Arbitrage Theory in Continuous Time

481 Pages·2007·13.19 MB·English

Arbitrage Theory in
Continuous Time
Second Edition
OXFORD
UNIVERSITY PRESS
LJ
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PREFACE TO THE SECOND EDITION
One of the main ideas behind the first edition of this book was to provide
a reasonably honest introduction to arbitrage theory without going into abstract
measure and integration theory. This approach, however, had some clear draw-
backs: some topics, like the change of numeraire theory and the recently
developed LIBOR and swap market models, are very hard to discuss without
using the language of measure theory, and an important concept like that of
a martingale measure can be fully understood only within a measure theoretic
framework.
, For the second edition I have therefore decided to include some more advanced
material, but, in order to keep the book accessible for the reader who does not
want to study measure theory, I have organized the text as follows:
The more advanced parts of the book are marked with a star *.
1' The main parts of the book are virtually unchanged and kept on an
elementary level (i.e. not marked with a star).
f The reader who is looking for an elementary treatment can simply skip
1'
the starred chapters and sections. The nonstarred sections thus constitute
a self-contained course on arbitrage theory.
The organization and contents of the new parts are as follows:
I have added appendices on measure theory, probability theory, and mar-
tingale theory. These appendices can be used for a lighthearted but honest
introductory course on the corresponding topics, and they define the pre-
requisites for the advanced parts of the main text. In the appendices there
is an emphasis on building intuition for basic concepts, such as measur-
ability, conditional expectation, and measure changes. Most results are
given formal proofs but for some results the reader is referred to the
literature.
8 There is a new chapter on the martingale approach to arbitrage theory,
where we discuss (in some detail) the First and Second Fundamental The-
orems of mathematical finance, i.e. the connections between absence of
arbitrage, the existence of martingale measures, and completeness of the
market. The full proofs of these results are very technical but I have tried
to provide a fairly detailed guided tour through the theory, including the
+
Delbaen-Schachermayer proof of the First Fundamental Theorem.
Following the chapter on the general martingale approach there is a sep
r
mate chapter on martingale representation theorems and Girsanov trans-
formations in a Wiener framework. Full proofs are given and I have also
added a section on maximum likelihood estimation for diffusion processes.
viii PREFACE TO THE SECOND EDITION
As the obvious application of the machinery developed above, there is
a chapter where the Black-Scholes model is discussed in detail from the
martingale point of view. There is also an added chapter on the martingale
approach to multidimensional models, where these are investigated in some
detail. In particular we discuss stochastic discount factors and derive the
Hansen-Jagannathan bounds.
The old chapter on changes of numeraire always suffered from the restric-
tion to a Markovian setting. It has now been rewritten and placed in its
much more natural martingale setting.
I have added a fairly extensive chapter on the LIBOR and swap market
models which have become so important in interest rate theory.
Acknowledgements
Since the publication of the first edition I have received valuable comments
and help from a large number of people. In particular I am very grateful to
Raquel Medeiros Gaspar who, apart from pointing out errors and typos, has
done a splendid job in providing written solutions to a large number of the exer-
cises. I am also very grateful to Ake Gunnelin, Mia Hinnerich, Nuutti Kuosa,
Roger Lee, Trygve Nilsen, Ragnar Norberg, Philip Protter, Rolf Poulsen, Irina
Slinko, Ping Wu, and K.P. Garnage. It is a pleasure to express my deep grati-
tude to Andrew Schuller and Stuart Fowkes, both at OUP, for transforming
the manuscript into book form. Their importance for the final result cannot be
overestimated.
Special thanks are due to Kjell Johansson and Andrew Sheppard for providing
important and essential input at crucial points.
Tomas Bjork
Stockholm
30 April 2003
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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 gradu-
ate 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 know-
ledge 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 con-
cerning 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 know-
ledge of the powerful mathematical tool known as It6 calculus. We treat basic
SDE techniques, including Feynman-KaE 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.
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 the-
ory, 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 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
i I have included a chapter on stochastic optimal control and its applications to
optimal portfolio selection. i
1 Interest rate theory constitutes a large pfU3 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 object-
ive 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
k
x PREFACE TO THE FIRST EDITION
theory, to give a fairly complete treatment of the general change of numeraire
technique which is so essential to modern interest rate theory. 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 (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.
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
(multidime~sional)f ramework. 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.
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 informa-
tion the reader is referred to Duffie (1996) and to Musiela and Rutkowski (1997)
which both contain encyclopedic bibliographies.
On the more technical side the following facts can be mentioned. I have tried to
present a reasonably honest picture of SDE theory, including Feynman-Kat rep
resentations, 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 arguments are the
"correct" ones in the sense that they can beaompleted to formal proofs. In the
rest of the text I try to 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 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
Feynrnan-KaE 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.
C PREFACE TO THE FIRST EDITION
Acknowledgements
Bertil Nblund, 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 Biihlman, 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.
