A Course in Financial Calculus

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A Course in Financial Calculus

A Course in

Financial Calculus
Alison Etheridge University of Oxford

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521890779 © Cambridge University Press 2002 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 eBook (EBL) ISBN-13 978-0-511-33725-3 ISBN-10 0-511-33725-6 eBook (EBL) ISBN-13 ISBN-10 paperback 978-0-521-89077-9 paperback 0-521-89077-2

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface 1 Single period models Summary 1.1 Some deﬁnitions from ﬁnance 1.2 Pricing a forward 1.3 The one-step binary model 1.4 A ternary model 1.5 A characterisation of no arbitrage 1.6 The risk-neutral probability measure Exercises Binomial trees and discrete parameter martingales Summary 2.1 The multiperiod binary model 2.2 American options 2.3 Discrete parameter martingales and Markov processes 2.4 Some important martingale theorems 2.5 The Binomial Representation Theorem 2.6 Overture to continuous models Exercises Brownian motion Summary 3.1 Deﬁnition of the process 3.2 L´ vy’s construction of Brownian motion e 3.3 The reﬂection principle and scaling 3.4 Martingales in continuous time Exercises Stochastic calculus Summary

page vii 1 1 1 4 6 8 9 13 18 21 21 21 26 28 38 43 45 47 51 51 51 56 59 63 67 71 71 v 2

3

4

vi

contents

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Stock prices are not differentiable Stochastic integration Itˆ ’s formula o Integration by parts and a stochastic Fubini Theorem The Girsanov Theorem The Brownian Martingale Representation Theorem Why geometric Brownian motion? The Feynman–Kac representation Exercises

72 74 85 93 96 100 102 102 107 112 112 112 118 122 126 131 132 134 139 139 139 141 144 149 150 154 159 159 160 163 175 181 185 189 191 193

5

The Black–Scholes model Summary 5.1 The basic Black–Scholes model 5.2 Black–Scholes price and hedge for European options 5.3 Foreign exchange 5.4 Dividends 5.5 Bonds 5.6 Market price of risk Exercises Different payoffs Summary 6.1 European options with discontinuous payoffs 6.2 Multistage options 6.3 Lookbacks and barriers 6.4 Asian options 6.5 American options Exercises Bigger models Summary 7.1 General stock model 7.2 Multiple stock models 7.3 Asset prices with jumps 7.4 Model error Exercises Bibliography Notation Index

6

7

Preface

Financial mathematics provides a striking example of successful collaboration between academia and industry. Advanced mathematical techniques, developed in both universities and banks, have transformed the derivatives business into a multi-trillion-dollar market. This has led to demand for highly trained students and with that demand comes a need for textbooks. This volume provides a ﬁrst course in ﬁnancial mathematics. The inﬂuence of Financial Calculus by Martin Baxter and Andrew Rennie will be obvious. I am extremely grateful to Martin and Andrew for their guidance and for allowing me to use some of the material from their book. The structure of the text largely follows Financial Calculus, but the mathematics, especially the discussion of stochastic calculus, has been expanded to a level appropriate to a university mathematics course and the text is supplemented by a large number of exercises. In order to keep the course to a reasonable length, some sacriﬁces have been made. Most notable is that there was not space to discuss interest rate models, although many of the most popular ones do appear as examples in the exercises. As partial compensation, the necessary mathematical background for a rigorous study of interest rate models is included in Chapter 7, where we brieﬂy discuss some of the topics that one might hope to include in a second course in ﬁnancial mathematics. The exercises should be regarded as an integral part of the course. Solutions to these are available to bona ﬁde teachers from solutions@cambridge.org. The emphasis is on stochastic techniques, but not to the exclusion of all other approaches. In common with practically every other book in the area, we use binomial trees to introduce the ideas of arbitrage pricing. Following Financial Calculus, we also present discrete versions of key deﬁnitions and results on martingales and stochastic calculus in this simple framework, where the important ideas are not obscured by analytic technicalities. This paves the way for the more technical results of later chapters. The connection with the partial differential equation approach to arbitrage pricing is made through both delta-hedging arguments and the Feynman– Kac Stochastic Representation Theorem. Whatever approach one adopts, the key point that we wish to emphasise is that since the theory rests on the assumption of vii viii

preface

absence of arbitrage, hedging is vital. Our pricing formulae only make sense if there is a ‘replicating portfolio’. An early version of this course was originally delivered to ﬁnal year undergraduate and ﬁrst year graduate mathematics students in Oxford in 1997/8. Although we assumed some familiarity with probability theory, this was not regarded as a prerequisite and students on those courses had little difﬁculty picking up the necessary concepts as we met them. Some suggestions for suitable background reading are made in the bibliography. Since a ﬁrst course can do little more than scratch the surface of the subject, we also make suggestions for supplementary and more advanced reading from the bewildering array of available books. This project was supported by an EPSRC Advanced Fellowship. It is a pleasure and a privilege to work in Magdalen College and my thanks go to the President, Fellows, staff and students for making it such an exceptional environment. Many people have made helpful suggestions or read early drafts of this volume. I should especially like to thank Ben Hambly, Alex Jackson and Saurav Sen. Thanks also to David Tranah at CUP who played a vital rˆ le in shaping the project. His input has o been invaluable. Most of all, I should like to thank Lionel Mason for his constant support and encouragement. Alison Etheridge, June 2001

1 Single period models

Summary
In this chapter we introduce some basic deﬁnitions from ﬁnance and investigate the problem of pricing ﬁnancial instruments in the context of a very crude model. We suppose the market to be observed at just two times: zero, when we enter into a ﬁnancial contract; and T , the time at which the contract expires. We further suppose that the market can only be in one of a ﬁnite number of states at time T . Although simplistic, this model reveals the importance of the central paradigm of modern ﬁnance: the idea of a perfect hedge. It is also adequate for a preliminary discussion of the notion of ‘complete market’ and its importance if we are to ﬁnd a ‘fair’ price for our ﬁnancial contract. The proofs in §1.5 can safely be omitted, although we shall from time to time refer back to the statements of the results.

1.1

Some deﬁnitions from ﬁnance
Financial market instruments can be divided into two types. There are the underlying stocks – shares, bonds, commodities, foreign currencies; and their derivatives, claims that promise some payment or delivery in the future contingent on an underlying stock’s behaviour. Derivatives can reduce risk – by enabling a player to ﬁx a price for a future transaction now – or they can magnify it. A costless contract agreeing to pay off the difference between a stock and some agreed future price lets both sides ride the risk inherent in owning a stock, without needing the capital to buy it outright. The connection between the two types of instrument is sufﬁciently complex and uncertain that both trade ﬁercely in the same market. The apparently random nature of the underlying stocks ﬁlters through to the derivatives – they appear random too.

Derivatives

Our central purpose is to determine how much one should be willing to pay for a derivative security. But ﬁrst we need to learn a little more of the language of ﬁnance.
1

2 Deﬁnition 1.1.1

single period models

A forward contract is an agreement to buy (or sell) an asset on a speciﬁed future date, T , for a speciﬁed price, K . The buyer is said to hold the long position, the seller the short position.

Forwards are not generally traded on exchanges. It costs nothing to enter into a forward contract. The ‘pricing problem’ for a forward is to determine what value of K should be written into the contract. A futures contract is the same as a forward except that futures are normally traded on exchanges and the exchange speciﬁes certain standard features of the contract and a particular form of settlement. Forwards provide the simplest examples of derivative securities and the mathematics of the corresponding pricing problem will also be simple. A much richer theory surrounds the pricing of options. An option gives the holder the right, but not the obligation, to do something. Options come in many different guises. Black and Scholes gained fame for pricing a European call option.
Deﬁnition 1.1.2

A European call option gives the holder the right, but not the obligation, to buy an asset at a speciﬁed time, T , for a speciﬁed price, K . A European put option gives the holder the right to sell an asset for a speciﬁed price, K , at time T .

In general call refers to buying and put to selling. The term European is reserved for options whose value to the holder at the time, T , when the contract expires depends on the state of the market only at time T . There are other options, for example American options or Asian options, whose payoff is contingent on the behaviour of the underlying over the whole time interval [0, T ], but the technology of this chapter will only allow meaningful discussion of European options.
Deﬁnition 1.1.3

The time, T , at which the derivative contract expires is called the exercise date or the maturity. The price K is called the strike price.

The pricing problem

So what is the pricing problem for a European call option? Suppose that a company has to deal habitually in an intrinsically risky asset such as oil. They may for example know that in three months time they will need a thousand barrels of crude oil. Oil prices can ﬂuctuate wildly, but by purchasing European call options, with strike K say, the company knows the maximum amount of money that it will need (in three months time) in order to buy a thousand barrels. One can think of the option as insurance against increasing oil prices. The pricing problem is now to determine, for given T and K , how much the company should be willing to pay for such insurance. For this example there is an extra complication: it costs money to store oil. To simplify our task we are ﬁrst going to price derivatives based on assets that can be held without additional cost, typically company shares. Equally we suppose that there is no additional beneﬁt to holding the shares, that is no dividends are paid.

3

1.1 some deﬁnitions from ﬁnance

Payoff

K

ST

K

ST

K

ST

(a)

(b)

(c)

Figure 1.1

Payoff at maturity of (a) a forward purchase, (b) a European call and (c) a European put with strike K as a function of ST .

Assumption

Unless otherwise stated, the underlying asset can be held without additional cost or beneﬁt.

This assumption will be relaxed in Chapter 5. Suppose then that our company enters into a contract that gives them the right, but not the obligation, to buy one unit of stock for price K in three months time. How much should they pay for this contract?
Payoffs

As a ﬁrst step, we need to know what the contract will be worth at the expiry date. If at the time when the option expires (three months hence) the actual price of the underlying stock is ST and ST > K then the option will be exercised. The option is then said to be in the money: an asset worth ST can be purchased for just K . The value to the company of the option is then (ST − K ). If, on the other hand, ST < K , then it will be cheaper to buy the underlying stock on the open market and so the option will not be exercised. (It is this freedom not to exercise that distinguishes options from futures.) The option is then worthless and is said to be out of the money. (If ST = K the option is said to be at the money.) The payoff of the option at time T is thus (ST − K )+ max {(ST − K ), 0} .

Figure 1.1 shows the payoff at maturity of three derivative securities: a forward purchase, a European call and a European put, each as a function of stock price at maturity. Before embarking on the valuation at time zero of derivative contracts, we allow ourselves a short aside.
Packages

We have presented the European call option as a means of reducing risk. Of course it can also be used by a speculator as a bet on an increase in the stock price. In fact by holding packages, that is combinations of the ‘vanilla’ options that we have described so far, we can take rather complicated bets. We present just one example; more can be found in Exercise 1.

4 Example 1.1.4 (A straddle)

single period models

Suppose that a speculator is expecting a large move in a stock price, but does not know in which direction that move will be. Then a possible combination is a straddle. This involves holding a European call and a European put with the same strike price and maturity.

Explanation: The payoff of this straddle is (ST − K )+ (from the call) plus (K −

ST )+ (from the put), that is, |ST − K |. Although the payoff of this combination is always positive, if, at the expiry time, the stock price is too close to the strike price then the payoff will not be sufﬁcient to offset the cost of purchasing the options and the investor makes a loss. On the other hand, large movements in price can lead to substantial proﬁts. ✷

1.2

Pricing a forward
In order to solve our pricing problems, we are going to have to make some assumptions about the way in which markets operate. To formulate these we begin by discussing forward contracts in more detail. Recall that a forward contract is an agreement to buy (or sell) an asset on a speciﬁed future date for a speciﬁed price. Suppose then that I agree to buy an asset for price K at time T . The payoff at time T is just ST − K , where ST is the actual asset price at time T . The payoff could be positive or it could be negative and, since the cost of entering into a forward contract is zero, this is also my total gain (or loss) from the contract. Our problem is to determine the fair value of K .

Expectation pricing

At the time when the contract is written, we don’t know ST , we can only guess at it, or, more formally, assign a probability distribution to it. A widely used model (which underlies the Black–Scholes analysis of Chapter 5) is that stock prices are lognormally distributed. That is, there are constants ν and σ such that the logarithm of ST /S0 (the stock price at time T divided by that at time zero, usually called the return) is normally distributed with mean ν and variance σ 2 . In symbols: P ST ∈ [a, b] = P log S0 = log b log a

ST ∈ [log a, log b] S0 (x − ν)2 1 exp − √ 2σ 2 2π σ

d x.

Notice that stock prices, and therefore a and b, should be positive, so that the integral on the right hand side is well deﬁned. Our ﬁrst guess might be that E[ST ] should represent a fair price to write into our contract. However, it would be a rare coincidence for this to be the market price. In fact we’ll show that the cost of borrowing is the key to our pricing problem.
The risk-free rate

We need a model for the time value of money: a dollar now is worth more than a dollar promised at some later time. We assume a market for these future promises (the bond market) in which prices are derivable from some interest rate. Speciﬁcally:

5

1.2 pricing a forward

Time value of money We assume that for any time T less than some horizon τ the value now of a dollar promised at T is e−r T for some constant r > 0. The rate r is then the continuously compounded interest rate for this period.

Such a market, derived from say US Government bonds, carries no risk of default – the promise of a future dollar will always be honoured. To emphasise this we will often refer to r as the risk-free interest rate. In this model, by buying or selling cash bonds, investors can borrow money for the same risk-free rate of interest as they can lend money. Interest rate markets are not this simple in practice, but that is an issue that we shall defer.
Arbitrage pricing

We now show that it is the risk-free interest rate, or equivalently the price of a cash bond, and not our lognormal model that forces the choice of the strike price, K , upon us in our forward contract. Interest rates will be different for different currencies and so, for deﬁniteness, suppose that we are operating in the dollar market, where the (risk-free) interest rate is r . • Suppose ﬁrst that K > S0 er T . The seller, obliged to deliver a unit of stock for $K at time T , adopts the following strategy: she borrows $S0 at time zero (i.e. sells bonds to the value $S0 ) and buys one unit of stock. At time T , she must repay $S0 er T , but she has the stock to sell for $K , leaving her a certain proﬁt of $(K − S0 er T ). If K < S0 er T , then the buyer reverses the strategy. She sells a unit of stock at time zero for $S0 and buys cash bonds. At time T , the bonds deliver $S0 er T of which she uses $K to buy back a unit of stock leaving her with a certain proﬁt of $(S0 er T − K ). Unless K = S0 er T , one party is guaranteed to make a proﬁt.
Deﬁnition 1.2.1

•

An opportunity to lock into a risk-free proﬁt is called an arbitrage

opportunity. The starting point in establishing a model in modern ﬁnance theory is to specify that there is no arbitrage. (In fact there are people who make their living entirely from exploiting arbitrage opportunities, but such opportunities do not exist for a signiﬁcant length of time before market prices move to eliminate them.) We have proved the following lemma.
Lemma 1.2.2

In the absence of arbitrage, the strike price in a forward contract with expiry date T on a stock whose value at time zero is S0 is K = S0 er T , where r is the risk-free rate of interest. The price S0 er T is sometimes called the arbitrage price. It is also known as the forward price of the stock.

6

single period models Remark: In our proof of Lemma 1.2.2, the buyer sold stock that she may not own.

This is known as short selling. This can, and does, happen: investors can ‘borrow’ stock as well as money. ✷ Of course forwards are a very special sort of derivative. The argument above won’t tell us how to value an option, but the strategy of seeking a price that does not provide either party with a risk-free proﬁt will be fundamental in what follows. Let us recap what we have done. In order to price the forward, we constructed a portfolio, comprising one unit of underlying stock and −S0 cash bonds, whose value at the maturity time T is exactly that of the forward contract itself. Such a portfolio is said to be a perfect hedge or replicating portfolio. This idea is the central paradigm of modern mathematical ﬁnance and will recur again and again in what follows. Ironically we shall use expectation repeatedly, but as a tool in the construction of a perfect hedge.

1.3

The one-step binary model
We are now going to turn to establishing the fair price for European call options, but in order to do so we ﬁrst move to a simpler model for the movement of market prices. Once again we suppose that the market is observed at just two times, that at which the contract is struck and the expiry date of the contract. Now, however, we shall suppose that there are just two possible values for the stock price at time T . We begin with a simple example.

Pricing a European call

Example 1.3.1

Suppose that the current price in Japanese Yen of a certain stock is 2500. A European call option, maturing in six months time, has strike price 3000. An investor believes that with probability one half the stock price in six months time will be 4000 and with probability one half it will be 2000. He therefore calculates the expected value of the option (when it expires) to be 500. The riskless borrowing rate in Japan is currently zero and so he agrees to pay 500 for the option. Is this a fair price?

Solution: In the light of the previous section, the reader will probably have guessed

that the answer to this question is no. Once again, we show that one party to this contract can make a risk-free proﬁt. In this case it is the seller of the contract. Here is just one of the many possible strategies that she could adopt.
Strategy: At time zero, sell the option, borrow

2000 and buy a unit of stock.

•

•

Suppose ﬁrst that at expiry the price of the stock is 4000; then the contract will be exercised and so she must sell her stock for 3000. She then holds (−2000+3000). That is 1000. If, on the other hand, at expiry the price of the stock is 2000, then the option will not be exercised and so she sells her stock on the open market for just 2000. Her

7

1.3 the one-step binary model x2 (x1, x 2 ) x1 x 1 + 4000x2 = 1000 x 1 + 2000x2 = 0

Figure 1.2

The seller of the contract in Example 1.3.1 is guaranteed a risk-free proﬁt if she can buy any portfolio in the shaded region.

net cash holding is then

(−2000 + 2000). That is, she exactly breaks even.

Either way, our seller has a positive chance of making a proﬁt with no risk of making a loss. The price of the option is too high. So what is the right price for the option? Let’s think of things from the point of view of the seller. Writing ST for the price of the stock when the contract expires, she knows that at time T she needs (ST − 3000)+ in order to meet the claim against her. The idea is to calculate how much money she needs at time zero, to be held in a combination of stocks and cash, to guarantee this. Suppose then that she uses the money that she receives for the option to buy a portfolio comprising x1 Yen and x2 stocks. If the price of the stock is 4000 at expiry, then the time T value of the portfolio is x1 er T + 4000x2 . The seller of the option requires this to be at least 1000. That is, since interest rates are zero, x1 + 4000x2 ≥ 1000. If the price is 2000 she just requires the value of the portfolio to be non-negative, x1 + 2000x2 ≥ 0. A proﬁt is guaranteed (without risk) for the seller if (x1 , x2 ) lies in the interior of the shaded region in Figure 1.2. On the boundary of the region, there is a positive probability of proﬁt and no probability of loss at all points other than the intersection of the two lines. The portfolio represented by the point (x 1 , x 2 ) will provide exactly the wealth required to meet the claim against her at time T . Solving the simultaneous equations gives that the seller can exactly meet the claim if x 1 = −1000 and x 2 = 1/2. The cost of building this portfolio at time zero is (−1000 + 2500/2), that is 250. For any price higher than 250, the seller can make a risk-free proﬁt.

8

single period models

If the option price is less than 250, then the buyer can make a risk-free proﬁt by ‘borrowing’ the portfolio (x 1 , x 2 ) and buying the option. In the absence of arbitrage then, the fair price for the option is 250. ✷ Notice that just as for our forward contract, we did not use the probabilities that we assigned to the possible market movements to arrive at the fair price. We just needed the fact that we could replicate the claim by this simple portfolio. The seller can hedge the contingent claim (ST − 3000)+ using the portfolio consisting of x1 and x2 units of stock.
Pricing formula for European options

One can use exactly the same argument to prove the following result.
Lemma 1.3.2 Suppose that the risk-free dollar interest rate (to a time horizon τ > T ) is r . Denote the time zero (dollar) value of a certain asset by S0 . Suppose that the motion of stock prices is such that the value of the asset at time T will be either S0 u or S0 d. Assume further that

d < er T < u. At time zero, the market price of a European option with payoff C(ST ) at the maturity T is 1 − de−r T ue−r T − 1 C (S0 u) + C (S0 d) . u−d u−d Moreover, the seller of the option can construct a portfolio whose value at time T is exactly (ST − K )+ by using the money received for the option to buy φ C (S0 u) − C (S0 d) S0 u − S0 d (1.1)

units of stock at time zero and holding the remainder in bonds. The proof is Exercise 4(a).

1.4

A ternary model
There were several things about the binary model that were very special. In particular we assumed that we knew that the asset price would be one of just two speciﬁed values at time T . What if we allow three values? We can try to repeat the analysis of §1.3. Again the seller would like to replicate the claim at time T by a portfolio consisting of x1 and x2 stocks. This time there will be three scenarios to consider, corresponding to the three possible values of ST . If interest rates are zero, this gives rise to the three inequalities i i x1 + ST x2 ≥ (ST − 3000)+ ,

i = 1, 2, 3,

i where ST are the possible values of ST . The picture is now something like that in Figure 1.3.

9

1.5 a characterisation of no arbitrage x2 x1i + ST x2 = (STi – 3000)+ x1

i

Figure 1.3

If the stock price takes three possible values at time T , then at any point where the seller of the option has no risk of making a loss, she has a strictly positive chance of making a proﬁt.

In order to be guaranteed to meet the claim at time T , the seller requires (x1 , x2 ) to lie in the shaded region, but at any point in that region, she has a strictly positive probability of making a proﬁt and zero probability of making a loss. Any portfolio from outside the shaded region carries a risk of a loss. There is no portfolio that exactly replicates the claim and there is no unique ‘fair’ price for the option. Our market is not complete. That is, there are contingent claims that cannot be perfectly hedged.
Bigger models

Of course we are tying our hands in our efforts to hedge a claim. First, we are only allowing ourselves portfolios consisting of the underlying stock and cash bonds. Real markets are bigger than this. If we allow ourselves to trade in a third ‘independent’ asset, then our analysis leads to three non-parallel planes in R3 . These will intersect in a single point representing a portfolio that exactly replicates the claim. This then raises a question: when is there arbitrage in larger market models? We shall answer this question for a single period model in the next section. The second constraint that we have placed upon ourselves is that we are not allowed to adjust our portfolio between the time of the selling of the contract and its maturity. In fact, as we see in Chapter 2, if we consider the market to be observable at intermediate times between zero and T , and allow our seller to rebalance her portfolio at such times (without changing its value), then we can allow any number of possible values for the stock price at time T and yet still replicate each claim at time T by a portfolio consisting of just the underlying and cash bonds.

1.5

A characterisation of no arbitrage
In our binary setting it was easy to ﬁnd the right price for an option simply by solving a pair of simultaneous equations. However, the binary model is very special and, after our experience with the ternary model, alarm bells may be ringing. The binary model describes the evolution of just one stock (and one bond). One solution to our

10

single period models

difﬁculties with the ternary model was to allow trade in another ‘independent’ asset. In this section we extend this idea to larger market models and characterise those models for which there are a sufﬁcient number of independent assets that any option has a fair price. Other than Deﬁnition 1.5.1 and the statement of Theorem 1.5.2, this section can safely be omitted.
A market with N assets

Our market will now consist of a ﬁnite (but possibly large) number of tradable assets. Again we restrict ourselves to single period models, in which the market is observable only at time zero and a ﬁxed future time T . However, the extension to multiple time periods exactly mirrors that for binary models that we describe in §2.1. Suppose then that there are N tradable assets in the market. Their prices at time zero are given by the column vector 
1 2 N S0 = S0 , S0 , . . . , S0 t

 2  S0   .  .   .  .
N S0

1 S0



Notation

For vectors and matrices we shall use the superscript ‘t’ to denote

transpose.

Uncertainty about the market is represented by a ﬁnite number of possible states in which the market might be at time T that we label 1, 2, . . . , n. The security values at time T are given by an N × n matrix D = (Di j ), where the coefﬁcient Di j is the value of the ith security at time T if the market is in state j. Our binary model corresponds to N = 2 (the stock and a riskless cash bond) and n = 2 (the two states being determined by the two possible values of ST ). In this notation, a portfolio can be thought of as a vector θ = (θ1 , θ2 , . . . , θn )t ∈ N , whose market value at time zero is the scalar product S · θ = S 1 θ + S 2 θ + R 0 0 1 0 2 N · · · + S0 θ N . The value of the portfolio at time T is a vector in Rn whose ith entry is the value of the portfolio if the market is in state i. We can write the value at time T as  D11 θ1 + D21 θ2 + · · · + D N 1 θ N     = D t θ.  

 D θ + D θ + ··· + D θ 22 2 N2 N  12 1  .  .  . D1n θ1 + D2n θ2 + · · · + D N n θ N

11

1.5 a characterisation of no arbitrage

Notation

For a vector x ∈ Rn we write x ≥ 0, or x ∈ Rn , if x = + (x1 , . . . , xn ) and xi ≥ 0 for all i = 1, . . . , n. We write x > 0 to mean x ≥ 0, x = 0. Notice that x > 0 does not require x to be strictly positive in all its coordinates. We write x 0, or x ∈ Rn , for vectors in Rn that are ++ strictly positive in all coordinates.

In this notation, an arbitrage is a portfolio θ ∈ R N with either S0 · θ ≤ 0, Dt θ > 0 or S0 · θ < 0, D t θ ≥ 0.

Arbitrage pricing

The key to arbitrage pricing in this model is the notion of a state price vector.
Deﬁnition 1.5.1

A state price vector is a vector ψ ∈ Rn such that S0 = Dψ. ++         .  (1.2)

To see why this terminology is natural, we ﬁrst expand this to obtain   S2  0   .  . . N S0
1 S0



    = ψ1     

D11 D21 . . . DN 1

     + ψ2   

D12 D22 . . . DN 2

     + · · · + ψn   

D1n D2n . . . DN n

The vector, D (i) , multiplying ψi is the security price vector if the market is in state i. We can think of ψi as the marginal cost at time zero of obtaining an additional unit of wealth at the end of the time period if the system is in state i. In other words, if at the end of the time period, the market is in state i, then the value of our portfolio increases by one for each additional ψi of investment at time zero. To see this, suppose that we can ﬁnd vectors θ (i) ∈ R N 1≤i≤n such that θ (i) · D ( j) = 1 0 if i = j, otherwise.

That is, the value of the portfolio θ (i) at time T is the indicator function that the market is in state i. Then, using equation (1.2), the cost of purchasing θ (i) at time n ( j) · θ (i) = ψ . Such portfolios {θ (i) } zero is precisely S0 · θ (i) = i 1≤i≤n j=1 ψ j D are called Arrow–Debreu securities. We shall ﬁnd a convenient way to think about the state price vector in §1.6, but ﬁrst, here is the key result.
Theorem 1.5.2

For the market model described above there is no arbitrage if and only if there is a state price vector.

12

single period models

R

n

K

M

R

1

Figure 1.4

There is no arbitrage if and only if the regions K and M of Theorem 1.5.2 intersect only at the origin.

This result, due to Harrison & Kreps (1979), is the simplest form of what is often known as the Fundamental Theorem of Asset Pricing. The proof is an application of a Hahn–Banach Separation Theorem, sometimes called the Separating Hyperplane Theorem. We shall also need the Riesz Representation Theorem. Recall that M ⊆ Rd is a cone if x ∈ M implies λx ∈ M for all strictly positive scalars λ and that a linear functional on Rd is a linear mapping F: Rd → R. Suppose M and K are closed convex cones in Rd that intersect precisely at the origin. If K is not a linear subspace, then there is a non-zero linear functional F such that F(x) < F(y) for each x ∈ M and each non-zero y ∈ K .
Theorem 1.5.3 (Separating Hyperplane Theorem)

This version of the Separating Hyperplane Theorem can be found in Dufﬁe (1992). Any bounded linear functional on Rd can be written as F(x) = v0 · x. That is F(x) is the scalar product of some ﬁxed vector v0 ∈ Rd with x.
Theorem 1.5.4 (Riesz Representation Theorem) Proof of Theorem 1.5.2: We take d = 1 + n in Theorem 1.5.3 and set

M=

−S0 · θ, D t θ : θ ∈ R N ⊆ R × Rn = R1+n ,

K = R + × Rn . + Note that K is a cone and not a linear space, M is a linear space. Evidently, there is no arbitrage if and only if K and M intersect precisely at the origin as shown in

13

1.6 the risk-neutral probability measure

Figure 1.4. We must prove that K ∩ M = {0} if and only if there is a state price vector. (i) Suppose ﬁrst that K ∩ M = {0}. From Theorem 1.5.3, there is a linear functional F: Rd → R such that F(z) < F(x) for all z ∈ M and non-zero x ∈ K . The ﬁrst step is to show that F must vanish on M. We exploit the fact that M is a linear space. First observe that F(0) = 0 (by linearity of F) and 0 ∈ M, so F(x) ≥ 0 for x ∈ K and F(x) > 0 for x ∈ K \{0}. Fix x0 ∈ K with x0 = 0. Now take an arbitrary z ∈ M. Then F(z) < F(x0 ), but also, since M is a linear space, λF(z) = F(λz) < F(x0 ) for all λ ∈ R. This can only hold if F(z) = 0. z ∈ M was arbitrary and so F vanishes on M as required. We now use this actually to construct explicitly the state price vector from F. First we use the Riesz Representation Theorem to write F as F(x) = v0 · x for some v0 ∈ Rd . It is convenient to write v0 = (α, φ) where α ∈ R and φ ∈ Rn . Then F(v, c) = αv + φ · c for any (v, c) ∈ R × Rn = Rd .

Since F(x) > 0 for all non-zero x ∈ K , we must have α > 0 and φ 0 (consider a vector along each of the coordinate axes). Finally, since F vanishes on M, −αS0 · θ + φ · D t θ = 0 for all θ ∈ R N .

Observing that φ · D t θ = (Dφ) · θ, this becomes −αS0 · θ + (Dφ) · θ = 0 for all θ ∈ R N ,

which implies that −αS0 + Dφ = 0. In other words, S0 = D(φ/α). The vector ψ = φ/α is a state price vector. (ii) Suppose now that there is a state price vector, ψ. We must prove that K ∩M = {0}. By deﬁnition, S0 = Dψ and so for any portfolio θ , S0 · θ = (Dψ) · θ = ψ · (D t θ). (1.3)

Suppose that for some portfolio θ , (−S0 · θ, D t θ) ∈ K . Then D t θ ∈ Rn and + 0, if D t θ ∈ Rn , then ψ · (D t θ) ≥ 0 which, by −S0 · θ ≥ 0. But since ψ + equation (1.3), tells us that S0 · θ ≥ 0. Thus it must be that S0 · θ = 0 and D t θ = 0. That is, K ∩ M = {0}, as required. ✷

1.6

The risk-neutral probability measure
The state price vector then is the key to arbitrage pricing for our multiasset market models. Although we have an economic interpretation for it, in order to pave the way for the full machinery of probability and martingales we must think about it in a different way. Recall that all the entries of ψ are strictly positive.

14 State prices and probability

single period models

Writing ψ0 =

n i=1 ψi ,

we can think of ψ ψ1 ψ2 ψn , ,... , ψ0 ψ0 ψ0 t (1.4)

as a vector of probabilities for being in different states. It is important to emphasise that they may have nothing to do with our view of how the markets will move. First of all, What is ψ0 ? Suppose that as in our binary model (where we had a risk-free cash bond) the market allows positive riskless borrowing. In this general setting we just suppose that we can replicate such a bond by a portfolio θ for which   1  1    Dt θ =  .  ,  .  . 1 i.e. the value of the portfolio at time T is one, no matter what state the market is in. Using the fact that ψ is a state price vector, we calculate that the cost of such a portfolio at time zero is n S0 · θ = (Dψ) · θ = ψ · (D t θ) = i=1 ψi = ψ0 .

That is ψ0 represents the discount on riskless borrowing. In our notation of §1.2, ψ0 = e−r T .
Expectation recovered

Now under the probability distribution given by the vector (1.4), the expected value of the ith security at time T is n i E ST = j=1

Di j

ψj 1 = ψ0 ψ0

n

Di j ψ j = j=1 1 i S , ψ0 0

where in the last equality we have used S0 = Dψ. That is i i S0 = ψ0 E ST ,

i = 1, . . . , n.

(1.5)

Any security’s price is its discounted expected payoff under the probability distribution (1.4). The same must be true of any portfolio. This observation gives us a new way to think about the pricing of contingent claims.
Deﬁnition 1.6.1

We shall say that a claim, C, at time T is attainable if it can be hedged. That is, if there is a portfolio whose value at time T is exactly C.

Notation

When we wish to emphasise the underlying probability measure, Q, we write EQ for the expectation operator.

15

1.6 the risk-neutral probability measure

If there is no arbitrage, the unique time zero price of an attainable Theorem 1.6.2 claim C at time T is ψ0 EQ [C] where the expectation is with respect to any i i probability measure Q for which S0 = ψ0 EQ [ST ] for all i and ψ0 is the discount on riskless borrowing.
Remark: Notice that it is crucial that the claim is attainable (see Exercise 11).

✷

Proof of Theorem 1.6.2: By Theorem 1.5.2 there is a state price vector and this leads i i to the probability measure (1.4) satisfying S0 = ψ0 E ST for all i. Since the claim can be hedged, there is a portfolio θ such that θ · ST = C. In the absence of arbitrage, the time zero price of the claim is the cost of this portfolio at time zero, N

θ · S0 = θ · (ψ0 E[ST ]) = ψ0 i=1 i θi E[ST ] = ψ0 E[θ · ST ].

The same value is obtained if the expectation is calculated for any vector of i i probabilities, Q, such that S0 = ψ0 EQ ST since, in the absence of arbitrage, there is only one riskless borrowing rate and this completes the proof. ✷
Risk-neutral pricing

In this language, our arbitrage pricing result says that if we can ﬁnd a probability vector for which the time zero value of each underlying security is its discounted expected value at time T then we can ﬁnd the time zero value of any attainable contingent claim by calculating its discounted expectation. Notice that we use the same probability vector, whatever the claim.
Deﬁnition 1.6.3

If our market can be in one of n possible states at time T , then 0, of probabilities for which each security’s any vector, p = ( p1 , p2 , . . . , pn ) price is its discounted expected payoff is called a risk-neutral probability measure or equivalent martingale measure. The term equivalent reﬂects the condition that p 0; cf. Deﬁnition 2.3.12. Our simple form of the Fundamental Theorem of Asset Pricing (Theorem 1.5.2) says that in a market with positive riskless borrowing there is no arbitrage if and only if there is an equivalent martingale measure. We shall refer to the process of pricing by taking expectations with respect to a risk-neutral probability measure as risk-neutral pricing.

Example 1.3.1 revisited Let us return to our very ﬁrst example of pricing a European call option and conﬁrm that the above formula really does give us the arbitrage price. Here we have just two securities, a cash bond and the underlying stock. The discount on borrowing is ψ0 = e−r T , but we are assuming that the Yen interest rate is zero, so ψ0 = 1. The matrix of security values at time T is given by

D=

1 4000

1 . 2000

16

single period models

Writing p for the risk-neutral probability that the security price vector is (1, 4000)t , if the stock price is to be equal to its discounted expected payoff, p must solve 4000 p + 2000(1 − p) = 2500, which gives p = 0.25. The contingent claim is 1000 if the stock price at expiry is 4000 and zero otherwise. The expected value of the claim under the risk-neutral probability, and therefore (since interest rates are zero) the price of the option, is then 0.25 × 1000 = 250, as before. An advantage of this approach is that, armed with the probability p, it is now a trivial matter to price all European options on this stock with the same expiry date (six months time) by taking expectations with respect to the same probability measure. For example, for a European put option with strike price 3500, the price is E (K − ST )+ = 0.75 × 1500 = 1125. Our original argument would lead to a new set of simultaneous equations for each new claim. ✷
Complete markets

We now have a prescription for the arbitrage price of a claim if one exists, that is if the claim is attainable. But we must be a little cautious. Arbitrage prices only exist for attainable claims – even though the prescription may continue to make sense.
Deﬁnition 1.6.4

A market is said to be complete if every contingent claim is attainable, i.e. if every possible derivative claim can be hedged. A market consisting of N tradable assets, evolving according to a single period model in which at the end of the time period the market is one of n possible states, is complete if and only if N ≥ n and the rank of the matrix, D, of security prices is n.

Proposition 1.6.5

Proof: Any claim in our market can be expressed as a vector v ∈ Rn . A hedge for

that claim will be a portfolio θ = θ (v) ∈ R N for which D t θ = v. Finding such a θ amounts to solving n equations in N unknowns. Thus a hedging portfolio exists for every choice of v if and only if N ≥ n and the rank of D is n, as required. ✷ Notice in particular that our single period binary model is complete. Suppose that our market is complete and arbitrage-free and let Q and Q be any two equivalent martingale measures. By completeness every claim is attainable, so for every random variable X , using that there is only one risk-free rate, EQ [X ] = EQ [X ] .

In other words Q = Q . So in a complete arbitrage-free market the equivalent martingale measure is unique.

17 The main results so far

1.6 the risk-neutral probability measure

Let us summarise the results for our single period markets. They will be reﬂected again and again in what follows.

Results for single period models

• The market is arbitrage-free if and only if there exists a martingale measure, Q. • The market is complete if and only if Q is unique. • The arbitrage price of an attainable claim C is e−r T EQ [C] .

Martingale measures are a powerful tool. However, in an incomplete market, if a claim C is not attainable different martingale measures can give different prices. The arbitrage-free notion of fair price only makes sense if we can hedge.
Trading in two different markets

We must sound just one more note of caution. It is important in calculating the risk-neutral probabilities that all the assets being modelled are tradable in the same market. We illustrate with an example. Suppose that in the US dollar markets the current Sterling exchange rate is 1.5 (so that £100 costs $150). Consider a European call option that offers the holder the right to buy £100 for $150 at time T . The riskless borrowing rate in the UK is u and that in the US is r . Assuming a single period binary model in which the exchange rate at the expiry time is either 1.65 or 1.45, ﬁnd the fair price of this option.
Example 1.6.6 Solution: Now we have a problem. The exchange rate is not tradable. Nor, in dollar markets, is a Sterling cash bond – it is a tradable instrument, but in Sterling markets. However, the product of the two is a dollar tradable and we shall denote the value of this product by St at time t. Now, since the riskless interest rate in the UK is u, the time zero price of a Sterling cash bond, promising to pay £1 at time T , is e−uT and, of course, at time T the bond price is one. Thus we have S0 = e−uT 150 and ST = 165 or ST = 145. Let p be the risk-neutral probability that ST = 165. Then, since the discounted price (in the dollar market) of our ‘asset’ at time T must have expectation S0 , we obtain

150e−uT = e−r T (165 p + 145(1 − p)) , which yields p= 150e(r −u)T − 145 . 20

The price of the option is the discounted expected payoff with respect to this

18

single period models

probability which gives V0 = e−r T 15 p = 3 150e−uT − 145e−r T . 4 ✷

Exercises
1 What view about the market is reﬂected in each of the following strategies? (a) Bullish vertical spread: Buy one European call and sell a second one with the same expiry date, but a larger strike price. (b) Bearish vertical spread: Buy one European call and sell a second one with the same expiry date but a smaller strike price. (c) Strip: Buy one European call and two European puts with the same exercise date and strike price. (d) Strap: Buy two European calls and one European put with the same exercise date and strike price. (e) Strangle: Buy a European call and a European put with the same expiry date but different strike prices (consider all possible cases). 2 A butterﬂy spread represents the complementary bet to the straddle. It has the following payoff at expiry:

Payoff

E1

E2

ST

Find a portfolio consisting of European calls and puts, all with the same expiry date, that has this payoff. 3 Suppose that the price of a certain asset has the lognormal distribution. That is log (ST /S0 ) is normally distributed with mean ν and variance σ 2 . Calculate E[ST ].

4 (a) Prove Lemma 1.3.2. (b) What happens if we drop the assumption that d < er T < u? 5 Suppose that at current exchange rates, £100 is worth e160. A speculator believes that by the end of the year there is a probability of 1/2 that the pound will have fallen to e1.40, and a 1/2 chance that it will have gained to be worth e2.00. He therefore buys a European put option that will give him the right (but not the obligation) to

19

exercises

sell £100 for e1.80 at the end of the year. He pays e20 for this option. Assume that the risk-free interest rate is zero across the Euro-zone. Using a single period binary model, either construct a strategy whereby one party is certain to make a proﬁt or prove that this is the fair price. 6 7 How should we modify the analysis of Example 1.3.1 if we are pricing an option based on a commodity such as oil? Show that if there is no arbitrage in the market, then any portfolio constructed at time zero that exactly replicates a claim C at time T has the same value at time zero.

8 Put–call parity: Denote by Ct and Pt respectively the prices at time t of a European call and a European put option, each with maturity T and strike K . Assume that the risk-free rate of interest is constant, r , and that there is no arbitrage in the market. Show that for each t ≤ T , Ct − Pt = St − K e−r (T −t) .

9 Use risk-neutral pricing to value the option in Exercise 5. Check your answer by constructing a portfolio that exactly replicates the claim at the expiry of the contract. 10 What is the payoff of a forward at expiry? Use risk-neutral pricing to solve the pricing problem for a forward contract. 11 Consider the ternary model for the underlying of §1.4. How many equivalent martingale measures are there? If there are two different martingale measures, do they give the same price for a claim? Are there arbitrage opportunities? 12 Suppose that the value of a certain stock at time T is a random variable with distribution P. Note we are not assuming a binary model. An option written on this stock has payoff C at time T . Consider a portfolio consisting of φ units of the underlying and ψ units of bond, held until time T , and write V0 for its value at time zero. Assuming that interest rates are zero, show that the extra cash required by the holder of this portfolio to meet the claim C at time T is C − V0 − φ (ST − S0 ) . Find expressions for the values of V0 and φ (in terms of E [ST ], E [C], var [ST ] and cov (ST , C)) that minimise E 2 , and check that for these values E [ ] = 0. Prove that for a binary model, any claim C depends linearly on ST − S0 . Deduce that in this case we can ﬁnd V0 and φ such that = 0. When the model is not complete, the parameters that minimise E 2 correspond to ﬁnding the best linear approximation to C (based on ST − S0 ). The corresponding value of the expectation is a measure of the intrinsic risk in the option.

20

single period models

13 Exchange rate forward: Suppose that the riskless borrowing rate in the UK is u and that in the USA is r . A dollar investor wishes to set the exchange rate, C T , in a forward contract in which the two parties agree to exchange C T dollars for one pound at time T . If a pound is currently C0 dollars, what is the fair value of C T ? 14 The option writer in Example 1.6.6 sells a digital option to a speculator. This amounts to a bet that the asset price will go up. The payoff is a ﬁxed amount of cash if the exchange rate goes to $165 per £100, and nothing if it goes down. If the speculator pays $10 for this bet, what cash payout should the option writer be willing to write into the option? You may assume that interest rates are zero. 15 Suppose now that the seller of the option in Example 1.6.6 operates in the Sterling markets. Reexpress the market in terms of Sterling tradables and ﬁnd the corresponding risk-neutral probabilities. Are they the same as the risk-neutral probabilities calculated by the dollar trader? What is the dollar cost at time zero of the option as valued by the Sterling trader? This is an example of change of numeraire. The dollar trader uses the dollar bond as the reference risk-free asset whereas the Sterling trader uses a Sterling bond.

2 Binomial trees and discrete parameter martingales

Summary
In this chapter we build some more sophisticated market models that track the evolution of stock prices over a succession of time periods. Over each individual time period, the market follows our simple binary model of Chapter 1. The possible trajectories of the stock prices are then encoded in a tree. A simple corollary of our work of Chapter 1 will allow us to price claims by taking expectation with respect to certain probabilities on the tree under which the stock price process is a discrete parameter martingale. Deﬁnitions and basic properties of discrete parameter martingales are presented and illustrated in §2.3, and we see for the ﬁrst time how martingale methods can be employed as an elegant computational tool. Then, §2.4 presents some important martingale theorems. In §2.5 we pave the way for the Black–Scholes analysis of Chapter 5 by showing how to construct, in the martingale framework, the portfolio that replicates a claim. In §2.6 we preview the Black–Scholes formula with a heuristic passage to the limit.

2.1

The multiperiod binary model
Our single period binary model is, of course, inadequate as a model of the evolution of an asset price. In particular, we have allowed ourselves to observe the market at just two times, zero and T . Moreover, at time T , we have supposed the stock price to take one of just two possible values. In this section we construct more sophisticated market models by stringing together copies of our single period model into a tree. Once again our ﬁnancial market will consist of just two instruments, the stock and a cash bond. As before we assume that unlimited amounts of both can be bought and sold without transaction costs. There is no risk of default on a promise and the market is prepared to buy and sell a security for the same price (that is, there is no bid–offer spread).
21

22

trees and martingales

S3 S2 S1 S2
01 0 00

000

S3

001

S3 011 S3
100

010

S0 S2 S1 S2 time
Figure 2.1 The tree of stock prices.
11 111 1 10

S3 101 S3
110

S3 S3

We suppose the market to be observable at times 0 = t0 < t1 < · · · < t N = T .
The stock

Over each time period [ti , ti+1 ] the stock follows the binary model. This is illustrated in Figure 2.1. After i time periods, the stock can have any of 2i possible values. However, given its value at time ti there are only two admissible possibilities for the stock price at time ti+1 . It is not necessary, but it is conventional, to suppose that all time periods have the same length and so we shall write ti = iδt where δt = T /N . In our simple model, the cash bond behaved entirely predictably. There was a known interest rate, r , and the cash bond increased in value over a time period of length T by a factor er T . Now, we do not have to impose such a stringent condition. The interest rate can itself be random, varying over different time periods. Our work will generalise immediately provided that we insist that the interest rate over the time interval [ti , ti+1 ) is known at the start of that interval, although it may depend on which of the 2i nodes our market is in. In this way, we admit the possibility of randomness in our cash bond. Notice however that it is a very different sort of randomness from that of the stock. The value of the bond at time ti+1 is already known to us at time ti . This is certainly not true for the stock. In spite of our newfound freedom, for simplicity, we shall continue to suppose that the interest rate is the constant, r .

