=

1.4 Problem: Cox, Ross and Rubinstein model

=

The Cox-Ross-Rubinstein model is a discrete-time version of the Black-Scholes model. It considers only one risky asset whose price is Sn at time n, 0 ~ n ~ N, ' and a riskless asset whose return is r over one period of time. To be consistent with the previous sections, we denote S~ (1 + r)". The risky asset is modelled as follows; between two consecutive periods the relative price change is either a or b, with -1 < a < b:

=

Sn(l+a) Sn+l = { Sn(1 + b).

n=

The initial stock price So is given. The set of possible states is then {I + a, 1 + b}N, Each N -tuple represents the successive values of the ratio Sn+d Sn, n 0,1, , N - 1. We also assume that.1'o {0, n} and .1' pen). For n 1, , N, the a-algebra .1'n is equal to a(SI"'" Sn) generated by the random variables SI ,... ,Sn. The assumption that each singleton in has a strictly positive probability implies that P is defined uniquely up to equivalence. We now ,N. If (XI, ... ,XN) is introduce the variables Tn = Sn/Sn-I, for n = 1, one element of n, P{(XI, ... , XN)} P(TI Xl, ,TN XN). As a result, knowing P is equivalent to knowing the law of the N -tuple (T I, T2 ;' ... , TN). We' also remark that for n 2: 1,.1'n = a(TI, ... ,Tn).

= =

=

=

=

=

1. Show that the discounted price (Sn) is a martingale under P if and only if

E(Tn+II.1'n) = 1"+ r, '
s:

The equality E(Sn+I'IFn) = is equivalent to E(Sn+dSnIFn) == 1, since Sn is, Fn-measurable and this last equality is actually equivalentto E(Tn+dFn) = 1 + r . 2. Deduce 'that r must belong to[c, b[ for the market to be arbitrage-free. If the market is viable, there exists a probability P* equivalent to P, under which (Sn) is a martingale. Thus, according to Question 1. '

E*(Tn+IIFn)

=1+r

and therefore E*(Tn+d = 1 + r . Since Tn+l is either equal to 1 + a or 1 + b with non-zero probability, we necessarily have (1 + r) E]1 + a, 1 + b[.

=

IfTi are independent and satisfy P(Ti = 1 + a) = P = 1 - P(Ti = 1 + b), we have E(Tn+dFn) = E(Tn+d = p(1 + a) + (1 - p)(1 + b) = 1 + r and thus, (Sn) is a P-martingale, according to Question I. ' CO,nversely, if for n = 0,1, .. : , N - I, E(Tn+ljFn) = 1 + r, we can write

p

+ a)E (1{T n + 1 = l+ a } IF n ) Then, the following equality

E(1{Tn+

+ (1 + b)E (1{Tn+ 1=1+b}IFn) = 1 + r.

n)

1=l+a}IF

+E

(1{Tn+ 1=1+b}IFn)

= 1,:

~mplie~ that E (l{Tn+l=l+a}jF~) = P and E (1{Tn+ 1=1+b}jFn) induction, we prove that for any xi E {I + a, 1 + b}, P (TI

n

=

= =

= Xl, ... .t; =

x;,)

=

1.- p. By

= II Pi i=l

where Pi = P if Xi = 1 + a and Pi = 1 - pif Xi = 1 + b. That shows that the variables T, are lID under measure P and that P(Ti = 1 + a) = p. We have shown that the very fact that (Sn) is a P-martingale uniquely deterrni~es the distribution of the N-tuple (TI, T 2 , • • • , TN) under P, hence the measure P itself. Therefore, the market is arbitrage-freeand complete. 5. We denote by C; (resp. Pn ) the value at time n, ofa European call (resp. put) on a share of stock, with strike price K and maturity N. (a)

Derive the put/call parity equation

C; - Pn

= Sn -

K(1

+ r)-(N-n),

knowing the put/call prices in their conditional expectation form. If we deEote E* the expectation with respect to the probabilitymeasure P* under

which (Sri) is a martingale, we have

len -

Pn

= = =

"

(1 + r)-(N-n)E* ((SN - K)+ - (K - SN )+IFn)

(1 + r)-(N-n)E* (SN - KIFn) Sn-K(1+r)-(N-n),

14

Discrete-time models

Problem: Cox, Ross and Rubinstein model

the last equality comes from the fact that (Sn) is a P" -martingale. (b), Show that we can write en = c(n, Sn) where c is a function of K, a; b, r andp. When we write SN = S« n::n+l Ti, we get

c; = (1 + r)-(N-n)E"

((Sn

,IT K) : t: -

.=n+l

where, for each N, the random variables

and their mean is equal to J.LN, with limN400(N J.LN) = J.L. Show that the sequence (YN ) converges in law towards a Gaussian variable with mean J.L and variance a 2 •

Fn).

Wejust need to study the con,:ergenceof the characteristic function tPYN of YN. We obtain

+

N

= E(exp(iuYN» =

tPYN(U)

(x

N-n ""'

= (E (exP(iuXf»)t

IT K)

,r

t: -

i=n+l

(N - n).! .

L.J (N-n-J)!J!

"

+

rl (1 _ p)N-n- i

(x(I

+ a)i (1 + b)N-n- i

-

K) .

i=O

6. Show that the replicating strategy of a call is characterised by' a quantity H n b.(n, Sn-l) at time n, where b. wi11 be expressed in terms offunction c.

+

Hence, limj,..... oo tPYN (u) in law.

H~(I

=

+ r)" + HnSn = c(n, Sn).

'Since H~ and H; are Fn_1-measurable, they are functions of Sv,. . . ,Sn-l only and, since is equal to Sn-l (l + a) or Sn-l (1 + b), the previous equality implies

s:

H~(l

+ r)n + HnSn..:.1(I + a) = c(n, Sn-l(I + a»

and

H~(I + rt + HnSn-1(I + b) = c(n, Sn--:l(I Subtracting one from the othtr, it turns out that D.(n, x )

=

+ b».

pJN)

(a) Let

(YN )N~l

=

be a sequence ofrandom variables equal to

YN=xi'+xf+'''+x~

(1 + RT/N)-N'E"

=

E",((l

(K':'" ~o IT Tn)

+ RT/N)-N K

+

- So exp(YN») +

with Y N ' = L::=I log(Tn/(I + r». According to the assumptions, ~he variables = log(Ti/(l + are valued in {-a/VN,a/VN}, and are liD under probability P", Moreover

r»

Xf"

,

E"(XiN ) = (1 _ 2p)~ =

2

trlVN

-trlVN

- e - e ~. VN etrl VN - e- trl VN VN Therefore, the sequence (YN) satisfies the conditions of Question 7.(a), with J1. 2 _a /2. If we write 'Ij;(y) = (Ke- R T - Soe Y)+, we are able to write

IPJN)

E" ('Ij;(YN»

=

< K

RT

=

I'

IE" (((1 + RT/N)-N K - (Ke-

=

=

=

n=l

c(n,x(I+b»-c(n,x(I+a» x(b _ a) :

, 7. We can now use the model to price a call or a put with maturity T on a single stock. In order to do that, we study the asymptotic case when N converges RT/N, log((l + a)j(l + r)) -ajVN and log((l + to 'infinity, and r b)j(l + r)) ajVN. The real number R is interpreted as the instantaneous rate at al1 times between 0 and T, because e RT limN4oo(1 + r)N. a 2 can be seen as the limit variance, under measure P", of the variable log(SN), when N converges to infinity. -, '

(1 + iUJ1.N - a 2 u 2 / 2N + a(I/N») N. = exp (iuJ1. - a 2 u 2 /2), which proves the convergence

(b) Give explicitly the asymptotic prices of the put and the call at time O. For a certain N, the put price at time 0 is given by

We denote H~ the number of riskless assets in the replicating portfolio. We have

=

IIE (exp(iuXf"») i=l

c(n, x) (l +r)-(N-n) E"

XI' areIll), belong to

{-ajVN, ajVN},

Since under the probability P", the random variable n::n+l T, is independent of F« and since S« is Fn-measurable, Proposition A.2.5 in the Appendix allows us to write: C« = c( n, Sn), where c is the function defined by

=

15

So exp(YN») +

-Soexp(yN»)+)1

1(1 + RT/N)-N _

e-RTI.

Since 'Ij; is a bounded t, continuous function and because the sequence (YN) converges in law, we conclude that I'm p'(N)

N~oo

0

=

lim E" ('Ij;(YN»

N .....oo

t It is precisely to be able to work with a bounded function that we studied the put first.

Discrete-time models

16

=

_1_1+ .j21i

2

00

(Ke-RT _ Soe-u2/2+UY)+e-y2/2dy .

-00

"The integral can be expressed easily i~ terms of the cumulative normaldistribution F, so that lim pJN) = Ke- RTF(-d 2) - SoF(-d 1) , N-+oo 2/2)/a, where dl = (log(x/ K) + RT + a d 2 = d 1 - a and

=

F(d)

_1_1 .j21i

d

Optimal stopping problem and American options

e-,,2/ 2dx .

-00

The priceofthecallfollows easilyfromput/call paritylimN-+oo C~N) K e- RT F(d 2 ) .

= SoF(d

1) -

Remark 1.4.1 We note that the only non-directly observable parameter is .a .. Its interpretation as a variance suggests that it should be estimated by statistical methods. However, we shall tackle this question in Chapter 4. Notes: We have assumed throughout this chapter that the risky assets were not offering any dividend. Actually, Huang and Litzenbe~ger (19~8) a~p.ly the same ideas to answer the same questions when the stock IS carrying dIvId.en~s. 1?e theorem of characterisation of complete markets can also be proved WIth infinite probability spaces (cf. Da1ang, Morton and Willinger (1990) and ~orton (1989)). In continuous time, the problem is much more tricky (cf. Hamson and Kreps (1979), Stricker (1990) and Delbaen and Schachermayer (~994)). Th~ theory of complete markets in continuous-time was developed by H~rns~n and Ph~ka .( 198.1, 1983). An elementary presentation of the Cox-Ross-Rubmstem model IS given m the book by I.e. Cox and M. Rubinstein (1985).

The purpose of this chapter is to address the pricing and hedging of American options and to establish the link between these questions and the optimal stopping problem. To do so, we will need to define the notion of optimal stopping time, which will enable us to model exercise strategies for American options. We will also define the Snell envelope", which is the fundamental concept used to solve. the optimal stopping problem. The application of these concepts to American options will be described in Section 2.5. 2.1 Stopping time The buyer of an American option can exercise its right at any time until maturity. The decision to exercise or not at time n will be made according to the information available at time n. In a discrete-time model built on a finite filtered space (S1, F, (Fn)O:::;n:::;N , P), the exercise date is described by a random variable called stopping time. Definition 2.1.1 A random variable IJ taking values in {O,1,2, ... , N} i~'a stopping time if, for any nE {O, 1", . ,N},

{IJ = n} E (

F~.

Remark 2.1.2 As in the previous chapter, we assume that F = P(S1) and P( {w}) > 0, Vw E S1.~ This hypothesis is nonetheless not essential: if it does not hold, the results presented in this chapter remain true almost surely. However, we will not assume F o = {0, S1} and F N = F, except in Section 2.5, dedicated to finance. . ~

,

Remark 2.'1.3 The reader can verify, as an exercise; that IJ is a stopping time if and only if, for any n E {O, 1, ... ,Nt

{IJ :::; n} E F n . We will use this equivalent definition to generalise the concept of stopping time to the continuous-time setting. .

18

Optimal stopping problem and American options

Let us introduce now the concept of a 'sequence stopped at a stopping time'. Let (Xn)O
n

if j if j

~

and for k

~

{VO = ~} = {Uo = Zo} E F o,

1

r;

To demonstrate that (U~o) is a martingale, we write as in the proof of Proposition 2.1.4: n

U:;o = Un/\vo ==

1, we have

ti; + L ¢>j 6.Uj, j=1

n

= x, + L

.

IS

{vo = k} = {Uo > Zo} n··· n {Uk-I> Zk-d n {Uk = Zd E

The stopped sequence (X~)O
Xv/\n

n/\vo O
n

> n.

Proposition 2.1.4 Let (X n ) be an adapted sequence and v be a stopping time.

~

(2.1)

. l a martmga e. Proof. Since UN = ZN, Vo is a well-defined element of {O 1 N} and have ' , ... , . we

Note that X N(w) = Xv(w)(w) (= X j on {v = j}).

Proof. We see that, for n

= inf {n ~ 0IUn = Zn}

Vo .

i.e., on the set {v = j} we have J

19

Proposition 2.2.1 The random variable defined by

is a stopping time and the stopped sequence ( U )

(w) = Xv(w)/\n (w),

X X~ = { X

The Snell envelope

¢>j

ix, -

X j- 1),

where ¢>j = I{vo~j}. So that, for n E {OJ 1, ... ,N - I},

u» n+l - u» = . .¢>n+l (Un+. 1 - Un) = l{n+l5.vo} (Un+l - Un).

j=1

n

where ¢>j = I {j~v}' Since {j ~ v} is the complement of the set {v < j} = {v ~ j - I}, the process (¢>n)Oj )05.j 5. N. 0

By definition, U'; = max (Zn, E (Un+lIFn)) and on the set {n Zn. Consequently Un = ;E (Un+lIFn) and we deduce .

U vo

n+l -

uvo

n =

+ 1 ~ VO}, U; >

1 . {n+l5.vo} (Un+l - E (Un+lIFn))

and taking the conditional expectation on both sides of the equality E ((U::+ 1 --: because {n

n}).

U~o)

+ 1 ~ vol

IFn)

= l{n+l5. vo}E ((Un+l - E(Un+I!Fn))IFn)

E Fn (since the complement of

{n + 1 < vol is {v < -

0 -

Hence

E ((U:;+1 - U:;o) \Fn) .

2.2 The Snell-envelope . In this section, we consider an adapted sequence (Zn)05.n5.N, and define the sequence (Un)05.n5.N as follows:

=

=

In the remainder, we s?all note

o

Tn,N the set of stopping times taking values in

{n, n + 1.'... , N}. Nonce that Tn,N is a finite set since f! is assumed to be finite.

ZN max (Zn, E'(Un+lIFn))

= 0,

which proves that U'» is a martingale.

"In ~ N:"-1.

The study of this sequence is motivated by our first approachof American options (Section 1.3.3 of Chapter 1). We already know, by Proposition 1.3.6 of Chapter 1, that (Un)O
The martingale property of the sequence U'» gives the following result which relates the concept of Snell envelope to the optimal stopping problem. Corollary 2.2.2 The stopping time Vo satisfies

Uo = E (ZvoIFo) = sup) E (ZvIFo). . .

vETo.N

If we. think ~f Zn as the total winnings of a gambler after n games, we see that stopping ~t tlmevvo. maximises the expected gain given F o. Proof. Since U ° IS a martingale, we have

Optimal stopping problem and American options

20

On the other hand, if 1/ E Io,N, the stopped sequence U" is a supermartingale. So that

> E (UNI.ro) = E (UIII.ro) > E (ZIII.ro),

Uo which yields the result.

0

Remark 2.2.3 An immediate generalisation of Corollary 2.2.2 gives

u;

sup E (ZIII.rn) liE/noN

=

.

E (ZlIn

l.rn) ,

where I/n = inf {j ~ nlUj = Zj}. Definition 2.2.4 A stopping time 1/ is called optimalfor the sequence (Zn)o< n < N if - -

E (ZIII.ro) = sup E (ZII!.rO) ~ We can see that I/o is optimal. The following result gives a characterisation of optimal stopping times that shows that I/o is the smallest optimal stopping time. Theorem 2.2.5 A stopping time v is optimal ifand only if (2.2)

Proof. If the stopped sequence U" is a martingale, Uo = E(UIII.ro) and consequently, if (2.2) holds, Uo = E(ZIII.ro). Optimality of 1/ is then ensured by Corollary 2.2.2. Conversely, if 1/ is optimal, we have

Uo

= E (ZIII.ro)

2.3 Decomposition of supermartingales The following decomposition (commonly called 'Doob decomposition') is used in viable complete market models to associate any supermartingale with a trading strategy for which consumption is allowed (see Exercise 5 for that matter). Proposition 2.3.1 Every supermartingale (Un)05,n5,N has the unique following decomposition: Un = M n - An, where (Mn ) is a martingale and (An) is a non-decreasing, predictable process, null at O. Proof. It is clearly seen that the only solution for n Then we must have

Un+1 - Un = M n+1 - M n - (A n+1 - An) . So that, conditioning both sides with respect to F.n and using the properties of M

= E (Un+ll.rn) -

Un

and

M n+1 - M n = Un+1 - E (Un+11.rn) . (Mn ) and (An) are entirely determined using the previous equations and we see that (Mn ) is a martingale and that (An) is predictable and non-decreasing (because (Un) is a supermartingale). 0 Suppose then that (Un) is the Snell envelope of an adapted sequence (Zn)' We can then give a characterisation of the largest optimal stopping time for (Zn) using the non-decreasing process (An) ~f the Doob decomposition of (U~): Proposition 2.3.2 The largest optimal stopping time for (Zn) is given by

S E (UIII.ro) .

_{Ninf in,. A

1/_ -

n+1 =I- O}

if AN = 0 if AN =I- O.

Proof. It is straightforward to see that I/m", is a stopping time using the fact that \A n)o5,n5,N is predictable. From Un = M n - An and because Aj = 0, for J S ~, we deduce that U"fn", = M"fnu and conclude that U"fn", is a martingale. To show the optimality of !{n'"" it is sufficient to prove

Therefore

E (UIII.ro) = E (ZIII.ro) and since UII ~ ZII' UII = ZII' Since E (UIII.ro) = Uo and from the following inequalities

U"fn'" =Z"fn""

Uo ~ E (UlIl\nl.fo) ~ E (U1I1l"0)

We note that

. , (based on the supermartingale property of (U~) we get

N-l

v.: - "L: 1{~",=j}Uj

= E (UIII.ro) = E (E (UIII.rn)l.ro).

But we have UlIl\n ~ E (UIII.rn), therefore that (U::) is a martingale.

.

- (A n+1 - An)

But, since U" is a supermartingale,

E (UlIl\nl.ro)

= 0 is M o = Uo and A o = O.

~A..

Io,N

ZII = UII { and (UlIl\n)05,n5,N is a martingale.

21

Decomposition ofsupe rmartingales

u-; =' E (VIII.rn), which proves 0

)

+

1{"fn",";N}UN

j=O N-l

L: 1{"fnu j=O

=j}

max (Zj, E (Uj+d.rj))

+

1{"fnu=N}ZN,

Application to American options

Optimal stopping problem and American options

22

We have E (Uj+IIFj) = M, - Aj+l and, on the set {vrnox = j}, A j = 0 and Aj+l > 0, so Uj = M j and E (Uj+IIFj) = M j - Aj+l < Uj. It follows that U, = max (Zj, E (Uj+IIFj)) = Zj. So that finally Ulfuv. = Z lfuv. •

I I,

of the sequence (Zn) is given by Un = u(n, X n), where the function u is defined by u(N, x) = 'ljJ(N, x) '
It remains to show that it is the greatest optimal stopping time. If v is a stopping time such that v 2: Vrnax and P (v > vrnax) > 0, then

E(Uv)

= E(Mv) -

E(A v)

= E(Uo) -

E(A v) < E(Uo)

and UV cannot be a martingale, which establishes the claim.

23

= max ('ljJ(n, .), Pu(n + 1, .)) .

2.5 Application to American options

o

From now on, we will work in a viable complete market. The modelling will be

(fl,

F, (Fn)O:::;n:::;N ,P) and, as in Sections 1.3.1 and based on the filtered space 1.3.3 of Chapter 1, we will denote by P* the unique probability under which the discounted asset prices are martingales. 2.4 Snell envelope and Markov chains The aim of this section is to compute Snell envelopes in a Markovian setting. A sequence (Xn)n~O of random variables taking their values in a finite set E is called a Markov chain if, for any integer n 2: 1 and any elements xo, Xl,' .. , Xn-l, X, Y of E, we have

= y!Xo = Xo, ... ,Xn- l = Xn-l, X n = x) = P(Xn+l = ylXn = x) . The chain is said to be homogeneous if the value P(x, y) = P (Xn+l = ylXn = x)

2.5.1 Hedging American options In Section 1.3.3 of Chapter 1, we defined the value process (Un) of an American option described by the sequence (Zn), by the system

, UN { Un

P(Xn+l

does not depend on n. The matrix P = (P(x, y))(X,Y)EEXE' Indexed byE x .E, is then called the transition matrix of the chain. The matrix P has non-negative , entries and satisfies: LYEE P(x, y) = 1 for all x E' E; itis said to be a stochastic matrix. On a filtered probability space ( n, F, (F";)O:::;n~N ,P), we can define tlie notion of a Markov chain with respect to the filtration:

column indexed by E, then P f is indeed the product of the two matrices P and f. It can also be easily seen that a Markov chain, as defined at the beginning of the section, is a Markov chain with respect to.its natural filtration, definedby F n = a(Xo, ... ,Xn ) . ' . C The following proposition is an immediate consequence of the latter definition and the definition of a Snell envelope. ' Proposition 2.4.2 Let (Zn) be an adapted sequence defined by Zn = 'ljJ(n, X n), where (X n) is a homogeneous Markov chain with transition matrix P, taking values in E, and ib is afunctionfrom N x E to JR. Then, the Snell envelope (Un)

ZN

max (Zn, S~E* (Un+l/ S~+lIFn))

'
Thus, the sequence CUn) 'defined by ii; = Un/ S~ (discounted price of the option) is,the Snell envelope, under P*, of the sequence (Zn). We deduce from the above Section 2.2 that . ' ii; = sup' E* ( ZvlFn)

,

vETn.N

and consequently

u; = S~

Definition 2.4.1 A sequence (Xn)O:::;n:::;N of random variables taking values in

E is a homogeneousMarkov chain with respectto thefiltration (Fn)O
= =

sup E* vETn.N

(SZ~v IFn) .

From Section 2.3, we can write .

ii; = Mn -

An,

where (Mn ) is a Pt-martingale and (An) is an increasing predictable process, null at O. Since the market is complete, there is a self-financing strategy ¢such that o VN (¢) = SNM N, i.e.,

VN

(¢) =

MN. For the seque?ce (Vn Vn(¢)

(¢)) is a P* -martingale,

E* (VN(¢)!Fn ) E* (MNIFn )

w~ have

24

Optimal stopping problem and American options

Therefore

Un = Vn(cP) - An, where An = S~ An. From the previous equality, it is obvious that the writer of the option can hedge himself perfectly: once he receives the premium Uo = Vo(cP), he can generate a wealth equal to Vn(cP) at time n which is bigger than Un and a fortiori Zn. What is the optimal date to exercise the option? The date of exercise is to be chosen among all the stopping times. For the buyer of the option, there is no point in exercising at time n when U« > Zn, because he would trade an asset worth Un (the option) for an amount Zn, (by exercising the option). Thus an optimal date T of exercise is such that UT ~ ZT' On the other hand, there is no point in exercising after the time /I"", = inf {j, A j +! i- O} (which is equal to inf

{j,

Aj +! i- 0})

eF

2.5.2 American options and European options

'

Proposition 2.5.1 Let Cn be the value at time n ofan American option described by an adapted sequence (Zn)O
=C n

\In E {O,I, ... ,N}.

-

The inequality Cn ~ Cn makes sense since the American option entitles the holder '. to more rights than its European counterpart. c.. Proof. For the discounted value

Cn

(C

n)

Cn~C;'

is a supermartingale ~nder p. , we have

~ E· (CNIFn) = E· (cNIFn) = cn.

\lnE{O,I, ... ,N}.

o Remark 2.5.2 One checks readily that if the relationships of Proposition 2.5.1 did not hold, there would be some arbitrage opportunities by trading the options. To illustrate the last proposition, let us consider the case of a market with a single risky asset, with price Sn at time n and a constant riskless interest rate, equal to r ~ 0 on each period, so that S~ = (I + r)". Then, with notations of Proposition 2.5.1, if we take Zn = (Sn - K)+, Cn is the price at time n of a European call with maturity N and strike price K on one unit of the risky asset and Cn is the price of the corresponding American call. We have

cn

because, at that time, selling the optia,n

provides the holder with a wealth UJ.iruu = VJ.iruu (cP) and, following the strategy cP from that time, he creates a portfolio whose value is strictly bigger than the .option's at times /1m.. + I, /1max + 2, ... , N. Therefore we set; as a second condition, T :::; /I""" which allows us to say that is a martingale. As..a result, optimal dates of exercise are optimal stopping times for the sequence (Zn), under probability p ". To make this point clear, let us considerthe writer's point of view. If he hedges himself using the strategy cP as defined above and if the buyer exercises at time T which is not optimal, then UT > ZT or AT > O. In both cases, the writer makes a profit VT(cP) - ZT = UT + AT - Zro which is positive.

c, ~ Cn.

25

If Cn ~ Zn for any n then the sequence (cn), which is a martingale under P", appears to be a supermartingale (under P") and an upper bound of the sequence (Zn) and consequently

and consequently

Hence

Exercises

+ r)-NE· ((SN - K)+IFn) > E· (SN - K(I + r)-NIFn) , (I

=

-N Sn-K(I+r) ,

s; -

using the martingale property of (Sn). Hence: ~n ~ K(I + r)-(N-n) ~ Sn - K, for r ~ O. As Cn .~ 0, we also have c., 2: (Sn - K) + and by Proposition 2.5.1,Cn = Cn. There is equality between the price of the European call and the price of the corresponding American call. , This property does not hold for the put, nor in the case of calls on currencies or dividend paying stocks. Notes: For further discussions .on the Snell envelope and optimal stopping, one may consult Neveu (1972), Chapter VI and Dacunha-Castelle and Duflo (1986), Chapter 5, Section 1. For the theory of optimal stopping in the continuous case, see EI Karoui (1981) and Shiryayev (1978). 2.6 Exercises Exercise 1 Let /I be a stopping time with respect to a filtration (Fn)O
E(XIFv) =

L j=O

l{v=j}E(XIFj).

Optimal stopping problem and American options

26

4. Let T be a stopping time such that

T

2: v, Show that:Fv

C

Fr.

5. Under the same hypothesis, show that if (M n ) is a martingale, we have

(Hint: first consider the case

T

= N.)

Exercise 2 Let (Un) be the Snell envelope of an adapted sequence (Zn). Without assuming that :Fo is trivial, show that E (Uo) = sup E (Zv) , vETo,N

and more generally

vETn,N

Exercise 3 Show that v is optimal according to Definition 2.2.4 if and only if E(Zv)= sup E(ZT)' TETo,N

Exercise 4 The purpose of this exercise is to study the American put in the model of Cox-Ross-Rubinstein. Notations are those of Chapter 1. 1. Show that the price P n, at time n, of an American put on a share with maturity N and strike price K can be written as

P n= Pam(n, Sn)

= (K -

Pam(n,x) = max ( (K - x)+, with f(n + 1,x) = pPam(n + 1,.:r(1 andp = (b- r)/(b - a).

27

6. Show that the hedging strategy of the American put is determined by a quantity H n = ~(n, Sn-d of the risky asset to be held at time n, where ~ can be written as a function of Pam.

Exercise 5 Consumption strategies. The self-financing strategies defined in Chapter 1 ruled out any consumption. Consumption strategies can be introduced in the following way: at time n, once the new prices S~, . . . ,S~ are quoted, the investor readjusts his positions from ¢n to ¢n+1 and selects the wealth 1'n+1 to be consumed at time n + 1. Any endowment being excluded and the new positions being decided given prices at time n, we deduce

¢n+1,Sn

E (Un) = sup E (Zv).

where Pam(n, x) is defined by Pam(N, x)

Exercises

= ¢n,Sn -

1'n+l.

(2.3)

So a trading strategy with consumption will be defined as a pair (¢,1'), where ¢ is a predictable process taking values in IR d+1, representing the numbers of assets held in the portfolio and l' = bnh
s;»:

(a) The pair (¢, 1') defines a trading strategy with consumption. (b) For any n E {l, .. :,N}, n

n

x)+ and, for n :::; N - 1,

f(n+1,x)) 1+r , '

+ a)) + (1 ., p)P~;.,,(n + 1,x(1 + b))

2. Show that the function Pam(O,.) can be expressed as

Pam(O,x) = sup E' ((1 + r)-V(K - xVv)+) , vETo,N

where the sequence of random 'variables (Vn)O::;n::;N is defined by: Vo 1 and, for n 2: 1, Vn = TI7=1 Ui, where the U/s aresome random variables. ' Give their joint law under P". 3. From the last formula, show that the function x !-t' Pam(O,x) is convex and non-increasing. " 4. We assume a < 0. Show that there is' a real number x' E [0, K] such that, for x:::; x', Pam (0, x) = (K - x)+ and; for x E ]x', K/(l + a)N[, Pam(O, x) >

(K - x)+. 5. An agent holds the American put at time 0. For which values of the spot So would he rather exercise his option immediately?

Vn (¢ ) = Vo(¢) + L

j=l

¢j.~Sj - L 1'j. j=l

(c) For any n E {I, ... ,N}, n

n

Vn(¢ ) = Vo(¢ ) + L¢j·~Sj - L1'j/SJ-1' j=l

j=l

2. In the remainder, we assume that the market is viable and complete and we denote by P' the unique probability under which the assets discounted prices are martingales. Show that if the pair (¢, 1') defines a trading strategy with consumption, then (V~(¢)) is a supermartingale under P". 3. Let (Un) be an adapted sequence such that (Un) is a supermartingale under P". Using the Doob decomposition, show that there is a trading strategy with consumption (¢,1') such that Vn (¢ ) = Un for any n E {O,... , N}. 4. Let (Zn) be an adapted sequence. We say that a trading strategy with consumption (¢, 1') hedges the American option defined by (Zn) if Vn (¢) 2: Zn for any n E {O,1, .. '. , N}. Show that there is at least one trading strategy with consumption that hedges (Zn), whose value is precisely the value (Un) of the American option. Also, prove that any trading strategy with consumption (¢, 1') hedging (Zn) satisfies Vn(¢) 2: Un, for any n E {O,1, ... , N}. '

Optimal stopping problem and American options

28

5. Let x be a non-negative number representing the investor's endowment and let 'Y = bnh
(L::=l 'Yj / SJ-l) ~ x.

3

Brownian motion and stochastic differential • equations

The first two chapters of this book were dealing with discrete-time models. We had the opportunity to see the importance of the concepts of martingales, selffinancing strategy and Snell envelope. We are going to elaborate on these ideas in a continuous-time framework. In particular, we shall introduce the mathematical tools needed to model financial assets and to price options. In continuous-time, the technical aspects are more advanced and more difficult to handle than in discrete-time, but the main ideas are fundamentally the same. Why do we consider continuous-time models? The primary motivation comes from the nature of the processes that we want to model. In practice, the price changes in the market are actually so frequent that a discrete-time model can barely follow the moves. On the other hand, continuous-time models lead to more explicit computations, even if numerical methods are sometimes required. Indeed, the most widely used model is the continuous-time Black-Scholes model which leads to an extremely simple formula. As we mentioned in the Introduction, the connections between stochastic processes and finance are not recent. Bachelier (1900), in his dissertation called Theorie de la speculation, is not only among the first to look at the properties of Brownian motion, but he also derived optjon pricing formulae. We will be giving a few mathematical definitions in order to understand continuous-time models. In particular, we shall define the Brownian motion since it is the core concept orthe Black-Scholes model and appears in most financial asset models. Then we shall state the concept of martingale in a continuous-time set-up and, finally, we shall construct the stochastic integral and introduce the differential calculus associated with it, namely the Ito calculus. It is advisable that, upon first reading, the reader passes over the proofs in small print. as they-are very technical. 3.1 General comments on continuous-time processes What do we exactly mean by continuous-time processes?

30

Brownian motion and stochastic differential equations

31

Brownian motion

Definition 3.1.1 A continuous-time stochastic process in a space E endowed with a a- algebra E is afamily (Xt)tEIR+ of random variables defined on a probability space (n, A, P) with values in a measurable space (E, £).

The r7-algebra associated with r is defined as

Remark 3.1.2

This r7-algebra represents the information available before the random time r. One can prove that (refer to Exercises 8, 9, 10, 11 and 14):

• In practice, the index t stands for the time.

F = {A T

E

A, for any t

~ 0 ,A

n {r

~

t} E Ftl .

• A process can also be considered as a random map: for each w in n we associate the map from IR+ to E: t -t Xt(w), called a path of the process.

Proposition 3.1.6

• A process can be considered as a map from IR+ x n into E. We shall always consider that this map is measurable when we endow the product set IR+ x n with the product e-algebra B(IR+) x A and when the set E is endowed with

• If S is a stopping time, finite almost surely, and (Xt)t~O is a continuous, adapted process, then X s is F s measurable.

E. • We will only work with processes that are indexed on a finite time interval

[O,T].

.

• If S is a stopping time, S is F s measurable.

~ T

• If Sand T are two stopping times such that S

P a.s., then F s eFT.

• If Sand T are two stopping times, then S 1\ T = inf(S, T) is a stopping time. In particular; if S is a stopping time and t is a deterministic time S 1\ t is a stopping time.

As in discrete-time, we introduce the concept offiltrat(on.

Definition 3.1.3 Consider the probability space (n, A, P), a filtration (Ft)t>o is an increasing family of a-algebras included in A The o-algebra F t represents the information available at time t. We say that a process (Xtk:~o is adapted to (Ftk~o, if for any t, X, is Ft-measurable.

Remark 3.1.4 From now on, we will be working with filtrations which have the following property If A E A and if P(A)

= 0, 'then for any t, A EFt ..

In other words F t contains all the P-null sets of A. The importance of this technical assumption is that if X = Y P a.s. and Y is Ft-measurable then we can show that X is also Ft-measurable. We can build a filtration generated by a process (Xt)t>o and we write F t = r7(X s , S ~ t). In general, this filtration does not satisfy -the previous condition. However, if we replace F t by :Ft which is the o-algebra generated by both F t and N (the o-algebra generated by all the P-null sets of A), we obtain a proper filtration satisfying the desired condition. We call it the natural filtration of the process (Xth~o. When we talk about a filtration without mentioning anything, it is assumed that we are dealing with the natural filtration of the process that we are considering. Obviously, a process is adapted to its natural filtration. As in discrete-time, the concept of stopping time will be useful. A stopping time . is a random time that depends on the underlying process in a non- anticipative way. In other words, at a 'given time t, we know if the stopping time is smaller than t. Formally, the definition is the following: .

Definition 3.1.5 r is a stopping time with respect to the filtration (Fdt>o if r is a random variable in IR+ U {+oo}, such that for any t 2: 0 {r~t}EFt.

3.2 Brownian motion A particularly important example of stochastic process is the Brownian motion. It will be the core of most financial models, whether we consider stocks, currencies or interest rates. Definition 3.2.1 A Brownian motion is a real-valued, continuous stochastic process (Xdt~o, with independent and stationary increments. In other words:

X, (w) is continuous. • independent increments: If S ~ t, X, - X s is independent ofF s = r7(Xu , U ~

• continuity: P a.s. the map

S I--t

s). • stationary increments: if'S ~ t, X, - X; and X t - s - X o have the same probability law. This definition induces the distribution of the process X t , but the result is difficult to prove and the reader ought to consult the book by Gihman and Skorohod (1980) for a proof of the following theorem. '

Theorem 3.2.2 If (Xt)t>o is a Brownian motion, then X; - X o is a normal random variable with mean rt and variance r7 2 t, where rand o are constant real numbers. Remark 3;2.3 A Brownian motion is standard if X o = 0 P a.s.

E(Xd

= 0,

E

(Xi) = t.

From now on, ,a Brownian motion is assumed to be standard if nothing else is mentione~. In that case, the distribution of X, is the following:

(X

2)

1 --exp --

V2ii

where dx is the Lebesgue measure on IR. '

dx

2t'

32

Brownian motion and stochastic differential equations

The following theorem emphasises the Gaussian property of the Brownian motion. We have just seen that for any t, X, is a normal random variable. A stronger result is the following: Theorem 3.2.4 If (Xdt?o is a Brownian motion and if 0 ~ (Xt1, ... , X t n ) is a Gaussian vector.

tt < ... < t«

then

The reader ought to consult the Appendix, page 173, to recall some properties of Gaussian vectors. Proof. Consider 0 ~ tl < ... < t«. then the random vector (X t1, X t2 X t1, ... , X t n - X tn_1) is composed of normal, independent random variables (by Theorem 3.2.2 and by definition of the Brownian motion). Therefore, this vector is Gaussian and so is (Xt1, ... ,Xt n ) . · 0 We shall also need a definition of a Brownian motion with respect to a filtration

(Ft ) . Definition 3.2.5 A real-valued continuous stochastic process is an (Ft)-Brownian motion if it satisfies: ' • For any t

2:

33

Continuous-time martingales

3. exp (aX t - (a 2/2)t) is an Frmartingale. Proof. If s ~ t then X, - X, is independent of the a-algebra F s . Thus E(Xt XsIFs) = E(Xt - X s)' Since a standard Brownian motion has an expectation equal to zero, we have E(Xt - X s) = O. Hence the first assertion is proved. To show the second one, we remark that

E ((X t - X s)2

=

+ 2X s(Xt -

E ((X t - Xs)2IFs)

Xs)IFs)

+ 2X sE (Xt -

XsIFs) ,

and since (Xdt?o is a martingale E (X t - XsIFs) = 0, whence

Because the Brownian motion has independent and stationary increments, it follows that E ((X t - X s)2JFs ) = E (X't-s) t - s. The last equality is due to the fact that X, has a normal distribution with mean zero and variance t. That yields E (Xl - tlFs ) = X; - s, if s < t. Finally, let us recall that if 9 is a standard normal random variable, we know

0, X, is Ft-measurable.

• Ifs ~ t, X t - Xs.isindependimtofthea-algebraFs. • If s ~ t, X t - X; and X t- s - X o have the same law.

that

Remark 3.2.6 The first point of this definition shows that a(X u , u ~ t) eFt. Moreover, it is easy to check that an Ft-Brownianmotion is also a Brownian motion with respect to its natural filtration.·

E (e-X9) On the other hand, if s


=/+00 e-x:r. e-:r.2/2 dx =e-X 2/2. -00 ~

E (eO'X.-0'2t/2IFs) ="eO'X.-0'2t/2E (i(X'-X')IFs) 3.3 Continuous-time martingales As in discrete-time models, the concept of martingale is a crucial tool to explain the notion of arbitrage. The following definition is an extension of the one in discrete-time.

because X; is Fs-measurable. Since X, - X, is independent of F s ' it turns out that

E (eO'(X'-X')IFs)

Definition 3.3.1 Let us consider a probability space (n, A, P) and a filtration (Fdt?o on this space. An adaptedfamily (Mt)t?o ofintegrable random variables, i.e. E(IMtI) < +00 for any tis:

• a submartingale if, for any s ~ t, E (MtIFs)

E (e0'9vr=s) exp

u;

2: Ms.

