Introduction to stochastic calculus applied to finance, lamberton lapeyre

2.4 Snell envelope and Markov chains 22 2.5 Application to American options Visit the CRC Press Web site at www.crepress.com 2.6 Exercises Ppa P " 23 25 First CRC reprint 2000 ik 3 Brow

Centre for Quantitative Finance Imperial College, London

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and ' Francois Mantion

Centre for Quantitative Finance Imperial College

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vi

4 The Black-Scholes model

4.1 Description of the model :

4.2 Change of probability Representation of martingales

4.3 Pricing and hedging options in the Black-Scholes model

4.4 American options in the Black-Scholes model

4.5 Exercises

5 Option pricing and partial differential equations

5.1 European option pricing and diffusions

5.2 Solving parabolic equations numerically

7.2 Dynamics of the risky asset

7.3 Pricing and hedging options

7.4 Exercises

8 Simulation and algorithms for financial models

8.1 Simulation and financial models

8.2 Some useful algorithms

8.3 Exercises

Appendix

A.1 Normal random variables

A.2 Conditional expectation

A.3 Separation of convex sets

Introduction

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 modelling 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 mathematically more rigorous The theory seems to be advanced enough to attempt to make it accessible to students

Options

Our presentation concentrates on options, because they have been the main motiva- tion in the construction of the theory and still 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 sell a certain amount of a financial asset,

by a certain date, for a certain strike price

The writer of the option needs to specify:

e the type of option: the option to buy is called a call while the option to sell is a

e the underlying asset: typically, it can be a stock, a bond, a currency and so on;,

"

vill Introduction

e the amount of an underlying asset to be purchased or sold;

e 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;

e 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 ofa European call option on a stock, whose price at

time ¢ is denoted by S; Let us call T the expiration date and K the exercise

price Obviously, if K is greater than S, the holder of the option has no interest

whatsoever in exercising the option But, if Sy; > K, the holder makes a profit of

Sr — K by exercising the option, i.e buying the stock for A’ and selling it back

on the market at Sr Therefore, the value of the call at maturity is given by

(Sp — K)4 = max (Sp — K,0)

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 (S7 — A), at maturity At the

time of writing the option, which will be considered as the origin of time, Sy 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 (Sy — A); at time T’? That is the problem

of pricing the option

2 How should the writer, who earns the premium initially, generate an amount

(Sp — K)x 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 that.into 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 S; at

time ¢ We shall assume that it is possible to borrow or invest money at a constant

rate r

Let us denote by C; 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

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:

e Sr > 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 + e"(7-*)(C, — P, — S;) > 0

e Sr < K: we exercise the put and clear our bank account as before to finish

with the wealth K + e'Œ~Ð9(Œ; ~ P, — S,) > 0

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 (Sr — K)4 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, Ité

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-Scholes.model, we will attempt to give to the reader the means to understand those extensions.

x 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 However, some

basic results are also proved in the Appendix l

Acknowledgements

This book is based on the lecture notes taught at /’Ecole Nationale des Ponts

et Chaussées since 1988 The-organisation of this lecture series would not have

been possible without the encouragement of N 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 Internationale 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 read the earlier draft of our book and helped us with their remarks

Amongst them are S Cohen, O Faure, C Philoche, M Picqué 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.F Le Gall and D Talay.

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: Fo,F1, , nN Fn can be seen as the information available at time n and

is sometimes called the o-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 Fo = {0,2}, Fy = F = P(Q) and Ww € N, P({w}) > 0 The market

consists in (d + 1) fifancial assets, whose prices at time n are given by the non- negative random variables S), S),, ,S4, measurable with respect to F, (investors know, past and present prices but ‘obviously not the future ones) The vector S, = (S°,S1, , S42) is the vector of prices at time n The asset indexed

by 0 is the riskless asset and we have S° = 1 If the return of the riskless asset

over one period is constant and equal to r, we will obtain Sẽ = (1+r)” "The

coefficient 8, = 1/S° is interpreted as the discount factor (from time n to time 0): if an amount 8, is invested in_the riskless asset at time 0, then one dollar will

be available at time n The assets indexed by 1 = 1 d are called risky assets

1.1.2 Strategies

A trading strategy is defined as a stochastic process (i.e a sequence in the discrete case) @ = (($2,¢4, -, :64))ocaey in IR“? where ¢3, denotes the number of

2 Discrete-time models

shares of asset 7 held in the portfolio at time n @ is predictable, i.e

độ 1s Zo-measurable

Vi € {0,1, , đ}

This assumption means that the positions in the portfolio at time ø (Ó2., ó5, , 64)

are decided with respect to the information available at time (n — 1) and kept until

time n when new quotations are available

The value of the portfolio at time n is the scalar product

d Va(6) = dn.Sa = À dan : —

Its discounted value is

quoted, the investor readjusts his positions from ¢n to @n+1 without bringing or

consuming any wealth

Remark 1.1.1 The equality on-Sn = $n41-Sn 1s obviously equivalent to

bnti-(Sn41'— Sn) = bngi-Sn4i — bn-Sn,

or to

Vn41(¢) — Val) = n+i-(Sn+1 —.Ốn)

At time n + 1, the portfolio is worth dng1-Sn41 and Ony1-Sn41 — On41-Sn is

the net gain caused by the price changes between times n and n + 17 Hence, the

profit or loss realised by following a self-financing strategy is only due to the price

_ moves

The following proposition makes this clear in terms of discounted prices

Proposition 1.1.2 The following are equivalent

(i) The strategy ¢ is self-financing

where AS; is the vector S; ~ §;-1 = B;S; — Bj-1S;~1

Proof The equivalence between (i) and (ii) results from Remark 1.1.1 The equivalence between (i) and (iii) follows from the fact that @n.S, = dn41-Sn if and only if @n-Sp = On41-Sn Oo

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

initial wealth and the strategy ($),, ., 64) pen< n (this is only justified because AS? = 0) More precisely, we can prove the following proposition

Proposition 1.1.3 For any predictable process ((bn -164) gene and for Tt? `

any Fo-measurable variable Vo, there exists a unique predictable process ($9) <„›<y such that the strategy o= (9, gi, , j2) is self-financing and its initial value

which defines $y, We just have to check that ớP is predictable, but this is obvious

if we consider’ the equation

d1 =W+S (9A8) + +-4/45/)+(sk (S§t j=1 ¡) 4-468 (-SL,))

QO

1.1.3 Admissible strategies and arbitrage

We did not make any assumption on the sign of the quantities ¢i, If 6° <0, we

have borrowed the amount |#9| in the riskless asset If 61, < 0 for ¿ > 1, we say

that we are short a number ¢}, of asset i Short-selling and borrowing is allowed but the value of our portfolio must be positive at all times

Definition 1.1.4 A'strategy ¢ is admissible if it is self-financing and if V,,(¢) > 0

for any n € {0,1, , N} Ỷ

Definition 1.1.5 An arbitrage strategy is an admissible strategy with zero initial

value and non-zero final value

Most models exclude any arbitrage opportunity and the objective of the next

section is to characterise these models with the notion of martingale

1.2 Martingales and arbitrage opportunities

In order to analyse the connections between martingales and arbitrage, we must

first define a martingale on a finite 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 ‘

1.2.1 Martingales and martingale transforms

In this section, we consider a finite probability space (Q, F, P), with F = P(Q)

and Ww € 2, P({w}) > 0, equipped with a filtration (F,)o<n<n (without

necessarily assuming that Fy = F, nor Fy = {0, 2}) A sequence (X;,)o<n<n

of random variables is adapted to the filtration if for any n, Xp, is 7„-measurable

Definition 1.2.1 An adapted sequence (Mn)o<n< N Of real random variables is:

© amartingale ifE (Mn+1|Fn) = Mn foralln < N -1;

© asupermartingale ifE(MnzilFn) < Mn forall n < N — 1;

° a submartingale ifE(Mysi|Fn) > Mn foralln < N — 1

These definitions can be extended to the multidimensional case: for instance, a

sequence (M,,)o<n<wn of IR?-valued random variables is a martingale if each

component is.a real-valued.martingale

In a financial context, saying that the price (S4)o<n<w of the asset 2 is a

martingale implies that, at each time n, the best estimate (in the least-square

sense) of S7 „¡ is given by Sto

The following properties are easily derived from the previous definition and stand

as a good exercise to get used to the concept of conditional expectation

1 (M,)o<n<n is a martingale if and only if

E(Ma+;|#a) = Mạ Vj >0

2 If (Mn noo i is a martingale, thus for any n: E (M,,) = E (Mo)

3 The sum of two martingales is a martingale

4 Obviously, similar properties can be shown for supermartingales and submartin-

Proposition 1.2:3 Let (Mn)g<,<y be a martingale and (Hn)geney 4 pre- dictable sequence with respect to the filtration (Fn)yc,cy Denote AM, =

Mr, — Mn-1 The sequence (Xn)ocn<n defined by ` -

Xo = HoMo „

is a martingale with respect to (#a)s<a< N-

(Xa) is sometimes called the martingale transform of (M„) by (Hạ): A conse-

quence 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, (X,,) is an adapted sequence Moreover, for n > 0

That shows that (X,,) isa martingale , có n

The following proposition is a very useful characterisation of martingales

Proposition 1.2.4 An adapted sequence of real random variables (Mạ) is amar-

_ tingale if and only if for any predictable sequence (Hn), we have

E (>: Hoan.) =0

Proof If (M,) is a martingale, the sequence (X,) defined by Xo = 0 and, forn > 1, X, = ye 1 HnAM,, for any predictable process (H,) is also a martingale, by Proposition 1.2.3 Hence, E(X yy) = E(Xo) = 0 Conversely, we notice that if 7 € {1, ,N}, we can associate the sequence (H,,) defined by

Hạ = 0forn # 7 + 1 and Hj41 = 14; for any F;-measurable A Clearly, (Ha)

1s predictable and E (or, HẠAM, n) = = 0 becomes ˆ

E (14 (Mjsi1 — M;)) = 0

Therefore E (Mj41|F;) = Mj ¬ `

6 Discrete-time models

1.2.2 Viable financial markets

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 opportunity eee

Lemma 1.2.6 If the market is viable, any predictable process Ce rests $°) satis-

‹Gn(9) £ I

Proof Let us assume that Gv (¢) € I First, if Gr(¢) > Oforalln € {0, ,N}

the market is obviously not viable Second, if the G,,(@) are not all non-negative,

where A is the event {Gn(¢) < 0} Because ¢ is predictable and A is 7a-

measurable, w is also predictable Moreover

- 0 ifj<n

G;0) =| 1s (Gj ()-Ga()) iff >n

thus, G; () > 0 for all j € {0, ,N} and Gn (W) > 0 on A That contradicts

the assumption of market viability and completes the proof of the lemma o

: *

measure P* equivalent | to P such that the discounted prices of assets are P*-

Proof (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

(bn), (1.1.2) implies

Ủa(ó) = Vo(d) + >, )-A5;

