Mathematics of Financial Markets

355 Pages·2005·1.602 MB·English

by  Robert J. Elliott, P. Ekkehard Kopp

Springer Finance
Editorial Board
M. Avellaneda
G. Barone-Adesi
M. Broadie
M.H.A. Davis
E. Derman
C. Klu¨ppelberg
E. Kopp
W. Schachermayer
Robert J. Elliott and P. Ekkehard Kopp
Mathematics of
Financial Markets
Second edition
RobertJ.Elliott P.EkkehardKopp
HaskayneSchoolofBusiness DepartmentofMathematics
UniversityofCalgary UniversityofHull
Calgary,Alberta HullHU67RX
CanadaT2N1N4 Yorkshire
[email protected] UnitedKingdom
[email protected]
With7figures.
LibraryofCongressCataloging-in-PublicationData
Elliott,RobertJ.(RobertJames),1940–
Mathematicsoffinancialmarkets/RobertJ.ElliottandP.EkkehardKopp.—2nded.
p.cm.—(Springerfinance)
Includesbibliographicalreferencesandindex.
ISBN0-387-21292-2
1. Investments—Mathematics. 2. Stochasticanalysis. 3. Options
(Finance)—Mathematicalmodels. 4. Securities—Prices—Mathematicalmodels.
I. Kopp,P.E.,1944– II. Title. III. Series.
HG4515.3.E37 2004
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Preface
This work is aimed at an audience with a sound mathematical background
wishingtolearnabouttherapidlyexpandingfieldofmathematicalfinance.
Its content is suitable particularly for graduate students in mathematics
who have a background in measure theory and probability.
The emphasis throughout is on developing the mathematical concepts
requiredforthetheorywithinthecontextoftheirapplication. Noattempt
is made to cover the bewildering variety of novel (or ‘exotic’) financial in-
struments that now appear on the derivatives markets; the focus through-
out remains on a rigorous development of the more basic options that lie
at the heart of the remarkable range of current applications of martingale
theory to financial markets.
The first five chapters present the theory in a discrete-time framework.
Stochastic calculus is not required, and this material should be accessible
to anyone familiar with elementary probability theory and linear algebra.
The basic idea of pricing by arbitrage (or, rather, by non-arbitrage)
is presented in Chapter 1. The unique price for a European option in a
single-period binomial model is given and then extended to multi-period
binomial models. Chapter 2 introduces the idea of a martingale measure
for price processes. Following a discussion of the use of self-financing trad-
ing strategies to hedge against trading risk, it is shown how options can
be priced using an equivalent measure for which the discounted price pro-
cess is a martingale. This is illustrated for the simple binomial Cox-Ross-
Rubinsteinpricingmodels, andtheBlack-Scholesformulaisderivedasthe
limit of the prices obtained for such models. Chapter 3 gives the ‘funda-
mental theorem of asset pricing’, which states that if the market does not
contain arbitrage opportunities there is an equivalent martingale measure.
Explicit constructions of such measures are given in the setting of finite
marketmodels. CompletenessofmarketsisinvestigatedinChapter4; ina
completemarket,everycontingentclaimcanbegeneratedbyanadmissible
self-financing strategy (and the martingale measure is unique). Stopping
times, martingale convergence results, and American options are discussed
in a discrete-time framework in Chapter 5.
Thesecondfivechaptersofthebookgivethetheoryincontinuoustime.
ThisbeginsinChapter6withareviewofthestochasticcalculus. Stopping
times, Brownian motion, stochastic integrals, and the Itoˆ differentiation
v
vi Preface
rule are all defined and discussed, and properties of stochastic differential
equations developed.
Thecontinuous-timepricingofEuropeanoptionsisdevelopedinChap-
ter 7. Girsanov’s theorem and martingale representation results are de-
veloped, and the Black-Scholes formula derived. Optimal stopping results
are applied in Chapter 8 to a thorough study of the pricing of American
options, particularly the American put option.
Chapter 9 considers selected results on term structure models, forward
and future prices, and change of num´eraire, while Chapter 10 presents the
basic framework for the study of investment and consumption problems.
Acknowledgments Sectionsofthebookhavebeenpresentedincourses
at the Universities of Adelaide and Alberta. The text has consequently
benefited from subsequent comments and criticism. Our particular thanks
go to Monique Jeanblanc-Piqu´e, whose careful reading of the text and
valuable comments led to many improvements. Many thanks are also due
to Volker Wellmann for reading much of the text and for his patient work
in producing consistent TEX files and the illustrations.
