Table Of ContentArbitrage Theory in Continuous Time
This page intentionally left blank
Arbitrage Theory in Continuous Time
Tomas Björk
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford
ItfurtherstheUniversity'sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
OxfordNewYork
AucklandBangkokBuenosAiresCapeTownChennai
Dar esSalaamDelhiHongKongIstanbulKarachiKolkata
KualaLumpurMadridMelbourneMexicoCityMumbaiNairobi
SãoPauloShanghaiTaipeiTokyoToronto
Oxfordisaregisteredtrademark ofOxfordUniversityPress
intheUK andincertainothercountries
PublishedintheUnitedStatesby
OxfordUniversityPressInc., NewYork
©TomasBjörk1998
Themoralrightsoftheauthorshavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished1998
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,or transmitted,inanyform orbyanymeans,
withoutthepriorpermissioninwriting ofOxfordUniversityPress,
oras expresslypermittedbylaw, or under termsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethissameconditiononanyacquirer
BritishLibraryCataloginginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
ISBN0–19–877518–0
Preface
The purpose of this book is to present arbitrage theory and its applications to pricing problems for financial
derivatives. It is intended as a textbook for graduate and advanced undergraduate students in finance, economics,
mathematics, and statistics and I also hope that it will be useful for practitioners.
Because of its intended audience, the book does not presuppose any previous knowledge of abstract measure theory.
The only mathematical prerequisites are advanced calculus and a basic course in probability theory. No previous
knowledge in economics or finance is assumed.
The book starts by contradicting its own title, in the sense that the second chapter is devoted to the binomial model.
After that, the theory is exclusively developed in continuous time.
The main mathematical toolused in the book is thetheory of stochastic differential equations (SDEs), and instead of
goingintothetechnicaldetailsconcerningthefoundationsofthattheoryIhavefocusedonapplications.Theobjectisto
give the reader, as quickly and painlessly as possible, a solid working knowledge of the powerful mathematical tool
known as Itö calculus. We treat basic SDE techniques, including Feynman-Kač representations and the Kolmogorov
equations. Martingales are introduced at an early stage. Throughout the book there is a strong emphasis on concrete
computations, and the exercises at the end of each chapter constitute an integral part of the text.
Themathematicsdevelopedinthefirstpartofthebookisthenappliedtoarbitrage pricing offinancialderivatives. We
cover thebasic Black-Scholes theory, including deltahedging and “thegreeks”, and we extenditto thecase of several
underlying assets (including stochastic interest rates) as well as to dividend paying assets. Barrier options, as well as
currency and quanto products, are given separate chapters. We also consider, in some detail, incomplete markets.
American contracts are treated only in passing. The reason for this is that the theory is complicated and that few
analytical results are available. Instead I have included a chapter on stochastic optimal control and its applications to
optimal portfolio selection.
Interestratetheoryconstitutesalargepartofthebook,andwecoverthebasicshortratetheory,includinginversionof
the yield curve and affine term structures. The Heath-Jarrow-Morton theory is treated, both under the objective
measure and under a martingale measure, and we also present the Musiela parametrization. The basic framework for
most chapters is that of a multifactor model, and this allowsus, despite the fact that we do not formally use measure
theory, to give a fairly complete treatment of the general change of numeraire
vi PREFACE
technique whichis so essentialtomoderninterest ratetheory. Inparticular wetreat forward neutral measures insome
detail. This allows us to present the Geman-El Karoui-Rochet formula for option pricing, and we apply it to the
general Gaussian forward rate model, as well as to a number of particular cases.
Concerning the mathematical level, the book falls between the elementary text by Hull (1997), and more advanced
texts such as Duffie (1996) or Musiela and Rutkowski (1997). These books are used as canonical references in the
present text.
Inorder tofacilitateusingthebookforshortercourses, thepedagogicalapproachhasbeenthatoffirstpresenting and
analyzing a simple (typically one-dimensional) model, and then to derive the theory in a more complicated
(multidimensional) framework. The drawback of this approach is of course that some arguments are being repeated,
but this seems to be unavoidable, and I can only apologize to the technically more advanced reader.
Notestotheliteraturecanbefoundattheendofmostchapters.Ihavetriedtokeepthereferencelistonamanageable
scale, but any serious omission is unintentional, and I will be happy to correct it. For more bibliographic information
the reader is referred to Duffie (1996) and to Musiela and Rutkowski (1997) which both contain encyclopedic
bibliographies.
Onthemoretechnicalsidethefollowingfactscanbementioned.Ihavetriedtopresentareasonablyhonestpictureof
SDE theory, including Feynman-Kač representations, while avoiding the explicit use of abstract measure theory.
Because of the chosen technical level, the arguments concerning the construction of the stochastic integral are thus
forced to be more or less heuristic. Nevertheless I have tried to be as precise as possible, so even the heuristic
argumentsare the“correct” onesinthesensethattheycanbecompleted toformal proofs. IntherestofthetextI try
give full proofs of all mathematical statements, with the exception that I have often left out the checking of various
integrability conditions.