- j=l

Thus by Proposition 1.2.3, (Va (¢)) is a P*- martingale Therefore Vw (¢) and

E* (Vw (¢)) = E* (Yo(#))

ili i if ly if for any event

† robability measures P; and P2 are equivalent if and only 1

ee (A) " é 2 Po (A) = 0 Here, P* equivalent to P means that, for any w € 9,

P* ({w}) > 0

Martingales and arbitrage opportunities 7

If the strategy is admissible and its initial value is zero, then E* (Yw(¢)) = 0, with Vy (¢) >-0 Hence Vy (¢) = 0 since P* ({w}) > 0, for allw € 2

(b) The proof of the converse implication is more tricky Let us call I’ the convex cone of strictly positive random variables The market is viable if and only if for

any admissible strategy ¢: Vo (¢) = 0 > Vy (¢) ¢ I

(b1) To any admissible process ($1,, , 62) we associate the process defined by

value Gn (¢) is the discounted value of this strategy at time n and because the market is viable, the fact that this value is positive at any time, i.e Gn(#) > 0 for

n=1, ,N, implies that Gy(¢) = 0 The following lemma shows that even if

we do not assume thạt Ớ„ (ó) are non-negative, we still have Gu(¢) ¢ I (b2) The set V of random variables Gn (¢), with ¢ predictable process in IR%, is

clearly a vector subspace of IR® (where IR® is the set of real random variables

defined on 92) According to Lemma 1.2.6, the subspace V does not intersect I Therefore it does not intersect the convex compact set K = {X €T| 3)„ X(œ) = 1} which is included in I’ As a result of the convex sets separation theorem (see

) wen A(w")

is equivalent to P Moreover, if we denote by E* the expectation under measure P*, Property 2

means that, for any predictable process (¢,) in R¢,

N

E* {5° 4,48; | =0

J=1

We shall define a European option* of maturity N by giving its payoff h > 0,

Fy-measurable For instance, a call on the underlying S! with strike price K

will be defined by setting: h = (S} — K) , A put on the same underlying asset

with the same strike price K will be defined by h = (K — Sy) ,- In those two

examples, which are actually the two most important in practice, h is a function of

Sy 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 -

Definition 1.3.1 The contingent claim defined by h is attainable if there exists an

admissible strategy worth h at time N

Remark 1.3.2 In a viable financial market, we just need to find a self-financing

strategy worth A at maturity to say that h is attainable Indeed, if ¢ is a self-

financing strategy and if P* is a probability measure equivalent to P under which

discounted prices are martingales, then (V.(#)) is also a P*-martingale, being

a martingale transform Hence, for n € {0, .,N} Va(¢) = E* (x@)1#:):

Clearly, if (ở) > 0 ứn particular if Vw (ó) = A), the strategy ¢ is admissible

‘ Definition 1.3.3 The market is complete if every contingent claim is attainable

’ 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

* Or more generally a contingent claim

Complete markets and option pricing 9 Theorem 1.3.4 A viable market is complete if and only if there exists a unique probability measure P* equivalent to P under which discounted prices are mar- tingales

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-negative, Fjy-measurable random variable h can be written as h = Vy (¢)

where ¢ is an admissible strategy that replicates the contingent claim h Since ¢

is self-financing, we know that

oaual nee | is arbitrary, P; = P»2 on the whole o-algebra Fy 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 _ _ en call V the set of random

It follows from Proposition 1.1.3 and Remark 1.3.2 that the variable h /S%, does

not belong to V Hence, V is a strict subset of the set of all random variables on

(2, F )- Therefore, if P* is a probability equivalent to P under which discounted

Prices are martingales and if we define the following scalar product on the set

of random variables (X,Y) + E* (XY), we notice that there exists a non-zero

random variable X orthogonal to V We also write

** Ww _ Xứ) *

P™ (()) = (14 go) Pr qu)

10 Discrete-time models

with ||_X |]oo = SUP eg |X (w)| Because E* (X) = 0, that defines a new proba-

bility measure equivalent to P and different from P* Moreover

N

E** (> áo a6) =0

n=1

for any predictable process ((¢4, ,4%)) ycney- It follows from Proposition

1.3.2 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 prices of financial assets are

martingales Let h be an Fyy-measurable, non-negative random variable and @ be

an admissible strategy replicating the contingent claim hence defined, i.e

Va(ó) = SE" (= Fn) , n=0,1, ,N

ˆ N

At any time, the value of an admissible strategy replicating his completely deter-

mined by h It seems quite natural to call V,,(¢) the price of the option: that is the

wealth needed at time ? to replicate A at time N by following the strategy ¢ If, at

time 0, an investor sells the option for ' ‘

h

» B(x): E* | =>

he can follow a replicating strategy ¢ in order to generate an amount hat time N

In other words, the investor is perfectly hedged

Remark 1.3.5 It is important to 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 (2, F) equipped with the filtration (F,,) 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

define it as a positive sequence (Z,,) adapted to (F,), where Z,, is the immediate

profit made by exercising the option at time n In the case of an American option

on the stock S! with strike price K, Z, = (S} — K) ,) in the case of the put,

Zn = (K -S}) +: In order to define the price of the option associated with

(Zn)o<n<w, we shall think in terms of a backward induction starting at time

Indeed, the value of the option at maturity is obviously equal to Uy = Zy At what price should we sell the option at time N ~ 1? If the holder exercises straight away he will earn Zy_, or he might exercise at time N in which case the writer must be ready to pay the amount Zy Therefore, at time N — 1, the writer has

to earn the maximum between Zy_1 and the amount necessary at time N — 1 to

generate Zy at time N In other words, the writer wants the maximum between

Zn-1 and the value at time N — 1 of an admissible strategy paying off Zy at time

0 « {> : 5 `

N, i Sy_,E (ZxIZu¬) with Zwy = ZN/SÂ As we see, it makes sense to

price the option at time N — 1 as

Un_-1 = max (Zv-1,5%_1E" (Zn Fy-1))

By induction, we define the American option price forn = 1, ,N by

We should note that, as opposed to the European case, the discounted price of the American option is generally not a martingale under P* _

Proof From the equality °

Ủ„_¡ = max (Z.-1,E" (0 IZa~)) ›

1t follows that (Ủa)o<»<N is a supermartingale dominating (Zn)ocnen Let us

T,-1 > max (Z.-1,E* (Gn \Fn—1 )) = Unt

A backward induction proves the assertion that (T;,) dominates (Úa) oO

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 S, attimen,O<n<QN,

and a riskless asset whose return is r over one period of time To be consistent

with the previous sections, we denote So =(1+r)"

The risky asset is modelled as follows: between two consecutive periods the

relative price change is either a or 6, with -1 <a < b:

- Jf Sn(1 +a)

Sata = { Sa +0)

The initial stock price So is given The set of possible states is then N= {1+

a,1+}% Each N-tuple represents the successive values of the ratio Sp4i/Sn,

n = 0,1, ,N — 1 We also assume that Fp = {0,9} and F = P(Q) For

n = 1, ,N, the o-algebra F,, is equal to o(S1, , Sn) generated by the

random variables S,, ,S, The assumption that each singleton in 2 has a strictly

positive probability implies that P is defined uniquely up to equivalence We now

introduce the variables T, = Sn/Sn-1, forn = 1, ,N If (11, ,2N) is

one element of 2, P{(11, ,2N)} = P(T,; = 11, ,Tn = xn) Asa result,

knowing P is equivalent to knowing the law of the N-tuple (T,,T2; ,Tv) We’

also remark that forn > 1, ¥, = 0(T1, -,Tn)-

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

E(Tn4i|Fa) = Ptr, Wn € {0,1, ,N — 1}

The equality B(§n41|Fn) = §, is equivalent to E(§n41/Sn|Fn) = 1, since Sn is

#,-measurable and this last equality is actually equivalent to E(Tn+1 |Fn) = 1+r

2 Deduce that r must belong to Ja, b[ for the market to be arbitrage-free

If the market is viable, there exists a probability P* equivalent to P, under which (Sn)

E'(TeailZa)=1+r 7

and therefore B*(Tn41) = 1+ Since Tn+1 is either equal to 1 + a or 1 + 6 with

non-zero probability, we necessarily have (1 +r) €]1 + a,1+ 6]

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 < a By borrowing an amount 5» 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 Sw — %a(1 + r)Ÿ which is always positive, since

Sw > So(1 +a)” Moreover, it is strictly positive with non-zero probability There is arbitrage opportunity If r > 6 we can make a riskless profit by short-selling the risky

4 From now on, we assume that r € Ja, b[ and we write p = (b — r)/(b — a)

Show that (Sn) isa P-martingale if and only if the random variables 7}, To, tes Tw are independent, identically distributed (ID) and their distribution is

given by: P(T; =1+a) =p=1-P(T, = 1+5) Conclude that the market

is arbitrage-free and complete

If T; are independent and satisfy P(T; = 1+a)=p=1- P(T; = 1+6), we have

- Efn+i|7fa) = B(n+i) = p( +4) +(1— p)(1 +6) =1+z

and thus, (S,,) is a P-martingale, according to Question 1

Conversely, if forn = 0,1, ,N — 1, E(Tn41|Fn) = 1417, wecan write

ạ + a)E (1¢7, 4,=14+0}!Fn) + lại + b)E (17, 4:=148)|Fa) =l+r

Then, the following equality

E (10,,i=i+a)l#f2) + B (,¿y=i+e 7n) = 1,

induction, we prove that for any 2; € {1+a,1+ (i wnt) |Fo) — P By

P(N =Z1, ,Ía = Zn) =|»:

i=1

T; are IID under measure P and that P(T; = 1+) =p _—

We have shown that the very fact that (S,,) is a P-martingale uniquely determines the distribution of the N-tuple (Ti, T2, , Ty ) under P, hence the measure P itself

Therefore, the market is arbitrage-free and complete

We denote by C,, (resp P,) the value at time n, of a European call (resp put) ona share of stock, with strike price K and maturity N

(a) Derive the putcall parity equation 7

Ca — Py =S,~—K(14+r)7 8-9),

knowing the put/call prices in their conditional expectation form

If we denote E” the expectation with respect to the probability measure P* under

which (S,.) is a martingale, we have ‘

ICn— Pa = (1+) EY (Sy — K)4 — (K - Sw) 4|Fa)

= (L+r) O° ET (Sw — K|Fa)

5= K(+r) 0N,

14 Discrete-time models

the last equality comes from the fact that (52) is a P*-martingale

(b) Show that we can write C,, = c(n, S,) where c is a function of K, a; br

and p

When we write Sv = Sp Tas T;, we get

Since under the probability P*, the random variable Tin 41 T; is independent of

Fp 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

c{n, z)

(t+r)~N-")

nhám 7

= treat Goren enna),

6 Show that the replicating strategy of a call is characterised by a quantity H, =

A(n, S,—1) at time n, where A will bé expressed in terms of function c

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

HẠ(1L +r)” + AnSn = c(n, Sn)

Since Hp and H„ are Fn—1-measurable, they are functions of 5i, „0„—1 Only and,

since S,, is equal to S,-1(1 + @) or Sn-1(1 + 6), the previous equality implies

Ho(i +r)" + AnSn-1(1 +a) = cín, S2—1(1 + đ)) and

Ho(itr)” + AnSn-i(1 + 6) = e(n, Sn—i(1 +9)

c{n, z(1 + b)) — c{n, z(1+ 3)

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

to infinity, and r = R7/N, log((1 +a)/(1+r)) = —ơ/VN and log((1 +

b)/(1+r)) = o/VN The real number R is interpreted as the instantaneous

rate at all times between 0 and T, because e®? = limy_, (1 + r)% 0? can

be seen as the limit variance, under measure P*, of the variable log(Sj), when

N converges to infinity %5

(a) Let (Yw)w>¡ be a sequence of random variables equal to

Yy= XÈ +X? +-::+XN

where, for each N, the random variables X N are HD, belong to

{-ø/VN, o/VN}, and their mean is equal to wy, with limny_,.o(Nun) = p Show that the sequence (Yn) converges in law towards a Gaussian variable with mean u and variance ø?