Finally, the authors wish to express their sincere thanks to the Social
Sciences and Humanities Research Council of Canada for its financial sup-
port of this project.
Edmonton, Alberta, Canada Robert J. Elliott
Hull, United Kingdom P. Ekkehard Kopp
Preface to the Second
Edition
This second, revised edition contains a significant number of changes and
additions to the original text. We were guided in our choices by the com-
mentsofanumberofreadersandreviewersaswellasinstructorsusingthe
text with graduate classes, and we are grateful to them for their advice.
Any errors that remain are of course entirely our responsibility.
Inthefiveyearssincethebookwasfirstpublished,thesubjecthascon-
tinued to grow at an astonishing rate. Graduate courses in mathematical
finance have expanded from their business school origins to become stan-
dardfareinmanymathematicsdepartmentsinEuropeandNorthAmerica
and are spreading rapidly elsewhere, attracting large numbers of students.
Texts for this market have multiplied, as the rapid growth of the Springer
Finance series testifies. In choosing new material, we have therefore fo-
cused on topics that aid the student’s understanding of the fundamental
concepts, while ensuring that the techniques and ideas presented remain
up to date. We have given particular attention, in part through revisions
toChapters5and6,tolinkingkeyideasoccurringinthetwomainsections
(discrete- and continuous-time derivatives) more closely and explicitly.
Chapter 1hasbeen revised toincludeadiscussion of risk and return in
theone-stepbinomialmodel(whichisgivenanew, extendedpresentation)
andthisiscomplementedbyasimilartreatmentoftheBlack-Scholesmodel
inChapter7. DiscussionofelementaryboundsforoptionpricesinChapter
1 is linked to sensitivity analysis of the Black-Scholes price (the ‘Greeks’)
in Chapter 7, and call-put parity is utilised in various settings.
Chapter 2 includes new sections on superhedging and the use of ex-
tended trading strategies that include contingent claims, as well as a more
elegant derivation of the Black-Scholes option price as a limit of binomial
approximants.
Chapter3includesasubstantialnewsectionleadingtoacompleteproof
of the equivalence, for discrete-time models, of the no-arbitrage condition
and the existence of equivalent martingale measures. The proof, while not
original, is hopefully more accessible than others in the literature.
This material leads in Chapter 4 to a characterisation of the arbitrage
vii
viii Preface to the the Second Edition
interval for general market models and thus to a characterisation of com-
plete models, showing in particular that complete models must be finitely
generated.
The new edition ends with a new chapter on risk measures, a subject
thathasbecomeamajorareaofresearchinthepastfiveyears. Weincludea
briefintroductiontoValueatRiskandgivereasonswhytheuseofcoherent
risk measures (or their more recent variant, deviation measures) is to be
preferred. Chapter 11 ends with an outline of the use of risk measures in
recent work on partial hedging of contingent claims.
The changes we have made to the text have been informed by our
continuing experience in teaching graduate courses at the universities of
Adelaide, Calgary and Hull, and at the African Institute for Mathematical
Sciences in Cape Town.
Acknowledgments Particular thanks are due to Alet Roux (Hull) and
AndrewRoyal(Calgary)whoprovidedinvaluableassistancewiththecom-
plexities of LaTeX typesetting and who read large sections of the text.
ThanksarealsoduetotheSocialSciencesandHumanitiesResearchCoun-
cil of Canada for continuing financial support.