Over the years of writing, I have received valuable comments and advice from
a large number of people. My greatest debt is to Camilla Landen 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 Engstrom, Jan Ericsson, Damir
FilipoviE, Andrea Gombani, Stefano Herzel, David Lando, Angus MacDonald,
Alexander Matros, Ragnar Norberg, Joel Reneby, Wolfgang Runggaldier, Per
Sjoberg, Patrik Siifvenblad, Nick Webber, and Anna Vorwerk.
The main part of this book has been written while I have been at the Fin-
ance Department of the Stockholm School of Economics. I am deeply indebted
to the school, the department and the st& working there for support and
Parts of the book were written while I was still at the mathematics depart-
ment of KTH, Stockholm. It is a pleasure to acknowledge the support I got from
the department and from the persons within it.
I
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,, Me freelance copy-editor who worked on the book. The help given
(and patience shown) by these people has been remarkable and invaluable.
Tomas Bjork
CONTENTS
1 Introduction
1.1 Problem Formulation
2 The Binomial Model
2.1 The One Period Model
2.1.1 Model Description
2.1.2 Portfolios and Arbitrage
2.1.3 Contingent Claims
2.1.4 Risk Neutral Valuation
2.2 The Multiperiod Model
2.2.1 Portfolios and Arbitrage
2.2.2 Contingent Claims
2.3 Exercises
2.4 Notes
3 A More General One Period Model
3.1 The Model
3.2 Absence of Arbitrage
3.3 Martingale Pricing
3.4 Completeness
3.5 Stochastic Discount Factors
3.6 Exercises
4 Stochastic Integrals
4.1 Introduction
4.2 Information
4.3 Stochastic Integrals
4.4 Martingales
4.5 Stochastic Calculus and the It8 Formula
4.6 Examples
4.7 The Multidimensional It6 Formula
4.8 Correlated Wiener Processes
4.9 Exercises
4.10 Notes
5 Differential Equations
5.1 Stochastic Differential Equations
5.2 Geometric Brownian Motion
5.3 The Linear SDE
5.4 The Infinitesimal Operator
CONTENTS xiii
5.5 Partial Differential Equations 68
5.6 The Kolmogorov Equations
5.7 Exercises
1
Y 5.8 Notes
6 Portfolio Dynamics
6.1 Introduction
6.2 Self-financing Portfolios
-
'p' 6.3 Dividends
6.4 Exercise
Arbitrage Pricing
7.1 Introduction
7.2 Contingent Claims and Arbitrage
7.3 The Black-Scholes Equation
7.4 Risk Neutral Valuation
7.5 The Black-Scholes Formula
7.6 Options on Futures
7.6.1 Forward Contracts
7.6.2 Futures Contracts and the Black Formula
7.7 Volatility
7.7.1 Historic Volatility
7.7.2 Implied Volatility
7.8 American options
7.9 Exercises
7.10 Notes
8 Completeness and Hedging
8.1 Introduction
8.2 Completeness in the Black-Scholes Model
8.3 Completeness-Absence of Arbitrage
1 8.4 Exercises
8.5 Notes
i
9 Parity Relations and Delta Hedging
-
- 9.1 Parity Relations
9.2 The Greeks
9.3 Delta and Gamma Hedging
1 E 9.4 Exercises
I
10 The Martingale Approach to Arbitrage Theory*
10.1 The Case with Zero Interest Rate
10.2 Absence of Arbitrage
10.2.1 A Rough Sketch of the Proof
10.2.2 Precise Results
10.3 The General Case 4 6
10.4 Completeness
10.5 Martingale Pricing
10.6 Stochastic Discount Factors
10.7 Summary for the Working Economist
10.8 Notes
11 The Mathematics of the Martingale Approach*
11.1 Stochastic Integral Representations
11.2 The Girsanov Theorem: Heuristics
11.3 The Girsanov Theorem
11.4 The Converse of the Girsanov Theorem
11.5 Girsanov Transformations and Stochastic Differentials
11.6 Maximum Likelihood Estimation
11.7 Exercises
11.8 Notes
12 Black-Scholes from a Martingale Point of View*
12.1 Absence of Arbitrage
12.2 Pricing
12.3 Completeness
13 Multidimensional Models: Classical Approach
13.1 Introduction
13.2 Pricing
13.3 Risk Neutral Valuation
13.4 RRducing the State Space
13.5 Hedging
13.6 Exercises
14 Multidimensional Models: Martingale Approach*
14.1 Absence of Arbitrage
14.2 Completeness
14.3 Hedging
14.4 Pricing
14.5 Markovian Models and PDEs
14.6 Market Prices of Risk
14.7 Stochastic Discount Factors
14.8 The Hansen-Jagannathan Bounds
14.9 Exercises
14.10 Notes
15 Incomplete Markets
15.1 Introduction
15.2 A Scalar Nonpriced Underlying Asset
15.3 The Multidimensional Case