The cash bond

23 Replicating portfolios

2.1 the multiperiod binary model

At ﬁrst sight it is not clear that we can make progress with our new model. For a tree consisting of k time steps there are 2k possible values for the stock price. If we now look back at Proposition 1.6.5, this suggests that we need at least 2k stocks to be traded in our market if we want it to be complete. For k = 20, this requires over a million ‘independent’ assets, far more than we see in any real market. But things are not so bad. More claims become attainable if we allow ourselves to rebalance our replicating portfolio after each time period. The only restriction that we impose is that this rebalancing cannot involve any extra input of cash: the purchase of more stock must be funded by the sale of some of our bonds and vice versa. This will be formalised later as the self-ﬁnancing property. The key to understanding pricing and hedging in this bigger model is backwards induction on the tree of stock prices.
Example 2.1.1 (Pricing a European call)

Backwards induction on the tree

Suppose again that we are pricing a European option with maturity time T . As above, we set δt = T /N so that T corresponds to N time periods and we write Si for the stock price at time iδt. The payoff of the option at time T is denoted by C N .

the stock at time (N − 1)δt. Then our previous analysis would tell us the value, (N ) C N −1 , of the option at time (N − 1)δt. Speciﬁcally, C N −1 = ψ0 E N −1 [C N ] where the expectation is with respect to a probability measure for which S N −1 = (N ) (N ) ψ0 E N −1 [S N ] and ψ0 = e−r δt . (In a world of varying interest rates r must be replaced by the rate at the node of the tree corresponding to the known value of S N −1 .) Moreover, using Lemma 1.3.2, we know how to construct a portfolio at time (N − 1)δt that will have value exactly C N at time N δt. In this way, for each of the 2 N −1 nodes of the tree at time (N − 1)δt, we calculate the amount of money, C N −1 , that we require to construct a portfolio that exactly replicates the claim C N at time T . We now think of C N −1 as a claim at time (N − 1)δt and we repeat the process. If we know S N −2 , we can construct a portfolio at time (N − 2)δt whose value at time (N − 1)δt will be exactly C N −1 , and this portfolio will cost us (N −1) E N −2 [C N −1 ], where the expectation is with respect to a measure such that ψ0 (N −1) (N −1) S N −2 = ψ0 E N −2 [S N −1 ]. Here again ψ0 = e−r δt . Continuing in this way, we successively calculate the cost of a portfolio that, after appropriate readjustment at each tick of the clock, but without any extra input of wealth and without paying dividends, will allow us to meet exactly the claim against us at time N δt = T . We’ll illustrate the method in Example 2.1.2. ✷
Binomial trees

Method: The key idea is as follows. Suppose that we know the price, S N −1 , of

It is useful to consider a special form of the binary tree in which over each time step [ti , ti+1 ] the stock price either increases from its current value, Si , to Si u or decreases to Si d for some constants 0 < d < u < ∞. In such a tree the same stock price can be attained in many different ways. For example the value S0 ud at time t2 can be attained as the result of an upward stock movement followed by a downward stock

24

trees and martingales

S 0 uuu S 0 uu S0 u S 0 ud S0 S0 d S 0 dd S 0 ddd
Figure 2.2 A recombinant or binomial tree of stock prices.

S 0 uud

S 0 udd

movement or vice versa. The tree of stock prices then takes the form of Figure 2.2. Such a tree is said to be recombinant (different branches can recombine). These special recombinant trees are also known as binomial trees since (provided u, d, and r remain constant over time) the risk-neutral probability measure will be the same on each upward branch and so the stock price at time tn = nδt is determined by a binomial distribution. Such trees are computationally much easier to work with than general binary trees and, as we shall see, are quite adequate for our purposes. The binomial model was introduced by Cox, Ross & Rubinstein (1979) and has played a key rˆ le in the derivatives industry. o We now illustrate the method of backwards induction on a recombinant tree.
Example 2.1.2

Suppose that stock prices are given by the tree in Figure 2.3 and that δt = 1. If interest rates are zero, what is cost of an option to buy the stock at price 100 at time 3?

Solution: It is easy to ﬁll in the value of the claim at time 3. Reading from top to bottom, the claim has values 60, 20, 0 and 0. Next we need to ﬁnd the risk-neutral probabilities for each triad of nodes of the form

Si+1(u)

Si Si+1(d)
Evidently in this example the risk-neutral probability of stepping up is 1/2 at every node. We can now calculate the value of the option at the penultimate time, 2, to be,

25

2.1 the multiperiod binary model

100 (15)

120 (25) 80 (5)

140 (40) 100 (10) 60 (0)

160 (60) 120 (20)

80 (0) 40 (0)

Figure 2.3

The tree of stock prices for the underlying stock in Example 2.1.2. The number in brackets is the value of the claim at each node.

again reading downward, 40, 10, 0. Repeating this for time 1 gives values 25 (if the price steps up from time 0) and 5 (if the price has stepped down). Finally, then, the value of the option at time 0 is 15. Having ﬁlled in the option prices on the tree, we can now construct a portfolio that exactly replicates the claim at time 3 using the prescription of Lemma 1.3.2. We write (φi , ψi ) for the amount of stock and bond held in the portfolio over the time interval [(i − 1)δt, iδt). • • • • At time 0, we are given 15 for the option. We calculate φ1 as (25 − 5)/(120 − 80) = 0.5. So we buy 0.5 units of stock, which costs 50, and we borrow 35 in cash bonds. Suppose that S1 = 120. The new φ is (40 − 10)/(140 − 100) = 0.75, so we buy another 0.25 units of stock, taking our total bond borrowing to 65. Suppose that S2 = 140. Now φ = (60 − 20)/(160 − 120) = 1, so we buy still more stock, to take our holding up to 1 unit and our total borrowing to 100 bonds. Finally, suppose that S3 = 120. The option will be in the money, so we must hand over our unit of stock for 100, which is exactly enough to cancel our bond debt. The table below summarises our stock and bond holding if the stock price follows another path through the tree.

Time i 0 1 2 3

Last jump — down up down

Stock price Si 100 80 100 80

Option value Vi 15 5 10 0

Stock holding φi — 0.50 0.25 0.50

Bond holding ψi — −35 −15 −40

26

trees and martingales

Notice that all of the processes {Si }0≤i≤N , {Vi }0≤i≤N , {φi }1≤i≤N , {ψi }1≤i≤N depend on the sequence of up and down jumps. In particular, {φi }1≤i≤N and {ψi }1≤i≤N are random too. We do not know the dynamics of the portfolio at time 0. However, we do know that our portfolio is self-ﬁnancing. The portfolio that we hold over [i +1, i +2) can be bought with the proceeds of liquidating (at time i + 1) the portfolio that we held over the time interval [i, i + 1) – there is no need for any extra input of cash. Moreover, we know how to adjust our portfolio at each time step on the basis of knowledge of the current stock price. There is no risk. ✷ In the single period binary model, we saw that any claim at time T was attainable and its price at time zero could be expressed as an expectation. The same is true in the multiperiod setting (see Exercise 1). The proof that any claim is attainable is just backwards induction on the tree. To recover the pricing formula as an expectation, we deﬁne a probability distribution on paths through the tree.
Path probabilities

Notice that our backwards induction argument has speciﬁed exactly one probability on each branch of the tree. For each path through the tree that the stock price could follow we deﬁne the path probability to be the product of the probabilities on the branches that comprise it. In Exercise 2 you are asked to show that the price of a claim at time T that we obtained by backwards induction is precisely the discounted expected value of the claim with respect to these path probabilities (in which the discounted claim at each node is weighted according to the sum of the probabilities of all paths that end at that node). Let’s just check this prescription for our preceding example. In the recombining tree of Example 2.1.2, there are a total of eight paths, one ending at the top node, one at the bottom and three at each of the other nodes. Each path has equal probability, 1/8, and the expectation of the claim is therefore 1/8 × 60 + 3/8 × 20 = 15, which is the price that we calculated by backwards induction.

2.2

American options
Our somewhat more sophisticated market model is sufﬁcient for us to take a ﬁrst look at options whose payoff depends on the path followed by the stock price over the time interval [0, T ]. In this section we concentrate on the most important examples of such options: American options.
Deﬁnition 2.2.1 (American calls and puts)

An American call option with strike price K and expiry time T gives the holder the right, but not the obligation, to buy an asset for price K at any time up to T . An American put option with strike price K and expiry time T gives the holder the right, but not the obligation, to sell an asset for price K at any time up to T . Evidently the value of an American option should be more than (or at least no less than) that of its European counterpart. The question is, how much more?

27 Calls on nondividendpaying stock

2.2 american options

First let us prove the following oft-quoted result.
Lemma 2.2.2 It is never optimal to exercise an American call option on nondividend-paying stock before expiry. Proof: Consider the following two portfolios.

• •

Portfolio A: One American call option plus an amount of cash equal to K e−r (T −t) at

time t.
Portfolio B: One share.

Writing St for the share price at time t, if the call option is exercised at time t < T , then the value of portfolio A at time t is St − K + K e−r (T −t) < St . (Evidently the option will only be exercised if St > K .) The value of portfolio B is St . On the other hand at time T , if the option is exercised then the value of portfolio A is max{ST , K } which is at least that of portfolio B. We have shown that exercising prior to maturity gives a portfolio whose value is less than that of portfolio B whereas exercising at maturity gives a portfolio whose value is greater than or equal to that of B. It cannot be optimal to exercise early. ✷ This result only holds for non-dividend-paying stock. An alternative proof of Lemma 2.2.2 is Exercise 5. In Exercise 7 the result is extended to show that if the underlying stock pays discrete dividends, then it can only be optimal to exercise at the ﬁnal time T or at one of the dividend times (see also Exercise 8). More generally, the decision whether to exercise early depends on the ‘cost’ in terms of lost dividend income.
Put on nondividendpaying stock

The case of American put options is harder (even without dividends). We illustrate with an example.
Example 2.2.3

Suppose once again that our asset price evolves according to the recombinant tree of Figure 2.3. To illustrate the method, again we suppose that the risk-free interest rate is zero (but see the second paragraph of Remark 2.2.4). What is the value of a three month American put option with strike price 100?

Solution: As in the case of a European option, we work our way backwards through

the tree. • • The value of the claim at time 3, reading from top to bottom, is 0, 0, 20, 60. At time 2, we must consider two possibilities: the value if we exercise the claim, and the value if we do not. For the top node it is easy. The value is zero either way. For the second node, the stock price is equal to the strike price, so the value is zero if we exercise the option. On the other hand, if we don’t, then from our analysis of the single step binary model, the value of the claim is the expected value under the risk-neutral probabilities of the claim at time 3. We already calculated the risk-neutral

28

trees and martingales

0 0 5 15 25 40 60
Figure 2.4 The evolution of the price of the American put option of Example 2.2.3.

0 10 20

•

•

probabilities to be 1/2 on each branch of the tree, so this expected value is 10. For the bottom node, the value is 40 whether or not we exercise the claim. Now consider the two nodes at time 1. For the top one, if we exercise the option it is worthless whereas if we hold it then, again by our analysis of the single period model, its value is 5. For the bottom node, if we exercise the option then it is worth 20, whereas if we wait it is worth 25. Finally, at time 0, if we exercise, the value is zero, whereas if we wait the value is 15. The option prices are shown in Figure 2.4. ✷
Remark 2.2.4

1 Notice that in the above example it was not optimal to exercise the option at time 1, even when it was ‘in the money’. If S1 = 80, we make 20 from exercising immediately, but there is 25 to be made from waiting. 2 In this example there was never a strictly positive advantage to early exercise of the option. It was always at least as good to wait. In fact if interest rates are zero this is always the case, as is shown in Exercise 6. For non-zero interest rates, early exercise can be optimal, see Exercise 9. ✷

2.3

Discrete parameter martingales and Markov processes
Our multiperiod stock market model still looks rather special. To prepare the ground for the continuous time world of later chapters we now place it in the more general framework of discrete parameter martingales and Markov processes. First we recall the concepts of random variables and stochastic processes.

29 Random variables

2.3 discrete parameter martingales and markov processes

Formally, when we talk about a random variable we must ﬁrst specify a probability triple ( , F, P), where is a set, the sample space, F is a collection of subsets of , events, and P speciﬁes the probability of each event A ∈ F. The collection F is a σ -ﬁeld, that is, ∈ F and F is closed under the operations of countable union and taking complements. The probability P must satisfy the usual axioms of probability: • • • • 0 ≤ P[A] ≤ 1, for all A ∈ F, P[ ] = 1, P[A ∪ B] = P[A] + P[B] for any disjoint A, B ∈ F, if An ∈ F for all n ∈ N and A1 ⊆ A2 ⊆ · · · then P[An ] ↑ P
Deﬁnition 2.3.1

n

An as n ↑ ∞.

A real-valued random variable, X , is a real-valued function on that is F-measurable. In the case of a discrete random variable (that is a random variable that can only take on countably many distinct values) this simply means {ω ∈ : X (ω) = x} ∈ F,

so that P assigns a probability to the event {X = x}. For a general real-valued random variable we require that {ω ∈ : X (ω) ≤ x} ∈ F,

so that we can deﬁne the distribution function, F(x) = P[X ≤ x]. This looks like an excessively complicated way of talking about a relatively straightforward concept. It is technically required because it may not be possible to deﬁne P in a non-trivial way on all subsets of , but most of the time we don’t go far wrong if we ignore such technical details. However, when we start to study stochastic processes, random variables that evolve with time, it becomes much more natural to work in a slightly more formal framework.
Stochastic processes

To specify a (discrete time) stochastic process, we typically require not just a single σ -ﬁeld, F, but an increasing sequence of them, Fn ⊆ Fn+1 ⊆ · · · ⊆ F. The , F, {Fn }n≥0 , P collection {Fn }n≥0 is then called a ﬁltration and the quadruple is called a ﬁltered probability space.
Deﬁnition 2.3.2

A real-valued stochastic process is just a sequence of realvalued functions, {X n }n≥0 , on . We say that it is adapted to the ﬁltration {Fn }n≥0 if X n is Fn -measurable for each n.

One can then think of the σ -ﬁeld Fn as encoding all the information about the evolution of the stochastic process up until time n. That is, if we know whether each event in Fn happens or not then we can infer the path followed by the stochastic process up until time n. We shall call the ﬁltration that encodes precisely this information the natural ﬁltration associated to the stochastic process {X n }n≥0 .

30

trees and martingales
00

X p0 X0 p1

0 1

p00 p01 p10

X2

p000 p001 p010 p011 p100

X3

000

X3 X3 X3

001 010

01 X2

011

X2 X1
1

10

p101 p110

X3 101 X3 X3 X3
110 111

100

p11

11 X2 p 111

Figure 2.5

Tree representing the stochastic process of Example 2.3.3 and its distribution.

There is an important consequence of the very formal way that this is set up. Notice that we have deﬁned the process {X n }n≥0 as a sequence of measurable functions on without reference to P. This is exactly analogous to the situation in our tree models. We speciﬁed the possible values that the stock price could take at time n, corresponding to prescribing the functions {X n }n≥0 , and superposed the probabilities afterwards. Even if we had a preconception of what the probabilities of up and down jumps might be, we then changed probability (to the risk-neutral probabilities) in order actually to price claims. This process of changing probability will be fundamental to our approach to option pricing, even in our most complex market models.
Conditional expectation

When we constructed the probabilities on paths through our binary (or binomial) trees, we ﬁrst speciﬁed the probability on each branch of the tree. This was done in such a way that the expected value of e−r δt Sk+1 given that the value of the stock at time kδt is known to be Sk is just Sk . This condition speciﬁes the probabilities on the two branches emanating from the node corresponding to Sk at time kδt. We should like to extend this idea, but ﬁrst we need to remind ourselves about conditional expectation. This is best explained through an example.
Example 2.3.3

Consider the stochastic process represented by the tree in Figure 2.5. Its distribution is given by the probabilities on the branches of the tree, where, as in §2.1, we assume that the probability of a particular path through the tree is the product of the probabilities of the branches that comprise that path. , implicitly speciﬁes P. In later examples we shall be less pedantic, but here we write down explicitly. There are many possible choices, but an obvious one is the set of

Calculation: Our tree explicitly speciﬁes {X n }n≥0 and, for a given

31

2.3 discrete parameter martingales and markov processes

all possible sequences of ‘up’ and ‘down’ jumps. If ω = (u, u, d), say, then 0 00 001 X 1 (ω) = X 1 , X 2 (ω) = X 2 and X 3 (ω) = X 3 . First let us calculate the conditional expectation E [ X 3 | F1 ] . Using our interpretation of Fn as ‘information up to time n’, our problem is to determine the conditional expectation of X 3 given all the information up to time one. Notice that what we are calculating is an F1 -measurable random variable. It depends only on what happened up until time one. There are just two possibilities: the ﬁrst jump is up, or the ﬁrst jump is down. •
000 001 010 011 If the ﬁrst jump is up, the possible values of X 3 are X 3 , X 3 , X 3 and X 3 . The probability of each value is determined by the path probabilities but restricted to paths emanating from the upper node at time one. The conditional expectation then takes the value 000 001 010 011 E [ X 3 | F1 ] (u) = p00 p000 X 3 + p00 p001 X 3 + p01 p010 X 3 + p01 p011 X 3 .

•

This happens with probability p0 . If the ﬁrst jump is down, which happens with probability p1 , the conditional expectation takes the value
100 101 110 111 E [ X 3 | F1 ] (d) = p10 p100 X 3 + p10 p101 X 3 + p11 p110 X 3 + p11 p111 X 3 .

Similarly, we can calculate E [ X 3 | F2 ]. This random variable will be F2 -measurable – its value depends on the ﬁrst two jumps of the process. Its distribution is given in the table below.

Value E[X |F2 ](uu) = E[X |F2 ](ud) = E[X |F2 ](du) = E[X |F2 ](dd) =
000 p000 X 3 + 010 p010 X 3 + 100 p100 X 3 + 110 p110 X 3 + 001 p001 X 3 011 p011 X 3 101 p101 X 3 111 p111 X 3

Probability p0 p00 p0 p01 p1 p10 p1 p11

Of course, since E [ X 3 | F2 ] is an F2 -measurable random variable and F1 ⊆ F2 , we can calculate the conditional expectation E [ E [ X 3 | F2 ]| F1 ] . E [ E [ X 3 | F2 ]| F1 ] (u) = E [ E [ X 3 | F2 ]| F1 ] (d) = p00 E [ X 3 | F2 ] (uu) + p01 E [ X 3 | F2 ] (ud), p10 E [ X 3 | F2 ] (du) + p11 E [ X 3 | F2 ] (dd).

32

trees and martingales

Substituting the value of E [ X 3 | F2 ] from the table above it is easily checked that this reduces to E [ E [ X 3 | F2 ]| F1 ] = E [ X 3 | F1 ] . Here is the formal deﬁnition.
Deﬁnition 2.3.4 (Conditional expectation)

(2.1) ✷

Suppose that X is an F-measurable random variable with E [|X |] < ∞. Suppose that G ⊆ F is a σ -ﬁeld; then the conditional expectation of X given G, written E [ X | G], is the G-measurable random variable with the property that for any A ∈ G E [[ X | G] ; A]
A

E [ X | G] dP =
A

X dP

E [X ; A] .

The conditional expectation exists, but is only unique up to the addition of a random variable that is zero with probability one. This technical point will be important in Exercise 17 of Chapter 3. Equation (2.1) is a special case of the following key property of conditional expectations.
The tower property of conditional expectations: Suppose that Fi ⊆ F j ; then

E[E[X |F j ]|Fi ] = E[X |Fi ]. In words this says that conditioning ﬁrst on the information up to time j and then on the information up to an earlier time i is the same as conditioning originally up to time i. ✷ In calculations with conditional expectations, it is often useful to remember the following fact.
Taking out what is known in conditional expectations: Suppose that E [X ] and E [X Y ] <

∞; then if Y is Fn -measurable,

E [ X Y | Fn ] = Y E [ X | Fn ] .

This just says that if Y is known by time n, then if we condition on the information up to time n we can treat Y as constant. ✷
The martingale property

The probability measure on the tree that we used in §2.1 to price claims was chosen ˜ ˜ so that if we deﬁne { Sk }k≥0 to be the discounted stock price, that is Sk = e−kr δt Sk , ˜ ˜ ˜k+1 given that we know Sk is just Sk . We use the notation then the expected value of S ˜ ˜ ˜ E Sk+1 | Sk = Sk . Because in our model the stock price has ‘no memory’, so that the movement of the stock over the next tick of the clock is not inﬂuenced by the way in which it reached

33

2.3 discrete parameter martingales and markov processes

˜ its current value, conditioning on knowing Sk is actually the same as conditioning on knowing all of Fk , so that ˜ ˜ E Sk+1 |Fk = Sk . The property (2.2) is sufﬁciently important that it has a name.
Deﬁnition 2.3.5

(2.2)

Suppose that , {Fn }n≥0 , F, P is a ﬁltered probability space. The sequence of random variables {X n }n≥0 is a martingale with respect to P and {Fn }n≥0 if E [|X n |] < ∞, and E X n+1 | Fn = X n , If we replace equation (2.4) by E X n+1 | Fn ≤ X n , ∀n, ∀n. (2.4) ∀n, (2.3)

then {X n }n≥0 is a P, {Fn }n≥0 -supermartingale. If instead we replace it by E X n+1 | Fn ≥ X n , then {X n }n≥0 is a P, {Fn }n≥0 -submartingale. These deﬁnitions are not exhaustive. There are plenty of processes that fall into none of these categories. A martingale is often thought of as tracking the net gain after successive plays of a fair game. In this setting a supermartingale models net gain from playing an unfavourable game (one we are more likely to lose than to win) and a submartingale is the net gain from playing a favourable game. It is extremely important to note that the notion of a martingale is really that of a P, {Fn }n≥0 -martingale. Recall that our deﬁnition of stochastic process has divorced the rˆ les of the sequence {Fn }n≥0 , the F-measurable functions {X n }n≥0 on o and the probability measure P deﬁned on elements of F. In the setting of §2.1, our view of the market may be that the discounted stock price is not a martingale (indeed it probably isn’t or no one would ever speculate on stocks – they could get the same money, risk-free, by buying cash bonds). We change the probability measure to one which makes the discounted stock price a martingale for the purposes of pricing and, as we shall see, hedging. We shall refer to the probability measure that represents our view of the market as the market measure. The new probability measure, which we use for pricing and hedging, is known as the equivalent martingale measure.
Remark: (Martingales indexed by a subset of N) Although we have deﬁned martingales indexed by n ∈ N, we shall often talk about martingales indexed by {0 ≤ n ≤ N }. They are deﬁned by restricting conditions (2.3) and (2.4) to {0 ≤ n ≤ N }. We shall state our key results for martingales indexed by {n ≥ 0}; they can be modiﬁed in the obvious way to apply to martingales indexed by {0 ≤ n ≤ N }. ✷

∀n,

34

trees and martingales

It is often useful to observe that, by the tower property, if {X n }n≥0 is a P, {Fn }n≥0 martingale then for i < j, E X j Fi = X i .
The Markov property

Calculations can also be simpliﬁed if our martingales have an additional property: the Markov property.
Deﬁnition 2.3.6 (Markov process)

The stochastic process {X n }n≥0 (with its natural ﬁltration, {Fn }n≥0 ) is a discrete time Markov process if P X n+1 ∈ B| Fn = P X n+1 ∈ B| X n ,

for all B ∈ F. In words this says that the probability that X n+1 ∈ B given that we know the whole history of the process up to time n is the same as the probability that X n+1 ∈ B given only the value of X n . A Markov process has no memory. Many of our examples of martingales (and all our examples of market models) will also have the Markov property. However, not all martingales are Markov processes and not all Markov processes are martingales (see Exercise 11).

Notation:

When we wish to emphasise that a ﬁltration is ‘generated by’ the X stochastic process {X n }n≥0 we use the notation {Fn }n≥0 . Unless otherwise stated, {Fn }n≥0 will always be understood to mean the natural ﬁltration associated with the stochastic process under consideration.

It would be excessively pedantic always to insist upon an explicit speciﬁcation of and so, generally, we won’t. We shall also use ‘{X n }n≥0 is a P-martingale’ to mean X {X n }n≥0 is a P, {Fn }n≥0 -martingale’.
Examples

A one-dimensional simple random walk, {Sn }n≥0 , is a Markov process such that Sn+1 = Sn + ξn+1 where (for each n) ξn ∈ {−1, +1} and, under P, {ξn }n≥0 are independent identically distributed random variables. Thus
Example 2.3.7 (Random walk)

P Sn+1 = k + 1| Sn = k = p,

P Sn+1 = k − 1| Sn = k = 1 − p,

where p ∈ [0, 1]. If p = 0.5, then {Sn }n≥0 is a P-martingale. If p < 0.5 (resp. p > 0.5), then {Sn }n≥0 is a P-supermartingale (resp. P-submartingale).
Justiﬁcation: To check this, notice that since the random walk can be a distance at

most n from its starting point at time n, the expectation E [|Sn |] < ∞ is evidently

35

2.3 discrete parameter martingales and markov processes

ﬁnite. Moreover, E Sn+1 | Fn = E Sn + ξn+1 | Fn = = Sn + E ξn+1 | Fn Sn + E ξn+1 ,

where we have used independence of the {ξn }n≥0 in the last line. It sufﬁces then to observe that   < 0, p < 0.5, E ξn+1 = 0, p = 0.5,  > 0, p > 0.5. ✷ Suppose that and a ﬁltration {Fn }n≥0 are given. (The example that we have in mind is that Fn encodes the history of a ﬁnancial market up until time nδt.) Let C N be any bounded F N -measurable random variable. (This we are thinking of as a claim against us at time N δt.) Then for any probability measure P, the conditional expectation process, {X n }0≤n≤N , given by X n = E [ C N | Fn ] ,
Example 2.3.8 (Conditional expectation of a claim)

is a P, {Fn }0≤n≤N -martingale. In solving our pricing problem for a European option with value C N at the expiry time N δt in the multiperiod binary model of stock prices of §2.1, we found a probability measure, which we denote by Q, under which the discounted stock price is a martingale. For any claim, C N , at time N δt, provided EQ [|C N |] < ∞, the fair price at time nδt of an option with payoff C N at time N δt was found to be
Example 2.3.9 (The discounted price of a claim)

Vn = e−r (N −n)δt EQ [ C N | Fn ] . ˜ ˜ Deﬁne the discounted claim process by Vn = e−r nδt Vn . Then {Vn }0≤n≤N is a Qmartingale. This would remain true even if we dropped the assumption of constant interest rates, provided that we knew the risk-free rate over the time interval [iδt, (i + 1)δt) at the beginning of the period.
New martingales from old

Our last example shows that the discounted price process of a European option is a martingale. In other words, the discounted value of our replicating portfolio is a martingale. As before, we write (φn , ψn ) for the amount of stock and bond held in the replicating portfolio over the nth time interval, that is [(n − 1)δt, nδt). The value of the portfolio at time nδt is then Vn = φn+1 Sn + ψn+1 Bn , where Bn is the value of the cash bond at time nδt. The portfolio is self-ﬁnancing, that is the cost of constructing the new portfolio at time (n + 1)δt is exactly offset

36

trees and martingales

by the proceeds of selling the portfolio that we have held over [nδt, (n + 1)δt). In symbols, φn+1 Sn+1 + ψn+1 Bn+1 = φn+2 Sn+1 + ψn+2 Bn+1 . The discounted price is ˜ ˜ Vn = φn+1 Sn + ψn+1 ,

and since, using the self-ﬁnancing property, ˜ ˜ φn+1 Sn+1 + ψn+1 = φn+2 Sn+1 + ψn+2 , we have ˜ ˜ Vn+1 − Vn = = That is ˜ Vn = V0 + n−1 j=0

˜ ˜ φn+2 Sn+1 + ψn+2 − φn+1 Sn − ψn+1 ˜ ˜ φn+1 Sn+1 − Sn .

˜ ˜ φ j+1 S j+1 − S j .

(2.5)

˜ From our earlier remarks, {Vn }0≤n≤N is a Q-martingale, so what we have checked ˜ is that under the probability measure Q for which { Sn }0≤n≤N is a martingale, the expression on the right hand side of equation (2.5) is also a martingale. This is part of a general phenomenon. To state a precise result we need a deﬁnition. Recall that we knew φi at time (i − 1)δt.
Deﬁnition 2.3.10

Given a ﬁltration {Fn }n≥0 , the process {An }n≥1 is {Fn }n≥0 previsible or {Fn }n≥0 -predictable if An is Fn−1 -measurable for all n ≥ 1. Note that this is the sort of randomness that we have permitted for our cash bond.

Discrete stochastic integrals

Proposition 2.3.11

Suppose that {X n }n≥0 is adapted to the ﬁltration {Fn }n≥0 and that {φn }n≥1 is {Fn }n≥0 -previsible. Deﬁne n−1 Zn = Z0 + j=0 φ j+1 X j+1 − X j ,

(2.6)

where Z 0 is a constant. If {X n }n≥0 is a P, {Fn }n≥0 -martingale, then so is {Z n }n≥0 .
Remark: If {θn }n≥0 is adapted to {Fn }n≥0 , then the process {φn }n≥1 deﬁned by φn =

θn−1 is previsible. Thus for an {Fn }n≥0 -adapted process {θn }n≥0 , if {X n }n≥0 is a P, {Ft }t≥0 -martingale then so is n−1 Zn = Z0 + j=0 θ j X j+1 − X j . ✷

37

2.3 discrete parameter martingales and markov processes Proof of Proposition 2.3.11: This is an exercise in the use of conditional expectations.

E Z n+1 | Fn − Z n

= = = =

E Z n+1 − Z n | Fn E φn+1 (X n+1 − X n )| Fn φn+1 E (X n+1 − X n )| Fn 0. ✷

= φn+1 E X n+1 | Fn − X n

We can think of the sum in equation (2.6) as a discrete stochastic integral. When we turn to stochastic integration in Chapter 4, we shall essentially be passing to limits in sums of this form.
The Fundamental Theorem of Asset Pricing

It is not just our binomial models that can be incorporated into the martingale framework. The same argument that allows us to pass from the single period to the multiperiod binary model allows us to pass from the single period models of §1.5 and §1.6 to a multiperiod model. We now recast Theorems 1.5.2 and 1.6.2 in this language. Suppose that our market consists of K stocks and that the possible values that the stock prices S 1 , . . . , S K can take on at times δt, 2δt, 3δt, . . . , N δt = T are known. We denote by the set of all possible ‘paths’ that the stock price vector can K follow in R+ . Theorem 1.5.2 tells us that the absence of arbitrage is equivalent to the existence of a probability measure, Q, on that assigns strictly positive mass to every ω ∈ and such that (r ) Sr −1 = ψ0 EQ [Sr |Sr −1 ], where Sr is the vector of stock prices at time r and ψ0 is the discount on riskless borrowing over [(r − 1)δt, r δt]. ˜ ˜ If, as above, we consider the discounted stock prices, { S j }0≤ j≤N , given by S j = j (i) i=1 ψ0 S j , then ˜ ˜ ˜ ˜ ˜ EQ [ Sr | S1 , . . . , Sr −1 ] = EQ Sr Fr −1 = Sr −1 . In other words, the discounted stock price vector is a Q-martingale.
Deﬁnition 2.3.12
(r )

Two probability measures P and Q on a space equivalent if for all events A ⊆ Q(A) = 0 if and only if P(A) = 0.

are said to be

Suppose then that we have a market model in which the stock price vector can follow K one of a ﬁnite number of paths through R+ . We may even have our own belief as to how the price will evolve, encoded in a probability measure, P, on . Theorem 1.5.2 and Theorem 1.6.2 combine to say:

38

trees and martingales

For the multiperiod market model described above, there is no Theorem 2.3.13 arbitrage if and only if there is an equivalent martingale measure Q. That is, there is a measure, Q, equivalent to P, such that the discounted stock price process is a Q-martingale. In that case, the time zero market price of an attainable claim C N (to be delivered at time N δt) is unique and is given by EQ [ψ0 C N ], where ψo =
N 1

ψ0 is the discount factor over N periods.

(i)

Although there are extra technical conditions, this fundamental theorem has essentially the same statement for markets that evolve continuously with time.

2.4

Some important martingale theorems
Phrasing everything in the martingale framework places many powerful theorems at our disposal. In this section, we present some of the most important results in the theory of discrete parameter martingales. However, our coverage is necessarily cursory. An excellent and highly readable account is Williams (1991).

Stopping times

One of the most important calculational tools in martingale theory is the Optional Stopping Theorem. Before we can state it, we need to introduce the notion of a stopping time.
Deﬁnition 2.4.1

Given a sample space equipped with a ﬁltration {Fn }n≥0 , a stopping time or optional time is a random variable T : → Z+ with the property that for all n ≥ 0. {T ≤ n} ∈ Fn , This just says that we can decide whether or not T ≤ n on the basis of the information available at time n – we don’t need to look into the future.

Example 2.4.2

Consider the simple random walk of Example 2.3.7. Deﬁne T to be the ﬁrst time that the random walk takes the value 1, that is T = inf {i ≥ 0 : Si = 1} ; then T is a stopping time. On the other hand, U = sup {i ≥ 0 : Si = 1} is not a stopping time.

Optional stopping

An equivalent deﬁnition of stopping time is that the random variable θn 1{T ≥n+1} , for n ≥ 0, is adapted (see Deﬁnition 2.3.2). Consequently, from the remark following

39

2.4 some important martingale theorems

Proposition 2.3.11, if {X n }n≥0 is a martingale, then so is the process n−1 Zn j=0 θ j X j+1 − X j .

(2.7)

Notice that we can rearrange this expression, n−1 Zn

= j=0 n−1

θ j X j+1 − X j 1{T ≥ j+1} X j+1 − X j j=0 = =

X T ∧n − X 0 ,

where T ∧ n denotes the minimum of T and n.
Theorem 2.4.3 (Optional Stopping Theorem)

Let , F, {Fn }n≥0 , P be a ﬁltered probability space. Suppose that the process {X n }n≥0 is a P, {Fn }n≥0 -martingale, and that T is a bounded stopping time. Then E [ X T | F0 ] = X 0 , and hence E [X T ] = X 0 .

Proof: The proof is a simple application of the calculation that we did above. If we

know that T ≤ N , then in the notation of (2.7), Z N = X T − X 0 and since {Z n }n≥0 is a martingale, E [ Z N | F0 ] = Z 0 = 0, i.e. E [ X T | F0 ] = X 0 . Taking expectations once again yields E [X T ] = X 0 . ✷ It is essential in this result that the stopping time be bounded. In practice this will be the case in all of our ﬁnancial applications, but Exercise 15 shows what can go wrong. More general versions of the theorem are available; see for example, Williams (1991). Here we satisfy ourselves with an application (see also Exercise 14).
Proposition 2.4.4

Let {Sn }n≥0 be the (asymmetric) simple random walk of Example 2.3.7 with p > 1/2. For x ∈ Z we write Tx = inf {n : Sn = x} ,

40

trees and martingales

and deﬁne φ(x) = Then for a < 0 < b, P [Ta < Tb ] =

1− p p

x

.

1 − φ(b) . φ(a) − φ(b)

Proof: We ﬁrst show that {φ(Sn )}n≥0 is a P-martingale. Since the walk can only take

one step at a time, −n ≤ Sn ≤ n. Using also that 0 < (1 − p)/ p < 1 for p > 1/2, we evidently have that E [|φ(Sn )|] < ∞, ∀n.

To check that we really have a martingale is reduced to another exercise in conditional expectations. We must calculate E φ(Sn+1 )| Fn . Recall that Sn+1 = n+1 ξ j = Sn + ξn+1 , where, under P, the random variables ξ j j=1 are independent and identically distributed with P[ξ j = 1] = p This gives E φ(Sn+1 )| Fn = = = = E φ(Sn ) φ(Sn )E φ(Sn ) p φ(Sn ). 1− p p 1− p p 1− p p ξn+1 and

P[ξ j = −1] = 1 − p.

Fn

ξn+1

1

+ (1 − p)

1− p p

−1

We should now like to apply the Optional Stopping Theorem to the stopping time T = Ta ∧ Tb , the ﬁrst time that the walk hits either a or b. The difﬁculty is that T is not bounded. Instead then, we apply the theorem to the stopping time T ∧ N for an arbitrary (deterministic) N . This gives 1 = = E [φ(S0 )] = E [φ(ST ∧N )] φ(a)P [ST = a, T ≤ N ] + φ(b)P [ST = b, T ≤ N ] + E [φ(S N ), T > N ] . (2.8) Now 0 ≤ E [φ(S N ), T > N ] = ≤ E [ φ(S N )| T > N ] P [T > N ] 1− p p b +

p 1− p

a

P [T > N ] ,

41

2.4 some important martingale theorems

and since P [T > N ] → 0 as N → ∞, we can let N → ∞ in (2.8) to deduce that φ(a)P [ST = a] + φ(b)P [ST = b] = 1. (2.9)

Finally, since P [ST = a] = 1 − P [ST = b], and P [Ta < Tb ] = P [ST = a], equation (2.9) becomes φ(a)P [Ta < Tb ] + φ(b) (1 − P [Ta < Tb ]) = 1. Rearranging, P [Ta < Tb ] = as required.
A convergence theorem

1 − φ(b) , φ(a) − φ(b) ✷

Often one can deduce a great deal about martingales from apparently scant information. An example is the result of Exercise 12 which says that a previsible martingale is constant. Another example is provided by the following result.
Theorem 2.4.5 (Positive Supermartingale Convergence Theorem)

If {X n }n≥0 is a (P, {Fn }n≥0 )-supermartingale and X n ≥ 0 for all n, then there exists an F∞ measurable random variable, X ∞ , with E [X ∞ ] < ∞ such that with P-probability one X n → X ∞ as n → ∞. A proof of this result is beyond our scope here, but can be found, for example, in Williams (1991).

Compensation Before returning to some ﬁnance, we record just one more result. Recall that

submartingales tend to rise on the average and supermartingales fall on the average. The following result, sometimes called compensation, says that we can subtract a non-decreasing process from a submartingale to obtain a martingale and we can add a non-decreasing process to a supermartingale to obtain a martingale. In both cases, the interesting thing is that the non-decreasing processes are previsible.
Proposition 2.4.6

1 Suppose that {X n }n≥0 is a (P, {Fn }n≥0 )-submartingale. Then there is a previsible, non-decreasing process {An }n≥0 such that {X n − An }n≥0 is a (P, {Fn }n≥0 )martingale. If we insist that A0 = 0, then {An }n≥0 is unique. 2 Suppose that {X n }n≥0 is a (P, {Fn }n≥0 )-supermartingale. Then there is a previsible, non-decreasing process {An }n≥1 such that {X n + An }n≥0 is a (P, {Fn }n≥0 )martingale. If we insist that A0 = 0, then {An }n≥0 is unique.
Proof: The proofs of the two parts are essentially identical, so we restrict our attention to 1.

42

trees and martingales

Deﬁne A0 = 0 and then for n ≥ 1 set An − An−1 = E X n − X n−1 | Fn−1 . By deﬁnition {An }n≥0 will be previsible and non-decreasing (since {X n }n≥0 is a submartingale). We must check that {X n − An }n≥0 is a martingale. First we check that E [|X n − An |] < ∞ for all n. E [|X n − An |] ≤ E [|X n |] + E [An ] n = E [|X n |] + E A0 + j=1 n

A j − A j−1 (by deﬁnition of A j )

= E [|X n |] + j=1 n

E E X j − X j−1 F j−1 E E j=1 n

≤ E [|X n |] + = E [|X n |] + j=1 X j + X j−1 F j−1 (tower property), < ∞ for

E X j + X j−1

and evidently this ﬁnal expression is ﬁnite since by assumption E X j all j. Now we check the martingale property, E X n+1 − An+1 | Fn = E X n+1 − E X n+1 − X n | Fn − An Fn = = E X n+1 − X n+1 + X n − An | Fn X n − An .

(by deﬁnition of An+1 )

It remains to check that if A0 = 0 then the process {An }n≥0 is unique. Suppose that there were another predictable process {Bn }n≥0 with the same property. Then {X n − An }n≥0 and {X n − Bn }n≥0 are both martingales and, therefore, so is the difference between them, {An − Bn }n≥0 . On the other hand {An − Bn }n≥0 is predictable and predictable martingales are constant (see Exercise 12). Since A0 = 0 = B0 , the proof is complete. ✷
American options and supermartingales

Let’s see what these concepts correspond to in a ﬁnancial example.
Example 2.4.7 (American options revisited)

Assume the binomial model and notation of §2.2 and let Q be the probability measure on the tree under which the ˜ ˜ discounted stock price { Sn }0≤n≤N is a martingale. We denote by {Vn }0≤n≤N the discounted value of an American call or put option with strike K and maturity T = N δt and deﬁne ˜ Bn = e−nδt (Sn − K )+ e−nδt (K − Sn )+ in the case of the call, in the case of the put.

43

2.5 the binomial representation theorem

˜ (The ﬁltration is always that generated by {Sn }0≤n≤N .) Then {Vn }0≤n≤N is the ˜ n }0≤n≤N . smallest Q-supermartingale that dominates { B In Exercise 16 it is shown that this characterisation provides yet another simple proof of Lemma 2.2.2.
Explanation for example: We know from §2.2 that

˜ ˜ ˜ Vn−1 = max Bn−1 , EQ Vn Fn−1

,

0 ≤ n ≤ N,

˜ ˜ ˜ and VN = B N . Evidently {Vn }0≤n≤N is a supermartingale that dominates ˜ { Bn }0≤n≤N . To check that it is the smallest supermartingale with this property, ˜ ˜ suppose that {Un }0≤n≤N is any other supermartingale that dominates { Bn }n≥0 . Then ˜ ˜ ˜ ˜ N ≥ VN , and if Un ≥ Vn , then U ˜ ˜ ˜ Un−1 ≥ EQ Un Fn−1 ≥ EQ Vn Fn−1 , and so ˜ ˜ ˜ Un−1 ≥ max Bn−1 , EQ Vn Fn−1 ˜ = Vn−1 .

˜ The result follows by backwards induction. The process {Vn }0≤n≤N is called the ˜ n }0≤n≤N . ✷ Snell envelope of { B
Remark: Proposition 2.4.6 tells us that we can write

˜ ˜ ˜ Vn = Mn − An ˜ ˜ where { Mn }n≥0 is a martingale and { An }n≥0 is a non-decreasing process, with A0 = 0. Since the market is complete, we can hedge M N exactly by holding a portfolio that consists over the nth time step of φn units of stock and ψn units of cash bond. The seller of the American option would more than meet her liability by holding ˜ such a portfolio. The holder of the option will exercise at the ﬁrst time j when A j+1 ˜ n }n≥0 is previsible), since at that time it is is non-zero (recall that the process { A better to sell the option and invest the money according to the hedging portfolio ✷ {(φn , ψn )} j≤n≤N .

2.5

The Binomial Representation Theorem
Pricing a derivative in the martingale framework corresponds to taking an expectation. But arbitrage prices are only meaningful if we can construct a hedging portfolio. If we know the hedging portfolio then we saw in the discussion preceding Deﬁnition 2.3.10 that we can express the discounted value of the portfolio, and therefore of the derivative, as a ‘discrete stochastic integral’ of the stock holding in the portfolio with respect to the discounted stock price. In order to pass from the discounted price of the derivative to a hedging portfolio we need the following converse to Proposition 2.3.11. We work in the context of our binomial model of stock prices.

44

trees and martingales

Suppose that the measure Q Theorem 2.5.1 (Binomial Representation Theorem) ˜ is such that the discounted binomial price process { Sn }n≥0 is a Q-martingale. ˜ If {Vn }n≥0 is any other (Q, {Fn }n≥0 )-martingale, then there exists an {Fn }n≥0 predictable process {φn }n≥1 such that ˜ ˜ Vn = V0 + n−1 j=0

˜ ˜ φ j+1 S j+1 − S j .

(2.10)

Proof: We consider a single time step for our binomial tree. It is convenient to write

˜ ˜ ˜ Vi+1 = Vi+1 − Vi

and

˜ ˜ ˜ Si+1 = Si+1 − Si .

˜ ˜ Given their values at time iδt, each of Vi+1 and Si+1 can take on one of two possible ˜ ˜ ˜ ˜ values that we denote by {Vi+1 (u), Vi+1 (d)} and { Si+1 (u), Si+1 (d)} respectively. ˜ ˜ We should like to write Vi+1 = φi+1 Si+1 + ki+1 , where φi+1 and ki+1 are both known at time iδt. In other words we seek φi+1 and ki+1 such that ˜ ˜ ˜ ˜ Vi+1 (u) − Vi = φi+1 Si+1 (u) − Si + ki+1 , and ˜ ˜ ˜ ˜ Vi+1 (d) − Vi = φi+1 Si+1 (d) − Si + ki+1 . Solving this gives φi+1 = ˜ ˜ Vi+1 (u) − Vi+1 (d) ˜ ˜ Si+1 (u) − Si+1 (d)

˜ ˜ ˜ ˜ and ki+1 = Vi+1 (u) − Vi − φi+1 Si+1 (u) − Si , both of which are known at time iδt. ˜ ˜ Now {Vi }i≥0 and { Si }i≥0 are both martingales so that E ˜ Vi+1 Fi = 0 = E ˜ Si+1 Fi

from which it follows that ki+1 = 0. In other words, ˜ ˜ Vi+1 = φi+1 Si+1 , where φi+1 is known at time iδt. Induction ties together all these increments into the result that we want. ✷ ˜ From our previous work, we know that if {Vi }i≥0 is the discounted price of a claim, then such a predictable process {φi }i≥1 arises as the stock holding when we construct our replicating portfolio. We should like to go the other way. Given {φi }i≥1 , can we construct a self-ﬁnancing replicating portfolio? Not surprisingly, the answer is yes.

From martingale representation to replicating portfolio

45

2.6 overture to continuous models Construction strategy: At time i, buy a portfolio that consists of φi+1 units of stock

˜ ˜ and Vi − φi+1 Si units of cash bond. We must check that this strategy really works. It is convenient to write Bi for the value of the bond at time iδt. S ˜ Suppose that at time iδt we have bought φi+1 units of stock and Vi − φi+1 Bii units of cash bond. This will cost us Si ˜ ˜ φi+1 Si + Vi − φi+1 Bi = Vi Bi = Vi . Bi

The value of this portfolio at time (i + 1)δt is then Si Si+1 Si ˜ ˜ Bi+1 = Bi+1 φi+1 + Vi φi+1 Si+1 + Vi − φi+1 − Bi Bi+1 Bi ˜ = Vi+1 Bi+1 (by the binomial representation) = Vi+1 , which is exactly enough to construct our new portfolio at time (i + 1)δt. Moreover, at time N δt we have precisely the right amount of money to meet the claim against us.

