Arbitrage theory in continuous time

584 Pages·2020·4.803 MB·English

by  Tomas Björk

OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
Arbitrage Theory in Continuous Time
Fourth Edition

OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
Arbitrage Theory in
Continuous Time
fourth edition
tomas bjo¨rk
Stockholm School of Economics
1
OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
3
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OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
To Agneta, Kajsa, and Stefan
OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
PREFACE TO THE FOURTH EDITION
The fourth edition differs from the third edition by the fact that I have added
chapters on the following subjects.
• Incomplete markets. This includes the theory of Esscher transforms,
minimal martingale measures, f-divergences, portfolio optimization in
incomplete markets, indifference pricing, and good deal bounds.
• Equilibrium theory. This is an introduction to the (vast) area of dynamic
equilibrium theory. It includes the Cox–Ingersoll–Ross production factor
model as well as the basic theory for unit net supply endowment models.
DuetospacelimitationsIhavedeletedachapterontheBlack–Scholesapproach
to multi asset models, and the chapter on barrier options. The interested reader
can find an excellent exposition of barrier options (and many other topics) in
Joshi (2008).
Prerequisites. For Chapters 1–10 the necessary mathematical background is a
good knowledge of advanced calculus and elementary probability theory. From
Chapter11andonwards,thetheoreticallevelishigher,andinordertoreadmost
of the following chapters the reader should be familiar with basic measure and
integration theory, as well as with abstract probability theory. A self-contained
introduction to these topics can be found in Appendices A–C.
I have corrected a large number of typos and other errors from the third
edition, and I am very grateful for comments from Nicholas Amuyedo, Andrey
Ashikhmin, Adnan Buyukbilgin, Jiakai Chen, JosephClark, ArrigoCoen Coria,
AndersDahlner,MartinJo¨nsson,EdwardKao,AlexKarpenko,YavorKovachev,
Katka Lucivjanska, Andrey Lizyayev, Glenn Mickelsson, Asad Munir, Kevin
Schmid, Alexander Szimayer, Bill Thygerson, A.M. Underwood, Sebastian
Wagner, and Russell Yang Gao. In all probability there are still several typos
left. If you find any of these, I would be very grateful if you could inform me by
e-mail <[email protected]>.
I am much indebted to Mariana Khapko, Ema Iancu, and Simon Wehrmu¨ller.
They persuaded me to start writing lecture notes on equilibrium theory and
theytookaveryactivepartinaninformalseminarseriesonthesubject.Special
thanks are due to Mariana Khapko, who has discussed most of the equilibrium
topics in detail with me.
Tomas Bjo¨rk
Stockholm
March 1, 2019
OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
PREFACE TO THE FIRST EDITION
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,andstatisticsandIalsohopethatitwillbeusefulforpractitioners.
Becauseofitsintendedaudience,thebookdoesnotpresupposeanyprevious
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 tool used in the book is the theory of stochastic
differential equations (SDEs), and instead of going into the technical details
concerning the foundations of that theory I have focused on applications. The
object is to give the reader, as quickly and painlessly as possible, a solid
working knowledge of the powerful mathematical tool known as Itˆo calculus.
WetreatbasicSDEtechniques,includingFeynman–Kaˇcrepresentationsandthe
Kolmogorovequations.Martingalesareintroducedatanearlystage.Throughout
thebookthereisastrongemphasisonconcretecomputations,andtheexercises
at the end of each chapter constitute an integral part of the text.
The mathematics developed in the first part of the book is then applied
to arbitrage pricing of financial derivatives. We cover the basic Black–Scholes
theory, including delta hedging and “the Greeks”, and we extend it to the case
of several underlying assets (including stochastic interest rates) as well as to
dividend-payingassets.Barrieroptions,aswellascurrencyandquantoproducts,
aregivenseparatechapters.Wealsoconsider,insomedetail,incompletemarkets.
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.
Interest rate theory constitutes a large part of the book, and we cover
the basic short rate theory, including inversion of 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 allows us, despite the fact that we do not formally
use measure theory, to give a fairly complete treatment of the general change
of numeraire technique which is so essential to modern interest rate theory.
OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
viii PREFACE
In particular we treat forward neutral measures in some 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 (2003) and more advanced texts such as Duffie (2001) or Musiela and
Rutkowski (1997). These books are used as canonical references in the present
text.
In order to facilitate using the book for shorter courses, the pedagogical
approach has been that of first presenting 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
someargumentsarebeingrepeated,butthisseemstobeunavoidable,andIcan
only apologize to the technically more advanced reader.
Notes to the literature can be found at the end of most chapters. I have
tried to keep the reference list on a manageable scale, but any serious omission
is unintentional, and I will be happy to correct it. For more bibliographic
informationthereaderisreferredtoDuffie(1996)andtoMusielaandRutkowski
(1997) which both contain encyclopedic bibliographies.
On the more technical side the following facts can be mentioned. I have tried to
presentareasonablyhonestpictureofSDEtheory,includingFeynman–Kaˇcrep-
resentations,whileavoidingtheexplicituseofabstractmeasuretheory.Because
of the chosen technical level, the arguments concerning the construction of the
stochasticintegralarethusforcedtobemoreorlessheuristic.NeverthelessIhave
tried to be as precise as possible, so even the heuristic arguments are the “cor-
rect”onesinthesensethattheycanbecompletedtoformalproofs.Intherestof
the text I try to give full proofs of all mathematical statements, with the excep-
tion that I have often left out the checking of various integrability conditions.
Since the Girsanov theory for absolutely continuous 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–Kaˇc 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
completenessinfullgenerality,butrestrictourselvestoaMarkovianframework.
For most applications this is, however, general enough.
Tomas Bjo¨rk
Stockholm
July 1998
OUP CORRECTED PROOF – FINAL, 18/11/2019, SPi
ACKNOWLEDGEMENTS
Bertil N¨aslund, Staffan Viotti, Peter Jennergren, and Ragnar Lindgren per-
suaded me to start studying financial economics, and they have constantly and
generously shared their knowledge with me.
Hans Bu¨hlman, Paul Embrechts, and Hans Gerber gave me the opportunity
to give a series of lectures for a summer school at Monte Verita in Ascona 1995.
This summer school was for me an extremely happy and 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.
Overtheyearsofwriting,Ihavereceivedvaluablecommentsandadvicefrom
alargenumberofpeople.MygreatestdebtistoCamillaLand´en,whohasgiven
memoregoodadvice(andpointedoutmoreerrors)thanIthoughtwashumanly
possible. I am also highly indebted to Flavio Angelini, Pia Berg, Nick Bingham,
SamuelCox,DarrellDuffie,OttoElmgart,MalinEngstro¨m,JanEricsson,Damir
Filipovi´c, Andrea Gombani, Stefano Herzel, David Lando, Angus MacDonald,
Alexander Matros, Ragnar Norberg, Joel Reneby, Wolfgang Runggaldier, Per
Sjo¨berg, Patrik S¨afvenblad, Nick Webber, and Anna Vorwerk.
ThemainpartofthisbookhasbeenwrittenwhileIhavebeenattheFinance
Department of the Stockholm School of Economics. I am deeply indebted
to the school, the department, and the staff working there for support and
encouragement.
Parts of the book were written while I was still at the mathematics
department of KTH, Stockholm. It is 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.
This book would never have come into existence without the hard work
of Judith Acevedo, Kumar Anbazhagan, Katie Bishop, and Kathleen Gill. All
of these have worked on the book through the production process with care,
professionalism, and efficiency. For this I am truly grateful.