Since the Girsanov theory for absolutelycontinuous changes of measures is outside the scope of this text, martingale
measures are introduced by the use of locally riskless portfolios, partial differential equations (PDEs) and the
Feynman-Kač representation theorem. Still, the approach to arbitrage theory presented in the text is basically a
probabilistic one, emphasizing the use of martingale measures for the computation of prices.
The integral representation theorem for martingales adapted to a Wiener filtration is also outside the scope of the
book. Thus we do not treat market completeness in full generality, but restrict ourselves to a Markovian framework.
For most applications this is, however, general enough.
Acknowledgements:
Bertil Näslund, Staffan Viotti, Peter Jennergren and Ragnar Lindgren persuaded me to start studying financial
economics, and they have constantly and generously shared their knowledge with me.
Hans Bühlman, Paul Embrechts and Hans Gerber gave me the opportunity to give a series of lectures for a summer
schoolat MonteVeritainAscona1995. This summer schoolwas for me an extremelyhappyand fruitful time, as well
as the start of a partially new career. The set of lecture notes produced for that occasion is the basis for the present
book.
Over the years of writing, I have received valuable comments and advice from large number of people. My greatest
debt is to Camilla Landén who has given me more good advice (and pointed out more errors) than I thought was
humanly possible.I am also highly indebted to Flavio Angelini, Pia Berg, Nick Bingham, Samuel Cox, Darrell Duffie,
Otto Elmgart, Malin Engström, Jan Ericsson, Damir Filipović, Andrea Gombani, Stefano Herzel, David Lando,
Angus MacDonald, Alexander Matros, Ragnar Norberg, Joel Reneby, Wolfgang Runggaldier, Per Sjöberg, Patrik
Säfvenblad, Nick Webber and Anna Vorwerk.
ThemainpartofthisbookhasbeenwrittenwhileIhavebeenattheFinanceDepartmentoftheStockholm Schoolof
Economics. I am deeply indebted to the school, the department and the staff working there for support and
encouragement.
Parts ofthebookwere writtenwhile I was stillat themathematicsdepartmentof KTH, Stockholm. Itis a pleasure to
acknowledge the support I got from the department and from the persons within it.
Finally I would like to express my deeply felt gratitude to Andrew Schuller, James Martin and Kim Roberts, all at
Oxford University Press, and Neville Hankins, the freelance copy-editor who worked on the book. The help given
(and patience shown) by these people has been remarkable and invaluable.
Stockholm, July 1998.
Tomas Björk
Contents
1 Introduction 1
1.1 Problem Formulation 1
2 The Binomial Model 6
2.1 The One Period Model 6
2.1.1 Model Description 6
2.1.2 Portfolios and Arbitrage 7
2.1.3 Contingent Claims 10
2.1.4 Risk Neutral Valuation 12
2.2 The Multiperiod Model 15
2.2.1 Portfolios and Arbitrage 15
2.2.2 Contingent Claims 18
2.3 Exercises 26
2.4 Notes 26
3 Stochastic Integrals 27
3.1 Introduction 27
3.2 Information 29
3.3 Stochastic Integrals 30
3.4 Martingales 33
3.5 Stochastic Calculus and the Itö Formula 35
3.6 Examples 40
3.7 The Multidimensional Itö Formula 43
3.8 Correlated Wiener Processes 45
3.9 Exercises 49
3.10 Notes 51
4 Differential Equations 52
4.1 Stochastic Differential Equations 52
4.2 Geometric Brownian Motion 53
4.3 The Linear SDE 56
4.4 The Infinitesimal Operator 57
4.5 Partial Differential Equations 58
4.6 The Kolmogorov Equations 62
4.7 Exercises 64
4.8 Notes 68
5 Portfolio Dynamics 69
5.1 Introduction 69
5.2 Self-financing Portfolios 72
CONTENTS ix
5.3 Dividends 74
5.4 Exercises 75
6 Arbitrage Pricing 76
6.1 Introduction 76
6.2 Contingent Claims and Arbitrage 77
6.3 The Black-Scholes Equation 82
6.4 Risk Neutral Valuation 85
6.5 The Black-Scholes Formula 88
6.6 Options on Futures 90
6.6.1 Forward Contracts 90
6.6.2 Futures Contracts and the Black Formula 91
6.7 Volatility 93
6.7.1 Historic Volatility 93
6.7.2 Implied Volatility 94
6.8 American options 94
6.9 Exercises 96
6.10 Notes 98
7 Completeness and Hedging 99
7.1 Introduction 99
7.2 Completeness in the Black-Scholes Model 100
7.3 Completeness--Absence of Arbitrage 105
7.4 Exercises 106
7.5 Notes 107
8 Parity Relations and Delta Hedging 108
8.1 Parity Relations 108
8.2 The Greeks 110
8.3 Delta and Gamma Hedging 113
8.4 Exercises 117
9 Several Underlying Assets 119
9.1 Introduction 119
9.2 Pricing 121
9.3 Risk Neutral Valuation 126
9.4 Reducing the State Space 127
9.5 Hedging 131
9.6 Exercises 134
10 Incomplete Markets 135
10.1 Introduction 135
10.2 A Scalar Nonpriced Underlying Asset 135
10.3 The Multidimensional Case 144
10.4 A Stochastic Rate of Interest 148
10.5 Summing Up 149