Three steps to replication: There are three steps to pricing and hedging a claim C T against us at time T . • Find a probability measure Q under which the discounted stock price (with its natural ﬁltration) is a martingale. • Form the discounted value process,

˜ Vi = e−riδt Vi = EQ e−r T C T Fi . • Find a predictable process {φi }1≤i≤N such that ˜ ˜ Vi = φi Si .

2.6

Overture to continuous models
Before rigorously deriving the acclaimed Black–Scholes pricing formula for the value of a European option, we are going to develop a substantial body of material. As an appetiser though, we can use our discrete techniques to see what form our results must take in the continuous world. It is easy to believe that we should be able to use a discrete model with very small time periods to approximate a continuous model. The Black–Scholes model is based on the lognormal model that we mentioned in §1.2. With this in mind, we choose our approximation to have constant growth rate and constant ‘noise’.

46 Model with constant stock growth and noise

trees and martingales

The model is parametrised by the time period, δt, and three ﬁxed constant parameters, ν, σ and the riskless rate r . • The cash bond has the form Bt = er t , which does not depend on the interval size. • The stock price process follows the nodes of a binomial tree. If the current value of the stock is s, then over the next time period it moves to the new value √ if up, s exp νδt + σ δt √ s exp νδt − σ δt if down. Suppose our belief is that the jumps are equally likely to be up or down. So under the market measure, P up jump = 1/2 = P down jump at each time step. For a ﬁxed time t, set N to be the number of time periods until time t, that is N = t/δt. Then √ 2X N − N , St = S0 exp νt + σ t √ N where X N is the total number of the N separate jumps which were up jumps. To see what happens as δt → 0 (or equivalently N → ∞) we call on the Central Limit Theorem.
Theorem 2.6.1 (Central Limit Theorem)

Let ξ1 , ξ2 , . . . be a sequence of independent identically distributed random variables under the probability measure P with ﬁnite mean µ and ﬁnite non-zero variance σ 2 and let Sn = ξ1 + . . . + ξn . Then Sn − nµ √ nσ 2

converges in distribution to an N (0, 1) random variable as n → ∞. Now X N is the sum of N independent random variables {ξi }1≤i≤N taking the value +1 with probability 1 and 0 otherwise. This means E [ξi ] = 1 and var [ξi ] = 1 so 2 2 4 that √ the Central Limit Theorem, the distribution of the random variable (2X N − by N )/ N converges to that of a normal random variable with mean zero and variance one. In other words, as δt gets smaller (and so N gets larger), the distribution of St converges to that of a lognormal distribution. More precisely, in the limit, log St is normally distributed with mean log S0 + νt and variance σ 2 t.
Under the martingale measure

This is what happens under the original measure P. What happens under the martingale measure, Q, that we use for pricing? By Lemma 1.3.2, under the martingale measure, the probability of an up jump is √ exp(r δt) − exp(νδt − σ δt) p= √ √ , exp(νδt + σ δt) − exp(νδt − σ δt) which is approximately √ ν + 1σ2 − r 1 2 1 − δt 2 σ .

47

exercises

So under the martingale measure, Q, X N is still binomially distributed, but now has mean N p and variance N p(1 − p). √ √ Thus, under Q, (2X N − N )/ N has mean that tends to − t ν + 1 σ 2 − r /σ 2 and variance that approaches one as δt tends to zero. Again using the Central Limit √ Theorem the random variable (2X N − N )/ N converges to a normally distributed √ random variable, with mean − t ν + 1 σ 2 − r /σ and variance one. Under Q 2 then, St is lognormally distributed with mean log S0 + (r − 1 σ 2 )t and variance σ 2 t. 2 This can be written √ 1 St = exp σ t Z + r − σ 2 t , 2 where, under Q, the random variable Z is normally distributed with mean zero and variance one.
Pricing a call option

If our discrete theory carries over to the continuous limit, then in our continuous model the price at time zero of a European call option with strike price K at time T will be the discounted expected value of the claim under the martingale measure, that is EQ e−r T (ST − K )+ , where r is the riskless rate. Substituting, we obtain EQ √ 1 S0 exp σ T Z − σ 2 T 2 − K exp (−r T ) . (2.11)

+

We’ll derive this pricing formula rigorously in Chapter 5 where we’ll also show that equation (2.11) can be evaluated as  S0 where  log
S0 K

 + r + 1σ2 T 2  − K e−r T √ σ T

 

log

S0 K

 + r − 1σ2 T 2 , √ σ T

is the standard normal distribution function, (z) = Q [Z ≤ z] = z −∞

1 2 √ e−x /2 d x. 2π

Exercises
1 Notice that, like the single period ternary model of Chapter 1, the two-step binomial model allows the stock to take on three distinct values at time 2. Show, however, that every claim can be exactly replicated by a self-ﬁnancing portfolio, that is, the market is complete. More generally, show that if the market evolves according to a k-step binomial model then it is complete.

48

trees and martingales

2 Show that the price of a claim obtained by backwards induction on the binomial tree is precisely the value obtained by calculating the discounted expected value of the claim with respect to the path probabilities introduced in §2.1. 3 Consider two dates T0 , T1 with T0 < T1 . A forward start option is a contract in which the holder receives at time T0 , at no extra cost, an option with expiry date T1 and strike price equal to ST0 (the asset price at time T0 ). Assume that the stock price evolves according to a two-period binary model, in which the asset price at time T0 is either S0 u or S0 d, and at time T1 is one of S0 u 2 , S0 ud and S0 d 2 with d < min er T0 , er (T1 −T0 ) ≤ max er T0 , er (T1 −T0 ) < u, where r denotes the risk-free interest rate. Find the fair price of such an option at time zero. 4 A digital option is one in which the payoff depends in a discontinuous way on the asset price. The simplest example is the cash-or-nothing option, in which the payoff to the holder at maturity T is X 1{ST >K } where X is some prespeciﬁed cash sum. Suppose that an asset price evolves according to the binomial model in which, at each step, the asset price moves from its current value Sn to one of Sn u and Sn d. As usual, if T denotes the length of each time step, d < er T < u. Find the time zero price of the above option. You may leave your answer as a sum. 5 Let Ct denote the value at time t of an American call option on non-dividend-paying stock with strike price K and maturity T . If the risk-free interest rate is r > 0, prove that Ct ≥ St − K e−r (T −t) > St − K , and deduce that it is never optimal to exercise this option prior to the maturity time, T. 6 Let Ct be as in Exercise 5 and let Pt be the value of an American put option on the same stock with the same strike price and maturity. By comparing the values of two suitable portfolios, show that Ct + K ≥ Pt + St . Using put–call parity for European options and the result of Exercise 5, show that Pt ≥ Ct + K e−r (T −t) − St . Combine these results to see that, if r > 0 and t < T , St − K ≤ Ct − Pt < St − K e−r (T −t) and deduce that if interest rates are zero, there is no advantage to early exercise of the put.

49

exercises

7 If a stock price is S just before a dividend D is paid, what is its value immediately after the payment? Suppose that a stock pays dividends at discrete times, T0 , T1 , . . . , Tn . Show that it can be optimal to exercise an American call on such a stock prior to expiry. 8 Suppose that the stock in Figure 2.3 will pay a dividend of 5% of its value at time 2. As before, interest rates are zero and between times 2 and 3 the value of the stock will either increase or decrease by 20. Find the time zero price of an American call option on this stock with strike 100 and maturity 3. Is it ever optimal to exercise early? 9 Consider the American put option of Example 2.2.3, but now suppose that interest rates are such that a $1 cash bond at time iδt is worth $1.1 at time (i + 1)δt. Find the value of the put. At what time will it be exercised? 10 Suppose that an asset price evolves according to the binomial model. For simplicity suppose that the risk-free interest rate is zero and T is 1. Suppose that under the probability P, at each time step, stock prices go up with probability p and down with probability 1 − p. The conditional expectation Mn E[S N |Fn ], 1 ≤ n ≤ N,

is a stochastic process. Check that it is a P-martingale and ﬁnd the distribution of the random variable Mn . 11 (a) Find a Markov process that is not a martingale. (b) Find a martingale that is not a Markov process. 12 Show that a previsible martingale is constant. 13 Let {Sn }n≥0 be simple random walk under the measure P. Calculate E [Sn ] and var [Sn ]. 14 Let {Sn }n≥0 be a symmetric simple random walk under the measure P, that is, in the 2 notation of Example 2.3.7, p = 1/2. Show that {Sn }n≥0 is a P-submartingale and 2 − n} that {Sn n≥0 is a P-martingale. / Let T = inf{n : Sn ∈ (−a, a)}, where a ∈ N. Use the Optional Stopping Theorem (applied to a suitable sequence of bounded stopping times) to show that E [T ] = a 2 . 15 As in Exercise 14, let {Sn }n≥0 be a symmetric simple random walk under P and write X n = Sn + 1. (Note that {X n }n≥0 is a simple random walk started from 1 at time zero.) Let T = inf{n : X n = 0}. Show that T is a stopping time and that if Yn = X T ∧n , then {Yn }n≥0 is a non-negative martingale and therefore, by Theorem 2.4.5, converges to a limit, Y∞ as n → ∞. Show that E [Yn ] = 1 for all n, but that Y∞ = 0. Why does this not contradict the conclusion of the Optional Stopping Theorem?

50

trees and martingales

16

Recall Jensen’s inequality: if g is a convex function and X a real-valued random variable then E [g(X )] ≥ g (E [X ]) . Combine this with the characterisation (Example 2.4.7) of the discounted price of an American call option on non-dividend-paying stock as the smallest Qsupermartingale that dominates {e−r nδt (Sn − K )+ }n≥0 to prove that the price of an American call on non-dividend-paying stock is the same as that of a European call with the same strike and maturity.

3 Brownian motion

Summary
Our discrete models are only a crude approximation to the way in which stock markets actually move. A better model would be one in which stock prices can change at any instant. As early as 1900 Bachelier, in his thesis ‘La th´ orie de la e sp´ culation’, proposed Brownian motion as a model of the ﬂuctuations of stock e prices. Even today it is the building block from which we construct the basic reference model for a continuous time market. Before we can proceed further we must leave ﬁnance to deﬁne and construct Brownian motion. Our ﬁrst approach will be to continue the heuristic of §2.6 by considering Brownian motion as an ‘inﬁnitesimal’ random walk in which smaller and smaller steps are taken at ever more frequent time intervals. This will lead us to a natural deﬁnition of the process. A formal construction, due to L´ vy, will be given in §3.2, e but this can safely be omitted. Next, §3.3 establishes some facts about the process that we shall require in later chapters. This material too can be skipped over and referred back to when it is used. Just as discrete parameter martingales play a key rˆ le in the study of random o walks, so for Brownian motion we shall use continuous time martingale theory to simplify a number of calculations; §3.4 extends our deﬁnitions and basic results on discrete parameter martingales to the continuous time setting.

3.1

Deﬁnition of the process
The easiest way to think about Brownian motion is as an ‘inﬁnitesimal random walk’ and that is often how it arises in applications, so to motivate the formal deﬁnition we ﬁrst study simple random walks.

A characterisation of simple random walks

We declared in Example 2.3.7 that the stochastic process {Sn }n≥0 is a simple random n walk under the measure P if Sn = i=1 ξi where the ξi can take only the values {−1, +1} and are independent and identically distributed under P. We concentrate
51

52

brownian motion

on the symmetric case when P [ξi = −1] = 1 = P [ξi = +1] . 2

This process is often motivated as a model of the gains from repeated plays of a fair game. For example, suppose I play a game with a friend in which each play is equivalent to ﬂipping a fair coin. If it comes up heads I pay her a dollar, otherwise she pays me a dollar. For each n, Sn models my net gain after n plays. Recall from Exercise 13 of Chapter 2 that E[Sn ] = 0 and var(Sn ) = n.
Lemma 3.1.1

{Sn }n≥0 is a P-martingale (with respect to the natural ﬁltration) cov(Sn , Sm ) = n ∧ m.

and

Proof: We checked in Example 2.3.7 that {Sn }n≥0 is a P-martingale. It remains to

calculate the covariance. cov(Sn , Sm ) = = = = E[Sn Sm ] − E[Sn ]E[Sm ] E [E[Sn Sm |Fm∧n ]]
2 E[Sm∧n ]

(tower property)

= E [Sm∧n E [Sm∨n |Fm∧n ]] (martingale property) ✷ As a result of the independence of the random variables {ξi }i≥1 , if 0 ≤ i ≤ j ≤ k ≤ l, then S j − Si is independent of Sl − Sk . More generally, if 0 ≤ i 1 ≤ i 2 ≤ · · · ≤ i n , then {Sir − Sir −1 : 1 ≤ r ≤ n} are independent. Moreover, if j − i = l − k = m, say, then S j − Si and Sl − Sk both have the same distribution as Sm . var(Sm∧n ) = m ∧ n.

Notation:

For two random variables X and Y we write X =Y
D

to mean that X and Y have the same distribution. We also write X ∼ N (µ, σ 2 ) to mean that X is normally distributed with mean µ and variance σ 2 .

Combining the observations above we have
Lemma 3.1.2

Under the measure P the process {Sn }n≥0 has stationary, indepen-

dent increments.

53

3.1 deﬁnition of the process

Lemmas 3.1.1 and 3.1.2 are actually enough to characterise symmetric simple random walks.
Rescaling random walks

Recall that we want to think of Brownian motion as an inﬁnitesimal random walk. In terms of our gambling game, the time interval between plays is δt and the stake is δx say, and we are thinking of both of these as ‘tending to zero’. In order to obtain a non-trivial limit, there has to be a relationship between δt and δx. To see what this must be, we use the Central Limit Theorem (stated in §2.6). In our setting, √ µ = E[ξi ] = 0 and σ 2 = var(ξi ) = 1. Thus, taking δt = 1/n and δx = 1/ n, Sn P √ ≤x → n More generally, P S[nt] √ ≤x → n x −∞ x −∞

1 2 √ e−y /2 dy as n → ∞. 2π

√

1 2π t

e−y

2 /2t

dy as n → ∞,

where [nt] denotes the integer part of nt (Exercise 1). For the limiting process, at time t our net gain since time zero will be normally distributed with mean zero and variance t.
Deﬁnition of Brownian motion

Just as in our deﬁnition of a discrete time stochastic process, to deﬁne a continuous time stochastic process {X t }t≥0 (formally) requires a probability triple ( , F, P) such that X t is F-measurable for all t. However, as in the discrete case, we shall rarely specify explicitly. Heuristically, passage to the limit in the random walk suggests that the following is a reasonable deﬁnition of Brownian motion.
Deﬁnition 3.1.3 (Brownian motion)

A real-valued stochastic process {Wt }t≥0 is a P-Brownian motion (or a P-Wiener process) if for some real constant σ , under P,

1 for each s ≥ 0 and t > 0 the random variable Wt+s −Ws has the normal distribution with mean zero and variance σ 2 t, 2 for each n ≥ 1 and any times 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn , the random variables {Wtr − Wtr −1 } are independent, 3 W0 = 0, 4 Wt is continuous in t ≥ 0.
Remarks: Conditions 1 and 2 ensure that, like its discrete counterpart, Brownian

motion has stationary independent increments. Condition 3 is a convention. Brownian motion started from x can be obtained as {x + Wt }t≥0 . In a certain sense condition 4 is a consequence of the ﬁrst three, but we should like to insist once and for all that all paths that our Brownian motion can follow are continuous. ✷

54

brownian motion

The parameter σ 2 is known as the variance parameter. By scaling of the normal distribution it is immediate that {Wt/σ }t≥0 is a Brownian motion with variance parameter one.
Deﬁnition 3.1.4

The process with σ 2 = 1 is called standard Brownian motion.

Assumption:

Unless otherwise stated we shall always assume that σ 2 = 1.

Combining conditions 1 and 2 of Deﬁnition 3.1.3, we can write down the transition probabilities of standard Brownian motion. P Wtn ≤ xn |Wti = xi , 0 ≤ i ≤ n − 1 = P Wtn − Wtn−1 ≤ xn − xn−1 = xn −xn−1 −∞

1 2π(tn − tn−1 )

exp −

u2 du. 2(tn − tn−1 )

Notation:

We write p(t, x, y) for the transition density (x − y)2 1 exp − p(t, x, y) = √ 2t 2π t .

This is the probability density function of the random variable Wt+s conditional on Ws = x. For 0 = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tn , writing x0 = 0, the joint probability density function of Wt1 , . . . , Wtn can also be written down explicitly as n f (x1 , . . . , xn ) =
1

p(t j − t j−1 , x j−1 , x j ).

The joint distributions of Wt1 , . . . , Wtn for each n ≥ 1 and all t1 , . . . , tn are called the ﬁnite dimensional distributions of the process. The following analogue of Lemma 3.1.1 is immediate.
Lemma 3.1.5

For any s, t > 0,

1 E Wt+s − Ws | {Wr }0≤r ≤s = 0, 2 cov(Ws , Wt ) = s ∧ t. In fact since the multivariate normal distribution is determined by its means and covariances and normally distributed random variables are independent if and only

55

3.1 deﬁnition of the process

Figure 3.1

Zooming in on Brownian motion.

if their covariances are zero, this, combined with continuity of paths, characterises standard Brownian motion.
Behaviour of Brownian motion

Just because the sample paths of Brownian motion are continuous, it does not mean that they are nice in any other sense. In fact the behaviour of Brownian motion is distinctly odd. Here are just a few of its strange behavioural traits. Although {Wt }t≥0 is continuous everywhere, it is (with probability one) differentiable nowhere.

1

2 Brownian motion will eventually hit any and every real value no matter how large, or how negative. No matter how far above the axis, it will (with probability one) be back down to zero at some later time. 3 Once Brownian motion hits a value, it immediately hits it again inﬁnitely often (and will continue to return after arbitrarily large times). 4 It doesn’t matter what scale you examine Brownian motion on, it looks just the same. Brownian motion is a fractal. Exercise 9 shows that the process cannot be differentiable at t = 0. We shall discuss some properties related to the hitting probabilities in §3.3 and in Exercise 8. The scaling alluded to in our last comment is formally proved in Proposition 3.3.7. It is really a consequence of the construction of the process. Figure 3.1 illustrates the result for a particular realisation of a Brownian path. That such a bizarre process actually exists is far from obvious and so it is to this that we turn our attention in the next section.

56

brownian motion

3.2

L´vy’s construction of Brownian motion e
We have hinted that Brownian motion can be obtained as a limit of random walks. However, rather than chasing the technical details of the random walk construction, in this section we present an alternative construction due to L´ vy. This can be omitted e by readers willing to take existence of the process on trust.

A polygonal approximation

The idea is that we can simply produce a path of Brownian motion by direct polygonal interpolation. We require just one calculation.
Lemma 3.2.1

Suppose that {Wt }t≥0 is standard Brownian motion. Conditional on Wt1 = x1 , the probability density function of Wt1 /2 is pt1 /2 (x) 2 1 exp − πt1 2 x − 1 x1 2 t1 /4
2

.

In other words, the conditional distribution is a normally distributed random variable with mean x1 /2 and variance t1 /4. The proof is Exercise 11.
The construction: Without loss of generality we take the range of t to be [0, 1]. L´ vy’s e

construction builds (inductively) a polygonal approximation to the Brownian motion from a countable collection of independent normally distributed random variables with mean zero and variance one. We index them by the dyadic points of [0, 1], a generic variable being denoted by ξ k2−n where n ∈ N and k ∈ {0, 1, . . . , 2n }. The induction begins with X 1 (t) = tξ(1). Thus X 1 is a linear function on [0, 1]. The nth process, X n , is linear in each interval [(k − 1)2−n , k2−n ], is continuous in t and satisﬁes X n (0) = 0. It is thus determined by the values {X n (k2−n ), k = 1, . . . , 2n }.
The inductive step: We take

X n+1 2k2−(n+1) = X n 2k2−(n+1) = X n k2−n . We now determine the appropriate value for X n+1 (2k − 1)2−(n+1) . Conditional on X n+1 2k2−(n+1) − X n+1 2(k − 1)2−(n+1) , Lemma 3.2.1 tells us that X n+1 (2k − 1)2−(n+1) − X n+1 2(k − 1)2−(n+1) should be normally distributed with mean 1 X n+1 2k2−(n+1) − X n+1 2(k − 1)2−(n+1) 2 and variance 2−(n+2) .

57

3.2 l´vy’s construction of brownian motion e

X3

X2

X1

0

1/4

1/2

3/4

1

Figure 3.2

L´vy’s sequence of polygonal approximations to Brownian motion. e

Now if X ∼ N (0, 1), then a X + b ∼ N (b, a 2 ) and so we take X n+1 (2k − 1)2−(n+1) − X n+1 2(k − 1)2−(n+1) = 2−(n/2+1) ξ (2k − 1)2−(n+1) + In other words X n+1 (2k − 1)2−(n+1) = 1 X n (k − 1)2−n 2 1 + X n k2−n + 2−(n/2+1) ξ (2k − 1)2−(n+1) 2 X n (2k − 1)2−(n+1) + 2−(n/2+1) ξ (2k − 1)2−(n+1) , where the last equality follows by linearity of X n on [(k − 1)2−n , k2−n ]. The construction is illustrated in Figure 3.2.
Convergence to Brownian motion

1 X n+1 2k2−(n+1) − X n+1 2(k − 1)2−(n+1) 2

.

=

(3.1)

Brownian motion will be the process constructed by letting n increase to inﬁnity. To check that it exists we need some technical lemmas. The proofs are adapted from Knight (1981).
Lemma 3.2.2

P lim X n (t) exists for 0 ≤ t ≤ 1 uniformly in t = 1. n→∞ Proof: Notice that maxt |X n+1 (t) − X n (t)| will be attained at a vertex, that is for

58

brownian motion

t ∈ {(2k − 1)2−(n+1) : k = 1, 2, . . . , 2n } and using (3.1) P max |X n+1 (t) − X n (t)| ≥ 2−n/4 t = ≤

P

1≤k≤2n

max ξ (2k − 1)2−(n+1) ≥ 2n/4+1

2n P ξ(1) ≥ 2n/4+1 .

Now using the result of Exercise 7 (with t = 1), for x > 0 1 2 P [ξ(1) ≥ x] ≤ √ e−x /2 , x 2π and combining this with the fact that exp −2(n/2+1) < 2−2n+2 , we obtain that for n ≥ 4 2n P ξ(1) ≥ 2n/4+1 ≤ 2n 2n/4+1 2n 1 √ exp −2(n/2+1) ≤ n/4+1 2−2n+2 < 2−n . 2 2π

Consider now for k > n ≥ 4 P max |X k (t) − X n (t)| ≥ 2−n/4+3 = 1 − P max |X k (t) − X n (t)| ≤ 2−n/4+3 t t

and P max |X k (t) − X n (t)| ≤ 2−n/4+3 t k−1

≥

P j=n max X j+1 (t) − X j (t) ≤ 2−n/4+3 t ≥ P max X j+1 (t) − X j (t) ≤ 2− j/4 , j = n, . . . , k − 1 t k−1 j=n

≥

1−

2− j ≥ 1 − 2−n+1 .

Finally we have that P max |X k (t) − X n (t)| ≥ 2−n/4+3 ≤ 2−n+1 , t for all k ≥ n. The events on the left are increasing (since the maximum can only increase by the addition of a new vertex) so P max |X k (t) − X n (t)| ≥ 2−n/4+3 for some k > n ≤ 2−n+1 . t In particular, for n→∞ > 0,

lim P [For some k > n and t ≤ 1, |X k (t) − X n (t)| ≥ ] = 0, ✷

which proves the lemma.

59

3.3 the reﬂection principle and scaling

To complete the proof of existence of the Brownian motion, we must check the following.
Lemma 3.2.3

Let X (t) = limn→∞ X n (t) if the limit exists uniformly and 0 otherwise. Then X (t) satisﬁes the conditions of Deﬁnition 3.1.3 (for t restricted to [0, 1]).

imation X n (t) restricted to Tn = {k2−n : k = 0, 1, . . . , 2n }. Since we don’t change X k on Tn for k > n, the same must be true for X on ∞ Tn . A uniform limit of n=1 continuous functions is continuous, so condition 4 holds and now by approximation of any 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ 1 from within the dense set ∞ Tn we see that in n=1 fact all four properties hold without restriction for t ∈ [0, 1]. ✷

Proof: By construction, the properties 1–3 of Deﬁnition 3.1.3 hold for the approx-

3.3

The reﬂection principle and scaling
Having proved that Brownian motion actually exists, we now turn to some calculations. These will amount to no more than a small bag of tricks for us to call upon in later chapters. There are many texts devoted exclusively to Brownian motion where the reader can gain a more extensive repertoire.

Stopping times

By its very construction, Brownian motion has no memory. That is, if {Wt }t≥0 is a Brownian motion and s ≥ 0 is any ﬁxed time, then {Wt+s − Ws }t≥0 is also a Brownian motion, independent of {Wr }0≤r ≤s . What is also true is that for certain random times, T , the process {WT +t − WT }t≥0 is again a standard Brownian motion and is independent of {Ws : 0 ≤ s ≤ T }. We have already encountered such random times in the context of discrete parameter martingales.
Deﬁnition 3.3.1

A stopping time T for the process {Wt }t≥0 is a random time such that for each t, the event {T ≤ t} depends only on the history of the process up to and including time t. In other words, by observing the Brownian motion up until time t, we can determine whether or not T ≤ t.

We shall encounter stopping times only in the context of hitting times. For ﬁxed a, the hitting time of level a is deﬁned by Ta = inf{t ≥ 0 : Wt = a}. We take Ta = ∞ if a is never reached. It is easy to see that Ta is a stopping time since, by continuity of the paths, {Ta ≤ t} = {Ws = a for some s, 0 ≤ s ≤ t}, which depends only on {Ws , 0 ≤ s ≤ t}. Notice that, again by continuity, if Ta < ∞, then WTa = a.

60

brownian motion

Just as for random walks, an example of a random time that is not a stopping time is the last time that the process hits some level.
The reﬂection principle

Not surprisingly, there is often much to be gained from exploiting the symmetry inherent in Brownian motion. As a warm-up we calculate the distribution of Ta .
Lemma 3.3.2

a > 0; then

Let {Wt }t≥0 be a P-Brownian motion started from W0 = 0 and let P [Ta < t] = 2P [Wt > a] .

Proof: If Wt > a, then by continuity of the Brownian path, Ta < t. Moreover, since

Ta is a stopping time, Wt+Ta − WTa t≥0 is a Brownian motion, so, by symmetry, P[Wt − WTa > 0|Ta < t] = 1/2. Thus P[Wt > a] = = P[Ta < t, Wt − WTa > 0] P[Ta < t]P[Wt − WTa > 0|Ta < t] 1 P[Ta < t]. = 2 ✷ A more reﬁned version of this idea is the following.
Lemma 3.3.3 (The reﬂection principle)

Let {Wt }t≥0 be a standard Brownian mo-

tion and let T be a stopping time. Deﬁne ˜ Wt = Wt , t ≤ T, 2WT − Wt , t > T ;

˜ then {Wt }t≥0 is also a standard Brownian motion. ˜ Notice that if T = Ta , then the operation Wt → Wt amounts to reﬂecting the portion of the path after the ﬁrst hitting time on a in the line x = a (see Figure 3.3). We don’t prove the general form of the reﬂection principle here. Instead we put it into action. The following result will be the key to pricing certain barrier options in Chapter 6.
Lemma 3.3.4 (Joint distribution of Brownian motion and its maximum)

Let Mt = max0≤s≤t Ws , the maximum level reached by Brownian motion in the time interval [0, t]. Then for a > 0, a ≥ x and all t ≥ 0, P[Mt ≥ a, Wt ≤ x] = 1 − 2a − x √ t ,

where

1 2 √ e−u /2 du 2π −∞ is the standard normal distribution function. (x) =

x

61

3.3 the reﬂection principle and scaling

~ Wt
Wt

a

0 t

Figure 3.3

The reﬂection principle when T = Ta . Proof: Notice that Mt ≥ 0 and is non-decreasing in t and if, for a > 0, Ta is deﬁned

to be the ﬁrst hitting time of level a, then {Mt ≥ a} = {Ta ≤ t}. Taking T = Ta in the reﬂection principle, for a ≥ 0, a ≥ x and t ≥ 0, P[Mt ≥ a, Wt ≤ x] = = P[Ta ≤ t, Wt ≤ x]

˜ P[Ta ≤ t, 2a − x ≤ Wt ] ˜ = P[2a − x ≤ Wt ] 2a − x . = 1− √ t ˜ In the third equality we have used the fact that if Wt ≥ 2a − x then necessarily ˜ s }s≥0 , and consequently {Ws }s≥0 , has hit level a before time t. ✷ {W
Hitting a sloping line

For pricing a perpetual American put option in Chapter 6 we shall use the following result.
Proposition 3.3.5

Set Ta,b = inf{t ≥ 0 : Wt = a + bt}, where Ta,b is taken to be inﬁnite if no such time exists. Then for θ > 0, a > 0 and b ≥ 0 E exp −θ Ta,b = exp −a b + b2 + 2θ .

Proof: We defer the proof of the special case b = 0 until Proposition 3.4.9 when we

shall have powerful martingale machinery to call upon. Here, assuming that result, we deduce the general result. Fix θ > 0, and for a > 0, b ≥ 0, set ψ(a, b) = E e−θ Ta,b .

62 x brownian motion x = a1 + a 2 + bt x = a1 + bt a2 + b(Ta1 + a2, b – Ta1, b ) a1 + a 2 a1 0 Ta1, b a2

~ Ta2, b

t Ta1 + a2, b

Figure 3.4

˜ ˜ In the notation of Proposition 3.3.5, Ta1 +a2 ,b = Ta1 ,b + Ta2 ,b where Ta2 ,b has the same distribution as Ta2 ,b .

Now take any two values for a, a1 and a2 say, and notice (see Figure 3.4) that ˜ Ta1 +a2 ,b = Ta1 ,b + Ta1 +a2 ,b − Ta1 ,b = Ta1 ,b + Ta2 ,b , ˜ where Ta2 ,b is independent of Ta1 ,b and has the same distribution as Ta2 ,b . In other words, ψ(a1 + a2 , b) = ψ(a1 , b)ψ(a2 , b), and this implies that ψ(a, b) = e−k(b)a ,
D

for some function k(b). Since b ≥ 0, the process must hit level a before it can hit the line a + bt. We use this to break Ta,b into two parts; see Figure 3.5. Writing f Ta for the probability density function of the random variable Ta and conditioning on Ta , we obtain ψ(a, b) =
0 ∞ ∞ ∞

f Ta (t)E e−θ Ta,b Ta = t dt f Ta (t)e−θt E e−θ Tbt,b dt f Ta (t)e−θt e−k(b)bt dt

=
0

=
0

= =

E e−(θ+k(b)b)Ta exp −a 2(θ + k(b)b) .

We now have two expressions for ψ(a, b). Equating them gives k 2 (b) = 2θ + 2k(b)b. Since for θ > 0 we must have ψ(a, b) ≤ 1, we choose k(b) = b + which completes the proof. b2 + 2θ, ✷

63

3.4 martingales in continuous time

bTa a

0 t

Figure 3.5

˜ ˜ In the notation of Proposition 3.3.5, Ta,b = Ta + TbTa ,b where TbTa ,b has the same distribution as TbTa ,b . Deﬁnition 3.3.6

For a real constant µ, we refer to the process Wt = Wt + µt as a Brownian motion with drift µ.

µ

In the notation above, Ta,b is the ﬁrst hitting time of the level a by a Brownian motion with drift −b.
Transformation We conclude this section with the following useful result. and scaling of Brownian Proposition 3.3.7 motion If {Wt }t≥0 is a standard Brownian motion, then so are

1 {cWt/c2 }t≥0 for any real c, 2 {t W1/t }t≥0 where t W1/t is taken to be zero when t = 0, 3 {Ws − Ws−t }0≤t≤s for any ﬁxed s ≥ 0.
Proof: The proofs of 1–3 are similar. For example in the case of 2, it is clear that

t W1/t has continuous sample paths (at least for t > 0) and that for any t1 , . . . , tn , the random variables {t1 W1/t1 , . . . , tn W1/tn } have a multivariate normal distribution. We must just check that the covariance takes the right form, but E sW1/s t W1/t = stE W1/s W1/t = st and the proof is complete. 1 1 ∧ s t = s ∧ t, ✷

3.4

Martingales in continuous time
Just as in discrete time, the notion of a martingale plays a key rˆ le in our continuous o time models.

64

brownian motion

Recall that in discrete time, a sequence X 0 , X 1 , . . . , X n for which E[|X r |] < ∞ for each r is a martingale with respect to the ﬁltration {Fn }n≥0 and a probability measure P if for all r ≥ 1. E X r |Fr −1 = X r −1 We can make entirely analogous deﬁnitions in continuous time.
Filtrations Deﬁnition 3.4.1

Let F be a σ -ﬁeld. We call {Ft }t≥0 a ﬁltration if

1 Ft is a sub-σ -ﬁeld of F for all t, and 2 Fs ⊆ Ft for s < t. As in the discrete setting we are primarily concerned with the natural ﬁltration, {FtX }t≥0 , associated with a stochastic process {X t }t≥0 . As before, FtX encodes the information generated by the stochastic process X on the interval [0, t]. That is A ∈ FtX if, based upon observations of the trajectory {X s }0≤s≤t , it is possible to decide whether or not A has occurred.

Notation:

If the value of a stochastic variable Z can be completely determined given observations of the trajectory {X s }0≤s≤t then we write Z ∈ FtX .

More than one process can be measurable with respect to the same ﬁltration.
Deﬁnition 3.4.2

If {Yt }t≥0 is a stochastic process such that we have Yt ∈ FtX for all t ≥ 0, then we say that {Yt }t≥0 is adapted to the ﬁltration {FtX }t≥0 .

Example 3.4.3

1 The stochastic process Zt =
0

t

X s ds

is adapted to {FtX }t≥0 . 2 The process Mt = max0≤s≤t Ws is adapted to the ﬁltration {FtW }t≥0 . 2 3 The stochastic process Z t Wt+1 − Wt2 is not adapted to the ﬁltration generated by {Wt }t≥0 . Notice that just as in the discrete world we have divorced the rˆ les of the stochastic o process and the probability measure. Thus a process may be a Brownian motion under the probability measure P, but the same process not be a Brownian motion under a different measure Q.
Martingales Deﬁnition 3.4.4

Let ( , F, P) be a probability space with ﬁltration {Ft }t≥0 . A family {Mt }t≥0 of random variables on this space with E[|Mt |] < ∞ for all t ≥ 0 is

65

3.4 martingales in continuous time

a (P, {Ft }t≥0 )-martingale if it is adapted to {Ft }t≥0 and for any s ≤ t, EP [ Mt | Fs ] = Ms . By restricting the conditions to t ∈ [0, T ], we deﬁne martingales parametrised by [0, T ]. Generally we shall be sloppy about specifying the ﬁltration. In all of our examples there will be a Brownian motion around and it will be implicit that the ﬁltration is that generated by the Brownian motion. A more general notion is that of local martingale.
Deﬁnition 3.4.5

A process {X t }t≥0 is a local P, {Ft }t≥0 -martingale if there is a sequence of {Ft }t≥0 -stopping times {Tn }n≥1 such that {X t∧Tn }t≥0 is a P, {Ft }t≥0 martingale for each n and P lim Tn = ∞ = 1. n→∞ All martingales are local martingales but the converse is false. It is because of this distinction that we impose boundedness conditions in many of our results of Chapter 4.
Lemma 3.4.6

Let {Wt }t≥0 generate the ﬁltration {Ft }t≥0 . If {Wt }t≥0 is a standard Brownian motion under the probability measure P, then Wt is a (P, {Ft }t≥0 )-martingale, Wt2 − t is a (P, {Ft }t≥0 )-martingale, exp σ Wt − σ2 t 2

1 2 3

is a (P, {Ft }t≥0 )-martingale, called an exponential martingale.
Proof: The proofs are all rather similar. For example, consider Mt = Wt2 − t.

Evidently E[|Mt |] < ∞. Now E Wt2 − Ws2 Fs = = = Thus E Wt2 − t Fs

E (Wt − Ws )2 + 2Ws (Wt − Ws ) Fs E (Wt − Ws )2 Fs + 2Ws E [ (Wt − Ws )| Fs ] t − s.

= =

E Wt2 − Ws2 + Ws2 − (t − s) − s Fs (t − s) + Ws2 − (t − s) − s = Ws2 − s. ✷

66 Optional stopping

brownian motion

What we should really like is the continuous time analogue of the Optional Stopping Theorem. In general, we have to be a little careful (see Exercise 17 for what can go wrong). Problems arise if the sample paths of our martingale are not sufﬁciently ‘nice’. In all our examples the stochastic process will have c` dl` g sample paths. a a
Deﬁnition 3.4.7

The function f : R → R is c` dl` g if it is right continuous with a a

left limits. ` In particular, continuous functions are automatically c` dl` g (continues a droite, a a ` limites a gauche). If {Mt }t≥0 is a c` dl` g martingale a a with respect to the probability measure P and the ﬁltration {Ft }t≥0 and if τ1 and τ2 are two stopping times such that τ1 ≤ τ2 ≤ K where K is a ﬁnite real number, then
Theorem 3.4.8 (Optional Stopping Theorem)

E Mτ2 and

0, E e−θ Ta = e−
√ 2θ|a|

.

Proof: We assume that a ≥ 0. (The case a < 0 follows by symmetry.) We should

like to apply the Optional Stopping Theorem to the martingale 1 Mt = exp σ Wt − σ 2 t 2 and the random time Ta , but we encounter a familiar obstacle. We cannot apply the Theorem directly to Ta as it may not be bounded. Instead we take τ1 = 0 and τ2 = Ta ∧ n. This gives us that E MTa ∧n = 1.

67

exercises

So 1 = E MTa ∧n = E MTa ∧n Ta < n P [Ta < n] + E MTa ∧n Ta > n P [Ta > n] . Now, by Lemma 3.3.2 and the result of Exercise 7, P [Ta < n] → 1 as n → ∞. (3.2)

Also, if Ta < ∞, limn→∞ MTa ∧n = MTa , whereas if Ta = ∞, Wt ≤ a for all t and so limn→∞ MTa ∧n = 0. Letting n → ∞ in equation (3.2) then yields E MTa = 1. Taking σ 2 = 2θ completes the proof.
Dominated Convergence Theorem

✷

Arguments of this type are often simpliﬁed by an application of the Dominated Convergence Theorem.
Theorem 3.4.10 (Dominated Convergence Theorem)

Let {Z n }n≥1 be a sequence of random variables with limn→∞ Z n = Z . If there is a random variable Y with |Z n | < Y for all n and E[Y ] < ∞, then E[Z ] = lim E[Z n ]. n→∞ In the proof of Proposition 3.4.9, since 1 0 ≤ MTa ∧n = exp σ WTa ∧n − σ 2 (Ta ∧ n) ≤ exp (σ a) , 2 we could take the constant eσ a as the dominating random variable Y .

Exercises
1 Suppose that {Sn }n≥0 is a symmetric simple random walk under P. Show that P S[nt] √ ≤x → n x −∞

y2 1 exp − √ 2t 2π t

dy

as n → ∞ where [nt] is the integer part of nt. 2 Let Z be normally distributed with mean zero and variance one under the measure √ √ P. What is the distribution of t Z ? Is the process X t = t Z a Brownian motion? 3 ˜ Suppose that Wt and Wt are independent Brownian motions under the measure P and ˜ let ρ ∈ [−1, 1] be a constant. Is the process X t = ρWt + 1 − ρ 2 Wt a Brownian motion?

68

brownian motion

4 Let {Wt }t≥0 be standard Brownian motion under the measure P. Which of the following are P-Brownian motions? (a) (b) (c) (d) {−Wt }t≥0 , {cWt/c2 }t≥0 , where c is a constant, √ { t W1 }t≥0 , {W2t − Wt }t≥0 .

Justify your answers. 5 Suppose that X is normally distributed with mean µ and variance σ 2 . Calculate E eθ X and hence evaluate E X 4 . 6 Prove Lemma 3.4.6.3. 7 Prove that if {Wt }t≥0 is standard Brownian motion under P then, for x > 0, √ ∞ 1 t 2 2 e−y /2t dy ≤ √ e−x /2t . P [Wt ≥ x] ≡ √ 2π t x 2π x [Hint: Integrate by parts.] 8 Let {Wt }t≥0 be standard Brownian motion under P. Let Z = supt Wt . Evidently, for any c > 0, cZ has the same distribution as Z . Deduce that, with probability one, Z ∈ {0, ∞}. Let p = P [Z = 0]. By conditioning on the event {W1 ≤ 0}, prove that P [Z = 0] ≤ P [W1 ≤ 0] P [Z = 0] , and hence p = 0. Deduce that P sup Wt = +∞, inf Wt = −∞ = 1. t t

9 Deduce from the result of Exercise 8 and the result of Proposition 3.3.7.2 that P [For each > 0, ∃s, t ≤ such that Ws < 0 < Wt ] = 1.

Deduce that if {Wt }t≥0 is differentiable at zero, then the derivative must be zero and ˜ sW1/s , arrive at a hence |Wt | ≤ t for all sufﬁciently small t. By considering Ws contradiction and deduce that Brownian motion is not differentiable at zero. 10 Brownian motion is not going to be adequate as a stock market model. First, it has constant mean, whereas the stock of a company usually grows at some rate, if only due to inﬂation. Moreover, it may be too ‘noisy’ (that is the variance of the increments may be bigger than those observed for the stock) or not noisy enough. We can scale to change the ‘noisiness’ and we can artiﬁcially introduce a drift, but this still won’t be a good model. Here is one reason why. Suppose that {Wt }t≥0 is standard Brownian motion under P. Deﬁne a new process {St }t≥0 by St = µt + σ Wt where σ > 0 and µ ∈ R are constants. Show that for all values of σ > 0, µ ∈ R and T > 0 there is a positive probability that ST is negative.

69

exercises

11

Suppose that {Wt }t≥0 is standard Brownian motion. Prove that conditional on Wt1 = x1 , the probability density function of Wt1 /2 is 2 1 exp − πt1 2 x − 1 x1 2 t1 /4
2

.

12 Let {Wt }t≥0 be standard Brownian motion under P. Let Ta be the ‘hitting time of level a’, that is Ta = inf{t ≥ 0 : Wt = a}. Then we proved in Proposition 3.4.9 that √ E exp (−θ Ta ) = exp −a 2θ . Use this result to calculate (a) E[Ta ], (b) P[Ta < ∞]. 13 Let {Wt }t≥0 denote standard Brownian motion under P and deﬁne {Mt }t≥0 by Mt = max Ws .
0≤s≤t

Suppose that x ≥ a. Calculate (a) P [Mt ≥ a, Wt ≥ x], (b) P [Mt ≥ a, Wt ≤ x]. 14 Let {Wt }t≥0 be standard Brownian motion under P. Let Ta,b denote the hitting time of the sloping line a + bt. That is, Ta,b = inf{t ≥ 0 : Wt = a + bt}. We proved in Proposition 3.3.5 that for θ > 0, a > 0 and b ≥ 0 E exp −θ Ta,b = exp −a b + b2 + 2θ .

The aim of this question is to calculate the distribution of Ta,b , without inverting the Laplace transform. In what follows, φ(x) = (x) and (x) = x −∞

y2 1 √ exp − 2 2π

dy.

(a) Find P[Ta,b < ∞]. (b) Using the fact that sW1/s has the same distribution as Ws , show that P Ta,b ≤ t = P [Ws ≥ as + b for some s with 1/t ≤ s < ∞] .

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brownian motion

(c) By conditioning on the value of W1/t , use the previous part to show that P Ta,b ≤ t = b+a/t −∞

P Tb−x+a/t,a < ∞ φ bt − a √ t

√

tx dx + 1 − a + bt √ t

a + bt √ t

.

(d) Substitute for the probability in the integral and deduce that P Ta,b ≤ t = e−2ab +1− .

15 Let {Wt }t≥0 be standard Brownian motion under the measure P and let {Ft }t≥0 denote its natural ﬁltration. Which of the following are (P, {Ft }t≥0 )-martingales? (a) exp (σ Wt ), (b) cWt/c2 , where c is a constant, t (c) t Wt − 0 Ws ds. 16 Let {Ft }0≤t≤T denote the natural ﬁltration associated to a standard P-Brownian motion, {Wt }0≤t≤T . The result of Lemma 3.4.6.3 can be rewritten as 1 1 E exp σ Wt − σ 2 t ; A = exp σ Ws − σ 2 s 1 A , 2 2 for all A ∈ Fs .

Use differentiation under the integral sign to provide another proof that {Wt2 − t}t≥0 is a (P, {Ft }t≥0 )-martingale and show that the following are also (P, {Ft }t≥0 )martingales: (a) Wt3 − 3t Wt , (b) Wt4 − 6t Wt2 + 3t 2 . 17 Let ( , F, P) be a probability space. Suppose that the real random variable T : R is uniformly distributed on [0, 1] under the measure P. Deﬁne {X t }t≥0 by X t (ω) = 1, 0, T (ω) = t, T (ω) = t. →

Check that {X t }t≥0 is a P-martingale with respect to its own ﬁltration. [Hint: Conditional expectation is only unique to within a random variable that is almost surely zero.] Show that T is a stopping time for which the Optional Stopping Theorem fails. 18 As before, let Ta , Tb denote the ﬁrst hitting times of levels a and b respectively of a P-Brownian motion, {Wt }t≥0 , but now W0 is not necessarily zero (see the remarks after Deﬁnition 3.1.3). Prove that if a < x < b then (b − x) . P [ Ta < Tb | W0 = x] = (b − a) [Hint: Mimic the proof of the corresponding result for random walk, cf. Proposition 2.4.4.] 19 Using the notation of Exercise 18, let T = Ta ∧ Tb . Prove that if a < 0 < b then E [ T | W0 = 0] = −ab.