That completes the proof.

Remark 3.3.2 It follows from this definition that, if (Mt)t>o is a martingale, then E(Md = E(Mo) for any t. .Here are some examples of martingales. Proposition 3.3.3 If(Xt}t?o is a standard FrBrownian motion: I. X, is an Ft-martingale.

2.

Xl -

t is an Fi-martingale.

E (eO'(X'-X')) E (eO'x,-,)

• a martingale if, for any s ~ t, E (MtIFs) = M s; • asupermartingaleif,foranys ~ t, E(MtIFs) ~

=

(

(~a2(t - S)) o

If (Mt)t>o is a martingale, the property E (M; IFs) = M s, is also true if t and s are bounded .stopping times. This result is actually an adaptation of Exercise 1 in Chapter 2 to the continuous case and it is called the optional sampling theorem. We will not prove this theorem, but the reader ought to refer to Karatzas and Shreve (1988), page 19.

34

Brownian motion and stochastic differential equations

Theorem 3.3.4 (optional sampling theorem) If (M t )r2.o is a continuous martingale with respect to the filtration (Ft)t>o, and if 71 and 72 are two stopping times such that 71 ~ 72 ~ K, where K- is a finite real number, then M T 2 is integrable and

Stochastic integral and Ito calculus

By letting a converge to 0, we show that P(Ta < +00) = 1 (which means that the Brownian motion reaches the level a almost surely). Also

The case a

< 0 is easily solved if we notice that

Remark 3.3.5

= inf.{s 2: 0, -X. = -a},

Ta

• This result implies that if 7 is a bounded stopping time then E(MT ) = E(Mo ) (apply the theorem with 71 = 0,72 = 7 and take the expectation on both sides). • If M, is a submartingale, the same theorem is true if we replace the previous equality by .

We shall now apply that result to study the properties of the hitting time of a point by a Brownian motion. Proposition 3.3.6 Once again, we consider (Xt)t>o an Ft-Brownian motion. If a is a real number, we call T a = inf {s 2: 0, X, = ~} or +00 if that set is empty. Then, T a is a stopping time, finite almost surely, and its distribution ischaracterised by its Laplace transform

where (-Xdt~o is an Ft-Brownian motion because it is a continuous stochastic process with zero mean and variance t and with stationary, independent increments.

o The optional sampling theorem is also very useful to compute expectations involving the running maximum of a martingale. If M, is a square integrable martingale, we can show that the second-order moment of sUPo9:S: T IMti can be bounded. This is known as the Doob inequality. Theorem 3.3.7 (Doob inequality) If(M t)o9:S:T is a continuous martingale, we have

E ( sup IMtl2)

~ 4E(IMT I2).

°9:S:T The proof of this theorem is the purpose of Exercise 13. ,.'l

3.4 Stochastic integral and Proof. We will assume that a 2: O. First, we show that T a is a stopping time. ,. Indeed, since X, is continuous

35

Ito calculus

In a discrete-time model, if we follow a self-financing strategy ¢ = (Hn)o
s>

That last set belongs to F t , and therefore the result is'proved. In the following, we write x 1\ y = inf(x, y). Let us apply the sampling theorem to the martingale M; = exp (aX t - (a 2 / 2)t ). We cannot apply the theorem to T a which is not necessarily bounded; however, if n is a positive integer, T a 1\ n is a bounded stopping time (see Proposition 3.1.6), and from the optional sampling theorem

E (MTa/\n) = 1. On the one hand MTa/\n = exp (aXTa/\n - a 2 (Ta 1\ n)/2) ~ exp (aa). On the other hand, if T a < +00, limn--Hoo MTa/\n = M Ta and if T a = +00, X, ~ a at any t, therefore limn-Hoo MTai\n = O. Lebesgue theorem implies that E(I{Ta<+oo}MTJ = 1, i.e. since X Ta = a when T~< +00 E (I{Ta<+oo} exp ( -

~2 T a ) ) =e-: u a .

YO +

L H (5 j

j -

5j -

1) .

j=l

That wealth appears to be a martingale transform under a certain probability measure such that the discounted price of the stock is a martingale. As far as continuous-tim~ models are concerned, integr~ls of the form J H.dS. will help us to describe the same idea. However, the processes modelling stock prices are normally functions of one or several Brownian motions. But one of the most important properties of a Brownian motion is that, almost surely, its paths are not differentiable at any point. In other words, if (X t ) is a Brownian motion, it can be proved that for almost every wEn, there is not any time t in lR+ such that dX t / dt exists. As a result, we are not able to define the integral above as

r t

r t

dX io f(s)dX. = io f(s) ds' ds. Nevertheless, we are able to define this type of integral with respect to a Brownian

Brownianmotion and stochasticdifferential equations

36

motion, and we shall call them stochastic integrals. That is the whole purpose of this section.

Stochastic integraland Ito calculus

37

If we include sand t to the subdivision to = 0 < t 1 < ... < t p = T, and if we . t call M n = f on HsdWs and 9n = F tn for 0 :::; n :::; p, we want to show that M n is a 9n-martingale. To prove it, we notice that

3.4.1 Constructionofthe stochastic integral Suppose that (Wt}t~O is a standard Ft-Brownian motion defined on a filtered probability space (fl, A, (Ft)t>o, P). We are about to give a meaning to the expression f~ f (s, w)dWs for; certain class of processes f (s, w) adapted to the filtration (Ft}t~o, To start with, we shall construct this stochastic integral for a set of processes called simple processes. Throughout the text, T will be a strictly positive, finite real number. Definition 3.4.1 (Ht}O
with r/>i 9i-1 -measurable. Moreover, X n = W tn is a 9n-martingale (since (Wt)t>o is a Brownian motion). (Mn)nE[O,pj turns out to be a martingale transform of (Xn)nE[O,pj' The Proposition 1.2.3 of Chapter 1 allows us to conclude that (Mn)nE[O,pj is a martingale. The second assertion comesfrom the fact that

if it can be written as

p

Ht(w) = ~::>t>i(w)lJt'_l,t.j(t) i=l

n

where 0 = to < t 1 < ... < t p = T and r/>i is F t._ 1 -measurableand bounded. Then, by definition, the stochastic integral of a simple process H is the continuous process (I(H)t)O~t~T defined for any t E '~k, tk+d as c

I(H)t

L

=

r/>i(Wt• - Wt._J

+ r/>k+dWt -

L L E (r/>ir/>j(Xi < i. we have

Also, if io

E (r/>ir/>j(Xi - Xi-d(Xj ~ Xi-d)

W tk)·

E (E (r/>ir/>j (Xi - Xi-I) (Xi - X j- 1 ) 19j-1))

~

Note that I(H)t can be written as

=

L

r/>i(Wt.f\t - Wt._1f\t),

l~i~p

which proves the continuity oft t--+ I(Hk We shall write f~ HsdWs for I(Hk The following proposition is fundamental. Proposition 3.4.2

•

(f~ HsdWs)

• E ( ([

• E

If(Ht)o~t~T

09~T

(3.1)

Xi-d(Xj - Xj-d)·

i=l j=l

1~i9

I(H)t

n

=

E (r/>ir/>j(X i - Xi..:.dE (Xi - X j- 119j-df

Since X, is a martingale, E(Xj - X j- 11 9i-d = O. Therefore, if i < i. E (r/>ir/>j (Xi - Xi-d (Xj - X j- 1 )) = o. If j > i we get the same thing. Finally, if i = i.

=

E(r/>:(Xi-Xi-iY)

is a simple process:

is a continuous Ft-martingale.

H~dW.) ') ~ E (1,' H;d'). .

E(E(r/>:(Xi-:-Xi-d

219i-I))

E (r/>:E((X i - Xi.-d219i~I))

1

as a result E ( (Xi ~ X i - i) 219i-I) = E ((Wt• - Wti_l )2) = t; - ti-1.

(3.2)

From (3.1) and (3.2) we conclude that

(i~?I[ H.dW-I') < 4E ( [ H;d')

In order to prove this proposition, we are going to use discrete-time processes. Indeed, to show that (J~ HsdWs) is a 'martingale, w~ just need to check that, for any t > s, t : E HudWulFs) = HudWu. Proof.

(I

1;

The continuity, of t -t f~HsdWs is clear if we look at the definition. The third assertion is just a consequence ofDoob inequality (3.3.7) applied to the continuous martingale

(f~ Hs_dWs) '.

0

t~O

r

r

IJ\

38

Brownian motion and stochastic differential equations

iT H.dW.

=

I

HsdWs

-I

Stochastic integral and Ito calculus

39

A proof of this result can be found in Karatzas and Shreve (1988) (page 134, problem 2.5). If H E Hand (Hn)n?o is a sequence of simple processes converging to H in the previous sense, we have

Remark 3.4.3 We write by definition T

rI

t HsdWs-

If t ~ T, and if A E Ft. then s -t 1 Al {t<.} H s is still a simple process and it is easy to check from the definition of the integral that T lAHs1{t<s}dWs = lA iT n.sw.. (3.3)

I

Therefore, there exists a subsequence HcP(n) such that

Now that we have defined the stochastic integral for simple processes and stated some of its properties, we are going to extend the concept to a larger class of adapted processes H H = { (H t)05,t5,T,

(Ft}~?o -

adapted process, E

(I

T H;dS)

< +00} .

Proposition 3.4.4 Co~sider (Wt}t?o an Ft-Browriian motion. There exists a unique linear mapping J from H to the space of continuous Ft-martingales defined on

[0, T], suc~ that:

1. If (Ht}t5,T is a simple process, P a.s. for any 2. If t

thus, the series whose general term is I(HcP(n+l)) - I(HcP(n)) is uniformly convergent, almost surely. Consequently I(HcP(n))t converges towards a continuous function which will be, by definition, the map t t-+ J(H}t. Taking the limit in (3.6), we obtain

s T, E (J(H);) = E (I

a~ t

. (3.7)

if both

J and J' satisfy the

That implies that (J(H)t}O
'10 ~ t ~ T, J(H)t = J'(H}t.

P a.s. t

I

T, J(H)t = I(Hk

H;dS).

This linear mappingis unique in the following sense, previous properties then

We denote,for HE H,

~

t

H.dW.

Moreover, for any t, lim n -+ + oo I(Hn)t = J(H}t in L 2(HiP) norm and, because the conditional expectation is continuous in L 2 (n, P), we can conclude.

= J(H}t.

On top of that, the stochastic integral satisfies the following properties: Proposition 3.4.5 If (Ht}O
2

From (3.7) and from the fact that E(I(Hnm = E (JoT IH';I. dS} it follows that E(J(H);) E(SUPt5,T I(Hnm

2. 1fT is an Fi-stopping time

r-]

HsdWs =

I

T l{s5,T}HsdWs'

Proof.

(3.5)

We shall use the fact that if (H s)s5,T is in,H, there. exists a sequence (H';) s5,T of simple processes such that ' , lim E (iT IHs

n-++oo

0

2

-

H';1 dS) = O.

= E (JoT IHs l2 dS). In the same way, from (3.7) arid since ~ 4E

(JOT

IH';1 2 dS), we prove (3.4).

The uniqueness of the extension results from the fact that the set of simple processes is dense in H. : We now prove (3.5). First of all, we notice that (3.3) is still true if H E H. This is justified by the fact that the simple processes are dense in H and by (3.7). We first consider stopping times of the form T = L:1
40

Brownian motion and stochastic differential equations

also, each l{s>qlA.Hs is adapted because this process is zero if s ~ t, and is equal to lA; H, otherwise, therefore it belongs to 1t. It follows that T

JoT l{s>T}HsdWs

=

L lA; l:Si:Sn

1 u.sw, 1 T

T

=

HsdWs,

T

t,

and then JoT l{s:ST}HsdWs = J; HsdWs. In order to prove this result for an arbitrary stopping time T, we must notice that T can be approximated by a decreasing sequence of stopping times of the previous form. If

Tn converges almost surely to T. By continuity of the map t ~ J6 HsdWs we can affirm that, almost surely, J;n HsdWs converges to JOT HsdWs. On the other hand

(

41

Remark 3.4.7 It is crucial to notice that in this case (J6 HsdWs) necessarily a martingale.

O
is not

L i l A . l{s>qHsdWs l
E

Stochastic integral and Ito calculus

T ' . T 11{s:ST}HSdW. -11{s:STn}HsdWs

2)

= E

(T ) 11{T<s:STn}H;dS .

Proof. It is easy to deduce from the extension property and from the continuity property that if H E 1t then P a.s. Vt ~ T, J(H)t = J(Hk Let H E ic. and define t; = inf {O ~ s ~ T, J; H~du 2: n} (+00 if that set is empty), and H;' = H, l{s:STn}' Firstly, we show that Tn is a stopping time. Since {Tn ~ t} = H~du 2: n},

U6

we just need to prove that J6 H~du is Frmeasurable. This result is true if H is a simple process and, by density, it is true if H E 11.. Finally, if H E il, J6 H~du is also Ft-measurable because it is the limit of J6(Hu A K)2du almost surely as K tends to infinity. Then, we see immediately that the processes H;' are adapted . and bounded, hence they belong to 1t. Moreover

it

H;'dWs

=

it

l{s:STn}H;,+ldWs,

and relation (3.5) implies that

l:

This last term converges to 0 by dominated co~vergence, therefore 1{S:STn} HsdWs T ' converges to Jo l{s:ST}HsdWs in L 2(0, P) (a subsequence converges almost surely). That completes the proof of (3.5) for an arbitrary stopping time. 0 In the modelling, we shall need processes which only satisfy a weaker integrability condition than the processes in 1t, that is why we define '

il

= {(Hs)o:SS:ST

(Fth~o -

adapted process, iT H;ds

< +00 P

Un~oUoT H~du < n}

= Uo H~du < +oo}, we can define almost surely a ·T process J(H)t by: on the set o H~du < n},

U

Vt ~ T

\

il into the vector P

.

(SUPt:STIJ(H)tl2:~) <

P

(I; H;ds 2: liN)

+P (l{foT H;'du
Pa.s.\fO ~ t ~ T, J(H)t 7' I(Hit.

2. Continuity property: If (Hn)n>o is a sequence of processes in il such that JoT (H,;)2 ds converges to 0 in ;robability then SUPt:s'T IJ(Hnhl converges to o in probability.

J(H)t = J(Hnk

The process t ~ J(H)t is almost surely continuous, by definition. The extension property is satisfied by construction. We just need to prove the continuity property of J. To do so, we first notice that

1. Extension property: 1f(Ht)09:ST is a simple process then

Consistently, we write J6 HsdWs for J(H)t.

U; H~du
a.s. } .

The following proposition defines an extension of the stochastic integral from 1t to ii. ' ,

Proposition 3.4.6 There exists a unique linear mapping I from space of continuous processes defined on [0, TJ,such that:

Thus, on the set

If we call TN = inf {s ~ T, Jos H~du 2: on the set {J; r-,

liN}

€).

(+00 if this set is empty), then

H~du < liN}, it follows from (3.5) that, for any t ~ r.

Brownian motion and stochastic differential equations

42

(~~~ IJ(H)tl ~ E)

< P

(iT H;ds

+4/E 2E

< P

(iT

H~ds ~ ~ )

W?

= 2 f~ W. dW

s-

Indeed, from the previous section we know that

is a martingale (because E TN

}ds)

+

= 2 f~ f (s )j(s )ds =

2 f~ f(s)df(s). In the Brownian case, it is impossible to have a similar formula

~ ~)

(1: H; I {.5:.

43

differentiable function f(t) null at the origin, we have f( t)2

whence, by applying (3.4) to the process s t-t H. I {.5,TN} we get

P

Stochastic integral and Ito calculus

4 N E2 '

As a result, if f: (H;l ds converges to 0 in probability, then SUPt..Hn + J.LKn)t >"J(Hn)t +J.LJ(Kn)t, to prove the continuity of 1. Finally, the fact that if HE it then f: (Ht-H[,)2dt converge to 0 in probability 0 and the continuity property yields the uniqueness ofthe extension.

=

(f~ W,;ds) < +00),

f~ W. dW.

null at zero. If it were equal to

W? /2 it would be non-negative, and a non-negative martingale vanishing at zero can only be identically zero. We shall define precisely the class of processes for which the Ito formula is applicable. Definition 3.4.8 Let (0, .1', (Fdt~o, P) be ajilteredprobability space and (Wt)t>o an FtcBrownian motion. (Xt )05, t5,T is an IR-valuedIto process ifit can be written as

where

• X o is Fo-measurable. • (Kt)05,t5,T and (Ht)095,T are Fe-adapted processes. • f: IK.lds T

< +00 P a.s. < +00 P a.s.

• 2ds

We are about to summarise the conditions needed to define the stochastic integral with respect to a Brownian motion and we want to specify the assumptions that make it a martingale.

• fo IH.1 We can prove the following proposition (see Exercise 16) which underlines the of the previous decomposition. uniqueness ;J ,

Summary:

Proposition 3.4.9

Let 'us consider an Ft-Brownian motion (Wt)t>o and an Ft-adapted process (Hd095,T. We are able to define the stochastic-integral (J~ H.dW. )05,t5,T as soonasf: H;ds

< +00

P a.s. By construction.the process Ij'[ H.dW.)o5,t5,T T is a martingale ifE (fo H;ds) < +00. This condition is not necessary. Indeed, the inequality E

(1: II.;ds) < +00 is satisfied if and only if E

(SUp

05,t5:.T

(t Jo

If (M t )05, t5,T is a continuous martingale such that

T M t = i t K..ds, with P a.s.i IK.lds

< +00,

then

P a.s. 'lit

~

T,

u,

= O.

This implies that: - An Ito process decomposition is unique. That means that if

H.dW.)

2) < +00.

This is proved in Exercise IS.

3.4.2 Ito calculus It is now time to introduce a differential calculus based on this stochastic integral. It will be called the Ito calculus and the main ingredient is the famous Ito formula. In particular, the Ito formula allows us to differentiate such a function as t t-t f (Wd if f is twice continuously differentiable. The following example will simply show that a naive extension of the classical differential calculus is bound to fail. Let us try to differentiate the function t --+ W? in terms of 'dWt'. Typically, for a

x, then

= X o + i t K.ds 0: '

X o = x;

dP a.s.

H.

+

r

H.dW. =' Xb

+

Jo

= ti;

rK~ds + rH~dW.

Jo ds x dP a.e.

K.

Jo

= K;

- If (X t )05, t5,T is a martingale of the form X o + f~ K.ds K, = 0 dt x dP a.e.

ds x dPa.e.

+ f~ H.dW.,

then

We shall state Ito formula for continuous martingales. The interested reader should refer to Bouleau (1988) for an elementary proof in the Brownian case, i.e. when (Wd is a standard Brownian motion, or to Karatzas and Shreve (1988) for a complete proof.

44

Stochastic integral and Ito calculus

Brownian motion and stochastic differential equations

J; Ssds and J~ SsdWs exist and at any time t

Theorem 3.4.10 Let (X t )09 $ T be an Ito process

x,

= X o + i t Ksds

+ i t HsdWs,

P a.s. ,

and f be a twice continuously differentiable function, then

I

It

+ i t f'(Xs)dX s + ~ i t j"(Xs)d(X,X)s

f(X t) = f(X o)

[

,~ I

whe re, by definition . (X,Xk and

it

Ii

it

log(St) = log(So)

It turns out that 2

Wt Since E

(J; W

2 s ds )

..:..

r + '2 '1

a WsdW s

+ it

(

1)

-2

Ss

a 2S;ds .

+ i t adWt,

+ (J.L 2/2)

a 2/2) t

t

+ aWt.

+ aWt)

f(t,x)=xoexp((J.L-a 2/2)t+ax).

St

=

f(t, Wd

=

f(O, W o)

+

it

f:(s, Ws)ds

+ i t f~(s, Ws)dWs +.~ it'f~/x(S, Ws)d(W, W)S'

io 2ds.

t t = 2 i WsdWs'

Furthermore, since (W, W)t = t, we can write

< +00, it confirms the fact that W? -

t is a martingale.

r

s, ~ x'o + i t s. (J.Lds + adW:).

(3.8)

+ adWt),

So

= xo·

. it. St = Xo + a SsJ.Lds

it

+ a

SsadWs"

Remark 3.4.11 We could have obtained the same result (exercise) by applying Ito formula to St = ¢(Zd, with Zt = (J.L-a 2 /2)t+aWt (which is an Ito process) and ¢(x) =nxo exp(x).

This is often written in the symbolic form ~ St (J.Ldt

2 a

(J.L - a 2/2) dt

We now want to.tackle the problem of finding the solutions (Sdt~o of

as,

+ -l i t

Ito formula is now applicable and yields

Let us start by giving an elementary example. If f(x) = x 2 and X t = W t, we identify K, = and H, = 1, thus

i

dS _s

is a solution of equation (3.8). We must check that conjecture rigorously. We have St = f(t, Wd with

3.4.3 Examples: Ito formula in practice

=2

t

a Ss

Yt = log(St) = log(So) Taking that into account, it seems that

roJ

2

i

St = Xo exp ((J.L - a

+ i t f~(s,Xs)dXs + ~ i t f~lx(S:Xs)d(X.,X)s.

Wt

+

and finally

+ i t f:(s, Xs)ds

t

+ i t J.LSsds + i t aSsdWs.

Using (3.9), we get }Ii = Yo

Likewise, if(t, x) -+ f(t, x) is a function which is twice differentiable with respect to x and once with respect to t, and ifthese partial derivatives are continuous with respectto (t,x) (i.e. f is a junction of class C 1 ,2), ltd formulabecomes

°

Xo

II

a f'(Xs)dX s = a f'(Xs)Ksds + a f'(Xs)HsdWs'

f(O, X o)

s, =

To put it in a simple way, let us do a formal calculation. We write }Ii = log(St) where St is a solution of (3.8). St is an Ito process with K; = J.LSs and H; = aSs. Assuming that St is non-negative, we apply Ito formula to f(x) =log(x) (at least formally, because f(x) is not a C 2function!), and we obtain

tf

= iat H;ds,.

'it

45

(3.9)

We are actually looking for an adapted process (St)t~O such that the integrals

We have just proved the existence of a solution to equation (3.8). We are about to prove its uniqueness. To do that, we shall use the integration by parts formula.

46

Brownian motion and stochastic differential equations

Proposition 3.4.12 (integration by parts formula) Let X, and Yi be two Ito processes, x, = X o + J; K.ds + J; H.dW. and Yi = Yo + J; K~ds + J; H~dW•. Then

XtYi = XoYo +

I

t

+

X.dY.

I

In this case, we have

(X, Z)t =

t

Y.dX.

+ (X, Y)t

(1" X.adW., - 1" Z.adW.)t = -l,t a 2x.z.u.

Therefore

with the following convention

=

d(XtZt)

I H.H~ds.

+

t

=

(X, Y)t Proof. By

"It ~ 0, P a.s. x, = XOZ;I = St.

(X t + Yi)2 =

(Xo + YO)2 +2J;(X. + J;(H.

The processes X, and Zt being continuous, this proves that

, P a.s. "It ~ 0, X, = XOZ;I = St.

+ Y.)d(X. + Y.)

We have just proved the following theorem:

+ H~)2ds'

Theorem 3.4.13 If we consider two real numbers a, J.L and a Brownian motion (Wth>o and a strictly positive constant T, there exists a unique Ito process (SdO;:::;T which satisfies,for any t ~ T,

xg + 2 J; X.dX. + J; H;ds

=

2

Yo

r rt ,2 + 2 Jo Y.dY. + Jo H. ds. t

s, =

By subtracting equalities 2 and 3 from the first one, it turns out that

XtYi = XoYo +

I

t X.dY.

+

I

t Y"dX.

+

I H.H~ds.

+ aW t)

is a solution of (3.8) and assume that (Xt)t>o is another one. We attempt to . ' compute the stochastic differential of the quantity XtS;I. Define

~~

= exp (( -J.L +a 2/2) t -q-Wt) ,

+ a 2 and a'

= -a, Then Zt = exp((J.L' - a,2/ 2)t verification that we have just Gone shows that

Zt

=1+

+ a'Wt)

+

I

t S. (J.Lds

+ adW.).

'0

We now have the tools to show that equation (3.8) has a unique solution. Recall th~ ,

Sc = Xo exp ( (J.L - a 2/2) t

Xo

This process is given by

t

o

J.L' =' -J.L

X tZt{(-J.L+a 2)dt-adWt) XtZt (J.Ldt + adWd - X tZta 2dt = o.

Hence, XtZ t is equal to XoZo, which implies that

Ito formula

, Zt =

47

Stochastic integral and Ito calculus

and the

r Z.(J,L'ds + a'dW.) = + ~r z, ((-J.L + ( 2) ds - adW.). ~

Remark 3.4.14 • The process St that we just studied will model the evolution of a stock price in the Black-Scholes model. • When J.L = 0, St is actually a martingale (see Proposition 3.3.3) called the exponential martingale of Brownian motion.

Remark 3.4.15 Let e be an open set in IR and (Xt)O:::;t:::;T an Ito process which stays in e at all times. If we consider a function f from e to lR which is twice continuously differentiable, we can derive an extension of Ito formula in that case

f(Xt).~ f(;o) + I This result allows us to apply positive process.

t !'(X.)dX.

+~

I

t !"(X.)H;ds.

Ito formula to log(Xd for instance, if X t is a strictly "

.

1

,

From the integration by parts formula, we can compute the differential of X;Zt

3.4.4 Multidimensional Ito formula We apply a multidimensional version of Ito formula when f is a function of'several Ito processes which are themselves functions of several Brownian motions. This

Brownian motion and stochastic differential equations

48

Stochastic differential equations

49

• (J~ HsdWt , J~ H~dW/)t = 0 if i ;i j. • (J~ tt.aw], J~ H~dWDt = J~ HsH~ds. i

version will prove to be very useful when we model complex interest rate structures for instance.

Definition 3.4.16 We call standard p-dimensional FrBrownian motion an lRP valued process (Wt = (Wl, . . . , Wi) k:~o adapted to F t, where all the (Wnt~O are independent standard FrBrownian motions.

This definition leads to the cross-variation stated in the previous proposition.

Along these lines, we can define a multidimensional Ito process.

In Section 3.4.2, we studied in detail the solutions to the equation

Definition 3.4.17

(Xt)09~T

is an Ito process

3.5 Stochastic differential equations

if

x, = x +

it

Xs(/Lds

+ adWs)'

We can now consider some more general equations of the type

x, = Z +

where: • K t and all the processes (Hi) are adapted to (Ft). • JoT IKslds

b(s,Xs)ds

+

it

(3.10)

a(s,Xs)dWs.

These equations are called stochastic differential equations and a solution of (3.10) is called a diffusion. These equations are useful to model most financial assets, whether we are speaking about stocks or interest rate processes. Let us first study some properties of the solutions to these equations.

< +00 ~ a.s.

• JoT (H;) 2ds

it

< +00 P a.s.

Ito formula becomes:

Proposition 3.4.18 Let (Xl, ... , X;') be n Ito processes X ti = Xi0

+

i

t

Kids s

P

+ '" L..J

o

3.5.I Ito theorem

it ,

Whatdowe mean by a solution of(3.1O)?

Hi,idWi s s

j:=l

0

then, if 1 is twice differentiable with respect to x and once differentiable with respect to t, with continuous partial derivatives in (t, x) 1(0,XJ, ... ,Xfn

+tit i=l

+~2 ..t

0

it ~~ (s,X~,;

.. ,X:)ds'

::.(s,x~, ... ,X:)dX;

it

',J=l

+

0

t

S

5

• d(X i , Xi) s =

,",P

L...,,-J==l

,",P L...,,-m=l

llb(s,Xs),ds

21 88 xixi

(s,X~, ... ,X:)d(Xi.,Xi)s .

Hi,idWj 5 5' Hi,m Hi,mds 5 s .

Remark3.4.19 If (Xs)O
• (X, Y)t is bilinear and symmetric.

•

.

,

with:

• ax: = Kids +

Definition 3.5:1 We consider a probability space (n, A,.P) equipped with a filtratioh (Ft)t~o, We also have functions b : lR+ x lR -+ IR., a : lR+ x lR -+ IR., a Fo-measurable randomvariable Z and finally an Ft-Brownian motion (Wdt~o. -1 solution to equation (3./0) is an Fi-adapted stochastic process (Xt)t~O that satisfies: • For any t ~ 0, the integrals J~ b(s, Xs)ds and J~ a(s, Xs)dWs exist

(J~ Ksds, x')t = 0 if (Xt)09~T is an Ito process.

t

< +00 andila(s,XsWds < +00 P a.s.

• (Xt)t~O satisfies (3./0), i.e.

_.

'It

~0

P a.s.

.'

.x,t = Z +

1

1 t

t

b (s,Xs) ds

+

"'

a (s, X s ) dWs'

Remark 3.5.2 Formally, we often write equation (3.10) as

.{ ex. Xo

b (t, X t) dt Z

+ a (t, X t) dWt

The following theorem gives sufficient conditions on b and a to guarantee the existence and uniqueness of a solution of equation (3.10).

Theorem·3.5.3 Ifb and a are continuous functions and if there exists a constant K < +00 such that

50

Brownian motion and stochastic differential equations

J. Ib(t,x) - b(t, y)1

51

Stochastic differential equations

+ loo(t, x) - oo(t,Y)1 ::; Klx - yl

Ib(t,x)1 + loo(t,x)l::; K(1 + Ixl) 3. 'E(Z2) < +00 then, for any T 2: 0, (3.10) admits a unique solution in the interval [0, T].

2.

Moreover, this solution (Xs)O~s~T satisfies E ( sup IXsI2) O~s~T

< +00

The uniqueness of the solution means that if (Xt)09~T and (Yi)09~T are two solutions oj(3.1O), then P a.s. '10 ::; t ::; T, X, = Yi, Proof. We define the set

E

= {(XS)O~S~T' Ft-adapted continuous processes,

sU~h that E (~~~IXsI2)

<

+oo} .

Together with the norm IIXII = (E (sUPO<s
~(X)t = Z + it b(s, Xs)d~ + it oo(s, Xs)dWs'

We deduce that ~ is a mapping from E to E with a Lipschitz norm bounded from . above by k(T) = (2(K 2T2 + 4K 2T)) 1/2. If we assume that T is small enough so that k(T) < 1, it turns out that ~ is a contraction from E into E, Thus it has a fixed point in E, Moreover, if X is a fixed point of ~, it is a solution of (3.10). That completes the proof of the existence forT small enough. On the other hand, a solution of (3.10) which belongs to E is a fixed point of~. That proves the uniqueness of a solution of equation (3.10) in the class E, In order to prove the uniqueness in the whole class of Ito processes, we just need to show that a solution of (3.10) always belongs to E, Let X be a solution of (3.10), and define T; = inf{s 2: 0, IXsl > n} andr(t) = E (suPO~S
E

(suPO~u~tIlTn IX

uI

If X belongs to E, ~(X) is well defined and furthermore if X and Y are both in E we can use the fact that (a + b)2 ::; 2(a2 + b2) to write that

1~(X)t - ~(Y)tI2

<' 2 (suPO~t~T'lf~(b(S,Xs) - b(s, Ys)) ds I

+sUPO~t~T If~(oo(s,~s) -

-sr

(J~IITn K(1 + IXsl)ds) 2

r(t) ::; a + bit r(s)ds.

1~(X)t - ~(Y)tI2)

In order to complete the proof, let us recall the Gronwall lemma.

< 2E (sup (it Ib(s, X s) .: b(s, Ys)ldS) 09~T

< 3 ( E(Z2) + E

This yields the following inequality

and therefore by inequality (3.4)

E (sup

)

+4E (J~IITn K2(1 + IX sI)2 ds)) < 3 (E(Z2) + 2(K 2T + 4K 2) x f~ (1 + E (sUPO~U~SIlTn IXuI 2 ) ) dS) .

2

oo(s, Ys))dWsI2)

2

0

,

2)

Lemma 3.5.4 (Gronwalll~mma)

0::; t

Proof. Let us write u(t)

+8E (iT (oo(s, X s) - oo(s, Ys))2dS) i

< 2(K 2T2 + 4K 2T)E ,( sup IXt - Yi1 2 ) 09~T

Iff is a continuous function

::; T, f(t) ::; a + b fo f(s)ds, then f(T) ::; a(1

+ ebT).

such that for any

= e- bt f; f(s)ds. Then,

u'(t) = e-bt(f(s) - bit f.(s)ds) ::; ae- bt. o .

By first-order integration we obtain u(T) ::; alb and f(T) ::; a(1 + ebT).

0

Brownian motion and stochastic differential equations

52

In our case, we have t" (T) < K < +00, where K is a function ofT independent of n. It follows from Fatou lemma that, for any T,

E(

sup IXsI2) O$;s$;T

< K < +00.

Stochastic differential equations

, An are real numbers and if 0 ::; t l < ... < t-: the random variable To convince ourselves, we just notice that

that if AI, Al Xtl +

+ AnXtn is normal.

x; = xe- eti + O'e- et•

Therefore X belongs to [; andthat completes the proof for small T. For an arbitrary T, we consider a large enough integer n and think successively on the intervals [0, Tin), [Tin, 2TIn), ...,[(n - l)TIn, T). 0

53

1+

00

l{s$;t;}eCSdWs

= mi +

it

Ji(s)dWs.

Then AIXtt + '" + AnXtn = E7=I Aimi + J~ (E7=1 Adi(S)) dWs which is indeed a normal random variable (since it is a stochastic integral of a deterministic function of time).

3.5.3 Multidimensional stochastic differential equations 3.5.2 The Omstein-Ulhenbeck process Omstein-Ulhenbeck process is the unique solution of the following equation:

ax, { Xo

= =

-cXtdt x

+ O'dWt

x,»

It can be' written explicitly. Indeed, if we consider yt = and integrate by parts, it yields dyt = dXte ct + Xtd(eet) + d(X, eC'}t. Furthermore (X,eC')t = 0 because d(e ct) = ceetdt. It follows that dyt O'ectdWt and thus .

X;

= xe -ct + ae -ct

it o

e cSdWs-

The analysis of stochastic differential equations can be extended to the case when . processes evolve in lR n . This generalisation proves to be useful in finance when we want to model baskets of stocks or currencies. We consider • W = (WI, ... , W1') an lR1' -valued Ft-Brownian motion.

• b : lR+ x lR n ---+ lR n. b(s , x) = (b1 ( s, x), .. : , b"(s , x) ). • a: lR+

X

lR n ---+ lR nx1' (which is the set of n x p matrices), O'(s,x)

= (O'i,j(s,x)h$;i$;n ,I$;j$;1"

• z = (ZI, ... , Z") an Fo-measurable random variable in lR n . We are also interested in the following stochastic differential equation: -v

"

(3.11)

This enables us to compute the mean and variance of Xt:

E(Xd= xe- et + O'e-ctE (it eCSdws) = xe- et (since E

In other words, we are looking for a process (Xt)O
(J;(e~S)2ds) < +00, J~ ecsdWs is a martingale null at time 0 and

therefore its expectation is zero). Similarly

,

.

E ((X t- E(Xt))2)

.'E

(e-'" (I,' e"dW')')

,.O'2~-2ctE =

0'2

(lte2csdi) '

1 - e- 2ct_

2c We can also prove that X, is a normal random variable, since X; can be written as J~ j(s)dWs where j(.) is a deterministic function oftim~ and J~ !2(s)d~ < +00 (see Exercise 12). More precisely, the process (Xtk~o IS GaUSSIan. ThIS means

The theorem of existence and uniqueness of a solution of (3.11) can be stated as:

Theorem 3.5~5 If x E lR n , we denote by [z] the Euclidean norm of x and n x1' 2 a E lR , 10'1 = EI
if

+ IO'(t,x)l::; K(1 + Ixl)

3. E(IZI 2) < +00 then, there exists a unique solution to (3.11). Moreover. this solution satisfies for anyT

E(

sup IXsI2) < +00. O$;s$;T The proof is very ,similar to the one in the scalar case.

Brownian motion and stochastic differential equations

54

3.5.4 The Markov property of the solution ofa stochastic differential equation

The intuitive meaning of the Markov property is that the future behaviour of the process (Xtk~o after t depends only on the value X; and is not influenced by the history of the process before t. This is a crucial property of the Markovian model and it will have great consequences in the pricing of options. For instance, it will allow us to show that the price of an option on an underlying asset whose price is Markovian depends only on the price of this underlying asset at time t. Mathematically speaking, an Fradapted process (Xt)t>o satisfies the Markov property if, for any bounded Borel function f and for any ~ and t such that s t, we have E (f (X t) IFs) = E (f (X t) IX s) .

s

We are going to state this property for a solution of equation (3.10). We shall denote by (X;'X, s ;:::- t) the solution of equation (3.10) starting from x at time t and by X" = Xo,x the solution starting from x at time O. For s ;::: t, X;'x satisfies s s b (u a (u 'Xt,X) Xt,x = x Xt,X) du + dWu· s " u u

~1

_1

t

t

A priori, X.t,x is defined for any (t, x) almost surely. However, under the assumptions of Theorem 3.5.3, we can build a process depending on (t, x, s) which is almost surely continuous with respect to these variables and is a solution of the previous equation. This result is difficult to prove and the interested reader should refer to Rogers and Williams (1987) for the proof. The Markov property is a consequence of the flow property of a solution of a stochastic differential equation which is itself an extension of the flow property of solutions of ordinary differential equations.

Lemma 3.5.6 Under the assumptions of Theorem 3.5.3,

if s ;::: t

55

Stochastic differential equations

Indeed, if t ::; s X~·x

x

+ J; b (u, X~) du + J; a (u, X~) dWu

x: + J/ b (u, X~) du + J/ a (u, X~) dWu' The uniqueness of the solution to this equation implies that X~,x = X;'x, for t ::; s. 0 In this case, the Markov property can be stated as follows:

Theorem 3.5.7 Let (Xtk~o be a solution of (3. 10). It is a Markov process with respect to the Brownian filtration (Ftk~o. Furthermore, for any bounded Borel function f we have P a.s. E (f (X t) IFs) = ¢(Xs), with ¢(x)

= E (f(X;·X)).