We just need to study the convergence of the characteristic function gyy of Yn We

@yx (u) =E (exp(tuYn)) = H E (exp (iux}’))

(E (exp (ux?)))”

= (1 + tUpin —07u?/2N + o(1/N)) ”

Hence, limyoo $¥y(u) = exp (iup — o?u?/ 2), which proves the convergence

in law

(b) Give explicitly the asymptotic prices of the put and the call at time 0

For a certain N, the put price at time 0 is given by

Therefore, the sequence (Yw) satisfies the conditions of Question 7.(a), with u=

—ø”/2 If we write (w) = (Ke~*T — Soe")+, we are able to write

Ke~*TF(da)

Remark 1.4.1 We note that the only non-directly observable parameter 1s ø Íts

interpretation as a variance suggests that it should be estimated by statistical

methods However, we shall tackle this question in Chapter 4

2

e? (dz

Notes: We have assumed throughout this chapter that the risky assets were not

offering any dividend Actually, Huang and Litzenberger (1988) apply the same

ideas to answer the same questions when the stock is carrying dividends The

theorem of characterisation of complete markets can also be proved with infinite

probability spaces (cf Dalang, Morton and Willinger (1990) and Morton (1989))

In continuous time, the problem is much more tricky (cf Harrison and Kreps

(1979), Stricker (1990) and Delbaen and Schachermayer (1994)) The theory of

complete markets in continuous-time was developed by Harrison and Pliska (1981 ,

1983) An elementary presentation of the Cox-Ross-Rubinstein model Is given in

the book by J.C Cox and M Rubinstein (1985) -

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 informa- tion available at time n In a discrete-time model built on a finite filtered space (2, F, (Fn)ocn<n > P), the exercise date is described by arandom variable called stopping time

Definition 2.1.1 A random variable v taking values in {0,1,2, , N} isa Stop-

ping time if, for any n€ {0,1, -, N},

{v=n}e Fy Remark 2.1.2 As in the previous chapter, we assume that F = P() and P({w}) > 0, Wwe ©ˆ 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 Fo = {0,2} and Fy = F, except in Section 2.5, dedicated

Remark 2.1.3 The reader can verify, as an éxercise, that v is a stopping time if

and only if, for any n € {0,1, ,N},

{U < n} € 7a

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

(Xa)s<„< be a sequence adapted to the filtration (Fn)ocncy and let v be a

stopping time The sequence stopped at time v is defined as

Xã (2) = Xy(w)an (w) )

i.e., on the set {⁄ = 7} we have

„_( X; Wj<n

x ={ n ifj>n

Note that X¥, (w) = Xiu (w) (= X; on {v = j})

Proposition 2.1.4 Let (Xn) be an adapted sequence and v be a stopping time

The stopped sequence (Xx )ocnen #5 adapted Moreover, if (X,,) is a martingale

(resp a supermartingale), then (Xy) is a martingale (resp a supermartingale)

Proof We see that, form > 1, we have

Xvan = Xo + » $; (X; — Xj-1) ’

where $; = Ì{;<u}: Since {j < v} is the complement of the set {y<j} =

It is clear then that (X„An)o<a<w 1S adapted to the filtration (Ta)o<n<N-

Furthermore, if (X,,) is a martingale, (X_,n) is also a martingale with respect to

(Fp), since it is the martingale transform of (Xn) Similarly; we can show that

if the sequence (X,,) is a supermartingale (resp a submartingale), the stopped

sequence is still'a supermartingale (resp a submartingale) using the predictability

2.2 The Snell:envelope `

In this section, we considér an adapted sequence (Zn)o<n<n> and define the

sequence (U,,)o<n<wn as follows:

Un = Zn

U, = max(Zn,E(Unt|Fn)) Vn<N~—1

The study of this sequence is motivated by our first approach of American options

(Section 1.3.3 of Chapter 1) We already know, by Proposition 1.3.6 of Chapter

1, that (Un)o<n<wn is the smallest supermartingale that dominates the sequence

(Za)o<n<N We call ít the Snell envelope of the sequence (Zn)o<n<n-

a strict inequality, Un = E(Un4i|Fn) It suggests that, by stopping adequately the

sequence (U,), it is possible to obtain a martingale, as the following proposition

shows ,

Proposition 2.2.1 The random variable defined by

vp = inf {n > 0|Un = Zn} (2.1)

se sopping time and the stopped sequence (Unarrg)o <n<wn #8 4 martingale

ros - Since Uy = Zn, UY is a well-defined element of {0,1, , N} and we

{Uo = 0} = {Uo = Zo} € Fo,

and fork > 1

214: (Ux) is amartingale, we write as in the proof of Proposition

Un? = Unave = Uo,+ >> b;AU;,

where $; = 11,,>,} So that, forn € {0,1, ,N—1},

Una ~ U,° nti (On+1 — Un)

= l{n+i<ze} (Ua+n — Un) *

By definition, U, = max (Z,, E (Un41|Fn)) and on the set {n+ 1 < v9}, U, >

- Zn Consequently U, = E (Un41|Fn) and we deduce

Uniti — Un? = Mnticin} (Unt1 — E (Un41|Fn)) and taking the conditional expectation on both sides of the equality

E((U73, ~ U29) [Za) = lexa<a)E ((U»¿i — B Until Fa))| Fa)

{n+ 1 < uọ} € Fy (since the complement of {n +1 < vo} is {vp <

Hence

| B (Uns ~ Us") Fa) =0,

which proves that U” is a martingale , n

In the remainder, we shall note Jn, the set of stopping times taking values in

in, n+ 1, .,N} Notice that 7;, xv is a finite set since © is assumed to be finite

e 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(Z,,|Fo) = sup, E(Z,|Fo)

If we think of Z,, as the total winnings of a gambler after n games, we see that stopping at time vp) maximises the expected gain given Fo

Proof Since U”° is a martingale, we have

Up = US? = E(U%?|Fo) = E(Uv,|Fo) = E(Zio|Fo).

20 Optimal stopping problem and American options

On the other hand, if v € 7o,n, the stopped sequence U” is a supermartingale So

that

E (Z,|Fo) ; which yields the result n

where v,, = inf {j > n|U; = Z;}

Definition 2.2.4 A stopping time v is called optimal for the sequence (Zn)gcn<n

if

E(Z,|Fo) = sup E(Z,|Fo)

Ton

We can see that vo is optimal, The following result gives a characterisation of

optimal stopping times that shows that vo is the smallest optimal stopping time

Theorem 2.2.5 A stopping time v is optimal if and only if

Proof If the stopped sequence U” is a martingale, Up = E(U,|Fo) and con-

sequently, if (2.2) holds, Up = E(Z,|Fo) Optimality of v is then ensured by

(based on the supermartingale property of (UY )) we get

E (Uvan|Fo) = E (U,|Fo) = E(E (Us| Fn)| Fo)

But we have Uyan > E(U.|F,), therefore „A„ = E(U_| Fn), which proves

that (U27) is a martingale n

Decomposition of supermartingales 21 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)o<n<n has the unique following decomposition:

Unt — Un = Mnsi — Mn — (Anti — An)

So that, conditioning both sides with respect to F,, and using the properties of M

_ (Anti — An) =E (Un+1|Frn) -U, and

3 Mansi — Mn = Una — E(Da+i|Za)

(M,) and (A,) are entirely determined using the previous equations and we see

that (M,,) is a martingale and that (A,.) is predictable and non-decreasing (because (U,,) is a supermartingale) ¬ Oo Suppose then that (U,,) is the Snell envelope of an adapted sequence (Z,,) We

can then give a characterisation of the largest optimal stopping time for (Z,,) using

the non-decreasing process (A,,.) of the Doob decomposition of (U,):

Proposition 2.3.2 The largest optimal stopping time for (Za) is given by

y -{m if An =0

me inf {n, An41 40} ifAn #0

Proof It is straightforward to see that v,,, is a stopping time using the fact that (An)o<n<w is predictable From U, = M, — A, and because A; = 0, for

j < Max, we deduce that U* = M= and conclude that U= is a martingale To

22 - Optimal stopping problem and American options

We have E (Uj41|F;) = M; — 4;+¡ anđ, on the set {„ = 7}, 4; = 0 and

4;+i >0,so Ủy = M; and E( Uj41|F;) = Mj - Ajai < Uj It follows that

U;= max (Z;,E (UyailF;)) = Z; So that finally

BI = Zeina

It remains to show that it is the greatest optimal stopping time If v is a stopping

time such that vy > y,,, and P(v > „) > 0, then

E(U,) = E(M,) — B(A,) = E(Uo) — B(A,) < E(Uo)

and U” cannot be a martingale, which establishes the claim n

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 Eis

called a Markov chain if, for any integer n > 1 and any elements Zo, 2j, ,

In-1,2,y of EB, we have

P(Xn4i = y|Xo = Tọ; »Xn-1 = In-1,Xn = z)=P(X„¿¡ = y|Xn = #) `

The chain is said to be homogeneous if the value P(z, y) = P (Xn+i = yl|Xn = 2)

does not depend on n The matrix P = (P(z,))(„, ye EXE indexed by Ex E,

is then called the transition matrix of the chain The matrix P has non-negative

entries and satisfies: yeE P(z,y) = 1 forall x €'E; it is said to be a stochastic

matrix On a filtered probability space (9, Fy (Fridoenen > P) , we can define the

notion of a Markov chain with respect to the filtration: ,

Definition 2.4.1 A sequence (Xn)o<n<n of random variables taking values in

E is a homogeneous Markov chain with respect to the filtration (Fn) ycn<y» with

transition matrix P, if (Xn) is adapted and if for any real-valued function f on

E, we have `

B( (Xa¿i) [#a) = Pƒ (Xa),

where P f represents the function which mapsx € Eto P f(x) = yee P(z,y)f(y)

Note that, if one interprets real-valued functions on E as matrices with a single

column indexed by E, then Pf 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, defined by

Fn = 0(Xo, -,Xn) ` ˆ

The following proposition is an immediate consequence of the latter definition

and the definition of a Snell envelope ˆ

Proposition 2.4.2 Let (Z,,) be an adapted sequence defined by Zn = b(n, Xn),

where (Xp) is a homogeneous Markov chain with transition matrix -P, taking

values in E, andy is a function from N x E to R Then, the Snell envelope (Un)

Application to American options 23

of the sequence (Z,,) is given by U, = u(n, Xn), where the function u is defined

by

u(N,z) =u(N,z) VieE

and, form < N - 1,

u{n, :) = max (#(n,-), Pu(n + 1,-))

2.5 Application to American options

From now on, we will work in a viable complete market The modelling will be based on the filtered space (9, Z,(Ta)o<n<ụ P) ‘and, as in Sections 1.3.1 and

1.3.3 of Chapter 1, we will denote by P* the unique probability under which the

discounted asset prices are martingales

2.5.1 Hedging American options

In Section 1.3.3 of Chapter 1, we defined the value process (U,,) of an American

option described by the sequence (Z,,), by the system

Un = Zn

U, = max (Zn, S2E* (Unai/S Suil#a)) Vn<N-—I1

Thus, the sequence (U,,) defined by U,, = U,,/S® (discounted price of the option)

is the Snell envelope, under “ Of the sequence (Za) We deduce from the above Section 2.2 that

Un = sup Et (Zuz.)

