Financial Mathematics An Introduction to Derivatives Pricing

234 Pages·2000·1.05 MB·English

by  Lane P. Hughston

KING’S COLLEGE LONDON
DEPARTMENT OF MATHEMATICS
Financial Mathematics
An Introduction to Derivatives Pricing
Lane P. Hughston
Christopher J. Hunter
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Financial Mathematics
An Introductory Guide
Lane P. Hughston1
Department of Mathematics
King’s College London
The Strand, London
WC2R 2LS, UK
Christopher J. Hunter2
NatWest Group
135 Bishopsgate, London
Postal Code, UK
and
Department of Mathematics
King’s College London
The Strand, London
WC2R 2LS, UK
copyright c 2000 L.P. Hughston and C.J. Hunter
(cid:176)
1email: lane [email protected]
2email: [email protected]
i
Preface
This book is intended as a guide to some elements of the mathematics of
flnance. Had we been a bit bolder it would have been entitled ‘Mathematics
for Money Makers’ since it deals with derivatives, one of the most notorious
ways to make (or lose) a lot of money. Our main goal in the book is to
develop the basics of the theory of derivative pricing, as derived from the
so-called ‘no arbitrage condition’. In doing so, we also introduce a number
of mathematical tools that are of interest in their own right. At the end of it
all, while you may not be a millionaire, you should understand how to avoid
‘breaking the bank’ with a few bad trades.
In order to motivate the study of derivatives, we begin the book with a
discussion of the flnancial markets, the instruments that are traded on them
and how arbitrage opportunities can occur if derivatives are mispriced. We
then arrive at a problem that inevitably arises when dealing with physical
systems such as the flnancial markets: how to deal with the ‘(cid:176)ow of time’.
There are two primary means of parametrizing time|the discrete time pa-
rameterization, where time advances in flnite steps; and the continuous time
parameterization, where time varies smoothly. We initially choose the former
method, and develop a simple discrete time model for the movements of asset
prices and their associated derivatives. It is based on an idealised Casino,
where betting on the random outcome of a coin toss replaces the buying and
selling of an asset. Once we have seen the basic ideas in this context, we then
expand the model and interpret it in a language that brings out the analogy
with a stock market. This is the binomial model for a stock market, where
time is discrete and stock prices move in a random fashion. In the second
half of the notes, we make the transition from discrete to continuous time
models, and derive the famous Black-Scholes formula for option pricing, as
well as a number of interesting extensions of this result.
Throughoutthebookweemphasisetheuseofmodernprobabilisticmeth-
ods and stress the novel flnancial ideas that arise alongside the mathematical
innovations. Some more advanced topics are covered in the flnal sections|
stocks which pay dividends, multi-asset models and one of the great simpli-
flcations of derivative pricing, the Girsanov transformation.
This book ia based on a series of lectures given by L.P. Hughston at
King’s College London in 1997. The material in appendix D was provided by
Professor R.F. Streater, whom we thank for numerous helpful observations
on the structure and layout of the material in these notes.
ii
For lack of any better, yet still grammatically correct alternative, we will
use ‘he’ and ‘his’ in a gender non-speciflc way. In a similar fashion, we will
use ‘dollar’ in a currency non-speciflc way.