Remark 3.5.8 The previous equality is often written as

Proof. Yet again, we shall only sketch the proof of.this theorem. For a full proof, the reader ought to refer to Friedman (1975). x' The-flow property shows that, if s ::; t, Xf = X:' '. On the other hand, we can prove that X;,x is a measurable function of x and the Brownian increments (Ws +u - W s , u ;::: 0) (this result is natural but it is quite tricky to justify (see Friedman (1975)). If we use this result for fixed sand t we obtain X;·x (x, W s+u ~ w., u ;::: 0) and thus

x: = (X:, W s+u Proof. We are only going to sketch the proof of this lemma. For any x, we have s s P a.s. X;'x = x + b (u, du + dWu. a (u,

1

1

X~,X)

X~·X)

+ ~s b.(u,X~'Y) du + l

u;::: 0),

where X: is Fs-measurable and (Ws+u, - Ws)u~o is independent of F s. If we apply the result of Proposition A.2.5 in the Appendix to X s , (Ws +u Ws)u~o, .and F" it turns out that E (f ( (X: , W s +u

It follows that, P a.s. for any y E JR, X;·y = y

w.,

-

w.,

-

u;::: 0))1 F s )

E (f ((x, W s+u - W s; u;::: O)))lx=x; s a

(u,X~;Y) dWu,

=

E (f (X:·X))lx=x~ . -0

and also

1b(u,X~'X;)dU~ 1a(u,X~·X;)dWu. s

X;·x; =X:+

s

These results are intuitive, but they can be proved rigorously by using the continuity of H We can also notice that is also a solution of the previous equation.

y x»,

X:

The previous result can be extended to the case when we consider a function of the whole path of a diffusionafter time s. In particular, the following theorem is useful when we do computations involving interest rate models.

Theorem 3.5.9 Let (Xt)t>o be a solution of(3.1O) and r(s, x) be a non-negative

Brownian motion and stochastic differential equations

56

measurable function. For t > s P a.s. E (e-

with

¢(x)

=E

Exercises

57

Exercise 9 Let S be a stopping time, prove that Sis Fs-measurable.

f.' r(u,X,,)du 1 (Xt) IFs)

Exercise 10 Let Sand T be two stopping times such that S thatFs eFT.

= ¢(Xs)

~

T P a.s. Prove

Exercise 11 Let S be a stopping time almost surely finite, and (Xt)t>o be an adapted process almost surely continuous. -

(e-f.' r(u,X:"~)du I(Xt'X)) .

l. Prove that, P a.s., for any s

It is also written as

E

(e- f.'

r(u,X,,)du

1 (Xt) IFs) = E

Remark 3.5.10 Actually, one can prove a more general result. Without getting into the technicalities, let us just mention that if ¢ is a function of the whole path of X, after time s, the following stronger result is still true: Pa.s. E(¢.(X:, t~s)IFs)= E(¢(Xt'X, t~s))lx=x.'

E (J(X:;t)) = E

(J(X~'X)).

(e-I." r(X:"~)du1(X:;t) ) = E (e-f; r(X~'~)du I(X~'X)) .

In that case, the Theorem 3.5.9 becomes

(e-f.'

r(X,,)du

1 (Xt) IFs) = E

lim

n-t+oo

2. Prove that the mapping

([0, t] x n, B([O, t]) x F t ) (s,w)

--+

(lR,B(lR))

f-----+

Xs(w)

3. Conclude that if S ~ t, X s is Fcmeasurable, and thus that X s is F smeasurable. Exercise 12 This exerciseis an introduction to the copcept of stochastic integration. We want to build an integral of the form l(s)dX s, where (Xt)t>o is an Ft-Brownian'motion and I(s) is a measurable function from (lR+, B(lR+)) into (lR, B(lR)) such that j2 (s )ds < +00. This type of integral is called Wiener integral and it is a particular case of Ito integral which is studied in Section 3.4. We~recall that the set 11. of functionsof the form LO
s:

(e-fo'-' r(X~'~)du I(X~~~)) Ix=x.

with the norm l. Consider

II/ilL2 = (Jo+oo j2(s)ds) 1/2.

a; E lR, and 0 = to ~ t1 ~ ... ~ i», and call 1=

L

ai 1 j ti,t.+d·

0:Si:SN-1

3.6 Exercises Exercise 6 Let (Mtk:~.o be a martingale such that for any t, E(Ml) < Prove that if s ~ t .

E ((Mt - M s)2IFs)

+00.

= E (M t2 -'M;IFs) .

Exercise 7 Let X, be a process with independent stationary increments and zero initial value such that for any t, E (Xl) < +00. We shall also assume that the map t f-t E (Xl) is continuous. Prove thatE (Xt ) = ct and that Var(Xt) = c't, where c and c' are two constants. Exercise 8 Prove that, if

Fr is a a-algebra.

'L..J " l[k/n' (k+1)/n[(S)X k/n (w).

k~O

s:

We can extend this result and show that if r is a function of x only then

E

=

is measurable.

Remark 3.5.11 When' b and a are independent of x (the diffusion is said to be homogeneous), we can show that the law of X:;t is the same as the one of X~,x, which implies that if 1 is a bounded measurable function, then

E

x,

(e-f.' r(u,X:"~)du I(X;'X)) Ix=x.

T

is a stopping time,

= {A E A,

for all t ~ 0 , A

n {T ~ t} E F t }

We define

Ie(f) =

L

ai(Xt .+1

-

XL;).

0:Si:SN -1

Prove that I, (f) is a normal random variable and compute its mean and variance. In particular, show that

2. ~rom ~his, show that there exists a unique linear mapping I from L 2(lR+, dx) into Z (fl,F,P),suchthatI(f) = Ie(f),when 1 belongs to 11. andE(I(f)2) = 11f1l£2, for any 1 in L 2(lR+). 3. Prove that if (Xn)n>O is a sequence of normal random variables 'with zeromean which converge to X in L 2(fl, .1',P), then X is also a normal random

Brownian motion and stochastic differential equations

58

variable with zero-mean. Deduce that if f E dx) then 1(1) is a normal random variable with zero mean and a variance equal to f2 (s )ds.

L 2(IR+,

s:

4. We consider f E L 2(IR+, dx), and we define

z, =

i t f(s)dX s =

J

I]O,tJ(s)f(s)dX s,

prove that Zt is adapted to F t, and that Zt - Zs is independent of F, (hint: begin with the case f E H).

5. Prove that the processes Zt, Z; - f; P(s)ds, exp(Zt - ~ f; f2(s)ds) are Frmartingales.

Exercise 13 Let T be a positive real number and (M t)095,T be a continuous

Exercises

59

Exercise 15

T

1. Let (H t)05,t5,T be an adapted measurable process such that fo H'fdt < 00, a.s. Let u, = f; HsdWs. Show that if E (sUPO
H; dt) < 00. Hint: introduce the sequence ~f~topping times Tn = inf{t ~ a If; H;ds = n} andsho~thatE(Mj.I\TJ= E (foTl\Tn H;ds). T

E (fo

2. Letp(t,x) = 1/vT=texp(-x2/2(1- t)), for a ~ t < 1 and x E IR, and p(l, x) = O. Define M; = p(t, W t), where (Wt)O
Ft-martingale. We assume that E(Mj.) is finite.

1. Prove that (IMtI)095,T is a submartingale. (b) Let

2. Show that, if M* ., sUP05,t5,T IMtl, AP (M* ~ A) ~ E (IMTll{M·~.q). (Hint: apply the optional sampling theorem to the submartingale IMtl between T /\ T and T where T = inf {t ~ T, IMti ~ A} (if this set is empty T is equal to +00).) 3. From the previous result, deduce that for positive A

E((M* /\ A)2) ~ 2E((M* /\ A)IMTI). M·I\A (Use the fact that (M* /\ A)P = f o pxp-1dx for p = 1,2.) 4. Prove that E(M*) is finite and E ( sup IMtI

2

)

°95,T

~ 4E(IMT I2 ) .

H, = l Prove that fo Hldt

op

ox (t, Wd·

< 00, a.s. and E

(fol Hldt)

=+00.

Exercise 16 Let (M t)05,t5,T be a continuous Ft-martingale equal to f;

tc,e«

T where (Kt)O.::;t5,T is an Ft-adapted process such that fo IKslds < +00 P a.s, '.

T

1. Moreover, we assume that P a.s. f o IKslds ~ C write tf = Ti/n for a ~ i ~ n, then lim E

n---t+oc>

< +00.

Prove that if we

(~' L....J (Mtn - Mt~ ) 2) = O. 1-1

1.

i=1

2. Under the same assumptions, prove that ~

Exercise 14 1. Prove that if 5 and 5' are two Frstopping times then 5 /\ 5~ = inf(5, 5') and 5 V 5' = sup(5, 5') are also two Ft-stopping times. 2. By applying the sampling theorem to the stopping time 5 V s prove that

~ (Ms l{s>s}IFs) 3. Deduce that for s

~

= u, l{s>s}'

t E (Msl\t1{s>s}IFs) = Msl{s>s}'

4. Remembering that MSl\s is Fs-measurable, show that t -t MSl\t is an Ftmartingale.

Conclud~ that

MT =

a P a.s. ,and thus

P a.s. 'Vt ~ T, M, = O.

T 3. fo IKslds is now assumid to be finite'almost surely as opposed to bounded. We shall accept the fact that the random variable f; IKslds is Ft-m~asurable. Show that Tn defined by

t;

= inf{O

~ s ~ T, i t IKslds ~ n}

a

,(or T if this set is empty) is stopping time. Prove that P a.s. limn-Hex> Tn = T. Considering the sequence of martingales (MtI\TJt~O' prove that

-P a.s. 'Vt ~ T, M t

= O.

60

Brownian motion and stochastic differential equations

Ksds

4. Let M t be a martingale of the form J~ n.sw, + J~ with J~ H'[ds < . +00 P a.s. and J~ IKslds < +00 P a.s. Define the sequence of stopping times P a.s.

Tn = inf{t :::; T,J~ H;ds

2: n}, in order to prove that K;

= 0 dt

x

Exercise 17 Let us call X, the solution of the following stochastic differential equation

ax,

{ Xo

=

(p.x t + fJ.')dt + (oXt + a')dWt O.

= 2

We write St = exp ((fJ. - a /2)t

+ aWt).

1. Derive the stochastic differential equation satisfied by St- 1 . 2. Prove that

d(XtS;l)

= St- 1 ((fJ.' -

aa')dt + a'dWd·

3. Obtain the explicit representation of X t . Exercise 18 Let (Wt k:~o be an F t - Brownian motion. The purpose of this exercise is to compute the law of (W t , sUPs9 W s ) . 1. Consider 5 a bounded stopping time. Apply the optional sampling theorem to the martingale M t ' = exp(izWt + z 2 t / 2), where z is a real number to prove that if 0 u :::; v then

s

E (exp (iz(Wv+s - W u+ s)) IFu+s) = exp (-Z2(V - u)/2) . 2. Deduce that w,f = Wu+s - Ws is an 'Fs+u-Brownian motion independent of the a-algebra F s3. Let (Yi)t>o be a continuous stochastic process independent of the a-algebra B such that-E(suPO<s
E (YTIB)

= E (Yi)lt=T'

We shall start by assuming that T can be written as I:l
o < tl < ... < t« = K, and the Ai are disjoint B-measurablesets. 4. We denote by r>' the inf{s 2: 0, W s function we have '

> A},

prove that if

f

is a bounded Borel

(i(Wt)I{T>'~t}) = E (I{T>'~t}¢(t - r>.)) , = E(f(Wu + A)). NoticethatE(f(Wu +A)) = E(f(-Wu+A)) E

where¢(u) and prove that

E (f(Wt)l{ T>'9

1} )

= E (i(2A -

Wt) l{ T>'9}) .

5. Show that if we write Wt = sUPs9 W s and if A 2: 0

P(Wt :::; A, Wt 2: A) = P(Wt 2: A, wt 2: A) ::;:; P(Wt 2: A). Conclude that wt and IWtl have the same probability law.

Exercises

61

6. If A 2: fJ. and A 2: 0, prove that

P(Wt :::; u; wt 2: A) = P(Wt 2: 2A - fJ., wt 2: A) = P(Wt 2: 2A - fJ.), and if A :::; fJ. and A 2: 0 P(Wt :::; u; wt 2: A) = 2P(Wt 2: A) - P(Wt 2: fJ.). 7. Finally, check that the law of (W t , Wt) is given by l{o~y} l{x~y}

2(2y-x)

((2 Y

...,fi;i3 exp -

- X)2) dxdy.

2t

4

The Black-Scholes model

I

I·

I

Black and Scholes (1973) tackled the problem of pricing and hedging a European option (call or put) on a non-dividend paying stock. Their method, which is based on similar ideas to those developed in discrete-time in Chapter 1 of this book, leads to some formulae frequently used by practitioners, despite the simplifying character of the model. In this chapter, we give an up-to-date presentation of their work. The case of the American option is investigated and some extensions of the model are exposed.

4.1 Description of the model

4.1.1lhe behaviour ofprices The model suggested by Black and Scholes to describe the behaviour of prices is a continuous-time model with one risky asset (a share with price St at time t) and a riskless asset (with price Sp at time t). We suppose the behaviour of Sp to be encapsulated by the following (ordinary) differential equation:

dSP = rSpdt,

(4.1)

where r is a non-negative constant. Note that r is an instantaneous interest rate and should not be confused with the one-period rate in discrete-time models. We set sg = 1, so that Sp = eft for t ~ O. . We assume that the behaviour of the stock price is determined by the following stochastic differential equation:

,

dS t

= St (J.tdt + adBt) ,

(4.2)

where J.t and a are two constants and (B t) is a standardBrownian motion. The model is valid on the interval [0, T] where T is the maturity of the option. As we saw (Chapter 3, Section 3.4.3), equation (4.2) has a closed-form solution . ~

s, = So exp (J.tt -

~2 t + aB t) ,

The Black-Scholes model

64

where So is the spot price observed at time O. One particular result from this model is that the law of St is lognormal (i.e. its logarithm follows a normal law). More precisely, we see that the process (St) is a solution of an equation of type (4.2) if and only if the process (log(St)) is a Brownian motion (not necessarily standard). According to Definition 3.2.1 of Chapter 3, the process (Sd has the following properties: • continuity of the sample paths;' . • independence of the relative increments: if u :s t, StlS« or (equivalently), the relative increment (St - Su) / Su is independent of the O"-algebra O"(Sv, v :s u);

:s

• stationarity of the relative increments: if u identical to the law of (St-u - So)/ So·

t, the law of, (St - Su)/ Su is

These three properties express in concrete terms the hypotheses of Black and Scholes on the behaviour of the share price.

A strategy will be defined as it process ¢ = (¢t)09::;T=( (H?, H t)) with values in IR?, adapted to the natural filtration (Ft ) of the Brownian motion; the components H? and H, are the quantities of riskless asset and risky asset respectively, held in the portfolio at time t. The value of the portfolio at time t is then given by Vi (¢)

= Hf S?+tt.s;

This equality is extended to give the self-financing condition in the continuoustime case dVi (¢)

= HfdSf + HtdSt·

To give a meaning to this equality, we set the condition

< +00 a.s.

and

iT H;dt

< +00 a.s.

Then the integral

1T . iT 1 1 (H~St/-L) + 1 HfdSf =

0

Hfrertdt

is well-defined, as is the stochastic integral T "T HtdS t = dt f-t

I H~dS~ I n.as; t

2. Hfsf +

u.s, =

HgSg

+ tt-s; +

65

t

+

a.s.

for all t E [0, T]. We denote by St =' e- rt S, the discounted price of the risky asset. The following proposition is the counterpart of Proposition 1.1.2 of Chapter 1. Proposition 4.1.2 Let ¢ = ((H?, H t)')09::;T be an adapted process with vdlues inIR 2 , satisfying JoT IH?ldH JoT Hldt < +ooa.s. Weset: Vi(¢) = HfS?+HtSt and frt(¢) = e"':rtVi(¢). Then, ¢ defines a self-financing strategy if and only ,if frt(¢) = Vo(¢)

+

it

HudSu a.s.

(4.3)

for all t E [0, T]. Proof. Let us consider the self-financing strategy ¢. From equality :

+ e-rtdVi(¢)

which results from the differentiation of the product of the processes (e~rt) arid (Vi (¢)) (the cross-variation term d{e- r . , (¢) kis null), we deduce dfrt(¢) . _re- rt (Hfe rt + HtSt) dt + e-rtHfd(e rt) + «
v:

HtdSt,

In the discrete-time models, we have characterised self-financing strategies by the equality: Vn+l (¢) - Vn(¢) = ¢n+dSn+l - Sn) (see Chapter 1, Remark 1.1.1).

since the map t

l

dVt(¢) = -rfrt(¢)dt

4.1.2 Self-financing strategies

iT IHfldt

Change ofprobability. Representation of martingales T 1. iT IHfldt + H;dt < +00 a.s.

which yields equality (4.3). The converse is justified similarly.

0

Remark 4.1.3 We have not imposed any condition of predictability on strategies unlike in Chapter 1. Actually, it is still possible to define a predictable process in continuous-time but, in the case of the filtration of a Brownian motion, it-does not restrict the class of adapted processes significantly (because of the continuity of sample paths of Brownian motion). In our study of complete discrete models, we had to consider at some stage a probability measure equivalent to the initial probability and under which discounted prices of assets are martingales. We were then able to design self-financing strategies replicating the option. The following section provides the tools which allow us to apply these methods in continuous time. 4.2 Change of probability. Representation of martingales

.T

O"HtStr/B t,

St is continuous, th~s bounded on [0, T] almost surely.

Definition 4.1.1 A self-financing sstrategy is defined by a pair ¢ of adapted processes (Hf)o::;t::;T and (Ht)09::;T satisfying:

4.2.1 Equivalent probabilities Let (fl, A, P) be a probability sp.ace. A probability measure absolutely continuous relative to P if VA E A

P (A) = 0 ~

Q (A) =0.

Q on (fl, A) ' .

IS

The Black-Scholes model

66

Theorem 4.2.1 Q is absolutely continuous relative to P ifand only if there exists a non-negative random variable Z on (n, A) such that VA E A

Q(A) =

i

Pricing and hedging options in the Black-Scholes model

Theorem 4.2.4 Let (Mt)09~T be a square-integrable martingale, with respect to the filtration (Ft)09~T. There exists an adapted process (Ht)O
Z(w)dP(w).

Z is called density ofQ relative to P and sometimes denoted dQ/ dP. The sufficiency of the proposition is obvious, the converse is a version of the Radon-Nikodym theorem (cf. for example Dacunha-Castelle and Duflo (1986), Volume 1, or Williams (1991) Section 5.14). The probabilities P and Q are equivalent if each one is absolutely continuous relative to the other. Note that if Q is absolutely continuous relative to P, with density Z, then P and Q are equivalent if and only if P (Z > 0) = 1.

u, = M o +

Vt E [0, T]

it n.s»,

+

iT u.s»,

4.2.2 The Girsanov theorem

(n, F, (Ft)o~t~; , p) be a probability space equipped with the natural fil-

tration of a standard Brownian" motion (Bt)O
Theorem 4.2.2 Let ({1t)09~T be an adapted process satisfying a.s. and such that the process (Lt)O
t;

= exp

l{ O;ds < 00

(-it e.s», - ~ it O;dS)

is a martingale. Then, under the probability p(L) with density LT relative to P, the process (Wt)o~t~T"defined by W t = B; + f~ Osds, is a standard Brownian motion. Remark 4.2.3 A sufficient condition for (Lt)09~T to be a martingale is:

T

Let (Bt)O
<"

00,

the process

(J~ lIs-dB s)

is a square-integrable martin-

gale, null at o. The following theorem shows that any Brownian martingale can be represented in terms of a stochastic" integral.

<

+00.

To prove it,

consider the martingale M; = E (UIFt ) . It can also be shown (see, for example, Karatzas and Shreve (1988» that if (Mt)O~t~T is a martingale (not necessarily square-integrable) there is a representation similar to (4.4) with a process satisfying

T

only fo Hlds

< 00, a.s.

We will use this result in Chapter 6.

4.3 Pricing and hedging options in the Black-Scholes model 4.3.1 A probability under which (St) is a martingale We now consider the model introduced in Section 4.1. We will prove that there exists a probability equivalent to P, under which the discounted share price St = e-rtSt is a martingale. From the stochastic differential equation satisfied by (St), we have

Consequently, if we set W t

= =

-re- rt St dt

+ «r'as,

St ((J.L - r)dt

= 13t + (J.L -

+ adBt).

r)t/a,

dSt = StadWt. 4.2.3 Representation ofBrownian martingales

(4.4)

a.s.,

where (Ht) is an adapted process such that E (fo Hlds)

dSt (see Karatzas and Shreve (1988), Dac~nha-Castelle and Duflo (1986». The proof of Girsanov theorem when (Ot) is constant is the purpose of Exercise 19:

a.s.

Note that this representation only applies to martingales relative to the natural filtration of the Brownian motion (cf. Exercise 26). From this theorem, it follows that if U is an FT-measurable, square-integrable random variable, it can be written as

U = E (U)

Let

67

=

(4.5)

From Theorem 4.2.2, with Ot (J.L - r) / a, there exists a probability P" equivalent to P under which (Wt)O<:t"
St = So exp(aWt - a 2t/2).

The Black-Scholes model

68

69

Pricing and hedging options in the Black-Scholes model and consequently

4.3.2 Pricing In this section, we will focus on European options. A European option will be defined by a non-negative, FT-measurable, random variable h. Quite often, h can be written as f(ST)' (f(x) (x-K)+ inthecaseofacall,f(x) (K-:z)+ in the case of a put). As in the discrete-time setting, we will define the option value by a replication argument. FOf technical reasons, we will limit our study to the following admissible strategies: Definition 4.3.1 A strategy
=

=

=

and if the discounted value "Ct(
\It

So we have proved that if a portfolio (HO, H) replicates the option defined by remains to show that the option is indeed replicable, i.e. to find some processes (H2) and (Ht) defining an admissible strategy, such that

H~S~ + n.s, = E* (e-r(T-t) hlFt) . Under the probability P*, the process defined by M; = E*(e-rThIFt} is a square-integrable martingale. The filtration (Ft ) , which is the natural filtration of (B t ) , is also the natural filtration of (Wt ) and, from the theorem ofrepresentation of Brownian martingales, there exists an adapted process (Kt)O
(JoT K~ds)

<

+ u.s;

= h. Let "Ct = Vte- rt be the discounted value

and, by hypothesis, we haveVr

-

Vt =

n;0 + u.s;

=

I +I

=

Vo

u,

o·

= Mo +

=

I -

t KsdWs a.s. 0

=

-

.

\It(
Remark4.3.3 When the random variable hces: be written as h

=

f(ST)' we can express the option value V t at time t as a function oft and St. We have indeed ',\It

E*

(e-r(T~t) f(ST )IFt)

I

E* (e':"r(T-t) f ( Ster(T-t)e"(WT-W, )~(,,2/2)(T-t)) F t) .

= b

t

Vo +

"Ct

- -

The strategy
Since the strategy is self-financing, we get from, Proposition 4.1.2 and equality (4.5)

+00 and

\It E [0, T]

Thus, the option value at time t can be. naturally defined by the expression

\It == H~ S~

(4.6)

h, its value is given by equality (4.6). To complete the proof of the theorem, it

Vt = E* (e-r(T-t) hlFt) . " ..' . o. . E* (e-r(T-t) hJFt}. Proof. First, let us assume that there IS an admissible strategy (H ,H), replicating the option. The value at time t of the portfolio (H2, H t) is given by

= E* (e-r(T-:-t) hlFt) .

HudSu

The random variable S, is .r.:t-measurable and, under P*, W T - W t is independent of Ft. Therefore, from Proposition A.2.5 of the Appendix, we deduce

t

Vt

HuaSudWu.

Under the probability p:;', SUPtE[O,Tj Vt is square-integrable, by definition of admissible strategies. Furthermore, the preceding equality shows that the pro.c~ss (lit) is a stochastic integral r~la~ve to (Wt ): It follows (cf..Chapter 3, ProyoSItlOn 3.4.4 and Exercise 15) that (Vt ) ISa square-mtegrablemartmgale under P . Hence

where F(t, x) = E* (e-r(T-t) f (xe r(T-t)e,,(WT-W,j_(,,2/2)(T-t)) ) .

(4.7)

Since, under P*, WT - W t is a zero-mean normal variable with variance T - t F(t, x)

V t = E* (VTIFt) ,

= F(t, St),

,

= e-r(T-:t) 1+~ f -~

(xe(r-,,2/ 2)(T-t)+"Yv'T-t) e

-~~d

~

y.

The Black-Scholes model

70

F can be calculated explicitly for calls and puts. If we choose the case of the call, where f(x) = (x - K)+, we have, from equality (4.7) F(t,x)

(xe(r-0'2/2)(T-t)~0'(WT-W.)-

=

E· (e-r(T-t)

=

E ( x e O'Y6g- 0'26/2 _ K e- r6) +

K)+)

log (xl K)

+ (r + (2/2) 0

aVO

1-

and' d 2 = d 1

-

In

avO.

_ =

In the proof of Theorem 4.3.2, we referred to the theorem of representation of Brownianmartingales to show the existence of a rep,licating portfolio. In practice, a theorem of existence is not satisfactory and it is essential to be able to build a real replicating portfolio to hedge an option. When the option is defined by a random variable h = f (ST ), we show that it is possible to find an explicit hedging portfolio. A replicating portfolio must have, at any time t, a discounted value equal to

E [( x e O'Y6g- 0'26/ 2 _Ke- r6) l{g+d22: 0 } ] y2 +oo ) e- / 2 ( x eO'Y6y-0'2 6/2 _ Ke- r6 ---dy -d2 ~ y2 2 x e- 0'Y6y- 0'26/ 2 _ Ke- r6 ---'-dy. / e(

j

I.

.

-00

)

~

Writing this expression as the difference of two integrals and in the first one using , the change of variable z = y + aVO, we obtain r6 F(t, x) = xN(d 1 ) - Ke- N(d 2), (4.8) where

jd

1 2 N(d) == -,-, e- X /2dx. ~ -00 Using identical notations and through similar calculations, the price of the put is

equal to

F(t, x)

2. The 'implied' method: some options are quoted on organised markets; the price of options (calls and puts) being an increasing function of a (cf. Exercise 21), we can associate an 'implied' volatility to each quoted option, by inversion of the Black-Scholes formula. Once the model is identified, it can be used to elaborate hedging schemes.

4.3.3 Hedging calls and puts

Using these notations, we have

F(t,x)

71

In those problems concerning volatility, one is soon confronted with the imperfections of the Black-Scholes model. Important differences between historical volatility and implied volatility are observed, the latter seeming to depend upon the strike -price and the maturity. In spite of these incoherences, the model is considered as a reference by practitioners.

where 9 is a standard Gaussian variable and 0 = T - t. Let us set

d -

Pricing and hedging options in the Black-Scholes model

= tee:" N(-d 2) -

xN( -dd·

(4.9)

Vi = e- rt F(t, St), where F is the function defined by equality (4.7). Under large hypothesis on f (and, in particular, in the case of calls and puts where we have the closed-form solutions of Remark 4.3.3), we see that the function F is of class Coo on [0, T[xIR.. If we set F'(~, x) = e- rt F (t, xe rt) , ,

we have

the variables 10g(STISo), log(S2TIST), ... , 10g(SNTIS(N-l)T) are independent and identically distributed. Therefore, a can be estimated by statisti~al means using the asset prices observed in the past (for example ~y calculating empirical variances; cf. Dacunha-Castelle and Duflo (1986), Chapter 5).

Vi = fi'(t, St) and, for tcT; from the Ito formula

'-, P (t, St)

=

P

it

(0, So) + ~~' (u, Su) dSu t

The reader will find efficient methods to compute N(d) in Chapter 8. Remark 4.3.4 One of the main features of the Black-Scholes model (and one of the reasons for its success) is the fact that the pricing formulae, as well as the hedging formulae we will give later, depend on only one non-o?servable parameter: a, called 'volatility' by practitioners (the drift parameter J.L disappears by change of probability). In practice, two methods are used to evaluate a: I. The historical method: in the present model, a 2T is the variance oflog(ST) and

I

+

-

1at 0;

12 t

8F ( w.S« -) du+

0

1 8 2 F- ( -) - 8x 2 U,Su d(S,S)u

,

,

From equality dSt = aStdWt, we deduce -

-

d(S, S)u so that

-2 = a 2 ~udu,

P-(t, St) ca~ be written as

1a~~ (u,~u)SudWu+ 1,K t

P(t,St)'=p(o,so}+

-

, /

t

udu .

Since p(t, St) is a martingale under P", the process K, is necessarily null (cf.

72

The Black-Scholes model

American options in the Black-Scholes model

S~ ) SudWu

Definition 4.4.1 A trading strategy with consumption is defined as an adapted pro.cess if> = ((Hf, H t))095,T, with values in IR?, satisfying thefollowing properties: T 1. iT IHfldt + Htdt < +00 a.s.

Chapter 3, Exercise 16). Hence

P (t, St)

=

P (0, SO) +

it ~~ a

(u,

-) es; = F- (O,So-) + it aP ax («,s;

o The natural candidate for the hedging process H, is then

aP ( t, St-) H, =

s:

= aF ax (t, St) .

= P (t, St) - tt.s; the portfolio (Hf, H t) is self-financing and its discounted value is indeed Vi = P (t, St).

If we set Hf

Remark 4.3.5 The preceding argument shows that it is not absolutely necessary to use the theorem of representation of Brownian martingales to deal with options of the fomi f (ST). . . Remark 4.3.6 In the' case of the call, we have, using the same notations as in Remark 4.3.3,

and in the case of the put

aF -a x (t,x) = -N(-d . 1) . This is left as an exercise (the easiest way is to differentiate under the expectation). This quantity is often called the 'delta' of the option by practitioners. More generally, when the value at time t of a portfolio can be expressed as ll1(t, St), the quantity (alII/ ax) (t, St), which measures the sensitivity of the portfolio with respect to the variations of the asset price at time t, is called the 'delta' of the portfolio. 'gamma' refers to the second-order derivative (a 2 1l1 / ax 2 ) (t, St), 'theta' to the derivative with respect to time and 'vega' to the derivative of III with respect to the volatility a.

4.4 American options in the Black-Scholes model

4.4.1 Pricing American options We have seen in Chapter 2 how the pricing of American options and the optimal stopping problem are related in discrete-time setting. The theory of optimal stopping in continuous-time is based on the same ideas as in discrete-time but is far more complex technically speaking. The approach we proposed in Section 1.3.3 of Chapter 1, based on an induction argument, cannot be used directly to price American options. Exercise 5 in Chapter 2 shows that, in a discrete model, it is possible to associate any American option to a hedging scheme with consumption.

73

I

2. H?sf + I

.

u.s; =

ir H~dS~ + inr HudSu - c, for all t

HgSg + HoSo +

t

o

t E [0, Tj, where (Ct )05,t5,T is an adapted, continuous, ~on-decreasing process null at t = 0; Ct corresponds to the cumulative consumption up to time t.

An American option is naturally defined by an adapted non-negative process (h t )09 5, -r:' To sim?lify, we wil.l only study processes of the form ht = 'lj;(St) , where w IS a .;:ontmuous function from IR+ to IR+, satisfying: 'lj;(x) ~ A + B«, 'Vx E IR ,for some non-negative constants A and B. For a call, we have: 'lj;(x) = (x - K)+ and for a put: 'lj;(x) = (K - x)+. The tra~ing str~tegy with consumption if> = ((Hf, H t) )05,t5,T is said to hedge the Amencan option defined by h t = 'lj;(St) if, setting Vt (if» = Hf Sf + HtSt, . . we have 'Vt E [O,Tj

Vt(if» 2': 'lj;(St} a.s.

De~ote by <1>'" the set of tradi~g strategies with consumption hedging the American option defined by h t = 'lj;(St). If the writer of the option follows a strategy if> E <1>"', he or she p~ssesses a~ an~ time t, a wealth at least equal to 'lj;( St), which is precisely the payoff If the opuon IS exercised at time t. The following theorem introduces the minimal value of a hedging scheme for an American option: Theorem 4.4.2 Let u be the map from [0, Tj x IR+ to IR defined by u(t, x) =

sup E··[e-r(r-t)'lj;(xexp((r_ (a 2/2))(T-t)+a(Wr -Wd))]

rE7't,T

wh~re Ti,T represents the set ofstopping times taking values in [t, Tj. There exists a strategy if> E <1>'" such that Vt (¢) = u(t, St),for all t E [0, Tj. Moreover; for any strategy if> E <1>"', we have: Vt(if» 2': u(t,St),forallt E [O,Tj. To overcome technical difficulties, we give only the outlines of the proof (see ~aratzas and Shreve (1988)Jor details). First, we show that the process (e-rtu( t, St)) IS t~e Snell env~lope of the process (e-rt'lj;(St)), i.e. the smallest supermartingale which bounds It from above under p ". As it can be proved that the discounted value of a ~ading ~trategy 'with consumption is a supermartingale under P", we de~uce the inequality: ~(if» 2': u(t, St), for any strategy if> E <1>"'. To show the existence of a strategy- if> such that Vt(¢) = u(t, St), we have to use a theorem of decomposition of supermartingales similar to Proposition 2.3.1 of Chapter 2 as well as a theorem of representation of Brownian martingales. . It i~ ~atural t~ ~onsider u(t; St) as a price for the American option at time t, smce.lt IS the minimal value of a strategy hedging the option.

74

The Black-Scholes model

Remark 4.4.3 Let 7 be a stopping time taking values in [0, T]. The value at time 0 of an admissible strategy in the sense of Definition 4.3.1 with value 'l/J(Sr) at time 7 is given by E* (e- rr 'l/J(Sr)), since the discounted value of any admissible strategy is a martingale under P*. Thus the quantity u(O, So) = sUPrE70,T E* (e- rr'l/J(Sr)) is the initial wealth that hedges the whole range of possible exercises.

As in discrete models, we notice that the American call price (on a non-dividend paying stock) is equal to the European call price: Proposition 4.4.4 If in Theorem 4.4.2, 'l/J 'is given by 'l/J(x) real x, then we have

u(t,x)

= (z -

u(O,x)= sup E*(Ke- rr-xexp(aWr-(a 27/2)))+. rE70,T .

u(O, x)

sup E(Ke-rr-xexp(aBr-(a27/2)))+

rE70,T

<

sup E [(KerE70,<X>

= E*(ST

>

0). Then it is

- e-rTK)+.

On the other hand, we have

~ E* ((ST - e-rTK)IFr) ~

Uoo(x)

s, - e-rTK'.,

rr = rE70.<X> sup E.[(Ke- -

E* ((ST - e-rTK)+IFr)

Uoo(x) = K - x

s;

~ (Sr - e- rr K) + .

We obtain the desired inequality by computing the expectation of both sides. , 0

Uoo(x) = (K - z")

In the case of the put, the American option price is not equal to the European one and there is no closed-form solution for the function u. One has to use numerical methods; we present some of-them in Chapter 5. In this section we will only use the formula .

- xexp (:--a 2(7 - t)/2 + a(Wr - Wd)) + .

. (4.10) to deduce some properties of the function u. To make our point clearer, we assume t = O. In fact, it is always possible to come down to this case by replacing Twith

(4.13)

(

X) x*

-"'I

for

x:::; z"

for

x > x"

=

K,/(l +,) and, = 2r/a 2 . with z" '. Proof. From formula (4.13) we deduce that the function u oo is convex, decreasing on [O,oo[ and satisfies: Uoo(x) ~ (K - x)+ and, for any T > 0, Uoo(x) ~ rT E(Ke- - xexp (aBT - (a 2T/2)))+, which implies: Uoo(x) > 0, for all x ~ O. Now we note x" = sup{x ~ Oluoo(x) = K - x}. From the properties of u oo we have just stated, it follows that "Ix:::; z"

4.4.2 Perpetual puts, critical price

(K~-r(r-t)

xexp (aB r - (a 27/2)))+ l{r
is given by the formulae

~ e- rr K

0 and by non-negativity of the left-hand term,

u(t, x) = sup E* rET.,T

- xexp (aB r - (a 27/2)))+ l{r
noting To,oo the set of all stopping times of the filtration of (Bt)t~o and To,T the set of all elements of To,oo with values in [0, T]. The right-hand term in inequality (4.12) can be interpreted naturally as the value of a 'perpetual' put (i.e. it can be exercised at any time). The following proposition gives an explicit expression for the upper bound in (4.12). Proposition 4.4.5 The function

since (St) is a martingale under P*. Hence,

~

rr

(4.12)

E*(e-rr(Sr - K)+) :::; E*(e-rT(ST - K)+)

E* ((ST - e- rT K)+ IFr ) .~

(4.11)

Let us consider a probability space (0, F, P), and let (Bt)o
= F(t,x)

Proof. We assume here that t = 0 (the proof is the same for t sufficient to show that, for any stopping time 7,

since r

75

T - t. Equation (4.10) becomes

K)+, for all

where F is the function defined by equation.iaB) corresponding to the European call price.

E* ((ST - e-rTK)+!Fr)

American options in the Black-Scholes model

Uoo(x) = K - x

and

"Ix >x*

Uoo(x) > (K - x)+. (4.14)

On the other hand, the Spell envelope theory in continuous time (cf. E1 Karoui (1981)(Kushner (1977), aswell as Chapter 5)enables us to show

Uoo(x) = E [(Ke,,"':'rrz

-

xexp (aB r

%

-

(a 27x/2)))+ 1h<00}]

where 7x is the stopping time defined by 7x = inf{t ~ 01 e-rtuoo(Xt) Xt)+} (with inf 0 = 00), the process (Xf) being defined by: Xf = 2 x exp ((r - a / 2)t + a B t ) . The stopping time 7 x is therefore an optimal stopping time (note the analogy with the results in Chapter 2). It follows from (4.14) that

«<;« -

7x = inf{t ~ 0IX: :::; z"}

= inf {t ~ Ol(r -

a 2/2)t + aB t :::; log(x* /x)}.

The Black-Scholes model

76

where,

Introduce, for any z E IR+, the stopping time Tx,z defined by

77

Exercises

= 2r1

a2 .

The derivative of this function is given by

z'Y- 1

Tx,z = inf{t ~ 0IX: ~ z}.

¢'(z) = -

. .. ,IS giiven by T x With these notations, the optimal stoppmg ume and we note ¢ the function of z defined by

..1.( ) =. E ( e - r T " Z

'I' Z

= T x,x'"We fix x

'1{Tz,.
••

=

I'S' optimal the function ¢ attains its maximum at point z x": We . S mce T x x " ' . . . d . * and are going to calculate ¢ explicitly, then we will maximise It to etermme x

UOO(x) = ¢(x*). ( ) _ (K _ ) If z < x we have If z > x, it is obvious that Tx,z = a and ¢ z x +. -' , by the continuity of the paths of (Xnt"~o' Tx,z

= inf{t ~OIX: = z}

.

x'Y

(K, -

b + l)z).