- vETaN and consequently e

Zz

U, = S09 2 sup B* (Sir) E* (| (7, )

From Section 2.3, we can write -

Un = Mạ — Ấn,

where (M,,) is a P*-martingale and (A,) is an increasing predictable process,

null at 0 Since the market is complete, there is a self-financing strategy @’such that

-2A4 -~ Optimal stopping problem and American options

and consequently

Un = Va(d) - A

Therefore

U, = Vn(¢) _ where A, = S° A,, From the previous equality, it is obvious that the writer of the

option can hedge himself perfectly: once he receives the premium Up = Vo(¢),

he can generate a wealth equal to V,,(@) at time n which is bigger than U,, and a

fortiori 2a

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,, > Z,,, because he would trade an asset worth U,,

(the option) for an amount Zy (by exercising the option) Thus an optimal date r

of exercise is such that U; = Z, On the other hand, there is no point in exercising

after the time

= inf {j, Ajzi AO}

(which is equal to inf {3 3 Aj41 z o}) because, at that time, selling the option

provides the holder with a wealth Uj = V„„(Ó) and, following the strategy

¢ from that time, he creates a portfolio whose value is strictly bigger than the

‘option’s at times „„ + 1, Y%ax + 2, ,N Therefore we set; as a second condition,

T < v,,, which allows us to say that Oi isa martingale As aresult, optimal dates

of exercise are optimal stopping times for the sequence (Zp ), under probability

P* To make this point clear, let us consider the writer’s point of view If he hedges

himself using the strategy ¢ as defined above and if the buyer exercises at time 7

which is not optimal, then U; > Z, or A; > 0 In both cases, the writer makes a

profit V,(¢) — Z, = U; + Ar — Z,, which is positive

2.5.2 American options and European options |"

Proposition 2.5.1 Let Cy be the value at time n of an American option described

by an adapted sequence (Zn)gcncn and let Cn be the value at time n of the

European option defined by the Fy-measurable random variable h = Zy Then,

we have Cy > Cn

Moreover, if Cn > Zn for any n, then

n=Cn Vn€({0,1, ,N}

The inequality C,, > c, makes sense since the American option entitles the holder

to more rights than its European counterpart ©

Proof For the discounted value (Cn) i is a supermartingale under P*, we have

é, > E* (đ„IZ-) =E* (ẽw|#a) = ẽ

Hence Ca > cạ

If cn, > Z, for any n then the sequence (é,), which is a martingale under P*,

appears to be a supermartingale (under P*) and an upper bound of the sequence

(Zn) and consequently

Cn SG, Wn {0,1, , 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 S, at time n and a constant riskless interest rate,

equal to r > 0 on each period, so that S? = (1 +1)” Then, with notations of Proposition 2.5.1, if we take Z, = (S, — K)4, cy 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 C,, is the price of the corresponding American call We have

E* (Sn —~K(1+ ry" |F a) |

= S,-—K(1+r)-%,

using the martingale property of (S,,) Hence: cn > Sy — K(1+1r)7(N-™) >

S, — K, for r > 0.As cy’> 0, we also have c, > (S, — K)4 and by Proposition

2.5.1, Cy = 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 El Karoui (1981) and Shiryayev (1978) :

2.6 Exercises

Exercise 1 Let v be a stopping time with respect to a filtration (F,)o<n<n-

We denote by F, the set of events A such that AN {v = n} € F, for any neé {0, ,N}

1 Show that F, is a sub-o-algebra of Fy F, is often called ‘o-algebra of events

determined prior to the stopping time v’

2 Show that the random variable v is ¥,-measurable

3 Let X bea real-valued random variable Prove the equality

N

Yo l= E(X1F))

j=0

E(X|F,) =

26 Optimal stopping problem and American options

4 Let r be a stopping time such that r > v Show that F, C F;

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

M, = B(M.|7.)

(Hint: first consider the case r = N.)

Exercise 2 Let (U,) be the Snell envelope of an adapted sequence (Z,) Without

assuming that Fo is trivial, show that

E(Ua)= sup E(Z,), U€To,N

and more generally

E (Un) = sup E (Z.)

` 1U€Ta,N Exercise 3 Show that v is optimal according to Definition 2.2.4 if and only if

E(Z,)= sup E(Z;) T€To,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,,, at time n, of an American put on a share with maturity

N and strike price K can be written as

where the sequence of random variables (Vạ )o<n<ụ is defned by: Vọ = l

and, forn > 1, V, = [iL Ui, where the U;’s are’ some random variables

Give their joint law under P*

3 From the last formula, show that the function 2 ++ Pam(0,2) is convex and

non-increasing ,

4 We assume a < 0 Show that there is a real number x* € (0, K] such that, for

+ <S #*, Pzm(0,+) = (K — +)+ and, for z c]z ,K/(+a)%[, Pam(0,2) >

5 An agent holds the American put at time 0 For which values of the spot So

would he rather exercise his option immediately?

6 Show that the hedging strategy of the American put is determined by a quantity

Hạ = A(n, S,-1) of the risky asset to be held at time n, where A 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 So sẻ st are quoted, the

investor readjusts his positions from ¢, to ¢n4, and selects the wealth yn, to be consumed at time n + 1 Any endowment being excluded and the new positions being decided given prices at time n, we deduce

on41-5 = on-Sn — Yn4+1- (2.3)

So a trading strategy with consumption will be defined as a pair (¢,y), where

@ is a predictable process taking values in IR, representing the numbers of

assets held in the portfolio and y = (Ya)i<n< N 1S a predictable process taking

values in IRt, representing the wealth consumed at any time Equation (2.3) gives the relationship between the processes ¢ and and replaces the self-financing

condition of Chapter 1

1 Let @ bea predictable process taking values in IR*? and let y be a predictable

process taking values in IR* We set Va(¢) = @n-Sn and Va(d) = da-Šn

Show the equivalence between the following conditions:

(a) The pair (4, 7) defines a trading strategy with consumption

(b) For any n € {1, , N},

Va($) = Vo(d) + 3 9/.A6; — Ð 2a,

j=l

j=l (c) Foranyn € {1, ,N},

Va(d) = Vo(d) + 3 2ó;.AŠ; — Nxj/S9

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 (¢,y) defines a trading strategy with consumption, then (V,,(¢)) is a supermartingale under P*

3 Let (U,) be an adapted sequence such that (U,,) is a supermartingale under

P* Using the Doob decomposition, show that there is a trading strategy with

consumption (¢, y) such that V,(¢) = U, for any n € {0, , N}

4 Let (Z,,) be an adapted sequence We say that a trading strategy with consump-

tion (¢, y) hedges the American option defined by (Z,) if Va(¢) > Zn for

any n € {0,1, ,.N} Show that there is at least one trading strategy with

consumption that hedges (Z,,), whose value is precisely the value (U,) of the

American option Also, prove that any trading strategy with consumption (4, +) `

hedging (Z,,) satisfies V,(¢) > Un, for any n € {0, 1, oN}

Hl

28 Optimal stopping problem and American options

5 Let x be a non-negative number representing the investor’s endowment and let

= (Yn)i<n<n be a predictable strategy taking values in IR* The consump- tion process (Yn) is said to be budget-feasible from endowment z if there is a

‘predictable process ¢ taking values in TR#?!, such that the pair (ở, y) defines

a trading strategy with consumption satisfying: Vo(¢) = 2 and V,(@) 2 0, for

any n € {0, , N} Show that (ya) is budget-feasible from endowment z if

and only if E* (oa 74/591) <z

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 Théorie de la spéculation, is not only among the first to look at the properties of Brownian motion, but he also derived option 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 ofthe 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

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

Definition 3.1.1 A continuous-time stochastic process in a space E endowed with

ao- algebra E is a family (Xt)em+ of random variables defined on a probability

space (2, A, P) with values in a measurable space (E, €)

Remark 3.1.2

e In practice, the index ¢ stands for the time

e A process can also be considered as arandom map: for each w in 2 we associate

the map from IRt to E: t > X;(w), called a path of the process

e A process can be considered as a map from IR* x 2 into E We shall always

consider that this map is measurable when we endow the product set Rt x

with the product o-algebra B(IR*) x A and when the set E is endowed with

€

e We will only work with processes that are indexed on a finite time interval

(0, T]

As in discrete-time, we introduce the concept of filtration

Definition 3.1.3 Consider the probability space (Q, A, P), a filtration (Fideso is

an increasing family of o-algebras included in A

The o-algebra Z7; represents the information available at time ¢ We say that a

process (X¢}z>0 is adapted to (Fz)t>0, if for any t, Xz is F-measurable

Remark 3.1.4 From now on, we will be working with filtrations which have the

following property

If A € Aand if P(A) =0, then for any t, A€ F; -

In other words F; contains all the P-null sets of A The importance of this technical

assumption is that if X = Y Pas and Y is F,-measurable then we can show

that X is also #;-measurable "

We can build a ñltration generated by a process (Ã;);>o and we write F, =

ơ(X.,s < t) In general, this filtration does not satisfy the previous condition

However, if we replace F; by #; which is the o-algebra generated by both F,

and WV (the g-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 (X;):>0 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 7 is a stopping time with respect to the filtration (Fe)e>o if TIS

a random variable in R* U {+00}, such that for any t s0

{r<t)}€7

The o-algebra associated with 7 is defined as

Fr, = {AE A, foranyt >0,AN{r<t}e€ Fi}

This o-algebra represents the information available before the random time r One

can prove that (refer to Exercises 8, 9, 10, 11 and 14):