4 Stochastic calculus

Summary
Brownian motion is clearly inadequate as a market model, not least because it would predict negative stock prices. However, by considering functions of Brownian motion we can produce a wide class of potential models. The basic model underlying the Black–Scholes pricing theory, geometric Brownian motion, arises precisely in this way. It will inherit from the Brownian motion very irregular paths. In §4.1 we shall see why a stock price model with rough paths is forced upon us by arbitrage arguments. This is not in itself sufﬁcient to justify the geometric Brownian motion model. However in §4.7 we provide a further argument that suggests that it is at least a sensible starting point. A more detailed discussion of the shortcomings of the geometric Brownian motion model is deferred until Chapter 7. In order to study models built in this way, we need to develop a calculus based o on Brownian motion. The Itˆ stochastic calculus is the main topic of this chapter. In §4.2 we deﬁne the Itˆ stochastic integral and then in §4.3 we derive the corresponding o chain rule of stochastic calculus and learn how to integrate by parts. Just as in the discrete world, there will be two key ingredients to pricing and hedging in the Black–Scholes framework. First we need to be able to change the probability measure so that discounted asset prices are martingales. The tool for doing this is the Girsanov Theorem of §4.5. The construction of the hedging portfolio depends on the continuous analogue of the Binomial Representation Theorem, the Martingale Representation Theorem of §4.6. Again as in the discrete world, the pricing formula will be in the form of the discounted expected value of a claim. Black and Scholes obtained this result via a completely different argument (see Exercise 5 of Chapter 5) in which the price is obtained as the solution of a partial differential equation. The connection with the probabilistic approach is via the Feynman–Kac stochastic representation formula of §4.8 which exposes the intricate connection between stochastic differential equations and certain second order parabolic (deterministic) partial differential equations. Once again our coverage of this material is necessarily rather sketchy. Even so readers eager to get back to some ﬁnance may wish to skip the proofs in this
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stochastic calculus

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Figure 4.1

1 Two graphs of non-dividend-paying stock over long period (6 3 years) and short period (1 year).

chapter. There is no shortage of excellent stochastic calculus texts to refer to. Some suggestions are included in the bibliography.

4.1

Stock prices are not differentiable
Figure 4.1 shows the Microsoft share price over 6 1 year and 1 year periods. It 3 certainly doesn’t look like a particularly nice function of time. Even over short time scales, the path followed by the price looks rough. There are many statistical studies that investigate the irregularity of paths of stock prices. In this section we explore through a purely mathematical argument of Lyons (1995) just how rough paths of our stock price model should be, at least under the assumption that we can trade continuously without incurring transaction costs and, as usual, that there are no arbitrage opportunities. We continue to suppose that our market contains a riskless cash bond.

3/5/01

73 Quantifying roughness

4.1 stock prices are not differentiable

First we need a means of quantifying ‘roughness’. For a function f : [0, T ] → R, its variation is deﬁned in terms of partitions. Let π be a partition of [0, T ], N (π ) the number of intervals that make up π and δ(π) be the mesh of π (that is the length of the largest interval in the partition). Write 0 = t0 < t1 < · · · < t N (π) = T for the endpoints of the intervals of the partition. Then the variation of f is
Deﬁnition 4.1.1
N (π) δ→0

lim

π:δ(π )=δ j=1

sup

f (t j ) − f (t j−1 )

.

If the function is ‘nice’, for example differentiable, then it has bounded variation. Our ‘rough’ paths will have unbounded variation. To quantify roughness we can extend the idea of variation to that of p-variation.
Deﬁnition 4.1.2

In the notation of Deﬁnition 4.1.1, the p-variation of a function f : [0, T ] → R is deﬁned as
N (π) δ→0

lim

π:δ(π)=δ j=1

sup

f (t j ) − f (t j−1 )

p

.

Notice that if p > 1 the p-variation will be ﬁnite for functions that are much rougher than those for which the variation is bounded. For example, roughly speaking, ﬁnite 2-variation will follow if the ﬂuctuation of the function over an interval of order δ is √ order δ.
Bounded variation and arbitrage

We now argue that if stock prices had bounded variation, then either they would be constant multiples of the riskless cash bond or (provided we can trade continuously and there are no transaction costs) there would be unbounded arbitrage opportunities. In the discrete time world of Chapter 2 we showed (equation (2.5)) that if a portfolio consisting of φi+1 units of stock and ψi+1 cash bonds over the time interval [iδt, (i + 1)δt) is self-ﬁnancing, then its discounted value at time N δt = T is ˜ VN = V0 +
N −1 j=0

˜ ˜ φ j+1 S j+1 − S j .

(4.1)

˜ Here φ j+1 is known at time jδt, but is typically a function of S j . In our continuous world, we can let the trading interval δt tend to zero and, if the discounted stock price process has bounded variation, as δt ↓ 0 the Riemann sum in (4.1) will converge to the Riemann integral
T

˜ ˜ φt St d St

where φt denotes our stock holding at time t. This says that for any choice of {φt (·)}0≤t≤T , we can construct a self-ﬁnancing portfolio whose discounted value at time T is T ˜ ˜ V0 + φt ( St )d St .
0

0

74

stochastic calculus

Now choose a differentiable function F(x) that is small near x = S0 and very large everywhere else. Then by investing F(S0 ) at time zero and holding a self-ﬁnancing ˜ portfolio with φ( St ) units of stock at time t, where φ(x) = F (x), we generate a portfolio at time T whose discounted value is F(S0 ) +
0 T

˜ ˜ F ( St )d St ,

˜ which, by the Fundamental Theorem of Calculus, is F( ST ). We only have to wait for the discounted stock price to move away from S0 to ˜ ˜ generate a lot of wealth. For example, the strategy that holds St − S0 units of stock at time t generates er T
0 T

˜ ˜ ˜ ˜ ˜ St − S0 d St = er T ST − S0

2

units of wealth at time T (where we have multiplied by er T to ‘undo’ the discounting). In the absence of arbitrage then we do not expect the paths of our stock price to have bounded variation. In fact, as Lyons points out, arguments of L C Young extend this. Again assuming continuous trading and no transaction costs, if the paths of the stock price have ﬁnite p-variation for some p < 2, then there are arbitrarily large proﬁts to be made.

4.2

Stochastic integration
The work of §4.1 suggests that we should be looking for models in which the stock price has inﬁnite p-variation for p < 2. A large class of such models can be constructed using Brownian motion as a building block, but this will require a new calculus. The paths of Brownian motion are too rough for the familiar Newtonian calculus to help us and, indeed, if it did the Fundamental Theorem of Calculus would once again lead us to discard Brownian motion as a basis for our models.

A differential equation for the stock price

The processes used to model stock prices are usually functions of one or more Brownian motions. Here, for simplicity, we restrict ourselves to functions of just one Brownian motion. The ﬁrst thing that we should like to do is to write down a differential equation for the way in which the stock price evolves. Suppose that the stock price is of the form St = f (t, Wt ). Using Taylor’s Theorem (and assuming that f at least is ‘nice’), f (t + δt, Wt+δt ) − f (t, Wt ) = δt f˙ (t, Wt ) + O(δt 2 ) + (Wt+δt − Wt ) f (t, Wt ) 1 + (Wt+δt − Wt )2 f (t, Wt ) + · · · 2! ∂f ∂2 f (t, x). (t, x) and f (t, x) = ∂x ∂x2

where we have used the notation ∂f f˙ (t, x) = (t, x), ∂t f (t, x) =

75

4.2 stochastic integration

Now in our usual derivation of the chain rule, when {Wt }t≥0 is replaced by a bounded variation function, the last term on the right hand side is order O(δt 2 ). However, for Brownian motion, we know that E[(Wt+δt − Wt )2 ] is δt. Consequently we cannot ignore the term involving the second derivative. Of course, now we have a problem, because we must interpret the term involving the ﬁrst derivative. If (Wt+δt − Wt )2 √ is O(δt), then (Wt+δt − Wt ) should be O( δt), which could lead to unbounded changes in {St }t≥0 over a bounded time interval. However, things are not hopeless. The expected value of Wt+δt − Wt is zero, and the ﬂuctuations around zero are on √ the order of δt. By comparison with the Central Limit Theorem, it is plausible that St − S0 is a well-deﬁned random variable. Assuming that we can make this rigorous, the differential equation governing St = f (t, Wt ) will take the form 1 d St = f˙(t, Wt )dt + f (Wt )d Wt + f (Wt )dt. 2 It is convenient to write this in integrated form, St = S0 +
0 t

f˙(s, Ws )ds +
0

t

f (Ws )d Ws +
0

t

1 f (Ws )ds. 2

(4.2)

Quadratic variation

In order to make sense of a calculus based on Brownian motion, we must ﬁnd a rigorous mathematical interpretation of the stochastic integral (that is, the ﬁrst integral) on the right hand side of equation (4.2). The key is to study the quadratic variation of Brownian motion. For a typical Brownian path, the 2-variation will be inﬁnite. However, a slightly weaker analogue of 2-variation does exist.
Theorem 4.2.1

Let Wt denote Brownian motion under P and for a partition π of S(π ) =
N (π ) j=1

[0, T ] deﬁne

Wt j − Wt j−1 .

2

Let πn be a sequence of partitions with δ(πn ) → 0. Then E |S(πn ) − T |2 → 0 as n → ∞. (4.3)

We say that the quadratic variation process of Brownian motion, denoted by {[W ]t }t≥0 , is [W ]t = t. More generally, we can deﬁne the quadratic variation process associated with any bounded continuous martingale. Suppose that {Mt }t≥0 is a bounded continuous P-martingale. The quadratic variation process associated with {M}t≥0 is the process {[M]t }t≥0 such that for any sequence of partitions πn of [0, T ] with δ(πn ) → 0,
Deﬁnition 4.2.2

E

N (π) j=1

Mt j − Mt j−1

2

2

− [M]T

→0

as n → ∞.

(4.4)

Remark: We don’t prove it here, but the limit in (4.4) will be independent of the sequence of partitions. ✷

76

stochastic calculus Proof of Theorem 4.2.1: We expand the expression inside the expectation in (4.3) and N (π ) make use of our knowledge of the normal distribution. Let {tn, j } j=0n denote the endpoints of the intervals that make up the partition πn . First observe that

|S(πn ) − T | =
2

N (πn ) j=1

2

Wtn, j − Wtn, j−1
2

2

− tn, j − tn, j−1

.

It is convenient to write δn, j for Wtn, j − Wtn, j−1 |S (πn ) − T |2 =
N (πn ) j=1

− (tn, j − tn, j−1 ). Then δn, j δn,k . j 0. By analogy with the Itˆ integral with respect to Brownian motion, o for a simple function n f (s, ω) = i=1 ai (ω)1 Ii (s), n we deﬁne f (s, ω)d Ms

ai (ω) Msi+1 − Msi . i=1 Passing to limits we can then deﬁne J M ( f )t =
0 M M for all f ∈ HT where HT is the set of predictable functions f : R+ × R → R such that T t

f (s, ω)d Ms

E f (s, ω)2 d[M]s < ∞.

0

By redeﬁning J M ( f ) to be zero on a set of P-measure zero if necessary, we obtain the following analogue of Theorem 4.2.7. Assume that {Mt }t≥0 is a bounded continuous (P, {Ft }t≥0 )martingale with E Mt2 < ∞ for each t > 0. Then there exists a linear mapping M J M from HT to the space of continuous (P, {Ft }t≥0 )-martingales deﬁned on [0, T ] such that
Theorem 4.2.9

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stochastic calculus

1 if f is simple, then J M ( f )t =
0 t

f (s, ω)d Ms =

f (s, ω)1[0,t] (s)d Ms ,

as deﬁned above, 2 if t ≤ T E J M ( f )2 = E t
0

t

f (s, ω)2 d[M]s ,

where {[M]t }t≥0 is the quadratic variation process associated with {Mt }t≥0 , and 3 E
0≤t≤T

sup J M ( f )2 ≤ 4E t

T 0

f (s, ω)2 d[M]s .

Except possibly on a set of P-probability zero, t 0

f (s, ω)d Ms = lim

N (π)−1 i=0

δ(π)→0

f (si , ω) Msi+1 − Msi .

We can now extend the deﬁnition still further to deﬁne the integral with respect to any process {X t }t≥0 that can be written as X t = Mt + At for a continuous martingale {Mt }t≥0 (with E Mt2 < ∞) and a process {At }t≥0 of bounded variation. We’ll exploit this greater generality in proving Lemma 4.2.11, a useful result that tells us that martingales of bounded variation are constant. Suppose that {Mt }t≥0 is a continuous martingale and {At }t≥0 is a process of bounded variation; then the process {X t }t≥0 deﬁned by X t = Mt + At is said to be a semimartingale.
Deﬁnition 4.2.10

A continuous semimartingale is any process that can be decomposed in this way. If we insist that A0 = 0 then the decomposition is unique.

Warning:

Strictly we should replace ‘martingale’ by ‘local martingale’ in Deﬁnition 4.2.10. See, for example, Ikeda & Watanabe (1989) or Chung & Williams (1990) for a more general treatment.

Lemma 4.2.11

Let {At }0≤t≤T be a continuous (P, {Ft }0≤t≤T )-martingale with E A2 < ∞ for each 0 ≤ t ≤ T . If {At }0≤t≤T has bounded variation on [0, T ] then t P [At = A0 , ∀t ∈ [0, T ]] = 1.

ˆ ˆ Proof: Let At = At − A0 . Since { At }0≤t≤T is a continuous process of bounded variation we can deﬁne the integral t 0

ˆ ˆ As d As in the classical way, and by the

85

4.3 itˆ ’s formula o

Fundamental Theorem of Calculus, ˆ0 ˆt ˆt A2 − A2 = A2 = 2 t 0

ˆ ˆ As d As .

The integral will be the same whether viewed as a classical or as a stochastic integral and so Theorem 4.2.9 tells us that it is a martingale and hence E 2
0 t

ˆ ˆ As d As = 0.

ˆ ˆ ˆt Thus E A2 = 0 for all t and so by continuity of { At }0≤t≤T , P At = 0, ∀t ∈ [0, T ] = 1 as required. ✷

4.3

Itˆ ’s formula o
Having made some sense of the stochastic integral, we are now in a position to establish some of the rules of Itˆ stochastic calculus. We begin with the chain rule o and some of its ramiﬁcations.

The stochastic Theorem 4.3.1 (Itˆ ’s formula) o For f such that the partial derivatives ∂t , ∂ x and ∂2 f f chain rule exist and are continuous and ∂ x ∈ H, almost surely for each t we have ∂ ∂x2

∂f

∂f

f (t, Wt ) − f (0, W0 )

=

t ∂f ∂f (s, Ws )d Ws + (s, Ws )ds 0 ∂x 0 ∂s 1 t ∂2 f + (s, Ws )ds. 2 0 ∂x2 t

Notation:

Often one writes Itˆ ’s formula in differential notation as o 1 d f (t, Wt ) = f (t, Wt )d Wt + f˙(t, Wt )dt + f (t, Wt )dt. 2

Outline of proof: To avoid too many cumbersome formulae, suppose that f˙ ≡ 0. (The t

proof extends without difﬁculty to the general case.) The formula then becomes f (Wt ) − f (W0 ) =
0 t

∂f 1 (Ws )d Ws + ∂x 2

t 0

∂2 f (Ws )ds. ∂x2

(4.8)

Let π be a partition of [0, t] and as usual write ti , ti+1 for the endpoints of a generic interval. Then f (Wt ) − f (W0 ) =
N (π )−1 j=0

f (Wt j+1 ) − f (Wt j ) .

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stochastic calculus

We apply Taylor’s Theorem on each interval of the partition. f (Wt ) − f (W0 ) =
N (π)−1 j=0

f (Wt j ) Wt j+1 − Wt j +

1 2

N (π)−1 j=0

f (ξ j ) Wt j+1 − Wt j

2

,

for some points ξ j ∈ Wt j ∧ Wt j+1 , Wt j ∨ Wt j+1 . By continuity of the Brownian path, we can write this as f (Wt ) − f (W0 ) =
N (π)−1 j=0

f (Wt j )(Wt j+1 − Wt j ) +

1 2

N (π)−1 j=0

f (Wη j )(Wt j+1 − Wt j )2 ,

where η j ∈ [t j , t j+1 ]. We rewrite the second term as 1 2 where j N (π )−1 j=0

f (Wt j ) +

j

Wt j+1 − Wt j

2

,

= f (Wη j ) − f (Wt j ).
∂2 f

j → 0 as the mesh of the partition tends to zero. Now we mimic our proof of Theorem 4.2.1.

A special case: Suppose that ∂ x 2 is bounded. Then, for each ﬁxed T , since r → 2
∂ f (Wr ) is uniformly continuous on [0, T ], sup j ∂x2

 E =

N (π )−1 j=0

2

 
2

f (Wt j )

Wt j+1 − Wt j

2

− t j+1 − t j
2

E

N (π )−1 j=0

( f (Wt j ))2

Wt j+1 − Wt j

− t j+1 − t j Wti+1 − Wti
2

+ 2E
0≤i< j≤N (π)−1

f (Wti ) f (Wt j ) × Wt j+1 − Wt j

2

− (ti+1 − ti ) . (4.9)

− t j+1 − t j

f expectations, coupled now with boundedness of ∂ x 2 , shows that the right hand side ∂ of (4.9) tends to zero as the mesh of the partition tends to zero. Finally, using that
2

Exactly as before, conditioning on Ft j and using the tower property of conditional

N (π)−1 j=0

∂f (Wt j ) Wt j+1 − Wt j → ∂x
2

t 0

∂f (Ws )d Ws , ∂x

f and exploiting continuity, we see that if ∂ x 2 is bounded, the formula (4.8) holds ∂ except possibly on a set of P-measure zero.

87

4.3 itˆ ’s formula o The general case: In order to drop the assumption that ∂ x 2 is bounded, we can use
∂2 f

what is called a ‘localising sequence’. Let

τn = inf {t ≥ 0 : |Wt | > n} . f f Then replacing {Ws }s≥0 by {Ws∧τn }s≥0 , since ∂ x 2 is continuous, ∂ x 2 (Ws∧τn ) ∂ ∂ s≥0 is uniformly bounded. Our proof goes through with {Ws }s≥0 replaced by {Ws∧τn }s≥0 throughout. (Note that we need the fact that τn is a stopping time to make this work.) The full result then follows by letting n → ∞. ✷
2 2

For full details of the proof of Itˆ ’s formula, see, for example, Ikeda & Watano abe (1989) or Chung & Williams (1990).
Example 4.3.2

Use Itˆ ’s formula to compute E[Wt6 ]. o

Solution: Let us deﬁne {Z t }t≥0 by Z t = Wt6 . Then by Itˆ ’s formula o

d Z t = 6Wt5 d Wt + 15Wt4 dt, and, of course, Z 0 = 0. In integrated form, Zt − Z0 =
0 t

6Ws5 d Ws +

t 0

15Ws4 ds.

The expectation of the stochastic integral vanishes (by the martingale property) and so t E[Z t ] =
0

15E Ws4 ds.

Now from Exercise 5 of Chapter 3 (or Exercise 9 of this chapter), E Ws4 = 3s 2 , and so substituting E Wt6 = E [Z t ] = 15 t 0

3s 2 ds = 15t 3 . ✷

Geometric Brownian motion

The basic reference model for stock prices in continuous time is geometric Brownian motion, deﬁned by St = S0 exp (νt + σ Wt ) , (4.10)

where, as usual, {Wt }t≥0 is a standard P-Brownian motion. Applying Itˆ ’s formula, o 1 d St = σ St d Wt + ν + σ 2 St dt. 2 This expression is called the stochastic differential equation for St . It is common to write such symbolic equations even though it is the integral equation that makes sense.

88 Lemma 4.3.3

stochastic calculus

Writing µ = ν + σ 2 /2, the geometric Brownian motion process deﬁned above is a P, {Ft }t≥0 -martingale if and only if µ = 0. Moreover E [St ] = S0 exp (µt) .

Proof: Writing the stochastic differential equation for geometric Brownian motion in integrated form,

St = S0 +
0

t

1 ν + σ 2 Ss ds + 2

t 0

σ Ss d Ws .

(4.11)

Notice that {St }t≥0 is a P, {Ft }t≥0 -semimartingale. Proving that it is a martingale if and only if µ = 0 amounts to proving uniqueness of the decomposition of a semimartingale into a martingale and a bounded variation process in this special case.
Suppose µ = 0: The classical integral in (4.11) then vanishes. Since {St }t≥0 inherits

continuity from the Brownian motion and by (4.10) it is adapted, by the remark after Deﬁnition 4.2.3 it is predictable and so by Theorem 4.2.7 the stochastic integral in (4.11) is a martingale.
Suppose that {St }t≥0 is a (P, {Ft }t≥0 )-martingale: Since the difference of two martin-

gales (with respect to the same ﬁltration) is again a martingale, we see that {At }t≥0 deﬁned by At = St − S0 −
0 t

σ Ss d Ws =
0

t

µSs ds

is a martingale. This classical integral has bounded variation and so by Lemma 4.2.11 with probability one it is equal to A0 = 0. Since Ss > 0 for all s, it follows that µ = 0.
The expectation: To verify the second claim, we take expectations in (4.11) and use

once again that the stochastic integral term is a mean zero martingale to obtain E [St ] − S0 = E
0 t

µSs ds =
0

t

µE [Ss ] ds.

(The interchange of time integral and expectation is justiﬁed by classical integration theory.) Solving this integral equation gives E [St ] = S0 exp (µt)
Itˆ ’s formula o for geometric Brownian motion

as required.

✷

It is convenient to have a version of Itˆ ’s formula that allows us to work directly with o {St }t≥0 (so that we can write down a stochastic differential equation for f (t, St )). We now know how to make our original heuristic calculations rigorous, so with a

89

4.3 itˆ ’s formula o

clear conscience we proceed as follows: f (t + δt, St+δt ) − f (t, St ) ≈ f˙(t, St )δt + f (t, St ) (St+δt − St ) 1 + f (t, St ) (St+δt − St )2 2 ≈ f˙(t, St )dt + f (t, St )d St 1 + f (t, St ) σ 2 St2 d Wt2 + µ2 St2 dt 2 + 2σ µSt2 d Wt dt . 2 That is 1 d f (t, St ) = f˙(t, St )dt + f (t, St )d St + f (t, St )σ 2 St2 dt, 2 where we have used the multiplication table × d Wt dt d Wt dt 0 dt 0 0

Writing this version of Itˆ ’s formula in integrated form gives then o f (t, St ) − f (0, S0 ) = t ∂f ∂f (u, Su )du + (u, Su )d Su 0 ∂u 0 ∂x 1 t ∂2 f 2 + (u, Su )σ 2 Su du 2 0 ∂x2 t ∂f t ∂f (u, Su )du + (u, Su )σ Su d Wu = 0 ∂u 0 ∂x t ∂f 1 t ∂2 f 2 (u, Su )µSu du + + (u, Su )σ 2 Su du. 2 0 ∂x2 0 ∂x t

Warning:

Be aware that the stochastic integral with respect to {St }t≥0 will not be a martingale with respect to the probability P under which {Wt }t≥0 is a martingale except in the special case when {St }t≥0 is a P-martingale, that is when µ = 0. To actually calculate it is often wise to separate the martingale part by expanding the ‘stochastic’ integral as in the last line.

Example 4.3.4

Suppose that {St }t≥0 is a geometric Brownian motion satisfying d St = µSt dt + σ St d Wt , (4.12)

where {Wt }t≥0 is standard Brownian motion under P. Calculate E Stn for n ∈ N.
Solution: From our calculations above,

St = exp (νt + σ Wt )

90

stochastic calculus

where ν = µ − σ 2 /2. Thus {Stn }t≥0 is also a geometric Brownian motion with parameters ν (n) = nν and σ (n) = nσ . By Lemma 4.3.3 we then have n E Stn = S0 exp

1 nν + n 2 σ 2 t 2

n = S0 exp

1 nµ + n(n − 1)σ 2 t . 2 (4.13)

To gain some more practice with stochastic calculus, suppose that we did not know how to express the solution to the stochastic differential equation (4.12) explicitly as a function of {Wt }t≥0 . An alternative approach to calculating E Stn , which we now sketch, is to apply Itˆ ’s lemma. o d Stn = 1 nStn−1 d St + n(n − 1)Stn−2 σ 2 St2 dt 2 1 n = St nµ + n(n − 1)σ 2 dt + nσ Stn d Wt . 2

Writing this equation in integrated form and taking expectations yields n E Stn − E S0 = t 0

1 n nµ + n(n − 1)σ 2 E Ss ds. 2 ✷

This leads us once again to the expression (4.13).
L´vy’s chare acterisation of Brownian motion

The Itˆ formula provides a quick route to a useful characterisation of Brownian o motion due to L´ vy. We have seen that Brownian motion is a martingale. It is useful e to be able to identify when a martingale is in fact Brownian motion.
Theorem 4.3.5

Let {Wt }0≤t≤T be a continuous (P, {Ft }0≤t≤T )-martingale such that W0 = 0 and [W ]t = t for 0 ≤ t ≤ T . Then {Wt }0≤t≤T is a (P, {Ft }0≤t≤T )Brownian motion.

Proof: We must check that for any 0 ≤ s < t ≤ T , Wt − Ws is normally distributed

with mean zero and variance t − s and is independent of Fs . Let θ2 Mtθ exp θ Wt − t . 2 Applying Itˆ ’s formula we see that {Mt }0≤t≤T is a P, {Ft }0≤t≤T -martingale and o so for 0 ≤ s ≤ t ≤ T , Mt Fs = 1. E Ms Substituting and rearranging gives E exp (θ (Wt − Ws ))| Fs = exp 1 (t − s)θ 2 . 2

Since the normal distribution is characterised by its moment generating function, the result follows. ✷

91 Stochastic differential equations

4.3 itˆ ’s formula o

Let us return to processes of the form Z t = f (t, Wt ). Using Itˆ ’s formula o d Zt = 1 f˙(t, Wt ) + f (t, Wt ) dt + f (t, Wt )d Wt . 2

Suppose that f is invertible; then we may write this as d Z t = µ(t, Z t )dt + σ (t, Z t )d Wt .
Deﬁnition 4.3.6

(4.14)

An equation of the form (4.14) for some deterministic functions µ(t, x) and σ (t, x) on R+ ×R is called a stochastic differential equation for {Z t }t≥0 . It is often easier to write down a stochastic differential equation for {Z t }t≥0 than to produce the function f (t, x) explicitly.

Warning:

Just as for (Newtonian) ordinary differential equations, in general a stochastic differential equation may not have a solution and, even if it does, the solution may not be unique.

If the functions µ(t, x) and σ (t, x) are, for example, bounded and uniformly Lipschitz-continuous in x then a unique solution does exist, but these conditions are certainly not necessary (see, for example, Chung & Williams (1990) and Ikeda & Watanabe (1989) for more details). Of course, we should really understand equation (4.14) in integrated form: Zt − Z0 =
0 t

µ(s, Z s )ds +
0

t

σ (s, Z s )d Ws .

It is left to the reader to justify the following version of the Itˆ formula. o
Theorem 4.3.7

If Z t = f (t, Wt ) satisﬁes d Z t = σ (t, Z t )d Wt + µ(t, Z t )dt,

and Yt = g(t, Z t ), for some twice differentiable functions f and g, then 1 dYt = g(t, Z t )dt + g (t, Z t )d Z t + g (t, Z t )σ 2 (t, Z t )dt. ˙ 2
Remark: Notice that

(4.15)

Mt = Z t − Z 0 −
0

t

µ(s, Z s )ds

92

stochastic calculus

is a martingale with mean zero. From the Itˆ isometry, we know that its variance is o E[Mt2 ] = E t 0

σ (s, Z s )2 ds .

The expression σ 2 (t, Z t )dt appearing in (4.15) is just d[M]t where {[M]t }t≥0 is the quadratic variation associated with {Mt }t≥0 . If {Z t }t≥0 is deﬁned by Z t = Mt + At where {Mt }t≥0 is a continuous martingale with E Mt2 < ∞ and {At }t≥0 has bounded variation, then setting Yt = g(t, Z t ) we have 1 dYt = g(t, Z t )dt + g (t, Z t )d Z t + g (t, Z t )d[M]t . ˙ 2 In particular, by applying this to Yt = Mt2 , one shows (Exercise 15) that Mt2 − [M]t is a martingale. ✷
Solving stochastic differential equations

Even when a stochastic differential equation has a unique solution it is rare to be able to express it in closed form as a function of Brownian motion. However, if this can be done, then Itˆ ’s formula provides a route to ﬁnding the solution. o
Example 4.3.8

Solve d X t = X t3 dt − X t2 d Wt , X 0 = 1. (4.16)

Solution: If X t = f (t, Wt ), then

1 d X t = f˙(t, Wt )dt + f (t, Wt )d Wt + f (t, Wt )dt, 2 and, substituting in (4.16), d X t = f (t, Wt )3 dt − f (t, Wt )2 d Wt . Equating coefﬁcients we obtain − f (t, Wt )2 = f (t, Wt ) and 1 f˙(t, Wt ) + f (t, Wt ) = f (t, Wt )3 , 2 f (t, x) =

which yields 1 x +c where c is a constant. Using the initial condition, we ﬁnd Xt = 1 . Wt + 1 ✷

Notice that this solution blows up in ﬁnite time.

93

4.4 integration by parts and a stochastic fubini theorem

4.4

Integration by parts and a stochastic Fubini Theorem
We shall need two more rules for manipulating stochastic integrals: the integration by parts formula and a ‘stochastic Fubini Theorem’. The ﬁrst is the product rule of stochastic differentiation, the second is used to justify interchange of order of stochastic and classical integrals. Suppose that we have two stochastic differential equations: dYt = µ(t, Yt )dt + σ (t, Yt )d Wt , d Z t = µ(t, Z t )dt + σ (t, Z t )d Wt . ˜ ˜ We assume for now that these equations are driven by the same Brownian motion {Wt }t≥0 . Consider the (P, {Ft }t≥0 )-martingales deﬁned by MtY = t 0

σ (s, Ys )d Ws

and

MtZ =

t 0

σ (s, Z s )d Ws , ˜

with associated quadratic variation processes [M Y ]t =
0 t

σ 2 (s, Ys )ds

and

[M Z ]t =
0

t

σ 2 (s, Z s )ds. ˜

Covariation

Clearly {MtY }t≥0 and {MtZ }t≥0 are not independent. One way of quantifying the dependence between them is through their covariance. Evidently {(MtY + MtZ )}t≥0 and {(MtY − MtZ )}t≥0 are also (P, {Ft }t≥0 )-martingales with quadratic variation [M Y ± M Z ]t =
0 t

˜ (σ (s, Ys ) ± σ (s, Z s ))2 ds.

We’re interested in the process {MtY MtZ }t≥0 . We attack it via polarisation: MtY MtZ = Taking expectations, E MtY MtZ = 1 E 4 t 0 t 0

1 4

MtY + MtZ

2

− MtY − MtZ

2

.

˜ (σ (s, Ys ) + σ (s, Z s ))2 ds −
0

t

˜ (σ (s, Ys ) − σ (s, Z s ))2 ds

=E We write

σ (s, Ys )σ (s, Z s )ds . ˜ t 0

MY , M Z

t

=

σ (s, Ys )σ (s, Z s )ds. ˜

Since for a (P, {Ft }t≥0 )-martingale {Mt2 − [M]t }t≥0 is a (P, {Ft }t≥0 )-martingale (Exercise 15), again exploiting polarisation we see that {MtY MtZ − [M Y , M Z ]t }t≥0 is a (P, {Ft }t≥0 )-martingale.

94

stochastic calculus

More generally, we could consider stochastic differential equations driven by different (but not necessarily independent) Brownian motions. For example, suppose that dYt = µ(t, Yt )dt + σ (t, Yt )d Wt , ˜ d Z t = µ(t, Z t )dt + σ (t, Z t )d Wt , ˜ ˜ ˜ ˜ where E (Wt − Ws )(Wt − Ws ) = ρ(t − s) for some 0 < ρ < 1. This means that the Brownian motions driving the two equations are positively correlated, an increase in one tends to be associated with an increase in the other, but they are not identical. We deﬁne MtY and MtZ exactly as before and again study MtY MtZ via polarisation. Writing 1 MY + M Z t − MY − M Z t , MtY MtZ = 4 and using the deﬁnition of quadratic variation, MY , M Z t = lim

N (π)−1 j=0

δ(π )→0

MtYj+1 − MtYj

MtZ − MtZ . j j+1

In our example, provided σ and σ are continuous say, we have ˜ MY , M Z t = lim

N (π)−1 j=0

δ(π)→0

σ (t j , Yt j )σ (t j , Z t j ) Wt j+1 − Wt j ˜

˜ ˜ Wt j+1 − Wt j ,

and by mimicking the argument of the proof of Theorem 4.2.1 we obtain MY , M Z
Deﬁnition 4.4.1 t =
0

t

σ (s, Ys )σ (s, Z s )ρds. ˜

For continuous (P, {Ft }t≥0 )-martingales {Mt }t≥0 and {Nt }t≥0 [M, N ]t

1 [M + N ]t − [M − N ]t 4 is called the mutual variation or covariation process of M and N . Of course {[M, M]t }t≥0 is just the quadratic variation process associated with {Mt }t≥0 . In the notation of Deﬁnition 4.4.1, if we write δ(π ) for a generic partition of [0, t] then [M, N ]t = lim
N (π)−1 j=0

δ(π )→0

Mt j+1 − Mt j

Nt j+1 − Nt j .

We now have the technology required for manipulating products of semimartingales.
Y If Yt = MtY + At and Z t = MtZ + AtZ , Y where {MtY }t≥0 and {MtZ }t≥0 are continuous (P, {Ft }t≥0 )-martingales and {At }t≥0 Z} and {At t≥0 are continuous processes of bounded variation, then

Theorem 4.4.2 (Integration by parts)

d(Yt Z t ) = Yt d Z t + Z t dYt + d M Y , M Z

t

.

95

4.4 integration by parts

o Proof: We apply the Itˆ formula to (Yt + Z t )2 , Yt2 and Z t2 , and subtract the second two from the ﬁrst to obtain the result.
Example 4.4.3

✷

Suppose that the Sterling price of an asset follows the stochastic differential equation d St = µ1 St dt + σ1 St d Wt ˜ d E t = µ2 E t dt + σ2 E t d Wt . ˜ Here {Wt }t≥0 and {Wt }t≥0 are P-Brownian motions with E ˜ ˜ Wt − Ws Wt − Ws = ρ(t − s)

and the dollar cost of £1 at time t is E t where

for some constant ρ > 0. If the riskless interest rate is r in the UK and s in the U S A, ﬁnd the stochastic differential equation for the discounted asset price in the Sterling and dollar markets respectively. For what values of the parameters are the discounted asset prices martingales in each market? ˜ Solution: In the Sterling market, write { S}t≥0 for the discounted stock price. That is ˜ St = e−r t St . Since the function e−r t has bounded variation, our integration by parts formula gives ˜ d St = = −r e−r t St dt + e−r t d St ˜ ˜ (µ1 − r ) St dt + σ1 St d Wt .

In Sterling markets, this is the stochastic differential equation governing the discounted asset price. The solution is a martingale if and only if µ1 = r . Let us write {X t }t≥0 for the dollar price of the asset. Then X t = E t St and, again by integration by parts, d Xt = = E t d St + St d E t + σ1 σ2 E t St ρdt ˜ (µ1 + µ2 + ρσ1 σ2 ) X t dt + σ1 X t d Wt + σ2 X t d Wt .

˜ The discounted asset price in the dollar market, denoted by { X t }t≥0 , then follows ˜ ˜ ˜ ˜ ˜ d X t = (µ1 + µ2 + ρσ1 σ2 − s) X t dt + σ1 X t d Wt + σ2 X t d Wt . The discounted price in the dollar market is a martingale if and only if µ1 + µ2 + ρσ1 σ2 − s = 0. Notice that it is perfectly possible for the discounted asset price to be a martingale in one market but not the other. It is important to keep this in mind when valuing options in the foreign exchange market (see §5.3) or when valuing quantos (see §7.2). ✷

96 A stochastic Fubini Theorem

stochastic calculus

We ﬁnish this section with one more useful result. This is a ‘stochastic Fubini Theorem’ that will allow us to interchange the order of a stochastic and a classical integral. We shall need this result in valuation of certain path-dependent exotic options in Exercise 23 of Chapter 6. We state the theorem in a very special form that will be sufﬁcient for our needs. For a more general version see, for example, Ikeda & Watanabe (1989).
Theorem 4.4.4

Let , F, {Ft }t≥0 , P be a ﬁltered probability space and let {Mt }t≥0 be a continuous P, {Ft }t≥0 -martingale with M0 = 0. Suppose that (t, r, ω) : R+ × R+ × → R is a bounded {Ft }t≥0 -predictable random variable. Then for each ﬁxed T > 0, t 0

R

(s, r, ω) 1[0,T ] (r )dr d Ms =

t

R 0

(s, r, ω) 1[0,T ] (r )d Ms dr.

Example 4.4.5

Suppose that {Wt }t≥0 is a P, {Ft }t≥0 -Brownian motion. Evaluate the mean and variance of t Yt
0

Wr dr.

Solution: The classical Fubini Theorem tells us that

E [Yt ] = E
0

t

Wr dr =
0

t

E [Wr ] dr = 0.

The difﬁculty is to calculate E Yt2 and this is where we exploit our stochastic Fubini Theorem with (s, r, ω) = 1[0,r ] (s). Yt =
0 t

Wr dr

=
0

t 0 t 0 s t 0

r

1d Ws dr t = =

1dr dWs (t − s)d Ws .

(Fubini’s Theorem)

Now using the Itˆ isometry we can calculate E Yt2 to be o E Yt2 = E t 0 2

(t − s)dWs

=
0

t

(t − s)2 ds =

1 3 t . 3 ✷

4.5

The Girsanov Theorem
In order to price and hedge in the Black–Scholes framework we shall need two fundamental results. The ﬁrst will allow us to change probability measure so that the discounted asset prices are martingales. Recall that in our discrete time world, once

97

4.5 the girsanov theorem

we had such a martingale measure, the pricing of options was reduced to calculating expectations under that measure. In the continuous world it will no longer be possible to ﬁnd the martingale measure by linear algebra. Nonetheless, before stating the continuous time result, we revert to our binomial trees for guidance.
Changing probability on a binomial tree

We use the notation of Chapter 2. Suppose that, under the probability measure P, if the value of an asset at time iδt is known to be Si then its value at time (i + 1)δt is Si u with probability p and it is Si d with probability 1 − p. As we saw in Chapter 2, if we let Q be the probability measure under which the probability of an up jump is q = (1−d)/(u−d) and of a down jump is (u−1)/(u−d), then the process {Si }0≤i≤N is a Q-martingale. We can regard the measure Q as a reweighting of the measure P. For example, consider a path S0 , S1 , . . . , Si through the tree. Its probability under P is p N (i) (1 − p)i−N (i) , where N (i) is the number of up jumps that the path makes. Under Q its probability is L i p N (i) (1 − p)i−N (i) where Li = q p
N (i)

1−q 1− p

i−N (i)

.

Evidently L i depends on the path that the stochastic process takes through the tree and can itself be thought of as a stochastic process adapted to the ﬁltration {Fi }1≤i≤N . Moreover, L i /L i−1 is q/ p if Si /Si−1 = u and is (1 − q)/(1 − p) if Si /Si−1 = d, so that EP L i | Fi−1 = L i−1 p q 1−q + (1 − p) p 1− p = L i−1 .

In other words, {L i }0≤i≤N is a P, {Fi }1≤i≤N -martingale with E [L i ] = L 0 = 1. If we wish to calculate the expected value of a claim in the Q-measure, it is given by EQ [C] = EP [L N C] .

We have obtained the Radon–Nikodym derivative of Q with respect to P. It is customary to write
Notation:

Li =

dQ . dP Fi

Change of measure in the continuous world

We have shown that the process of changing to the martingale measure can be viewed as a reweighting of the probabilities of paths under our original measure P according to a positive, mean one, P-martingale. This procedure of reweighting according to a positive martingale can be extended to the continuous setting. Our aim now is to investigate the effect of such a reweighting on the distribution of the P-Brownian

98

stochastic calculus

motion. Later this will enable us to choose the right reweighting so that under the new measure obtained in this way the discounted stock price is a martingale.
Theorem 4.5.1 (Girsanov’s Theorem)

Suppose that {Wt }t≥0 is a P-Brownian motion with the natural ﬁltration {Ft }t≥0 and that {θt }t≥0 is an {Ft }t≥0 -adapted process such that 1 T 2 θ dt < ∞. E exp 2 0 t Deﬁne t 1 t 2 L t = exp − θs d Ws − θ ds 2 0 s 0 and let P(L) be the probability measure deﬁned by P(L) [A] =
A

L t (ω)P(dω).
(L)

Then under the probability measure P(L) , the process {Wt Wt is a standard Brownian motion.
(L)

}0≤t≤T , deﬁned by

= Wt +
0

t

θs ds,

Notation:

We write

dP(L) dP

= Lt .
Ft

(L t is the Radon–Nikodym derivative of P(L) with respect to P.)

Remarks:

1 The condition

1 T 2 θ dt < ∞, 2 0 t known as Novikov’s condition, is enough to guarantee that {L t }t≥0 is a (P, {Ft }t≥0 )martingale. Since L t is clearly positive and has expectation one, P(L) really does deﬁne a probability measure. 2 Just as in the discrete world, two probability measures are equivalent if they have the same sets of probability zero. Evidently P and P(L) are equivalent. 3 If we wish to calculate an expectation with respect to P(L) we have E exp EP More generally, EP
(L) (L)

[φt ] = E [φt L t ] . Lt Fs . Ls ✷

[ φt | Fs ] = EP φt

This will be fundamental in option pricing.

99

4.5 the girsanov theorem Outline of proof of theorem: We have already said that {L t }t≥0 is a (P, {Ft }t≥0 )-

martingale. We don’t prove this in full, but we ﬁnd supporting evidence by ﬁnding the stochastic differential equation satisﬁed by L t . We do this in two stages. First, deﬁne t 1 t 2 θs d W s − θ ds. Zt = − 2 0 s 0 Then 1 d Z t = −θt d Wt − θt2 dt. 2 Now we apply Itˆ ’s formula to L t = exp(Z t ). o d Lt = = 1 exp(Z t )θt2 dt 2 −θt exp(Z t )d Wt = −θt L t d Wt . exp(Z t )d Z t +

Now we use our integration by parts formula of Theorem 4.4.2 to ﬁnd the stochastic (L) differential equation satisﬁed by Wt L t . Since d Wt d(Wt
(L) (L)

= = = =

d Wt + θt dt, Wt Wt
(L) (L)

Lt )

d L t + L t d Wt
(L)

(L)

+ d[M W

(L)

, M L ]t

d L t + L t d Wt + L t θt dt − θt L t dt d Wt .

L t − θt L t Wt

Granted enough boundedness (which is guaranteed by our assumptions), (L) {Wt L t }t≥0 is then a P-martingale and has expectation zero. Thus, under the (L) measure P(L) , {Wt }t≥0 is a martingale. We proved in Theorem 4.2.1 that with P-probability one, the quadratic variation of {Wt }t≥0 is given by [W ]t = t. The probability measures P and P(L) are equivalent (L) and so have the same sets of probability one. Therefore {Wt }t≥0 also has quadratic e variation given by [W (L) ]t = t with P(L) -probability one. Finally, by L´ vy’s (L) characterisation of Brownian motion (Theorem 4.3.5) we have that {Wt }t≥0 is a ✷ P(L) -Brownian motion as required. We now try this in practice.
Example 4.5.2

Let {X t }t≥0 be the drifting Brownian motion process X t = σ Wt + µt,

where {Wt }t≥0 is a P-Brownian motion and σ and µ are constants. Find a measure under which {X t }t≥0 is a martingale.
Solution: Taking θ = µ/σ , under P(L) of Theorem 4.5.1 we have that Wt
(L) σ Wt

Wt + µt/σ is a Brownian motion, and X t = motion.

= is then a scaled Brownian

(L)

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stochastic calculus

Notice that, for example, EP X t2 = EP σ 2 Wt2 + 2σ µt Wt + µ2 t 2 = σ 2 t + µ2 t 2 , whereas EP
(L)

X t2 = EP

(L)

σ 2 (Wt

(L) 2

)

= σ 2 t. ✷

4.6

The Brownian Martingale Representation Theorem
Let , F, {Ft }t≥0 , P be a ﬁltered probability space and let {Wt }t≥0 be a (P, {Ft }t≥0 )-Brownian motion. We have seen that if f (t, ω) is an {Ft }t≥0 predictable random variable and E f 2 (t, ω) < ∞ for each t ≥ 0, then t Mt
0

f (s, ω)d Ws

is a P, {Ft }t≥0 -martingale. It is natural to ask if there are any others. Just as in the discrete world the binomial representation theorem allowed us to represent martingales as ‘discrete stochastic integrals’ so here the Brownian martingale representation theorem tells us that all (nice) P, {Ft }t≥0 -martingales can be represented as Itˆ integrals. This result is also sometimes called the predictable o representation property. It will be the key to hedging in our continuous world.
Deﬁnition 4.6.1

A

P, {Ft }t≥0 -martingale {Mt }t≥0 is said to be squareE |Mt |2 < ∞ for each t > 0.

integrable if

Theorem 4.6.2 (Brownian Martingale Representation Theorem)

Let {Ft }t≥0 denote the natural ﬁltration of the P-Brownian motion {Wt }t≥0 . Let {Mt }t≥0 be a square-integrable P, {Ft }t≥0 -martingale. Then there exists an {Ft }t≥0 -predictable process {θt }t≥0 such that with P-probability one, Mt = M0 +
0 t

θs d Ws .

Outline of proof: We restrict our attention to t ∈ [0, T ] for some ﬁxed T . The ﬁrst

step is to show that any F ∈ FT for which E F 2 < ∞ can be represented as F = E [F] +
0 T

θs d Ws

(4.17)

for some predictable process {θs }0≤s≤T . Write G for the linear space of such F that can be represented in this way. For any F ∈ G, E F 2 = E [F]2 + E
0 T

θs2 ds .