It results that if x ~ K,I(,+l),max z ¢(z) = ¢(x) = K -xandifx > K,/b+ I), max, ¢(z) = ¢ (K, Ib + 1)), and we recognise the required expressions. 0

Remark 4.4.6 Let us come back to the American put with finite maturity T. Following the same arguments as in the beginning of the proof of Proposition 4.4.5, we see that, for any t E [0, T[, there exists a real s(t) satisfying "Ix ~ s(t) u(t,x) = K - x and "Ix> s(t) u(t,x)

> (K - x)+.

(4.15)

Taking inequality (4.12) into account, we obtain s(t) ~ z", for all t E [0, T[. The real number s(t) is interpreted as the 'critical price' at time t: if the price of the underlying asset at time t is less than s( t), the buyer of the option should exercise his or her option immediately; in the opposite case, he should keep it.

and consequently

.

.

WIth, by convention, e see that, for z ~ x,

Tx,z

rTz

=

¢(z)

(K - z)+E (e- ,· l h ,.
-roo _

= ria -

expression of Xt in terms of B t , we ,

inf{t~OI(r-a;)t+aBt=IOg(ZIX)}

=

inf {t with JL

a Using the

-.

~ 0l/.tt + e, = ~ IOg(ZIX)} ,

a 12. Thus, if we note, for any real b, T b =inf{t ~ OIJLt + B;

4.5 Exercises

= b},

we get

~(z)

(K - x)+ (K z)E (exp (-rT1og(z/x)/u)) = {

a

if z > x . if Z E [0, x] n [0, K] if z E [O,x] n [K,+oo[.

The maximum of ¢ is attained on the interval [0, x] n [0, K]. Using the following formula (proved in Exercise 24)

.

E

(e-aT&)

= exp

(~b - IbJv!JL2 + 2a) ,

it can be seen that

Vz E [O,x] n [O,K] ¢(z)

= (K- z)

Notes: The presentation we have used, based on Girsanov theorem, is inspired by Harrison and Pliska (1981) (also refer to Bensoussan (1984) and Section 5.8 in Karatzas and Shreve (1988». The initial approach of Black-Scholes (1973) and Merton (1973) consisted in deriving a partial differential equation satisfied by the the call price as a function of time and spot price. It is based on an arbitrage argument and the Ito formula. For more information on statistical estimation of the models' parameters, the reader should refer to Dacunha-Castelle and Duflo (1986) and Dacunha-Castelle and Duflo (1986) and to the references inthese books.

(~)'Y,

Exercise 19 The objective of this exercise is to prove the Girsanov theorem 4.2.2 in the special case where the process (Ot)is constant. Let (Bt)o9~T be a standard Brownian motion with respect to the filtration (Ft)O
=

1. Sh6w that (Lt)O
t

~elative to

(i.t

the filtration (Ft) and that

.

2. We note p(L.) the density probability L t with respect to the initial probability P. Show that the probabilities p(L T ) and p(L.) coincide on the a-algebra Ft. 3. Let Z be an FT-measurable, bounded random.variable, Show that theconditional expectation of Z, under the probability p(LT), given F«, is

E'(LT)(ZIFt)

= E (ZLTIFt). Lt

Exercises

The Black-Scholes model 78 4. We set W = J-Lt + Bi, for all t E [0, T]. Show that for all real-valued u and for t all sand t in E [0,T], with s ~ t, we have

E(LT) (eiU(w.-w.) \.rs)

= e-

79

Exercise 25

1. Let P and Q be two equivalent probabilities on a measurable space (n, A). Show that if a sequence (X n ) of random variables converges in probability under P, it converges in probability under Q to the same limit.

u 2(t-s)/2.

2. Notations and hypothesis are those of Theorem 4.2.2. Let (Ht)O
Conclude using Proposition A.2.2 of the Appendix. Exercise 20 Show that the portfolio replicating a European option in the BlackScholes model is unique (in a sense to be specified).

x, =

1. Show that if f is non-decreasing (resp. non-increasing), F(t, x) is a nondecreasing function (resp. ,non-increasing) of x. 2 We assume that f is convex. Show that F(t,x) is a convex ~unction ~f z, a . d . function of t if r - 0 and a non-decreasing function of a many ecreasmg fJ ,. ality: . case. (Hint: first consider equation (4.7) and make us~ 0 ensen ~ mequ cI>(E(X)) ~ E (cI>(X)), where cI> is a convex function and X IS a random variable such that X and cI>(X) are integrable.) We note Fe (resp. Fp) the function F obtained when f(x) = (x ~ K)+ (resp. 3. f(x) = (K _ x)+). Prove that Fe(t,.) and Fp(t,.) are non-negatlvefor t < T. Study the functions Fe(t,.) and Fp(t, .) in the neighbourhood of 0 and +00.

lit =

.and b, we set

= inf{t 2: 0 I J-Lt + B; = b} with the convention: inf 0 = 00. Tt

1. Use the Girsanov theorem to show the following equality:

'Va,t>O

E(e-a(T~t\t)exP(J-LBT~I\t-~2T~t\t)l {t
3. Deduce from above and Proposition 3.3.6 that

.

'Va> 0 E (e-aTt: 1{Tt:
< 00).

I

<

00

p(L)-a.s. and we

t HsdWs.

converges to 0 in probability.

Exercise 24 Let (Btk?o be a standard Brownian motion. For any real-valued J-L

2. Prove the inequality'

Hs(}sds.

The question is to prove the equality of the two processes X and Y. To do so, it is advised to consider first the case of simple processes; and to use the fact that if (Ht)o9~T is an adapted process satisfying J; H;ds < 00 a.s., there is a sequence (Hn) of elementary processes such that JoT (H 5 - H;') 2 ds

ropean call is exercised. . . Exercise 23 Justify formulae (4.8) and (4.9) and calculate for a call and a put the delta, the gamma, the theta and the vega (cf. Remark 4.3.6).

(e-a{T~I\t) exp (J-LBT~1\t - J-L; T~ t\ t) )

t

Since p(L) and P are equivalent, we have J; H;ds can define; under p(L), the process

Exercise 22 Calculate under the initial probability P, the probability that a Eu-

'Va, t > 0 E (e-a(Tt: I\t)) = E

I n.e», + I t

Exercise 21 We consider an option described by ~ = f.(ST) and w~ note F the function of time and spot corresponding to the option pnce (cf. equation (4.7)).

.

Exercise 26 Let (Bt)O
9t

T

by.rt

t\ t. '

1. Show that (9t)O1}19t). Show that M, is equal to ,.. e->'(l-t) l{T>t} a.s. The following property can be used: if 8 1 and 8 2 are two sub-a-algebras and X a non-negative random variable such that the aalgebra generated by 8 2 and X are independent of the a-algebra 8 1, then E ('~18l V 82) = E (XI82 ) , where 8 1 V 82 represents the o-algebragenerated by 8 1 and 8 2. . 3. Showthat there exists-no path-continuous process (X t ) such that for all t E [0,1]' P (M t = X t ) =,1 (remark that we would necessarily have

P ('Vt E [0, IJ M,

= X t ) = 1).

Deduce that the martingale (M t ) cannot be represented as a stochastic integral with respect to (Bd. . Exercise 27 The reader may use the results of Exercise '18 of Chapter 3. Let (Wtk~o be an .rt-Brownian motion.

The Black-Scholes model

Exercises

81

3. Prove that there exists a probability P" equivalent to P, under which the discounted stock price is a martingale. Give its density with respect to P. 4. In the remainder, we will tackle the problem of pricing and hedging a call with maturity T and strike price K. Deduce that if

E (e

aWT1

x ~ JL

. (a

. } ) -exp -.

{WT~~,inf.::;TW.~A

2T

--

.

2

+ 2a),) N (2), - + aT) JL

1m

vT

(a) Let (H?, Hl) be a self-financing strategy, with value \It at time t. Show that if (\It/ SP) is a martingale under P" and if VT (ST - K)+, then

=

.

'TIt E [0, T) Vi = F(t, St),

2. Let H ~ K; we are looking for an analytic formula for

C

= E (e-rT(XT -

where F is the function defined by

K)+1{inf.::;T X.~H}) ,

F(!,

where X, = x exp ((r /2) t+ uWt) . Give a finan~iaUnterpretati.?n to this value and give an expression for the probability p that makes W t (r / o - o /2) t + W t ~ standard Brownian motion. _

x) = E" (x exp (J.T u(, )dW. - ~ J.T U'U)d,) -: K e- J.T .(.)d.) +

'u2

and (Wd is a standard Brownian motion under p ". (b) Give an expression for the function F and compare it to the Black-Scholes formula. (c) Construct a hedging strategy for the call (find H?and H t ; check the selffinancing c o n d i t i o n ) . l

3. Write C as the expectation under P of a random variable function only ofWT and sUP~~s~T Ws .

. ,.

4. Deduce an analytic formula for C.

Problem 2 Garman-Kohlhagen model

Problem 1 Black-Scholes model with time~dependent parameters ~e consider once again the Black-Scholes model, assuming that ~e ass~t p~ces are described by the following equations (we keep the same notations as In this J

ter)

dSP

ch~~.

= r(t)Spdt

{ dS t = St(JL(t)dt + u(t)dBd . where ret), JL(t), u(t) are detenninistic functions of time, continuous on [0, T). Furthermore, we assume that inftE[o,T] u(t) > 0: . 1. Prove that

s,

= So exp

(~t JL(~)ds + ~t u(s)dB ~ ~t u

2(S)dS)

s _

I

.

. : JL(s)ds

dS t

.

S

.

= JLdt

t.

+ udWt,

where (Wt}tE[O,Tj is a standard Brownian motion on a probability space (O,\F, P), JL and o are real-valued, with o > 0. We note (Ft)tE[O,Tj the filtration generated by (WdtE[O,Tj and assume that F t represents the accumulated information up to time t '( . . ..'

J

You may consider the process

z, = s, exp --:-

!

The Garman-Kohlhagen model (1983) is the most commonly used model to price and hedge foreign-exchange options. It derives directly from the Black-Scholes model. To clarify, we shall concentrate on 'dollar-franc' options. For example, a European call on the dollar, with maturity T and strike price K, is the right to buy, at time T, one dollar for K francs. We will note S; the price of the dollar at time t, i.e. the number of francs-per dollar. The behaviour of S, through time is modelled by the following stochastic differential equation:

+ ~t u(s)dB s

:-

~ ~t u

2(s)dS)

I

.

1. Express S, as a function of So, t and W t. Calculate the expectation of St.

2.

2. Show that if JL (a) .Let (X n ) be a sequence of real-valued, zero-mean normal random vari.ables converging to X in mean-square. Show that X is a normal random variable. (b) By approximating o by simple .functi?ns, show that random variable and calculate ItS varIance. 0

J~ u(s )dB s is anormal •

.

.

> 0, the process (St)tE[O,Tj is asubmartingale.

3. Let U; = 1/ St be the exchange rate of the franc' against the dollar. Show that Uc satisfies the following stochastic differential equation dU t (2 . = u - JL)dt - udWt.

u,

82

Exercises

The Black-Scholes model

83

Deduce that if 0 < J.L < a 2, both processes (St)tE[O,T] and (Ut)tE[O,TJ are submartingales. In what sense does it seem to be paradoxical?

(The symbol E stands for the expectation under the probability P.) 5. Show (through detailed calcUlation) that

II We would like to price and hedge a European call on one dollar, with maturity T and strike price K, using a Black-Scholes-type method. From his premium, which represents his initial wealth, the writer of the option elaborates a strategy, defining at any time t a portfolio made of HP francs and H, dollars, in order to create, at time T, a wealth equal to (ST - K)+ (in francs). At time t, the value in francs of a portfolio made of Hp francs and H, dollars is obviously Vt = H~ + (4.16) We suppose that French francs are invested or borrowed at the domestic rate TO and US dollars are invested or borrowed at the foreign rate TI' A self-financing strategy will thus be defined by an adaptedprocess ((HP, Ht»tE[O,Tj, such that

F(t,x) = e-r,(T-t)xN(dd - Ke-ro(T-t)N(d2) where N is the distribution function of the standard norrnallaw, an d

n.s;

dVt = ToHpdt + TIHtStdt

+ HtdSt

+ (a 2j2»(T a..;T=t

dl

10g(xjK) + (TO -

d2

10g(xjK)

-

(a) We set St = e h -ro)t St. Derive the equality

es, = aStdWt.

(4.17)

(b) Let

~ be the function defined by F(t, x)

F(t, St}. Derive the equality

_

-

TO t

a

is non-negative for all t and if SUPtE[O,Tj

Vi

(Vi) is square-integrable under

P. Show that the discounted value of an admissible strategy is a martingale under

P.

4. Show that if an admissible strategy replicates the call, in other words it is worth VT = (ST - K)+ at time T, then for any t ~ T the value of the strategy at time t is given by where

F(t,x)

=

E(xexP(-(TI.+(a 2j2»(T-t)+a(WT - Wt») - K e-~o(T-t)) . +

-rotc

t =

7. fors« down a put-call parity relationship, similar to the relations~iP we gave or stocksh' alndd grve an example of arbitrage opportunity when this relationship does not 0 .

Proble~ 3 Option to exchange one asset for another

+ UT t't't

is a standard Brownian motion. (b) A self-financing strategy is.said to be admissible if its discounted value

t - e

(c) renlicati that the c.all is replicable and give an explicit expression for the rep icatmg portfolio (( Hp, Ht».

To)dt + Hte-rotStadWt.

J.L + TI

, t

sc, = -aF ( t St)ae-rotS dW'; ax ' t t-

(a) Show that there exists a probability P, equivalent to P, under which the process t't't -

= e- rot F(t, xe(ro-r,jt) (F is the

f~nctl_on defined in Question 4). We set C, = F( t S) and G -

3.

TiT

(a 2j2»(T - t)

a..;r-=-t

Vi

+ TI

TI -

t)

6. The next step is to show that the option is effectively replicable.

where Vt is defined by equation (4.16). 1. Which integrability' conditions must be imposed on the processes (HP) and (Hd so that the differential equality (4.17) makes sense? 2., Let = e-rotVt be the discounted value of the (self-financing) portfolio (HP, Ht}. Prove the equality

,dVt = Hte-rotSt(J.L

+ (TO -

TI

1 I'

I

.

W~ tO~~der a fin2anci~1 market in which there are two risky assets with respective pnces

t a~d St at tlI~e t and a riskless asset with price So

= ert at time

t

~~::S~~~~~e~:nt~;1~~~:~~!s and Slover time are modelled by the followin~ dS!

dS~

S! (J.Lldt

+ aldB!)

{ / Sl (J.L2dt + a2a B l) where (BI) [ . and (B2) .' . d t tE O,T] t tE[O,Tj are two Independent standard Brownian mo nons efined on a probability space (0 F P). with :> 0 . d ' , ,J.LI,J.L2,al anda2arerealnumbers abiesa11 an ~2 > O. We note!t the a-algebra generated by the random vari~ • and B. for s ~ t. Then the processes (B I) d (B2) e (Fd-Brownian motions and, for t ~ s, the vecto/ (~IO~];~ B2 ~ ~JO)?J :rrIndependent of F.. t·. , t . • IS .J'

The Black-Scholes model

84

85

I

where the function F is defined by

We study the pricing and hedging of an option giving the right to exchange one of the risky assets for the other at time T. 1. We set by

(h = (ILl - r) /0"1

u,

and

= exp (

Exercises

(h = (IL2 - r) /0"2, Show that the process defined

-e.e; - B2B; - ~(B~ + B~)t)

,

is a martingale with respect to the filtration (Ft)tE[o,T]'

2. Let P be the probability with density MT with respect to P. We introduce the processes WI and W 2 defined by Wl = Bf +B1t and Wl = B; +B2t. Derive, under the probability P, the joint characteristic function of (Wl , Wl). Deduce that, for any t E [0, T], the random variables Wl and wl ar~ independent normal random variables with zero-mean and variance t under P. In the remainder, we' will admit that, under the probability P, the processes (Wl )o:9~T"and(Wl)O:9~T are (Ft)-indepe~dent st;n?~d Brownian motions and that, for t 2: s, the vector (Wl - W s1, W t - W s ) IS independent of F s •

3. Write Sland Sl as functions of SJ, S5,!Vl and Wl and show that, under P, the discounted prices Sl = e:" Sl and Sl == e:" S; are martingales. We want to price and hedge a European option, with maturity T, giving to the holder the right to exchange one unit of the asset 2 for cine unit of the as~et.1: do so we use the same method as in the Black-Scholes model. From hIS initial wealth, the premium, the writer ?f the opt~on builds a strategi' defini~g a~ any time t a portfolio made of Units of the riskless asset and H; and H; Units of the assets 1 and 2 respectively, in order to generate, at time T, a wealth equal to (St - Sf)+· A trading strategy will be defined by the three adapted processes HO, HI and H 2.

:0

HP

~ ,,2 ) +' F(t, Xl, X2) = E- ( x1eO'I (WIr - W•I) "'-T(T-t) _ x2e0'2(Wf-W,2)-=f(T-t) (4.19) the ~ymb~l E repres~nting the expectation under P. The existence of a strategy having this value will be proved later on. We will consider in the remainder that the value of the option (St - Sj)+ at time t is given by F(t, Sf, Sl). 4. Find a parity relationship between the value of the option with payoff (Sl _ sj)+ and the symmetrical option with payoff (Sf - St)+, similar to :the put-call parity relationship previously seen and give an example of arbitrage opportunity when this relationship does not hold.

ill The objective of this section is to find an explicit expression for the function F defined by (4.19) and to establish a strategy replicating the option.

1. Let 91 and 92 be two independent standard normal random variables and let A be a real number. (a) Show that under the probability p(A), with density with respect to P given ._ by dP(A) 2

__ = eAgI-A

rtSl dW l -r d1Yt t 0"1 t - HIt e -

= e-rtVt is

2 + H t2e-rtS20" t 2 dW t .

2. Show that if the processes (HI )o~t~~ and (Hl)o:9~T of a self-financing strategy are uniformly bounded (which means that: 30 > 0, Vet,w) E [O,~] x fl, IHt(w)1 ::; 0, for i 1,2), then the. discounted value of the strategy IS a martirigale under P.

=

3. Prove that if a self-financing strategy satisfies the hypothesis of the previous question and has a terminal value equal to VT (St .:... Sf)+ then its value at any time t < T is given by

=

(4.18)

'

the random Gaussian variables 91- A arid 92 are independent standard variables. (b) Deduce that for all real-valued Y1, Y2, Al and A2' we have

E (exp(Yl

+ A19d' -

exp(Y2

=eYI+A~/2N-(Yl -

+ A292))+

Y2 + A?) _ JA~+A~

·11 1. Define precisely th~ self- financing strategies' and prove that, if Vt the discounted value of a self-financing strategy, we have

/2

dP

.

eY2+A~/2N (Y1 -

Y2

-A~)

JA~+A~

,

where N is the standard normal distribution function.

2. Deduce from the previous question an expression for F using the function N. 3. We set c, = «:" F(t, Sf, S;). Noticing that : /

.

c, =

F(t, sf, S;) =

E (e- rT (S} -

Sf)+ 1F t ) ,

prove the equality

-

of

- -

of

_ _

ac, = !lx (t,SI,S;)O"l e- rtSf dwl + ~(t,SI,S;)0"2e-rtSt2dWr u

1

uX2

=

Hint: use the fact that if (Xt ) is an Ito process which can be written as X t t t X o + f o J1dW; .: f0 7 ; dw ; + f~ Ksds and if it is a martingale under P, then K, = 0, dtdP-almost everywhere.

The Black-Scholes model

86

,

Exercises

show that condition (ii) is satisfied if and only if we have, for all t E [0, Tj.

4. Build a hedging scheme for the option.

t:t = Vo +

Problem 4 A study of strategies with consumption

We consider a financial market in which there is one riskless asset, with price S~ = ert at time t (with r ~ 0) and one risky asset, with price St at time t: The model is studied on the time interval [0, Tj (0 ~ T < 00). In the following, (St)O~t~T is a stochastic process defined on a probability space (0, F, P), equipped with a filtration (Ft)o9~T' We assume that (Ft)O~t~T is the natural filtr~tion of a standard Brownian motion (Bt)o9~T and that the process (St)09~T IS adapted to this filtration. .

I

,

We want to study strategies in :ovhich consu~ption is allowed. The dynamic of (St)09~T is given by the Black-Scholes-model

dS t with JL E IR and a

= St(JLdt + adBt) ,

with

it

HudBu - i t c(u)du,

a.s.

s. = «r:s; and c(u) = e-ruc(u).

2. We suppose that conditions (i) to (iv) are satisfied and we still note t:t = e-rtVi = e:" (H2 S2 + HtSt). Prove that the process (t:t)O
3. Let (c(t))09~T be an adapted process with non-negative values such that

E· (JOTc(t)dtf ~ 00 and let x > O. We say that (c(t))09~T is a budgetfeasible consumption process from the initial endowment x if there exist some processes (H2)o~t~T and (Ht)O~t~T such that conditions (i) to (iv) are satisfied, and furthermore Vo = HgSg + HoSo = x. . (a) Show that if the process (c(t))O
> O. yve note P" the probability with density exp (-(JBT - (J2T /2)

endowment x then E· (JoT e-rtc(t)dt)

with respect to P, with (J = (JL -r)/a. Under p •. the process (Wt)09~T. defined by W t = (JL - r)t/a + Be, is a standard Brownian motion. A strategy with consumption is defined by three stochastic processes: (H~)09~T, (Ht)O
iT

87

(IH~I + H~ + Ic(t)1) dt < oo,a.s.

.~ x.

'

(b) Let (c(t))09~T be an adapted process. with non-negative values and such that

E' ( { C(t)dt)' < 00 and E' ( { e:"'C(t)dt) '" x .. Prove that (c(t))O~t::;T is a budget-feasible consumption process with an initial endowmentx. Hint: introduce themartingale (Mt)O::;t::;T defined by

M t = E·

(x + foT e-rsc(s)dsIFt) and apply the theore~ of martingales

representation. (c) An investor with initial endowment x wants to consume a wealth corresponding to the sale of p risky assets by unit of time whenever.S, crosses ~. some. ?arrier K upward (that corresponds to c(t) pSt1{S,>K}). What conditions on p and x are necessary for this consumption process to be budget-feasible?

=

(ii) For all t E [0, Tj,

HOSo+HtSt = HgSg+HoSo+ t

t

uc(u)du, . 10r H~dS~+ . 10r HudS 1r0

a.s.

(iii) For all t E [0, Tj, c(t) ~ 0 a.s. (iv) For all t E [0, Tj.

th~ rand.om variable H2 S~ + HtS t is non-negative and

II

/

We now suppose that the volatility is stochastic. i.e. that the process (St)O
dS t = St(JLdt sup (Itiffi.+HtSt+ tC(S)dS)

tE[O,T]

10

is square-integrable under the probability p ", 1. Let (H2)09~T. (Ht)O~t~T and (c(t))09~T be three adapt~dpro~~~ses satisfying condition (i) above. We set Vi = H2 S~ + HtS t and Vt = e Vi, Then

+ a(t)dBt),

(4.20)

where JL E IR and (a(t) )O::;t~T is 'an adapted process. satisfying I

Vt E

[O,Tj

al

~

a(t)

~ (,2,

for so~e constants.o, and az such that 0 < al < az- We consider a European call With maturity T and strike price K on one unit of the risky asset. We know

88

The Black-Scholes model

(see Chapter 5) that if the process (a(t) )O
ec at

-(t,x)

a 2x2 a2c

'C(T, x) = (x - K)+. We note C 1 the function C corresponding to the case a = a1 and C 2 the function C corresponding to the case a = a2. We want to show that the price of the call at time 0 in the model with stochastic volatility belongs to [C1(0, So), C2(0, So)l. Recall that if (OdO
e.se, - ~ I; O;ds) i~ a martingale.

with Il E IR and a

2. Show that the solution of equation (4.20) is given by

S~ exp (Ilt + it a(s)dB ~ it a 2(S)ds) ..

h = (C(T1, STl) - Kd+, where C(t, x) is the price of the underlying call, given by the Black-Scholes formula. 1. (a) Graph function x I-t C(T1, x). Show that the line y = x - Ke-r(T-Tt} is an asymptote (hint: usethe put/call parity).

4. Explain why the price of the call at time

a is given by

. Co = E* (e:-rT(ST - K)+) . 5, We set

S~

(b) Show that the equation C(T1 , x) 2. Show that at time t · dfi e ne db y G IS

s -

3. Determine a probability P* equivalent to P under which the process defined by W t = B, + I;(Il- r)/a(s)ds is a standard Brownian motion.

= e-rtSt. Show that E*

(Sn s S6elT~t.

G(O, xl

M, =

0

e-

7. Using Ito formula and Questions 1 and 6, show that «rc, (t, Sd is a submartingale under probability .P*. Deduce that C 1 (0, So) ~ Co. . , 8. Derive the inequality Co

s C2(0, So).

Problem 5 Compound option ) We consider a financial market offering two investment opportunities.. The first traded security is a riskless asset whose price is equal to Sp = e rt at time t (with r ~ 0) and the second security is risky and its price is denoted by S, at time t E [0, T1. Let (SdO~t~T be a stochastic process defined on a probability space (n,:F, P), equipped with a filtration (:Ft)O~t~T, We assume that (:Fd09~T is

_ t S) h , t,were

~ E [e-'~ (C ( T" xe(.- "n';-,"",) - K,

i.

= E [e-rIlC (n,xe(r-lT

n'

2/2)lI+lT

V69)

1{9>~d}]

_ K1e-rIlN(d),

where

d = 10g(X/!1) + (r- a2 /2) 0

(u, Su)a(u)SudWu

is a martingale under probability P* . .

1

,

(a) ~how that x I-t G(O, x) is an increasing convex function . (b) We now want to compute G explicitly. Let us denote by· N the standard ' , cumulative normal distribution. Prove that

G(O,x)

.

< T 1 , the compound option is 'worth G(T

with 9 bei~g a standard normal random variable.

6. Prove that the process defined by

ru

= K 1 has a unique solution Xl.

3.

I

. l't . eca}

> O.

.We wan~ to study an example of compound option. We consider a call option WI.th matunty T I E10, T[ and strike price K 1 on a call of maturity T and strike pnce K. The.value of this option at time T1 is equal to

1. Prove (using the price formulae written as expectations) that the, functions x I-t C1(t,x) and x I-t C 2(t,x) are convex.

s, =

89

the natural filtration generated by a standard Brownian motion (Bdo
ec

+ ---;:;-z(t,x) + rx-;:)(t,x) - rC(t,x) = a 2 ox ox on [0, T[x10, oo[

d~fined by i, =

Exercises

a../O

(c) Show that if 91 is a st~ndard normal variable independent of 9, we can write O~/= T - T 1 and characterise G by, ,"

. G(O, x)

+ K 1 e- r llN(d)

E [( xe lT( V69+V919t) -

"22

(1I+IIt) _

K e-r(II+II l

») lA] ,

where the event A is defined by

(log(X/ Kr) and 9

>

+

-d}.

(r _~2) (0 + (1))

The Black-Scholes model

90

(d) From this, derive a formula for G(B, x) in terms of Nand N 2 the twodimensional cumulative normal distribution defined by

= P(g < y,g+ pg1

N 2(Y,Y1,P)

< yd

y,Y1,P E JR.

for

4. Show that we can replicate the compound option payoff by trading the underlying call and the riskless bond.

Problem 6 Behaviour of the critical price close to maturity

Exercises

91

Problem 7 Asian option We conside~ a ?nan~ial market offering two investment opportunities. The first traded secunty 1S a nskless asset whose price is equal to Sp = ert at time t (with r ~ 0) and the second security is risky and its price is denoted by St at time t E [0, T]. Let .(St)O$;t?T be a stochastic process defined on a probability space (f!, F, P), equipped with a filtration (Ft)O
We consider an American put maturing at T with strike price K on a share of risky asset S. In the Black-Scholes model, its value at time t < T is equal to P(t, St), when P is defined by

P(t,x)

=

sup'

E

.(

Ke

-rT

- xe

I1W~_.,.2T)+ 2

,

TE70.T-t

To T-t

is the set of stopping times with values in

[0, T - t] and (Wt)O$;t$;T is a

dS t

with./L E JR and a > ~. We shall denote by p. the probability measure with de.ns1ty exp (-BBT - B T/2) with respect to P, where B = (/L - r)/a. Under P , th~ proce~s (Wt)O$;t$;T' defined by W t = (/L - r)t/a + B t is a standard Brownian motion. We are going to study the option whose payoff is equal to

standard P·-Brownian' motion. We also assume thatr > O. For t E [0, T[, we denote by s( t) the critical ~rice defined as,

h

s(t) = inf{x > 0 I P(t,x) > K - z}. we recall that limt-+T s(t)

= St(/Ldt + adBd,

~ (~ {

S,dt - K

r

where K is a positive constant. '

= K.

I

1. Let P, be the function pricing the European put with maturity T and strike

rrJ

1. Explain briefly why the Asian option price at time t (t ~ T) is given by

price K

Pe(t, x)

= E( e-r(T-t) K

.. xeI1VT-tg- .,.; (T-t)) + , V,

where 9 is a standard normal variable. Show that if t E [0, T[, the equation P; (t, x) = K - x has a unique solution in ]0, K[. Let us call it se(t).

2. Sh(j.w that on the event {

2. Show that s(t) ~ se(t), for any t E [0, T[.

3. Show that liminf K A ) > E (liminf t-+T . T - t t-+T

~ E' [,-d

K~) T - t

n.

Vi

=

aKg) +

We shall need Fatou l~mma: for any sequence (Xn)nEN of non-negative random variables, E(lim inf n -+ oo X n ) ~ lim inf n -+ oo E(X n ) .

T

- ')

(~ S.da -K

J;, Sudu ~ KT}, we have

e-r(T-t) it 1 ~ e-r(T-t) T' Sudu + S, - K e-r(T-t). o rT

3. We define s, = e:" s.. for t E [0,T]. (a) Derive the inequality (

4.

E·(St:Ke-rT)+ (a) Show that for any real number 1/,

(b) Deduce that

Vo

(b) Deduce that

.

t-+T

K-se(t)

VT -

t

~E·[e-rT(ST-K)+].

(Use conditional expectations given F t ) .

E(1/ - Kag)+ > 1/.

lim

{

. = hm

t-+T

K-s(t)

VT -

t

= +00.

s E· [e- rT (ST -

K)+] ,

i.e, the Asian option price.is smaller than its European counterpart. (c) For t ~ u, we denote by Ct,u the value at time t of a European call maturing

The Black-Scholes model

92

< e-r(T-t)t (-l i t S du - K -

T

t

0

)+ + - iT e-r(T-u)Ct 1

T

U

t

duo

Sudu -

3. Prove the inequality

K) .

1. Show that (~t)O~t~T is the solution of the following stochastic differential

equation:

2. (a) Show that

v. ~ ;-'(T-'IS,E'

[((.+ ~ t S~dU) \1"] ,

with S~ = exp ((r - (12 /2)(u - t) (b) Conclude that

Vi

+ (1(Wu

-

Wt}) .

= e-r(T-t) StF(t, ~t), with F(t,O

~E' (u ~ t

'>:'duf

3. Find a replicating strategy to hedge the Asian option. We shall assume that the function F introduced earlier is of class C 2 on [0, T[ x IR and we shall use Ito formula.

ill The purpose of this section is to suggest an approximation of Vo obtained by considering the geometric average as opposed to the arithmetic one. We define

Vo

- K) + ,

where 9 is a standard normal variable. Give a closed-form formulator Vt in terms of the normal distribution function. 0

We denote by (~t)09~T the process defined by t

Vo = e-rTE (So exp ((r - (12 /2)(T/2) + (1VT /39)

,u

n

~t = ~t (~ I

93

(b) Deduce that

at time u with strike price K. Prove the following inequality Vt

Exercises

~ e-,TE' (exp (~ [tn(S,)dt) -K) +

1. Show that Vo ~ Vo. 2. (a) Show that under measure P", the random variable JoT Wtdt is normal with zero mean and a variance equal to T 3/3.

Vo

-va, ~ Soe- rT (e

rT

rT- 1 - exp ((rT /2) - (12T /12) ) .

5

Option pricing and partial differential equations

In the previous chapter, we saw how we could derive a closed-form formula for the price of a European 'option in the Black-Scholes environment. But, if we are working with more complex models or even if we want to price American options, we are not able to find such explicit expressions. That is why we will often require numerical methods. The purpose of this chapter is to give an introduction to some concepts useful for computations. Firstly, we shall show how the problem of European option pricing is related to a parabolic partial differential equation (PDE). This link is basedon the concept of the infinitesimal generator of a diffusion. We shall also address the problem of solving the PDE numerically. ' The pricing of American options is rather difficult and we will not attempt to address it in its whole generality. We shall concentrate on the Black-Scholes model and, in particular, we shall underline the natural duality between the Snell envelope and a parabolic system of partial differential inequalities. We shall also explain how we can solve this kind of system numerically. We shall only use classical numerical methods and therefore we will just recall the few results that we need. However, an introduction to numerical methods to solve parabolic PDEs can be found in Ciarlet and Lions(1990) or Raiviart and Thomas (1983). i-:

5.1 European option pricing and diffusions In a Black-Scholes environment, the European option price is given by

Vi

= E (e-r(T-t) I(ST)!.rt)

with I(x) = (x - K)+ (for a call), (K - x)+ (for a put) and 2/2)T+uW T S T -- x 0 e(r-u

.

96

Optionpricing and partial differential equations

In fact, we should point out that the pricing of a European option is only a particular case of the following problem. Let (Xtk~o be a diffusion in ffi., solution of

Europeanoption pricing and diffusions

97

Proof. Ito formula yields

(5.1) where band (1 are real-valued functions satisfying the assumptions of Theorem 3.5.3 in Chapter 3 and ret, x) is a bounded continuous function modelling the riskless interest rate. We generally want to compute

Vt

=E

Hence

(e-J.T r(s,X.)dsf(XT )IFt) .

f(Xo) + i t f'(X s)(1(X s )dWs

=

f(Xt)

+it

In the same way as in the Black-Scholes model, Vt can be written as

~nd ther~sult follows from the fact that the stochastic integral J~ f'(X s)(1(Xs )dW s IS a martingale, Indeed, ~ccording to Theorem 3.5.3 and since 1(1(x)1 is dominated by K(1 + Ix!), we obtain .

Vi = F(t, X t ) where

F(t,:x)

(e-J.T r(s,X;'%)ds f(X~X)) ,

=E

[~(12(Xs)JII(Xs)+ b(Xs)f'(Xs)] ds

,

:

and X;'x is the solution of (5.1) starting from x at time t. Intuitively

F(t, x)

~ E (e-J.T r(s,X.)dsJ(XT )!Xt '= x) .

Mathematically, this result is a consequence of Theorem 3.5.9 in Chapter 3. The computation of Vt is therefore equivalent to the computation of F(t,x):'Under some regularity assumptions that we shall specify, this function F( t, x) is the unique solution of the following partial differential equatiori

{

"Ix E ffi.

u(T, x)

o Remark 5.1.2 If we denote by X{ the solution of (5.3) such that Xx = . 0 x, It follows from Proposition 5.1.1 that

= f(x) (5.2)

(au/at

+ Atu -

E (J (Xt)) = f(x)

+E (It Af (X:) dS) .

ru) (t, x) = 0 Vet, x) 'E [0, T] x ffi.

where

(Atf)(x) = (12(t, x) f"(x)

+ bet, x)f'(x).

.2, .. Before we prove this result, let us explain why the operator At appears naturally when we solve stochastic differential equations.

Moreover, since the derivatives of f are bounded by a constant K] and since Ib(x)1 + 1(1(x)1 :::; K(1 + Ixl) we can say that

E(~~~ IAf(X:)I) s Kj (1 + E(~~~ IX:1 2) ) < +00. !heref?re, since x .H Af(x) and s H IS applicable and yields

5.1.1 Infinitesimal generator ofa diffusion

We assume that band (1 are time independent and we denote by (Xtk:~o the solution of ' dXt = b (Xt) dt + (1 (.Xt ) dWt. (5.3)

Proposition 5.1.1 Let f bea 0 2 function with bounded derivativesand A be the

differentialoperator that maps a 0 2 function f to Af such that

l

d"

dt E (f (Xt))lt=o

+ b(x)J'(x).

Then, the process M, = f(Xd - J~ Af(Xs)ds is an Ft-l'TIfrtingale.

(1 r Af(X:)dS) = Af(x).

= l~ E t 1

0

The differential operator A is called the infinitesimal generator of the diffusion (Xt ) . The re~der can refer to Bouleau (1988) or Revuz and Yor (1990) t t d some properties of the infinitesimal generator of a diffusion 0 s u Y J

(AI) (x) = (12 (x) f",(x) _ 2 '

X: ,are continuous, the Lebesgue theorem

.

•

The ProPositio,n 5.1.1 can also be extended to the time-dependent case. We assume that b ~nd (1 satisfy th.e assumptions ofTheorem 3.5.3 in Chapter 3 which guarantee the existence and unIqueness of a solution of equation (5.1).

98

Option pricing and partial differential equations

Proposition 5.1.3 If u( t, x) is a C 1 ,2 function with bounded derivatives in x and if X, is a solution of(5./), the process t u, = u(t, X t) + Asu) (s, Xs)ds

I (~~

is a martingale. Here, As is the operator defined by _ 0'2 (s, x) 8 2u 8u (Asu) (x) 2 8x2 + b(s, x) Bx'

=

The proof is very similar to that of Proposition 5.1.1: the only difference is that we apply the Ito formula for a function of time and an Ito process (see Theorem 3.4.10). . In order to deal with discounted quantities, we state a slightly more general result in the following proposition. Proposition 5.1.4 Under the assumptions of Proposition 5.1.3, and ifr(t, x) is a bounded continuous function defined on IR+ x IR, the process t = e- r(s,X.)dsu(t, Xt)-l e- r(v,Xu)dv + Asu - ru) (s, Xs)ds

u,

I;

(~~

I:

is a martingale. Proof. This proposition can be proved by using the integration by parts formula to differentiate the product (see Proposition 3.4.12 in Chapter 3)

e - Jof'r(s,X.)ds u (t , X t )i,

o

and then applying Ito formula to the process u(t, X t ) .

This result is still true in a multidimensional modeJ. Let us consider the stochastic

{

bl (t, X t) dt

=

bn (t, X t) dt

dXi'

+ :E~=I alj (t, X t) dW/

(5.4)

+ :E~=I anj (t, X t) dW/.