Proposition 3.1.6

e ffSisa stopping time, S is Fs measurable

e If S is a stopping time, finite almost surely, and (Xt)e>0 is a continuous,

adapted process, then Xs is Fs measurable

e If S andT are two stopping times such that S<T Pas then Fs C Fr

e If S andT are two stopping times, then S AT = inf(S, T) is a stopping time

In particular, if S is a stopping time and t is a deterministic time S At is a

Definition 3.2.1 A Brownian motion is a real-valued, continuous stochastic pro-

cess (Xt)s>0, with independent and stationary increments In other words:

e continuity: Pas the map s+ X,(w) is continuous

e independent increments: If s < t, X_— X is independent of Fz =o(Xu,us

8)

e stationary increments: if:s < t, Xt —- Xs and X;_„ — Xo have the same

probability law sả This definition induces the distribution of the process X¢, 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>0 is @ Brownian motion, then X, — Xo is a normal random variable with mean rt and variance o*t, where r and o are constant real numbers

Remark 3.2.3 A Brownian motion is standard if

From now on, a Brownian motion is assumed to be standard if nothing else is

mentioned In that case, the distribution of X; is the following:

V2zt P \ 2

where dz is the Lebesgue measure on R ˆ

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

Proof Consider 0 < í¡ < - < tn, then the random vector (Xt, Xt -

Xt,, -,Xt, — Xt,_,) 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 (Xt,, -,Xe,) D

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 (F;)-Brownian

motion if it satisfies:

e Foranyt > 0, X¢ is 7?¡-measurable ,

® ifs <t, X, — X,.is independent of the o-algebra F,

© ifs <t, X, — X, and Xt_, — Xo have the same law

Remark 3.2.6 The first point of this definition shows that o(X,,u < t) C Fy

Moreover, it is easy to check that an 7;-Brownian motion is also a Brownian

motion with respect to its natural filtration.-

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

Definition 3.3.1 Let us consider a probability space (Q,A,P) and a filtration

(Ft)t>0 on this space An adapted family (M;z)1>0 of integrable random variables,

Le E(|Mt|) < +00 for any t is:

© amartingale if foranys <t, E (M.|Z,) = M,;

® asupermartingale if for anys < t, E(Mi|F;) < M,,

® a submartingale # for any s < t, Ð(M;|Z;) > M,

Remark 3.3.2 It follows from this definition that, If (Mf;);¿>o is a martingale, then

ĐB(M,) = E(Mo) for any t

Here are some examples of martingales

Proposition 3.3.3 If (Xz)t>0 is a standard F,-Brownian motion:

1 X is an 7ị-martingale

2 X}? —t is an F,-martingale C

Continuous-time martingales 33

3 exp (aX: — (a? /2)t) is an F,-martingale

Proof If s < ¢ then X; — X, is independent of the o-algebra F, Thus E(X;—

X,|Fs) = E(Xt — Xs) Since a standard Brownian motion has an expectation equal to zero, we have E(X; — X,) = 0 Hence the first assertion is proved To

show the second one, we remark that

Tf (M:):>o Ìs a inaningale, the property E (M;|Zs) = Ms, is also true if ¢ 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

KK

34 Brownian motion and stochastic differential equations

Theorem 3.3.4 (optional sampling theorem) /f (4;):>0 is @ continuous mar-

tingale with respect to the filtration (F,)t>0, and if 7 and T2 are two stopping

times such that T) < To < K, where K is a finite real number, then M,, is

integrable and

E(M,,|F,,) = Mr, Pas

Remark 3.3.5

e This result implies that if 7 is a bounded stopping time then E(M,) = E(Mo)

(apply the theorem with 7, = 0,72 = 7 and take the expectation on both sides)

e If M; is a submartingale, the same theorem is true if we replace the previous

equality by

E(M,,|F,,) > M,, Pas

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 (X+z):>0 an F,.-Brownian motion If

ais a real number, we callT, = inf {s > 0,X, = a} or +00 if that set is empty

Then, T, is a stopping time, finite almost surely, and its distribution is charac-

terised by its Laplace transform

E (eT) _ 7 V2Alal,

Proof We will assume that a > 0 First, we show that T, isa | stopping time

Indeed, since X, is continuous

{To < t} = Meeqt« {sup X, >a- eS = Meeqt: Useqt,s<t {X, >a- e}

ss

That last set belongs to ¥;, and therefore the result is proved In the following, we

write x A = inf(œ, 9)

Let us apply the sampling theorem to the martingale M; = exp (0X; — (0? / 2)t)

We cannot apply the theorem to T, which is not necessarily bounded; however, if

n is a positive integer, T, An is a bounded Stopping t time (see Proposition 3.1.6),

and from the optional sampling theorem

E (MT, An) = ]1

On the one hand M,A„ = exp (ØXT,An — Ø "ức An)/2) < exp(ơa) On

the other hand, if 7¿ < +oo, lima-;+ee ẤMr,An = Mr, and if Ty = +00,

E(ltr,<+eeyMr,) = 1, i.e since Xr„, = a when Tz’ < +00

ơ2 `

the Brownian motion reaches the level a almost surely) Also

2

E (exp (-$*)) =e"

The case a < 0 is easily solved if we notice that

T, = inf {s >0,-X, = —a}, where (— Xz)t>0 is an F,-Brownian motion because it is a continuous stochastic

process with zero mean and variance t and with stationary, independent increments

oO

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 supg<t<r |Ä;| can be

bounded This is known as the Doob inequality

Theorem 3.3.7 (Doob inequality) [f (M:)o<i<r is a continuous martingale, we have

O<t<T

The proof of this theorem is the purpose of Exercise 13

D 3.4 Stochastic integral and Ito calculus

In a discrete-time model, if we follow a self-financing strategy ¢ = (Hn)ocncn> the discounted value of the portfolio with initial wealth Vo is ~~

ˆ n

Vo + SAS; — 5-1)

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-time models are concerned, integrals of the form f 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,) is a Brownian motion, it can be proved that for almost every w € Q, there is not any time ¢ in IR* such that dX; /dt exists As a result, we are not able

to define the integral above as

[ toax.= [ 10954

Nevertheless, we are able to define this type of integral with respect to a Brownian

36 Brownian motion and stochastic differential equations

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

this section

3.4.1 Construction of the stochastic integral

Suppose that (W;)¢>0 is a standard F,-Brownian motion defined on a filtered

probability space (2, A, (F:):>0,P) We are about to give a meaning to the

expression J f(s, w)dW, for a certain class of processes f(s,w) adapted to the

filtration (F;):>0 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 (Htosesr is called a simple process if it can be written as

=3 al œ)1I¿,_, ¿,])

where Ö = tọ < tị < - < tp = T and ộ¡ is 7ị,_, -measurable and bounded

Then, by definition, the stochastic integral of a simple process H is the continuous

process (I(H):)o<t<r defined for any t € Ite, te41] as

1<i<k

Note that ƒ(JJ); can be written as

I(H),= D> ®(WùAr— Wu, cay),

1<t<p

which proves the continuity of t » I(H), We shall write J H,dW, for I(H):

The following proposition is fundamental

Proposition 3.4.2 If (H:)o<t<r is a simple process:

processes Indeed, to show that ( J H,aW,) is a martingale, we just need to

check that, for any t > s, `

Stochastic integral and It6 calculus 37

If we include s and ¢ to the subdivision tj = 0 < t) < - < t, = T, and if we call M, = f,” H.dW, and Gn, = 7¡„ for 0 < n < p, we want to show that My

is a G,-martingale To prove it, we notice that +

Mạ = |" Haw, = » We, — Wei.)

i=1 with @; G;-1-measurable Moreover, X, = W;,, isaG,-martingale (since (We) ido

is a Brownian motion) (Mn),,¢/o,p] 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,p) 18 a martingale The second assertion comes from the fact that

=_ E(E(2¡i2;(X:¡ — X;—i)(X; — X;~—¡)| đy~1))

2= E(đ9,(X;¡- — Kini JE (X5 ~ X5-119;- 1))-

Since X; is a martingale, E(X; - X3-1|G;-1) = 0 Therefore, ifi < j,

E (¢:6;(X: — Xi-1)(X; — Xj-1)) = 0 If j > i we get the same thing Fi-

nally, if = 7,

as a result

B((X¡ > Xi-¡)?|[đ—¡) = B (0, — Wyj_,)2) =0 T—b.a — 2)

From (3.1) and (3.2) we conclude that cu ca

38 Brownian motion and stochastic differential equations

Remark 3.4.3 We write by definition

T T t : hi

| Haw, = | Haw, [ H,dM, Ỷ

Ift < 7T, and If A € 7¡, then s 1A1(¿<;yH; is sull a simple process and it is

easy to check from the definition of the integral that

0 t ,

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

Proposition 3.4.4 Consider (W:):>o an 7,-Brownian motion There exists a

unique linear mapping J from H to the space of continuous F,-martingales

defined on [0,T), such that:

1 If (Ht)t<r isa simple process, Pas foranyO<t<T, J(H), =I(H)t

t

2 Ift<T,E(J(H)?) =E (| a

This linear mapping is unique in the following sense, if both J and J' satisfy the

previous properties then

Pas W<t<T, (UH) =J'(H)

We denote, for H € H, [ H,dW;, = J(H);

On top of that, the stochastic integral satisfies the following properties:

Proposition 3.4.5 If (H:)o<t<r belongs to H then

1 We have

®(mp|[ ) <4E ự sta) (3.4) t<T

2 Ifr is an F,-stopping time

T T

0 0 —

Proof We shall use the fact that if (H,)s<r is in H, there exists a sequence

(H2).<r of simple processes such that :

T

n—+cœo 0

Stochastic integral and Ité calculus 39

A proof of this result can be found in Karatzas and Shreve (1988) (page 134,

That implies that (J(H)t)o<e<r does not depend on the approximating sequence

On the other hand, (J(H)t Joxesr i is a martingale, indeed

E(I (H"),|Z,) = I(H"),

Moreover, for any t, liMn++oo1(H"), = J(H), in L*(9;P) norm and, because

the conditional expectation is continuous in L?(, P), we can conclude

From (3.7) and from the fact that E([(H")?) = E (1P ds), it fol-

lows that E(J(H)?) =E bì \H.|° ds) In the same way, from (3.7) and since

EGup,<~ J(H")ÿ) < 4E (5 IH?i 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 € 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 7 = = Vicicn t;14,;, where

0<t <-+ <t, = T and the A;’s are disjoint and F;, measurable, and we

prove (3 5) i in that case First

c

T T

0 0 l<i<n

40 Brownian motion and stochastic differential equations

also, each 1,,5+;}14;Hs is adapted because this process is zero if s < t¢, and is

equal to 14,H, otherwise, therefore it song to H It follows that

In order to prove this result for an arbitrary stopping time 7, we must notice that

7 can be approximated by a decreasing sequence of stopping times of the previous

form If (k+ Nr,

™m™ = » on — Mage, <<“)

o<i<2"