(4.18)

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4.6 the brownian martingale representation theorem

This guarantees that if we take a sequence of random variables {Fn }n≥1 in G for which E (Fn − Fm )2 → 0 as n, m → ∞, then they converge to a limit that also lies in G. Now by Itˆ ’s formula, for any simple o function n f (s) = i=1 ai (ω)1(ti−1 ,ti ] (s), t 0 f

if we deﬁne Et then f t

exp
0 f

f (s)d Ws − t 0

1 2

f (s)2 ds ,

Et = 1 + f f (s)Es d Ws .

In other words ET ∈ G. We now approximate any F ∈ FT for which E F 2 < ∞ by f linear combinations of the ET to see that all such F are in G and so can be represented as in (4.17). The identity (4.18) guarantees that if the representation holds with two different predictable processes, {θs }0≤s≤T and {ψs }0≤s≤T say, then E
0 T

(θs − ψs )2 ds = 0.

Now we replace F ∈ FT by the martingale {Mt }0≤t≤T to complete the proof. This step is elementary. Since Mt = E [ MT | Ft ], applying the representation to MT and then taking (conditional) expectations of both sides we obtain Mt = E [ MT | Ft ] = = E [MT ] + E
0 T

θs d Ws Ft

E [MT ] +
0

t

θs d W s . ✷

For full details of this proof see Revuz & Yor (1998).
Remarks:

1 The Martingale Representation Theorem tells us that such an {Ft }t≥0 -predictable process {θt }t≥0 exists. Unfortunately, unlike the Binomial Representation Theorem, the proof is not constructive. When we call upon it in hedging options in Chapter 5, we are going to have to work harder to actually produce an explicit expression for the predictable process. 2 Notice that the quadratic variation of the martingale {Mt }t≥0 satisﬁes d[M]t = θt2 dt. (1) (2) If we have two Brownian martingales, {Mt }t≥0 and {Mt }t≥0 , then provided (i) ] /dt is non-vanishing for i = 1, 2, the Martingale Representation Theorem d[M t tells us that each is a locally scaled version of the other. ✷

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stochastic calculus

4.7

Why geometric Brownian motion?
We now have the main results in place that will allow us to price and hedge in stock market models based on Brownian motion. Other than suggesting that the paths of stock prices in an arbitrage-free world should be rough, we have thus far provided no justiﬁcation for such models. In this short section we use L´ vy’s characterisation e of Brownian motion to motivate the basic reference model in mathematical ﬁnance: geometric Brownian motion. We begin by sketching Bachelier’s argument for the Brownian motion model. Bachelier argued that stock markets cannot have any consistent bias in favour of either buyers or sellers: ‘L’esp´ rance math´ matique du sp´ culateur est nulle’. e e e This is almost the martingale property. Assuming the stock price process to have the Markov property, he introduced the transition density P St ∈ [y, y + dy] Ss = x p (s, t; x, y) dy.

If the dynamics are homogeneous in space and time, then p(s, t; x, y) = q(t −s, y − x) for some function q. Bachelier then ‘derived’ what is now known as the Chapman– Kolmogorov equation for q and showed that this is solved by the probability density function of Brownian motion. Bachelier’s argument is not rigorous, but from L´ vy’s characterisation of Browe nian motion we know that if the stock price process is a martingale under P whose increments have stationary conditional variance then the stock price process must be Brownian motion under P. It is remarkable that Bachelier’s argument pre-dates Einstein’s famous work on Brownian motion and, of course, Wiener’s rigorous construction of the process. Although we would not take issue with the mathematical conclusions of Bachelier’s analysis, we have already discarded Brownian motion as a model. A modern approach makes different assumptions, but we need not completely abandon Bachelier’s argument. His key assumption was that the increments of the stock price process were stationary. Suppose that instead we assume that the relative increments, (St − Ss )/Ss , measuring the returns are stationary. Taking logarithms, the process {log St }t≥0 should have stationary increments. We don’t know that {log St }t≥0 is a martingale, so this time we can only deduce that this is Brownian motion plus a constant drift. This gives d St = µSt dt + σ St d Wt , where {Wt }t≥0 is a P-Brownian motion and µ and σ are constants. This is the geometric Brownian motion model, originally championed by Samuelson (1965).

4.8

The Feynman–Kac representation
Our probabilistic approach to pricing options will result in a price expressed as the discounted expected value of a claim with respect to a probability measure under

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4.8 the feynman–kac representation

which the discounted stock price is a martingale. In the simple case of European calls and puts we’ll be able to ﬁnd an explicit expression for the price. However, for more complicated claims numerical methods have to be brought to bear. One approach is to revert to our binomial tree model. Another is to express the price as the solution to a partial differential equation and employ, for example, ﬁnite difference methods. In fact for European options the binomial method amounts to a ﬁnite difference method for solving the Black–Scholes partial differential equation. We refer to Wilmott, Howison & Dewynne (1995) for an account of the numerical methods. Here we simply make the connection between the partial differential equation approach and the probabilistic approach to pricing derivatives.
Solving pde’s probabilistically

The fact that the price can be expressed as the solution to a partial differential equation is a consequence of the deep connection between stochastic differential equations and certain parabolic partial differential equations.
Theorem 4.8.1 (Feynman–Kac stochastic representation)

Assume that the function

F solves the boundary value problem ∂F 1 ∂F ∂2 F (t, x) + µ(t, x) (t, x) + σ 2 (t, x) 2 (t, x) = 0, 0 ≤ t ≤ T, (4.19) ∂t ∂x 2 ∂x F(T, x) = (x). Deﬁne {X t }0≤t≤T to be the solution of the stochastic differential equation d X t = µ(t, X t )dt + σ (t, X t )d Wt , 0 ≤ t ≤ T,

where {Wt }t≥0 is standard Brownian motion under the measure P. If
T 0

E

σ (t, X t )

∂F (t, X t ) ∂x

2

ds < ∞,

(4.20)

then F(t, x) = EP [ (X T )| X t = x] .
Proof: We apply Itˆ ’s formula to {F(s, X s )}t≤s≤T . o

F(T, X T ) = F(t, X t ) T ∂F ∂F 1 ∂2 F (s, X s ) + µ(s, X s ) (s, X s ) + σ 2 (s, X s ) 2 (s, X s ) ds + ∂s ∂x 2 ∂x t + t T

σ (s, X s )

∂F (s, X s )d Ws . ∂x

(4.21)

Now using assumption (4.20) and Theorem 4.2.7 E t T

σ (s, X s )

∂F (s, X s )d Ws X t = x = 0. ∂x

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stochastic calculus

Moreover, since F satisﬁes (4.19), the deterministic integral on the right hand side of (4.21) vanishes, so, taking expectations, E [ F(T, X T )| X t = x] = F(t, x), and substituting F(T, X T ) =
Example 4.8.2

(X T ) gives the required result.

✷

Solve 1 ∂2 F ∂F + = 0, ∂t 2 ∂x2 F(T, x) = (x).

(4.22)

Solution: The corresponding stochastic differential equation is

d X t = d Wt so, by the Feynman–Kac representation, F(t, x) = E [ (WT )| Wt = x] . In fact we knew this already. In §3.1 we wrote down the transition density of Brownian motion as p(t, x, y) = √ This gives E [ (WT )| Wt = x] = p (T − t, x, y) (y)dy. 1 2π t exp − (x − y)2 2t . (4.23)

To check that this really is the solution, differentiate and use the fact that p(t, x, y) given by (4.23) is the fundamental solution to the equation 1 ∂ 2u ∂u = , ∂t 2 ∂x2 to obtain (4.22).
Kolmogorov equations

✷

We can use the Feynman–Kac representation to ﬁnd the partial differential equation solved by the transition densities of solutions to other stochastic differential equations. Suppose that d X t = µ(t, X t )dt + σ (t, X t )d Wt . For any set B let p B (t, x; T, y) P [ X T ∈ B| X t = x] = E [ 1 B (X T )| X t = x] . (4.24)

105

4.8 the feynman–kac representation

By the Feynman–Kac representation (subject to the integrability condition (4.20)) this solves ∂ pB (t, x; T, y) + Ap B (t, x; T, y) = 0, ∂t p B (T, x) = 1 B (x), where (4.25)

1 ∂2 f ∂f (t, x) + σ 2 (t, x) 2 (t, x). ∂x 2 ∂x Writing |B| for the Lebesgue measure of the set B, the transition density of the process {X s }s≥0 is given by A f (t, x) = µ(t, x) p(t, x; T, y)
|B|→0

lim

1 P [ X T ∈ B| X t = x] . |B|

Since the equation (4.25) is linear, we have proved the following lemma.
Lemma 4.8.3 Subject to satisfying (4.20), the transition density of the solution {X s }s≥0 to the stochastic differential equation (4.24) solves

∂p (t, x; T, y) + Ap(t, x; T, y) ∂t p(t, x; T, y)

=

0, as t → T.

(4.26)

→ δ y (x)

Equation (4.26) is known as the Kolmogorov backward equation (it operates on the ‘backward in time’ variables (t, x)). The operator A is called the inﬁnitesimal generator of the process {X s }s≥0 . We can also obtain an equation acting on the forward variables (T, y).
Lemma 4.8.4

In the above notation, ∂p (t, x; T, y) = A∗ p(t, x; T, y) ∂T (4.27)

where A∗ f (T, y) = − ∂ 1 ∂2 σ 2 (t, Y ) f (T, y) . (µ(T, y) f (T, y)) + ∂y 2 ∂ y2

Heuristic explanation: We don’t prove this, but we provide some justiﬁcation. By the

Markov property of the process {X t }t≥0 , for any T > r > t p(t, x; T, y) = p(t, x; r, z) p(r, z; T, y)dz.

Differentiating with respect to r and using (4.26),
∞ −∞

∂ p(t, x; r, z) p(r, z; T, y) + p(t, x; r, z)Ap(r, z; T, y) dz = 0. ∂r

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stochastic calculus

Now integrate the second term by parts to obtain
∞ −∞

∂ p(t, x; r, z) − A∗ p(t, x; r, z) p(r, z; T, y)dz = 0. ∂r

This holds for all T > r , which, if p(r, z; T, y) provides a sufﬁciently rich class of functions as we vary T , implies the result. ✷ Equation (4.27) is the Kolmogorov forward equation of the process {X s }s≥0 .
Example 4.8.5

Find the forward and backward Kolmogorov equations for geometric Brownian motion.

Solution: The stochastic differential equation is

d St = µSt dt + σ St d Wt . Substituting in our formula for the forward equation we obtain 1 ∂2 ∂p ∂ (t, x; T, y) = σ 2 2 y 2 p(t, x; T, y) − µ (yp(t, x; T, y)) , ∂T 2 ∂y ∂y and the backward equation is ∂p ∂2 p ∂p 1 (t, x; T, y) = − σ 2 x 2 2 (t, x; T, y) − µx (t, x; T, y). ∂t 2 ∂x ∂x The transition density for the process is the lognormal density given by  p(t, x; T, y) = 1  exp − √ σ y 2π(T − t) log(y/x) − µ − 1 σ 2 (T − t) 2 2σ 2 (T − t)
2

  . ✷

Example 4.8.6

Suppose that {X t }t≥0 solves d X t = µ(t, X t )dt + σ (t, X t )d Wt ,

where {Wt }t≥0 is a P-Brownian motion. For k : R+ × R → R and : R → R given deterministic functions, ﬁnd the partial differential equation satisﬁed by the function F(t, x) for 0 ≤ t ≤ T . E exp − t T

k(s, X s )ds

(X T ) X t = x ,

107 Solution: Evidently F(T, x) =

exercises

(x). By analogy with the proof of the Feynman– Kac representation, it is tempting to examine the dynamics of Z s = exp − t s

k(u, X u )du F(s, X s ).

Notice that for this choice of {Z s }t≤s≤T we have that if X t = x Z t = F(t, x) = E [ Z T | X t = x] . Thus the partial differential equation satisﬁed by F(t, x) is that for which {Z t }0≤t≤T is a martingale. Our strategy now is to ﬁnd the stochastic differential equation satisﬁed by {Z s }t≤s≤T . We proceed in two stages. Remember that t is now ﬁxed and we vary s. First notice that d exp − t s

k(u, X u )du

= −k(s, X s ) exp − t s

k(u, X u )du ds

and by Itˆ ’s formula o d F(s, X s ) = = ∂F ∂F 1 ∂2 F (s, X s )ds + (s, X s )d X s + (s, X s )σ 2 (s, X s )ds ∂s ∂x 2 ∂x2 ∂F ∂F 1 ∂2 F (s, X s ) + µ(s, X s ) (s, X s ) + σ 2 (s, X s ) 2 (s, X s ) ds ∂s ∂x 2 ∂x + σ (s, X s )

∂F (s, X s )d Ws . ∂x Now using our integration by parts formula we have that d Z s = exp − t s

k(u, X u )du ∂F ∂F 1 ∂2 F (s, X s ) + µ(s, X s ) (s, X s ) + σ 2 (s, X S ) 2 ds ∂s ∂x 2 ∂x

×

−k(s, X s )F(s, X s ) + + σ (s, X s )

∂F (s, X s )d Ws . ∂x

We can now read off the solution: in order for {Z s }t≤s≤T to be a martingale, F must satisfy ∂F 1 ∂2 F ∂F (s, x) + µ(s, x) (s, x) + σ 2 (s, x) 2 (s, x) − k(s, x)F(s, x) = 0. ∂s ∂x 2 ∂x

✷

Exercises
1 Let {Ft }t≥0 denote the natural ﬁltration associated to a standard P-Brownian motion {Wt }t≥0 . Deﬁne the process {St }t≥0 by St = f (t, Wt ). What equation must f satisfy if St is to be a (P, {Ft }t≥0 )-martingale? Use your answer to check that St = exp (νt + σ Wt )

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stochastic calculus

is a martingale if ν + 1 σ 2 = 0 (cf. Lemma 4.3.3). 2 2 A function, f , is said to be Lipschitz-continuous on [0, T ] if there exists a constant C > 0 such that for any t, t ∈ [0, T ] | f (t) − f (t )| < C|t − t |. Show that a Lipschitz-continuous function has bounded variation and zero 2variation over [0, T ]. 3 Let {Wt }t≥0 denote a standard Brownian motion under P. For a partition π of [0, T ], write δ(π) for the mesh of the partition and 0 = t0 < t1 < t2 < · · · < t N (π ) = T for the endpoints of the intervals of the partition. Calculate (a) δ(π)→0 N (π)−1

lim

Wt j+1 Wt j+1 − Wt j ,

0 N (π)−1

(b)
T 0

Ws ◦ d Ws

δ(π )→0

lim

0

1 Wt j+1 + Wt j 2

Wt j+1 − Wt j .

This is the Stratonovich integral of {Ws }s≥0 with respect to itself over [0, T ]. 4 Suppose that the martingale {Mt }0≤t≤T has bounded quadratic variation and {At }0≤t≤T is Lipschitz-continuous. Let St = Mt + At . By analogy with Deﬁnition 4.2.2, we deﬁne the quadratic variation of {St }0≤t≤T over [0, T ] to be the random variable [S]T such that for any sequence of partitions {πn }n≥1 of [0, T ] with δ(πn ) → 0 as n → ∞,   2 E
N (π ) j=1

|St j − St j−1 |2 − [S]T  → 0

as n → ∞.

Show that [S]T = [M]T . 5 If f is a simple function and {Wt }t≥0 is a P-Brownian motion, prove that the process o {Mt }t≥0 given by the Itˆ integral Mt =
0 t

f (s, Ws )d Ws

is a (P, {FtW }t≥0 )-martingale. 6 Verify that if {Wt }t≥0 is a P-Brownian motion E
0 t 2

Ws d Ws

=
0

t

E Ws2 ds.

(If you need the moment-generating function of Wt , you may assume the result of Exercise 10.)

109

exercises

7 As usual, {Wt }t≥0 denotes standard Brownian motion under P. Use Itˆ ’s formula to o write down stochastic differential equations for the following quantities. (a) Yt = Wt3 , (b) Yt = exp σ Wt − 1 σ 2 t , 2 (c) Yt = t Wt . Which are (P, {FtW }t≥0 )-martingales? 8 Use a heuristic argument based on a Taylor expansion to check that for Stratonovich stochastic calculus the chain rule takes the form of the classical (Newtonian) one. 9 Mimic the calculation of Example 4.3.2 to show that if {Wt }t≥0 is standard Brownian motion under the measure P, then E Wt4 = 3t 2 .

10 Let {Wt }t≥0 denote Brownian motion under P and deﬁne Z t = exp(αWt ). Use Itˆ ’s formula to write down a stochastic differential equation for Z t . Hence ﬁnd an o ordinary (deterministic) differential equation for m(t) E[Z t ], and solve to show that α2 E exp (αWt ) = exp t . 2 11 The Ornstein–Uhlenbeck process Let {Wt }t≥0 denote standard Brownian motion under P. The Ornstein–Uhlenbeck process, {X t }t≥0 , is the unique solution to Langevin’s equation, d X t = −α X t dt + d Wt , X 0 = x.

This equation was originally introduced as a simple idealised model for the velocity of a particle suspended in a liquid. In ﬁnance it is a special case of the Vasicek model of interest rates (see Exercise 19). Verify that X t = e−αt x + e−αt
0 t

eαs d Ws ,

and use this expression to calculate the mean and variance of X t . 12 The Cox–Ingersoll–Ross model of interest rates assumes that the interest rate, r , is not deterministic, but satisﬁes the stochastic differential equation √ drt = (α − βrt )dt + σ rt d Wt , where {Wt }t≥0 is standard P-Brownian motion. This process is known as a squared √ Bessel process. Find the stochastic differential equation followed by { rt }t≥0 in the case α = 0. Suppose that {u(t)}t≥0 satisﬁes the ordinary differential equation σ2 du (t) = −βu(t) − u(t)2 , dt 2 u(0) = θ,

110

stochastic calculus

for some constant θ > 0. Fix T > 0. Assuming still that α = 0, for 0 ≤ t ≤ T ﬁnd the differential equation satisﬁed by E exp (−u(T − t)rt ) . Hence calculate the mean and variance of r T and P[r T = 0]. 13 The Black–Karasinski model of interest rates is 1 drt = σt rt d Wt + θt + σt2 − αt log rt rt dt, 2 where {Wt }t≥0 is a standard P-Brownian motion and σt , θt and αt are deterministic functions of time. In the special case where σ , θ and α are constants, ﬁnd rt as a t function of 0 eαs d Ws . 14 Suppose that, under the measure P, d St = σ St d Wt , where {Wt }t≥0 is a P-Brownian motion. Find the mean and variance of t Yt
0

Su du.

15

Suppose that {Mt }t≥0 is a continuous (P, {Ft }t≥0 )-martingale with E Mt2 ﬁnite for all t ≥ 0. Writing {[M]t }t≥0 for the associated quadratic variation process, show that Mt2 − [M]t is a (P, {Ft }t≥0 )-martingale.

16 Suppose that under the probability measure P, {X t }t≥0 is a Brownian motion with constant drift µ. Find a measure P∗ , equivalent to P, under which {X t }t≥0 is a Brownian motion with drift ν. 17 Let {Ft }t≥0 be the natural ﬁltration associated with a P-Brownian motion, {Wt }t≥0 . Show that if X is an FT -measurable random variable with E[|X |] < ∞ and P∗ is a probability measure equivalent to P, then the process Mt is a (P∗ , {Ft }0≤t≤T )-martingale. 18 Use the Feynman–Kac stochastic representation formula to solve ∂F 1 ∂2 F (t, x) + σ 2 2 (t, x) = 0, ∂t 2 ∂x subject to the terminal value condition F(T, x) = x 4 . EP [X |Ft ]
∗

111

exercises

19

Suppose that the interest rate, r , is not deterministic, but is itself a random process, {rt }t≥0 . In the Vasicek model, {rt }t≥0 is assumed to be a solution of the stochastic differential equation drt = (b − art )dt + σ d Wt , where, as usual, {Wt }t≥0 is standard P-Brownian motion. Find the Kolmogorov backward and forward differential equations satisﬁed by the probability density function of the process. What is the distribution of rt as t → ∞?

20

Suppose that v(t, x) solves 1 ∂v ∂ 2v (t, x) + σ 2 x 2 2 (t, x) − r v(t, x) = 0, ∂t 2 ∂x Show that for any constant θ , vθ (t, x) θ2 x v t, θ x 0 ≤ t ≤ T.

is another solution. Use the Feynman–Kac stochastic representation to ﬁnd a probabilistic interpretation of this result. 21 Suppose that for 0 ≤ s ≤ T , d X s = µ(s, X s )ds + σ (s, X s )d Ws , X t = x,

where {Ws }t≤s≤T is a P-Brownian motion, and let k, : R → R be given deterministic functions. Find the partial differential equation satisﬁed by F(t, x) = E [ (X T )| X t = x] + t T

E [ k(X s )| X t = x] ds.

5 The Black–Scholes model

Summary
We now, ﬁnally, have all the tools that we need for pricing and hedging in the continuous time world of Black and Scholes. We shall begin with the most basic setting, in which our market has just two securities: a cash bond and a risky asset whose price is modelled by a geometric Brownian motion. In §5.1 we prove the Fundamental Theorem of Asset Pricing in this framework. In line with our analysis in the discrete world, this provides an explicit formula for the price of a derivative as the discounted expected payoff under the martingale measure. Just as in the discrete setting, we shall see that there are three steps to replication. In §5.2 we put this into action for European options. For simple calls and puts, the expectation that gives the price of the claim can be evaluated. We also obtain an explicit expression for the stock and bond holding in the replicating portfolio, via an application of the Feynman–Kac representation. The rest of the book consists of increasing the complexity of the derivative contracts and of the market models. Before embarking on this programme, we relax the ﬁnancial assumptions that we have made within the basic Black–Scholes framework. The risky asset that we have speciﬁed has a very simplistic ﬁnancial side. We have assumed that it can be held without additional cost or beneﬁt and that it can be freely traded at the quoted price. Even leaving aside the issues of transaction costs and illiquidity, not much of the ﬁnancial market is like that. Foreign exchange involves two assets that pay interest, equities pay dividends and bonds pay coupons. In §5.3–§5.5 we see how to apply the Black–Scholes technology in these more sophisticated ﬁnancial settings. Finally, in §5.6, we characterise tradable assets within a given market and we deﬁne the market price of risk.

5.1

The basic Black–Scholes model
In this section we provide a rigorous derivation of the Black–Scholes pricing formula obtained in §2.6. As in Chapter 2, our market consists of just two securities. The ﬁrst is our old friend the cash bond, {Bt }t≥0 . We retain (for now) our assumption that
112

113

5.1 the basic black–scholes model

the risk-free interest rate is constant, so that if B0 = 1 then Bt = er t . The second security in our market is a risky asset whose price at time t we denote by St . In this, our basic reference model, we suppose that {St }t≥0 is a geometric Brownian motion, that is it solves d St = µSt dt + σ St d Wt , for some constants µ and σ , where {Wt }t≥0 is a P-Brownian motion. Notice that this corresponds to taking ν = µ − 1 σ 2 in our calculations of §2.6. We call P the 2 market measure. Exactly as in the discrete world, the market measure tells us which market events have positive probability, but we shall reweight the probabilities for the purposes of pricing and hedging.
Selfﬁnancing strategies

As in the discrete world, our starting point is that the market does not admit any arbitrage opportunities. Our strategy also parallels the work of Chapter 2: to obtain the time zero price of a claim, C T , against us at time T , we seek a self-ﬁnancing portfolio whose value at time T is exactly C T . In the absence of arbitrage, the value of the claim must be the same as the cost of constructing the replicating portfolio. Of course for this argument to work, the trading strategy for this portfolio must be previsible. Moreover, because we are now allowed to rebalance the portfolio as often as we like, rather than just at the ‘ticks’ of a clock, to avoid obvious arbitrage opportunities we must introduce a further restriction on admissible trading strategies for our portfolio. We illustrate with an example.
Example 5.1.1 (The doubling strategy)

Consider the following strategy for betting on successive (independent) ﬂips of a coin that comes up heads with probability p > 0. We bet $ K that the ﬁrst ﬂip comes up heads. If it does come up heads then we stop, having won $ K . If it does not come up heads, then we bet $2K that the second ﬂip comes up heads. If it does, then our net gain is $ K and we stop. Otherwise we have lost $3K and we bet $4K that the next ﬂip is a head. And so on. If the ﬁrst n − 1 ﬂips all come up tails, then we have lost $ n−1 2 j K = $(2n −1)K and we bet $2n K j=0 on the nth ﬂip. Since with probability one the coin will eventually come up heads, we are guaranteed to win $ K . Of course, this relies on our having inﬁnite credit. If we only have limited funds, then the apparent arbitrage opportunity disappears. With this example in mind, we make the following deﬁnition.

Deﬁnition 5.1.2 A self-ﬁnancing strategy is deﬁned by a pair of predictable processes {ψt }0≤t≤T , {φt }0≤t≤T , denoting the quantities of riskless and risky asset respectively held in the portfolio at time t, satisfying

1
T 0

|ψt | dt +
0

T

|φt |2 dt < ∞

(with probability one), and

114

the black–scholes model

2 ψt Bt + φt St = ψ0 B0 + φ0 S0 +
0

t

ψu d Bu +
0

t

φu d Su

(with probability one) for all t ∈ [0, T ].
Remarks: Condition 1 ensures that the integrals in condition 2 make sense. More-

over, 0 φu d Wu will be a P-martingale. In differential form, condition 2 says that the value, Vt (ψ, φ) = ψt Bt + φt St , of the portfolio at time t satisﬁes d Vt (ψ, φ) = ψt d Bt + φt d St , that is, changes of value of the portfolio over an inﬁnitesimal time interval are due entirely to changes in value of the assets and not to injection (or removal) of wealth from outside. ✷ As in the discrete setting, the key will be to work with a probability measure, Q, equivalent to the ‘market measure’ P and under which the discounted stock price, ˜ { St }t≥0 , is a Q-martingale. This means that it is convenient to think of the discounted asset price process as the object of central interest. With this in mind we prove the following continuous analogue of equation (2.5).
Lemma 5.1.3

t

Let {ψt }0≤t≤T and {φt }t≥0 be predictable processes satisfying
T 0

|ψt | dt +
0

T

|φt |2 dt < ∞

(with probability one). Set Vt (ψ, φ) = ψt Bt + φt St , ˜ Vt (ψ, φ) = e−r t Vt (ψ, φ).

Then {ψt , φt }0≤t≤T deﬁnes a self-ﬁnancing strategy if and only if ˜ ˜ Vt (ψ, φ) = V0 (ψ, φ) +
0 t

˜ φu d Su

with probability one for all t ∈ [0, T ].
Proof: Suppose ﬁrst that the portfolio {ψt , φt }0≤t≤T is self-ﬁnancing. Then

˜ d Vt (ψ, φ)

= = =

−r e−r t Vt (ψ, φ)dt + e−r t d Vt (ψ, φ) −r e−r t ψt er t + φt St dt + e−r t ψt d(er t ) + e−r t φt d St

φt −r e−r t St dt + e−r t d St ˜ = φt d St ✷

as required. The other direction is similar and is left as an exercise.

115 A strategy for pricing

5.1 the basic black–scholes model

Before going further, we outline our strategy. We write C T for the claim at time T that we are trying to replicate. It may depend on {St }0≤t≤T in more complex ways than just through ST . Suppose that somehow we can ﬁnd a predictable process {φt }0≤t≤T such that the claim C T , discounted, satisﬁes C˜T e−r T C T = φ0 +
0 T

˜ φu d Su .

Then we can replicate the claim by a portfolio in which we hold φt units of stock and ψt units of cash bond at time t, where ψt is chosen so that ˜ ˜ Vt (ψ, φ) = φt St + ψt e−r t = φ0 +
0 t

˜ φu d Su .

By Lemma 5.1.3, the portfolio is then self-ﬁnancing, and, moreover, VT = C T . The fair price of the claim at time zero is then V0 = φ0 . This is ﬁne if we know φ0 , but there is a quick and easy way to ﬁnd the right price without explicitly ﬁnding the strategy {ψt , φt }t≥0 . Suppose instead that we can ﬁnd a probability measure, Q, under which the discounted stock price is a martingale. T 2 Then, at least provided 0 φu du < ∞, t 0

˜ φu d Su

will be a mean zero Q-martingale and so ˜ EQ VT (ψ, φ) = φ0 + EQ
0 t

˜ φu d Su = φ0 .

So φ0 = EQ C˜T is the fair price. This then is entirely analogous to the pricing formula of Theorem 2.3.13. If there is a probability measure, Q, equivalent to P and under which the discounted stock price is a martingale, then, provided a replicating portfolio exists, the fair time zero price of the claim is EQ [C˜T ], the discounted expected value of the claim under this measure. We have assumed that the process {φt }t≥0 exists. We prove this (for this basic Black–Scholes market model) in Theorem 5.1.5 via an application of the Martingale Representation Theorem. First, if our pricing formula is to be of any use, we should ﬁnd the equivalent martingale measure Q.
An equivalent martingale measure ˜ Lemma 5.1.4 (A probability measure under which { St }t≥0 is a martingale)

There is a probability measure Q, equivalent to P, under which the discounted stock price ˜ { St }t≥0 is a martingale. Moreover, the Radon–Nikodym derivative of Q with respect to P is given by dQ 1 (θ) Lt = exp −θ Wt − θ 2 t , dP Ft 2 where θ = (µ − r )/σ .

116 Proof: Recall that

the black–scholes model

d St = µSt dt + σ St d Bt , so ˜ ˜ d St = St (−r dt + µdt + σ d Wt ) . Consequently, if we set X t = Wt + (µ − r )t/σ , ˜ ˜ d St = St σ d X t . ˜ Now from Theorem 4.5.1, {X t }t≥0 is a Q-Brownian motion and so { St }t≥0 is a Qmartingale. Moreover, ˜ ˜ St = S0 exp σ X t − σ 2 t/2 . ✷
The Fundamental Theorem of Asset Pricing

We can now prove the Fundamental Theorem of Asset Pricing in the Black–Scholes framework.
Theorem 5.1.5 Let Q be the measure given by Lemma 5.1.4. Suppose that a claim at time T is given by the non-negative random variable C T ∈ FT . If
2 EQ C T < ∞,

then the claim is replicable and the value at time t of any replicating portfolio is given by Vt = EQ e−r (T −t) C T Ft . In particular, the fair price at time zero for the option is V0 = EQ e−r T C T = EQ C˜T .
Proof: In the argument that followed Lemma 5.1.3 we showed that if we could ﬁnd a process {φt }0≤t≤T such that

˜ C T = φ0 +
0

T

˜ φu d Su ,

then we could construct a replicating portfolio whose value at time t satisﬁes ˜ Vt (ψ, φ) = φ0 +
0 t

˜ φu d Su ,

(5.1)

which, by the martingale property of the stochastic integral is precisely ˜ Vt (ψ, φ) = EQ φ0 +
0 T

˜ φu d Su Ft = EQ C˜T Ft = EQ e−r T C T Ft .

117

5.1 the basic black–scholes model

Undoing the discounting on [0, t] gives Vt (ψ, φ) = EQ e−r (T −t) C T Ft . Let’s just reassure ourselves that such a price is unique. Evidently any other ˆ ˆ ˆ ˆ replicating portfolio, {ψt , φt }0≤t≤T , has VT (ψ, φ) = C T and if it is self-ﬁnancing (by Lemma 5.1.3) it satisﬁes an equation of the form (5.1). Repeating the argument above we see that we obtain the same value for any self-ﬁnancing replicating portfolio. The proof of the theorem will be complete if we can show that there is a predictable process {φt }0≤t≤T such that ˜ C T = φ0 +
0 T

˜ φu d Su .

Now, by Exercise 2, Mt EQ e−r T C T Ft

is a square-integrable Q-martingale. The natural ﬁltration of our original Brownian motion is the same as that for the process {X t }t≥0 deﬁned in Lemma 5.1.4. That is, {Mt }t≥0 is a square-integrable ‘Brownian martingale’ and by the Brownian Martingale Representation Theorem 4.6.2 there exists an {Ft }0≤t≤T -predictable process {θt }0≤t≤T such that Mt = M0 +
0 t

θs d X s .

˜ ˜ Since d Ss = σ Ss d X s , we set φt = θt ˜ σ St and ˜ ψt = Mt − φt St .

Condition 1 of Deﬁnition 5.1.2 is easily seen to be satisﬁed and so the strategy corresponding to {ψt , φt }0≤t≤T deﬁnes a self-ﬁnancing replicating portfolio as required. ✷
Remark: The theorem that we have just proved is very general. Subject to our mild

boundedness condition, the claim C T could be almost arbitrarily complex provided it depends only on the path of the stock price up to time T . The price of the claim at time zero is EQ e−r T C T and this can be evaluated, at least numerically, even for complex claims C T . We have proved that not only does there exist a fair price, but moreover, we can hedge the claim. Its shortcoming is that although we have asserted the existence of a hedging strategy, we have not obtained an explicit expression for it. We shall ﬁnd such an expression for European options, that is claims that depend only on the stock price at maturity, in the next section. ✷

118

the black–scholes model

Just as in the discrete world, we have identiﬁed a procedure for valuing and replicating a claim.

Three steps to replication:

˜ 1 Find a measure Q under which the discounted asset price { St }t≥0 is a martingale. 2 Form the process Mt = EQ e−r T C T Ft . ˜ 3 Find a predictable process {φt }t≥0 such that d Mt = φt d St .

5.2

Black–Scholes price and hedge for European options
In the case of European options, that is options whose payoff depends only on the price of the underlying at the time of maturity, both the price of the option and the hedging portfolio can be obtained explicitly. First we evaluate the price of the claim. Our assumptions are exactly those of §5.1.
Proposition 5.2.1

The value at time t of a European option whose payoff at maturity is C T = f (ST ) is Vt = F(t, St ), where F(t, x) = e−r (T −t)
∞ −∞

√ f x exp (r − σ 2 /2)(T − t) + σ y T − t × exp(−y 2 /2) dy. √ 2π

Proof: From Theorem 5.1.5 we know that the value at time t is

EQ e−r (T −t) f (ST ) Ft ,

(5.2)

where Q is the martingale measure obtained in Lemma 5.1.4. Under this measure X t = Wt + (µ − r )t/σ is a Brownian motion and ˜ ˜ d St = σ St d X t . Solving this equation, 1 ˜ ˜ ST = St exp σ (X T − X t ) − σ 2 (T − t) . 2 We can now substitute into (5.2) to obtain Vt = EQ e−r (T −t) f 1 St er (T −t) exp σ (X T − X t ) − σ 2 (T − t) 2 Ft .

Since under Q, conditional on Ft , X T − X t is a normally distributed random variable

119

5.2 black–scholes price and hedge for european options

with mean zero and variance (T − t), we can evaluate this as Vt = = F(t, St )
∞ −∞

e−r (T −t) f ×√

1 St er (T −t) exp σ z − σ 2 (T − t) 2

z2 1 dz exp − 2(T − t) 2π(T − t) √ 1 r − σ 2 (T − t) + σ y T − t 2 −∞ y2 1 dy, × √ exp − 2 2π
∞

=

e−r (T −t)

f

St exp

as required.
Pricing calls and puts

✷

For European calls and puts, the function F of Proposition 5.2.1 can be calculated explicitly.
Example 5.2.2 (European call)

In the notation of Proposition 5.2.1, suppose that f (ST ) = (ST − K )+ . Then, writing θ = (T − t), F(t, x) = x (d1 ) − K e−r θ (d2 ), (5.3)

where

(·) is the standard normal distribution function, given by (y) d1 = = y −∞

1 2 √ e−y /2 dy, 2π x K

log

+ r+ √ σ θ

σ2 2

θ

√ and d2 = d1 − σ θ .
Proof: Substituting for f and x in the last line of the proof of Proposition 5.2.1 we have that

F(t, x) = E

xeσ

√ θ Z −σ 2 θ/2

− K e−r θ

+

,

(5.4)

where Z ∼ N (0, 1). First we establish for what range of values of Z the integrand is non-zero. Rearranging, xeσ is equivalent to Z> log
K x √ θ Z −σ 2 θ/2

> K e−r θ
2

+ σ2 θ − r θ . √ σ θ

120

the black–scholes model

Thus the integrand in (5.4) is non-zero if Z + d2 ≥ 0. Using this notation F(t, x) = = = = x E
∞ −d2 d2 −∞ d2 −∞

xeσ

√

θ Z −σ 2 θ/2 √

− K e−r θ 1 Z +d2 ≥0 − K e−r θ − K e−r θ √ e−y /2 dy √ 2π
2

xeσ

θ y−σ 2 θ/2

xe−σ e

√

θ y−σ 2 θ/2

e−y /2 dy √ 2π
2

−y 2 /2 √ −σ θ y−σ 2 θ/2 e

2π

dy − K e−r θ (d2 ).

√ Substituting z = y + σ θ in the ﬁrst integral in the last line we ﬁnally obtain F(t, x) = x (d1 ) − K e−r θ (d2 ). ✷ Equation (5.3) is known as the Black–Scholes pricing formula for a European call option. The corresponding formula for a European put option can be found in Exercise 3.
Remarks:

1 Bachelier actually obtained a formula that looks very like this for the price of a European call option, except that the geometric Brownian motion is replaced by Brownian motion. This, however, was a ﬂuke. Bachelier was using expectation pricing and did not have the notion of dynamic hedging. 2 Notice that the pricing formula depends on just one unknown parameter, σ , called the volatility by practitioners. The same will be true of our hedging portfolio. The problem that then arises is how to estimate σ from market data. The commonest approach is to use the implied volatility. Some options are quoted on organised markets. The price of European call and put options is an increasing function of volatility and so we can invert the Black–Scholes formula and associate an implied volatility with each option. Unfortunately, the estimate of σ obtained in this way usually depends on strike price and time to maturity. We brieﬂy discuss the implications of this in §7.4. ✷
Hedging calls and puts

We now turn to the problem of hedging European options. That is, how should we construct a portfolio that replicates the claim against us? The Martingale Representation Theorem tells us that since the discounted option price and the discounted stock price are martingales under the same measure, one is locally just a scaled version of the other. It is this local scaling that we should like an expression for. In our discrete world of §2.5 we found φi+1 as the ratio of the change in value of the option to that of the stock over the (i + 1)st tick of the clock. It is reasonable to guess then that in the continuous world φt should be the partial

121

5.2 black–scholes price and hedge for european options

derivative of the option value with respect to the stock price and this is what we now prove. In the notation of Proposition 5.2.1, the process {φt }0≤t≤T that determines the stock holding in the replicating portfolio of Theorem 5.1.5 is given by ∂F (t, x) φt = . ∂x x=St
Proposition 5.2.3 Proof: In this notation, the result of Theorem 5.1.5 becomes

F(t, x) = EQ e−r (T −t) f (ST ) St = x where, under Q, ˜ ˜ d St = σ St d X t

and {X t }0≤t≤T is a Brownian motion. Evidently d St = r St dt + σ St d X t . Combining the Feynman–Kac representation and the usual product rule of differentiation, F(t, x) satisﬁes 1 ∂2 F ∂F ∂F (t, x) + σ 2 x 2 2 (t, x) − r F(t, x) + r x (t, x) = 0, ∂t 2 ∂x ∂x 0 ≤ t ≤ T.

This is the Black–Scholes equation. You are asked to verify that F(t, x) satisﬁes this equation via a different route in Exercise 4. ˜ ˜ ˜ ˜ Deﬁne the function F(t, x) = e−r t F(t, xer t ), then Vt = F(t, St ). Observing that ˜ ˜ 1 ∂F ∂2 F (t, x) = − σ 2 x 2 2 (t, x) ∂t 2 ∂x ˜ ˜ and applying Itˆ ’s formula to F(u, Su ) for 0 ≤ u ≤ T , o ˜ ˜ F(T, ST ) = = This gives φt = as required.
Example 5.2.4 (Hedging a European call)

F(0, S0 ) +
0

T

F(0, S0 ) +
0

T

˜ ˜ ˜ ∂ F (s, Ss )d X s σ Ss ∂x ˜ ∂F ˜ ˜ (s, Ss )d Ss . ∂x

˜ ∂F ∂F ˜ (t, St ) = (t, St ) ∂x ∂x

✷

Using the notation of Example 5.2.2,

for a European call option we obtain ∂F (t, x) = ∂x (d1 ).

122

the black–scholes model Proof: Using the same notation as in Example 5.2.2 we have

F(t, x) = E

√ x exp σ θ Z − σ 2 θ/2 − K

+

,

where Z ∼ N (0, 1) and θ = (T − t). Differentiating the integrand with respect to √ x gives exp σ θ Z − σ 2 /2 if the integrand is strictly positive and zero otherwise. Then, again using the notation of Example 5.2.2, ∂F (t, x) = ∂x √ E exp σ θ Z − σ 2 θ/2 1 Z +d2 ≥0
∞

√ 1 exp σ θ y − σ 2 θ/2 − y 2 /2 √ dy. 2π −d2 √ Substituting ﬁrst u = −y and then z = u + σ θ as before this reduces to = ∂F (t, x) = ∂x For the European put one calculates ∂F (t, x) = − (−d1 ). ∂x (d1 ).

(d1 ). So

✷
Remark: (The Greeks) The quantity ∂ F/∂ x is often called the delta of the option by

practitioners. For a portfolio π of assets and derivatives, the sensitivities of the price to the parameters of the market are determined by the Greeks. If we write π(t, x) for the value of the portfolio if the asset price at time t is x, then in addition to the delta, given by ∂π , we have the gamma, the theta and the vega: ∂x = ∂ 2π , ∂x2 = ∂π , ∂t V= ∂π . ∂σ ✷

5.3

Foreign exchange
In this section we begin our programme of increasing the ﬁnancial sophistication in our models by looking at the foreign exchange market. Holding currency is a risky business, and with this risk comes a demand for derivatives. To operate in this market we should like to be able to value claims based on the future value of one unit of currency in terms of another. The pricing problem for an exchange rate forward was solved in Exercise 13 of Chapter 1. In contrast to the pricing problem for a forward contract based on an underlying stock that pays no dividends, which we solved in §1.2, for an exchange rate forward we needed to take into account interest rates in both currencies. Similarly, in valuing a European call option based on the exchange rate between

123

5.3 foreign exchange

Sterling and US dollars in Example 1.6.6 we needed a model for both the Sterling and the US dollar cash bond. Our Black–Scholes model for foreign exchange markets too must incorporate cash bonds in both currencies. For deﬁniteness, we suppose that the two currencies are US dollars and pounds Sterling.

Black–Scholes currency model:

We write {Bt }t≥0 for the dollar cash bond and {Dt }t≥0 for its Sterling counterpart. Writing E t for the dollar worth of one pound at time t, our model is Dollar bond Sterling bond Exchange rate Bt = er t , Dt = eut , E t = E 0 exp (νt + σ Wt ) ,

where {Wt }t≥0 is a P-Brownian motion and r , u, ν and σ are constants.

We now encounter exactly the problem that we had to overcome in the discrete world: the exchange rate is not tradable. We must conﬁne ourselves to operating within a single market. Let us work ﬁrst from the point of view of the dollar investor. In the dollar markets, neither the Sterling cash bond nor the exchange rate is tradable. However, the product of the two, St = E t Dt , can be thought of as a dollar tradable. The dollar investor can hold Sterling cash bonds and their dollar value is precisely St at time t. Moreover, any claim based on E T can be thought of as a claim based on ST . We now have a set-up that precisely mirrors the basic Black–Scholes model of §5.1. From the point of view of the dollar trader there are really two processes, the dollar cash bond, {Bt }t≥0 , and the dollar value of the Sterling cash bond, {St }t≥0 . We can now apply the Black–Scholes methodology in this setting. Let C T denote the claim value (in dollars) at time T .

Three steps to replication (foreign exchange):

˜ 1 Find a measure Q under which the (dollar bond) discounted process { St = −1 Bt St }t≥0 is a martingale. 2 Form the process Mt = EQ e−r T C T Ft . ˜ 3 Find an adapted process {φt }0≤t≤T such that d Mt = φt d St .

Since St = E t Dt = exp ((ν + u) t + σ Wt ) , the process {St }t≥0 is just a geometric Brownian motion and so our work of §5.1 ensures that we can indeed follow these steps. First we apply Itˆ ’s formula to obtain the stochastic differential equation satisﬁed o

124

the black–scholes model

by {St }t≥0 :

1 d St = ν + u + σ 2 St dt + σ St d Wt . 2 Now applying Lemma 5.1.4, we ﬁnd that the Radon–Nikodym derivative with ˜ respect to P of the measure Q under which the dollar-discounted process { St }t≥0 is a martingale is dQ (θ) = Lt dP Ft 1 exp −θ Wt − θ 2 t , 2

where θ = ν + u + 1 σ 2 − r /σ . Moreover, 2 Xt Wt + ν + u + 1σ2 − r 2 σ t

is a Q-Brownian motion. We follow the rest of the procedure in a special case (see also Exercise 11).
Example 5.3.1 (Forward contract)

At what price should we agree to trade Sterling

at a future date T ?
Solution: Of course we already solved this problem in Exercise 13 of Chapter 1, but now rather than guessing the hedging portfolio, we follow our three steps to replication. We have already found the measure Q. Now if we agree to buy a unit of Sterling for K dollars at time T , then the payoff of the contract will be

CT = ET − K . The value of the contract at time t is then Vt = = EQ e−r (T −t) C T Ft EQ e−r (T −t) (E T − K ) Ft .

A forward contract costs nothing at time zero and so we must choose K so that V0 = 0. In other words, K = EQ [E T ]. Expressing E T as a function of X T gives 1 E T = E 0 exp σ X T − σ 2 T + (r − u)T , 2 and so using that {X t }t≥0 is a Q-Brownian motion gives the fair value of K as K = EQ [E T ] = e(r −u)T E 0 . Finally we ﬁnd the hedging portfolio. With this choice of strike price, the value of the contract at time t is Vt = E e−r (T −t) E T − E 0 e(r −u)T Ft .