We assume that the assumptions of Theorem 3.5.5 are still satisfied. For any time 2 t we define the following differential operator At which maps a C function from . IRn to IR to a function characterised by

. 1

(At!) (x)

n

= -2 ..L

82 f ' a"j(t,x)8' -.8 (x)

.

.

x,

',J~I

XJ

8f

n

is a martingale.

The proof is based on the multidimensional Ito formula stated page 48.

Remark 5.1.6 The differential operator 8/ 8t + At is sometimes called the Dynkin operator of the diffusion. 5.1.2 Conditional expectations and partial differential equations

In this se,ction, we want to emphasise the link between pricing a European option and ~olvmg a parabolic partial differential equation. Let us consider (Xt)t>o a solution of system (5.4), f(x) a function from IRn to IR, and r(t, x) a bounded continuous function. We want to compute

Vt =E

r(s,X.)ds f(Xr) 1Ft) .'

where F(t,

z) = E (e-I,T

r(s,X;,Z)ds f(X~X)) ,

when we denote by xt,x the unique solution of (5.4) starting from x attime t. The following result characterises the function F as a solution of a partial differential equation. .

Theorem 5.1.7 Let u be a C 1 ,2 function with a bounded derivative in x defined on [0, T) x IRn . Ifu satisfies n

Vx E IR

u(T, x) = f(x),

and

(~~ + ~tU -

ru) (t,x) = 0

n

V(t,x)E [O,T) x IR

,

then

where (a,j (t, x)) is the matrix of components p

k=1

(e- r

In a similar way, as in the scalar case, we can prove that

0

a,j(t,x) = La'k(t,x)ajk(t,x)

where 0'* is the transpose of a(t, x) =

. Proposition 5.1.5 If(X t) is a solution ofsystem (5.4) andu(t, x) is a real-valued function of class C 1 ,2 defined on IR+ x IRn with bounded derivatives in x and also, r( t, x) is a continuous boundedfunction defined on IR+ x IR, then the proces; t M, = e - Io' r(s,X.)dsu(t, Xd-l e- Io' r(v,Xu)dv (~~ + Asu - ru) (s, Xs)ds

+ Lbj(t,x) 8x (x), . I J J=

= a(t, x)a* (t, x)

In other words a(t, x) (a,j(t, x)).

differential equation

d~: t =

99

European option pricing and diffusions

0'

V(t, x) E [0, T)

xIR

n

~(t, x)

= F(t, x) =I E

(e-r

r(s,X;,Z)ds f(X~X)) .

Option pricing and partial differential equations

100

Pr~of. Let us prove the equality u(t, x)

= F(t, x) at time t = 0. By Proposition

5.1.5, we know that the process

- eMt-

f

0

= sinceu(T,x)

E

101

The operator At is now time independent and is equal to (12 2 a a A t = A b. =-x -+ rx2 2 ax ax' 2

r(s,X?,Z)ds u.(t , Xo,X) t

is a martingale. Therefore the relation E(Mo)

u(O,x) '= E

European option pricing and diffusions

= E(MT) yields

It is straightforward to check that the call price given by F(t x) = xN(d ) _ - (1vT _ t) with ' I

K e-r(T-t) N(dl

(e- JoT r(s,X?'Z)dSU(T,X~'X)) (e- JoT r(s,X?,Z)ds j(X~'X))

= j(X). The proof runs similarly fort > 0.

N(d)

o

=

log(x/ K)

=

1. --

+ (r + (12 /2)(T - t) (1'1/'T - t

I V2i

d

e- X 2 /2dx

-00

'

is solution of the equation

Remark 5.1.8 Obviously, Theorem 5.1.7 suggests the following method to price the option. In order to fompute

F(t,x) for

a given

=E

(e- J,T r(s,X:,Z)ds j(X~~))

The same type of result holds for the put.

au at + A,u _ ru ~ ° in_ [0, T] n

u(T, x) = j(x), "Ix E IR

x JR" (5.5)

ex, =

Problem (5.5) is a parabolic equation with afinal condition (as soon as the function

since S - S e(r-0'2/ 2)t+O'w, I t -

For the problem to be well defined, we need to work in a very specific function space (see Raviart and Thomas (1983)). Then we can apply some theorems of existence and uniqueness, and if the solution u of (5.5) is smooth enough to satisfy the assumptions of Proposition 5.1.4 we can conclude that F = u. Generally speaking, we shall impose some regularity.assumptions on the parameters band (1 and the operator At will need to be elliptic, i.e.

3C >0, V(t,x)'E [O,Tj x IR

"1(6,.·.,

Note th~t th~ operato~ A b. doe~ not satisfy the ellipticity condition (5.6). However, the tnck IS to consider the diffusion X, = log (St), which is solution of

.

u(T,.) is given).

n

°

.'

~n) E IRn ~ aij(t, X)~i~j ~ C (t ~?) >=1

.

(5.6)

(r - ~2)

.-

2

Ab.-log

= (12 a + 2 ax 2

' .

It is clearly elliptic because We write

(12

dt.+ (1dWt , ,',

. fini . ts m mtesimal generator can be written as

"

>

(r _2

(12)

.

~ ax'

°and, moreover, it has constant coefficients.

a2 + ( r ~ ~2) ax 'a = '"2 aX 2'

'(12 Ab.-Iog

..

Q

in [0, T]x ]0, +oo[

, u(T, z) = (z - K)+, "Ix E]O, +00[.

j, we just ~eed to find u such tha~

{

au '. : '{ at +Ab,u - ru ~ °

- r.

(5.7)

The ~onnection b~tween the parabolIc problem asso~iatedtoAb8~log and the computation of ~e pnce of an option in the Black-Scholes model can be highlighted as follows:.lf we ~ant to compute the price F(t, x) at time t and for a spot price x of an option paying off j (ST) at time T, we need to find a regular solution v of

5.1.3 Application to the Black-Scholes model

:~ (~' x) +Ab.-1ogv(t, x) = °

We are working under probability p ". The process (Wt)t::::o is a standard Brownian motion and the asset price S; satisfies

{

(5.8)

v(T, x) ,= j(e then F(t, z)

in [0, TJ x IR

= v(t, log(x)).

X

) ,

"Ix E IR,

/

Optionpricing and partial differential equations

102

5.1.4

Dnrtial ditterentiai equationson a boundedopen set and computationof su:

Solving parabolic equationsnumerically is a bounded stopping time, because r" = T; 1\ T: 1\ T where

r u.

trI = inf {O <- s <- T ,

expectations . u hout the rest of this section, we shall assume that there is only one asset :°th:t b(x), u(x) and rex) are all time independent. rex) is the riskless rate and

A is the differential operator defined by . 2 1 a f(x) af(x) (Af)(x) = '2u(x)2aT + b(x)~, We denote by A the discount operator such that Af(x)

= Af(x) -

u(O,x)

=

E (e-

{

u(T,x)

= f(x),

Furthermore, f(x)

, 9) 0 ] b[ as opposed to nt, we need to If we want to solve problem (5. on a, . th consider boundary conditions at a and b. We are going to concentrat~on e ~~s~ when the function takes the value zero on the boundaries'.These are e so-ca e Dirichlet boundary conditions. The problem to be solved IS then

+ Au(t, z)

=

b

on

{3s

(5.10)

z

x

u r>,

E

r

Z

•

0 on the event

[t, T], X;,x It O} ;

consequently

u(O,x)

[0, T] x 0

T

= u(T, x) and u(-rx , X~~X) =

=

au (t x) at '

-J.T° r(s,X~,X)ds u (T , XO,X)) -r r(s,X~,Z)ds ( XO,X)) X:,zllO}e °

X:,xEO}e

{3sE[t,Tj,

(5.9)

"Ix E nt.

s: ':(S'X~'X)dSu("X,X~~X))

E(1 {\lsE[t,Tj, +E(1

r(x)f(x).

on [O,T] x nt

Xt,x = I} s

and indeed T{ is a stopping time according to Proposition 3.3.6. By applying the optional sampling theorem between 0 and r", we get E(Mo) = E(Mrx), thus by noticing that if s E [0, "X], Af(X~'X)= 0, it follows that

Equation (5.5) becomes

~~(t,~) + Au(t,x) = 0

103

Xo,Z)ds ° ,) = E ( 1{\lsE[t,TJ, X:,xEO}e - J.T° r(s'. f(XT'X).

That completes the proof for t

= O..

o

u(t,a) = u(t,b) = 0 . "It S T u(T,x)

= f(x)

Remark 5.1.10 An option on the FT-measurable random variable "Ix E O.

As we are about to explain, a regular solution of (5.10) can a~so be expr~ssed in terms of the diffusion Xt,x which is the solution of (5.3) starting at x at orne. t.

a

Theorem 5.1.9 Let u be C 1 ,2function withboundedderivativein x that satisfies

equation (5.10). We then have V(t,x) E [O,T] x 0, u(t,x)

= E ( 1{\lsE[t,T],.x:,xEo}e

' - I,T r(X:,Z)ds f(Xt,X))

M«

=

- J.'r(XO,X)ds (t Xo,X) eo' u, t

i° t

-

-

e

f r(X~,X)dv °

(au at

+ Au - ru) (s, X~,X)ds'

is a martingale. Moreover "x

o x d o} = inf { 0 S s S T , X s'-'f'

or T if this set is empty

- J.T (X"Z)d ,r

•

.

t X

s f(Xi

)

is called extinguishable. Indeed, as soon as the asset price exits the open set 0, the option becomes worthless. In the Black-Scholes model, if 0 is of the form ]0, l[ or ]1, +oo[ weare able to compute explicit formulae for the option price (see Cox and Rubinstein (1985) and exercise 27 for the pricing of Down and Out options).

T

Proof. We shaltprove the result for t ~ 0 s~nce the argument is simil~ ~ o~h:: times There exists an extension of the function u from [0, T] x 0 to [.' ] that is still of class C 1,2 . We shall continue to denote by u such an extension. From Proposition 5.1.4, we know that

, 1{\lsE[t,T], x:,xEo}e

5.2 Solving parabolic equations numerically , (

We saw under which conditions the option price coincided with the solution of the partial differential equation (5.9); We now want to address the problem of solving a PDE such as (5.9) numerically and we shall see how we can approximate its solution using the so-called finite difference method. This method is obviously useless in the Black-Scholes model since we are able to derive a closed-form solution, but it proves to be useful when we are dealing with more general diffusion models. We shall only state the most important results, but the reader can referto Glowinsky, Lions and Tremolieres (1976) or Raviart and Thomas (1983) for a detailed analysis.

Option pricing and partial differential equations

104

Solving parabolic equations numerically

5.2.1 Localisation Problem (5.9) is set on ffi. In order to discretise, we will have to work on a bounded open set VI =]- I, l[, where I is a constant to be chosen carefully in order to optimise the' algorithm. We also need to specify the boundary conditions (i.e. at I and -I). Typically, we shall impose Dirichlet conditions (i.e, u(l) u( -I) 0 or some more relevant constants) or Neumann conditions (i.e. (8u/8x)(I), (8u/8x)( "':'l)). If we specify Dirichlet boundary conditions, the PDE becomes

=

105

Thus

lu(t, x) - UI(t, x)1

-

+ Au(t, x)

< MP (suPO:S;s:S;T Ix + O'Wsl 2: I -lr'T/) . By Proposition 3.3.6we know that if we define T. = inf {s > 0 W - } th E(exp(-ATa )) = exp(-.J2>:la/). It infers that f~r any a> 0, a~d f~r:n~ ~ en

= 0 on [0,T] x VI

P (sup s:S;T

u(t, I) = u(t, -I) = 0 if t E [0, T] u(T, x)

w, 2: a)

= P (Ta

Minimising with respect to

= f(x) if x EVI.

~ T) s e~TE (e-~To) _< e~T e -a.,;'2I .

>. yields

w, 2: a) ~ exp (_ a -sr T 2

P (sup

We are going to show how we can estimate the error that we make if we restrict our state space to VI. We shall work in a Black-Scholes environment and, thus, the logarithm of the asset price solves the following stochastic differential equation

)

,

'

and therefore

P (SUP(x

" dX t = (r - O' 2/2)dt + O'dWt.

We want to compute the price of an option whose payoff can be written as f(ST) = f(Soe XT). We write f(x) = !(e). To simplify, we adopt Dirichlet boundary conditions. We can prove in that case that the solution u of (5.9) and the solutions UI of (5.10) are smooth enough to be able to say that

MP (suPO:5S~T_t Ix + O'Wsl 2: I -lr'T/)

=

=

8u(t x) 8;

< MP (sUPt~s;5T [z + O'(Ws - Wt)1 2: I -lr'TI)

s:S;T

+ O'Ws) 2: a) < exp -

(

ja - X I2) O'2T'

Since (- Ws) s~O is also a standard Brownian motion

p

(.~~(X +UW.) :5 -a)~p(:~~(-X - aW,) ~ a):5 exp (

These two results imply that

P

(:~~ Ix + O'Wsl 2: a) s exp ( _laO'~;12) + exp (

and therefore ,

and

UI'(t,X) =

E(1 {'v'sE[t,TJ, IX;,zl
where X;'x == xexp((r - O' 2/2)(s - t) + O'(Ws - Wt )). We assume that the function f (hence J) is bounded by a constant M and that r 2: O. Then, it is easy to show that '

lu(t, x) - UI(t, x)1

II - Ir'TI- X12) O' 2 T

+ exp ( _II ~ I~;~+ X12)) .

(5.1l)

!his proves that for given t and x, liml--++ UI (t x) = u(t x) Th iforrn i 00" • e convergence even um orrn in t and :: as long as x remains in a compact set of ffi.

IS

Remark 5.2.1 If we call r ' = r - 0'2/2

{3s E [t,T],

IX;,xl 2: I}

Ix + r'(s -- t) + O'(Ws - Wt)1 2: I}

C

{sUPt:S;s:S;T

C

{sUPt:S;s;5T Ix + O'(Ws - Wdl 2: I - Ir'TI} ,

• :t can be proved ~h~t P(suPs:S;T Ws 2: a) = 2P(WT 2: a) (see Exercise 18 n Chapter 3). This would lead to a slightly better approximation than the one " above. • The ~~damental advantage of the localisation method is that it can be used for pncmg American options, and in that case the numerical approximation is

Option pricing and partial differential equations

106

Solving parabolic equations numerically

compulsory. The estimate of the error will give us a hint to choose the domain of integration of the PDE. It is quite a crucial choice that determines how efficient . our numerical procedure will be.

5.2.2 Thefinite difference method

((A ij) is.s», iss-:«

(E)

(J a 0

'Y

0

(J a

'Y

0

(J

'Y

0 0 0

0 0

0

0

h

)

Once the problem has been localised, we obtain the following system with Dirichlet boundary conditions:

ou(t, x) ot

107

following matrix:

-

+ Au(t, x)

= 0 on [0, T] x 0,

a

(J a

0 0 0 'Y

(J

where

u(t" l) = u(t, -l)= 0 if t E [0, T] u(T, x)

The finite difference method is basically a discretisation in time and space of equation (E). We shall start by discretising the differential operator A on 0,. In order to do this, a function (f(X))XEO, taking values in an infinite space will be associated to a vector (fi)l
2h

We obtain an operator Ah defined on IR N .

(J'2)

'2

(J'2

(J

=

- h2

'Y

=

(J'2 2h 2

(J+a a 0

I

with

-

r

-

+

1 ( 2h . r -

(J'2)

'2 .

If we specify null Neumann conditions, it has the following form:

h

and replace

2h 2

~

U i;+-1 _ U i - 1

b(x.) ,

with

1 ( 2h r -

(J'2

a

= f(x) if x EO,.

0 0 0

. 0

'Y

0

(J a

'Y

0

(J

'Y

0 0 0 (5.12)

0 0

a 0

(J a

'Y

(J+'Y

This discretisation in space transforms (E) into an ordinary differential equation (Eh ) :

Remark 5.2.2 In the Black-Scholes case (after the usual logarithmic change of variables) - b s - log

A

u (') x

02u (X) =.' -(J'22 --+ (r ox 2

(J'2)

-

~

2

ou(x) ox

( )

- - - ru x

if 0

'

is associated with -

2

(J'

u;.(T)

+1

(AhUh)i = 2h 2 (u~

.

- 2ui.

. 1 ( + u~)+ r

-

(J'2) '2

If we specify null Dirichlet boundary conditions,

Ah

.

1 (+1

2h u~

. 1)

- u~-

.

- rui..

is then represented by the

stsT

= fh

where!h = (f~h$i$N\ is the vector f~ = f(Xi)' '

.

W~ are now going to discretise this equation using the so-called O-schemes. We consIder 0 E [0,1], k a time-step such that T = Mk and, we approximate the

Option pricing and partial differential equations

108

solution Uh of (Eh) at time nk by Uh,k solution of

Solving parabolic equations numerically

U;;:k = fh

lim u~ =

n decreasing, we solve for each n:

(Eh,k)

un+! _ un

. h,k -k if a:::;

U

£2 ([0, T] x 01)

in

lim Ju~ = au/ax

£2 ([0,T] x 01).

in

;

h,k

n:::;

109

• When 1/2 :::; 0 :::; 1, as h, k tend to a

O)AhU~:t1 =

+ OAhUh,k + (1 -

a

• When a:::; 0

< 1/2, as h,

M-1.

lim u~

Remark 5.2.3

k tend to 0, with lim k] h 2

=U

£2 ([0,T] x 01)

in

lim Ju~ = au/ax

• When 0 = a the scheme is explicit because Uh,k is computed directly from u~tl. But when 0 >' 0, we have to solve at each step a system of the form T~h,k = b, with

T= (I - OkAh) { b = (I + (1- '0) kA

.

h)

U~:t1

= 0, we get

£2 ([0, T] x 01)

in

Remark 5.2.5 • In the case a :::; ~ < 1/2 we say that the scheme is conditionally conver ent because the algonthm converges only if h, k and k/h 2 tend to 0 Th gl .th her tri . . ese a gon ms are rat er tricky to Implement numerically and therefore they are rarely used except when 0 = O.

where T is a tridiagonal matrix. This is obviously more complex and more time consuming. However, these schemes are often used in practice because of their good convergence properties, as we shall seeshortly,

• In the ca~e 1/2 :::; 0 :::; 1 we say that the scheme is unconditionally convergent because It converges as soon as hand k tend to O. '

• When 0 = 1/2, the algorithm is called the Crank and Nicholson scheme. It is often used to solve systems of type (E) when b = a and a is constant.

Finally, we shall examine in detail how we can solve problem (E) . all At h ti . h,k numenc y. eac time-step n we are looking for a solution of T X = G where -:

• When 0

= 1, the scheme is said to be completely implicit.

X

We shall now state convergence results of the solution Uh,k of (Eh,k) towards u(t, x) the solution of (E), assuming the ellipticity condition. The reader ought to refer to Raviart and Thomas (1983) for proofs. We denote by u~(t, x) the function M

T

L L(uh,k)i 1

x

jx . - h/ 2,x i+h / 2j

l](n-I)k,nkj.

(J¢)(x)

= h1 (¢(x + h/2) -

Theorem 5.2.4 We assumethat b and a are Lipschitz and that r is a non-negative continuous function. Let us recall that Af(x) is equal to 1/2a(x)2(a 2f(x)/ax 2)+ b(x)(af(x)/ax) - r(x)f(x). We assume that the operator A is elliptic -

' .

1- kOAh.

T is a tridiagonal matrix. The following algorithm, known as the Gauss method

¢(x - h/2)).

(-Au, U)£2(O,) 2: f(lul£2(O,)

=

..

(I + (-1 - O)kAh) un~i h,k

sol~es the system with a number of multiplications proportional to N. Denot: X - (xih~i~N, G = (gih'~i~N and

We also call J¢ the approximate derivative defined by

with e > O. Then:

:

G

N

n=1 i=1

.'

= " Uh,k

I

+ Iu 1£2(0,))

T=

bi

CI

a

a2'"

b2

C2

a

a3

b3

C3

a a a a\ a

a

a

aN-I

a

a a o

bN - I

CN-I

aN

bN

The algorithm runs as follows: first, we transform T into a lower triangular matrix

Option pricing and partial differential equations

110

(on a stock offering no dividend) is equal to the European call price. Nevertheless, there is no explicit formula for the put price and we require numerical methods. The problem to be solved is a particular case of the following general problem: given a good function I and a diffusion (X t )t>o in IR". solution of system (5.4), compute the function

using the Gauss method from.bottom to top. Upward:

= bN = gN For 1 ~ i ~ N -

b'rv g'rv

1,

. l

'.

decreasmg.

~(t, x)

b'. = b, - Ciai+! /b~+1 - c·g' l/b'·+1 g~i-- g. I 11+ 1 0

a2

b~

O

a3

1

T'=.

o o o

0 0 b'3

Noticethat ~(t, x)

0

aN-I 0

f

r(s,X;,z)ds I (X;'X)) .

= T we obtain ~(T, x) = I(x).

is the smallest martingale that dominates the process I(Xt} at all times.

.,

b'rv_1 aN

I(x) and for t

e- Jot r(s,X.)ds~ (t,Xt}

0 ~

~

Remark 5.3.1 It can be proved (see Chapter 2 for the analogy with discrete time models and Chapter 4 foqhe Black-Scholes case) that the process

·00 0 0

. 0 0

sup E (e -

=

rETt.T

. I tem T' X - G' , where We have obtained an equiva en tsys

~

111

American options

~ bN

To conclude, we just have to compute X starting frorri the top of the matrix. Downward: XI gUb~ c For 2 ~ i ~N, i increasing

We just stressed the fact that the European option price is the solution of a parabolic partial differential equation. As far as American options are concerned, we obtain a similar result in terms of a parabolic system of differential inequalities. The following theorem, stated in rather loose terms (see Remark 5.3.3), tries to explain that. Theorem 5.3.2 Let us assume that u is a regular solution ofthe following system ofpartial differential inequalities:

=

au . at

Xi = (g~ - aixi_I)/b~,

+ Atu - ru ~ 0, u ~ I

. e matrix T is not necessarily invertible. However,.we c~n prove Remark5.2.6 Th. I 1+1.\ < Ibl WheneverTis.notmvertlble,the th t it i s if for any l we have ai C, ,. h k a I I , .' k I the Black-Scholes case, it is easy to c ec previous algonthm does not wor ..n. I _ 2/21 < a 2 /h, i.e. for that T satisfies the preceding condition as soon as r a -

(~~ + Atu -

sufficiently small h.

u(T, x)

. Then

= ~(t,x) =

= I(x)

ru) (f - u) in

=0

in

[0,T] x IRn

in

[0, T] x IRn

(5.13)

IRn

u(t,x)

5.3.1 Statement of the problem . . notions in continuous time is not straightforward. In obtained the foll.owin g form(ul)a an American call (f(x) = (x - K)-t) or an American put (f X +

Proof. We shall only sketch the proof of this result. For a detailed demonstration, the reader ought to refer to Bensoussan and Lions (1978) (Chapter 3; Section 2) and Jaillet, Lamberton and Lapeyre (1990) (Section 3). We only consider the case t = 0 since the proof is, very similar for arbitrary t. Let us denote by Xt the solution of (5.4) starting at x at time O. Proposition 5.1.3 shows that the process

~e;l:~ts~c~~~;:~:l,

rETt.T

:e

~or(;;~p~\ce)of

Vt :;::: ~(t, St)

M; where ~(t,x)

=

sup E (e- J.T

r(s,X;'~)ds I

5.3 American options

sup E

tl ))

• ( -r(r-t)1 ( (r_
rETt,T

.

is th t of • ( ) is a standard Brownian motion and !t,T IS e se , and, under P , ~t t~O • [ T] W howed how the American call pnce stopping times taking values in t, . e s .

'T

=

e- J;'r(s,X;)dsu(t,

-ior '

(X;'X)) . .

Xn

e- J; r(v,X;)dv (au at

+ A u - ru) s.

(s XX)ds 's

is a martingale. By applying the optional sampling theorem (3.3.4) to this martingale between times 0 and T, we get E(Mr) = E(Mo ), and since au/at + Asu-

Option pricing and partial differential equations

112

s0

ru

U(O, x) 2: E (e- JOT r(s,X;ldSU(T,x:)) . We recall that U(t,x) 2: f(x), thus U(O,X) 2: E (e- J: r(s,X;ldsf(X proves that

U(O, x) 2: sup E

(e-JOT

rE70,T

r(s,X:lds f(X:)) .

n).

Now, we define Topt = inf{O $ s $ T, u(s,X:) = f(X:)}; we can show that Topt is a stopping time. Also, for s between 0 and Topt, we have (au/at + Asu - ru) (s, x:) O.The optional sampling theorem yields /

=

U(O,x)=.E Because at time

Topt.

(

0

av at (t, x) + Abs-1ogv(t, x) $ 0 a.e. in

[0, T] xIR

This

= F(O, x).

X) _1. TOp' r(s,X:lds U(Topt,Xrop.) e

American options id 113 If weconsl er¢(x) = (K -eX) h " " to the price of the American putis t e partial differential inequality corresponding

.

v(t,x)2:¢(x)

a.e.in

(v~t, x) -

(:~ (t, x) + A:bs-Iogv(t, x))

¢(x))

[O,T]xIR = 0 a.e. in

[0, T] x IR

veT, x) = ¢(x). The following theorem states the results o f existence . . and uniqueness of a(5.14) solu~ tiraI mequalttyand . . tion to, this partial diirreren establi h th " Amencan put price. IS es e connection With the

U(T~Pt, Xrxopt ) = f(XrXopt .), we can write

Theorem 5.3.4 The inequality (5 14) has a '

.

vet, x) such that its partial deriv~( . ~mq~e :ont~nuous bounded solution a2v/ ax2 are locally bounded. More~:s I;h~ e dlls~nbutlO.n sense av / aX,av / at,

U(O,x~ = E (e- JoTop, r(s,X;lds f(X:o~J) .

er,

That proves that u(O,x) $ F(O, z}, and that u(O, x) = F(O,x). We even proved that Topt is an optimal stopping time (i.e. the supremum is attained for T = Topt).

o

v(t,log(x))

= ~(t,x) =

IS so ution satisfies

sup E* (e-r(r-tlf (xe(r-u2/2)(r-tl+U(WT-Wtl)) rET.,T

.

The proof of this theorem can be found in Jaillet , L am b erton and Lapeyre (1990),

Remark 5.3.3 The precise definition of system (5.13) is awkward because, even 2 for a regular function f, the solution U is generally not C . The proper method consists in adopting a variational formulation of the problem (see Bensoussan and Lions (1978». The proof that we have just sketched turns out to be tricky because we cannot apply the Ito formula to a solution of the previous inequality.

Numerical solution to this inequality We are going to show how we can numericall I" . the method is similar to the one us d i th E so ve inequality (5.14). Essentially, problem to work in the interval O~ ~] _ ; l[uropean case, Fir~t, we localise the conditions at ±l. Here is the inequalit ith N' Then, we must Impose boundary y WI eumann boundary conditions

5.3.2 The American put in the Black-Scholes model

aVe

put in the Black-Scholes model. We are working under the probability measure P* such that the process (Wt k~:o is a standard Brownian motion and the stock price St satisfies

dS t = St (rdt + O"dWt) .

-b

~at t,x)+As-1ogv(t,x)$0

We are leaving the general framework to concentrate on the pricing ofthe American

v (t, x) 2: ¢(x) . a.e. in J

(A) ,

a.e.in

[0 ,T] X

o.

[0, T] x Ot •

(V-¢)(:~(t,;r)+A:bS-IOgV(t,x))

=0

a.e. in

[0, T]

X

o,

We saw in Section 5.1.3 how we can get an elliptic operator by introducing the process

x, =

log (St) = log (So) +

veT, x) = ¢~x) ,

(r - 0";) t + O"Wt·

av -a(t, ±l) = X

Its infinitesimal generator A is actually time-independent and

2 2 (- 2) -a -

0" a + r - .-0" -bs- Iog __ Abs-log - r __ A . 2 ax 2 . 2

ax

r.

o.

r

We. can now dirscretiise inequality , (A) using the fi 't diff . notations are the same as in Section 5 2 2 I , m e I, eren~es method. The _ ' . . n particular, M IS the integer such that

Option pricing and partial differential equations

114

Mk

= T,

Ih

,I,() where x • -- -I is the vector given by I h = If' Xi i

+ 2il/(N 1) . IRn+we

American options

115

with

. (5 12) If U and v are two vectors m , represented by matrix . F' 11 the method is the same as in the . . < "f'vl < t < n Ui < Vi. onna y, .' write U - case: V I - t'isa ' tion in time leads to the finite dimensional inequality I European t he-d'iscre -

and Ah

.

IS

(Ah,k):

_Ok(~+~(r_0-2)) 2h 2 2h 2 '

=

C

and if 0 ~ n ~ M - 1

(AD) is a finite dimensional inequality. We know how to solve this type of Uh,k ~

!h

O)AhU~~I) ~0 '

n+l _ unh,k + k (afhuh ,k + (1 uh,k

- Uh. k'- + k (OAhUh,k ( Un+! h,k,

+ (1 .-

n+l) ,Uh,k n 0 )A- hUh,k .

Ir -

I h) -- 0.

. given . by (5 . 12). If we note . ) is the scalar product in IR an d A- h IS (x,y were . h .

'.

N

inequality both theoretically and numerically if the matrix T is coercive (i.e. X.T X ~ aX.X, with a > 0). In our case, T will satisfy this assumption if 2 2/h Ir - 0- /21 ~ 0and if 0- 2/21 k/2h < 1. Indeed, this condition implies that a and c are negative and, therefore, by using the fact that (a + b)2 ~ 2 (a2 + b2) we show that n

n

x:Tx

=

n-l

+ L bx~ + L

L aXi-lxi i=2 n

i=I

CXiXi+!

i=1

+ ax~ + cx~

> (a/2) L (xLI + xn ~2

x =

.

+ E~=, bx~ + (c/2) E~:/ (x~ + x~+!) + ax~ + cx~

Uh.k

G F

=

fh,

we have to solve, at each time n, the system of inequalities TX~G

Under the coercitivity assumption, we can prove that there exists a unique solution to the problem (Ah,k) (see Exercise 28). The following theorem analyses explicitly the nature of the convergence of a solution of(Ah,k) to the solution of (A). We note M

(AD)

X \

u~(t,x)

~ F

(T X - G, X - F) = 0,

< 1, the convergence is conditional: " k/h 2 converges to 0 then

0

a+b a 0

C

0

b a

C

-0

b

C

0 0 0

0 0

0

'.'

a

b . a

0 0 0 C

'b + C

l](n-l)k,~k]' .

.

Theorem 5.3.5 Ifu is a solution of (A), J. when 0

where T is the tridiagonal matrix

T=

N

= L L(uh,k)il]:Z:i-h/2,:Z:i+i/2] x n=I i=1

lim u~ \

=U

if li and

k converge to 0 and

if

L 2 ([0, T) x (1)

in

.

. au lim aukh = -ax

in

L 2 ([0, T) x 01),

2. when 0 = 1, the convergence is unconditional, i. e. the previous convergence is true when hand k converge to 0 without restriction.

116

Option pricing and partial differential equations

The reader will find the proof of this result in Glowinsky, Lions and Tremolieres (1976). See also XL Zhang (19 94).

American options

117

The computation gives

Remark 5.3.6 In practice, we nonnally use () = 1 because the convergence is unconditional.

which is not a solution of (AD).

Numerical solution ofa finite dimensional inequality

Remark 5.3.9 An implementation of the Brennan and Schwartz algorithm is offered in Chapter 8.

In the American put case, when the step h is sufficiently small, we can solve the system (AD) very efficiently by modifying slightly the algorithm used to solve tridiagonal systems of equations. We shall proceed as follows (we denote by b the vector (a + b, b, . . . ,b + e))

Upward: ,b'tv ~ bN g'tv = s«

~For 1 ~ i ~ N - 1, decreasing i

b~ g~

= b, = gi -

ca/b~+l eg~+db~+l

'American' downward:

5.3.3 American put p'ricing by a bitnomta ial method We shall now explain another numerical method th . . American put in the Black-Scholes d 1 L at IS WIdely used to price the mo e. et r a b be thre I b at -1 < a < r < b Let (8) b the bi :' e rea num ers such th 8 _ 8 'T' hen n n~O e e binornial model defined by S, d n+l n.L n, were (1', ) > is a fI X an P(T 1 + a) p (bn ::;:)O/(b _seq)uence 0 ID random variables such that Ch n 2 E a and P(Tn 1 + b) - 1 p Wi . that . the Am encan . ' "In this model - could - . bee written saw In as apter , xercise 4' put pnce

=

°-

= =

=

Pn

= Pam(n, $n),

'

,

and that the function Pam (n , x) cou ld be computed by induction. according to the equation

=

Xl gUb~ For 2 ~ i ~ N, increasing i Xi = (g~ - aXk-d/bi

Xi

= SUp(Xi, Ii)'

Jaillet, Lamberton and ~apeyre (1990) prove that, under the previous assumptions, this algorithm does compute a solution of inequality (AD).

Remark 5.3.7 'The algorithm is exactly the same as in the' European case, apart ' , from the step Xi = SUp(Xi, Ji). That makes it very effective.

Pam(n, x)

= max ( (K -

+ I, (1+ .)x) + (1 -

r

=

RT/N

l+a

=

exp

.

boundary conditions, the previous algorithm is due to Brennan and Schwartz

example should erase any doubts:

M~(~' T~} F~O)' a=O)

+b)X»)

withthefinalconditionP (N x)'- ('K )+. (5.15) in Chapter 1 Section 1 4 that ifth - . - X . On the other hand, we proved , . , e parameters are chosen as follows: '

Remark 5.3.8 When we plug in () =' 1 in (Ah,k), and we impose Neumann (1977). We must emphasise the fact that the previous algorithm only computes the exact solution of system (AD) if the assumptions stated above are satisfied. In particular, it works specifically for the American put. There exist some cases where the result computed by the previous algorithm is not the solution of (AD). The following

p)Pom(n + I, (1

l+r

There exist other algorithms to solve inequalities in finite dimensions. Some exact methods are described in Jaillet, Lamberton and Lapeyre (1990), some iterative methods are exposed in Glowinsky, Lions and Tremolieres (1976), ,

x)+,

PP( om n

~

(-aVT/N)

(+~VT/N)

l+b

=

exp

p

=

(b-r)/(b-a),

(5.16)

then the European option price in this model a . . computed for a riskless rate equal to R d ppr~xl~ates the Black-Scholes price that in order to price the Am . an avo atility equal to a. This suggests Given di . . encan put, we shall proceed as follow some iscrensanon parameter N we fix th I ..' (5.16) and we compute the price p N ( ) ' eva ues r, a, b,p according to i < n b . d . am n,. at the nodes x(1 + a)n-'(1 + b)i 0 < a ,y ~n uction of (5,15). It seems quite natural to take pN ( ,pproxrmation of the American ' P(O, x). Indeed, am 0, X)weascan an 0 esipnce , Black -Sch

Exercises

Option pricing and partial differential equations . if . N 0 x) = P(O,x). This result is quite tricky to JUStI Y show that lim» -++<Xl Pam ( , d P , (1990» and we will not try to (see Kushner (1977) and Lamberton an ages

1. We denote by u.(t, x) the price of the European put in the Black-Scholes model. Derive the system of inequalities satisfied by v = u - u•.

called Cox-Ross-Rubinstein method and it is exposed in prove. it herthe. d i th ThIS me 0 IS e sodetails in Cox and Rubinstein (1985).

2. We are going to approximate the solution v = u - u. of this inequality by discretising it in time, using one time-step only. When we use a totally implicit method, show that the approximation v(x) of v(O, x) satisfies

118

119

-vex) + T Absv(x) ::; 0 5.4 Exercises . d b (X Y)thescalarproductoftwovectorsX = (xih~i~n ExerCISe 28 We eno~e y , . X > Y means that for all i between 1 and .n, and Y = (Yih9~n' The notaltI ~. ffin M satisfies (X M X) ~ a(X, X) with l Xi ~ Yi. We assume that for a m, a > O. We are going to study the system ~

\

v (t, x) ~ ¢(x)

-vex)

.,

(5.17)

Prove the uniqueness of a solution of (5.17). . . Show that if M is the identity matrix there exists a unique solution to (5.17). '. . that 4. Let p be positive: we denote by S p (X) the unique vector Y ~ F such .

+ p(M X -

G), V - Y) ~ O.

(!< -

a.e.in

{

(5.19)

¢(x) otherwise.

lalu~(O,x) + 1 + lal

6. Prove that vex) defined by (5.19) where x· is the solution of f(x) = x is a solution of (5.18). 7. Suggest an iterative algorithm (using a dichotomy argument) to compute x" with an arbitrary accuracy.

[0, T] x ]0, +oo[ =0

>'xOt if x ~ x"

5. Using the closed-form formula for u. (0, x) (see Chapter 4, equation 4.9), prove . that f(O) > 0, that f(K) < K (hint: use the convexity of the function u.) and that f(x) - x is non-increasing. Conclude that there exists a unique solution to the equation f(x) = x.

[O,T]x]O,+oo[ .

(U_(K_X)+).(~~(t,x)+AbSU(t,X)) u(T, x) =

+oof

and u~(t, x) = (au.(t, x)jax).

5 Derive the existence of a solution to (5.17). . '. . to a roximate the Black-Scholes American ~ut pnce Exercise 29 We arIel :I;~ is a ~~lution of the partial differential inequality u(t,x). Let us reca a .1

a.e. in

]0,

K-u.(O,x)

f(x) =

'Show that for sufficiently small p, Sp is a contraction.

u (t, x) ~ (K - x)+

a.e. in

Write down the equations satisfied by >. and a so that v is continuous with continuous derivative at z" . Deduce that if v is continuously differentiable then z" is a solution of f(x) = x where

~:

aU(t,x)+AbBu(t,x).::;O a.e.in at .

=0

]0, +oo[

+ T Absv(x) = O.

vex) =

(M X - G, V ., X) ~ O..

(Y - X

a.e. in

4. We look for a continuous solution of (5.18) with a continuous derivative at x"

1. Show that this is equivalent to find X ~ F such that

VV > F

x)+ - u.(O,x)

(5.18) 3. Find the unique negative value for a such that vex) = x" is a solution of

(Mi -G,X - F) =0.