T, converges almost surely to 7 By continuity of the map t 4 J H,dW, we

can affirm that, almost surely, So” H,dW, converges to So H,dW,, On the other

converges to i lys<r} HsdW, in L?(Q,P) (a subsequence converges almost

surely) That completes the proof of (3.5) for an arbitrary stopping time oO

In the modelling, we shall need processes which only satisfy a weaker integrability

condition than the processes in H, that is why we define ,

Proposition 3.4.6 There exists a unique linear mapping J from H into the vector

space of continuous processes defined on [0,7], such that:

1 Extension property: Jƒ (H:)o<:<7T is a simple process then

Pas WO<t<T, J(H): ~ 1(Hì

2 Continuity property: if (H ")a>o is a sequence of processes in H such that

fe ( (H™)? ds converges to 0 in probability then SUP¿< |J(H");| converges to

Stochastic tmiegral and ltô calculus 4I

0<t<T necessarily a martingale

Proof It is easy to deduce from the extension property and from the continuity

property that if H € ?( then P as V < T, J(H); = J(H):

is empty), and H? = H, l{s<T„}- Firstly, we show that T,, is a stopping time Since {7 < t} = { J HỆdu > n},

we just need to prove that J H?du is F,-measurable This result is true if H is a simple process and, by density, it is true if H € H Finally, if H € H, J H?du

~ is also F;-measurable because it is the limit of J( (Hụ„ A K)?du almost surely as

K tends to infinity Then, we see immediately that the processes H? are adapted and bounded, hence they belong to H Moreover

t t [ Hraw, = | 1:;<r„yH?!đ4W,,

Thus, on the set {fo HỆdu < nj, for anyt < T, J(H"), = J(H"*?), Since

Ua>o{f H?du < n} = {fo H?du < +00}, we can define almost surely a process J(H), by: on the set Uy HỆdu < n},

If we call ty = inf {s <T, f> H2du > 1/N} (+00 if this set is empty), then

on the set {Jo H2du < 1/N}, it follows from (3.5) that, for any t < T,

| { f° H2du<i/N} SUP <T 7H)

[ H,dW, = Ï(H); = J(H)), = Ỉ H}1t(,<„„)dW, = Ỉ H,1(s<„„)dW,,

0 0 ~

42 Brownian motion and stochastic differential equations

whence, by applying (3.4) to the process s ++ Hs lts<ry} We get

> ) P( #4: > 5)

+4/c?E (5 H?1(„<-„)45)

T

P(Í HỆ > À )* le Ne

As a result, if Jo ( (HT?) ds converges to 0 in probability, then Sup;< l7 (H"):|

converges to 0 in (ot sly

In order to prove the linearity of J, let us consider two processes belonging to

H, called H and K, and the two sequences H? and K?* defined at the beginning

of the proof such that Se (H? — H,)*ds and fe (KP - K,)?ds converge to 0

in probability By continuity of J we can take the limit in the equality J(AH” +

pK"), = AJ(A"): + pJ(K")t, to prove the continuity of J

Finally, the fact thatif H € 71 then ƒ\( (H,— H?)?dt converge to 0in probability

and the continuity property yields the uniqueness of the extension

IA

P (sup J(H):

t<T

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

Summary:

Let ‘us consider an F;- Brownian motion (W:):>o and an Z7¡-adapted process

(Aiosesr- We are able to define the stochastic integral (fo H,dW,)o<t<T as

soon as fe H?ds < +oo Pas By construction, the process (fo H,dW,)o<t<t

is a martingale if E ( Se H ?ds) < +00 This condition is not necessary Indeed,

the inequality E ( fe H Ras) < +00 is satisfied if and only if

2 (sp (ƒ H,aW, ) < +00

0<t<7T ,

This is proved in Exercise 15

3.4.2 It6 calculus : TA

It is now time to introduce a differential calculus based on this stochastic integral

Tt will be called the /t6 calculus and the main ingredient is the famous /t6 formula

In particular, the It6 formula allows us to differentiate such a function as t +

f (W,) 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 ‘dW,’ Typically, for a

Stochastic integral and Ité calculus 43

or itaft function f(t) null at the origin, we have f(t)? = 2 So F( s)ds =

2 So £( ) In the Brownian case, it is impossible to have a nae vernal W? =2 ed m dW; Indeed, from the previous section we know that J W,dW,

1s a martingale (because E ( J W?ds) < +œ), null at zero If it were equal to

W7? /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 Itô formula is

Definition 3.4.8 Let(Q, F, (Ft):>0, P) bea filtered probability space and (W:)1>0

an Fy-Brownian motion (X+)o<t<r is an IR-valued It6 process if it can be written

This implies that:

~ An It6 process decomposition is unique That means that if

X, = Xo +/ K,ds +/ H,dW, = Xọ + [ ‘Kids +/ HidW,

X= Xi dPas H,=H' dsxdPae K,=K' dsxdPae

— If (Xt)o<esr is a martingale of the form Xo+ J K,ds + J H,dW,, then

K, = 0 dt x dP ae

We shall state It6 formula for continuous martingales The interested reader should refer to Bouleau (1988) for an elementary proof in the Brownian case, i.e when (W,) is a standard Brownian motion, or to Karatzas and Shreve (1988) for a

complete proof

44 Brownian motion and stochastic differential equations

Theorem 3.4.10 Let (X)o<:<r be an Ité process

t t Xi=Xe+ | K,ds-+ [ H.dW.,

0 0 and f be a twice continuously differentiable function, then

to x and once with respect to t, and if these partial derivatives are continuous with »

respect to (t, x) (i.e f is a function of class C}:?), It6 formula becomes

3.4.3 Examples: Ité formula in practice

Let us start by giving an elementary example If f(z) = x? and X; = W:, we

identify K, = 0 and H, = 1, thus

It turns out that

Since E ( Ss W2ds) < +00, it confirms the fact that W2 — ¢ is a martingale

We ‘now want to.tackle the problem of finding the solutions (5;):>0 of

Stochastic integral and It6 calculus 45

i S,ds and Ss S,dW, exist and at any time ¢

t t Pas Sy = Zo + / pS,ds + [ oS,dw,

To put it in a simple way, let us do a formal calculation We write Y¿ = log(S;)

where 5S; is a solution of (3.8) S; is an It6 process with K, = 5; and H; = aSs

Assuming that S; is non-negative, we apply Ité formula to f(z) = log(z) (at least formally, because f(x) is not a C? function!), and we obtain

Taking that into account, it seems that

St = Zo exp ((u — ơ?/2)t+ ơW,)

1s a solution of equation (3.8) We must check that conjecture rigorously We have

St = fit, W,) with

(u — ø2/2)t+ ơWi

^ ƒŒ,#) = zo exp ((uT— ø?/2)t + ơz)

Itô formula is now applicable and yields

Remark 3.4.11 We could have obtained the same result (exercise) by applying

Ité formula to S; = j(Z,), with Z; = (u—o? /2)t-+-oW; (which is an Ité process) and j(z) —zo exp(z)

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 Yt be two lô pro-

cesses, Xt = Xot fy Ks ds +f, H,dW, andY, = Yo +f Kids +f HidwW,

Then

t t X% = Xo¥o+ | X.aY, + [ Y,dX, + (X,Y),

St = x0 exp ((u — 07/2) t+ o0W)

is a solution of (3.8) and assume that (Ã;);>o is another one We attempt to

compute the stochastic differential of the quantity X, 15, ) Define

L= 2 = exp ((—u +ø?/2) t — ơW;),

t

p' = —u + Ø2 and ơ' = —-o Then 2 = exp(w _ a” /2)t + ơ'W,) and the

verification that we have just đone shows that

Stochastic integral and Ité calculus 47

In this case, we have

+_ X;Z¡ (udt + ơdW/,) — X;Z.ơ?dt =

Hence, X;Z; is equal to XoZa, which implies that

VWt>0, Pas X,;=202Z,' =S

The processes X; and Z, being continuous, this proves that

“Pas Wt>0, Xp=20Z,! = S

We have just proved the following theorem:

Theorem 3.4.13 if we consider two real numbers a, 2 and a Brownian motion

(W:)t>0 and a strictly positive constant T, there exists a unique It process

(St)o<e<r which satisfies, for anyt < T,

e The process S; that we just studied will model the evolution of a stock price in

the Black-Scholes model

e When pu = 0, S; is actually a martingale (see Proposition 3.3.3) called the exponential martingale of Brownian motion

Remark 3.4.15 Let © be an open set in IR and (Xt)o<e<7 an It6 process which stays in © at all times If we consider a function f from © to R which is twice continuously differentiable, we can derive an extension of It6 formula in that case

;œ%: f(Xo) + [res s)dX; + ; J2 X,)Hds

This result allows us to apply Itô formula to log(X;) for instance, if Ã; 1s a strictly

positive process

3.4.4 Multidimensional Ité formula

We apply a multidimensional version.of It6 formula when f is a function of'several

Ité processes which are themselves functions of several Brownian motions This

48 Brownian motion and stochastic differential equations

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 F;-Brownian motion an RRP -

valued process (W, = (W}, ,W?))>0 adapted to F,, where all the (W, 2)t>o0

are independent standard 7ị- Brownian motions

Along these lines, we can define a multidimensional Ité process

Definition 3.4.17 (Xt)o<t<r is an Ité process if

t Pp t

Xi = Xe + K,ds+S” | HịdW)

0 i=1 49

where:

e K, and all the processes (H}) are adapted to (Ft)

ef |Ke|ds < +00 Pas

respect to t, with continuous partial derivatives in (t, r)

Remark 3.4.19 If (X, Jocrer and (Ÿ;)o<¿<7 are two Itô processes, we can de

fine formally the cross-variation of X and Y (denoted by (X,Y);) through the

following properties:

e (X,Y) is bilinear and symmetric

© ({j Keds, X.)t = Oif Mosesr | is an Ité process |

Stochastic differential equations 49

o (f; HedWij, fo HidW)): = Jo HeHids

This definition leads to the cross-variation stated in the previous proposition 3.5 Stochastic differential equations

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

‘ot X;=z+ [ X,(puds + odW,)

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 :

(3.10)

3.5.1 Hô theorem

What do-we mean by a solution of (3.10)?