L.P. Hughston and C.J. Hunter
January 1999
iii
Contents
1 Introduction 1
1.1 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Basic Assets . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Uses of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Derivative Payofi Functions . . . . . . . . . . . . . . . . . . . 9
2 Arbitrage Pricing 13
2.1 Expectation Pricing . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Replication Strategy . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Currency Swap . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 A Simple Casino 24
3.1 Rules of the Casino . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 No Arbitrage Argument . . . . . . . . . . . . . . . . . . . . . 26
4 Probability Systems 29
4.1 Sample Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Event Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Probability Measure . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Back to the Casino 35
5.1 The Casino as a Probability System . . . . . . . . . . . . . . . 35
5.2 The Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . . 35
5.3 A Non-Zero Interest Rate . . . . . . . . . . . . . . . . . . . . 37
6 The Binomial Model 41
6.1 Tree Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Money Market Account . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 One-Period Replication Model . . . . . . . . . . . . . . . . . . 45
6.5 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . 47
iv
7 Pricing in N-Period Tree Models 50
8 Martingales and Conditional Expectation 54
8.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 54
8.2 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.3 Adapted Process . . . . . . . . . . . . . . . . . . . . . . . . . 56
8.4 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . 56
8.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.6 Financial Interpretation . . . . . . . . . . . . . . . . . . . . . 58
9 Binomial Lattice Model 60
10 Relation to Binomial Model 63
10.1 Limit of a Random Walk . . . . . . . . . . . . . . . . . . . . . 63
10.2 Martingales associated with Random Walks . . . . . . . . . . 64
11 Continuous Time Models 68
11.1 The Wiener Model . . . . . . . . . . . . . . . . . . . . . . . . 68
11.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . 69
12 Stochastic Calculus 76
13 Arbitrage Argument 79
13.1 Derivation of the No-Arbitrage Condition . . . . . . . . . . . . 79
13.2 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 83
14 Replication Portfolios 86
15 Solving the Black-Scholes Equation 89
15.1 Solution of the Heat Equation . . . . . . . . . . . . . . . . . . 90
15.2 Reduction of the Black-Scholes Equation to the Heat Equation 92
16 Call and Put Option Prices 97
16.1 Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
16.2 Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
17 More Topics in Option Pricing 104
17.1 Binary Options . . . . . . . . . . . . . . . . . . . . . . . . . . 104
17.2 ‘Greeks’ and Hedging . . . . . . . . . . . . . . . . . . . . . . . 105
v
17.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 107
18 Continuous Dividend Model 109
18.1 Modifled Black-Scholes Equation . . . . . . . . . . . . . . . . 112
18.2 Call and Put Option Prices . . . . . . . . . . . . . . . . . . . 113
19 Risk Neutral Valuation 115
19.1 Single Asset Case . . . . . . . . . . . . . . . . . . . . . . . . . 115
20 Girsanov Transformation 121
20.1 Change of Drift . . . . . . . . . . . . . . . . . . . . . . . . . . 122
21 Multiple Asset Models 126
21.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . 126
21.2 No Arbitrage and the Zero Volatility Portfolio . . . . . . . . . 129
21.3 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . 131
22 Multiple Asset Models Continued 132
22.1 Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
22.2 Martingales and the Risk-Neutral Measure . . . . . . . . . . . 133
22.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Glossary 137
B Some useful formulae and deflnitions 140
B.1 Deflnitions of a Normal Variable . . . . . . . . . . . . . . . . . 140
B.2 Moments of the Standard Normal Distribution . . . . . . . . . 140
B.3 Moments of a Normal Distribution . . . . . . . . . . . . . . . 141
B.4 Other Useful Integrals . . . . . . . . . . . . . . . . . . . . . . 141
B.5 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.6 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 142
B.7 Black-Scholes Formulae . . . . . . . . . . . . . . . . . . . . . . 142
B.8 Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . 143
B.9 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . 143
B.10 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 143
C Solutions 145
vi
D Some Reminders of Probability Theory 194
D.1 Events, random variables and distributions . . . . . . . . . . . 194
D.2 Expectation, moments and generating functions . . . . . . . . 195
D.3 Several random variables . . . . . . . . . . . . . . . . . . . . . 196
D.4 Conditional probability and expectation . . . . . . . . . . . . 199
D.5 Filtrations and martingales . . . . . . . . . . . . . . . . . . . . 204
E The Virtues and
Vices of Options1 207
F KCL 1998 Exam 209
F.1 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
F.1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 210
F.2 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
F.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 213
F.3 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
F.3.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 216
F.4 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
F.4.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 219
F.5 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
F.5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 222
F.6 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
F.6.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 224
G Bibliography 225
vii
1 Introduction
Thestudyofmostsciencescanbeusefullydividedintotwodistinctbutinter-
relatedbranches, theoryandexperiment. Forexample, thebodyofknowledge
thatweconventionallylabel‘physics’consistsoftheoretical physics, wherewe
develop mathematical models and theories to describe how nature behaves,
andexperimental physics, whereweactuallytestandprobenaturetoseehow
it behaves. There is an important interplay between the two branches|for
example, theory might develop a model which is then tested by experiment,
or experiment might measure or discover a fact or feature of nature which
must then be explained by theory.