F

]0, +oo[

(v(x) - ¢(x)) ( -vex) + T AbSv(x))

:::G

VV ~

= (K -

a.e. in

8. From the previousresults, write an algorithm in Pascal to compute the American put price.

[O,T]x]O,+oo[

The algorithm that we have just studied is a marginally different version of the MacMillan algorithm (see MacMillan (1986) and Barone-Adesi and Whaley (1987».

x)+

where'

"

6

Interest rate models

,

'

Interest rate models are mainly used to price and hedge bonds and bond options. Hitherto, there has not been any reference model equivalent to the Black-Scholes model for stock options. In this chapter, we will present the main features of interest rate modelling (following essentially Artzner and Delbaen (1989», study three particular models and see how they are used in practice.

6.1 Modelling principles

6.1.1 The yield curve In most of the models that we have already studied, the interest rate was assumed to be constant. In the real world, it is observed that the loan interest rate depends both on the date t of the loan emission and on the date T of the end or 'maturity' of the loan. Someone borrowing one dollar at time t, until maturity T, will have to pay back an amount F(t, T) at time T, which is equivalent to an average interest rate R(t, T) given by the equality.

F(t, T)

= e(T-t)R(t,T).

If we consider the future as certain, i.e. if we assume that all interest rates (R(t, T))t
'TIt < u < s

F(t, s) = F(t, u)F(u, s).

Indeed, it is easy to derive arbitrage schemes when this equality does not hold. \ From this relationship and the equality F(t, t) = I, it follows that, if F is smooth, there exists a functio~r(t) such that

'TIt < T

F(t, T)

= exp

(iT

r(S)ds)

Interest , rate models

122 and consequently

R(t, T)

1 = -T·' - t

iT t

Modelling principles

r(s)ds.

F(t, u) = e- fa' r(s)dsP(t, u)

. The function r(s) is interpreted as the instantaneous interest rate. In an uncertain world, this rationale does not hold any more. At time t, the future interest rates R( u, T) for T > u > t are not known. Nevertheless, intuitively, it makes sense to believe that there should be some relationships between the different rates; the aim of the modelling is to determine them. Essentially, the issue is to price bond options. We call 'zero-coupon bond' a security paying 1 dollar at a maturity date T and we note P(t, T) the value of this security at time t. Obviously we have P(T, T) = 1 and in a world-where the future is certain

P(t, T)

123

u E [0, T], the process (F(t, u))oStSu defined by

= e- J.T r(s)ds.

is a martingale. This hypothesis has some interesting conse erty under P* leads to, using the equality p(~~~)e~ I;,deed, the martingale prop-

. .

~(t,U) = E* (F(u,u)/Ft)

= E*

(e- fa"r(S)ds!Ft)

and, eliminating the discounting,

P(t,u)

(e- J."r(S)ds!;:,)

= E*

.

(6.1)

.

(6.2)

t·

This equality, which could be compared to fo

P(t, u) only depend on the behaviour of th rmula (6.1), shows that the prices d he process (r(s))OSsST under the probability P*. The hypothesis w e rna e on t e filtration (;:, ) . 11 express the density of the probabilit P* ith t 09ST a ows us to

6.1.2 Yield curvefor an uncertainfuture For an uncertain future, one must think of the instantaneous rate in terms of a random process: between times t and t + dt, it is possible to borrow at the rate r(t) (in practice it corresponds to a short rate, for example the overnight rate). To make the modelling rigorous, we will consider a filtered probability space (n, F, P, (Fdo
S to -- e.f0 r(s)ds where (r(t))oStST is an adapted process satisfying JOT Ir(t)ldt < 00, almost surely. It might seem strange that we should call such an asset riskless since its price is random; we will see later why this asset is less 'risky' than the others. The risky assets here are the zero-coupon bonds with maturity less or equal to the horizon T. For each instant u ~ T, we define an adapted process (P(t, u))o
WI respect to P We denot b L hi . Y density, For any non-negative random . bl X .' e Y T t IS and, if X is Ft-measurable E* (X) _ v;(~~) ,w~ have E* (X) = E(X L T ) the random variable i; is th~ density P* t.' sedttIn g t; ~ E(LTIFt). Thus . . restncte to Ft WIth respect to P. Proposition 6.1.1 There is an ada d ( t E [0, T],' ', . . pte process q(t ))OStST such'that, for all

;rt.

i;

= exp (it q(s)dWs- ~ i t q(S)2dS)

a.s.

(6.3)

Proof. The process (L ) . . filtration of the Brown:a~~~~~~ (;art) I~~a\~ relative to (:F.t), which is the natural that th . t t0 ows (cf. Section 4.2.3 of Chapter 4) ere exists an adapted process (H ) . fvi rT for all t E [0, T] t °StST sans ying Jo Hldt. < 00 a.s. and

t;

= Lo + i t HsdWs

S'

a.s.

0

In~e L T is a probability density, we have E(L ) _ _ equivalem to P we have L > 0 d T - 1 - L o and, forP" is t, To obtain the 'formula (6~~), we :p.sp'tynth m~~efgenera11Y P(L t > O} = 1 for any . ( e 0 ormula to the log function To do . so, we need to chec.k that P "It E [0 T] L + r t H d W ) f hi ~ ~ , , 0 Jo s s > 0 = 1 The proof OtIS }act relies in a crucial way on the martin a . . . I' g le property and It IS the purpose of Exercise 30 Then the ItA ~ . 0 tormu a yields

10g('L t)

.

=

r L HsdWs"':' ~2 Jr £22. H 2ds 1

Jo

s

o which leads to equality (6.3) with q(t) = HtiL . t

s

s

a.s.

o

Interest rate models

124

. > Corollary 6.1.2 The price at time t of the zero-coupon bond of maturity u -

Modelling principles

t

riskier. Furthermore, the term r(t) - arq(t) corresponds intuitively to the average yield (i.e. in expectation) of the bond at time t (because increments of Brownian motion have zero expectation) and the term -a;:q(t) is the difference between the average yield of the bond and the riskless rate, hence the interpretation of -q(t) as a 'risk premium'. Under probability P*, the process (TVt ) defined by TVt = W t - f; q(s)ds is a standard Brownian motion (Girsanov theorem), and we have

can be expressedas P(t u) ,

= E (exp (_jU r(s)ds + jU q(s)dWs - ~ jU q(S~2dS) \ Ft) t

t

t

.

(6.4)

6.1.1 and fro.m the following P roo f . This follows immediately from Proposition . d able X' ' formula which is easy to derive for any non-negative ran om van.. . '

E· (XIFt ) =

E (XLTI F t ) L t

125

dP(t, u) P(t, u)

(65)

.

.

U

-

= r(t)dt + at dWt.

(6.7)

For this reason the probability P" is often called the 'risk neutral' probability.

o

The following proposition gives lm ec~nomic interpretation of the process (q(t))

6.1.3 Bond options

(cf. following Remark 6.1.4). ~ . U 't' 613 For'each maturity u there is an adapted process (at )o9~u ' P,roposl Ion . "

To make things clearer, let us first consider a European option with maturity 0 on the zero-coupon bond with maturity equal to the horizon T. If it is a call with strike price K, the value of the option at time 0 is obviously (P(O, T) - K)+ and it seems reasonable to hedge this call with a portfolio of riskless asset and zerocoupon bond with maturity T. A strategy is then defined by an adapted process ((HP, H t) )O~t~T with values in rn?, Hp representing the quantity of riskless asset and H, the number of bonds with maturity T held in the portfolio at time t. The value of the portfolio at time t is given by

such that, on [0, uj, dP(t, u) = (r(t) _ afq(t))dt P(t, u) ,

+ afdWt.

(6.6)

Proof. Since the process (F(t, u))o9~u is a martingale under P~, (F()t u)L~09~U is a martingale under P (see Exercise 31). Moreo~er~ we have: P(t, U .t. > 6 ~.si' f ofPr?po~ltI2on ->: ' uj Then , using the same rationaleUas In the proof or aII t E [0, . h h t t ( OU) dt < 00 we see that there exists an adapted process (Ot )o~t~u SUC t a Jo t

L

and

.

F(t, u)Lt

Ie'(O~·)2ds = P(O,u)e Ie'oO"dW._l ' . 2

0

•

P(O,u) exp (it r(s)ds

+ fat (O~ -

dVi = HPdS~

.

q(s))d~s

r(t)dt

+ (Of -

q(t))dWt -

~((Of)2 - ~(t)2)dt

=

+ q(t)2 -

which gives the equality (6.6) with ar

if it is self-

Proposition 6.1.6 We assume sUPO
Ofq(t))dt

= Or -

((HP,Ht))o~t~T is admissible

The following proposition shows that under some assumptions, it is possible to hedge all European options with maturity 0 < T.

+~(Of - q(t))2dt ,2 (r(t)

=

financing and ifthe discounted value Vi (¢) = Hp + H, F( t, T) ofthe corresponding portfolio is, for all t, non-negative and if SUPtE[O,Tj Vi is square-integrable underP*. .

q(S)2)dS).

Applying the Ito fo~ula with the exponential fun~tion, we get

=

+ HtdP(t, T).

Taking into account Proposition 6.1.3, we impose the following integrability conT T ditions: fo IHpr(t)ldt < 00 and fo (H ta;:)2dt < 00 a.s. As in Chapter 4, we define admissible strategies in the following manner: Definition 6.1.5 A strategy ¢

_~ it ((O~)2 dP(t,u) P(t,u)

= HPSp + HtP(t, T) = Hpefo' r(s)ds + HtP(t, T)

and the self-financing condition is written, as in Chapterd, as

_

Hence, using the explicit expression of L; and getting rid of the discounting factor

P(t, u)

Vi

+ (Of -

q(t).

q(t))dWt,

o

14 Th e formula (6.6) is to be related with the equality dS? = r(t)S?dt, R emark 6.. . . dW hi h akes the bond t w IC m satisfied by the so-called riskless asset. It is the term In

r(s),~s is s~uare-integrabl~ under p •. Then there exists an admissible thathe strategy whose value at time 0 is equal to h. The value at time t ~ 0 of such a strategy is given by 1

.C

Vi

= E*

(e-

f

r(S)dshl Ft) .

Some classical models

Interest rate models

126

Vi is the (discounted) value at time t of an admissible strategy ((HP, H t) )O
Proof. The method is the same as in Chapter 4. We first observe that if

~~u;t:~n~n~;)e:~~r(~h~ ~how t?at i~ order to calculate the price of bonds, we

pair (r(t), q(t)) under P. ri:a:::~~sm~d;l~t~under P*, or the d.ynamics of the dynamics of ret) under P by a diffusi e are about to examme describe the q(t~ should have to get a similar e~u~~~~nu~~~:~~ and determ~ne the form that options depend explicitly on 'risk parameters' which The~.~e p~lces o~ bonds and d 1 . are I cu t to estimate. One advantage of the Heath-Jarrow Mort '11 lai . on mo e which w paragraph 6.2.3, is to provide formulae that oniy dep d e ~~ exp am bnefly in en on e parameters of the dynamics of interest rates under P.

= HtdP(t, T) = n.i«, T)aT dWt . We deduce, bearing in mind that SUPtE[O,TJ Vi is square-integrable under P*, that (Vi) is a martingale under P*. Thus we have 'It 5, 8 Vi = E* ( Vol F t ) dVt

6.2.1 The Vasicek model

and, if we impose the condition Vo = h, we get

~=

In this model, we assume that the process ret) satisfies

e I; r(s)ds E* (e - I: r(s)ds hlFt) .

dr(t) = a (b - ret)). dt + adWt . . (6.8) where a, b, a are non-negative t Wi . a const~nt q(t) = _.\, with EC;.s ;::~: e also assume that the process q(t) is

To complete the proof, it is sufficient to find an admissible strategy having the same value at any time. To do so, one proves that there exists a process (Jt)o
o such that Io Jt < 00, a.s, and

x

- -

+

dr(t~ = a (b* - ret)) dt

*

1Jsd~s. 0

he- I: r(s)ds = .E* (he- I: r(S)ds)

127

6.2 Some classical models

=

X,

of martingales because we do not know whether he-I: r(s)ds is in the a-algebra generated by the Wt's, t 5, 8 (we only know it is in the a-algebra Fo which can be bigger (see Exercise 32 for this particular point). Once this property is proved,

>:

_

an d

H tO-- E* (h e-

1.

9

0

.

r(S)ds\:F.) t

dX t

00

holds) ;hose value at time 8 is indeed equal to h.

ret)

= r(O)e- at + b (1- e- at) ~ ae-at

it

·asdWt e:

°

~(~ ~at~i~)) fOl~OWS ~ normal law w~ose mean is given by E(r(t))

0

it is not clear that the risk process (q(t)) is defined without ambiguity. Actually, it can be shown (cf. Artzner arid Delbaen (1989)) that P* is the unique probability equivalent to P under which (p(t, T))O
aT

= -aXtdt + adWt.

at

(6.10)

= r(O)e- at +

an variance by Var(r(t)) - a2/2 (1 -2at) P(r(t) < 0) > 0, which is not very satisfacto ; - e It f~llows that (unless this probability is always very small) Nry r~m a practical point ?f vi.ew to converges in .law to a Gaussian random' t tebnds d a 12a. ean an variance

e

Remark 6.1.7 We have not investigated the uniqueness of the probability P* and

(an

b,

which means that (X) . 0 . 3.5.2). We deduce th;t rl(t):anr~:t:~~~~l:~beckprocess (cf. Chapter 3, Section

i:T

for t 5, 8. We check easily that ((HP,Ht))o
I~ Ir(s)H~lds <

= ret) -

we see that (Xt ) is a soluti~n of the stochastic differential equation

it is sufficient to set

Jt T pet, T)a t

(6.9)

where b b - .\a 1a and W - W .\ . according to this model let us -g'i t + t. Before calculating the price of bonds set .' ve some consequences of equation (6.8). If we

Note that this property is not a trivial consequence of the theorem of representation

Ht -

+ adWt

r

r~t)

v:i~blea~i;h:

,.

.

m~mty,

I

To calculate the price of zero c b d and we use equation (6.9). Fro~ e~:~i~y ~;2~: we proceed under probability P*

P(t,T)

=

E*(e-J.Tr(S)d~IFt)

Interest rate models

128

=

e-b·(T-t)E* (e-

r x;dsl

Ft)

(6.11)

, X*t -- r (t) - b". Since (X*) is a solution of the diffusion equation with where t coefficients independent of time '

dXt = -aXtdt + O"dWt, we can write E* (e-

T X;dS\ Ft) = F(T -

ft

=

E* ( e -

. . 0 f equaU' on (6 . 12) which satisfies X unique solution

t,r(t) - b*)

(6.13)

s fo X:dS) , (X"') t

being the

o= x (cf. Chapter 3, Remark

3.5.11). ) letel We kn ow (cf, Chapter 3) that It is possible to calculate F(B, x compe y. 9 '" h (X"') is Gaussian with continuous paths. It follow.s that fo X s ds

~ e pro::~ random variable

since the integra! is the limit of Riemann sums fof IS a n~ ts Thus' from the expression of the Laplace transform 0 a , Gaussian componen.

-r X~dS)

GaUSSian(, E* e o '

From equality E* (X:)

( E* = exp -

(1

9

X"'dS) s

o

+

~Var (1 2

9

0 ~

X:dS)).

1X:d~ 9

(

'

(I,' X:d')

· ' X'"t -- xe:" S!nce

=

~2e~a(t+U)E* (it easdWslu easdWs) ,

=

0"2 e-a(t+ u)

=

. 1

2

0" e

-a(t+u)

t)R(T - t, r(t))],

where R(T - t, ret)), which can be seen as the average interest rate on the period [t,T], is given by the formula

R(B,r)=Roo- alB [(R oo - r ) (I - e-

a9)_::2 (l_e- a9)2]

with Roo = lim9~oo R(B, r) = b" - 0"2/(2a2). The yield Roo can be interpreted as a long-term rate; note that it does not depend on the 'instantaneous spot rate' r. This last property is considered as an imperfection of the model by practitioners. Remark 6.2.1 In practice, parameters must be estimated and a value for r must be chosen. For r we will choose a short rate (for example, the overnight rate); then we will fit the parameters b, a, 0" by statistical methods to the historical data of the instantaneous rate. Finally .x will be determined from market data by inverting the Vasicek formula. What practitioners really do is to determine the parameters, including r, by fitting the Vasicek formula on market data.

t l\ u

. e2asds

o ( e 2a(tI\U) -

2a

~

1)

+

= (a - br(t))dt a~dWt (6.15) with 0" and a non-negative, s e JR., and the process (q(t)) being equal to q(t) =

9

+ O"e- at)0ft easdW s, we have

Cov(Xf,X~)

= exp [-(T -

dr(t)

COY

9

Going back to equations (6.11) and (6.13), we obtain the following formula

Cox; Ingersoll and Ross (1985) suggest modelling the behaviour of the instantaneous rate by the following equation:

(I,' X:d'.[ X:dS) = 11 Cov(Xf,X~)dudt..

=

s

6.2.2 The Cox-Ingersoll-Ross model

For the calculation of the variance, we write

Var

9 r X"'dS) = 0"2B _ 0"2 (1 _ e- a9) _ z: (1 _ e-a9)2 . Jo a2 a3 2a3

Remark 6.2.2 In the Vasicek model, the pricing of bond options is easy because of the Gaussian property of the Ornstein-Uhlenbek process (cf. Exercise 33).

, 1 e- a9 = x - a

)

Var (

.

= xe- as ~ we deduce E*

.t29

and in equality (6.14), we get

pet, T)

= F(T -

t,X;)

where F is the function defi~ed by F(B, x )

(6.12)

Some classical models

(6.14)

-aVT(t), with a E JR.. Note that we cannot apply the theorem of existence and uniqueness that we gave in' Chapter 3 because the square root function is only defined on JR.+ and is not Lipschitz. However, from the HOlder property of the square root function, one can show the following result, Theorem 6.~.3 We suppose that (Wt ) is a standard Brownian motion defined on [0,00[. For any real number x 2: 0, there is a unique continuous, adapted process (X t ), taking values in'JR.+, satisfying X o x and

=

~ dXt,= (a -' bXd dt + O"~dWt

on

[0,00[.

(6.16)

For a proof of this result, the reader is referred to Ikeda and Watanabe (1981), p. 221. To be able to study the Cox-Ingersoll-Ross model, we give some properties

.:="

Interest rate models 130 . x h of this equation. We denote by (Xt) the solution of (6.16) starting at x and TO t e stopping time defined by

-'lj;'(t) {

0 = 00.

Proposition 6.2.4

Thi oposition is proved in Exercise 34. . .' f IS p r . .. hich enables us to characterise the JOint law 0 The following proposition, w . d I x ft XXdS) is the key to any pricing within the Cox-Ingersoll-Ross mo e . t

'Jo

s

'

Proposition 6.2.5 Forany non-negative A and JL. we have

.,

E (e->'X; e -I' J: X;dS)

2

E (e->'X;)

= (

2/4b

2

= - a2 log

2,e¥

(

a2~(e1't _ 1) + "! - b + e1'tCT-+ b)

'lj;(t).

When applying Proposition 6.2.5 with JL = 0, we obtain the Laplace transform of Xi

=

= exp (-a¢>',I'(t)) exp (-X1p>-,1' (t))

b ) 2a/u a 2/2A(1 - e- bt) + b exp

(2AL

1

+ 1)2a/u

2

exp (

with,

,

ACT + b + e1' tCT -

'lj;>',I'(t) =

a2A (e1't _ 1) +

2

= Jb + 2a

2JL.

)

b)) + 2JL (e1't - 1)

bt

.

-aq,(t)-xt/J(t) is due to the Proof The fact that this expectation can be written as e . d th . iti I • .. (XX) I tive to the parameter a an e uu a t re a d y, (1990)) If additivity property of the process condition x (cf. Ikeda and Watanabe (l?81) , p. 225, ~evuz an or ., for A and JL fixed, we consider the function F(t, x) defined by F(t,x):= E (e->.X;e-I'_J: X;dS) , it is

(6.17)

n~tural t~ look for F as a soiution of the problem . {

2 a 2F aF _ ='~x-2 at. Q . ax

., v

.

+ (a -

F(O,x)=e

aF . bx)- - JLxF ax ->.x

.

Indeed, if F satisfies these equations and has bounded derivatives, the Ito formula shows that, for any T, the process (Mt)o9~T, defined by .r

, XT.ds Mt.='e -I' 1. ~ F(T 0

x

t, X t )

2AA~ 1) .

This function is the Laplace transform of the non-central chi-square law with 8 degrees of freedom and parameter ( (see Exercise 35 for this matter). The density of this law is given by the function [s.c. defined by .

i: b + e1'tCT + b)

.

Abe- bt ) -x a2 /2A(1 _ e-bt) + b

AL() 2AL + 1

96,((A) = (2A +11)6/2 exp ( -

and

(

with L = a (1 - e- ) and ( = 4xb/(a 2 (e bt - 1)). With these notations, the Laplace transform of Xi / L is given by the function 94a/u2,(, where 96,( is defined by

where thefunctions·¢~,1' and 'lj;>',1' are given by ¢>',I'(t)

=

¢'(t)

Solving these two differential equations gives the desired expressions for ¢ and 'lj;. 0

I If a > a 2/2, we have P(T~ = 00) = 1, for all x> O. - a < a 2/2 and b ~ 0, we have P( TOx < 00) -- 1, for all x > O. 2. If 0 < 3: If 0 ~ a < a 2/2 and b < 0, we have Ph) < 00) E ]0, 1[,jor all x> O.

(x

131

is a martingale and the equality E(MT) = M o leads to (6.17). If F can be written as F(t, x) = e-aq,(t)-xt/J(t) , the equations above become ¢(O) = 0, 'lj;(0) = Aand

= inf{t ~ 0IX: = O}

T~

with, as usual, inf

Some classical models

.

)

(r::;()

-(/2

I () e J6,( x - 2(6/4-1/2 e -x/2 x 6/4-1/2I6/2-1 V zt,

lor x > 0 ,

l'

where Iv is the first-order modified Bessel function with index u, defined by

Iv(x) = ~

(~)v 2

f: n=O

(x/2)2n n!r(v + n + 1) .

The reader can find many properties of Bessel functions and some approximations of distribution functions of non-central chi-squared laws in Abramowitz and Stegun (1970)rChapters 9 and 26. Let us go back to the Cox-Ingersoll-Ross model. From the hypothesis on the processes (r(t)) and (q'(t)), we get

.dr(t) '= (a - (b + aa)r(t)) dt

+ av0'ijdWt,

where, under probability P", the process (Wt)O
Interest rate modeLs

132

Some classicaL modeLs

133

denoting by PI and P 2 the probabilities whose densities relative to P" are given respectively by

by:

P(O,T)

E. (e- JOT r(S)dS)

__

e

dP l

_a(T)-r(O).p(T)

th.+ b .

)

¢(t) = - a 2 log

(

9

0

r(s)ds P(O,T) P(O,T)

dP 2 dP·

and

e

- J.9 r(s)ds 0

P(O,O)

.

(e"Y· o - 1) o £1 = 2'" 'Y. (e"Y· + 1) + (a 2 'ljJ (T - 0) + b·) (e"Y· o - 1) a2

+ e"Y·tb· + e-)

'Y. _ b.

J

We prove (cf. Exercise 36) that, if we set

)

2'Y· e ........,.-

e-

dP· -

(6.18)

where the functions ¢ and 'ljJ are given by the following formulae 2

_

and

a2 (e"Y· o -1) -..,.---=--'-..,.----',-----,:----:- 2 'Y. (e"Y· o + 1) + b: (e"Y· o - 1)'

£2 - -

and

. 2(e"Y· t - l ) .'ljJ(t) ,= 'Y. _ b· + e"Y·tb· + b·)

.

with b"

= b + atx and 'Y • -P(t, T)

J(b·)2 + 2a 2. The price at time t is given by

= exp (-a¢(T -

t) - r(t)'ljJ(T - t)) .

the law of r(O) / £1 under PI (resp. r(O) / £2 under Ps) is a non-central chi-squared law with 4a/a 2 degrees of freedom and parameter equal to (I (resp. (2), with

, 8r(Oh· 2e"Y· O . . (1 = a2 (e"Y·O - 1) b·(e"Y· o + 1) + (a 2'ljJ (T - 0) + b·)(e"Y· o - 1)) and

8r(Oh· 2e"Y· O (2 = a2 (e"Y·O - 1) b·(e"Y· o + 1) + b·(e"Y· o - 1)) .

Let us now price a European call with maturity 0 and exercise pr~ce K, on a ~e~o­ coupon bond with maturity ~. We .can sho~ that the hypothesIs of Proposition 6.1.6 holds; the call price at time

Co

=

E·

°ISthus given by

With these notations, introducing the distribution function Fo,( of the non-central chi-squared law with fJ degrees of freedom and parameter (, we have consequently

0

[e-J09r(s)ds (P(O,~) - K)+]

,

• [ _J.9 r(s)ds ( e _a(T_O)-r(O).p(T-O) _ K) ] =Eeo . + =

Co

E. (e- J: r(S)dS p(O,T)1{r(O)
J: r(S)dS 1{r(o)
where r" is defined by

r"

.

Notice that

=-

a¢(T - 0) + log(K) 'ljJ(T _ 0)

E. ( e - t r(s)ds P(O T)) o,

= P(O, T) , from the martingale property ,

. • (e - J.9 r(S)ds) _ of discounted prices. Similarly, E 0

= P(O, T)pI (r(O)

-

K P(O,O)F4a/u 2 ' ( 2

(~:)

.

6.2.3 Other modeLs

The main drawback of the Vasicek model and the Cox-Ingersoll-Ross model lies in the fact that prices are explicit functions of the instantaneous 'spot' interest rate so that these models are unable to take the whole yield curve observed on the market into account in the price , structure. Some authors have resorted to a two-dimensional analysis to improve the models in terms of discrepancies between short and long rates, cf. Brennan and Schwartz (1979), Schaefer and Schwartz (1984) and Courtadon (1982). These more complex models do not lead' to explicit formulae and require the solution of partial differential equations. More recently, Ho and Lee (1986) have proposed a discrete-time . \ .model describing the behaviour of the whole yield curve. The continuous-time model we present now is based on the same' idea and has been introduced by Heath, Jarrow and Morton (1987) and Morton (1989). First of all we define the forward interest rates f (t, s), for t ::; s, characterised ,

P(O 0) We can then write

- , .

the price of the option as

Co

= P(O,T)F4a/u 2 ' ( 1 (~:)

0

< r·) - K P(O,O)P2 (r(O) < r·) ,

Interest rate models

134

(-l

P(t,u) = exp

U

f(t,S)dS)

(6.19)

for any maturity u. So f(t, s) represents the instantaneous interest rate at time s as 'anticipated' by the market at time t. For each u, the process (J(t,u))o~t~u must then be an adapted process and it is natural to set f(t, t) = ret). Moreover, we constrain the map (t, s) H f(t, s), defined for t ~ s, to be continuous. Then the next step of the modelling consists in assuming that, for each maturity u, the process (J(t, u))o~t~u satisfies an equation of the following form:

= f(O, u) + ,ita(v, u)dv +

it

a(J(v, u))dWv,

dP(t, u) pet, u)

1 ex, + "2d(X, x),

~

If the hypothesis (H) holds, we must have, from Proposition 6.1.3 and equality f(t, t) = ret),

arq(t) U

l -l + l {is + l (is = -l + l (l +l = + it (l

l

u

a(v, S)dV) ds

u

(lua(J(v, S))dS) dW v

Xo

-i~' (t

a(J(v, s))dS) dWv ~

0

-1~ a(t,S)dS) dt -

(l

a(J(t, s'))ds

(l 'a (J (t , s))ds - q(t)) .

= a(J(t, u))

(l a(J(t, S))dS) dt + a'(J(t, u)) dWt.

~a(t, u) = a(J(t, u)) (l

a(V,S)dS}diJ' (6.21)

The fact that the integrals commute in equation (6.21) is justified in Exercise 37. We then have

ex, = (f(t,t)

u

U

a(t,u) = a(J(t,u'))

of the form (6.20): take a solution of (6.22) and set

U

f(S,s)dS-i U

l

(6.22)

Theorem 6.2.6 If the.function a is Lipschitz and bounded, for any continuous junction ¢ from [0, T] to IR+ thereexists a uniquecontinuous processwithtwo indices (J(t, u))O~t~u~T suchthat,forall u, theprocess (J(t, u))o
a(v:S)dS}dV

t

q(t)

I

U

f(s,s)ds

2 -

The following theorem, by Heath, Jarrow and Morton (1987), gives some sufficient conditions such that equation (6.22) has a unique solution.

a(J(v, S))dWv) ds u

a(J(t, S))dS)

U

u

u

~(lU a(J(t,S))ds}2,

and, differentiating with respect to u,

df(t, u)

u

f(s, s)ds

a(t,S)dS) -

a(t, s)ds =

(- f (s, S) + f (s, s) - f (t , s)) dS u

(lU U ~ (l

Equation (6.20) becomes, if written in differential form,

u

=

=

with ar =. :- (ft a(J(t, s))ds). Whence

t

X;

(f(" t) - ( [ Q(",)d,) + ~ ( [ U(f(",))d,)}t -(l a(J(t, S))dS) dWt . U

(6.20)

the process (a(t, u))o9~u being adapted, the map (t, u) H a(t, u) being continuous and a being a continuous map from IR into IR (a could depend on time as well, cf. Morton (1989». Then we have to make sure that this model is compatible with the hypothesis (H). This gives some conditions on the coefficients a and a of the model. To find them, we derive the differential dP(t, u)/ pet, u) and we compare it to equation (6.6).LetussetXt = - f U f(t,s)ds. WehaveP(t,u) = eX. and,fromequation (6.20),

135

and by the Ito formula

by the following equality:

f(t, u)

Some classicalmodels

U

a(J(t,S))dS) dW t

.

U

a(J(t, s))ds - q(t)) .

The striking feature of this model is that the law of forward rates under P" only depends on t~e function zr. This" is a consequence of equation (6.22), in which only a a~d (Wd appear. It follows that the price of the options only depends on the function d. This situation is similar to Black-Scholes'. The case where a is a constant is covered in Exercise 38. Note that the boundedness condition on a is essential since, for a(x) = x, there is nQ solution (cf. Heath, Jarrow and Morton (1987) and Morton (1989».

Interest rate models

136

Notes: To price options on bonds with coupons, the reader is referred to Jamshidian (1989) and EI Karoui and Rochet (1989).

Exercises

137

4. Using the previous question, show that under the probabilities whose densities are respectively exp ( - J; r(s)ds) / P(O, ()) andexp ( - JoT r(s)ds) / P(O, T) with respect to P *, the random variable r( ()) is normal. Deduce an expression for the price of the option in the form Co = P(O, T)PI - K P(O, ())P2' for some parameters PI and P2 to be calculated.

6.3 Exercises Exercise'30 Let (Mt)O~t~T be a continuous martingale such that, for any t E [0, T], P(Mt > 0) = 1. We set T

= (inf{t E [O,T] I M, = O}) AT.

Exercise 34 The aim of this exercise is to prove Proposition 6.2.4. For x, M we note TM the stopping time defined by TM = inf{t 2: I Xi = M}.

°

1. Let s be the function defined on

1. Show that T is a stopping time.

2. Using the optional sampling theorem, show that E (MT) Deduce that P ({\it E [O"T] M, > O}) = 1.

s(x)

= E (MT1{T=T}).

= jX e2by/u2 y-2a/u 2dy.

/72

~s

- x -2 2 dx 2. For e

Exercise 32 The notations are those of Section 6.1.3. L~t (Mt)O~t~T be a proce~s adapted to the filtration (Ft ) . We suppose that (Mt ) IS a martingale under P . Using Exercise 31, show that there exists an adapted process (Ht)o9~T such that 00

]0,oo[ by

Prove that s satisfies

Exercise 31 Let (n, F, (Ft)o9~T, P) be a filtered space and let Q be a probability measure absolutely continuous with respect to P. We denote by L, the density of the restriction of Qto Ft. Let (~t)09~T ~e an adapted process. Show that (Mt)09~T is a martingale under Q If and only If the process (LtMt)o~t~T is a martingale under P.

J;{ Hldt <

> 0,

< x < M, we set T:'M

ds

+ (a - bx)= 0. dx

= T: A TM. Show that, for any t > 0, we have

Deduce, taking the variance on both sides and using the fact that s' is bounded from below on the interval [e, M], that E (T:'M) < 00, which implies that T:,M is finite a.s.

a.s. and

< x <M, s(x) = s(e)P (T: < TM) + s(M)P (T: > TM)' We assume a 2: /72/2. Then prove that limx-..+~ s(x) = -00. Deduce that P (TO < TM) = for all M > 0, then that P (TO < 00) = 0. We now assume that ~ a < /72/2 and we set s(O) == limx-..+o s(x). Show that, for all M > x, we have s(x) = s(O)P (TO < TM) + s(M)P (TO> T M) and complete the proof of Proposition 6.2.4.

3. Show that if e for all t

4.

E [0, T].

E~ercise 33

We would like to price, at time 0, a call with mat~rity () and strike price K on a zero-coupon bond with maturity T > (), in the Vasicek model. 1. Show that the hypothesis of Proposition 6.1.6 does hold.

< r" , where _ /72 (1 - e-a(T-II))

2. Show the option is exercised if and only if r( ()) a(T - ())

r*

Roo ( 1 - 1 _ -log(K)

)

e-a(T II)

(1 _e~a(T-II)

. , 4a2

.

I

°

Exercise 35 Let d be an integer and let Xl, X 2 , .•. , X d , d be independent Gaussian random variables with unit variance and respective means mI, m2, ... , md· Show that the random variable X = E~=I Xl follows a non-central E~=I m~. chi-squared law with d degrees of freedom and parameter (

=

) .

3. Let (X, Y) be a Gaussian ~~ctor with values in .IR? tinde~ha probabtiltity:':~ let P be a probability measure absolutely contmuous WIt respec 0 , density

dP

5.

°

' e->'x

dP = E (e->'x)"

Show that, under P, Y is normal and give its mean and variance.

Exercise 36 Using Proposition 6.2.5, derive, for the Cox-Ingersoll-Ross model, the law of r(()) u.i'der the probabilities PI and P 2 introduced at the end of Section ' 6.2.2. Exercise 37 Let (n, F, (Ft)O
Interest rate models

138

(H (t, s) )O$t~T is adapted. We would like to justify the equality

Exercises

139

where N is the standard normal distribution function and

d = a.JO(T - 0) _ log (K P(O,0)/ P(O,T)) 2 a.JO(T - 0) For simplicity, we assume that

f: E (f: H

to justify equality (6.21».

2(t,

s)dt) ds <

00

(which is sufficient

.

I. Prove that

f E(If H(t"jdW,!) d,'; f [E (f H'(t,,)dt) J: U:

Deduce that the integral 2. Let 0

r

ds,

H(t, s)dWt) ds exists.

= to < t l < : .. < t N = T be a partition of interval [0, T]. Remark that

(I: nu; I: (iT

[T

Jo =

s) (W ti+1 -.WtJ) ds

.=0

H(ti,S)dS) (Wt;+l - Wti)

.=0

0

.

and justify why we can take the limit to obtain the desired equality. Exercise 38 In the Heath-Jarrow-Morton model, we assume that the function a is a positive constant. We would like to price a call with maturity' 0 and strike price K, on a zero-coupon bond with maturity T > O. 1. Show that the hypothesis of Proposition 6.1.6 holds. 2. Show that the solution of equation (6.22) is given by f(t, u) a 2 t (u - t/2) + aWt . Deduce, that ll

T)

P( u'.

= P(O,T)

(_ (T' _ P(O,O) exp a

3. Derive, for A E JR, E*

(e

-rJ'

ll)TXT U

.,

a

HII

f: W.ds e,xWB) ~ Deduce

20T(T.

2

=

f(O, u)

0))

+

.

the law of WII'under the

probability measures PI and P 2 with densities with respect to P* respectively given by

dP

l

dP* =

e-

JOB r(~)dSp(o, T) P(O,T)

and

dP e- foB ~(s)ds 2 -= ---dP* P(O,O)

4.. Show that the price of a call at time 0 is given by

Co = P(O,T)N(d) - K P(O,O)N (d - a.JO(T - 0)) ,

7

Asset models with jumps

In the Black-Scholes model, the share price is a continuous function of time and this property is one of the characteristics of the model. But some rare events (release of an unexpected economic figure, major political changes or even a natural disaster in a major economy) can lead to brusque variations in prices. To model this kind of phenomena, we have to introduce discontinuous stochastic processes. Most of these models 'with jumps' have a striking feature that distinguishes them from the Black-Scholes model: they are incomplete market models, and there is no perfect hedging of options in this case. It is no longer possible to price options using a replicating portfolio. A possible approach to pricing and hedging consists in defining a notion of risk and choosing a price and a hedge in order to minimise this risk. In this chapter, we will study the simplest models with jumps. The description of these models requires a review of the main properties of the Poisson process; this is the objective of the first section. 7.1 Poisson process (Tik~l be a sequence of independent, identically exponentially distributed random variables with parameter A, i. e. their density is equal to l{x>o}Ae->.x. We s~t Tn L~l t: We call Poisson process with intensity A the process N, defined by .

Definition 7.1.1 Let

=

Nt

=L n~l

l{Tn::;t}

= L nl{T

n::;t
n~l

Remark 7.1.2 N; represents the number of points of the sequence (Tn)n~ 1 which are smaller than or equal to t. We have Tn

= inf{t ;to,

N,

= n}.

Dynamics ofthe risky asset

Asset models with jumps

143

142 . The following proposition gives an explicit expression for the law of N; for a

be 'memoryless'. The independence of the increments is a consequence of this property of exponential laws.

., . given t. Proposition 7.1.3 If (Ntk~o is a Poisson process Wlt~ intensity A then, for any t > 0, the random variable N, follows a Poisson law witn parameter A

Remark 7.1.6 The law of a Poisson process with intensity A is characterised by either of the folIowing two properties: • (Ntk~o is a right-continuous homogeneous Markov process with left-hand limit, such that

P(Nt = n) = e

->.t

(At)n

-,-. n.

P(Nt

n. • (Nt)t>o is a process with independent and stationary increments, right-contin-

In particular we have

E(Nt ) Moreover, for s

>

= At,

°

= n) = e->'.t (At~n.

E (sN.)

= exp {At (s -

uous, non-decreasing, the amplitude of the jumps being one. For the first characterisation, cf. Bouleau (1988), Chapter III; for the second one, cf. Dacunha-Castelle and Duflo (1986), Section 6.3.

I)}.