125

5.3 foreign exchange
−1 ˜ Since E T = DT BT ST , under Q we have

˜ ˜ E [ E T | Ft ] = e(r −u)(T −t) Dt−1 Bt E ST Ft = e(r −u)(T −t) Dt−1 Bt St = e(r −u)(T −t) E t , so, substituting, Vt = e−u(T −t) E t − er t−uT E 0 = e−uT eut E t − er t E 0 . The dollar-discounted portfolio value is ˜ Mt = e−r t Vt = e−uT e−(r −u)t E t − e−uT E 0 = e−uT St − e−uT E 0 . The required hedging portfolio is constant, consisting of φt = e−uT Sterling bonds ✷ and ψt = −e−uT E 0 (dollar) cash bonds.
The Sterling investor

We now turn to the Sterling investor. From her point of view, tradables are quoted in pounds Sterling. Once again, in effect there are two Sterling tradables. The ﬁrst is the Sterling cash bond. The second is the Sterling value of the dollar bond given by Z t = E t−1 Bt . Once again we can follow our three-step replication programme. The Sterlingdiscounted value of the dollar bond is
−1 ˜ Z t = Dt−1 E t−1 Bt = E 0 exp (−σ Wt − (ν + u − r ) t) .

We use Lemma 5.1.4 to see that under the measure Q£ given by dQ£ (λ) = Lt dP Ft 1 exp −λWt − λ2 t , 2

˜ with λ = ν + u − r − σ 2 /2 /σ , { Z t }t≥0 is a martingale and X t = Wt + ν + u − r − σ 2 /2 t σ
£

is a Q£ -Brownian motion. For the Sterling investor then the option price is Ut = Dt EQ
Change of numeraire
−1 −1 DT E T C T Ft .

We now have the same worry that we encountered in Exercise 15 of Chapter 1. The risk-neutral measures Q and Q£ can be thought of as deﬁning a probability measure on the paths followed by {E t }t≥0 – the only truly random part of our model – and the two measures are different. So do they give the same price? To put our minds at rest we ﬁnd the dollar worth of the Sterling investor’s valuation, that is £ −1 −1 E t Ut = E t Dt EQ DT E T C T Ft . To compare this with our expression for Vt we express the expectation as a Q-expectation, again using Girsanov’s Theorem. Now X t = Wt + ν + u − r − 1σ2 2 σ t = Xt − σ t

126

the black–scholes model

and so ζt Now

dQ£ 1 = exp σ X t − σ 2 t . dQ Ft 2

1 E t = E 0 exp σ X t − σ 2 t + (r − u)t , 2

so we can write ζt = Bt−1 Dt E t . Substituting gives E t Ut = = = E t Dt EQ
£

−1 −1 DT E T C T Ft

−1 −1 E t Dt ζt−1 EQ DT E T ζT C T Ft −1 Bt EQ BT C T Ft .

In other words the dollar value obtained by the Sterling investor is precisely Vt . The difference in the measures is merely an artefact of the different choice of ‘reference asset’ or numeraire.

5.4

Dividends
Our assumption so far has been that there is no value in simply holding a stock. We should now like to relax that assumption to allow the pricing and hedging of options based on equities – stocks that make periodic cash payments.

Continuous payments

It is simplest to begin with a dividend that is paid continuously. Assume as before that the stock price follows a geometric Brownian motion given by St = S0 exp (νt + σ Wt ) , but now in the inﬁnitesimal time interval [t, t + dt) the holder of the stock receives a dividend payment of δSt dt where δ is a constant. As always, we also assume that the market contains a riskless cash bond, {Bt }t≥0 , and we denote the continuously compounded interest rate by r . The difﬁculty that we now face is that {St }t≥0 does not represent the true worth of the asset: if we buy stock for price S0 at time zero, when we sell it at time t, the value of having held it is not just St − S0 but also the total accumulated dividends. In this model these depend on all the values that are taken by the asset price in the time interval [0, t]. In this sense, {St }t≥0 is not tradable. Our solution, just as for foreign exchange, is to translate the process into something that is tradable. The simplest solution is as follows. Suppose that whenever a cash dividend is paid, we immediately reinvest it in stock. The inﬁnitesimal payout δSt dt will buy δdt units of stock. At time t, rather than holding one unit of stock, we hold eδt units with total worth Z t = S0 exp ((ν + δ)t + σ Wt ) .

127

5.4 dividends

We regard the simple portfolio obtained by holding stock and continuously reinvesting the dividends in this way as a single asset with value Z t at time t. There is no cost in holding this asset and it makes no dividend payments. We are back in familiar Black–Scholes country.
Remark: Because the dividend payments were a constant proportion of the stock price, it was natural to reinvest them in stock. Had payments been ﬁxed amounts of cash, it would have been more natural to construct our ‘tradable’ asset as a portfolio in which dividends were immediately reinvested in bonds. This will be the situation of §5.5. ✷

Any portfolio consisting of φt eδt units of our original dividend-paying stock and ψt cash bonds at time t can be thought of as a portfolio of φt units of our new tradable asset and ψt units of cash bond. We can now follow our familiar procedure.

Three steps to replication (continuous dividends):

˜ Let Z t = Bt−1 Z t = e−r t Z t . ˜ t }t≥0 (with its natural ﬁltra1 Find a probability measure Q under which { Z tion) is a martingale. 2 Form the discounted value process, ˜ Vt = EQ e−r T C T Ft .

3 Find a predictable process {φt }0≤t≤T such that ˜ ˜ d Vt = φt d Z t .

Notice that a portfolio consisting of φt units of our tradable asset and ψt units of cash bond at each time t ∈ [0, t] is self-ﬁnancing if the value, {Vt }t≥0 , satisﬁes d Vt = φt d Z t + ψt d Bt = φt d St + φt δSt dt + ψt d Bt . The change in value over [t, t + dt) is due not only to proﬁts and losses of trading but also to dividend payments.
Example 5.4.1 (Call option) Suppose that a call option with strike price K and maturity T is written on the dividend-paying stock described above. What is the value of the option at time zero and what is the replicating portfolio? Solution: We follow our three steps to replication. First we must ﬁnd the martingale ˜ measure Q. The stochastic differential equation satisﬁed by { Z t }t≥0 is

1 ˜ ˜ ˜ d Z t = ν + δ + σ 2 − r Z t dt + σ Z t d Wt . 2

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the black–scholes model

As usual we apply the Girsanov Theorem. Under the measure Q deﬁned by dQ 1 = exp −λWt − λ2 t dP Ft 2 with λ = ν + δ + 1 σ 2 − r /σ 2 X t = Wt + ν + δ + 1σ2 − r 2 σ t

˜ is a Brownian motion and hence { Z t }t≥0 is a martingale. We can now just read off the price and the hedge from the corresponding formulae in Example 5.2.2 and Example 5.2.4. The value of the portfolio at time t is Vt = = = e−r (T −t) EQ (ST − K )+ Ft e−r (T −t) EQ e−δT Z T − K
+

Ft
+

e−r (T −t) e−δT EQ

Z T − K eδT

Ft .

That is the value is that of e−δT copies of a call option on {Z t }t≥0 with maturity T and strike K eδt . We write Ft = e(r −δ)(T −t) St for the forward price of our underlying stock at time t (see Exercise 14). Substituting from Example 5.2.2 we obtain    2 t  log KZ δT + r + σ2 (T − t) e −δT  Vt = e Z  √  t σ T −t   2 t log KZ δT + r − σ2 (T − t)  e  − K eδT e−r (T −t)  √  σ T −t     log Ft + 1 σ 2 (T − t) K 2  = e−r (T −t) Ft  √  σ T −t   log Ft − 1 σ 2 (T − t)  K 2  . −K  √  σ T −t Now using Example 5.2.4 we see that the replicating portfolio should consist of e−δT φt units of our tradable asset Z t at time t where   log Ft + 1 σ 2 (T − t) K 2 , φt =  √ σ T −t corresponding to e−δ(T −t)  
Ft K

log

 + 1 σ 2 (T − t) 2  √ σ T −t

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5.4 dividends

units of the dividend-paying asset. The bond holding in the portfolio will be   log Ft − 1 σ 2 (T − t) K 2 . ψt = −K e−r T  √ σ T −t ✷
Example 5.4.2 (Guaranteed equity proﬁts)

Let {St }t≥0 denote the value of the UK FTSE stock index. Suppose that we buy a ﬁve-year contract that pays out Z deﬁned to be 90% of the ratio of the terminal and initial values of the FTSE if this value is in the interval [1.3, 1.8], 1.3 if Z < 1.3 and 1.8 if Z > 1.8. What is the value of this contract at time zero?

Solution: The claim C T is

C T = min max 1.3, 0.9

ST , 1.8 , S0

where T is ﬁve years. Since the claim is based on a ratio, without loss of generality we set S0 = 1. As FTSE is composed of one hundred different stocks, their separate dividend payments will approximate a continuously paying stream. We assume the following data: FTSE drift µ = 7%, FTSE volatility σ = 15%, FTSE dividend yield δ = 4%, UK interest rate r = 6.5%. We can rewrite the claim as the sum of some cash plus the difference in the payout of two FTSE calls, C T = 1.3 + 0.9 (ST − 1.444)+ − (ST − 2)+ . Now the forward price for St is Ft = e(r −δ)(T −t) S0 = 1.133, and so using the call price formula for continuous dividend-paying stocks of Example 5.4.1 we can value these calls at 0.0422 and 0.0067 (per unit) at time zero. The value of our contract at time zero is then 1.3e−r T + 0.9 (0.0422 − 0.0067) = 0.9712. ✷
Periodic dividends

In practice, an individual stock does not pay dividends continuously, but rather at regular intervals. Suppose that the times of the payments are known in advance to be T1 , T2 , . . . and that at each time Ti the current holder of the equity receives a payment

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the black–scholes model

of δSTi . As shown in Exercise 7 of Chapter 2, in the absence of arbitrage, the stock price must instantaneously decrease by the same amount. So at any of the times Ti , the dividend payout is exactly equal to the instantaneous decrease in the stock price. Between payouts we assume our usual geometric Brownian motion model.

Equity model with periodic dividends:

At deterministic times T1 , T2 , . . . the equity pays a dividend of a fraction δ of the stock price which was current just before the dividend is paid. The stock price itself is modelled as St = S0 (1 − δ)n[t] exp (νt + σ Wt ) , where n[t] = max{i : Ti ≤ t} is the number of dividend payments made by time t. There is also a riskless cash bond Bt = exp (r t).

At ﬁrst sight it looks as though we have two problems. First, although between the times Ti our stock price follows the usual geometric Brownian motion model, at those times it has discontinuous jumps. This doesn’t ﬁt our framework. Secondly, as for continuous dividends, the stock price process {St }t≥0 does not reﬂect the true value of the stock. However, by adapting our strategy for the continuous dividends case and reinvesting all dividend payments in stock, we’ll overcome both of these obstacles. We deﬁne {Z t }t≥0 to be the value of the portfolio that starts with one unit of stock at time zero and every time the stockholding pays a dividend it is reinvested by buying more stock. The ﬁrst dividend payment is δST1 − , δ times the stock price immediately prior to the payment. Immediately after the payment of the ﬁrst dividend, the stock price jumps to ST1 + = (1 − δ)ST1 − , so our dividend payment will buy us an additional δ/(1 − δ) units of stock, thereby increasing our total stock holding by a factor of 1/(1 − δ). At time t the portfolio will therefore consist of 1/(1 − δ)n[t] units of stock. Thus Z t = (1 − δ)−n[t] St = S0 exp (νt + σ Wt ) . As before we think of our portfolio {Z t }t≥0 as a non-dividend-paying asset and so our market consists of two tradable assets, the portfolio {Z t }t≥0 and a riskless cash bond {Bt }t≥0 , and we are back in familiar territory. We mimic exactly what we did for continuous dividend payments. A portfolio consisting of φt units of Z t and ψt in cash bonds at time t is equivalent to (1 − δ)−n[t] φt units of the dividend-paying underlying stock, St , and ψt units of cash bond. ˜ The measure Q that makes the discounted process { Z t }t≥0 a martingale satisﬁes dQ 1 = exp −λWt − λ2 t dP Ft 2

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5.5 bonds

with λ = ν + 1 σ 2 − r /σ . Rewriting the claim as a function of Z T we can use the 2 classical Black–Scholes analysis to price and hedge the option.
Example 5.4.3 (Forward price)

Find the fair price to be written into a forward contract on a stock that pays periodic dividends.

Solution: The value of the contract at time T is C T = ST − K . We seek K so that the

time zero value of the contract is zero. As usual, the value at time t is the discounted expected value of the claim under the martingale measure Q. That is Vt = = = = EQ e−r (T −t) (ST − K ) Ft EQ e−r (T −t) (1 − δ)n[T ] Z T − K (1 − δ) (1 − δ) Zt − n[T ]−n[t] n[T ]

Ft

Ke

−r (T −t)

St − K e−r (T −t) .

The K for which this is zero at time zero is K = er T (1 − δ)n[T ] S0 . (5.5) ✷

5.5

Bonds
A pure discount bond is a security that pays off one unit at some future time T . Most market bonds also pay off a series of smaller amounts, c, at predetermined times T1 , T2 , . . . , Tn . Such coupon payments resemble dividend payments except that the amount of the coupon is known in advance. So far we have considered only a riskless cash bond in which the interest rate too is known in advance. In real markets, uncertainty in interest rates causes the price of bonds to move randomly as well. In order to keep the book to a reasonable length we do not intend to enter into a full account of bond market models. An excellent introduction can be found in Baxter & Rennie (1996). So for the purposes of this section, we are going to take a schizophrenic attitude to interest rates. We’ll assume that we have a riskless cash bond following Bt = er t , but also a stochastically varying coupon bond whose price between coupon payments evolves as a geometric Brownian motion. Clearly there are links between the short term interest rate and bond prices, but over short time horizons, many practitioners ignore them. In effect we are thinking of a coupon bond as an asset paying predetermined cash dividends at times T1 , T2 , . . . , Tn where we assume Tn < T . Writing I (t) = min{i : t < Ti }, the bond price satisﬁes n St = i=I (t)

ce−r (Ti −t) + A exp (νt + σ Wt ) ,

for some constants A, ν and σ .

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the black–scholes model

As for dividend-paying stock, the price process {St }t≥0 is discontinuous at the coupon payment times. Once again however we can manufacture a continuous non-dividend-paying asset from {St }t≥0 . Whereas when our dividend payment was a fraction of the stock price it was natural to reinvest it in stock, now, since our coupons are ﬁxed cash payments, we invest them in the riskless cash bond. With this investment strategy the coupon paid at time Ti then has value ce−r (Ti −t) for all t ∈ [0, T ] and so the portfolio constructed in this way has value n Zt = i=1 ce−r (Ti −t) + A exp (νt + σ Wt ) .

Exactly as before we think of our market as consisting of the riskless cash bond {Bt }t≥0 and the tradable asset {Z t }t≥0 . ˜ As usual we want to ﬁnd Q under which the discounted asset price { Z t }t≥0 is n −r Ti plus the geometric ˜ a martingale. But Z t is just the constant cash sum i=1 ce Brownian motion A exp ((ν − r )t + σ Wt ). This will be a Q-martingale if 1 dQ = exp −λWt − λ2 t dP Ft 2 where λ = ν + 1 σ 2 − r /σ . Under Q, 2 ν + 1σ2 − r 2 t σ is a Brownian motion. The value at time t of an option with payoff C T is now X t = Wt + EQ e−r (T −t) C T Ft . Under Q, the price of the bond at time T is just ST = A exp 1 r − σ 2 T + σ XT 2 .

The forward price for ST at time zero is F = Aer T and the value of a call on ST with strike price K at maturity T is e−r T see Exercise 20. F log
F K

+ 1 σ 2T √ 2 σ T

−K

log

F K

− 1 σ 2T √ 2 σ T

;

5.6

Market price of risk
A deﬁnite pattern has emerged. Given a non-tradable stock, we have tied it to a portfolio that can be thought of as a tradable, found the martingale measure corresponding to that tradable process and used that measure to price the option. In deciding what is tradable and what is not we have used only common sense. Indeed it is not something that can be reduced purely to mathematics, but if we decide that an asset with price {St }t≥0 is truly tradable and we have a riskless cash bond, {Bt }t≥0 , we should like to determine the class of tradables within the market that they create.

133 Martingales and tradables

5.6 market price of risk

˜ Suppose that St = Bt−1 St is the discounted price of our tradable asset at time t. Let Q ˜ be a measure under which { St }t≥0 is a martingale. If we take another process, {Vt }t≥0 ˜ ˜ ˜ for which the discounted process {Vt }t≥0 given by Vt = Bt−1 Vt is a Q, {FtS }t≥0 martingale, then is {Vt }t≥0 tradable? Our strategy is to construct a self-ﬁnancing portfolio consisting of our tradable asset and the tradable discounting process whose value at time t is always exactly Vt . As usual, the ﬁrst step is an application of the Martingale Representation Theorem. ˜ Provided that Bt−1 St has non-zero volatility, we can ﬁnd an {FtS }t≥0 -previsible process {φt }t≥0 such that ˜ ˜ d Vt = φt d St . (5.6)

Taking our cue from the construction of the portfolio replicating an option in §5.1, we create a portfolio that at time t consists of φt units of the tradable St and ψt = ˜ ˜ Vt − φt St units of (the tradable) Bt . The value of this portfolio at time t is exactly Vt . We must check that it is self-ﬁnancing. Now d Vt = = = = = ˜ ˜ Bt d Vt + Vt d Bt ˜ ˜ Bt φt d St + Vt d Bt (integration by parts) (equation (5.6))

˜ ˜ Bt φt d St + ψt + φt St d Bt ˜ ˜ φt Bt d St + St d Bt + ψt d Bt φt d St + ψt d Bt (integration by parts)

and so the change in value of the portfolio over any inﬁnitesimal time interval is due to changes in asset prices. That is we have the self-ﬁnancing property and {Vt }t≥0 is indeed tradable. What about the other way round? Suppose that {Bt−1 Vt }t≥0 were not a ˜ Q, {FtS }t≥0 -martingale. Then there would have to be times s < T such that with positive probability ˜ −1 −1 Bs Vs = EQ BT VT FsS . Suppose that {Vt }t≥0 were tradable. We can construct a process {Ut }t≥0 by setting
−1 Ut = Bt EQ BT VT FtS . ˜

Since {Bt−1 Ut }t≥0 is a Q, {FtS }t≥0 -martingale, we know that {Ut }t≥0 is tradable. That is we have two tradables that take the same value at time T , but with positive probability take different values at an earlier time s. Exercise 21 shows that in the absence of arbitrage this is a contradiction. So if {Bt−1 Vt }t≥0 is not a martingale, then {Vt }t≥0 is not a tradable. ˜ Of course, since interest rates are deterministic, FtS = FtS and so we have proved the following lemma.
Lemma 5.6.1

Given a riskless cash bond {Bt }t≥0 and a tradable asset {St }t≥0 , a process {Vt }t≥0 represents a tradable asset if and only if the discounted value

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the black–scholes model

{Bt−1 Vt }t≥0 is a Q, {FtS }t≥0 -martingale where Q is the measure under which the discounted asset price {Bt−1 St }t≥0 is a martingale.
Tradables and the market price of risk

Suppose that we have two tradable risky securities {St1 }t≥0 and {St2 }t≥0 in a single Black–Scholes market – that is they are both functions of the same Brownian motion. We deﬁne them both via their stochastic differential equations, d Sti = µi Sti dt + σi Sti d Wt . In order for both to be tradable, Lemma 5.6.1 tells us that they must both be martingales with respect to the same measure Q. Assuming that Bt = er t we must have that µi − r X t = Wt + t σi is a Q-Brownian motion for i = 1, 2. This can only be the case if µ1 − r µ2 − r = . σ1 σ2 Economists attach a meaning to this quantity. If we think of µ as the rate of growth of the tradable, r as the rate of growth of the riskless bond and σ as a measure of the risk of the asset, then µ−r γ = σ is the excess rate of return (above the risk-free rate) per unit of risk. As such, it is often called the market price of risk. It is also known as the Sharpe ratio. In such a simple market, every tradable asset should have the same market price of risk, otherwise there would be arbitrage opportunities. Of course, γ is precisely the change of drift in the underlying Brownian motion when we change measure from P (the market measure) to Q (the martingale measure). However, this appealing economic interpretation of γ does not provide a new argument for using Q. It is replication that makes the Black–Scholes analysis work. Without a replicating portfolio our arbitrage arguments collapse.

Exercises
1 Suppose that an asset price St is such that d St = µSt dt + σ St d Wt , where {Wt }t≥0 is, as usual, standard P-Brownian motion. Let r denote the risk-free interest rate. The price of a riskless asset then follows d Bt = r Bt dt. We write {ψt , φt } for the portfolio consisting of ψt units of the riskless asset Bt and φt units of St at time t. For each of the following choices of φt , ﬁnd ψt so that the portfolio {ψt , φt } is self-ﬁnancing. (Recall that the value of the portfolio at time t is Vt = ψt Bt + φt St and that the portfolio is self-ﬁnancing if d Vt = ψt d Bt + φt d St .) (a) φt = 1, t (b) φt = 0 Su du, (c) φt = St .

135

exercises

2 Let {Ft }t≥0 be the natural ﬁltration associated with a P-Brownian motion {Wt }t≥0 . Show that if Q is a probability measure equivalent to P and HT is an FT -measurable 2 random variable with EQ HT < ∞ then Mt EQ [HT |Ft ]

deﬁnes a square-integrable Q-martingale. 3 Show that, in the notation of Example 5.2.2, the Black–Scholes price at time t of a European put option with strike K and maturity T is F(t, St ) where F(t, x) = K e−r θ (−d2 ) − x (−d1 ). 4 Suppose that the value of a European call option can be expressed as Vt = F(t, St ) ˜ ˜ (as we prove in Proposition 5.2.3). Then Vt = e−r t Vt , and we may deﬁne F by ˜ ˜ ˜ Vt = F(t, St ). ˜ ˜ Under the risk-neutral measure, the discounted asset price follows d St = σ St d X t , where (under this probability measure) {X t }t≥0 is a standard Brownian motion. ˜ ˜ (a) Find the stochastic differential equation satisﬁed by F(t, St ). ˜ (b) Using the fact that Vt is a martingale under the risk-neutral measure, ﬁnd the ˜ partial differential equation satisﬁed by F(t, x), and hence show that 1 ∂2 F ∂F ∂F + σ 2x2 2 + r x − r F = 0. ∂t 2 ∂x ∂x This is the Black–Scholes equation. 5 Delta hedging The following derivation of the Black–Scholes equation is very popular in the ﬁnance literature. We will suppose, as usual, that an asset price follows a geometric Brownian motion. That is, there are parameters µ, σ , such that d St = µSt dt + σ St d Wt . Suppose that we are trying to value a European option based on this asset. Let us denote the value of the option at time t by V (t, St ). We know that at time T , V (T, ST ) = f (ST ), for some function f . (a) Using Itˆ ’s formula express V as the solution to a stochastic differential equation. o (b) Suppose that a portfolio, whose value we denote by π, consists of one option and a (negative) quantity −δ of the asset. Assuming that the portfolio is selfﬁnancing, ﬁnd the stochastic differential equation satisﬁed by π . (c) Find the value of δ for which the portfolio you have constructed is ‘instantaneously riskless’, that is for which the stochastic term vanishes. (d) An instantaneously riskless portfolio must have the same rate of return as the risk-free interest rate. Use this observation to ﬁnd a (deterministic) partial differential equation for the V (t, x). Notice that this is the Black–Scholes equation obtained in Exercise 4.

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the black–scholes model

Of course δ is precisely the stock holding in our replicating portfolio. In fact this derivation is not entirely satisfactory as it can be checked that the portfolio that we have constructed is not self-ﬁnancing, violating our assumption in (b). A rigorous approach requires a portfolio consisting of one option, −δ assets and e−r t (V (t, St ) − δSt ) cash bonds at time t. 6 7 Calculate the values of the Greeks for a European call option with strike price K at maturity time T . An alternative approach to solving the Black–Scholes equation is to transform it via a change of variables into the heat equation. Suppose that F(t, x) : [0, T ] × [0, ∞) → R satisﬁes 1 ∂2 F ∂F ∂F (t, x) + σ 2 x 2 2 (t, x) + r x (t, x) − r F(t, x) = 0, ∂t 2 ∂x ∂x subject to the boundary conditions F(t, 0) = 0, F(t, x) → 1 as x → ∞, x F(T, x) = (x − K )+ .

That is F solves the Black–Scholes equation with the boundary conditions appropriate for pricing a European call option with strike price K at time T . (a) Show that the change of variables x = K ey , results in the equation ∂ 2v ∂v ∂v (τ, y) = 2 (τ, y) + (k − 1) (τ, y) − kv(τ, y), ∂τ ∂y ∂y 1 y ∈ R, τ ∈ 0, σ 2 T , 2 t=T− 2τ , σ2 F = K v(τ, y)

where k = 2r/σ 2 and v(0, y) = (e y − 1)+ . (b) Now set v(τ, y) = eαy+βτ u(τ, y). Find α and β such that ∂ 2u ∂u (τ, y) = 2 (τ, y), ∂τ ∂y y ∈ R,

and ﬁnd the corresponding initial condition for u. (c) Solve for u and retrace your steps to obtain the Black–Scholes pricing formula for a European call option. 8 Show that, for each constant A, V (t, x) = Ax and V (t, x) = Aer t are both exact solutions of the Black–Scholes differential equation. What do they represent and what is the hedging portfolio in each case? 9 Find the most general solution of the Black–Scholes equation that has the special form (a) V (t, x) = V (x),

137

exercises

(b) V (t, x) = f (t)g(x). These are examples of similarity solutions. The solutions in (a) give prices of perpetual options. 10 Let C(t, St ) and P(t, St ) denote the values of a European call and put option with the same exercise price, K , and expiry time, T . Show that C(t, x)− P(t, x) also satisﬁes the Black–Scholes equation with the ﬁnal data C(T, x) − P(T, x) = x − K . Deduce that x − K e−r (T −t) is also a solution of the Black–Scholes equation. Interpret these results ﬁnancially. Assuming the model of §5.3, ﬁnd the Black–Scholes price for a Sterling call option that gives us the right to buy a pound Sterling at time T for K dollars. What is the corresponding hedging portfolio?

11

12 Check that the Sterling and dollar investors of §5.3 use exactly the same replicating strategy. 13 Suppose that the US dollar/Japanese Yen exchange rate follows the stochastic differential equation d St = µSt dt + σ St d Wt for some constants µ and σ . You are told that the expected $/ and rates in one years time are both 2S0 . Is this possible? 14 /$ exchange

In our usual notation suppose that an asset price follows geometric Brownian motion with St = S0 exp (νt + σ Wt ) at time t. If in each inﬁnitesimal time interval the asset pays to its holder a dividend of δSt dt, ﬁnd an expression for the fair price in a forward contract based on the stock with maturity time T . What is the corresponding hedging portfolio? What is the ‘put–call parity’ relation for the market in Exercise 14? Suppose that in valuing the contract in Example 5.4.2 we had failed to take account of the dividend stream from the constituent stocks of the FTSE. Find a ﬁnancial argument to indicate whether the price obtained for the contract will be too high or too low. Find the exact value that we would have obtained for the contract. Suppose that Vt is the value of a self-ﬁnancing portfolio consisting of φt units of stock that pays periodic dividends, as in §5.4, and ψt units of cash bond. Find the differential equation that characterises the self-ﬁnancing property of Vt in this setting. Find a portfolio that replicates the forward of Example 5.4.3. Value and hedge a European call option with maturity T and strike K based on the periodic dividend-paying stock of §5.4. Express your answer in terms of the usual Black–Scholes formula evaluated on the forward price of equation (5.5). Check the forward price and the value of a call option claimed in §5.5 and ﬁnd the corresponding self-ﬁnancing replicating portfolios.

15 16

17

18 19

20

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the black–scholes model

21

Show that if two tradable assets have the same value at time T , but with positive probability take different values at time s < T , then there are arbitrage opportunities in the market.

6 Different payoffs

Summary
Most of the concrete examples of options considered so far have been the standard examples of calls and puts. Such options have liquid markets, their prices are fairly well determined and margins are competitive. Any option that is not one of these vanilla calls or puts is called an exotic option. Such options are introduced to extend a bank’s product range or to meet hedging and speculative needs of clients. There are usually no markets in these options and they are bought and sold purely ‘over the counter’. Although the principles of pricing and hedging exotics are exactly the same as for vanillas, risk management requires care. Not only are these exotic products much less liquid than standard options, but they often have discontinuous payoffs and so can have huge ‘deltas’ close to the expiry time making them difﬁcult to hedge. This chapter is devoted to examples of exotic options. The simplest exotics to price and hedge are packages, that is, options for which the payoff is a combination of our standard ‘vanilla’ options and the underlying asset. We already encountered such options in §1.1. We relegate their valuation to the exercises. The next simplest examples are European options, meaning options whose payoff is a function of the stock price at the maturity time. The payoffs considered in §6.1 are discontinuous and we discover potential hedging problems. In §6.2 we turn our attention to multistage options. Such options allow decisions to be made or stipulate conditions at intermediate dates during their lifetime. The rest of the chapter is devoted to path-dependent options. In §6.3 we use our work of §3.3 to price lookback and barrier options. Asian options, whose payoff depends on the average of the stock price over the lifetime of the option, are discussed brieﬂy in §6.4 and ﬁnally §6.5 is a very swift introduction to pricing American options in continuous time.

6.1

European options with discontinuous payoffs
We work in the basic Black–Scholes framework. That is, our market consists of a riskless cash bond whose value at time t is Bt = er t and a single risky asset whose price, {St }t≥0 , follows a geometric Brownian motion. In §5.2 we established explicit formulae for both the price and the hedging portfolio for European options within this framework. Speciﬁcally, if the payoff of
139

140

different payoffs

the option at the maturity time T is C T = f (ST ) then for 0 ≤ t ≤ T the value of the option at time t is Vt = = F(t, St ) = EQ e−r (T −t) f (ST ) Ft e−r (T −t)
∞ −∞

f

St exp

r−

√ σ2 (T − t) + σ y T − t 2 (6.1)

y2 1 × √ exp − dy, 2 2π

where Q is the martingale measure, and the claim f (ST ) can be replicated by a portfolio consisting at time t of φt units of stock and ψt = e−r t (Vt − φt St ) cash bonds where φt = ∂F (t, x) ∂x . x=St (6.2)

Mathematically, other than the issue of actually evaluating the integrals, that would appear to be the end of the story. However, as we shall see, rather more careful consideration of our assumptions might lead us to doubt the usefulness of these formulae when the payoff is a discontinuous function of ST .
Digitals and pin risk Example 6.1.1 (Digital options)

The payoff of a digital option, also sometimes called a binary option or a cash-or-nothing option, is given by a Heaviside function. For example, a digital call option with strike price K at time T has payoff CT = 1 if ST ≥ K , 0 if ST < K

at maturity. Find the price and the hedge for such an option.
Solution: In order to implement the formula (6.1) we must establish the range of y

for which St exp r−

√ σ2 (T − t) + σ y T − t 2

> K.

Rearranging we see that this holds for y > d where d= Writing obtain Vt √ 1 log K St − r− σ2 2 (T − t) .

σ T −t

for the normal distribution function and substituting in equation (6.1) we 1 2 √ e−y /2 dy = e−r (T −t) 2π d (−d) = e−r (T −t) (d2 ),
∞ −d −∞

= =

e−r (T −t) e−r (T −t)

1 2 √ e−y /2 dy 2π

141

6.2 multistage options

where d2 = σ T −t √ 1 log St K + r− σ2 (T − t) , 2

as in Example 5.2.2. Now we turn to the hedge. By (6.2), the stock holding in our replicating portfolio at time t is φt = e−r (T −t) 1 1 √ St 2π(T − t)σ St 1 log 2 K 2(T − t)σ + r− σ2 (T − t) 2
2

× exp −

.

Now as t ↑ T , this converges to 1/K times the delta function concentrated on ST = K . Consider what this means for the replicating portfolio as t ↑ T . Away from St = K , φt is close to zero, but if St is close to K the stock holding in the portfolio will be very large. Now if near expiry the asset price is close to K , there is a high probability that its value will cross the value St = K many times before expiry. But if the asset price oscillates around the strike price close to expiry our prescription for the hedging portfolio will tell us to rapidly buy and sell large numbers of the underlying asset. Since markets are not the perfect objects envisaged in our Black–Scholes model and we cannot instantaneously buy and sell, risk from small asset price changes (not to mention transaction costs) can easily outweigh the maximum liability that we are exposed to by having sold the digital. This is known as the pin risk associated with the option. ✷ If we can overcome our misgivings about the validity of the Black–Scholes price for digitals, then we can use them as building blocks for other exotics. Indeed, since the option with payoff 1[K 1 ,K 2 ] (ST ) at time T can be replicated by buying a digital with strike K 2 and maturity T and selling a digital with strike K 1 and maturity T , in theory we could price any European option by replicating it by (possibly inﬁnite) linear combinations of digitals.

6.2

Multistage options
Some options either allow decisions to be made or stipulate conditions at intermediate dates during their lifetime. An example is the forward start option of Exercise 3 of Chapter 2. To illustrate the procedure for valuation of multistage options, we ﬁnd the Black–Scholes price of a forward start.
Example 6.2.1 (Forward start option)

Recall that a forward start option is a contract in which the holder receives, at time T0 , at no extra cost, an option with expiry date T1 > T0 and strike price equal to ST0 . If the risk-free rate is r ﬁnd the Black–Scholes price, Vt , of such an option at times t < T1 .

142

different payoffs Solution: First suppose that t ∈ [T0 , T1 ]. Then by time t we know ST0 and so the

value of the option is just that of a European call option with strike ST0 and maturity T1 , namely Vt = e−r (T1 −t) EQ ST1 − ST0 + Ft , where Q is a probability measure under which the discounted price of the underlying is a martingale. In particular, at time T0 , using Example 5.2.2, VT0 = ST0 (d1 ) − ST0 e−r (T1 −T0 ) (d2 ) where d1 = In other words VT0 = ST0 = cST0 where c = c(r, σ, T0 , T1 ) is independent of the asset price. To ﬁnd the price at time t < T0 , observe that the portfolio consisting of c units of the underlying over the time interval 0 ≤ t ≤ T0 exactly replicates the option at time T0 . Thus for t < T0 , the price is given by cSt . In particular, the time zero price of the option is √ √ σ2 T1 − T0 T1 − T0 σ2 r− r+ − e−r (T1 −T0 ) . V0 = S0 2 σ 2 σ ✷
General strategy

r+

σ T1 − T0 σ2 r+ 2 √ T1 − T0 σ

√

σ2 2

(T1 − T0 )

and d2 =

r−

σ T1 − T0 σ2 r− 2

√

σ2 2

(T1 − T0 )

. T1 − T0 σ

−e

−r (T1 −T0 )

√

Notice that, in order to price the forward start option, we worked our way back from time T1 . This reﬂects a general strategy. For a multistage option with maturity T1 and conditions stipulated at an intermediate time T0 , we invoke the following procedure.

Valuing multistage options:

1 2 3 4

Find the payoff at time T1 . Use Black–Scholes to value the option for t ∈ [T0 , T1 ]. Apply the contract conditions at time T0 . Use Black–Scholes to value the option for t ∈ [0, T0 ].

We put this into action for two more examples.
Example 6.2.2 (Ratio derivative)

A ratio derivative can be described as follows. Two times 0 < T0 < T1 are ﬁxed. The derivative matures at time T1 when its payoff is ST1 /ST0 . Find the value of the option at times t < T1 .

143

6.2 multistage options Solution: First suppose that t ∈ [T0 , T1 ]. At such times ST0 is known and so

Vt =

1 Q −r (T1 −t) E e ST1 Ft ST0

where, under Q, the discounted asset price is a martingale. Hence Vt = St /ST0 . In particular, VT0 = 1. Evidently the value of the option for t < T0 is therefore ✷ e−r (T0 −t) . Both forward start options and ratio derivatives, in which the strike price is set to be a function of the stock price at some intermediate time T0 , are examples of cliquets.
Compound options

A rather more complex class of examples is provided by the compound options. These are ‘options on options’, that is options in which the rˆ le of the underlying is o itself played by an option. There are four basic types of compound option: call-oncall, call-on-put, put-on-call and put-on-put.
Example 6.2.3 (Call-on-call option)

To describe the call-on-call option we must specify two exercise prices, K 0 and K 1 , and two maturity times T0 < T1 . The ‘underlying’ option is a European call with strike price K 1 and maturity T1 . The call-on-call contract gives the holder the right to buy the underlying option for price K 0 at time T0 . Find the value of such an option for t < T0 .

Solution: We know how to price the underlying call. Its value at time T0 is given by

the Black–Scholes formula as C ST0 , T0 ; K 1 , T1 = ST0 − K0e where d1 ST0 , T1 −r (T1 −T0 ) − T0 , K 1 d2 ST0 , T1 − T0 , K 1
2

+ r + σ2 (T1 − T0 ) d1 St0 , T1 − T0 , K 1 = √ σ T1 − T0 √ and d2 St0 , T1 − T0 , K 1 = d1 St0 , T1 − T0 , K 1 − σ T1 − T0 . The value of the compound option at time T0 is then log V T0 , ST0 = C ST0 , T0 ; K 1 , T1 − K 0
+

ST0 K1

.

Now we apply Black–Scholes again. The value of the option at times t < T0 is V (t, St ) = e−r (T0 −t) EQ C(ST0 , T0 , K 1 , T1 ) − K 0
+

FtS

(6.3)

where the discounted asset price is a martingale under Q. Using that 1 ST0 = St exp σ Z T0 − t + r − σ 2 (T0 − t) , 2

144

different payoffs

where, under Q, Z ∼ N (0, 1), equation (6.3) now gives an analytic expression for the value in terms of the cumulative distribution function of a bivariate normal random variable. We write 1 f (y) = S0 exp σ y T0 − t + r − σ 2 (T0 − t) 2 and deﬁne x0 implicitly by x0 = inf {y ∈ R : C( f (y), T0 ; K 1 , T1 ) ≥ K 0 } . Now log and so writing f (y) K1 = log S0 K1 1 + σ y T0 − t + r − σ 2 (T0 − t) 2

√ log(S0 /K 1 ) + σ y T0 − t + r T1 − σ 2 T0 + 1 σ 2 T1 2 ˆ d1 (y) = √ σ T1 − T0 ˆ d2 (y) = √ log(S0 /K 1 ) + σ y T0 − t + r T1 − 1 σ 2 T1 2 √ σ T1 − T0
∞ x0

and

we obtain V (t, St ) = e−r (T0 −t) ˆ ˆ f (y) (d1 (y)) − K 0 e−r (T1 −T0 ) (d2 (y)) − K 0 1 2 × √ e−y /2 dy. 2π ✷

6.3

Lookbacks and barriers
We now turn to our ﬁrst example of path-dependent options, that is options for which the history of the asset price over the duration of the contract determines the payout at expiry. As usual we use {St }0≤t≤T to denote the price of the underlying asset over the duration of the contract. In this section we shall consider options whose payoff at maturity depends on ST and one or both of the maximum and minimum values taken by the asset price over [0, T ].

Notation:

We write S∗ (t) = min {Su : 0 ≤ u ≤ t} , S ∗ (t) = max {Su : 0 ≤ u ≤ t} .

145

6.3 lookbacks and barriers

A lookback call gives the holder the right to buy Deﬁnition 6.3.1 (Lookback call) a unit of stock at time T for a price equal to the minimum achieved by the stock up to time T . That is the payoff is C T = ST − S∗ (T ).
Deﬁnition 6.3.2 (Barrier options) A barrier option is one that is activated or deactivated if the asset price crosses a preset barrier. There are two basic types:

1

knock-ins

(a) the barrier is up-and-in if the option is only active if the barrier is hit from below, (b) the barrier is down-and-in if the option is only active if the barrier is hit from above; 2 knock-outs (a) the barrier is up-and-out if the option is worthless if the barrier is hit from below, (b) the barrier is down-and-out if the option is worthless if the barrier is hit from above.
Example 6.3.3

A down-and-in call option pays out (ST − K )+ only if the stock price fell below some preagreed level c some time before T , otherwise it is worthless. That is, the payoff is C T = 1{S∗ (T )≤c} (ST − K )+ . As always we can express the value of such an option as a discounted expected value under the martingale measure Q. Thus the value at time zero can be written as V (0, S0 ) = e−r T EQ [C T ] (6.4)

where r is the riskless borrowing rate and the discounted stock price is a Qmartingale. However, in order to actually evaluate the expectation in (6.4) for barrier options we need to know the joint distribution of (ST , S∗ (T )) and (ST , S ∗ (T )) under the martingale measure Q. Fortunately we did most of the work in Chapter 3.
Joint distribution of the stock price and its minimum

In Lemma 3.3.4 we found the joint distribution of Brownian motion and its maximum. Speciﬁcally, if {Wt }t≥0 is a standard P-Brownian motion, writing Mt = max0≤s≤t Ws , for a > 0 and x ≤ a P [Mt ≥ a, Wt ≤ x] = 1 − 2a − x √ t −2a + x √ t .

By symmetry, writing m t = min0≤s≤t Ws , for a < 0 and x ≥ a, P [m t ≤ a, Wt ≥ x] = 1 − or, differentiating, if a < 0 and x ≥ a P [m T ≤ a, WT ∈ d x] = pT (0, −2a + x)d x = pT (2a, x)d x ,

146

different payoffs

where pt (x, y) = √

exp −|x − y|2 /2t . 2πt Combining these results with (two applications of) the Girsanov Theorem will allow us to calculate the joint distribution of (ST , S ∗ (T )) and of (ST , S∗ (T )) under the martingale measure Q. As usual, under the market measure P, St = S0 exp (νt + σ Wt ) where {Wt }t≥0 is a P-Brownian motion. Let us suppose, temporarily, that ν = 0 so that St = S0 exp(σ Wt ) and moreover S∗ (t) = S0 exp(σ m t ) and S ∗ (t) = S0 exp(σ Mt ). In this special case then the joint distribution of the stock price and its minimum (resp. maximum) can be deduced from that of (Wt , m t ) (resp. (Wt , Mt )). Of course, in general, ν will not be zero either under the market measure P or under the martingale measure Q. Our strategy will be to use the Girsanov Theorem not only to switch to the martingale measure but also to switch, temporarily, to an equivalent measure under which St = S0 exp(σ Wt ).
Lemma 6.3.4

1

Let {Yt }t≥0 be given by Yt = bt + X t where b is a constant and {X t }t≥0 is a Q-Brownian motion. Writing Y∗ (t) = min{Yu : 0 ≤ u ≤ t}, Q [Y∗ (T ) ≤ a, YT ∈ d x] = pT (bT, x)d x e2ab pT (2a + bT, x)d x if x < a, if x ≥ a,

where, as above, pt (x, y) is the Brownian transition density function.
Proof: By the Girsanov Theorem, there is a measure P, equivalent to Q, under which

{Yt }t≥0 is a P-Brownian motion and dP 1 = exp −bX T − b2 T . dQ Fr 2 Notice that this depends on {X t }0≤t≤T only through X T . The Q-probability of the event {Y∗ (T ) ≤ a, YT ∈ d x} will be the P-probability of that event multiplied by dQ d P F evaluated at YT = x. Now r dQ 1 = exp bX T + b2 T dP 2 and so for a < 0 and x ≥ a Q [Y∗ (T ) ≤ a, YT ∈ d x] =

1 = exp bYT − b2 T 2

1 P [Y∗ (T ) ≤ a, YT ∈ d x] exp bx − b2 T 2 1 2 = pT (2a, x) exp bx − b T d x 2 2ab = e pT (2a + bT, x)d x. (6.5)

147

6.3 lookbacks and barriers

Evidently for x ≤ a, {Y∗ (T ) ≤ a, YT ∈ d x} = {YT ∈ d x} and so for x ≤ a Q [Y∗ (T ) ≤ a, YT ∈ d x] = = = and the proof is complete. Q [YT ∈ d x] Q [bT + X T ∈ d x] pT (bT, x)d x ✷

Differentiating (6.5) with respect to a, we see that, in terms of joint densities, for a 0. In Exercise 13 you are asked to ﬁnd the joint distribution of YT and Y ∗ (T ) under Q.
An expression for the price

From Chapter 5, under the martingale measure Q, St = S0 exp (σ Yt ) where Yt = (r − 1 σ 2 ) 2 t + Xt σ

and {X t }t≥0 is a Q-Brownian motion. So by applying these results with b = (r − 1 2 2 σ )/σ we can now evaluate the price of any option maturing at time T whose payoff depends just on the stock price at time T and its minimum (or maximum) value over the lifetime of the contract. If the payoff is C T = g(S∗ (T ), ST ) and r is the riskless borrowing rate then the value of the option at time zero is V (0, S0 ) = e−r T EQ [g (S∗ (T ), ST )]
0 a=−∞ ∞ x=a

= e−r T

g S0 eσ x , S0 eσ a Q [Y∗ (T ) ∈ da, YT ∈ d x] . Find the time zero price of a down-and-

Example 6.3.5 (Down-and-in call option)

in call option whose payoff at time T is C T = 1{S∗ (T )≤c} (ST − K )+ where c is a (positive) preagreed constant less than K . Solution: Using St = S0 exp(σ Yt ) we rewrite the payoff as C T = 1{Y∗ (T )≤ 1 log(c/S0 )} S0 eσ YT − K σ +

.

Writing b = (r − 1 σ 2 )/σ , a = 2 V (0, S0 ) = e−r T

1 σ

log(c/S0 ) and x0 =

1 σ

log(K /S0 ) we obtain

∞ x0

S0 eσ x − K Q (Y∗ (T ) ≤ a, YT ∈ d x) .

148

different payoffs

Using the expression for the joint distribution of (Y∗ (T ), YT ) obtained above yields V (0, S0 ) = e−r T
∞ x0

S0 eσ x − K e2ab pT (2a + bT, x)d x.