Finance is the science of the flnancial markets. Correspondingly, it has
an important ‘theoretical’ side, called flnance theory or mathematical flnance,
whichentailsboththedevelopmentoftheconceptualapparatusneededforan
intellectually sound understanding of the behaviour of the flnancial markets,
as well as the development of mathematical techniques and models useful in
flnance; and an ‘experimental’ side, which we might call practical or applied
flnance, that consists of the extensive range of trading techniques and risk
management practices as they are actually carried out in the various flnan-
cial markets, and applied by governments, corporations and individuals in
their quest to improve their fortune and control their exposure to potentially
adverse circumstances.
In this book we ofier an introductory guide to mathematical flnance, with
particular emphasis on a topic of great interest and the source of numerous
applications: namely, the pricing of derivatives. The mathematics needed for
a proper understanding of this signiflcant branch of theoretical and applied
flnance is both fascinating and important in its own right. Before we can
begin building up the necessary mathematical tools for analysing derivatives,
however, we need to know what derivatives are and what they are used for.
Butthisrequiressomeknowledgeoftheso-called‘underlyingassets’onwhich
these derivatives are based. So we begin this book by discussing the flnancial
markets and the various instruments that are traded on them. Our intention
here is not, of course, to make a comprehensive survey of these markets, but
to sketch lightly the relevant notions and introduce some useful terminology.
Unless otherwise stated, all dates in this section are from the year 1999, and
all prices are the relevant markets’ closing values. If no date is given for a
price, then it can be assumed to be January 11, 1999.
1
1.1 Financial Markets
The global flnancial markets collectively comprise a massive industry spread
over the entire world, with substantial volumes of buying and selling occur-
ring in one market or another at one place or another at virtually any time.
The dealing is mediated by traders who carry out trades on behalf of both
their clients (institutional and individual investors) and their employers (in-
vestment banks and other flnancial institutions). This world-wide menagerie
of traders, in the end, determines the prices of the available flnancial prod-
ucts, and is sometimes collectively referred to as the ‘market’. The most
‘elementary’ flnancial instruments bought and sold in flnancial markets can
be described as basic assets. There are several common types.
1.1.1 Basic Assets
A stock or share represents a part ownership of a company, typically on a
limited liability basis (that is, if the company fails, then the shareholder’s
loss is usually limited to his original investment). When the company is
profltable, the owner of the stock beneflts from time to time by receiving a
dividend, whichistypicallyacashpayment. Theshareholdermayalsorealize
a proflt or capital gain if the value of the stock increases. Ultimately, the
sharepriceisdeterminedbythemarketaccordingtothelevelofconfldenceof
investors that the flrm will be profltable, and hence pay further and perhaps
higher dividends in the future. For example, the value of a Rolls-Royce share
atthecloseoftheLondonStockExchangeonJanuary11was248.5p(pence),
which was down 0.5p from the prior day’s closing value. In the previous 52
weeks the highest closing value was 309p, while the lowest was 176.5p. The
company has declared a dividend of 6.15p per share for 1998 compared with
5.9p and 5.3p per share paid in the two years previous to that.
Abondis, inefiect, aloanmadetoacompanyorgovernmentbythebond-
holder, usually for a flxed period of time, for which the bond-holder receives
a fee, known as interest. The interest rate charged is typically flxed at the
time that the loan is made, but might be allowed to vary in time according
to market levels and certain prescribed rules. The interest payments, which
are typically made on an annual, semi-annual or quarterly basis, are called
‘coupon’ payments. If a 10-year bond with a ‘face-value’ of $1000 has a 6%
annual coupon, that means that an interest rate payment of $60 is made
every year for ten years, and then at the end of the ten year period the $1000
2