Proof. First we notice rhatthe law of Tn is 7.2 Dynamics of the risky asset

(AX)n-l . l{x>O}.Ae->'x (n _ I)! dx, :I.e. a gamma law with parameters , A and n. Indeed, the Laplace transform of

E thus the law of Tn

(e-

T1

is

A

ct T 1 )

= T 1 + ... + Tn is E(e- ctr n ) = E(e-

= A + 0:'

ct T l (

= (A ~ 0:) n

We recognise the Laplace transform of the gamma law with parameters A and n (cf. Bouleau (1986), Chapter VI, Section 7.12). Then we have, for n 2: 1

P(Nt = n)

=

P(Tn ~ t) - P(Tn+l ~ t)

=

)n-l { Ae->'x AX dx _ Jo (n - I)!

=

(Att ->.t -e . n!

t (\

it 0

(AX)n Ae->'x - - I-dx n.

o

t 't" 7 1 4 Let (Nt ) t >0 be a Poisson process with intensity A d and .F = Proposl Ion . . a(N ', s ~ t). The process (Ntk~o is a process with independent an stationary

The objective of this section is to model a financial market in which there is one = e'", at time t) and one risky asset whose price riskless asset (with price jumps in the proportions U1 , ... , Uj, ..., at some times Tl , ... , Tj, ... and which, between two jumps, folIows the Black-Scholes model. Moreover, we will assume that. the Tj'S correspond to the jump times of a Poisson process.To be more rigorous, let us consider a probability space (11, A, P) on which we define a standard Brownian motion (Wdt~o, a Poisson process (Nt)t~o with intensity A and a sequence (Uj)j~1 of independent, identicalIy distributed random variables taking values in ]-1, +00[. We will assume that the a-algebras generated respectively by. (Wdt~o, (Nt)t~o, (Uj)j~1 are independent. For all t 2: 0, let us denote by F t the a-algebra generated by the random variables Ws , N, for s ~ t and tt, 1{j~Nd for j 2: 1. It can be shown that (Wdt>o is a standard Brownian motion with respect to the filtration (Ft)t>o, that (Nt)t~o is a process adapted to this filtration and that, for all't > s, N, ~ N, is independent of the zr-algebra F s . Because the random variables UjJ{j~Nd are Frmeasurable, we deduce that, at timet, the relative amplitudes of the jumps taking place before t are known. Note as well that the Tj'S are stopping times of (Fdt~o,.sinc~ {Tj ~ t} = {Nt 2: j} EFt. _ The dynamics of Xt, price of the risky asset at time t, can nowbe described in the following manner, The process (Xt)t~O is an adapted, right-continuous process satisfying:

Sr

• On the time intervals

s

increments, i.e. • independence:

dX t

.

if s > 0, N t+ s

• stationarity: the law of N t+ s

-

-

h, Tj+l [

N, is independent of the a-algebra Ft.

N; is identical to the law of N, - No

= N s·

Remark 7.1.5 It is easy to see that the jump times Tn are stopping tim~s. In~eed~ { Tn < t} = {Nt 2: n} E Ft. A random variable T with .expon~nuallaw sat~sfie P(T- 2: t + siT 2: t) = P(T 2: s). The exponential variables are said to

• At time

Tj,

the jump of X, is given by

- \ 6,Xr .

thus X r;

= Xt(jldt + adWt).

= Xr~, (I + Uj ) .

J

,

=X

r J, -

XT j

=X

Tj

-U , j

,

Asset models with jumps

144

Dynamics ofthe risky asset

145

So we have, for t E [0, Tl [

is independent of the a-algebra generated by the random variables N < d UjIUSN.}. Let A be a Borel subset of JRk, B aBorel subset f JRd u'dV,C- s an r. an of the a-algebra a(Nu, V, <) _ s ." we have, USIng the independence of U d N dan event that the Uj 's are independent and identically distributed, an an the fact

°

consequently, the left-hand limit at Tl is given by X _ = X oe(/L-
T \

and

n C n {(U 1, .. . , Ud) E B} n {d $ N s } )

P ({(UN.+l, ... ,UN.+k) E A}

2 X T\ ,= X o(1 + U1)e(/L-
T\

00 •

Then, for t E [Tl,T2[,

X,

X T\ e(/L-
= = =

T\ )

X -(1 + Ude(/L-
T \ )

Repeating this scheme, we obtain

x, =Xo

(fl(1

+Uj ) )

e(/L-
p=d

with the convention I1~=1 = 1. The process (Xt)t>o is obviously right-continuous, adapted and has only finitely many discontinuities-on each interval [0, t). We can also prove that it satisfies, for all t ~ 0,

P a.s.

= Xo +

x,

1x,

.

'.

(J.Lds

N,

+ adW s) + L

0

j=1

.

XTjU j •

(7.1)

~

P ({(UNs+l, ... , UN.+k) E A}

=

0, the a-algebras

n C n {(U 1, ... , Ud) E B} n {d $ N s } )

P«UN.+1, ... ,UN.+k E A)P(Cn{(U1,,,,,Ud E B)}n{d$ N s } )

Whence the independence stated above. Suppose now that E(lU1 1)

.

We will see that, for this kind of model, it is generally impossible to hedge the options perfectly. This difficulty is due to the fact that for T < +00, there are infinitely many probabilities equivalent to P on :FT under which the discounted price (e- rt Xt)O
Lemma 7.2.1 For all s

.

From the last equality, we deduce (taking C = n and B - JRd) h ) l' t at the vector (uNs+l, ... , U N.+k ,olIowsthesamelawas(Ul " " , U) and th th k, en at

J=1

t

= p}).

P «U1, ... ,Uk) E A) LP «U1, ... , Ud) E B)P (Cn {Ns

E(X,J.r.)

~

0

< +00 and set Xt -- «<xt- Th en

X.E (,(.-,-u',,*,-,)+U(w.-w.) .

fi

(1 +

U;)j.r.)

J=Ns+l

X\E (e(/L_r-
2/2)(t_ S)+
N rr'-Ns(1

.

=

+U

')1)

Ns+J :Fs

j=1

XsE (e(/L_r-
I+U Ns+j )

) ,

j=1

using Lemma 7.2.1 and the fact that Wt - Wand N t the a-algebra :Fs . H e n c e ' s

N s

-

a (UN.H, UN.+2,·'·' UN.+k,"') and :Fs are independent. Proof. As thea-algebras W = a(W., s ~ O),N = a(N., s ~ 0) andU = a(Ui, i ~ 1) areindependent, it suffices to provethatthea-algebra a (UN.+l, UN.• +2, ... , UN.+k,' .. )

.

=

Xse(/L-r)(t.,-s)e),(t-s)E(U!l

,

are

. d In

d epen ent of

146

Asset models with jumps

Moreover. IZPI2 -< C(N.t an d even In L . We have

using Exercise 39. It is now clear that (X t ) is a martingale if and only if

.=r: -

f.L

Lemma 7.2.2 Let o be a left-continuous process, taking values in ffid. adapted to the filtration (Ft)Qo, We assume thai.for ~ll t > 0 , ' "

(I ~8 J t

v(dZ)
Then the process M; defined by

Mt

N. =L

2(:s,

z)) <

ds

.

(7.2) setting

8.

'

v(dz)
where ~i(Y) is defined by

0

j=l

~i(Y) = E (N';~N';
is a square-integrable martingale and

1 JV(~Z)
' t

M; - A

.

ds

J=l

z)

~i (y) is thus the expectation of a random sum and, from Exercise 40,

)

is a martingale.

2:~=1

~i(Y) = A(Si+l -

= 1.

Proof. We assume first that

C

=

\2:7::1
M, is square-integrable. Let us fix 8 and t, with Z

~

8

s.)

J

av(z)
Going back to equation (7.2), we deduce

I
sup (y,z)EHtd xlR

UJ.en we have

N 5).. c 11 ' It 10 ows that the convergence takes place in L 1

Using 7.2.1'and the fact that Y.5,. is F -measurabl e, we apply Proposition .. A 2 5 Lemma f th A . . 0 e ppendix to see that

+00.

'It J


Notice that by convention

-

147

AE(V1 ) ·

To deal with the terms due to the jumps in the hedging schemes, we will need two more lemmas, whose proofs can be omitted at first reading. We will denote by v the common law of the random variables Vj's.

E

Dynamics ofthe risky asset

< t, and

set

E( Z'IF.)

~ E (~ ~.(Y.J IF.) ~ E (~ >'(s,+, -

s.)

I

dv(z) 'I> (Y.;> z)

When the mesh of p tends to 0, we obtain

N.

L


j=N.+1 To a partition p = (80 =.8

< S'l < ... < 8m = t)

associate m-l

ZP =

of the interval [8, t], let us which proves that M; is a martingale. Now set can w n t e '

N~i+l

L L


i=O j=N••+I The left-continuity of (Ydt~o and the continuity of

ZP = "m-1 E(Z· IT) L.",=o

,+ 1 .r 5;

•

We

IF.) .

,

.. ~

Asset models with jumps

148

Dynamics ofthe risky asset

and equality (7.3) implies that M? - A f; du f dv(z)~2(yu, z) is a martingale. If we do not assume that ~ is bounded, but instead

Moreover,

E ((Zp - 2

p)2I

FS)

2

~ E [(~ [Z;+, -E(Z;+,IF.JI) =

E (it ds /

IF.]

+2 LE ((Zi+1 - E(Zi+1IFs,)) (Zi+1 - E(Zi+dFsJ) Irs) . i
Fs)

dv(z)~2(ys, z))

< +00,

for any t, we can introduce the (bounded) functions ~n's defined by ~n(y, z) = inf(n, sup( -n, ~(y, z))), and the martingales (M;')c?o defined by

E (~[Z,+' - E(Z;+,IF.JI' IF.)

E ((Zp - 2

149

It is easily seen that E

(~1 (Zi+1 - E(Zi+lIFsi ))2I Fs)

=

E

=

E (~ E ([Z,+' - E(Z,+dF.JI'1 F.;) IF}

(1; dsJ v(dz) (~n(Ys, z) -

~(Y., Z))2)

tends to 0 as n

tends to infinity. It follows that the sequence (M;')n'21 is Cauchy in L2 and as M;' tends to .M; a.s., M; is square-integrable and taking the limit, the lemma is satisfied for ~. 0 Lemma 7.2.3 We keep the hypothesis and notations ofLemma 7.2.2. Let (Atk:~o

(f; A;ds)

be an adapted process such that E

< +00 for any t. We set L,

= .', ~

t

fa AsdWs and, as in Lemma 7.2.2,

Using Lemma 7.2.1 once again

E [(Zi+1 - E(Zi+1IFs,))2IFsi] = V(ys,), where the function V is defined by

V(y) = Var

Then the product Li M, is a martingale.

(N'i~N'i ~(y, UN.i+i))

Proof. It is sufficient to prove the lemma for ~ bounded (the general case is proved by approximating ~ by some ~n = inf(n, sup( -n, ~ )), as in the proof of Lemma 7.2.2). Let us fix s < t and denote by p = (so = s < S1 < ... < Sm = t) a partition of the interval [s, t]. We have

]=1

J

and, from Exercise 40,

V(y) = A(Si+l - Si)

Jdv(z)~2(y,Z).

Therefore

E ((ZP:- 2p)2IFs) = E

(~A(Si+1 -

s.)

JdV(Z)~2(YsilZ)IFs),

On the other hand, since (Ltk;~o and (Mtk:~o are martingales

E ((LSi+! M S i + 1

and so when the mesh of the partition p tends to 0,

'E [(M' Since

(Mtk~.o

M,)'iF:j = E [A

l J du

dv(z)
-

LSiMsJIFs,)

= E ((L Si+

1 -

LSi)(MSi+ 1

Whence

Z)IF.] .

(7.3) with

is a square-integrable martingale, we obtain

m-1 AI' = L (LSi+ 1

E [(Mt - Ms)2IFs] = E (M; + M; - 2MtMs1Fs) = E (M t2 - M;IFs)

,

! I

I

i=O

-

Ls.)(Msi+! --; M s,).

-

Ms.)IFsi) .

--j'p=

Asset models with jumps

150 m-1

<

(SUPO
L IM',+I -

M., I

i=O

-, (i}I'(Y", U;)I + ), f: du f dv(z )I~(Y., z)1) .~

(SUPO~i~m-1) IL';+1 - L.,I (C(Nt

-

N.)

+ )"C(t

- s)),

-

N.)

+ )"C(t -

= H?rertdt + HtdXt, i.e., taking into account equation (7.1), dlit = Hprertdt + HtXt(J.Ldt + O'dWd between the jump times and at a jump time Tj, lit jumps by an amount ~ Vrj = dVt

with C = sUPy z 1(y,z)l. From the continuity of t H L t , we see that AP tends almost surely t~ 0 as the mesh of the partition p tends to O. Moreover

IAPI ~ 2 sup ILul (C(Nt

151

In the following, we fix a finite horizon T. A trading strategy will be defined, as in the Black-Scholes model, by an adapted process ¢ = ((HP, Hd )O
We deduce

IAPI

Pricing and hedging options

H rj ~Xrj = H rj UjX r-:-. Precisely, the condition of self-financing can be written 1 as

s)).

.~u9

The random variablesuP.
Vo +

it

H?rer'ds

+

it

H.X.(J.Lds

+ O'dW.)

N,

consequently

+ '" L.J HrUjXr,-.

(7.5)

1

o

j=l

For this equation to make sense, it suffices to impose the condition JOT IH~lds

J::

7.3 Pricing and hedging options

7.3.1 Admissible strategies 1

Let us go back to the model introduced at the beginning of the previous section, assuming that the U/,s are square-integrable and that.

=r

J.L

which

- )"E(Ud

implie~ that the process

=r

(.it) t>O = -

)

- )..

J

zv(dz),

that

H:ds < 00, a.s. (it is easily seen that s H X. is almost surely bounded). Actually, for a specific reason to be discussed later, we will impose a stronger condition of integrability on the process (Hdo~t~T, by restricting the class of admissible strategies as follows:

Definition 7.3.1 An admissible strategy is defined by a process

o

'

¢ = ((Ht , Ht))O~t~T

(7.4)

(e-rt:Xtk:~o is a martingale. Notice

+

adapted, left-continuous, with values in t E [0, T] and such that JOT

IH~lds < +00

~ ais.

rn?,

satisfying equality (7.5) a.s. for all

andE (JoT H;X;ds)

< +00.

Note that we do' not impose any condition of non-negativity on the value of admissible strategies. The following proposition is the counterpart of Proposition 4.1.2 of Chapter 4.

and consequently, from Exercise 39,

E (X;) Therefore the process'

= XJ exp ((0'2 + 2r)t) exp ()..tE(Ut)) .

(.it)

is a square-integrable martingale. t~O

Proposition 7.3.2 Let (Hdo9~T be an adapted, left-continuous process such that

E (iT H;X;dS)

< 00,

and let Vo E nt., There exists a unique process (HP)O~t~T such that the pair

Asset models with jumps

152

((HP, H t))O$t<5,T defines an admissible strategy with initial value Vo. The discounted value at time t ofthis strategy is given by

II,

~ Vo + J.' H,X,adW, + t,H.,ui X:;- - ~ J.'
Proof. If the pair (HP, H t)O$t<5,T defines an admissible strattegy, its value at time t is given by Vi = yt + Zt, with yt = Vo + f; H~rersds + fo HsX s tude + adWs) and Zt = EN~1 H; UjX -. Differentiating the product e-rtyt, J-

~

]

e-rtVi = Vo +

1 t

(-re-rs)Ysds +

=

Vo + it HsXs((J.L - r)ds + adWs) N,

+ LHrjUjXr-:-, j=1

'

which, taking into account equality (7.4), yields

.

1 t

153

Pricing and hedging options

rt e-rsdYs + e- Zt."

(7.6)

It is clear then that if Vo and (H t) are given, the unique process (HP) such that ((HP, H t) )O$t<5,T is an admissible strategy with initial value Vo is given by

H?

Moreover, the product crt Zt can be written as follows:

=.

Vi -

lftXt t

-u.x, +Vo + N,

(

L j=1

=

e- rrj +

r(_re-rs)ds t

n., u.x..

N,

j=1 N,"

r

in

t

.

0

= ~ - + L.-t e-rrjHr,U1'X ] r ,

.

1 t

'0

1=1.

N,

N .•

ds(-re- rs) ~ L HrjUjX . - t . r-:-} j=I' ' .

r .

1(-re-rs)Vs~s 1H~rds +1

N,

+ LHrjUjXr~, j=1

1

I

It is also obvious that fo IHPldt

HtX t = ert (Hp

+ HtX t)

'j

}

< 00 almost surely. Moreover, writing Hpe rt +

and integrating by parts as above we see that

((H~, Ht))O
o

defines an admissible strategy with initial value Vo. Remark 7.3.3 The condition E

(J:{ H; X;ds) <

00

implies that the discounted

value (Vt ) of an admissible strategy is a square-integrable martingale. This results from the expression in Proposition 7.3.2 and Lemma 7.2.2, applied with the continuous process with left-hand limit defined by yt = (H t , Xt - ) (note that in the integral with respect to ds, one can substitute x, for because there is only finitely many discontinuities).

t

(H~ + HsX s) ds + H~rds + HsX~(J.Lds + adWs)

Nt

J ]

x..

+ LHrjUjX~jVo - i t r

v(dz)z.

= -HrD.Xr· + Hr·UjX - = o.

T

.

= L e- rrj HrjUjXrj + io (-re-rs)Zsds. j=1 Writing this in (7.6) and expressing dYs, we obtain t t t Vt Vo + + HsXs(J.Lds + adWs)

=

H rJ - H T j-

dsl{rj<5,s}( _re-rS)HrjU1Xrj-

.

dsHsXs

i J

From this formula, we see that the process (HP) is adapted, has left-hand limit at any point and is such that HP = H~_. This last property is straightforward if t is not a jump time Tj and if t is some Tj, we have o 0 - .

Nt

Le-rrjHrjUjXrjj=1

j=1

-A

ir, ,

.+ L

1 t

).

Nt

HsXsadWs +LHrjUjXr-:o j=I'

t

7.3.2 Pricing

.

\

'Let us consider'a European option with maturity T, defined by a random variable h, FT-measurable and square-integrable. To clarify, let us stand from the writer's I

,------

with

point of view. He sells the option at a price Vo at time 0 and' then follows an admissible strategy between times 0 and T. From Proposition 7.3.2, this strategy is completely determined by the process (Ht)os;t5;T representing the amount of the risky asset. If Vi represents the value of this strategy at time t, the hedging mismatch at maturity is given by h - VT. If this quantity is non-negative, the writer of the option loses money, otherwise he earns some. A way. of measuring the risk " is to introduce the quantity

HI = E ((e-rT(h -

155

Pricing and hedging options

Asset models with jumps

154

F( t, x)

~

If If

~ E (e -"(T-,) f (xe('-u' I')(T-')+UWT_'

E (e-"(T-.) f (xe("->E(U,)_U' I')(T-')+UWT_,

VT))2) .

(1 +

(1 + Uj

Uj)) ) )) )

.

Note that if we introduce the function

Since, from Remark 7.3.3, the discounted value (ft) is a martingale, we have E (e-rTVT) = Vo. Applying the identity E(Z2) = (E(Z))2 + E ([Z :::- E(Z)]2) to the random variable Z e-rT(h - VT), we obtain

Fo(t,x) = E (e-r(T-t) f (xe(r-cr 2/2)(T-t)+crwT_')) ,

=

HI = (E(e-rTh) -

Vo)2 + E (e-rTh - E(e-rTh.) - (VT - Vo)f·

which gives the price of the option for the Black-Scholes model, we have (7.7)

Proposition 7.3.2 shows that the quantity VT' - Vo depends only on (Ht ) (and he will not on Vo). If the writer of the option tries to minimise the risk ask for a premium Vo E(e-rTh). So it appears that E(e-rTh) is the initial value of any strategy designed to minimise the risk at maturity and this is what we will take as a definition of the price of the option associated with h. By a similar argument, we see that an agent selling the option at time t > 0, who wants

=

to minimise the quantity premium

R; = E ( (e-r(T-t)(h -

Vi = E (e-r(T-t) hIFt).

VT) )2 1Ft ) , will ask for a '

Before tackling the problem of hedging, we try to give an explicit expression for the price of the call or the put with strike price K. We will assume therefore that h can be written as f(XT), with f(x) = (x - K)+ or f(x) = (K - x)+. As we saw earlier, the price of the option at time t is given by

(e-~(T-t) f(XT)IFt)

=

E (e-r(T-t) f (Xte(l'-cr2

E (e- d T -.) I

).

.

(7.8)

Since NT-t is a random variable independent of the Uj's, following a Poisson law with parameter >'(T - t), we can also write

F(t, x)

~ ~E (FO ~' xe-'(T-')E(U,)

Q

(1+ Uj ) )

)

e-'(T-')

~~(T -

t)"

/2)(T-:-t)+cr~WT - W,) . IT, .'

(1 + Uj ) )

Ft)

]=N,+l

(X .e('-u'/2)(T-')+U(WT - W,t~r' (1 +

From Lemma 7.2.1 and this equality, we deduce that

E (e-r(T-t)!(XT)!Ft)

7.3.4 Hedging of calls and puts Let us examine the hedging problem for an option h= f(XT), with f(x) = (K - x)+. We have seen that the initial value of any (x - K)+ or f(x) at maturity is given by admissible strategy aiming at minimising the risk Vo = E(e-rTh) = F(O; X o). For such a strategy, equality (7.7) yields

=

'.

,

(1+ Uj ) )

Each term of this series can be computed numerically if we know how to simulate the law of the Uj's. For some laws, the mathematical expectation in the formula' can be calculated explicitly (cf. Exercise 42).

7.3.3 Prices of calls and puts

=

If

We will take this quantity to define the price

of the option at time t.

E

F(t, x) = E (FO (t,xe-'(T-.)E(U,)

RJ,

= F(t,Xt),

R6

RJ = E (e-rTh UN,+

j))

.r}

VT

f·

Now we determine a process (H t )Os;t5;T for the quantities of the risky asset to be To do so, we need the following proposition. held in portfolio t? minimise

RJ.

~~o.positio~ 7.3.4 Let Vi ~;;'he value at time t of an admissible strategy with f(XT)) = F(O, X o), determined by a process initial value Vo = E (e (Ht)Os;t5;T for the quantities of the risky asset. The quadratic risk at maturity

ir== 156

RJ = R&T

Asset models with jumps

Pricing and hedging options

157

E (e-.rT(f(X T) - VT)) 2 is given by the following formula:

E ( Jo r T (aF ax (s, X) s

=

-

- 2 2 ds H; ) 2 Xsu

E (it ds !v(dz) (F(s, Xs(l

o

+ J: A J v(dz)e- 2rs (F(s,

x.n + z)) -

Proof. From Proposition 7.3.2, we have, for t

~

s

F(s, X s) - H szXs)2 dS).

+ z)) - F(s, X s)) 2)

(1 dsX; JV(dz)z2) t

E

< +00,

T,

which, from Lemma 7.2.2, implies that the process Nt

u,

L

=

j=1

F(Tj, X r;) - F(Tj, X rj-)

Al t ds F(t,x) = e-rtF(t,xe rt), so that F(t, X t) = E

(iiIFt ) . It emerges that F(t, X t) is the discounted price of

the option at timet. We deduce easily (exercise) from fonnula (7.8) that F(t, x) is C 2 on [0, T[xlR+ and, writing down the Ito formula between the jump times, we obtain

F(t,

Xt ) = F(O, X o)+ lit

+-

2

0

aF r aF - ior 8;(s, Xs)ds + io ax (s,Xs)X s(-AE(Vdds + udWs)

aF 2

-

2 - 2

-a 2 (s,Xs)u Xsds x . .. .

+ L F(Tj,XrJ - F(Tj,Xr~)· N,

j=1

-

-

-

-

]

(7.10)

Remark that the function F( t, x) is Lipschitz of order 1 with respect to z, since

J

(F(s,Xs(l+z))-F(s,Xs))dv(z)

is a square-integrable martingale. We also know that F(t, Xt ) is a martingale. Therefore the process F( t, X t ) - M, is also a martingale and, from equality (7.10), it is an Ito process. From Exercise 16 of Chapter 3, it can be written as a stochastic integral. Whence

--

F(t, X t) - M,

= F(O, X o) +

i 0

t aF ax (s, Xs)XsudWs.

(7.11)

Gathering equalities (7.9) and (7.11), we get

ii - VT = M~1) + M¥), with

and N,

U

IF(t,x) -(F(t'Y)1 ( N _, ) T < E e-r(T-t) jxe(r--XE(Utl- u 2 /2)(T-t)+UWT_t (1 + V j)

=

L (Fh,Xr;) j=1

I !

Fh,Xr~) - HrjVjXr~) ] ]

t

-A

ds

dv(z) (F(s, Xs(l

+ z)) - F(s, X s) - HszX s) .

From Lemma 7.2:3, MP) M t(2) is a martingale and consequently

E (MP) M1

2))

= J.!a 1) Ma2) = O.

Whence

E

=

Ix -yl·

It follows that

(ii - VT)

=,

E((M~1))2) + E((M~2))2)

• 'E

(J:{~~

(s, X.) -

HrX;U'dS) + (M~'»)'), E(

Asset models with jumps

158

Notes: The financial models with jumps were introduced by Merton (1976). The approach used in this chapter is based on Follmer and Sondermann (1986), CERMA (1988) and Bouleau and Lamberton (1989). The approach we have chosen.relies heavily on the assumption that the discounted stock price is a martingale. This assumption is rather arbitrary: Moreover, the use of variance as a measure of risk is questionable. Therefore, the reader is urged to consult the recent literature de.aling with incomplete markets, especially Follmer and Schweizer (1991), Schweizer (1992,1993,1994), El Karoui and Quenez (1995).

and applying Lemma 7.2.2 again

E(( M f2))2) =

E

.

(A !:;ds J v(dz) v- X s(l + z)) -

The risk at maturity is then given by

m

=

E

_

2) .

)2 - s a ds

T aF Jo ax (s, X s) ( (

+ JoT A J v(dz)

F(s, X s) - HszX s)

Hs

X

2

159

Exercises

2

7.4 Exercises

(F(s, X s(l

+ z)) - F(s, X s) - HszX s) 2 dS) .

o It follows that the minimal risk is obtained when H, satisfies P a.s.

~xercis~ 3~. Let (Vn)n~I be a sequence of non-negative, independent and identically distributed random variables and let N be a random variable with values in N, following a Poisson law with parameter A, independent of the sequence (Vn)n~I. Show that .

E

aF .) -2 2 ( ax (s, x.) - H, Xsa

.

(11 vn) ~

e'(E(V,j-')

It suffices indeed to minimise the integrand with respect to ds. It yields, since

~xercise 40 Let (Vn)~~1 be a sequence of independent, identically distributed, Integrable random variables and let N be a random variable taking values in N integrable and aindependent of the sequence (V,n ). We set S = ""N V, (with the L ..m =I n

(Hdt~iJ must be left-continuous,

convention En=I = 0).

+

AJ v(dz) (F(s,X s(l

+ z)) - F(s,Xs) - HszX s) zX s = O.

H,

= 6.(s,.X s - ) ,

. 1. Prove that S is integrable and that E(S) = E(N)E(Vi}

with

6.(s,x)

=

aF (2 a - (s x ) (12 +A J v(dz)z2 ax' 1

+A

. j. v (d)z z (F(S,X(l+Z))-F(S,X))) x

(JoT

In this way, we obtain a process which satisfies E H;X;ds) < +00 and which determines therefore an admissible strategy minimising the risk at maturity. Note that if there is no jump (A = ,0), we recover the hedging formula for the Black-Scholes model and, in this case, we know that the hedging is perfect, i.e, = O. But, when there are jumps, the minimal risk is generally positive (cf.

m

Exercise 43 and Chateau (1990». Remark 7.3.5 The formulae we obtain indicate that calculations are still possible for models with jump. It remains to identify parameters and the law of the U, 'so As for the volatility in the Black-Scholes model, we can distinguish two approaches: (1) a statistical approach, from the historical data and (2) an implied approach, from the market data, in other words from the prices of options quoted on an organised market. In the second approach, the models with jump, which-involve several parameters, give a better 'fit' to the market prices.

2. "!'Ie assume N and VI to be square-integrable. Then show that S is squareIntegrable and that its variance is Var(S) = E(N)Var(Vt) + Var(N) (E(Vt ))2. 3. Deduce that if N follows a Poisson law with parameter A, E(S) and Var(S) = AE (Vn.

= AE(Vt)

Exercise 41 The hypothesis and notations are those of Exercise 40. We suppose that the Vi's take values in {a, ,8}, with a, ,8 E IR and we set p = P(VI = a) = 1 - P(Vt = ,8). Prove that S has the same law as aNI + ,8N2 , where N I and N 2 are two independent random variables following a Poisson law with respective parameters AP and (1 - p)A. . Exercise 42 1. W,e suppose, with the notations of Section 7.3, that UI takes values in {a, b}, WIth P = P(UI = a) = 1 - P(UI = b). Write the price formula (7.8) as a double series where each term is calculated from the Black-Scholes formulae (hint: use Exercise 41). 2. Now we suypose that UI has the same law as e 9 - 1, where 9 is a normal variablewithmean m and variance a 2 • Write the price formula (7.8) as a series of terms calculated from the Black-Scholes formulae (for some interest rates and v?latilities to be given).

Asset models with jumps 160 Exercise 43 The objective of this exercise is to show that there is no perfect hedging of calls and puts for the models with jumps we studied in this chapter. We consider a model in which a > 0, >. > a and P (U1 1= 0) > O. 1. From Proposition 7.3.4, show that if there is a perfect hedging scheme then, for ds almost every. s and for v almost every z, we have

P a.s.

ZXs~~ (s,~s) = F(s,Xs(1 + z))

- F(s,X s)'

2. Show that the law of X; has (for s > 0) a positive density on ]0, ~[. It may be worth noticing that if Y has a density 9 and i~ Z is a random vana~le independent ofY with values in ]0,00[; the random van able Y ~ has the density JdJ-L(z)(l/z)g(y/z),whereJ-L is the lawofZ. . 3. Under the same assumptions as in the first question, ~ho"w that there eXl~ts z 1= a such that f~r s E [0,T[ and x E]0, 00[,

aF ax (s,x)

=

F(s,x(l+z))-F(s,x) zx .

Deduce (using the convexity of F with respect to x) that, for s E [0, T], the function x t-t F (s; x) is linear. . . " 4. Conclude. It may be noticed that, in the case of the put, the function x t-t F(s, x) is non-negative and decreasing on ]0, 00[.

8

Simulation and algorithms for financial models

8.1 Simulation and financial models In this chapter, we describe some methods which can be used to simulate financial models and compute prices. When we can write the option price as the expectation of a random variable that can be simulated, Monte Carlo methods can be used. Unfortunately these methods are inefficient and are only used if there is no closedform solution for the price of the option. Simulations are also useful to evaluate complex hedging strategies (example: find the impact of hedging a portfolio every ten days instead of every day, see Exercise 46).

8.1.1 The Monte Carlo method The problem of simulation can' be presented as follows. We consider a random variable with law J-L(dx) and we would like to generate a sequence of independent trials, Xl, .. : ,Xn, . . . with common distribution J-L. Applying the law of large numbers, we can assert that if f is a J-L-integrable function 1

lim N N-t+oo

o

" L..J

l~n~N

f(Xn)

=

J

f(x)J-L(dx).

(8.1)

To implement this method on a computer, we proceed as follows. We suppose that we know how to build at sequence of numbers (Un)n>1 which is the realisation . . of a sequence of independent, uniform random variables on the interval [0,1] and we look for a function (Ul' ... ,up) t-t F( Ul, ... ,up) such that the random variable F(U l, ... ,Up) has the desired law J-L(dx). The sequence of random variables (Xn)n~1 where X n = F(U(n-l)p+l"" ,Unp) is then a sequence of independent random variables following the required law J-L. For example, we can apply (8.1) to the functions f(x) = x and f(x) = x 2 to estimate the first and second-order moments of X (provided E(IXI 2 ) is finite). The sequence (Un)n~l is obtained in practice from successive calls to a pseudorandom number generator. Most languages available on modern computers provide a random' function, already coded, which returns either a pseudo-random number

Simulation and algorithms for financial models

162

between 0 and 1, or a random integer in a fixed interval (this function is called rand () in C ANSI, random in Turbo Pascal).

Remark 8.1.1 The function F can depend in some cases (in particular when it comes to simulate stopping times), on the whole sequence (Un)n;?:I, and not only on a fixed number of Ui 'so The previous method can still be used if we can simulate X from an almost surely finite number of Ui 's, this number being possibly random. This is the case, for example, for the simulation of a Poisson random variable (see page 163). 8.1.2 Simulation of a uniform law on [0, 1] We explain how to build random number generators because very often, those available with a certain compiler are not entirely satisfactory. The simplest and most common method is to use the linear congruential generator. A sequence (Xn)n;?:O of integers between 0 andm - 1 is generated as follows: Xo = initial value E {O, 1, ... ,m - I} { ~n+I = aXn + b (modulo m),

a, b, m being integers to be chosen cautiously in order to obtain satisfactory characteristics for the sequence. Sedgewick (1987) advocates the following choice:

163

The previous generator provides reasonable results in common cases. However it might happen that its period (here m = 108 ) is not big enough. Then it is possible to create random number generators with an arbitrary long period by increasing m. The interested reader will find much information on random number generators and computer procedures in Knuth (1981) and L'Ecuyer (1990).

8.1.3 Simulation of random variables The probability laws we have used for financial models are mainly Gaussian laws (in the case of continuous models) and exponential and Poisson laws (in the case of models with jumps). We give some methods to simulate each of these laws.

Simulation ofa Gaussian law A classical method to simulate Gaussian random variables is based on the observation (see Exercise 44) that if (U I , U2 ) are two independent uniform random variables on [0, 1]

V- 2 10g(Ud cos(27rU2 ) follows a standard Gaussian law (i.e. zero-mean and with variance 1). To simulate a Gaussian random variable with mean m and variance a, it suffices to set X = m + ag, where 9 is a standard Gaussian random variable. function Gaussian(m, sigma : real) : real; begin gaussian := m + sigma" sqrt(-2.0 " log(Random» " Random); end;

31415821 1

108 . This method enables us to simulate pseudo-random integers between 0 and m - 1; to obtain a random real-valued number between 0 and 1 we divide this random integer by m. const m ml b

Simulation and financial models

100000000; 10000; 31415821;

" cos(2.0 " pi

Simulation ofan exponential law We recall that a random variable X follows an exponential law with parameter f,L if its law is 1{x;?:O}f,Le u» dx. r-

We can simulate X ~oticing ~hat, if U follows a uniform law on [0,1], 10g(U) / f,L follows an exponential law with parameter u,

var a : integer;

function exponential( mu : real) : real; begin exponential := - log (Random) / mu; end;

function Mult(p, q: integer) : integer; (" Multiplies p by q, avo i.d i nqjvove r f Lows ' ") var pI, pO, ql, qO : integer; l. begin . p l, := p div ml;pO := p mod ml; ql := q div ml;ql :=.q mod ml; Mult := (((pO"ql + pI"qO) mod ml)"ml + pO"qO) mod m; end;

Remark 8.1.2 This method of simulation of the exponential law is a particular case of the so-called 'inverse distribution function' method (for this matter see Exercise 45). ' Simulation ofa Poisson random variable

"

o

-

A Poisson random variable is a variable with values in N such that

P(X

An

= n) = e->'" n.

ifn 2: O.

164

Simulation and algorithms for financial models

We have seen in Chapter 7 that if (Ti )i>l -iS a sequence of exponential random variables with parameter A, then the law oeNt = L:n>l nl {Tl +..+Tn9l 's are independent random variables following the uniform law on [0, 1). N: can be written as

N:

= Lnl{U1U2 ...Un+l~e-A
This leads to the following algorithm to simulate a Poisson random variable. : integer;

165

Z is therefore a Gaussian vector with zero-mean and a variance matrix equal to the identity. The law of the vector Z is the law of n independent standard normal variables. The law of the vector X = m + AZ can then be simulated in the following manner: • Derive the square root of the matrix I', say A. • Simulate n independent standard normal variables G = (gl' ... , gn)' • Compute m

n:2:1

function poisson(larnbda : real) var u : real; n : integer; begin a := exp(-larnbda); u := Random; . n := 0; while u > a do begin u : = u * Random; n := n + 1 end; Poisson := n end;

Simulation andfinancial models

+ AG.

Remark 8.1.3 To derive the square root of I', we may assume that A is uppertriangular; then there is a unique solution to the equation A x t A = r. This method of calculation of the square root is called Cholesky's method (for a complete algorithm see Ciarlet (1988)). 8.1.4 Simulation of stochastic processes

For the simulation of laws not mentioned above or for other methods of simulation of the previous laws, one may refer to Rubinstein (1981). Simulation of Gaussian vectors Multidimensional models will generally involve Gaussian processes with values in IR.". The problem in simulating Gaussian vectors (see Section A.l.2 of the Appendix for the definition of a Gaussian vector) is then essential. We give a method of simulation for this kind of random variables. We will suppose that we want to simulate a Gaussian vector (Xl,' .. , X n) whose law is characterised by the vector of its means m = (mI, ... , m n ) = (E(X1 ) , ... , E(Xn)) and its variance matrix r = (O'ij h~i~n.1~j~n whererr., = E(XiX j) - E(Xi)E(Xj). The matrix r is positive definite and we will assume that it is invertible. We can find the square root of I', in other words a matrix A, such that A x t A = r. As r is invertible so is A, and we can consider the vector Z = A-I (X - m). It is easily verified that this vector is a Gaussian vector with zero-mean. Moreover, its variance matrix is given by

The methods delineated previously enable us to simulate a random variable, in particular the value of a stochastic process at a given time. Sometimes we need to know how to simulate the whole path of a process (for example, when we are studying the dynamics through time of the value of a portfolio of options, see Exercise 47). This section suggests some simple tricks to simulate paths of processes. Simulation of a Brownian motion We distinguish two methods to simulate a Brownian motion (Wdt>o. The first one consists in 'renormalising' a random walk. Let (Xi)i>O be a ~equence of independent, identically distributed random walks with law-P (Xi = 1) = 1/2, P (Xi = ,-:-~) = 1/2. Then we have E (Xi) = 0 and E (Xl) = 1. We set Sn = Xl + .,. + X n; then we can 'approximate' the Brownian motion by the process (X;')t:2:o where

X;'

= JnS[nt 1

where [x) is the largest integer less than or equal to x. This method of simulation of the Brownian motion,is partially justified in Exercise 48. In the second method, we notice that, if (gi)i:2:0 is a sequence of independent standard normal random variables, if t1t > 0 and if we set

So = 0 ,{ Sn+l - Sn = g':' then the law of (ViSJ,So, ViSJ,Sl,' .. , ViSJ,sn) is identical to the law of

o

~

(W o, W~t, W2~t, ... , Wn~t).