We have used the fact that, since c < K , x0 ≥ a. First observe that e−r T
∞ x0

K e2ab pT (2a + bT, x)d x = K e−r T e2ab = K e−r T e2ab = K e−r T = K e−r T c S0 c S0

∞ √

(x0 −2a−bT )/ T √ (2a+bT −x0 )/ T −∞ 2r 2 −1 σ 1 2 √ e−y /2 dy 2π 1 2 √ e−y /2 dy 2π

2a + bT − x0 √ T log(F/K ) − 1 σ 2 T √ 2 σ T

2r −1 σ2

where F = er T c2 /S0 . Similarly, e−r T
∞ x0

S0 eσ x e2ab pT (2a + bT, x)d x = = S0 e−r T e2ab S0 e−r T e2ab × exp = e−r T 1 (x − (2a + bT ))2 − 2σ x T exp − √ 2T 2π T x0 1 −y 2 /2 e dy √ √ 2π (x0 −(2a+bT )−σ T )/ T
∞

dx

1 2 σ T + 2aσ + bσ T 2 c S0
2r −1 σ2

F

log(F/K ) + 1 σ 2 T √ 2 σ T

.

Comparing this with Example 5.2.2 V (0, S0 ) = c S0
2r −1 σ2

C

c2 , 0; K , T S0

,

where C(x, t; K , T ) is the price at time t of a European call option with strike K and maturity T if the stock price at time t is x. ✷ The price of a barrier option can also be expressed as the solution of a partial differential equation.
Example 6.3.6 (Down-and-out call)

A down-and-out call has the same payoff as a European call option, (ST − K )+ , unless during the lifetime of the contract the price of the underlying asset has fallen below some preagreed barrier, c, in which case the option is ‘knocked out’ worthless.

149

6.4 asian options

Writing V (t, x) for the value of such an option at time t if St = x and assuming that K > c, V (t, x) solves the Black–Scholes equation for (t, x) ∈ [0, T ] × [c, ∞) subject to the boundary conditions V (T, ST ) = (ST − K )+ , V (t, c) = 0, t ∈ [0, T ], V (t, x) → 1, as x → ∞. x The last boundary condition follows since as St → ∞, the probability of the asset price hitting level c before time T tends to zero. Exercise 16 provides a method for solving the Black–Scholes partial differential equation with these boundary conditions. Of course more and more complicated barrier options can be dreamt up. For example, a double knock-out option is worthless if the stock price leaves some interval [c1 , c2 ] during the lifetime of the contract. The probabilistic pricing formula for such a contract then requires the joint distribution of the triple (ST , S∗ (T ), S ∗ (T )). As in the case of a single barrier, the trick is to use Girsanov’s Theorem to deduce the joint distribution from that of (WT , m T , MT ) where {Wt }t≥0 is a P-Brownian motion and {m t }t≥0 , {Mt }t≥0 are its running minimum and maximum respectively. This in turn is given by P [WT ∈ dy, a < m T , MT < b] = n∈Z pT (2n(a−b), y)− p(2n(b−a), y−2a) dy;

see Freedman (1971) for a proof. An explicit pricing formula will then be in the form of an inﬁnite sum. In Exercise 20 you obtain the pricing formula by directly solving the Black–Scholes differential equation.
Probability or pde?

As we have seen in Exercise 7 of Chapter 5 and we see again in the exercises at the end of this chapter, the Black–Scholes partial differential equation can be solved by ﬁrst transforming it to the heat equation (with appropriate boundary conditions). This is entirely parallel to our probabilistic technique of transforming the expectation price to an expectation of a function of Brownian motion.

6.4

Asian options
The payoff of an Asian option is a function of the average of the asset price over the lifetime of the contract. For example, the payoff of an Asian call with strike price K and maturity time T is CT = 1 T
T 0

St dt − K

+

.

Evidently C T ∈ FT and so our Black–Scholes analysis of Chapter 5 gives the value of such an option at time zero as V0 = EQ e−r T 1 T
T 0

St dt − K

+

.

(6.6)

150

different payoffs

However, evaluation of this integral is a highly non-trivial matter and we do not obtain the nice explicit formulae of the previous sections. There are many variants on this theme. For example, we might want to value a claim with payoff 1 T St dt . C T = f ST , T 0 In §7.2 we shall develop the technology to express the price of such claims (and indeed slightly more complex claims) as solutions to a multidimensional version of the Black–Scholes equation. Moreover (see Exercise 12 of Chapter 7) one can also ﬁnd an explicit expression for the hedging portfolio in terms of the solution to this equation. However, multidimensional versions of the Black–Scholes equation are much harder to solve than their one-dimensional counterpart and generally one must resort to numerical techniques. The main difﬁculty with evaluating (6.6) directly is that, although there are 1 T explicit formulae for all the moments of the average process T 0 St dt, in contrast to the lognormal distribution of ST , we do not have an expression for the distribution function. A number of approaches have been suggested to overcome this, including simply approximating the distribution of the average process by a lognormal distribution with suitably chosen parameters. A very natural approach is to replace the continuous average by a discrete analogue obtained by sampling the price of the process at agreed times t1 , . . . , tn and averaging the result. This also makes sense from a practical point of view as calculating the continuous average for a real asset can be a difﬁcult process. Many contracts actually specify that the average be calculated from such a discrete sample – for example from daily closing prices. Mathematically, the continuous average n 1 T 1 i=1 Sti . Options based on a discrete sample can be T 0 St dt is replaced by n treated in the same way as multistage options, although evaluation of the price rapidly becomes impractical (see Exercise 21). A further approximation is to replace the arithmetic average by a geometric 1/n n n 1 . This quantity average. That is, in place of n i=1 Sti we consider i=1 Sti has a lognormal distribution (Exercise 22) and so the corresponding approximate pricing formula for the Asian option can be evaluated exactly. (You are asked to ﬁnd the pricing formula for an Asian call option based on a continuous version of the geometric average in Exercise 23.) Of course the arithmetic mean of a collection of positive numbers always dominates their geometric mean and so it is no surprise that this approximation consistently under-prices the Asian call option.

6.5

American options
A full treatment of American options is beyond our scope here. Explicit formulae for the prices of American options only exist in a few special cases and so one must employ numerical techniques. One approach is to use our discrete (binomial tree) models of Chapter 2. An alternative is to reformulate the price as a solution to a

151

6.5 american options

partial differential equation. We do not give a rigorous derivation of this equation, but instead we use the results of Chapter 2 to give a heuristic explanation of its form.
The discrete case

As we saw in Chapter 2, the price of an American call option on non-dividend-paying stock is the same as that of a European call and so we concentrate on the American put. This option gives the holder the right to buy one unit of stock for price K at any time before the maturity time T . As we illustrated in §2.2, in our discrete time model, if V (n, Sn ) is the value of the option at time nδt given that the asset price at time nδt is Sn then V (n, Sn ) = max (K − Sn )+ , EQ e−r δt V (n + 1, Sn+1 ) Fn ,

where Q is the martingale measure. In particular, V (n, Sn ) ≥ (K − Sn )+ everywhere. We saw that for each ﬁxed n the possible values of Sn are separated into two ranges by a boundary value that we shall denote by S f (n): if Sn > S f (n) then it is optimal to hold the option whereas if Sn ≤ S f (n) it is optimal to exercise. We call {S f (n)}0≤n≤N the exercise boundary. In Example 2.4.7 we found a characterisation of the exercise boundary. We ˜ ˜ ˜ showed that the discounted option price can be written as Vn = Mn − An where ˜ ˜ n }0≤n≤N is a Q-martingale and { An }0≤n≤N is a non-decreasing predictable {M ˜ process. The option is exercised at the ﬁrst time nδt when An+1 = 0. In summary, ˜ within the exercise region An+1 = 0 and Vn = (K − Sn )+ , whereas away from the exercise region, that is when Sn > S f (n), V (n, Sn ) = Mn . ˜ The strategy of exercising the option at the ﬁrst time when An+1 = 0 is optimal in the sense that if we write T N for the set of all possible stopping times taking values in {0, 1, . . . , N } then V (0, S0 ) = sup EQ e−r τ (K − Sτ )+ F0 . τ ∈T N

Since the exercise time of any permissible strategy must be a stopping time, this says that as holder of the option one can’t do better by choosing any other exercise strategy. That this optimality characterises the fair price follows from a now familiar arbitrage argument that you are asked to provide in Exercise 24.
Continuous time

Now suppose that we formally pass to the continuous limit as in §2.6. We expect that in the limit too V (t, St ) ≥ (K − St )+ everywhere and that for each t we can deﬁne S f (t) so that if St > S f (t) it is optimal to hold on to the option, whereas if St ≤ S f (t) it is optimal to exercise. In the exercise region V (t, St ) = (K − St )+ whereas away from the exercise region V (t, St ) = Mt where the discounted process ˜ { Mt }0≤t≤T is a Q-martingale and Q is the measure, equivalent to P, under which ˜ the discounted stock price is a martingale. Since { Mt }0≤t≤T can be thought of as the discounted value of a European option, this tells us that away from the exercise region, V (t, x) must satisfy the Black–Scholes differential equation. We guess then that for {(t, x) : x > S f (t)} the price V (t, x) must satisfy the Black–Scholes equation whereas outside this region V (t, x) = (K − x)+ . This

152

different payoffs

can be extended to a characterisation of V (t, x) if we specify appropriate boundary conditions on S f . This is complicated by the fact that S f (t) is a free boundary – we don’t know its location a priori. An arbitrage argument (Exercise 25) says that the price of an American put option should be continuous. We have checked already that V (t, S f (t)) = (K − S f (t))+ . Since it is clearly not optimal to exercise at a time t < T if the value of the option is zero, in fact we have V (t, S f (t)) = K − S f (t). Let us suppose now that V (t, x) is continuously differentiable with respect to x as we cross the exercise boundary (we shall omit the proof of this). Then, since V (t, x) = (K − x) for x ≤ S f and V (t, x) ≥ (K − x) for x > S f , we must have that at the exercise boundary ∂ V ≥ −1. Suppose that ∂ V > −1 at ∂x ∂x some point of the exercise boundary. Then by reducing the value of the stock price at which we choose to exercise from S f to S ∗ we can actually increase the value of the f option at (t, S f (t)). This contradicts the optimality of our exercise strategy. It must be that ∂ V = −1 at the exercise boundary. ∂x We can now fully characterise V (t, x) as a solution to a free boundary value problem: We write V (t, x) for the value of an American put option with strike price K and maturity time T and r for the riskless borrowing rate. V (t, x) can be characterised as follows. For each time t ∈ [0, T ] there is a number S f (t) ∈ (0, ∞) such that for 0 ≤ x ≤ S f (t) and 0 ≤ t ≤ T ,
Proposition 6.5.1 (The value of an American put)

V (t, x) = K − x

and

1 ∂2V ∂V ∂V + σ 2x2 2 + r x − r V < 0. ∂t 2 ∂x ∂x

For t ∈ [0, T ] and S f (t) < x < ∞ V (t, x) > (K − x)+ and 1 ∂2V ∂V ∂V + σ 2x2 2 + r x − r V = 0. ∂t 2 ∂x ∂x

The boundary conditions at x = S f (t) are that the option price process is continuously differentiable with respect to x, is continuous in time and V (t, S f (t)) = (K − S f (t))+ , In addition, V satisﬁes the terminal condition V (T, ST ) = (K − ST )+ . The free boundary problem of Proposition 6.5.1 is easier to analyse as a linear complementarity problem. If we use the notation LB S f = ∂2 f ∂f ∂f 1 + σ 2x2 2 + r x − r f, ∂t 2 ∂x ∂x ∂V (t, S f (t)) = −1. ∂x

153

6.5 american options

then the free boundary value problem can be restated as L B S V (t, x) (V (t, x) − (K − x)+ ) = 0, subject to L B S V (t, x) ≤ 0, V (t, x)−(K −x)+ ≥ 0, V (T, x) = (K −x)+ , V (t, x) → ∞ as x → ∞ and V (t, x), ∂ V (t, x) are continuous. ∂x Notice that this reformulation has removed explicit dependence on the free boundary. Variational techniques can be applied to solve the problem and then the boundary is recovered from that solution. This is beyond our scope here. See Wilmott, Howison & Dewynne (1995) for more detail.
An explicit solution

We ﬁnish this chapter with one of the rare examples of an American option for which the price can be obtained explicitly.
Example 6.5.2 (Perpetual American put)

Find the value of a perpetual American put option on non-dividend-paying stock, that is a contract that the holder can choose to exercise at any time t in which case the payoff is (K − St )+ .

Solution(s): We sketch two possible solutions to this problem, ﬁrst via the free boundary problem of Proposition 6.5.1 and second via the expectation price. Since the time to expiry of the contract is always inﬁnite, V (t, x) is a function of x alone and the exercise boundary must be of the form S f (t) = α for all t > 0 and some constant α. The option will be exercised as soon as St ≤ α. The Black–Scholes equation reduces to an ordinary differential equation: 1 2 2 d2V dV σ x − r V = 0, + rx 2 dx dx2 for all x ∈ (α, ∞). (6.7)

The general solution to equation (6.7) is of the form v(x) = c1 x d1 + c2 x d2 for some constants c1 , c2 , d1 and d2 . Fitting the boundary conditions V (α) = K − α, gives V (x) = where α= (K − α) (K − x),
−2 α 2r σ , x

lim x↓α dV = −1 dx

and lim V (x) = 0 x→∞ x ∈ (α, ∞), x ∈ [0, α],

2r σ −2 K . 2r σ −2 + 1

An alternative approach to this problem would be to apply the results of §3.3. As we argued above, the option will be exercised when the stock price ﬁrst hits level α for some α > 0. This means that the value will be of the form V (0, S0 ) = EQ e−r τα (K − α)+ ,

154

different payoffs

where τα = inf{t > 0 : St ≤ α}. We rewrite this stopping time in terms of the time that it takes a Q-Brownian motion to hit a sloping line. Since St = S0 exp 1 r − σ 2 t + σ Xt 2

where {X t }t≥0 is a standard Brownian motion under the martingale measure Q, the event {St ≤ α} is the same as the event 1 −σ X t − r − σ 2 t ≥ log 2 S0 α .

The process {−X t }t≥0 is also a standard Q-Brownian motion and so, in the notation of §3.3, the time τα is given by Ta,b with a= 1 log σ S0 α , b= r − 1σ2 2 . σ

We can then read off EQ e−r τα from Proposition 3.3.5 and maximise over α to yield the result. ✷

Exercises
1 Let K 1 and K 2 be ﬁxed real numbers with 0 < K 1 < K 2 . A collar option has payoff C T = min{max{ST , K 1 }, K 2 }. Find the Black–Scholes price for such an option. 2 What is the maximum potential loss associated with taking the long position in a forward contract? And with taking the short position? Consider the derivative whose payoff at expiry to the holder of the long position is C T = min{ST , F} − K , where F is the standard forward price for the underlying stock and K is a constant. Such a contract is constructed so as to have zero value at the time at which it is struck. Find an expression for the value of K that should be written into such a contract. What is the maximum potential loss for the holder of the long or short position now? 3 The digital put option with strike K at time T has payoff CT = 0, 1, ST ≥ K , ST < K .

Find the Black–Scholes price for a digital put. What is the put–call parity for digital options?

155

exercises

4 Digital call option In Example 6.1.1 we calculated the price of a digital call. Here is an alternative approach: (a) Use the Feynman–Kac stochastic representation to ﬁnd the partial differential equation satisﬁed by the value of a digital call with strike K and maturity T . (b) Show that the delta of a standard European call option solves the partial differential equation that you have found in (a). (c) Hence or otherwise solve the equation in (a) to ﬁnd the value of the digital. 5 An asset-or-nothing call option with strike K and maturity T has payoff CT = ST , 0, ST ≥ K , ST < K .

Find the Black–Scholes price and hedge for such an option. What happens to the stock holding in the replicating portfolio if the asset price is near K at times close to T ? Comment. 6 Construct a portfolio consisting entirely of cash-or-nothing and asset-or-nothing options whose value at time T is exactly that of a European call option with strike K at maturity T . 7 In §6.1 we have seen that for certain options with discontinuous payoffs at maturity, the stock holding in the replicating portfolio can oscillate wildly close to maturity. Do you see this phenomenon if the payoff is continuous? 8 Pay-later option This option, also known as a contingent premium option, is a standard European option except that the buyer pays the premium only at maturity of the option and then only if the option is in the money. The premium is chosen so that the value of the option at time zero is zero. This option is equivalent to a portfolio consisting of one standard European call option with strike K and maturity T and −V digital call options with maturity T where V is the premium for the option. (a) What is the value of holding such a portfolio at time zero? (b) Find an expression for V . (c) If a speculator enters such a contract, what does this suggest about her market view? 9 Ratchet option A two-leg ratchet call option can be described as follows. At time zero an initial strike price K is set. At time T0 > 0 the strike is reset to ST0 , the value of the underlying at time T0 . At the maturity time T1 > T0 the holder receives the payoff of the call with strike ST0 plus ST1 − ST0 if this is positive. That is, the payoff is (ST1 − ST0 )+ + (ST0 − K )+ . If (ST0 − K ) is positive, then the intermediate proﬁt (ST0 − K )+ is said to be ‘locked in’. Why? Value this option for 0 < t < T1 .

156

different payoffs

10 Chooser option A chooser option is speciﬁed by two strike prices, K 0 and K 1 , and two maturity dates, T0 < T1 . At time T0 the holder has the right to buy, for price K 0 , either a call or a put with strike K 1 and maturity T1 . What is the value of the option at time T0 ? In the special case K 0 = 0 use put– call parity to express this as the sum of the value of a call and a put with suitably chosen strike prices and maturity dates and hence ﬁnd the value of the option at time zero. 11 Options on futures In our simple model where the riskless rate of borrowing is deterministic, forward and futures prices coincide. A European call option with strike price K and maturity T0 written on an underlying futures contract with delivery date T1 > T0 delivers to the holder, at time T0 , a long position in the futures contract and an amount of money (F(T0 , T1 ) − K )+ , where F(T0 , T1 ) is the value of the futures contract at time T0 . Find the value of such an option at time zero. 12 Use the method of Example 6.2.3 to ﬁnd the value of a put-on-put option. By considering the portfolio obtained by buying one call-on-put and selling one put-on-put (with the same strikes and maturities) obtain a put–call parity relation for compound options. Hence write down prices for all four classes of compound option. 13 Let {Yt }t≥0 be given by Yt = bt + X t where b is a constant and {X t }t≥0 is a QBrownian motion. Writing Y ∗ (t) = max{Yu : 0 ≤ u ≤ t}, ﬁnd the joint distribution of (YT , Y ∗ (T )) under Q. 14 What is the value of a portfolio consisting of one down-and-in call and one downand-out call with the same strike price and maturity? 15 Find the value of a down-and-out call with barrier c and strike K at maturity T if c > K. 16 One approach to ﬁnding the value of the down-and-out call of Example 6.3.6 is to express it as an expectation under the martingale measure and exploit our knowledge of the joint distribution of Brownian motion and its minimum. Alternatively one can solve the partial differential equation directly and that is the purpose of this exercise. (a) Use the method of Exercise 7 of Chapter 5 to transform the equation for the price into the heat equation. What are the boundary conditions for this heat equation? (b) Solve the heat equation that you have obtained using, for example, the ‘method of images’. (If you are unfamiliar with this technique, then try Wilmott, Howison & Dewynne (1995).) (c) Undo the transformation to obtain the solution to the partial differential equation.

157

exercises

17 An American cash-or-nothing call option can be exercised at any time t ∈ [0, T ]. If exercised at time t its payoff is 1 if St ≥ K , 0 if St < K . When will such an option be exercised? Find its value. 18 Suppose that the down-and-in call option of Example 6.3.5 is modiﬁed so that if the option is never activated, that is the stock price never crosses the barrier, then the holder receives a rebate of Z . Find the price of this modiﬁed option. 19 A perpetual option is one with no expiry time. For example, a perpetual American cash-or-nothing call option can be exercised at any time. If exercised at time t, its payoff is 1 if St ≥ K and 0 if St < K . What is the probability that such an option is never exercised? 20 Formulate the price of a double knock-out call option as a solution to a partial differential equation with suitably chosen boundary conditions. Mimic your approach in Exercise 16 to see that this too leads to an expression for the price as an inﬁnite sum. Calculate the value of an Asian call option, with strike price K , in which the average of the stock price is calculated on the basis of just two sampling times, 0 and T , where T is the maturity time of the contract. Find an expression for the value of the corresponding contract when there are three sampling times, 0, T /2 and T .

21

22 Suppose that {St }t≥0 is a geometric Brownian motion under P. Let 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn be ﬁxed times and deﬁne n 1/n

Gn = i=1 Sti

.

Show that G n has a lognormal distribution under P. 23 An asset price {St }t≥0 is a geometric Brownian motion under the market measure P. Deﬁne 1 T YT = exp log St dt . T 0 Suppose that an Asian call option has payoff (YT − K )+ at time T . Find an explicit formula for the price of such an option at time zero. 24 Use an arbitrage argument to show that if V (0, S0 ) is the fair price of an American put option on non-dividend-paying stock with strike price K and maturity T , then writing TT for the set of all possible stopping times taking values in [0, T ] V (0, S0 ) = sup EQ e−r τ (K − Sτ )+ F0 . τ ∈TT

158

different payoffs

25

Consider the value of an American put on non-dividend-paying stock. Show that if there were a discontinuity in the option value (as a function of stock price) that persisted for more than an inﬁnitesimal time then a portfolio consisting entirely of options would offer an arbitrage opportunity. Remark: This does not mean that all option prices are continuous. If there is an instantaneous change in the conditions of a contract (as in multistage options) then discontinuities certainly can occur. Find the value of a perpetual American call option on non-dividend-paying stock.

26

7 Bigger models

Summary
Having applied our basic Black–Scholes model to the pricing of some exotic options, we now turn to more general market models. In §7.1 we replace the (constant) parameters that characterised our basic Black– Scholes model by previsible processes. Under appropriate boundedness assumptions, we then repeat our analysis of Chapter 5 to obtain the fair price of an option as the discounted expected value of the claim under a martingale measure. In general this expectation must be evaluated numerically. We also make the connection with a generalised Black–Scholes equation via the Feynman–Kac Stochastic Representation Theorem. Our models so far have assumed that the market consists of a single stock and a riskless cash bond. More complex equity products can depend on the behaviour of several separate securities and, in general, the prices of these securities will not evolve independently. In §7.2 we extend some of the fundamental results of Chapter 4 to allow us to manipulate systems of stochastic differential equations driven by correlated Brownian motions. For markets consisting of many assets we have much more freedom in our choice of ‘reference asset’ or numeraire and so we revisit this issue before illustrating the application of the ‘multifactor’ theory by pricing a ‘quanto’ product. We still have no satisfactory justiﬁcation for the geometric Brownian motion model. Indeed, there is considerable evidence that it does not capture all features of stock price evolution. A ﬁrst objection is that stock prices occasionally ‘jump’ at unpredictable times. In §7.3 we introduce a Poisson process of jumps into our Black–Scholes model and investigate the implications for option pricing. This approach is popular in the analysis of credit risk. In §1.5 we saw that, if a model is to be free from arbitrage and complete, there must be a balance between the number of sources of randomness and the number of independent stocks. We reiterate this here. We see more evidence that the Black–Scholes model does not reﬂect the true behaviour of the market in §7.4. It seems a little late in the day to condemn the model that has been the subject of all our efforts so far and so we ask how much it matters
159

160

bigger models

if we use the wrong model. We also very brieﬂy discuss models with stochastic volatility that have the potential to better reﬂect true market behaviour. This chapter is intended to do no more than indicate some of the topics that might be addressed in a second course in ﬁnancial calculus. Much more detail can be found in some of the suggestions for further reading in the bibliography.

7.1

General stock model
In our classical Black–Scholes framework we assume that the riskless borrowing rate is constant and that the returns of the stock follow a Brownian motion with constant drift. In this section we consider much more general models to which we can apply the Black–Scholes analysis although, in practice, even for vanilla options the prices that we obtain must now be evaluated numerically. The key assumption that we retain is that there is only one source of randomness in the market, the Brownian motion that drives the stock price (cf. §7.3).

The model

Writing {Ft }t≥0 for the ﬁltration generating the driving Brownian motion, we replace the riskless borrowing rate, r , the drift µ and the volatility σ in our basic Black–Scholes model by {Ft }t≥0 -predictable processes {rt }t≥0 , {µt }t≥0 and {σt }t≥0 . In particular, rt , µt and σt can depend on the whole history of the market before time t. Our market model is then as follows.

General stock model:

The market consists of a riskless cash bond, {Bt }t≥0 , and a single risky asset with price process {St }t≥0 governed by d Bt = rt Bt dt, B0 = 1,

d St = µt St dt + σt St d Wt , where {Wt }t≥0 is a P-Brownian motion generating the ﬁltration {Ft }t≥0 and {rt }t≥0 , {µt }t≥0 and {σt }t≥0 are {Ft }t≥0 -predictable processes.

Evidently a solution to these equations should take the form Bt = exp
0 t 0 t 0 t

rs ds ,

(7.1)

St = S0 exp

1 µs − σs2 ds + 2

σs d Ws ,

(7.2)

but we need to make some boundedness assumptions if these expressions are to make sense. So to ensure the existence of the integrals in equations (7.1) and (7.2) we T T T assume that 0 |rt |dt, 0 |µt |dt and 0 σt2 dt are all ﬁnite with P-probability one.

161

7.1 general stock model

A word of warning is in order. In order to ‘calibrate’ such a model to the market we must choose the parameters {rt }t≥0 , {µ}t≥0 and {σt }t≥0 from an inﬁnite-dimensional space. Unless we restrict the possible forms of these processes, this presents a major obstacle to implementation. In §7.4 we examine the effect of model misspeciﬁcation on pricing and hedging strategies. Now, however, we set this worry aside and repeat the Black–Scholes analysis for our general class of market models.
A martingale measure

We must mimic the three steps to replication that we followed in the classical setting. The ﬁrst of these is to ﬁnd an equivalent probability measure, Q, under which the ˜ discounted stock price, { St }t≥0 , is a martingale. Exactly as before, we use the Girsanov Theorem to ﬁnd a measure, Q, under ˜ which the process {Wt }t≥0 deﬁned by ˜ Wt = Wt +
0 t

γs ds

˜ ˜ is a standard Brownian motion. The discounted stock price, { St }t≥0 deﬁned as St = St /Bt , is governed by the stochastic differential equation ˜ d St = = ˜ ˜ (µt − rt ) St dt + σt St d Wt ˜ ˜ ˜ (µt − rt − σt γt ) St dt + σt St d Wt ,

˜ and so we choose γt = (µt − rt )/σt . To ensure that { St }t≥0 really is a Q-martingale we make two further boundedness assumptions. First, in order to apply the Girsanov Theorem, we insist that EP exp
0 T

1 2 γ dt 2 t

< ∞.

˜ Second we require that { St }t≥0 is a Q-martingale (not just a local martingale) and so we assume a second Novikov condition: EQ exp
0 T

1 2 σ dt 2 t

< ∞.

˜ Under these extra boundedness assumptions { St }t≥0 then is a martingale under the measure Q deﬁned by dQ (γ ) = L t = exp − dP Ft
Second step to replication t 0

γs d Ws −
0

t

1 2 γ ds . 2 s

That completes the ﬁrst step in our replication strategy. The second is to form the (Q, {Ft }t≥0 )-martingale {Mt }t≥0 given by
−1 Mt = EQ BT C T Ft .

162 Replicating a claim

bigger models

The third step is to show that our market is complete, that is any claim C T can be replicated. First we invoke the martingale representation theorem to write Mt = M0 +
0 t

˜ θu d Wu

and consequently, provided that σt never vanishes, Mt = M0 +
0 t

˜ φs d Ss ,

where {φt }t≥0 is {Ft }t≥0 -predictable. Guided by our previous work we guess that a replicating portfolio should consist of φt units of stock and ψt = Mt − φt St units of cash bond at time t. In Exercise 1 it is checked that such a portfolio is self-ﬁnancing. Its value at time t is Vt = φt St + ψt Bt = Bt Mt . In particular, at time T , VT = BT MT = C T , and so we have a self-ﬁnancing, replicating portfolio. The usual arbitrage argument tells us that the fair value of the claim at time t is Vt , that is the arbitrage price of the option at time t is
−1 Vt = Bt EQ BT C T Ft = EQ e−
T t ru du

C T Ft .

The generalised Black– Scholes equation

In general such an expectation must be evaluated numerically. If rt , µt and σt depend only on (t, St ) then one approach to this is ﬁrst to express the price as the solution to a generalised Black–Scholes partial differential equation. This is achieved with the Feynman–Kac Stochastic Representation Theorem. Speciﬁcally, using Example 4.8.6, Vt = F(t, St ) where F(t, x) solves 1 ∂F ∂2 F ∂F (t, x) + σ 2 (t, x)x 2 2 (t, x) + r (t, x)x (t, x) − r (t, x)F(t, x) = 0, ∂t 2 ∂x ∂x subject to the terminal condition corresponding to the claim C T , at least provided
T 0

EQ

σ (t, x)

∂F (t, x) ∂x

2

ds < ∞.

For vanilla options, in the special case when r , µ and σ are functions of t alone, the partial differential equation can be solved explicitly. As is shown in Exercise 3 the procedure is exactly that used to solve the usual Black–Scholes equation. The price can be found from the classical Black–Scholes price via the following simple rule: for the value of the option at time t replace r and σ 2 by 1 T −t respectively.
T t

r (s)ds

and

1 T −t

T t

σ 2 (s)ds

163

7.2 multiple stock models

7.2

Multiple stock models
So far we have assumed that the market consists of a riskless cash bond and a single ‘risky’ asset. However, the need to model whole portfolios of options or more complex equity products leads us to seek models describing several securities simultaneously. Such models must encode the interdependence between different security prices.

Correlated security prices

Suppose that we are modelling the evolution of n risky assets and, as ever, a single risk-free cash bond. We assume that it is not possible to exactly replicate one of the assets by a portfolio composed entirely of the others. In the most natural extension of the classical Black–Scholes model, considered individually the price of each risky asset follows a geometric Brownian motion, and interdependence of different asset prices is achieved by taking the driving Brownian motions to be correlated. Equivalently, we take a set of n independent Brownian motions and drive the asset prices by linear combinations of these; see Exercise 2. This suggests the following market model.

Multiple asset model:

Our market consists of a cash bond {Bt }0≤t≤T and n different securities with prices {St1 , St2 , . . . , Stn }0≤t≤T , governed by the system of stochastic differential equations d Bt = r Bt dt, n d Sti = Sti j=1 j

σi j (t)d Wt + µi (t)dt ,

j

i = 1, 2, . . . , n,

(7.3)

where {Wt }t≥0 , j = 1, . . . , n, are independent Brownian motions. We assume that the matrix σ = (σi j ) is invertible.

Remarks:

1 This model is called an n-factor model as there are n sources of randomness. If there are fewer sources of randomness than stocks then there is redundancy in the model as we can replicate one of the stocks by a portfolio composed of the others. On the other hand, if we are to be able to hedge any claim in the market, then, roughly speaking, we need as many ‘independent’ stocks as sources of randomness. This mirrors Proposition 1.6.5. 2 Notice that the volatility of each stock in this model is really a vector. Since the j Brownian motions {Wt }t≥0 , j = 1, . . . , n, are independent, the total volatility of n 2 the process {Sti }t≥0 is . ✷ j=1 σi j (t) t≥0 164

bigger models

Of course we haven’t checked that this model really makes sense. That is, we need to know that the system of stochastic differential equations (7.3) has a solution. In order to verify this and to analyse such multifactor market models we need multidimensional analogues of the key results of Chapter 4.
Multifactor Itˆ formula o

The most basic tool will be an n-factor version of the Itˆ formula. In the same way as o we used the one-factor Itˆ formula to ﬁnd a description (in the form of a stochastic o differential equation) of models constructed as functions of Brownian motion, here we shall build new multifactor models from old. Our basic building blocks will be solutions to systems of stochastic differential equations of the form n d X ti = µi (t)dt + j=1 j

σi j (t)d Wt ,

j

i = 1, . . . , n,

(7.4)

where {Wt }t≥0 , j = 1, . . . , n, are independent Brownian motions. We write {Ft }t≥0 j for the σ -algebra generated by {Wt }t≥0 , j = 1, . . . , n. Our work of Chapter 4 gives a rigorous meaning to (the integrated version of) the system (7.4) provided {µi (t)}t≥0 and {σi j (t)}t≥0 , 1 ≤ i ≤ n, 1 ≤ j ≤ n, are {Ft }t≥0 -predictable processes with E
0 t n

σi j (s) j=1 2

+ |µi (s)| ds < ∞,

t > 0, i = 1, . . . , n.

Let us write {X t }t≥0 for the vector of processes {X t1 , X t2 , . . . , X tn }t≥0 and deﬁne a new stochastic process by Z t = f (t, X t ). Here we suppose that f (t, x) : R+ ×Rn → R is sufﬁciently smooth that we can apply Taylor’s Theorem, just as in §4.3, to ﬁnd the stochastic differential equation governing {Z t }t≥0 . Writing x = (x1 , . . . , xn ), we obtain d Zt = n ∂f ∂2 f ∂f 1 n j (t, X t )dt + (t, X t )d X ti + (t, X t )d X ti d X t + · · · . ∂t ∂ xi 2 i, j=1 ∂ xi ∂ x j i=1

(7.5) Since the Brownian motions {Wti }t≥0 are independent we have the multiplication table × d Wti j d Wt dt j d Wti dt 0 0

d Wt 0 dt 0

j

dt 0 0 0 for i = j (7.6)

n and this gives d X ti d X t = k=1 σik σ jk dt. The same multiplication table tells us j i d X d X k is o(dt) and so substituting into equation (7.5) we have provided that d X t t t a heuristic justiﬁcation of the following result.

165

7.2 multiple stock models Theorem 7.2.1 (Multifactor Itˆ formula) o

Let {X t }t≥0 = {X t1 , X t2 , . . . , X tn }t≥0 j solve d X ti = µi (t)dt +

n j=1

σi j (t)d Wt ,

i = 1, 2, . . . , n,

where {Wti }t≥0 , i = 1, . . . , n, are independent P-Brownian motions. Further suppose that the real-valued function f (t, x) on R+ × Rn is continuously differentiable with respect to t and twice continuously differentiable in the x-variables. Then deﬁning Z t = f (t, X t ) we have d Zt = n ∂f ∂2 f ∂f 1 n (t, X t )dt + (t, X t )d X ti + (t, X t )Ci j (t)dt ∂t ∂ xi 2 i, j=1 ∂ xi ∂ x j i=1 n k=1 σik (t)σ jk (t).

where Ci j (t) =

Remark: Notice that if we write σ for the matrix (σi j ) then Ci j = σ σ t i j where σ t

is the transpose of σ .

✷

We can now check that there is a solution to the system of equations (7.3).
Example 7.2.2 (Multiple asset model)

Brownian motions. Deﬁne i Sti = S0 exp t 0

{St1 , St2 , . . . µi (s) − 1 2

Let {Wti }t≥0 , i = 1, . . . , n, be independent , Stn }t≥0 by n 2 σik (s) ds + t n

σi j (s)d Ws

j

;

k=1

0 j=1

then {St1 , St2 , . . . , Stn }t≥0 solves the system (7.3). i Justiﬁcation: Deﬁning the processes {X t }t≥0 for i = 1, 2, . . . , n by

d X ti = µi (t) −

1 2

n 2 σik (t) dt + k=1

n

σi j (t)d Wt j=1 j

we see that Sti = f i (t, X t ) where, writing x = (x1 , . . . , xn ), f i (t, x) Applying Theorem 7.2.1 gives i d Sti = S0 exp(X ti )d X ti +

i S0 e xi .

1 i S exp(X ti )Cii (t)dt 2 0 n 2 σik (t) dt + j=1 j

= Sti

µi (t) −

1 2

n k=1 n

σi j (t)d Wt +

j

1 2

n

σik (t)σik (t)dt k=1 = Sti µi (t)dt + j=1 σi j (t)d Wt

as required.

✷

166

bigger models Remark: Exactly as in the single factor models, although we can write down

arbitrarily complicated systems of stochastic differential equations, existence and uniqueness of solutions are far from guaranteed. If the coefﬁcients are bounded and uniformly Lipschitz then a unique solution does exist, but such results are beyond our scope here. Instead, once again, we refer to Chung & Williams (1990) or Ikeda & Watanabe (1989). ✷
Integration by parts

We can also use the multiplication table (7.6) to write down an n-factor version of the integration by parts formula.
Lemma 7.2.3

If n d X t = µ(t, X t )dt + i=1 σi (t, X t )d Wti

and dYt = ν(t, Yt )dt +

n

ρi (t, Yt )d Wti i=1 n

then d(X t Yt ) = X t dYt + Yt d X t +

σi (t, X t )ρi (t, Yt )dt. i=1 Change of measure

Pricing and hedging in the multiple stock model will follow a familiar pattern. First we ﬁnd an equivalent probability measure under which all of the discounted stock ˜ ˜ prices { Sti }t≥0 , i = 1, . . . , n, given by Sti = e−r t Sti , are martingales. We then use a multifactor version of the Martingale Representation Theorem to construct a replicating portfolio. Construction of the martingale measure is, of course, via a multifactor version of the Girsanov Theorem. Let {Wti }t≥0 , i = 1, . . . , n, be independent Brownian motions under the measure P generating the ﬁltration {Ft }t≥0 and let {θi (t)}t≥0 , i = 1, . . . , n, be {Ft }t≥0 -previsible processes such that
Theorem 7.2.4 (Multifactor Girsanov Theorem)

EP exp Deﬁne n 1 2

T 0

n

θi2 (s)ds i=1 ds < ∞.

(7.7)

L t = exp − i=1 0

T

θi (s)d Wsi +

1 2

T 0

θi2 (s)ds

and let

P(L)

be the probability measure deﬁned by dP(L) dP = Lt .
Ft

167

7.2 multiple stock models

Then under P(L) the processes {X ti }t≥0 , i = 1, . . . , n, deﬁned by X ti = Wti + are all martingales.
Sketch of proof: The proof mimics that in the one-factor case. It is convenient to write t 0

θi (s)ds

Lt =

n i=1

L i where t L i = exp − t t 1 t 2 θ (s)ds . 2 0 i 0 That {L t }t≥0 deﬁnes a martingale follows from (7.7) and the independence of the Brownian motions {Wti }t≥0 , i = 1, . . . , n. To check that {X ti }t≥0 is a (local) P(L) -martingale we ﬁnd the stochastic differential equation satisﬁed by {X ti L t }t≥0 . Since θi (s)d Wsi − d L i = −θi (t)L i d Wti , t t repeated application of our product rule gives n d L t = −L t i=1 θi (t)d Wti .

Moreover, d X ti = d Wti + θi (t)dt, and so another application of our product rule gives d(X ti L t ) = = X ti d L t + L t d Wti + L t θi (t)dt − L t θi (t)dt n −X ti L t i=1 θi (t)d Wti + L t d Wti .

Combined with the boundedness condition (7.7), this proves that {X ti L t }t≥0 is a P-martingale and hence {X ti }t≥0 is a P(L) -martingale. P(L) is equivalent to P so {X ti }t≥0 has quadratic variation [X i ]t = t with P(L) -probability one and once again L´ vy’s characterisation of Brownian motion conﬁrms that {X ti }t≥0 is a P(L) e Brownian motion as required. ✷
A martingale measure

As promised we now use this to ﬁnd a measure Q, equivalent to P, under which ˜ the discounted stock price processes { Sti }t≥0 , i = 1, . . . , n, are all martingales. The measure Q will be one of the measures P(L) of Theorem 7.2.4. We just need to identify the appropriate drifts {θi }t≥0 . ˜ ˜ The discounted stock price { Sti }t≥0 , deﬁned by Sti = Bt−1 Sti , is governed by the stochastic differential equation ˜ d Sti = = ˜ ˜ Sti (µi (t) − r ) dt + Sti ˜ Sti µi (t) − r − n j=1 n

σi j (t)d Wt j=1 j

˜ θ j (t)σi j (t) dt + Sti

n j=1

σi j (t)d X t ,

j

168

bigger models

where as in Theorem 7.2.4 d X t = θ j (t)dt + d Wt . The discounted stock price processes will (simultaneously) be (local) martingales under Q = P(L) if we can make all the drift terms vanish. That is, if we can ﬁnd {θ j (t)}t≥0 , j = 1, . . . , n, such that n j j

µi (t) − r − j=1 θ j (t)σi j (t) = 0 for all i = 1, . . . , n.

Dropping the dependence on t in our notation and writing µ = (µ1 , . . . , µn ) , this becomes µ − r 1 = θσ. (7.8) θ = (θ1 , . . . , θn ) , 1 = (1, . . . , 1) and σ = σi j ,

A solution certainly exists if the matrix σ is invertible, an assumption that we made in setting up our multiple asset model. In order to guarantee that the discounted price processes are martingales, not just local martingales, once again we impose a Novikov condition: EQ exp
0 t

1 2

n

σi2j (t)dt j=1 0,

172

bigger models

then, for 0 ≤ s ≤ t, EQ [ X t | Fs ] = EP

ζt X t Fs . ζs

We ﬁrst apply this to check that {Sti /Bt2 }t≥0 is a Q2 -martingale for each i = 1, . . . , n. EQ
2

Sti Fs Bt2

= = =

EQ EQ

1

1 Bt2 Bs Sti Fs 2 Bt1 Bs Bt2 1 Bs Sti Fs 2 Bs Bt1

1

1 i Bs Ss Si = s, 2 1 2 Bs Bs Bs

1 2 where the last line follows since Bs and Bs are Fs -measurable and {Sti /Bt1 }t≥0 is a Q1 -martingale. In other words, {Sti /Bt2 }t≥0 is a Q2 -martingale as required. That {Bt1 /Bt2 }t≥0 is a Q2 -martingale follows in the same way. The price for our derivative given that we chose {Bt2 }t≥0 as numeraire is then

Vt2

= = =

EQ EQ EQ

2

Bt2 C Ft 2 T BT
2 BT Bt1 Bt2 C Ft 1 2 T BT Bt2 BT

1

1

Bt1 C Ft 1 T BT

= Vt1 .

In other words, the choice of numeraire is unimportant – we always arrive at the same price.
Quantos

We now apply our multifactor technology in an example. We are going to price a quanto forward contract.
Deﬁnition 7.2.8

A ﬁnancial asset is called a quanto product if it is denominated in a currency other than that in which it is traded.

A quanto forward contract is also known as a guaranteed exchange rate forward. It is most easily explained through an example. BP, a UK company, has a Sterling denominated stock price that we denote by {St }t≥0 . For a dollar investor, a quanto forward contract on BP stock with maturity T has payoff (ST − K ) converted into dollars according to some prearranged exchange rate. That is the payout will be $ E(ST − K ) for some preagreed E, where ST is the Sterling asset price at time T .
Example 7.2.9

As in our foreign exchange market of §5.3 we shall assume that there is a riskless cash bond in each of the dollar and Sterling markets, but now we have two random

173

7.2 multiple stock models

processes to model, the stock price, {St }t≥0 and the exchange rate, that is the value of one pound in dollars which we denote by {E t }t≥0 . This will then require a two-factor model.

Black–Scholes quanto model:

We write {Bt }t≥0 for the dollar cash bond and {Dt }t≥0 for its Sterling counterpart. Writing E t for the dollar worth of one pound at time t and St for the Sterling asset price at time t, our model is Dollar bond Sterling bond Sterling asset price Exchange rate Bt = er t , Dt = eut , St = S0 exp νt + σ1 Wt1 , E t = E 0 exp λt + ρσ2 Wt1 + 1 − ρ 2 σ2 Wt2 ,

where {Wt1 }t≥0 and {Wt2 }t≥0 are independent P-Brownian motions and r , u, ν, λ, σ1 , σ2 and ρ are constants.

In this model the volatilities of {St }t≥0 and {E t }t≥0 are σ1 and σ2 respectively and {Wt1 , ρWt1 + 1 − ρ 2 Wt2 }t≥0 is a pair of correlated Brownian motions with correlation coefﬁcient ρ. There is no extra generality in replacing the expressions for St and E t by ˜ ˜ St = S0 exp νt + σ11 Wt1 + σ12 Wt2 , ˜ ˜ E t = E 0 exp λt + σ21 Wt1 + σ22 Wt2 , ˜ ˜ for independent Brownian motions {Wt1 , Wt2 }t≥0 .
Pricing a quanto forward contract

What is the value of K that makes the value at time zero of the quanto forward contract zero? As in our discussion of foreign exchange, the ﬁrst step is to reformulate the problem in terms of the dollar tradables. We now have three dollar tradables: the dollar worth of the Sterling bond, E t Dt ; the dollar worth of the stock, E t St ; and the dollar cash bond, Bt . Choosing the dollar cash bond as numeraire, we ﬁrst ﬁnd the stochastic differential equations governing the discounted values of the other two dollar tradables. We write Yt = Bt−1 E t Dt and Z t = Bt−1 E t St . Since 1 2 d E t = λ + σ2 E t dt + ρσ2 E t d Wt1 + 1 − ρ 2 σ2 E t d Wt2 , 2 application of our multifactor integration by parts formula gives 1 2 d(E t Dt ) = u E t Dt dt + λ+ σ2 E t Dt dt +ρσ2 E t Dt dWt1 + 1 − ρ 2 σ2 E t Dt d Wt2 2 and 1 2 dYt = λ + σ2 + u − r Yt dt + Yt ρσ2 d Wt1 + 1 − ρ 2 σ2 d Wt2 . 2

174

bigger models

Similarly, since 1 2 d St = ν + σ1 St dt + σ1 St d Wt1 , 2 d (E t St ) = 1 2 ν + σ1 E t St dt + σ1 E t St d Wt1 2 1 2 + λ + σ2 St E t dt + ρσ2 St E t d Wt1 2 + and so d Zt = 1 2 1 2 ν + σ1 + λ + σ2 + ρσ1 σ2 − r 2 2 + (σ1 + ρσ2 ) Z t d Wt1 + Z t dt 1 − ρ 2 σ2 St E t d Wt2 + ρσ1 σ2 St E t dt

1 − ρ 2 σ2 Z t d Wt2 .