Definition 3.5.1 We consider a probability space (Q, A,.P) equipped with a fil-

tration (F:)t>0 We also have functions b : IRỶxIR> Ro: RtxR>Ra Fo-measurable random-variable Z and finally an F,- Brownian motion (W:):>o

A solution to equation (3.10) is an 7-adapted stochastic process (Ä:):>o that Salisfies:

e Foranyt > 0, the integrals [ÿ b(s, X,)ds and So als, X,)dW, exist

[Ws xolas < eoana f° Iz(s,X,)Ÿ ds < +00 Pas

© (X1t)t>0 Satisfies (3.10), i.e

+

t t w29 Pas Xi=Z+ | b(s,X,)đs+ Í ơ(s,X,)dW

l ` 0 0,

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

dX, = b(t, Xt) dt +a (t, Xt) dw,

Xo = 2 The following theorem gives sufficient conditions on b and o to guarantee the

existence and uniqueness of a solition of equation (3.10)

Theorem 3.5.3 Jf b and o are continuous functions and if there exists a constant

K < +00 such that

50 Brownian motion and stochastic differential equations

1 |b(t, x) — b(t, y)| + lo(t, x) — ø(,w)| < K|z — vị

2 |b(,z)| + |ơ(Œ,z)| < K( + |z|)

3 E(Z2}< +oo

then, for any T > 0, (3.10) admits a unique solution in the interval [0, T]

Moreover, this solution (Xs)o<s<T satisfies

The uniqueness of the solution means that ƒ (X:)o<t<r and (Y:)o<t<T đre two

Proof We define the set

£ = {doses ¥,-adapted continuous processes,

such that E (sup XP) < +00}

s<T

Together with the norm ||X||= (E (supe<:<z |X:|?)) 1⁄2 £ 1s acomplete normed

vector space In order to show the existence of a solution, we are going to use

the theorem of existence of a fixed point for a contracting mapping Let ® be the

function that maps a process (X;)o<s<T into a process (®(X)s)o<s<7 defined

6(X), = z+ Ws, Xe)ds+ | a(s,X,)dW,

0 0

If X belongs to €, (X) is well defined and furthermore if X and are both in

€ we can use the fact that {a +b)? < 2(a? + 6?) to write that

lð(X),— 8(V)# < 2 (supeceer [Jo (0(s, Xs) ~ bs, Y,))ds[

1)

+ §UPo<;<7 li, (s, X, )— ø(s, Y;))

and therefore by inequality (3.4)

B (sap |#(X) - #24) s<T

< an (sp, (ƒ |b(s, X;) — Đ(s, —we,Y2Ie) +8E ([ (a(s, Xs) — o(s, Ys)’ 24)

2(K?T2? + #er)B ( sup |X¿ — vi)

0<t<T

Stochastic differential equations — 51

whenee ||&(X) #®(Y)|| < (2(K?T? + 4K?7))!/? ||X — Y|| On the other

hand, if we denote by 0 the process that is identically equal to zero and if we notice that (a + b +c)? < 3(a? + b? + c?)

_ so that k(T) < 1, ít turns out that ® is a contraction from € into € Thus it

has a fixed point in € Moreover, if X is a fixed point of ®, it is a solution of

(3.10) That completes the proof of the existence for T small enough On the other

hand, a solution of (3.10) which belongs to € is a fixed point of 6 That proves

the uniqueness of a solution of equation (3.10) in the class € In order to prove the uniqueness in the whole class of It6 processes, we just need to show that a solution of (3.10) always belongs to € Let X be a solution of (3.10), and define

= inf{s > 0, |X,| > n} and f"(t) = E (supg<,ciar, |X|”) It is easy to

check that f”(t) is finite and continuous Using the same comparison arguments

as before, we can say that So ,; :

E (supocuctat, Xul’) S$, 9(Be +E ( fy" K(1+ |X Dds)

+4B (f° K7(1 + |X.1)?ds) )

3 (B(Z?) + 2(K?T + 4K?)

x fo (1 +E (suPo<ucear, IXxl?)) ds) This yields the following inequality

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

Lemma 3.5.4 (Gronwall F0 If f is a continuous function such that for any

0<t<Tf, (ose s)ds, then f(T) < a(1 +e"), Proof Let us write u(t) = e~** ‘So £¢ s)ds Then,

a ul (t) = e~ (f(s) - of f(s)ds) < ae~™

`

By first-order integration we obtain u(T) < a/b and f(T) < a(1 + e°?) Oo

2

52 Brownian motion and stochastic differential equations

In our case, we have f"(T') < K < +00, where K is a function of T independent

of n It follows from Fatou lemma that, for any T’,

B ( sup Ix.f) < K < +00

0<s<T Therefore X belongs to £ and that completes the proof for small 7' For an arbitrary

T, we consider a large enough integer and think successively on the intervals

[0, T'/n], [T/n, 2T/n] |[[n — 1)T/n, TỊ Oo

3.5.2 The Ornstein-Ulhenbeck process

Ornstein-Ulhenbeck process is the unique solution of the following equation:

dX = —cX,dt + odW,

Xo = @£

It can be written explicitly Indeed, if we consider Y, = X,e and integrate by

parts, it yields

dY, = dX;e* + X;d(e°) + d(X, e°)y

Furthermore (X,e%); = 0 because đ(e°?) = ce°td¿ It follows that dY, =

ơe°tđjWW; and thus : :

(since E (Jo (e*)?as) < +00, J e°°dW, is a martingale null at time 0 and

therefore its expectation is zero) Similarly

Var(X:) = E (Ge — E(X.))ˆ)

We can also prove that X; is a normal random variable, since X; can be written as

J ƒ(s)dW, where ƒ(.) is a deterministic function of time and J f?(s)ds < +oo

(see Exercise 12) More precisely, the process (Xt)>0 is Gaussian This means

Then Xt, +++ + AnXe, = Dy Ams + fo (OL, Acfi(s)) dW, 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

The analysis of stochastic differential equations can be extended to the case when

’ processes evolve in IR” This generalisation proves to be useful in finance when

we want to model baskets of stocks or currencies We consider

e W = (W!, , WP) an IRP-valued Z;-Brownian motion

e b:IRT xIR” OIR”,b(s,z) = (b1(s,z), -, b*{(s, z))

e z:TRỶ x]IR*” — IR^XP (which is the set of n x p matrices),

Ø(3,) = (Ø,7(s, Z))1<i<n ,1<j<p-

e Z=(Z', ,Z") an Fy-measurable random variable in IR”

We are also interested in the following stochastic differential equation:

t t

Xi= 2+ b(3,X,)ds+ f ơ (s, X,) dW; (3.11)

0 0

In other words, we are looking for a process (X;)o<¿<r with values in IR”,

adapted to the filtration (Z;);>o and such that P as , for any ¢ and for any ¿ < n

t Prt xXp=H Zt [ bi(s, X,)ds + >| Ø:,;(s, X,;)dWỷ

0 jan J0

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

Theorem 3.5.5 If x € IR", we denote by |x| the Euclidean norm of x and if

gER™?, lo? = Di cien, 1<j<p Tj We assume that

54 Brownian motion and stochastic differential equations

3.5.4 The Markov property of the solution of a stochastic differential equation

The intuitive meaning of the Markov property is that the future behaviour of the

process (X;)¢>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 7;-adapted process (X:):>o satisfies the Markov

property if, for any bounded Borel function f and for any s and £ such that s < ,

we have

E(f (Xt) |Fs) = E (f (Xe) |Xs)-

We are going to state this property for a solution of equation (3.10) We shall

denote by (X!*, s > t) the solution of equation (3.10) starting from = at time t

and by X* = X°* the solution starting from z at time 0 For s > t, X* satisfies

Xt =24 / _b(u, XÉ®) du + / o (u, Xb") aW,, t t

A priori, X** is defined for any (t, x) almost surely However, under the assump-

tions 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, ifs >t

tXƑ XoF =X, ‘ Pas

Proof We are only going to sketch the proof of this lemma For any «x, we have

xi =Xƒ+ [i (u, x.) du + Ƒs (u, xu") dW |

These results are intuitive, but they can be proved rigorously by using the continuity

of y +» X*¥, We can also notice that X° is also a solution of the previous equation

In this case, the Markov property can be stated as follows:

Theorem 3.5.7 Let (X1)t>0 be a solution of (3.10) It is a Markov process with

respect to the Brownian filtration (F;)t>0 Furthermore, for any bounded Borel function f we have

P as E(f (Xt) |Fs) = (Xs),

with $(z) = B (ƒ(X?”))

Remark 3.5.8 The previous equality is often written as

E (f (%X;) \Fs) =E ((X?”))|,~x.,

Proof Yet again, we shall only sketch the proof of.this theorem For a full proof,

the reader ought to refer to Friedman (1975)

The-flow property shows that, ifs < t, Xf = xX; Ke On the other hand, we

can prove that X;°* is a measurable function of x and the Brownian increments (W.4u — Ws, u > 0) (this result is natural but it is quite tricky to justify (see

Friedman (1975)) If we use this result for fixed s and t we obtain X;/°* =

®(z,W;+„ —W;; u > 0) and thus

Xf = 0(X2,Weiu — We; u > 0),

where X} is ¥,-measurable and (W.4 — W;)„>o is independent of F,

If we apply the result of Proposition A.2.5 in the Appendix to X,, (Ws4u — W;)„>o, ®.and F,, it turns out that

E(f (B(XE, Wein — Wes w > 0))|Z,)

= EB(f (Ole, Wau — We; U> O)leaxs

.= E (f (XP) )eaxe

mu

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

Theorem 3.5.9 Le (X;);¿>o be a solution of (3.10) and r(s, x) be a non-negative

2

56 ˆ Brownian motion and stochastic differential equations

measurable function Fort > s

E (J r(uXu)du (Xt) Fe) -E (=! ror (xe)

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:

Pas E (2X, t > 8) |Zs) = E(¿(X?”, t 2 8))Ì„—x, `

Remark 3.5.11 When ö and ø are independent of z (the diffusion is said to be

homogeneous), we can show that the law of X:%, is the same as the one of X}*,

which implies that if f is a bounded measurable function, then

E (F(X) = E (f(x?)

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

s+t su t = l

E (J r( x3 (X38) =E (= + r(x? * (x2*))

E (=! r(Xu)du (X;) Fe) =E Gus rome (x08)

Exercise 7 Let Xz be a process with independent stationary increments and zero

initial value such that for any t, E (X; 2) < +00 We shall also assume that the

mapt > E (X2) 1s continuous Prove that E (X;) = c‡ and that Var(X;) = c’t,

where c and c’ are two constants

Exercise 8 Prove that, if r is a stopping time,

F, ={AEA, forallt>0,AN{r <t} Ee Fi}

is a o-algebra

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

Exercise 10 Let S and T be two stopping times such that S < T Pas Prove that Fs C Fr

Exercise 11 Let S be a stopping time almost surely finite, and (Xt)t>0 be an adapted process almost surely continuous

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

X,= lim S2 Ta (6+a)/n((9)XXk/a (6) r:i>-+co

Prove that I.(f) is a normal random variable and compute its mean and vari-

E(Ie(f)’) = lI/lla-

2 From this, show that there exists a unique linear mapping I from L?(IR*, dz) into L?(0, F, P), such that I(f) = IIfllz2, fot any f in L?(R*) I,(f), (f), when f belongs to H and E(1(ƒ)2) when ƒ belongs to 1 and E(I

3 Prove that if (Xa)n>o is a sequence of normal random variables with zero-

mean which converge to X in L*(Q,F,P), then X is also a normal random

58 Brownian motion and stochastic differential equations

variable with zero-mean Deduce that if f € L?(IR*, dz) then I(f) is a normal

random variable with zero mean and a variance equal to i ƒ?(s)ds

4 We consider f € L?(IR*, dx), and we define

Z,= [ ƒ(s)dX, = J 1o,g(s)f(9)dÄX,,

prove that Z; is adapted to Fz, and that Z, — Z, is independent of + (hint: -

begin with the case ƒ € H)

5 Prove that the processes Z,, Z? — J f?(s)ds, exp(2: — ÿ " ƒ?(3)ds) are

Z;-martingales

Exercise 13 Let T be a positive real number and (Mz)o<t<r be a continuous

F,-martingale We assume that E(M/?) is finite

1 Prove that (|M:z|)o<e<7 is a submartingale

2 Show that, if M* = supy<per |Mil,

AP (M* >) SE (|Mr|lar->a}) - (Hint: apply the optional sampling theorem to the submartingale | M;| between

+ AT and T where 7 = inf{t < T,|M,| > A} (if this set is empty 7 is equal

3 From the previous result, deduce that for positive A

E((M A 4)?) < 2B((M* A A)|Mrl)

(Use the fact that (Mƒ* A A)? = An

4 Prove that E(M*) is finite and

pzP~ ldz for p = 1,2.)