The Brownian motion can be approximated by X;' = ViSJ,S[t/ ~tl'

Simulation and algorithms for financial models

166

Simulation and financial models

167

Simulation ofstochastic differential equations

An application to the Black-Scholes model

There are many methods, some of them very sophisticated, to simulate the solution of a stochastic differential equation; the reader is referred to Pardoux and Talay (1985) or Kloeden and Platen (1992) for a review of these methods. Here we present only the basic method, the so-called 'Euler approximation' . The principle is the following: consider the stochastic differential equation

In the case of the Black-Scholes model, we want to simulate the solution of the equation

{

=

Xo

ex.

x

b(Xt)dt + a(Xt)dWt.

We discretise time by a fixed mesh b.t. Then we can construct a discrete-time process (Sn)n~O approximating the solution of the stochastic differential equation at times nb.t, setting

{;;t

Two approaches are available. The first consists in using the Euler approximation. We set

= x SO { Sn+l Sn(1 + rb.t + agn.,fl;i) , and simulate X; by Xl' = S[t/ ~t). The either method consists in using the explicit expression of the solution

x, =

x

So { Sn+l - Sn

{b(Sn)b.t

+ a(Sn)

(W(n+l)~t "- Wn~t)} .

If Xl' = S[t/~t), (Xl')t~O approximates (Xt)t~O in the following sense:

Theorem 8.1.4 For any T

t<S.T

s; = z exp ((, .; ,

C T being a constant depending only on T. A proof of this result (as well as other schemes of discretisation of stochastic differential equations) can be found in Chapter 7 of Gard (1988) . The law of the sequence (W(n+l)~t - Wn~t)n>O is the law of a sequence of independent normal random variables with zero-mean ana variance b.t. In a simulation, we substitute gn.,fl;i to (W(n+l)~t - Wn~t) where (gn)n~O is a sequence of independent standard normal variables. The approximating sequence (S~)n~O is in this case defined by

=

x S~

xexp (rt -

We always approximate X, by Xl'

q'

/2)nLlt +

q~t, g,) .

(8.2)

= S[t/~t).

Remark 8.1.6 We can also replace the Gaussian random variables gi by some Bernouilli variables with values + 1 or -1·with probability 1/2 in (8.2); we obtain a binomial-type model close to the Cox-Ross-Rubinstein model used in Section 5.3.3 of Chapter 5.

Simulation of models with jumps We have investigated in Chapter 7 an extension of the Black-Scholes model with jumps; we describe now a method. to simulate this process. We take the notations and the hypothesis of Chapter 7, Section 7.2. The process (Xt)t~O describing the dynamics, of the asset is . ., ,

x, : ;: : x

+ b.t b(S~) + a(S~)gn.,fl;i.

Remark 8.1.5 We can substitute to the sequence of independent Gaussian random variables (gi)i~O a sequence of independent random variables (Ui)i~O, such that P(Ui = 1) = P(Ui = -1) = 1/2. Nevertheless, in this case, it must be noticed that the convergence is different from that found in Theorem 8.1.4. There is still a theorem of convergence, but it applies to the laws of the processes. Kushner (1977) and Pardoux and Talay (1985) can be consulted for some explanations on this kind of convergence and many results on discretisation in law for stochastic differential equations.

~2t + awt)

and simulating the Brownian motion by one of the methods presented previously. In the case where we simulate the Brownian motion by .,fl;i ~:::l gi, we obtain

>0

E (sup IXl' -.Xt12) ~ CTb.t,

~t(rdt + adWt).

:

}1 (i + ~j) N'

. (

)

e(Il-

CT2 /

2

.

)t+CTW, ,

(8.3)

where (Wt)t~O is a standard Brownian motion, (Nt)t>o is a Poisson process with intensity and (Uj) j~ 1 is a sequence of independent, identically distributed random variables, with values in ] -:- 1, +oo[ and law lJ(dx). The a-algebras generated by (W~)t~O' (Nt)t~o, (Uj)j~l are supposed to be independent. To simulate this process at times rust, we notice that

.x,

Xn~t ~ X If we noteYk

X

(X~t/x)

x

(X2~t/X~t)

x ···x

(Xn~t/X(n-l)~t).

= (Xk~t/X(k-l)~t), we can prove, from the properties of (Nt)t~o,

Simulation and algorithms for financial models

168

(Wdt2:o and (Uj ) j2:l that (Yk h2:l is a sequence of independent random variables with the same law. Since Xn~t = xYl . , . Yn , the simulation of X at times

Some useful algorithms

169

Cl

n.6.t comes down to the simulation of the sequence (Yk h 2: l . This sequence

C2

being independent and identically distributed, it sufficesto know how to simulate Yl = X~tlx. Then' we operate as follows:

C3 C4

• We simulate a 'standard Gaussian random variable g.

1

N(x) ::::: 1 - 2(1

• We simulate a Poisson random variable with parameter )".6.t: N. • If N = n, we simulate n random variables following the law J.L(dx): Ul

, ... ,

8.2 Some useful algorlthms In this section, we have gathered some widely used algorithms for the pricing of options. 8.2.1 Approximation ofthe distributionfunction of a Gaussian variable We saw in 'Chapter 4 that the pricing of many classical options requires the calculation of ,,2 dx N(x) = P(X :::; x) = e--: T rs:':

I

'"

-00

a

y 27r

where X is standard Gaussian random variable: Due to the importance of this function in the pricing of options, we give two approximation formulae from Abramowitz and Stegun (1970) , The first approximation is accurate to 10- 7 , but it uses the exponential function, Ifx>O p -' 0.231641900 b1 = 0.319381530 b2 = -0.356563782 b3 = 1.781477937 -b 4 ~ 1.821255978 b5 1.330274429

,

N(x) ::::: 1 -

1

= ,,2

rrce-T (bit

y27r

0.196854 0.115194 0.000344 0.019527 '

+ CIX + C2x2 + C3X,3 + C4x4)-4.

8.2.2 Implementation of the Brennan and Schwartz method The following program prices an American put using the method described in Chapter 5, Section 5.3.2: we make a logarithmic change of variable, we discretise the parabolic inequality using a totally implicit method and finally solve the inequality in infinite dimensions using the algorithm described on page 116, CONST PriceStepNb = 200;' TimeStepNb = 200; Accuracy = 0,01; DaysInYearNb = 360;

.

t

= = = =

Un'

All these variables are assumedto be independent. Then, from equation (8.3), it is clear that the law of

is identical to the law 'of Yl

>0

opposed to an exponential. If x

1/(I+px)

+ b2t 2 + b3t3 + b4t4 + b5t5v ) .

The second approximation is accurate to 10- 3 but it involves only a ratio as

TYPE Date = INTEGER; Amount = REAL; AmericanPut = RECORD ContractDate : Date; (* in days *) MaturityDate : Date; (* in days *) StrikePrice : Amount; END; vector = ARRAY[l, ,PriceStepNbI OF REAL; Model = RECORD r REAL; (* annual riskless interest rate sigma REAL; (* annual volatility *) xO i REAL; (* initial value of the SDE *) END;

*)

FUNCTION PutObstacle(x : REAL;Opt : AmericanPut) :'REAL; VAR u : REAL; BEGIN u := Opt,StrikePrice - exp(x); IF u > 0 THEN PutObstacle := u ELSE PutObstacle := 0,0; END;

FUNCTION Price(t : Date; x : Amount; option : AmericanPut; 'model : Model) : REAL;

(* prices the 'option' for the 'model' at time 't' if the price o~th~ underlying at 'this time is "x ".

*) VAR

Obst,A,B,C,G : vector; alpha, beta ;gamina, h , k , VV, temp, r , Y» del t a , Time', 1 : REAL; Index,PriceIndex,TimeIndex : INTEGER; BEGIN '"" Time := (option,MaturityDate - ~) / Days InYearNb; k := Time / TimeStepNb; r := model.r;

Simulation and algorithms for financial models

170 vv :=

~odel.sigma

171

I::

* model. sigma; + abs(r - vv / 2)

1 := (model.sigma * sqrt(Temps) * sqrt(ln(l/Accuracy» Time) ; h := 2 * 1 / PriceStepNb; writeln(1:5:3.'·' ,In(2) :5:3); alpha := k * (- vv / (2.0 * h * h) + (r - vv / 2'.0) / beta := 1 + k * (r + vv / (h * h»; gamma := k * (- vv / (2.0 * h * h) - (r - vv / 2.0) / FOR PriceIndex:=l TO PriceStepNb DO BEGIN A[PriceIndex] := alpha; B[PriceIndex] := beta; C[PriceIndex] := gamma;

Exercises

*

f(x)dx. We setF(u) = f(x)dx. Prove thatifU is a uniform random variable oo on [0,1], then the law of F-l (U) is f(x)dx. Deduce a method of simulation of X. Exerci.se 46 We model a risky asset S, by the stochastic differential equation

as, { So

(2.0 * h»; (2.0 * h»;

=

x,

where (Wt)t>o is a standard Brownian motion, a the volatility and r is the riskless interest rate. Propose a method of simulation to approximate

END;

B[l] B[PriceStepNb] G[PriceIndex]

:= := :=

beta + alpha; beta + gamma; 0.0;

B[PriceStepNbl := B[PriceStepNb]; FOR PriceIndex:=PriceStepNb-l DOWNTO 1 DO B[PriceIndex] := B[Pri'ceIndex] - C[PriceIndex] * A[PriceIndex+l] / B[PriceIndex+l] ; FOR PriceIndex:= i TO PriceStepNb DO A[PriceIndex] := A[PriceIndex] /" B [PriceIndex] ; FOR PriceIndex:= 1 TO PriceStepNb - 1 DO C[PriceIndex] := C[Pricelndex] / B[PriceIndex+l] ; y := In(x); FOR PriceIndex:=l TO PriceStepNb DO Obst(PriceIndex] := PutObstacle(y - 1 + PriceIndex * h , option ); . FOR PriceIndex:=l TO PriceStepNb DO G[PriceIndex] := ,0bst[priceIndex]; FOR TimeIndex:=l TO TimeStepNb DO BEGIN FOR PriceIndex := PriceStepNb-l DOWNTO 1 DO G[PriceIndex) := G[PriceIndex] - '. C[PriceIndex] * G[PriceIndex+1];

G[l] := G[l] / at i i , FOR ,PriceIndex:=2 TO PriceStepNb DO BEGIN G [PriceIndex] : = G [PriceIndexl / B [PriceIndex] - A[Pric'~IndexJ * G [Price Index-l] ; temp := Obst[PriceIndex]; IF G[PriceIndex] < temp THEN G[PriceIndex] := temp; END; END; Index := PriceStepNb DIV 2; delta := (G[Indice+l] - G[Index]) / h; Prix':= G[Index]+ delta*(Index * h - 1); END;

Give an interpretation for the final value in terms of option. Exercise 47 The aim of this exercise is to study the influence of the hedging frequency on the variance of a portfolio of options. The underlying asset of the options is described by the Black-Scholes model

as, { So

x,

(Wtk~o represents a standard Brownian motion, a the annual volatility and r the riskless interest rate. Further on we will fix r = lO%jyear, a = 20%/ Jyear = 0.2 and x = 100. Being 'delta neutral' means that we compensate the total delta of the portfolio by trading the adequate amount of underlying asset. In the following, the options have 3 months to maturity and are contingent on one unit of asset. We will choose one of the following combinations of options:

• Bull spread: long a call with strike price 90 (written as 90 call) and short a 110 call with same maturity. • Strangle: short a 90 put and short a 110 call. • Condor: short a 90 call, long a 95 call and a 105 call and finally short a 110 call. • Put ratio backspread: short a 110 put and long 3 90 puts.

8.3 Exercises

..

.:

Exercise 44 Let X and Y be two standard Gaussian random variables; derive! the joint law of (JX2 + Y2,arctg(Y/X)). Deduce that, if U1 and U2 are two independent uniform random variables on [0,1], the random variables -210g(Ul ) cos(27l'U2 ) and -210g(Ud sin(27l'U2 ) are independent and folIowa standard Gaussian law. . 0

J

J

Exercise 45 Let f be a function from JR to JR, such that f(x) > 0 for all x, and such that f(x)dx = 1. We want to simulate a random variable X with-law

r:

First we suppose that f.L = r . Write a program which: • Simulates the asset described previously. • Calculates the mean and variance of the discounted final value of the portfolio in the following cases: (

,

We ,90 not hedge: we sell the option, get the premium, we wait for three months, we take into account the exercise of the option sold and we evaluate the portfolio. We hedge immediately after selling the option, then we do nothing.

172

Simulation and algorithms for financial models

_ We hedge immediately after seIling the option, then every month. _ We hedge immediately after selling the option, then every 10 days . . _ We hedge immediately after selling the option, then every day.

Appendix

Investigate the influence of the discretisation frequency. Now consider the previous simulation assuming that J.L =I r (take values of J.L bigger and smaller than r). Are there arbitrage opportunities? Exercise 48 We suppose that (Wt)t>o is a standard Brownian motion and that (Ui)i>l is a sequence of independent random variables taking values +1 or -1 with probability 1/2. We set Sn = Xl + .,. + X n. 1. Prove that, if

X? = S[ntj/"fii, X? converges in law to Wt.

2. Let t and s be non-negative.using the fact that the random variable X?+s - X? is independent of X?, prove that the pair (X?+s' X?) converges in law to

A.I Normal random variables

(Wt+ s, Wt). 3. IfO < tl < ... < t p, show that (X~, . . . ,X~) converges in law to (W t l , · .. , Wt p ) .

In this section, we recall the main properties of Gaussian variables. The following results are proved in Bouleau (1986), Chapter VI, Section 9.

A.i.i Scalar normal variables ~

I

~

A real random variable X is a standard normal variable if its probability density function is equal to n(x)

= _1_ exp .J2;

2)

(_ X

•

2 If X is a standard normal variable and m and a are two real numbers, then the variable Y = m + a X is normal with mean m and variance 0'2. Its law is denoted by N( m, 0'2) (it does not depend on the sign of a since X and - X have the same law). If a i= .0, the density of Y is

(x-m)2) 20'2'

_1_ exp (

I

J27fO'2

If a = 0, the law of Y is the Dirac measure in m and therefore it does not have a density. It is sometimes called 'degenerate normal variable'. If X is a standard normal variable, we can prove that for any complex number z, we have . .

E

(e z X )

= e4 . u2

o

Thus,the characteristic function of X is given by ifJx(u) = e- / 2 and for Y, ifJy(u) = eiUTne-u2(j2/2. It is sometimes useful to know that if X is a standard normal variable, we have P(IXI > 1,96,..) = 0,05 and P(IXI > 2,6 ...) = 0,01. For-large values of t > 0, the following approximation is handy: . 1 P(X > t) = _rrc , v 27f

1

00

t

e'--x

2/2dx

$

1 _rrc tv 27f

1

00

t

2

/2

xe- x2/2dx = _e__ . _t

.

t.J2;

174

Appendix

Finally, one.should know that there exist very good approximations of the cumulative normal distribution (cf. Chapter 8) as well as statistical tables.

=

Definition Ad.I A random variable X (XI, ... ,Xd) in lR d is a Gaussian vector if for any sequence ofreal numbers at. ... , ad, the scalar random variable 2:~=1 aiX is normal. The components Xl •...• X d of a Gaussian vector are obviously normal, but the fact that each component of a vector is a normal random variable does not imply that the vector is normal. However. if Xl, X 2 • . . . , X d are real-valued, normal. independent random variables. then the vector (Xl, ... ,Xd) is normal. The covariance matrix of-a random vector X (Xl, ... , X d ) is the matrix I'(X) = (aij h~i,j~d whose coefficients are equal to

=

= cov(Xi, Xj) = E [(Xi -

E(Xi))(Xj - E(Xj))].

It is well known that if the random variables Xl'•... , X d are independent. the matrix I'(X) is diagonal. but the converse is generally wrong. except in the Gaussian case: Theorem A.I.2 Let X (Xl,' .. , X d) be a Gaussian vector in lRd. The random variables Xl, ... , X d are independent if and only if the covariance matrix X is . , diagonal.

=

The reader should consult Bouleau (1986), Chapter VI. p. 155, for a proof ofthis result.

RemarkA.I.3 The importance of normal random variables in modelling comes partly from the central limit theorem (cf. Bouleau (1986), Chapter VI,I, Section 4). The reader ought to refer to Dacunha-Castelle and Dufto (1986) (Chapter 5) for problems of estimation and to Chapter 8 for problems of simulation.

A.2 Conditional expectation A.2.I Examples of a-algebras .Let us consider a ~~Race (P,A) and a P~.Q'I. B 2 • • • • , ~n; with n events in A. The set B containing the elements of ~ which are ei~pty or that can be written as Bit U B i2 U··· U B ik, where i~, ... , ik E {I, ... , n}, is a finite sub-a-alg;bra of It is-the a-algebra generated by the sequence of B; " Conversely, to any finite s.!Jb~a-=algebraI3 of .A, we can associ~ition (B I , ... , B n ) of 0 where is·g~.!'erated by theelements B, of A: B, are the non_.-"""'-~ ----. ---. ~ ~ ,........ empty elements ofB whichcontain onIy-tnemselves and the empty set. They are 'called--atoms of B. There is a one-to-one mapping-fffiffi-tfie ser'OCfinite sub-aalgeoras ofAonto the set of partitions of 0 by elements of A. One should notice that if B is a sub-a-algebra of A, a map from 0 to lR (and its Borel a-algebra) is B-measurable if and only. if it is constant on each atom of B.

A.

.--.. --:""_..

_-~._-----

B .,.r:---.-.-.-

175

Let us now consider a random variable X defined o!!-(O, A) with values in a m~able ~~~.JE, E). Th~ ~~l~bra generated ~X isthesmallest a-algebra f~~~~.Ich X ~~~~~~~ It IS denoted by a(X). It isolWiouslyincluaea-in A .~--.

and It IS easy to show that

A.I.2 Multivariate normal variables

aij

Conditional expectation

'

-----

~----=----- ~~--_.~

-.

-'-'---'-~(X)~ {A E AI3B E E,A =

X-I(B) = {X E B}}.

We can prove that a random variable Y from (0, A) to (F,:F) is a(X)-measurable if and only if it can be written as . .

Y = foX, where fJs a me~~b~~~aP._~~~(E~E) toJF,:F). (cf. Bouleau (1986), p. 101-102). In other words, a(X)-measurable random variables are the measurable functions of X.

A.2.2 Properties. ofthe co~dition~l expectation Let (0, A, P) be a probability space and B a a-algebra in~luded in A. The definition of the conditional expectation is based on the following theorem (refer to Bouleau (1986), Chapter 8):

!heorem A.2.1 For any real integrable random variable X, there exists a real Integrable Btmeasurable random variable Y such that VB E B

E(XI B) = E(YI B).

If Y is another randomvar~able with these properties then Y = Y P a.s. Y is th~ cond~tional expectation of X given B and it is denoted by E(XIB). If B IS a finite sub-a-algebra, with atoms B I , ... , B n , . . . E(XIB)

=L

E(XIB;)/P(Bi)IB;,

where we sum on the atoms with strictly positive probability..Consequently, on each atom B i, E(XIB) is the mean value of X on Bi, As far as the trivial a-algebra is concerned (B = {0, OJ), we have E(XIB) = E(X). . . The ~omputationsinvolving conditional expectations are based on the following properties:

~!.; If £~~~l![~ble,E(XIB) = X, a.s. .~ E (E (XIB)) = E (X). 3. For any bounded! B-measurablerandom variable Z,E (ZE(XIB)) 4. L i n e a r i t y : ' . ''-' ~

E,: (~~ I

+ JLYIB) =AE (XIB) + JLE (YIB)

= E(ZX). ,'--

a.s.

~_

5. Positivity: if X 2: 0, then E(XIB) 2: a a.s. and more generally, X 2: Y E(XIB) .~ E(YIB) a.s. It follows from this property that

~

IE (XIB)I $ E (IXIIB) a.s.

=>

, 176

Appendix

Conditional expectation

and therefore II E(XIB)II£lCfl) ~ IIXII£lCfl). 6. If C is a sub-a-algebra of B, then

Proposition A.2.S Let us consider a B-measurable random variable X taking values in (E, E) and Y, a random variable independent of B with values in (F, F). For any Borelfunction non-negative (or bounded) on (E x F, E I8i F), the function cp defined by ,

E (E (XIB) IC) = E (XIC) a.s. 7. If Z is B-measurable and bounded, E (ZXIB) = ZE (XIB) a.s. 8. If X is independent of B then E (XIB) = E (X) a.s. The converse property is not true but we have the following result. Proposition A.2.2 Let X be a real random variable. X is independent of the a-algebra B if and only if VU'E IR

E

(eiUXIB) ='E (e iuX) a:!.

Vx E E

E

(eiUX~) = E P(B) -, ~

{l(X:~p~~))

((x, Y))

= cp(X) a.s.

In other words, under the previous assumptions, we can compute E ( (X, Y) IB) as if X was a-constant. ' .., .

(A.l)

Proof. Let us denote by P y the law of Y; We have

cp(x) =

.

i

(x, y)dPy(y)

and the measurability of cp is a consequence of the,Fubini theorem. Let Z be a non-negative B-measurable random variable (for example Z = IB, with B E B). If we denote by P X,Z the law of (X, Z), it follows from the independence between Y and (X, Z) that,

E ((X, Y)Z) =.E (f(X)),

for any bo~nded Borelfunction j, hence the independence.

=;: E

E ( (X, Y)IB)

This equality means that the characteristic function of X is identical under measure P and measure Q where thedensity of Q with respectto P is equal to IB /P(B). The equality of characteristic functions implies the equality of probability laws , and consequently E

cp(x)

is a Borelfunction on (E, E) and we have

Proof. Given the Property 8..above, we just need to prove that (A.l) implies that X is independent of B. If E (e iuX IB) = E (e iUX) then, by definition of the conditional expectation, for all B E B, E (e iuX IB) = E (e iuX) P(B). If P(B) =j:. 0, we can write '

(e iuX)

177

= / /

o

/

Remark A.2.3 If X is square integrable, so is E(XIB), and E(XIB) coincides with the orthogonal projection of X on L2(n, B, P), which is a closed subspace of L 2 (n, A,P), together with the scalar product (X, Y) H E(XY) (cf. Bouleau (1986), CnapterVIII, Section 2). The conditional expectation of X given B is the least-square-best B-measurable predictor of X. In particular, if B is the a-algebra generated by a random variable €, the conditional expectation E(XIB) is noted E(XI€), and it is the best approximation of X by a function of €, since a(€)measurable random variables are the measurable functions of €- Notice that by Pythagoras' theorem, we know that IIE(XIB)IIL2Cfl) ~ IIXII£2Cfl).

(x, y)zdPx,z(x,

( / (x, y)dPy(y))

z)dPy(y) zdPx,z(x, z)

= / cp(x)zdPx,z(x,z) = E (cp(X)Z) , which completes the proof.

0

Remark A.2.6 In the Gaussian case, the computation of a conditional expectation is particularly simple. Indeed, if (Y, Xl, X 2 , ..• ,Xn ) is a normal vector (in n I IR + ) , the conditional expectation Z' == E (YIX I , . . . ,Xn ) has the following form " '

Remark A.2.4 We can define E(XIB) for any non-negative random variable X (without integrability condition). Then E(X Z) = E (E(XIB)Z), for any Bmeasurable non-negative random variable Z. The rules are basically the same as in the integrable case (see Dacunha-Castelle and Duflo (1982), Chapter 6).

n

Z

= Co + LCiXi, i=l

:J

where c, are real constant numbers. This means that the function of Xi which approximates.Y in the least-square sense is linear. On top of that, we can compute Z by p~?jecting the random variable Y in L 2 on the linear subspace generated by I and the X/s (cf. Bouleau (1986), Chapter 8, Section 5).

A.2.3 Computations of conditional expectations , The following proposition is crucial and is used quite often in this book. I

/'\

Appendix

178

A.3 Separation of convex sets . In this section, we state the theorem of separation of convex sets that we use in the first chapter. For more details, the diligent reader can refer to Dudley (1989) p. 152 or Minoux (1983).

References

Theorem A.3.1 Let C be a closed convex set which does not contain the origin. Then there exists a real linear functional ( defined on IRn and 0: > 0 such that '
In particular, the hyperplane ((x)

((x)

2:

0:.

= 0 does not intersect C.

Proof. Let>' be anon-negative real number such that the closed ball B(>') with centre at the origin and radius>' intersects C. Let Xo be the point where the map x ~ Ilxll achieves its minimum (where 11·11 is the Euclidean norm) on the compact set C n B(>'). It follows immediately that ., '
IIxll 2: II xoll·

The vector Xo is nothing but the projection of the origin on the closed convex set C. If we consider x E C, then for all t E [0,1], Xo + t(x - xo) E C, since Cis convex. By expanding the following inequality " .

IIxo + t(x - xo)I1 2 2: Il xoll 2, it yields xo.x 2: IIxoll2 > 0 for any x E C, where xo.x denotes the scalar product of Xo and x. This completes the p r o o f . ' 0

Theorem A.3.2 Let us consider a compact convex set K and a vector subspace Vof IRn . If V and K are disjoint, there exists a linear functional ( defined on IRn , satisfying the following conditions: J. ' O. 2. ' 0 such that

'
((x)

2:

0:.

Hence

'
2:

0:.

For a fixed y, we can apply (A.2) to >.z, with>. E IR to obtainxvz E V, ((z) thus '
(A.2)

= 0, 0

Abramowitz, M. and LA. Stegun (eds), Handbook ofMathematical Functions, 9th printing, 1970. Artzner, P. and F. Delbaen, Term structure of interest rates: The martingale approach, Advances in Applied Mathematics 10 (1989), pp. 95-129, Bachelier, L., Theorie de la speculation, Ann. Sci. Ecole Norm. Sup., 17 (1900), pp. 21-86. Barone-Adesi, G. and R. Whaley, Efficient analytic approximation of American option values, J. of Finance, 42 (1987), pp. 301-320. Bensoussan, A., On the theory of option pricing, Acta Appl. Math., 2 (1984) pp. 139-158. Bensoussan, A. and J.L. Lions, Applications des inequations variationnelles en contriile . stochastique, Dunod, 1978. Bensoussan, A. and J.L. Lions Applications of Variational Inequalities in Stochastic Control, North-Holland, 1982. Black, F. and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), pp. 635-654. Brennan, MJ. and E.S. Schwartz, The valuation of the American put option, J. of Finance, 32, 1977, pp. 449-462. Brennan, MJ. and E.S. Schwartz, A continuous time approach to the pricing of bonds, J. of Banking and Finance, 3 (1979), pp. 133-155. Bouleau, N., Probabilites de l'Ingenieur, Hermann, 1986. Bouleau, N., Processus Stochastiques et Applications, Hermann, 1988. Bouleau, N. and D.Lamberton, Residual risks and hedging strategies in Markovian markets, Stoch. Proc. and Appl. , 33 (1989), pp. 131-150. . CERMA, Sur les risques residuels des strategies de couverture d'actifs conditionnels, Comptes Rendus de l'Academie des Sciences, 307 (1988), pp. 625-630. Chateau, 0., Quelques remarques sur les processus a accroissements independants et stationnaires, et la subordination au sens de Bochner, These de l'Universite Paris VI, 1990. Ciarlet, P.G., Une Introduction aI 'analyse numerique matricielle et al'optimisation, Masson, 1988. Ciarlet, P.G. and J.L. Lions, Handbook of Numerical Analysis, Volume I, Part I (1990), North Holland. . Courtadon, G., The pricing of options on default-free bonds, J. of Financial and Quant. Anal., 17 (1982), pp. 301-329.

180

References

References

Cox, J.C., J.E. IngersoU and S.A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), pp. 385-407. Cox, J.e. and M. Rubinstein, Options Markets, Prentice-Hall, 1985. Dalang, Re., A. Morton and W. Willinger, Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stochastics and Stochastics Reports, vol. 29, (2) (1990), pp. 185-202. Dacunha-CasteUe, D. and M. Duflo, Probability and Statistics, volume 1, Springer-Verlag, 1986. Dacunha-Castelle, D. and M. Duflo, Probability and Statistics, volume 2, Springer-Verlag, 1986. Delbaen, E and W. Schachennayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), pp. 463-520. Dudley, RM., Real Analysis and Probability, Wadsworth & Brooks/Cole, 1989. Duffie, D., Security Markets, Stochastic Models, Academic Press, 1988. EI Karoui, N., Les aspectsprobabilistes du controle stochastique, Lecture Notes in Mathematics 876, pp. 72~238; Springer-Verlag, 1981. EI Karoui, N. and J.e. Rochet, Apricing formula for options on coupon-bonds, Cahier de . , recherche du GREMAQ-CRES, no. 8925,1989. EI Karoui, N. and M.C. Quenez, Dynamic programming and pricing of contingent claims in an incomplete market, S.I.A.M. J. Control and Optimization 33 (1995), pp. 29-66. Follmer, H. and D. Sondermann, Hedging of non-redundant contingent claims, Contributions to Mathematical Economics in Honor df Gerard Debreu, W.Hildebrand and A 'Mas-Colell (eds), North-Holland, Amsterdam, 1986. Follmer, H. and D. Sondermann, Hedging of contingent.claims under incomplete information, Applied Stochastic Analysis, M.H.A. Davis and R.I. Elliott (eds), Gordon & . Breach, 1990. . Follmer, H. and M. Schweizer, "Hedging of contingent claims under incomplete information", Applied Stochastic Analysis, M.H.A Davis and R.I. Elliott (eds),'Stochastics ',' Monographs, vol. 5; Gordon & Breach (1991): pp. 389-414. Fri~dman, A., Stochastic Differential Equation~and Applications, Academic Press, 1975. Gard, T., Introduction to Stochastic Differential Equations, Marcel Dekker, 1988. Garman, M.B. and S.W. Kohlhagen, Foreign currency option values, J. of International Money and Finance, 2 (1983), pp. 231-237. Gihman, 1.1. and A.V. Skorohod, Introduction ala Theorie des Processus ALeatoires, Mir, 1980. Glowinsky, R, ri, Lions and R Tremolieres, Analyse numerique des inequations variationnelles, Dunod, 1976. . Harrison, M.J. and D.M. Kreps, Martingales and arbitrage in multiperiod securities markets, J. of Economic Theory, 29 (1979), pp.381-408. Harrison, M.J. and S.R. Pliska, Martingales and stochastic integrals in the theory of conti. nuous trading, Stochasti~ Processes and their Applications, 11 (1981), pp, 215-260. Harrison, M.I. and S.RPliska; A stochastic calculusmodel of continuous trading: complete markets, Stochastic Proces;es and their Applications, 15 (1983), pp, 313-316. Heath, D., A. Jarrow and A Morton, Bond pricing and the term structure of interest rates, preprint, 1987. Ho, T.S. and S.B. Lee, Term structure movements and pricing interest rate contingent claims, J. of Finance, 41 (1986), pp. 1011-1029.

,

181

Huang, C.E and RH. Litzenberger, Foundationsfor Financial Economics, North-Holland, . 1988. Ikeda, N. and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland,1981. Jaillet, P., D. Lamberton and B. Lapeyre, Variational inequalities and the pricing of American options, Acta Applicandae Mathematicae, 21 (1990), pp. 263-289. Jamshidian, E, An exact bond pricing formula, Journal ofFinance, 44 (1989), pp. 205-209. Karatzas, I., On the pricing of American options, Applied Mathematics and Optimization, 17 (1988),pp. 37-60. . Karatzas, I. and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1988. Kloeden, P.E. and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer- Verlag, 1992. Kushner, H.J., Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, 1977. Knuth, D.E., The Art of Computer Programming, Vol. 2: Seminumerical algorithms, Addison- Wesley, 1981. Lamberton, D. and G. Pages, Sur I'approximation des reduites, Annales de l'/HP, 26 (1990), pp.331-355. L'Ecuyer, P., Random numbers for simulation, Communications of the ACM, October 1990, vol. 33 (10). MacMillan, L., Analytic approximation for the American put price, Advances in Futures and Options Research, 1 (1986), pp. 119-139. Merton, R.C., Theory of rational option pricing, Bell 1. of Econ. and Management Sci., 4 (1973), pp. 141-183. Merton, RC., Option pricing when underlying stock returns are discontinuous, J. ofFinancial Economics, 3 (1976), pp. 125-144. Minoux, M., Programmation mathematique, 2 vols, Dunod, 1983. Morton, AJ., Arbitrage and Martingales, Ph.D. thesis, Cornell University, 1989. Neveu, J., Martingales a temps discret, Masson, 1972. Pardoux, E. and D. Talay, Discretization and simulation of stochastic differential equations, Acta Applicandae Mathematicae, 3 (1985), pp. 23-47. Raviart, P.A and J.M. Thomas, Introduction a l'analyse numerique des equations aux derivees partielles, Masson, 1983. Rogers, L.C.G. and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 2, Ito Calculus, John Wiley & Sons, 1987. Revuz, A. andM. Yor, Continuous Martingale Calculus, Springer-Verlag, 1990. Rubinstein, RY., Simulation and the Monte Carlo Method, (1981), John Wiley and sons. Schaefer, S. and E.S. Schwartz, A two-factor model of the term structure: an approximate analytical solution, J. of Finance and Quant. Anal., 19 (1984), pp. 413-424. Schweizer, M., Option hedging for semi-martingales, Stoch. Proc. and Appl., 37 (1991), pp. 339-363. Schweizer, M., Mean-variance hedging for general claims, Annals ofApplied Probability, 2 (1992), pp.171-179. Schweizer, M., Approximating random variables by stochastic integrals, Annals of Probability, 22 (1993), pp. 1536-1575.

References

182

Schweizer, M., On the minimal martingale measure and the Follmer-Schweizer decomposition, Stochastic Analysis and Applications, 13 (1994), pp. 573-599. Sedgewick, R., Algorithms, Addison-Wesley, 1987. Shiryayev, A.N., Optimal Stopping Rules, Springer-Verlag, 1978. Stricker, c., Arbitrage et lois de martingales, Ann. Inst. Henri Poincarre, 26 (1990), pp. 451-460. Williams, D., Probability with Martingales, Cambridge University Press, 1991. Zhang, XL, Numerical analysis of American option pricing in. a jump diffusion model, CERMICS, Soc.Gen , 1994.

Index'

Adapted,4 Algorithm Brennan and Schwartz, 116, 169 Cox, Ross and Rubinstein, 117 American call pricing, 25 American option price, 11 American put hedging, 27 pricing, 26 Arbitrage, viii Asset ' financial, vii, 1 riskless, 1, 122 risky, 1 underlying, vii Atom, 174 Attainable, 8

.....

"

o

Bachelier, vii Bessel function, 131 Black, vii Black-Scholes formulae, 70 Black-Scholes model, ix, 12 Bond pricing, 124, 129, 132 Bond option pricing, 125,' 136 Brownian motion, vii, 31 Simulation of process, 165

Call, vii pricing, 154 CalIon bond pricing, 133, 138 Complete, 8 Conditional expectation Gaussian case, 177 of a non-negative random variable, 176 orthogonal projection, 176 w.r.t. a random variable, 176 Contingent claim, 8 Continuous-time process, 29 Cox-Ross-Rubinstein model, 12 Crank-Nicholson scheme, 108 Critical price, 77 Delta, 72 Diffusion, 49 Doob decomposition, 21 Doob inequality, 35 Dynkin operator, 99 Equivalent probabilities, 66 Equivalent probability, 6 European call pricing, 70, 74 European option pricing, 68, 154 European put pricing, 70 Exercise price, viii Expectation, 5

Index

184 conditional, 174 Expiration, viii Exponential martingale, 33 Filtration, I, 30 Forward interest rate, 133 Gamma, 72 Girsanov theorem, 66, 77 Greeks delta, 72 gamma, 72 theta, 72 vega, 72 Hedging, viii a can, 14 cans and puts, 71 no replication, 160 of cans and puts, 155-159 Infinitesimal generator, 112 Ito calculus, 42 Ito formula, 42 multidimensional, 47 Ito processes, 43 Law chi-square, 131 exponential, 141, 142 gamma, 142 lognormal, 64 Market complete, 8 incomplete, 141, 160 Markov Property; 54, 56 Martingale, 4 continuous-time, 32 exponential, 47 optional sampling theorem, 33 Martingale transform, 5 Martingales representation, 66 Merton, vii Method

Monte Carlo, 161 Model Black-Scholes, 12,47,63,80 simulation, 167 Cox-Ingersoll-Ross, 129-133 Cox-Ross-Rubinstein, 12 discrete-time, 1 interest rate, 121-139 Vasicek, 127-129 with jumps, 141-160 simulation, 167 Yield curve, 133 Natural filtration, 30 Normal variable, 173 degenerate, 173 standard, 173 Numerical methods Brennan and Schwartz algorithm,116 algorithm of Brennan and Schwartz, 169 Cox, Ross and Rubinstein method, 117 distribution function of a gaussian law, s

168

finite differences, 106 Gauss method, 109 inequality in finite dimension, 116 Mac Millan and Waley, 118 partial differential inequality, 113 Option American, viii Asian, 8 European, viii, ,8 replicable, 68 I Optional sampling theorem; 33 Parity put/call, 13 Partial differential equation numerical solution, 106 on a bounded open set. 102 parabolic, 95, 99 Partial differential inequalities.vl ll, 113 Perpetual put, 75 pricing, 75 Poisson process, 141

Index Portfolio value, 2 Position short, 3 Predictable, 5 Premium, viii Pricing, viii Process continuous-time, 29 Omstein-Ulhenbeck, 52 Poisson, 141 Put, vii partial differential inequalities, 113 pricing, 154 Put/Call parity, ix Radon-Nikodym, 66 Random number generators, 162 Replicating strategy, 14 Scholes, vii Separation of convex sets, 178 Short- selling, 3 Sigma-algebra, 174 Simulation of processes, 165 Black-Scholes model, 167 Brownian motion, 165 model with jumps, 167 stochastic differential equations, 166 Simulation of random variables, 163 exponential variable, 163 Gaussian, 163 Gaussian vector, 164 . Poisson variable, 163 Snell envelope, 18 Stochastic differential equations, 49, 52, 96 Stopped sequence, 18 ' Stopping time, 17,30 hitting time, 34 optimal,20 Strategy, 1 admissible, 3, 68, 125, 151 consumption, 27,73 self-financing, 2, 64, 125, 151 Strike price, vii Submartingale, 4 Supermartingale, 4

185 Theta, 72 Vega, 72 Viable, 6 Volatility, ix, 70 implied,71 Wiener integral, 57 Yield curve, 121, 133 Zero coupon bond, 122