Now we seek a change of measure to make these two processes martingales. Our calculations after the proof of Theorem 7.2.4 reduce this to ﬁnding θ1 , θ2 such that 1 2 λ + σ2 + u − r − θ1 ρσ2 − θ2 1 − ρ 2 σ2 = 0 2 and 1 2 1 2 ν + σ1 + λ + σ2 + ρσ1 σ2 − r − θ1 (σ1 + ρσ2 ) − θ2 1 − ρ 2 σ2 = 0. 2 2 Solving this pair of simultaneous equations gives θ1 = and θ2 =
2 ν + 1 σ1 + ρσ1 σ2 − u 2 σ1

2 λ + 1 σ2 + u − r − ρσ2 θ1 2

1 − ρ 2 σ2

.

Under the martingale measure, Q, {X t1 }t≥0 and {X t2 }t≥0 deﬁned by X t1 = Wt1 + θ1 t and X t2 = Wt2 + θ2 t are independent Brownian motions. We have St = S0 exp In particular, 1 2 1 ST = exp (−ρσ1 σ2 T ) S0 euT exp σ1 X T − σ1 T 2 and we are ﬁnally in a position to price the forward. Since {X t1 }t≥0 is a Q-Brownian motion, 1 2 1 = 1, EQ exp σ1 X T − σ1 T 2 1 2 u − ρσ1 σ2 − σ1 t + σ1 X t1 . 2

175

7.3 asset prices with jumps

so V0 = = e−r T EEQ [(ST − K )] e−r T E exp (−ρσ1 σ2 T ) S0 euT − K .

Writing F = S0 euT for the forward price in the Sterling market and setting V0 = 0 we see that we should take K = F exp (−ρσ1 σ2 T ) .
Remark: The exchange rate is given by

E t = E 0 exp

1 2 r − u − σ2 t + ρσ2 X t1 + 2

1 − ρ 2 σ2 X t2 .

It is reassuring to observe that ρ X t1 + 1 − ρ 2 X t2 is a Q-Brownian motion with variance one so that this expression for {E t }t≥0 is precisely that obtained in §5.3. Notice also that the discounted stock price process e−r t St is not a martingale; there is an extra term, reﬂecting the fact that the Sterling price is not a dollar tradable. ✷

7.3

Asset prices with jumps
The Black–Scholes framework is highly ﬂexible. The critical assumptions are continuous time trading and that the dynamics of the asset price are continuous. Indeed, provided this second condition is satisﬁed, the Black–Scholes price can be justiﬁed as an asymptotic approximation to the arbitrage price under discrete trading, as the trading interval goes to zero. But are asset prices continuous? So far, we have always assumed that any contracts written will be honoured. In particular, if a government or company issues a bond, we have ignored the possibility that they might default on that contract at maturity. But defaults do happen. This has been dramatically illustrated in recent years by credit crises in Asia, Latin America and Russia. If a company A holds a substantial quantity of company B’s debt securities, then a default by B might be expected to have the knock-on effect of a sudden drop in company A’s share price. How can we incorporate these market ‘shocks’ into our model?

A Poisson process of jumps

By their very nature, defaults are unpredictable. If we assume that we have absolutely no information to help us predict the default times or other market shocks, then we should model them by a Poisson random variable. That is the time between shocks is exponentially distributed and the number of shocks by time t, denoted by Nt , is a Poisson random variable with parameter λt for some λ > 0. Between shocks we assume that an asset price follows our familiar geometric Brownian motion model. A typical model for the evolution of the price of a risky asset with jumps is d St = µdt + σ d Wt − δd Nt , St (7.9)

176

bigger models

where {Wt }t≥0 and {Nt }t≥0 are independent. To make sense of equation (7.9) we write it in integrated form, but then we must deﬁne the stochastic integral with respect to {Nt }t≥0 . Writing τi for the time of the ith jump of the Poisson process, we deﬁne t 0 Nt

f (u, Su )d Nu = i=1 f τ (i)−, Sτ (i)− .

For the model (7.9), if there is a shock, then the asset price is decreased by a factor of (1 − δ). This observation tells us that the solution to (7.9) is St = S0 exp 1 µ − σ 2 t + σ Wt (1 − δ) Nt . 2

To deal with more general models we must extend our theory of stochastic calculus to incorporate processes with jumps. As usual, the ﬁrst step is to ﬁnd an (extended) Itˆ formula. o

Assumption:

We assume that asset price processes are c` dl` g, that is they are a a right continuous with left limits.

Theorem 7.3.1 (Itˆ ’s formula with jumps) o

Suppose

dYt = µt dt + σt d Wt + νt d Nt where, under P, {Wt }t≥0 is a standard Brownian motion and {Nt }t≥0 is a Poisson process with intensity λ. If f is a twice continuously differentiable function on R then f (Yt ) = f (Y0 ) +
0 Nt t

f (Ys− )dYs +

1 2

t 0 Nt

f (Ys− )σs2 ds f (Yτi ) − f (Yτi − ) , (7.10)

− i=1 f (Tτi − ) Yτi − Yτi − + i=1 where {τi } are the times of the jumps of the Poisson process. We don’t prove this here, but heuristically it is not difﬁcult to see that this should be the correct result. The ﬁrst three terms are exactly what we’d expect if the process {Yt }t≥0 were continuous, but now, because of the discontinuities, we must distinguish Ys− from Ys . In between jumps of {Nt }t≥0 , precisely this equation should apply, but we must compensate for changes at jump times. In the ﬁrst three terms we Nt have included a term of the form i=1 f (Yτi − ) Yτi − Yτi − and the ﬁrst sum in equation (7.10) corrects for this. Since Nt is ﬁnite, we do not have to correct the term involving f . Now we add in the actual contribution from the jump times and this is the second sum.

177

7.3 asset prices with jumps

o Compensation As usual a key rˆ le will be played by martingales. Evidently a Poisson process, {Nt }t≥0 with intensity λ under P is not a P-martingale – it is monotone increasing. But we can write it as a martingale plus a drift. In Exercise 13 it is shown that the process {Mt }t≥0 deﬁned by Mt = Nt − λt is a P-martingale. More generally we can consider time-inhomogeneous Poisson processes. For such processes the intensity {λt }t≥0 is a function of time. The probability of a jump in the time interval [t, t + δt) is λt δt + o(δt). Thus, for example, the probability that t there is no jump in the interval [s, t] is exp − s λu du . The corresponding Poisson martingale is Mt = Nt − 0 λs ds. The process { t }t≥0 given by t = 0 λs ds is the compensator of {Nt }t≥0 . In Exercise 14 it is shown that just as integration with respect to Brownian martingales gives rise to (local) martingales, so integration with respect to Poisson martingales gives rise to martingales.
Poisson exponential martingales t t

Let {Nt }t≥0 be a Poisson process with intensity {λt }t≥0 under P t where for each t > 0, 0 λs ds < ∞. For a given bounded deterministic function {αt }t≥0 , let
Example 7.3.2

L t = exp
0

t

αs d M s +
0

t

1 + αs − eαs λs ds

(7.11)

where d Ms = d Ns − λs ds. Find the stochastic differential equation satisﬁed by {L t }t≥0 and deduce that {L t }t≥0 is a P-martingale.
Solution: First write

Zt =
0

t

αs d Ms +
0

t

1 + αs − eαs λs ds

so that L t = e Z t . Then d Z t = αt d Nt − αt λt dt + 1 + αt − eαt λt dt and by our generalised Itˆ formula o d L t = L t− d Z t + −e Z t− αt + e Z t− +αt − e Z t− d Nt , where we have used the fact that if a jump in {Z t }t≥0 takes place at time t, then that jump is of size αt− . Substituting and rearranging give d Lt = = L t− αt d Mt + L t− 1 + αt − eαt λt dt − L t− 1 + αt − eαt d Nt L t− eαt − 1 d Mt . ✷

By Exercise 14, {L t }t≥0 is a P-martingale.
Deﬁnition 7.3.3

Processes of the form of {L t }t≥0 deﬁned by (7.11) will be called Poisson exponential martingales. Our Poisson exponential martingales and Brownian exponential martingales are examples of Dol´ ans–Dade exponentials. e

178 Deﬁnition 7.3.4

bigger models

For a semimartingale {X t }t≥0 with X 0 = 0, the Dol´ ans–Dade e exponential of {X t }t≥0 is the unique semimartingale solution {Z t }t≥0 to Zt = 1 +
0 t

Z s− d X s .

Change of measure

In the same way as we used Brownian exponential martingales to change measure and thus ‘transform drift’ in the continuous world, so we shall combine Brownian and Poisson exponential martingales in our discontinuous asset pricing models. A change of drift for a Poisson martingale will correspond to a change of intensity for the Poisson process {Nt }t≥0 . More precisely, we have the following version of the Girsanov Theorem.
Theorem 7.3.5 (Girsanov Theorem for asset prices with jumps)

Let {Wt }t≥0 be a standard P-Brownian motion and {Nt }t≥0 a (possibly time-inhomogeneous) Poisson process with intensity {λt }t≥0 under P. That is Mt = N t −
0 t

λu du

is a P-martingale. We write Ft for the σ -ﬁeld generated by FtW ∪ FtN . Suppose that {θt }t≥0 and {φt }t≥0 are {Ft }t≥0 -previsible processes with φt positive for each t, such that t t θs 2 ds < ∞ and φs λs ds < ∞. Then under the measure Q whose Radon–Nikodym derivative with respect to P is given by dQ = Lt dP Ft where L 0 = 1 and d Lt = θt d Wt − (1 − φt ) d Mt , L t− the process {X t }t≥0 deﬁned by X t = Wt − has intensity {φt λt }t≥0 . t 0 θs ds 0 0

is a Brownian motion and {Nt }t≥0

In Exercise 16 it is shown that {L t }t≥0 is actually the product of a Brownian exponential martingale and a Poisson exponential martingale. The proof of Theorem 7.3.5 is once again beyond our scope, but to check that the t processes {X t }t≥0 and Nt − 0 φs λs ds t≥0 are both local martingales under Q is an exercise based on the Itˆ formula. o
Heuristics: An informal justiﬁcation of the result is based on the extended multipli-

cation table: × d Wt d Nt dt d Wt dt 0 0 d Nt 0 d Nt 0 dt 0 0 0

179

7.3 asset prices with jumps

Thus, for example, d L t Nt −
0 t

φs λs ds

=

Nt −
0

t

φs λs ds d L t + L t (d Nt − φt λt dt)

− L t (1 − φt )(d Nt )2 = Nt −
0 t

φs λs ds d L t + L t (d Mt + λt dt)

− L t φt λt dt − L t (1 − φt ) (d Mt + λt dt) = Nt −
0 t

φs λs ds d L t + L t φt d Mt .

Since {Mt }t≥0 and {L t }t≥0 are P-martingales, subject to appropriate boundedness t assumptions, L t Nt − 0 φs λs ds should be a P-martingale and consequently t≥0 Nt −

t 0

φs λs ds

t≥0

should be a Q-martingale.

✷

Our instinct is to use the extended Girsanov Theorem to ﬁnd an equivalent probability measure under which the discounted asset price is a martingale. Suppose then that d St = µdt + σ d Wt − δd Nt . St Evidently the discounted asset price satisﬁes ˜ d St = (µ − r ) dt + σ d Wt − δd Nt . ˜ St But now we see that there are many choices of {θt }t≥0 and {φt }t≥0 in Theorem 7.3.5 that lead to a martingale measure. The difﬁculty of course is that our market is not complete, so that although for any replicable claim we can use any of the martingale measures and arrive at the same answer, there are claims that cannot be hedged. There are two independent sources of risk, the Brownian motion and the Poisson point process, and so if we are to be able to hedge arbitrary claims C T ∈ FT , we need two tradable risky assets subject to the same two noises.
Market price of risk

So if there are enough assets available to hedge claims, can we ﬁnd a measure under which once discounted they are all martingales? Remember that otherwise there will be arbitrage opportunities in our market. If the asset price has no jumps, we can write d St St = = µdt + σ d Wt (r + γ σ ) dt + σ d Wt ,

where γ = (µ − r )/σ is the market price of risk. We saw in Chapter 5 that in the absence of arbitrage (so when there is an equivalent martingale measure for our market), γ will be the same for all assets driven by {Wt }t≥0 . If the asset price has jumps, then investors will expect to be compensated for the additional risk associated with the possibility of downward jumps, even if we have

180

bigger models

‘compensated’ the jumps (replaced d Nt by d Mt ) so that their mean is zero. The price of such an asset is governed by d St St = = µdt + σ d Wt + νd Mt (r + γ σ + ηλν) dt + σ d Wt + νd Mt

where ν measures the sensitivity of the asset price to the market shock and η is the excess rate of return per unit of jump risk. Again if there is to be a martingale measure under which all the discounted asset prices are martingales, then σ and η should be the same for all assets whose prices are driven by {Wt }t≥0 and {Nt }t≥0 . The martingale measure, Q, will then be the measure Q of Theorem 7.3.5 under which t µ−r t ds and Mt − Wt + ηλds σ 0 0 are martingales. That is we take θ = γ and φ = −η.
Multiple noises

The same ideas can be extended to assets driven by larger numbers of independent noises. For example, we might have n assets with dynamics d Sti Sti n = µi dt + α=1 σiα d Wtα +

m β=1

νiβ d Mt

β

where, under P, {Wtα }t≥0 , α = 1, . . . , n, are independent Brownian motions and β {Mt }t≥0 , β = 1, . . . , m, are independent Poisson martingales. There will be an equivalent martingale measure under which all the discounted asset prices are martingales if we can associate a unique market price of risk with each source of noise. In this case we can write n m

µi = r + α=1 γα σiα + β=1 ηβ λβ νiβ .

All discounted asset prices will be martingales under the measure Q for which ˜ Wtα = Wtα + γα t is a martingale for each α and β ˜β Mt = Mt + ηβ λβ t

is a martingale for each β. As always it is replication that drives the theory. Note that in order to be able to hedge arbitrary C T ∈ FT we’ll require n + m ‘independent’ tradable risky assets driven by these sources of noise. With fewer assets at our disposal there will be claims C T that we cannot hedge. All this is little changed if we take the coefﬁcients µ, σ , λ to be adapted to the ﬁltration generated by {Wti }t≥0 , i = 1, . . . , n; see Exercise 15. Since we are not introducing any extra sources of noise, the same number of assets will be needed for market completeness. These ideas form the basis of Jarrow–Madan theory.

181

7.4 model error

7.4

Model error
Even in the absence of jumps (or between jumps) we have given only a very vague justiﬁcation for the Samuelson model d St = µSt dt + σ St d Wt . (7.12)

Moreover, although we have shown that under this model the pricing and hedging of derivatives are dictated by the single parameter σ , we have said nothing about how actually to estimate this number from market data. So what is market practice?
Implied volatility

Vanilla options are generally traded on exchanges, so if a trader wants to know the price of, say, a European call option, then she can read it from her trading screen. However, for an over-the-counter derivative, the price is not quoted on an exchange and so one needs a pricing model. The normal practice is to build a Black–Scholes model and then calibrate it to the market – that is estimate σ from the market. But it is not usual to estimate σ directly from data for the stock price. Instead one uses the quoted price for exchange-traded options written on the same stock. The procedure is simple: for given strike price and maturity, we can think of the Black–Scholes pricing formula for a European option as a mapping from volatility, σ , to price V. In Exercise 17, it is shown that for vanilla options this mapping is strictly monotone and so can be inverted to infer σ from the price. In other words, given the option price one can recover the corresponding value of σ in the Black–Scholes formula. This number is the so-called implied volatility. If the markets really did follow our Black–Scholes model, then this procedure would give the same value of σ , irrespective of the strike price and maturity of the exchange-traded option chosen. Sadly, this is far from what we observe in reality: not only is there dependence on the strike price for a ﬁxed maturity, giving rise to the famous volatility smile, but also implied volatility tends to increase with time to maturity (Figure 7.1). Market practice is to choose as volatility parameter for pricing an over-the-counter option the implied volatility obtained from ‘comparable’ exchange-traded options. This procedure can be expected to lead to a consistent price for exchange-traded and over-the-counter options and model error is not a serious problem. The difﬁculties arise in hedging. Even for exchange-traded options a model is required to determine the replicating portfolio. We follow Davis (2001). Suppose that the true stock price process follows d St = αt St d St + βt St d Wt where {αt }t≥0 and {βt }t≥0 are {Ft }t≥0 -adapted processes, but we price and hedge an option with payoff (ST ) at time T as though {St }t≥0 followed equation (7.12) for some parameter σ . Our estimate for the value of the option at time t < T will be V (t, St ) where

Hedging error

182

bigger models

27% 25% 23% 21% 19% 17% 15% 5950 20-Jul-2001 21-Sep-2001 19-Oct-2001 21-Dec-2001 5700 Strike 17-Aug-2001

Implied Volatility

5200 5450

Figure 7.1

Implied volatility as a function of strike price and maturity for European call options based on the FTSE stock index.

V (t, x) satisﬁes the Black–Scholes partial differential equation ∂V 1 ∂V ∂2V (t, x) + r x (t, x) + σ 2 x 2 2 (t, x) − r V (t, x) = 0, ∂t ∂x 2 ∂x V (T, x) = (x). Our hedging portfolio consists at time t of φt = ∂ V (t, St ) units of stock and cash ∂x bonds with total value ψt er t V (t, St ) − φt St . Our ﬁrst worry is that because of model misspeciﬁcation, the portfolio is not self-ﬁnancing. So what is the cost of following such a strategy? Since the cost of purchasing the ‘hedging’ portfolio at time t is V (t, St ), the incremental cost of the strategy over an inﬁnitesimal time interval [t, t + δt) is ∂V (t, St ) St+δt − St + ∂x ∂V (t, St )St (er δt − 1) ∂x − V (t + δt, St+δt ) + V (t, St ). V (t, St ) −

In other words, writing Z t for our net position at time t, we have d Zt = ∂V ∂V (t, St )d St + V (t, St ) − (t, St )St r dt − d V (t, St ). ∂x ∂x

Since V (t, x) solves the Black–Scholes partial differential equation, applying Itˆ ’s o

183

7.4 model error

formula gives d Zt = ∂V ∂V (t, St )d St + V (t, St ) − (t, St )St r dt ∂x ∂x ∂V ∂V 1 ∂2V (t, St )dt − (t, St )d St − (t, St )βt2 St2 dt − ∂t ∂x 2 ∂x2 1 2 ∂2V S σ 2 − βt2 dt. 2 t ∂x2

=

Irrespective of the model, V (T, ST ) = (ST ) precisely matches the claim against us at time T , so our net position at time T (having honoured the claim (ST ) against us) is T 1 ∂2V St2 2 (t, St ) σ 2 − βt2 dt. ZT = ∂x 0 2 For European call and put options ∂ xV > 0 (see Exercise 18) and so if σ 2 > βt2 ∂ 2 for all t ∈ [0, T ] our hedging strategy makes a proﬁt. This means that regardless of the price dynamics, we make a proﬁt if the parameter σ in our Black–Scholes model dominates the true diffusion coefﬁcient β. This is key to successful hedging. Our calculation won’t work if the price process has jumps, although by choosing σ large enough one can still arrange for Z T to have positive expectation. The choice of σ is still a tricky matter. If we are too cautious no one will buy the option, too optimistic and we are exposed to the risk associated with changes in volatility and we should try to hedge that risk. Such hedging is known as vega hedging, the Greek vega of an option being the sensitivity of its Black–Scholes price to changes in σ . The idea is the same as that of delta hedging (Exercise 5 of Chapter 5). For example, if we buy an over-the-counter option for which ∂ V = v, ∂σ then we also sell a number v/v of a comparable exchange traded option whose value V is V and for which ∂∂σ = v . The resulting portfolio is said to be vega-neutral.
2

Stochastic volatility and implied volatility

Since we cannot observe the volatility directly, it is natural to try to model it as a random process. A huge amount of effort has gone into developing so-called stochastic volatility models. Fat-tailed returns distributions observed in data can be modelled in this framework and sometimes ‘jumps’ in the asset price can be best modelled by jumps in the volatility. For example if jumps occur according to a Poisson process with constant rate and at the time, τ , of a jump, Sτ /Sτ − has a lognormal distribution, then the distribution of St will be lognormal but with variance parameter given by a multiple of a Poisson random variable (Exercise 19). Stochastic volatility can also be used to model the ‘smile’ in the implied volatility curve and we end this chapter by ﬁnding the correspondence between the choice of a stochastic volatility model and of an implied volatility model. Once again we follow Davis (2001). A typical stochastic volatility model takes the form d St dσt = µSt dt + σt St d Wt1 , 1 − ρ 2 d Wt2 ,

= a(St , σt )dt + b(St , σt ) ρd Wt1 +

184

bigger models

where {Wt1 }t≥0 , {Wt2 }t≥0 are independent P-Brownian motions, ρ is a constant in (0, 1) and the coefﬁcients a(x, σ ) and b(x, σ ) deﬁne the volatility model. As usual we’d like to ﬁnd a martingale measure. If Q is equivalent to P, then its Radon–Nikodym derivative with respect to P takes the form dQ = exp − dP Ft t 0

ˆ θs d Ws1 −

1 2

t 0

ˆ θs2 ds −

t 0

θs d Ws2 −

1 2

t 0

θs2 ds

ˆ for some integrands {θt }t≥0 and {θt }t≥0 . In order for the discounted asset price ˜t }t≥0 to be a Q-martingale, we choose {S ˆ θt = µ−r . σt

The choice of {θt }t≥0 however is arbitrary as {σt }t≥0 is not a tradable and so no arbitrage argument can be brought to bear to dictate its drift. Under Q, X t1 = Wt1 + and X t2 = Wt2 +
0 t 0 t

ˆ θs ds

θs ds

are independent Brownian motions. The dynamics of {St }t≥0 and {σt }t≥0 are then most conveniently written as d St = r St dt + σt St d X t1 and ˜ dσt = a(St , σt )dt + b(St , σt ) ρd X t1 + where ˆ a(St , σt ) = a(St , σt ) − b(St , σt ) ρ θt + ˜ 1 − ρ 2 θt . 1 − ρ 2 d X t2

We now introduce a second tradable asset. Suppose that we have an option written on {St }t≥0 whose exercise value at time T is (ST ). We deﬁne its value at times t < T to be the discounted value of (ST ) under the measure Q. That is V (t, St , σt ) = EQ e−r (T −t) (ST ) Ft . Our multidimensional Feynman–Kac Stochastic Representation Theorem (combined with the usual product rule) tells us that the function V (t, x, σ ) solves the partial differential equation ∂V ∂V 1 ∂2V ∂V (t, x, σ )+r x (t, x, σ )+a(t, x, σ ) ˜ (t, x, σ )+ σ 2 x 2 2 (t, x, σ ) ∂t ∂x ∂σ 2 ∂x 1 ∂2V ∂2V + b(t, x, σ )2 2 (t, x, σ )+ρσ xb(t, x, σ ) (t, x, σ )−r V (t, x, σ ) = 0. 2 ∂ x∂σ ∂σ

185

exercises

Writing Yt = V (t, St , σt ) and suppressing the dependence of V , a and b on (t, St , σt ) ˜ in our notation, an application of Itˆ ’s formula tells us that o ∂V ∂V ∂V 1 ∂2V 2 2 dt + d St + dσt + σ S dt ∂t ∂x ∂σ 2 ∂x2 t t 1 ∂2V 2 ∂2V ρbσt St dt + b dt + ∂ x∂σ 2 ∂σ 2 ∂V 1 ∂V ∂2V 1 ∂2V ∂2V −a ˜ − σt2 St2 2 − b2 2 − ρσt St b dt = r V − r St ∂x ∂σ 2 2 ∂σ ∂ x∂σ ∂x ∂V ∂V ∂V ∂V ∂V + r St dt + σt St d X t1 + a dt + bρ d X t1 + b 1 − ρ 2 d X t2 ˜ ∂x ∂x ∂σ ∂σ ∂σ 1 ∂2V ∂2V ∂2V 1 dt + b2 2 dt + σt2 St2 2 dt + ρbσt St 2 ∂ x∂σ 2 ∂σ ∂x ∂V ∂V ∂V 1 1 d X t + bρ d Xt + b 1 − ρ2 d X t2 . = r Yt dt + σt St ∂x ∂σ ∂σ If the mapping σ → y = V (t, x, σ ) is invertible so that σ = D(t, x, y) for some nice function D, then dYt = dYt = r Yt dt + c(t, St , Yt )d X t1 + d(t, St , Yt )d X t2 for some functions c and d. We have now created a complete market model with tradables {St }t≥0 and {Yt }t≥0 for which Q is the unique martingale measure. Of course, we have actually created one such market for each choice of {θt }t≥0 and it is the choice of {θt }t≥0 that speciﬁes the functions c and d and it is precisely these functions that tell us how to hedge. So what model for implied volatility corresponds to this stochastic volatility model? The implied volatility, σ (t), will be such that Yt is the Black–Scholes price ˆ ˆ evaluated at (t, St ) if the volatility in equation (7.12) is taken to be σ (t). In this way each choice of {θt }t≥0 , or equivalently model for {Yt }t≥0 , provides a model for the implied volatility. There is a huge literature on stochastic volatility. A good starting point is Fouque, Papanicolau and Sircar (2000).

Exercises
1 Check that the replicating portfolio deﬁned in §7.1 is self-ﬁnancing. 2 Suppose that {Wt1 }t≥0 and {Wt2 }t≥0 are independent Brownian motions under P and let ρ be a constant with 0 < ρ < 1. Find coefﬁcients {αi j }i, j=1,2 such that ˜ Wt1 = α11 Wt1 + α12 Wt2 and ˜ Wt2 = α21 Wt1 + α22 Wt2 = ρt. Is your

˜ ˜ deﬁne two standard Brownian motions under P with E Wt1 Wt2 solution unique?

186

bigger models

3 Suppose that F(t, x) solves the time-inhomogeneous Black–Scholes partial differential equation 1 ∂2 F ∂F ∂F (t, x) + σ 2 (t)x 2 2 (t, x) + r (t)x (t, x) − r (t)F(t, x) = 0, ∂t 2 ∂x ∂x (7.13)

subject to the boundary conditions appropriate to pricing a European call option. Substitute y = xeα(t) , v = Feβ(t) , τ = γ (t) and choose α(t) and β(t) to eliminate the coefﬁcients of v and ∂v in the resulting ∂y equation and γ (t) to remove the remaining time dependence so that the equation becomes 1 ∂ 2v ∂v (τ, y) = y 2 2 (τ, y). ∂τ 2 ∂y Notice that the coefﬁcients in this equation are independent of time and there is no reference to r or σ . Deduce that the solution to equation (7.13) can be obtained by making appropriate substitutions in the classical Black–Scholes formula. 4 Let {Wti }t≥0 , i = 1, . . . , n, be independent Brownian motions. Show that {Rt }t≥0 deﬁned by n Rt = i=1 (Wti )2

satisﬁes a stochastic differential equation. The process {Rt }t≥0 is the radial part of Brownian motion in Rn and is known as the n-dimensional Bessel process. 5 Recall that we deﬁne two-dimensional Brownian motion, {X t }t≥0 , by X t = (Wt1 , Wt2 ), where {Wt1 }t≥0 and {Wt2 }t≥0 are independent (one-dimensional) standard Brownian motions. Find the Kolmogorov backward equation for {X t }t≥0 . Repeat your calculation if {Wt1 }t≥0 and {Wt2 }t≥0 are replaced by correlated Brown˜ ˜ ˜ ˜ ian motions, {Wt1 }t≥0 and {Wt1 }t≥0 with E d Wt1 d Wt2 = ρdt for some −1 < ρ < 1. Use a delta-hedging argument to obtain the result of Corollary 7.2.7.

6

7 Repeat the Black–Scholes analysis of §7.2 in the case when the chosen numeraire, {Bt }t≥0 , has non-zero volatility and check that the fair price of a derivative with payoff C T at time T is once again Vt = Bt EQ CT Ft BT

for a suitable choice of Q (which you should specify). 8 Two traders, operating in the same complete arbitrage-free Black–Scholes market of §7.2, sell identical options, but make different choices of numeraire. How will their hedging strategies differ? 9 Find a portfolio that replicates the quanto forward contract of Example 7.2.9.

187

exercises

10

A quanto digital contract written on the BP stock of Example 7.2.9 pays $1 at time T if the BP Sterling stock price, ST , is larger than K . Assuming the Black–Scholes quanto model of §7.2, ﬁnd the time zero price of such an option and the replicating portfolio. A quanto call option written on the BP stock of Example 7.2.9 is worth E(ST − K )+ dollars at time T , where ST is the Sterling stock price. Assuming the Black–Scholes quanto model of §7.2, ﬁnd the time zero price of the option and the replicating portfolio.

11

12 Asian options Suppose that our market, consisting of a riskless cash bond, {Bt }t≥0 , and a single risky asset with price {St }t≥0 , is governed by d Bt = r Bt dt, and d St = µSt dt + σ St d Wt , where {Wt }t≥0 is a P-Brownian motion. An option is written with payoff C T = (ST , Z T ) at time T where Zt =
0 t

B0 = 1

g(u, Su )du

for some (deterministic) real-valued function g on R+ × R. From our general theory we know that the value of such an option at time t satisﬁes Vt = e−r (T −t) EQ [ (ST , Z T )| Ft ] where Q is the measure under which {St /Bt }t≥0 is a martingale. Show that Vt = F(t, St , Z t ) where the real-valued function F(t, x, z) on R+ ×R×R solves ∂F ∂F 1 ∂2 F ∂F + rx + σ 2x2 2 + g − r F = 0, ∂t ∂x 2 ∂z ∂x F(T, x, z) = (x, z). Show further that the claim C T can be hedged by a self-ﬁnancing portfolio consisting at time t of ∂F (t, St , Z t ) φt = ∂x units of stock and ∂F (t, St , Z t ) ψt = e−r t F(t, St , Z t ) − St ∂x cash bonds. 13 Suppose that {Nt }t≥0 is a Poisson process whose intensity under P is {λt }t≥0 . Show that {Mt }t≥0 deﬁned by Mt = N t −
0 t

λs ds

is a P-martingale with respect to the σ -ﬁeld generated by {Nt }t≥0 .

188

bigger models

14

Suppose that {Nt }t≥0 is a Poisson process under P with intensity {λt }t≥0 and {Mt }t≥0 is the corresponding Poisson martingale. Check that for an {FtM }t≥0 -predictable process { f t }t≥0 , t f s d Ms
0

is a P-martingale. 15 Show that our analysis of §7.3 is still valid if we allow the coefﬁcients in the stochastic differential equations driving the asset prices to be {Ft }t≥0 -adapted processes, provided we make some boundedness assumptions that you should specify.

16 Show that the process {L t }t≥0 in Theorem 7.3.5 is the product of a Poisson exponential martingale and a Brownian exponential martingale and hence prove that it is a martingale. 17 Show that in the classical Black–Scholes model the vega for a European call (or put) option is strictly positive. Deduce that for vanilla options we can infer the volatility parameter of the Black–Scholes model from the price. Suppose that V (t, x) is the Black–Scholes price of a European call (or put) option at 2 time t given that the stock price at time t is x. Prove that ∂ xV ≥ 0. ∂ 2

18

19 Suppose that an asset price {St }t≥0 follows a geometric Brownian motion with jumps occurring according to a Poisson process with constant intensity λ. At the time, τ , of each jump, independently, Sτ /Sτ − has a lognormal distribution. Show that, for each ﬁxed t, St has a lognormal distribution with the variance parameter σ 2 given by a multiple of a Poisson random variable.

Bibliography

Background reading: • • Probability, an Introduction, Geoffrey Grimmett and Dominic Welsh, Oxford University Press (1986). Options, Futures and Other Derivative Securities, John Hull, Prentice-Hall (Second edition 1993). Grimmett & Welsh contains all the concepts that we assume from probability theory. Hull is popular with practitioners. It explains the operation of markets in some detail before turning to modelling.

Supplementary textbooks: • • • • Arbitrage Theory in Continuous Time, Tomas Bj¨ rk, Oxford University Press (1998). o Dynamic Asset Pricing Theory, Darrell Dufﬁe, Princeton University Press (1992). Introduction to Stochastic Calculus Applied to Finance, Damien Lamberton and Bernard Lapeyre, translated by Nicolas Rabeau and Francois Mantion, Chapman and Hall (1996). ¸ The Mathematics of Financial Derivatives, Paul Wilmott, Sam Howison and Jeff Dewynne, Cambridge University Press (1995). These all represent useful supplementary reading. The ﬁrst three employ a variety of techniques while Wilmott, Howison & Dewynne is devoted exclusively to the partial differential equations approach.

Further topics in ﬁnancial mathematics: • • • • Financial Calculus: an Introduction to Derivatives Pricing, Martin Baxter and Andrew Rennie, Cambridge University Press (1996). Derivatives in Financial Markets with Stochastic Volatility, Jean-Pierre Fouque, George Papanicolau and Ronnie Sircar, Cambridge University Press (2000). Continuous Time Finance, Robert Merton, Blackwell (1990). Martingale Methods in Financial Modelling, Marek Musiela and Marek Rutkowski, SpringerVerlag (1998). 189

190

bibliography Although aimed at practitioners rather than university courses, Chapter 5 of Baxter & Rennie provides a good starting point for the study of interest rates. Fouque, Papanicolau & Sircar is a highly accessible text that would provide an excellent basis for a special topic in a second course in ﬁnancial mathematics. Merton is a synthesis of the remarkable research contributions of its Nobel-prize-winning author. Musiela & Rutkowski provides an encyclopaedic reference.

Brownian motion, martingales and stochastic calculus: • • • • Introduction to Stochastic Integration, Kai Lai Chung and Ruth Williams, Birkh¨ user (Second a edition 1990). Stochastic Differential Equations and Diffusion Processes, Nobuyuki Ikeda and Shinzo Watanabe, North-Holland (Second edition 1989). Brownian Motion and Stochastic Calculus, Ioannis Karatzas and Steven Shreve, SpringerVerlag (Second edition 1991). Probability with Martingales, David Williams, Cambridge University Press (1991). Williams is an excellent introduction to discrete parameter martingales and much more (integration, conditional expectation, measure, . . . ). The others all deal with the continuous world. Chung & Williams is short enough that it can simply be read cover to cover. A further useful reference is Handbook of Brownian Motion: Facts and Formulae, Andrei Borodin and Paavo Salminen, Birkh¨ user (1996). a

Additional references from the text: • Louis Bachelier, La th´ orie de la speculation. Ann Sci Ecole Norm Sup 17 (1900), 21–86. e English translation in The Random Character of Stock Prices, Paul Cootner (ed), MIT Press (1964), reprinted Risk Books (2000) J Cox, S Ross and M Rubinstein, Option pricing, a simpliﬁed approach. J Financial Econ 7 (1979), 229–63. M Davis, Mathematics of ﬁnancial markets, in Mathematics Unlimited – 2001 and Beyond, Bjorn Engquist and Wilfried Schmid (eds), Springer-Verlag (2001). D Freedman, Brownian Motion and Diffusion, Holden-Day (1971). J M Harrison and D M Kreps, Martingales and arbitrage in multiperiod securities markets. J Econ Theory 20 (1979), 381–408 J M Harrison and S R Pliska, Martingales and stochastic integrals in the theory of continuous trading. Stoch Proc Appl 11 (1981), 215–60. F B Knight, Essentials of Brownian Motion and Diffusion, Mathematical Surveys, volume 18, American Mathematical Society (1981). T J Lyons, Uncertain volatility and the risk-free synthesis of derivatives. Appl Math Finance 2 (1995), 117–33. P Protter, Stochastic Integration and Differential Equations, Springer-Verlag (1990). D Revuz and M Yor, Continuous Martingales and Brownian Motion, Springer-Verlag (Third edition 1998). P A Samuelson, Proof that properly anticipated prices ﬂuctuate randomly, Industrial Management Review 6, (1965), 41–50.

• • • • • • • • • •

Notation

Financial instruments and the Black–Scholes model T , maturity time. C T , value of claim at time T . {Sn }n≥0 , {St }t≥0 , value of the underlying stock. K , the strike price in a vanilla option. (ST − K )+ = max {(ST − K ), 0}. r , continuously compounded interest rate. σ , volatility. P, a probability measure, usually the market measure. Q, a martingale measure equivalent to the market measure. EQ , the expectation under Q. dQ the Radon–Nikodym derivative of Q with respect to P. dP ˜ ˜ { St }t≥0 , the discounted value of the underlying stock. In general, for a process {Yt }t≥0 , Yt = Yt /Bt where {Bt }t≥0 is the value of the riskless cash bond at time t. V (t, x), the value of a portfolio at time t if the stock price St = x. Also the Black–Scholes price of an option.

General probability ( , F , P), probability triple. P[A|B], conditional probability of A given B. , standard normal distribution function. p(t, x, y), transition density of Brownian motion. X = Y , the random variables X and Y have the same distribution. Z ∼ N (0, 1), the random variable Z has a standard normal distribution. E[X ; A], see Deﬁnition 2.3.4.
D

Martingales and other stochastic processes {Mt }t≥0 , a martingale under some speciﬁed probability measure. {[M]t }t≥0 , the quadratic variation of {Mt }t≥0 . {Fn }n≥0 , {Ft }t≥0 , ﬁltration. 191

192

notation
X {Fn }n≥0 (resp. {FtX }t≥0 ), ﬁltration generated by the process {X n }n≥0 (resp. {X t }t≥0 ). E [ X | F ], E X n+1 X n , conditional expectation; see pages 30ff. {Wt }t≥0 , Brownian motion under a speciﬁed measure, usually the market measure. X ∗ (t), X ∗ (t), maximum and minimum processes corresponding to {X t }t≥0 .

Miscellaneous =, deﬁned equal to. δ(π), the mesh of the partition π. f x , the function f evaluated at x. θ t (for a vector or matrix θ ), the transpose of θ. x > 0, x 0 for a vector x ∈ Rn , see page 11.

Index

adapted, 29, 64 arbitrage, 5, 11 arbitrage price, 5 Arrow–Debreu securities, 11 at the money, 3 attainable claim, 14 axioms of probability, 29 Bachelier, 51, 102 Bessel process, 186 squared, 109 bid–offer spread, 21 binary model, 6 Binomial Representation Theorem, 44 binomial tree, 24 Black–Karasinski model, 110 Black–Scholes equation, 121, 135, 136 similarity solutions, 137 special solutions, 136 variable coefﬁcients, 162, 186 Black–Scholes model basic, 112 coupon bonds, 131 dividends continuous payments, 126 periodic payments, 129 foreign exchange, 123 general stock model, 160 multiple assets, 163 quanto products, 173 with jumps, 175 Black–Scholes pricing formula, 45, 120, 135 bond, 4 coupon, 131 pure discount, 131 Brownian exponential martingale, 65 Brownian motion deﬁnition, 53 ﬁnite dimensional distributions, 54

hitting a sloping line, 61, 69 hitting times, 59, 66, 69 L´ vy’s characterisation, 90 e L´ vy’s construction, 56 e maximum process, 60, 69 path properties, 55 quadratic variation, 75 reﬂection principle, 60 scaling, 63 standard, 54 transition density, 54 with drift, 63, 99, 110 c` dl` g, 66 a a calibration, 181 cash bond, 5 Central Limit Theorem, 46 chain rule Itˆ stochastic calculus, see Itˆ ’s formula o o Stratonovich stochastic calculus, 109 change of probability measure continuous processes, see Girsanov’s Theorem on binomial tree, 97 claim, 1 compensation Poisson process, 177 sub/supermartingale, 41 complete market, 9, 16, 47 conditional expectation, 30 coupon, 131 covariation, 94 Cox–Ross–Rubinstein model, 24 delta, 122 delta hedging, 135 derivatives, 1 discounting, 14, 32 discrete stochastic integral, 36 distribution function, 29 standard normal, 47

193

194 dividend-paying stock, 49, 126 continuous payments, 126 periodic dividends, 129 three steps to replication, 127 Dol´ ans–Dade exponential, 177 e Dominated Convergence Theorem, 67 Doob’s inequality, 80 doubling strategy, 113 equities, 126 periodic dividends, 129 equivalent martingale measure, 15, 33, 115 equivalent measure, 15, 37, 98 exercise boundary, 151 exercise date, 2 expectation pricing, 4, 14 Feynman–Kac Stochastic Representation Theorem, 103 multifactor version, 170 ﬁltered probability space, 29 ﬁltration, 29, 64 natural, 29, 64 forward contract, 2 continuous dividends, 137 coupon bonds, 137 foreign exchange, 20, 124 periodic dividends, 131, 137 strike price, 5 forward price, 5 free boundary, 152 FTSE, 129 Fundamental Theorem of Asset Pricing, 12, 15, 38, 116 futures, 2 gamma, 122 geometric Brownian motion, 87 Itˆ ’s formula for, 88 o justiﬁcation, 102 Kolmogorov equations, 106 minimum process, 145 transition density, 106 Girsanov’s Theorem, 98 multifactor, 166 with jumps, 178 Greeks, 122 for European call option, 136 guaranteed equity proﬁts, 129 Harrison & Kreps, 12 hedging portfolio, see replicating portfolio hitting times, 59; see also Brownian motion implied volatility, see volatility in the money, 3 incomplete market, 17, 19

index inﬁnitesimal generator, 105 interest rate Black–Karasinski model, 110 continuously compounded, 5 Cox–Ingersoll–Ross model, 109 risk-free, 5 Vasicek model, 109, 110 intrinsic risk, 19 Itˆ integral, see stochastic integral o Itˆ isometry, 80 o Itˆ ’s formula o for Brownian motion, 85 for geometric Brownian motion, 88 for solution to stochastic differential equation, 91 multifactor, 165 with jumps, 176 Jensen’s inequality, 50 jumps, 175 Kolmogorov equations, 104, 110 backward, 105, 186 forward, 106 L 2 -limit, 76 Langevin’s equation, 109 L´ vy’s construction, 56 e Lipschitz-continuous, 108 local martingale, 65 localising sequence, 87 lognormal distribution, 4 long position, 2 market measure, 33, 113 market price of risk, 134, 179 market shocks, 175 Markov process, 34, 49 martingale, 33, 49, 64 bounded variation, 84 square-integrable, 100 martingale measure, 15 Martingale Representation Theorem, 100 multifactor, 168 maturity, 2 measurable, 29 mesh, 73 model error, 181 and hedging, 181 multifactor model, 163 multiple stock models, 10, 163 mutual variation, 94 Novikov’s condition, 98 numeraire, 126 change of, 20, 125, 171

195

index option, 2 American, 26, 42, 150 call on dividend-paying stock, 49 call on non-dividend-paying stock, 27, 50 cash-or-nothing, 157 exercise boundary, 151 free boundary value problem, 152 hedging, 43 linear complementarity problem, 152 perpetual, 157, 158 perpetual put, 153 put on non-dividend-paying stock, 27, 48, 157, 158 Asian, 149, 157, 187 asset-or-nothing, 155 barrier, 145, 148 binary, 140 call, 2, 127 coupon bonds, 137 dividend-paying stock, 127, 137 foreign exchange, 137 call-on-call, 143 cash-or-nothing, 48, 140 chooser, 156 cliquets, 143 collar, 154 compound, 143 contingent premium, 155 digital, 20, 48, 140, 154, 155 double knock-out, 149, 157 down-and-in, 145, 147, 157 down-and-out, 145, 148, 156 European, 2 hedging formula, 8, 25, 121 pricing formula, 8, 23, 45, 118 exotic, 139 foreign exchange, 17, 122 forward start, 48, 141 guaranteed exchange rate forward, 172 lookback call, 145 multistage, 142 on futures contract, 156 packages, 3, 18, 139 path-dependent, 144; see also (option) American, Asian pay-later, 155 perpetual, 137, 157 put, 2 put-on-put, 156 ratchet, 155 ratio, 142 up-and-in, 145 up-and-out, 145 vanilla, 3, 139 see also quanto Optional Stopping Theorem, 39, 49, 66 optional time, see stopping time Ornstein–Uhlenbeck process, 109 out of the money, 3 packages, 3, 18, 139 path probabilities, 26 payoff, 3 perfect hedge, 6 pin risk, 141 Poisson exponential martingale, 177 Poisson martingale, 177, 187 Poisson random variable, 175 positive riskless borrowing, 14 predictable, 36, 78 predictable representation, 100 previsible, see predictable probability triple, 29 put–call parity, 19, 137 compound options, 156 digital options, 154 quadratic variation, 75, 108 quanto, 172 call option, 187 digital contract, 187 forward contract, 172, 186 Radon–Nikodym derivative, 97, 98 random variable, 29 recombinant tree, 24 reﬂection principle, 60 replicating portfolio, 6, 23, 44 return, 4 Riesz Representation Theorem, 12 risk-neutral pricing, 15 risk-neutral probability measure, 13, 15 sample space, 29 self-ﬁnancing, 23, 26, 35, 113, 127, 137 semimartingale, 84 Separating Hyperplane Theorem, 12 Sharpe ratio, 134 short position, 2 short selling, 6 σ -ﬁeld, 29 simple function, 79 simple random walk, 34, 39, 49, 51 Snell envelope, 43 squared Bessel process, 109 state price vector, 11 and probabilities, 14 stationary independent increments, 52 stochastic calculus chain rule, see Itˆ ’s formula o Fubini’s Theorem, 96 integration by parts (product rule), 94 multifactor, 166

196 stochastic differential equation, 87, 91 stochastic integral, 75 discrete, 36 Itˆ , 78, 83 o integrable functions, 81 Stratonovich, 78, 108 with jumps, 176 with respect to semimartingale, 83 stochastic process, 29 stopping time, 38, 59 straddle, 4 Stratonovich integral, 78, 108 strike price, 2 submartingale, 33 compensation, 41 supermartingale, 33 and American options, 42 compensation, 41 Convergence Theorem, 41 theta, 122 three steps to replication basic Black–Scholes model, 118 continuous dividend-paying stock, 127

index discrete market model, 45 foreign exchange, 123 time value of money, 4 tower property, 32 tradable assets, 123, 126, 130 and martingales, 133 transition density, 54, 104–106 underlying, 1 vanillas, 139 variance, 54 variation, 73 and arbitrage, 73 p-variation, 73 vega, 122 vega hedging, 183 volatility, 120 implied, 120, 181 smile, 181, 183 stochastic, 183 and implied, 183 Wiener process, see Brownian motion

A Course in Financial Calculus

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