B( sup is) < 4E(|Mr|’)

O<t<T Exercise 14

s

1 Prove that if S and S’ are two Z-stopping times then Š A Š° = inf(S, 5”) and

SV S' =sup(S, S’) are also two F;-stopping times

2 By applying the sampling theorem to the stopping time S V s prove that

2 Let p(t,2) = 1//1 - texp(—2x?/2(1 — t)), forO < t < 1 and z € IR, and

p(1, 2) = 0 Define M; = p(t, W,), where (W;)o<¢<1 is standard Brownian motion

(a) Prove that

Prove that Lo H2dt < 00, a.s and E (6 Hat) = +,

Exercise 16 Let (M.)o<:<r be a continuous Z7;-martingale equal to J K,ds,

where (;)o<¿<7 is an 7¿-adapted process such that fe |K;|ds < +œ Pas

we T '

1 Moreover, we assume that Pais to |K,|ds < C < +00 Prove that if we write t? = Ti/n forO0 <i < n, then

2 Under the same assumptions, prove that

E (= (Muy - Me.,)') = B (M3 - M3) i=l

Conclude that M7; =0 Pas.,andthus Pas Vt < T, M;, = 0

3 J, |Ks|ds is now assumed to be finite‘almost surely as opposed to bounded n y as Opp |

We shall accept the fact that the random variable f |&,|ds is F,-measurable

Show that T,, defined by

‹

t

T„ =inf{0 < s< r, [ |Ks{ds > n}

- 0

(or T if this set is empty) is a stopping time Provethat P as limy.400 Tn =

T Considering the sequence of martingales (M7, )e>0, prove that

‘Pas Vi<T, M, =0

60 Brownian motion and stochastic differential equations

4 Let M; be a martingale of the form Ss H,dW, + Ss K,ds with Ss Hids <

.+00 Pas and Ss |K,|ds < +oo Pas Define the sequence of stopping

times T, = inf{t < T, fs H?ds > n}, in order to prove that K; = 0 dt x

Pas

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

equation

0

We write S; = exp ((u — ơ?/2)‡ + ơW/,)

1 Derive the stochastic differential equation satisfied by S,- 1

2 Prove that

{ dX, = (pXitp')dt+ (0X, + 0')dW,

d(X,S71) = S71 ((u! — o0')dt + o'dW)

3 Obtain the explicit representation of X;

Exercise 18 Let (W2)1>0 be an F;-Brownian motion The purpose of this exercise

is to compute the law of (W:, sup, <; Ws)

1 Consider S a bounded stopping time Apply the optional sampling theorem to

the martingale M, = exp(izW; + 27/2), where z is a real number to prove

that if 0 <u < v then

E (exp (iz(Wu4s5 — Wuts)) |Futs) = exp (—z?( — u)/2)

2 Deduce that WS = Wu4s — Ws is an -F54u-Brownian motion independent

of the ơ-algebra Fs :

3 Let(Y¿);>o be a continuous stochastic process independent of the ø-algebra

such that E(supo<.<x |¥s|) < +00 Let T be a non-negative B-measurable

random variable bounded from above by K: Show that

E(¥r|B) = E(Y)hier-

We shall start by assuming that J can be written as 3°, <;<, tila,, where

0< < - < tạ = K,and the A4; are disjoint B-measurable sets

4 We denote by 7? the inf{s > 0, W, > A}, prove that if f is a bounded Borel

function we have *

where ó(u) = E(f(W,, + A)) Notice that E(f(W + À)) = B(ƒ(—Wv +À))

and prove that =

B (FW) acy) = B(F2- Wel prcy)

5, Show that if we write W? = sup,c, W and if A > 0

P(W, <A, Wi >A) = P(W, 2A, We 3 À) = P(W, > )

Conclude that W; and |W;| have the same probability law ~

4.1 Description of the model

4.1.1 The behaviour of prices -

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 5; at time £) and

a riskless asset (with price S? at time t) We suppose the behaviour of S? to be encapsulated by the following (ordinary) differential equation:

where r is a non-negative constant Note that r is an instantaneous interest rate and should fot be confused with the one-period rate in discrete-time models We

set S? = 1, so that SP? = e”* fort > 0

We assume that the behaviour of the stock price is determined by the following stochastic differential equation:

where yz and o are two constants and (B;) isa standard Brownian motion

The model is valid on the interval [0,7] 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

2

St = So exp (4 - St + øB,) ,

64 The Black-Scholes model

where So is the spot price observed at time 0 One particular result from this model

is that the law of S; is lognormal (i.e its logarithm follows a normal law)

More precisely, we see that the process (S;) is a solution of an equation of type

(4.2) if and only if the process (log(5;)) is a Brownian motion (not necessarily

standard) According to Definition 3.2.1 of Chapter 3, the process (St) has the

following properties:

e continuity of the sample paths; |

e independence of the relative increments: if u < t, St /S or (equivalently), the

relative increment (S; — S,)/S., is independent of the o-algebrao(S.,v < u);

e stationarity of the relative increments: if u < t, the law of (S; — Su)/Su is

identical to the law of (S:-u — So)/So

These three properties express in concrete terms the hypotheses of Black and

Scholes on the behaviour of the share price

4.1.2 Self-financing Strategies

A strategy will be defined as a process ¢ = (Ó:)o<:e<r=((H?, H,)) with values in

IR?, adapted to the natural filtration (F;) 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 ¢ is then given by

Vi (¢) = Hp S? + HeSt

In the discrete-time models, we have characterise self financing strategies by the

equality: Visi(¢) — Va(¢) = on41-(Sn41 — Sn) (see Chapter 1, Remark 1.1.1)

This equality is extended to give the self-financing condition in the continuous-

since the map t ++ S; is continuous, thus bounded on [0, T] almost surely

Definition 4.1.1 A self-financing strategy is defined by a pair of adapted pro-

cesses (HP) c, cp and (Hi)o<t<T satisfying:

Change of probability Representation of martingales 65

T T

W IPla+ [ Hệ dt < +œ a.s

for allt € [0, TỊ

We denote by 5; = e~*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 ¢ = (CHP ,H:))o<tcr be an adapted process with values

in IR’, satisfying fy |HP|dt+ fr H?dt < +ooa.s We set: Vi(¢) = H°S9+H,S,

and V,() = e-**V,(¢) Then, ¢ defines a self-financing strategy if and only if

- t

Ủ,(9) = Wa(đ) + [ Hyd, as (4.3) for allt € [0,7]

Proof Let us consider the self-financing strategy ¢ From equality _

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 prob- ability measure equivalent to the initial probability and under which discounted prices of assets are martingales We were then able to design self-financing strate- gies 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 4.2.1 Equivalent probabilities

Let (2,.A,P) be a probability space A probability measure Q on (2, A) is

absolutely continuous relative to P if

VAEA P(4)=0>Q(4)=

66 , The Black-Scholes model

Theorem 4.2.1 Q is absolutely continuous relative to P if and only if there exists

a non-negative random variable Z on (1, A) such that

VAEA ata) = | Z(w)dP(w)

A

Z is called density of Q 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 ifP(Z>0)=1

4.2.2 The Girsanov theorem

Let (9, +, đo <:<T ,P) be a probability space equipped with the natural fil-

tration of a standard Brownian motion (B:)o<,<7» indexed on the time interval

[0, 7'] The following theorem, which we admit, is known as the Girsanov theorem

(cf Karatzas and Shreve (1988), Dacunha-Castelle and Duflo (1986), Chapter 8)

Theorem 4.2.2 Let (6:)o<e<r be an adapted process satisfying fe 62ds < 00

a.s and such that the process (Lt)o<t<T defined by

t 1 t

It = exp (- [ 6,dB, — >| 02s)

is a martingale Then, under the probability P() with density Lr relative to P,

the process (Wi )o<t<r defined by W, = Bet+ J 6,ds, is a standard Brownian

Remark 4.2.3 A sufficient condition for (Lt)o<e<r to be a martingale is:

E | exp 5 [ 62 dt < 00, : 0

(see Karatzas and Shreve (1988), Dacunha-Castelle and Duflo (1986)) The proof

of Girsanov theorem when (6;) is constant is the purpose of Exercise 19 _

4.2.3 Representation of Brownian martingales

Let (Bi)o<e<r be a standard Brownian motion built on a probability space

(0, F,P) and let (F:)o<e<r be its natural filtration Let us recall (see Chap-

ter 3, Proposition 3.4.4) that if (Hz)o<e<7 is an adapted process such that

E ( fe Hat) < oo, the process ( J H,dB,) is a square-integrable martin-

gale, null at 0 The following theorem shows that any Brownian martingale can be

represented in terms of a stochastic integral

Pricing and hedging options in the Black-Scholes model 67

Theorem 4.2.4 Let (M:)o<:<r be a square-integrable martingale, with respect

to the filtration (Fy)o<t<r There exists an adapted process (H,)o<t<r such that

0

where (H;) is an adapted process such that E ( fe H}ds) < +00 To prove it,

consider the martingale M, = E (U|F,) It can also be shown (see, for example,

Karatzas and Shreve (1988)) that if (M:)o<:<r is a martingale (not necessarily square-integrable) there is arepresentation similar to (4.4) with a proce$s satisfying

only f, H?ds < oo, 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 (5) 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 S; = e—*'S, is a martingale From the stochastic differential equation satisfied by (S;),

From Theorem 4.2.2, with 6, = (—r)/o, there exists a probability P* equivalent

to P under which (W;)o<t<7 is a standard Brownian‘ motion We will admit that the definition of the stochastic integral is invariant by change of equivalent probability (cf Exercise 25) Then, under the probability P*, we deduce from

equality (4.5) that (S;) is a martingale and that

St = So exp(aoW, — ơ2t/2).