Calgary, Alberta, Canada Robert J. Elliott
Hull, United Kingdom P. Ekkehard Kopp
May 2004
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . vii
1 Pricing by Arbitrage 1
1.1 Introduction: Pricing and Hedging . . . . . . . . . . . . . . 1
1.2 Single-Period Option Pricing Models . . . . . . . . . . . . . 10
1.3 A General Single-Period Model . . . . . . . . . . . . . . . . 12
1.4 A Single-Period Binomial Model . . . . . . . . . . . . . . . 14
1.5 Multi-period Binomial Models . . . . . . . . . . . . . . . . . 20
1.6 Bounds on Option Prices . . . . . . . . . . . . . . . . . . . 24
2 Martingale Measures 27
2.1 A General Discrete-Time Market Model . . . . . . . . . . . 27
2.2 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Martingales and Risk-Neutral Pricing . . . . . . . . . . . . 35
2.4 Arbitrage Pricing: Martingale Measures . . . . . . . . . . . 38
2.5 Strategies Using Contingent Claims. . . . . . . . . . . . . . 43
2.6 Example: The Binomial Model . . . . . . . . . . . . . . . . 48
2.7 From CRR to Black-Scholes . . . . . . . . . . . . . . . . . . 50
3 The First Fundamental Theorem 57
3.1 The Separating Hyperplane Theorem in Rn . . . . . . . . . 57
3.2 Construction of Martingale Measures . . . . . . . . . . . . . 59
3.3 Pathwise Description . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 General Discrete Models . . . . . . . . . . . . . . . . . . . . 71
4 Complete Markets 87
4.1 Completeness and Martingale Representation . . . . . . . . 88
4.2 Completeness for Finite Market Models . . . . . . . . . . . 89
4.3 The CRR Model . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 The Splitting Index and Completeness . . . . . . . . . . . . 94
4.5 Incomplete Models: The Arbitrage Interval . . . . . . . . . 97
4.6 Characterisation of Complete Models . . . . . . . . . . . . . 101
ix
x CONTENTS
5 Discrete-time American Options 105
5.1 Hedging American Claims . . . . . . . . . . . . . . . . . . . 105
5.2 Stopping Times and Stopped Processes . . . . . . . . . . . 107
5.3 Uniformly Integrable Martingales . . . . . . . . . . . . . . . 110
5.4 Optimal Stopping: The Snell Envelope . . . . . . . . . . . . 116
5.5 Pricing and Hedging American Options . . . . . . . . . . . 124
5.6 Consumption-Investment Strategies. . . . . . . . . . . . . . 126
6 Continuous-Time Stochastic Calculus 131
6.1 Continuous-Time Processes . . . . . . . . . . . . . . . . . . 131
6.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . 141
6.4 The Itˆo Calculus . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Stochastic Differential Equations . . . . . . . . . . . . . . . 158
6.6 Markov Property of Solutions of SDEs . . . . . . . . . . . . 162
7 Continuous-Time European Options 167
7.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.2 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . 168
7.3 Martingale Representation . . . . . . . . . . . . . . . . . . . 174
7.4 Self-Financing Strategies . . . . . . . . . . . . . . . . . . . . 183
7.5 An Equivalent Martingale Measure . . . . . . . . . . . . . . 185
7.6 Black-Scholes Prices . . . . . . . . . . . . . . . . . . . . . . 193
7.7 Pricing in a Multifactor Model . . . . . . . . . . . . . . . . 198
7.8 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.9 The Black-Scholes Equation . . . . . . . . . . . . . . . . . . 214
7.10 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
8 The American Put Option 223
8.1 Extended Trading Strategies. . . . . . . . . . . . . . . . . . 223
8.2 Analysis of American Put Options . . . . . . . . . . . . . . 226
8.3 The Perpetual Put Option . . . . . . . . . . . . . . . . . . . 231
8.4 Early Exercise Premium . . . . . . . . . . . . . . . . . . . . 234
8.5 Relation to Free Boundary Problems . . . . . . . . . . . . . 238
8.6 An Approximate Solution . . . . . . . . . . . . . . . . . . . 243
9 Bonds and Term Structure 247
9.1 Market Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 247
9.2 Future Price and Futures Contracts . . . . . . . . . . . . . 252
9.3 Changing Num´eraire . . . . . . . . . . . . . . . . . . . . . . 255
9.4 A General Option Pricing Formula . . . . . . . . . . . . . . 258
9.5 Term Structure Models . . . . . . . . . . . . . . . . . . . . 262
9.6 Short-rate Diffusion Models . . . . . . . . . . . . . . . . . . 264
9.7 The Heath-Jarrow-Morton Model . . . . . . . . . . . . . . . 277
9.8 A Markov Chain Model . . . . . . . . . . . . . . . . . . . . 282
CONTENTS xi
10 Consumption-Investment Strategies 285
10.1 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . 285
10.2 Admissible Strategies. . . . . . . . . . . . . . . . . . . . . . 287
10.3 Maximising Utility of Consumption . . . . . . . . . . . . . . 291
10.4 Maximisation of Terminal Utility . . . . . . . . . . . . . . . 296
10.5 Consumption and Terminal Wealth . . . . . . . . . . . . . . 299
11 Measures of Risk 303
11.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 304
11.2 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . 308
11.3 Deviation Measures. . . . . . . . . . . . . . . . . . . . . . . 316
11.4 Hedging Strategies with Shortfall Risk . . . . . . . . . . . . 320
Bibliography 329
Index 349