Interest Rate Models: An Introduction

289 Pages·2004·21.24 MB·English

by  Andrew J. G. Cairns

Interest Rate Models

Interest Rate Models
An Introduction
Andrew J. G. Cairns
Princeton University Press
Princeton and Oxford
Copyright © 2004 by Princeton University Press
Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
3 Market Place, Woodstock, Oxfordshire OX20 1SY
All rights reserved
Library of Congress Cataloguing-in-Publication Data
Cairns, Andrew (Andrew J. G.)
Interest rate models: an introduction / Andrew J. G. Cairns,
p.cm.
Includes bibliographical references and index.
ISBN 0-691-11893-0 (cl.: alk. paper) — ISBN 0-691-11894-9 (pbk.: alk. paper)
1. Interest rates—Mathematical models. 2. Bonds—Mathematical models.
3. Securities—Mathematical models. 4. Derivative securities—Prices—Mathematical models.
I. Title.
HG1621.C25 2004
332.8'01'51—dc22 2003062309
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
This book has been composed in Times and typeset by T&T Productions Ltd, London
Printed on acid-free paper @
www.pupress.princeton.edu
Printed in the United States of America
10 9876543
ISBN-13: 978-0-691-11894-9 (pbk.)
ISBN-10: 0-691-11894-9 (pbk.)
Contents
Preface ix
Acknowledgements xiii
1 Introduction to Bond Markets 1
1.1 Bonds 1
1.2 Fixed-Interest Bonds 2
1.3 STRIPS 10
1.4 Bonds with Built-in Options 10
1.5 Index-Linked Bonds 10
1.6 General Theories of Interest Rates 11
1.7 Exercises 13
2 Arbitrage-Free Pricing 15
2.1 Example of Arbitrage: Parallel Yield Curve Shifts 16
2.2 Fundamental Theorem of Asset Pricing 18
2.3 The Long-Term Spot Rate 19
2.4 Factors 23
2.5 A Bond Is a Derivative 23
2.6 Put-Call Parity 23
2.7 Types of Model 24
2.8 Exercises 25
3 Discrete-Time Binomial Models 29
3.1 A Simple No-Arbitrage Model 29
3.2 The Ho and Lee No-Arbitrage Model 30
3.3 Recombining Binomial Model 32
3.4 Models for the Risk-Free Rate of Interest 37
3.5 Futures Contracts 45
3.6 Exercises 48
4 Continuous-Time Interest Rate Models 53
4.1 One-Factor Models for the Risk-Free Rate 53
4.2 The Martingale Approach 55
4.3 The PDE Approach to Pricing 60
vi Contents
4.4 Further Comment on the General Results 64
4.5 The Vasicek Model 64
4.6 The Cox-Ingersoll-Ross Model 66
4.7 A Comparison of the Vasicek and Cox-Ingersoll-Ross Models 70
4.8 Affine Short-Rate Models 74
4.9 Other Short-Rate Models 77
4.10 Options on Coupon-Paying Securities 77
4.11 Exercises 78
5 No-Arbitrage Models 85
5.1 Introduction 85
5.2 Markov Models 86
5.3 The Heath-Jarrow-Morton (HJM) Framework 91
5.4 Relationship between HJM and Markov Models 96
5.5 Exercises 97
6 Multifactor Models 101
6.1 Introduction 101
6.2 Affine Models 102
6.3 Consols Models 112
6.4 Multifactor Heath-Jarrow-Morton Models 115
6.5 Options on Coupon-Paying Securities 116
6.6 Quadratic Term-Structure Models (QTSMs) 118
6.7 Other Multifactor Models 118
6.8 Exercises 119
7 The Forward-Measure Approach 121
7.1 A New Numeraire 121
7.2 Change of Measure 122
7.3 Derivative Payments 122
7.4 A Replicating Strategy 123
7.5 Evaluation of a Derivative Price 124
7.6 Equity Options with Stochastic Interest 126
7.7 Exercises 128
8 Positive Interest 131
8.1 Introduction 131
8.2 Mathematical Development 131
8.3 The Flesaker and Hughston Approach 134
8.4 Derivative Pricing 135
8.5 Examples 136
8.6 Exercises 142
9 Market Models 143
9.1 Market Rates of Interest 143
9.2 LIBOR Market Models: the BGM Approach 144
9.3 Simulation of LIBOR Market Models 152
9.4 Swap Market Models 153
9.5 Exercises 155
Contents vii
10 Numerical Methods 159
10.1 Choice of Measure 159
10.2 Lattice Methods 160
10.3 Finite-Difference Methods 168
10.4 Numerical Examples 178
10.5 Simulation Methods 184
10.6 Exercise 196
11 Credit Risk 197
11.1 Introduction 197
11.2 Structural Models 199
11.3 A Discrete-Time Model 201
11.4 Reduced-Form Models 206
11.5 Derivative Contracts with Credit Risk 218
11.6 Exercises 222
12 Model Calibration 227
12.1 Descriptive Models for the Yield Curve 227
12.2 A General Parametric Model 228
12.3 Estimation 229
12.4 Splines 234
12.5 Volatility Calibration 238
12.6 Exercises 239
Appendix A Summary of Key Probability and SDE Theory 241
A.l The Multivariate Normal Distribution 241
A.2 Brownian Motion 241
A.3 Itô Integrals 242
A.4 One-Dimensional Ito and Diffusion Processes 243
A.5 Multi-Dimensional Diffusion Processes 244
A.6 The Feynman-Kac Formula 245
A.7 The Martingale Representation Theorem 246
A.8 Change of Probability Measure 246
Appendix B The Vasicek and CIR Models: Proofs 249
B.1 The Vasicek Model 249
B.2 The Cox-Ingersoll-Ross Model 253
References 265
Index 271

Preface
The past thirty years or so have seen considerable development in the field of financial
mathematics: first, in the field of equity derivatives following on from the work of
Black, Scholes and Merton; and then in the theory of bond pricing and derivatives
following on, for example, from Vasicek's work. If we wish to model a stock market
in which prices evolve in a way that is free of arbitrage, the move from equities to
bonds adds a whole new level of complexity and interest for the modeller. This is
because we have a large number of tradable assets whose price dynamics typically
depend upon the same (small number of) random factors. As a result, we must
ensure with any model that the prices of these assets all evolve in a way that avoids
arbitrage.
In recent years a considerable number of textbooks have been written that cover
this now broad field. These range widely in their level of comprehensiveness and
technical difficulty.
The origin of this book lies within a graduate-level lecture course on bond pricing
given to students on the MSc in Financial Mathematics at Heriot-Watt University
and Edinburgh University. While there exist textbooks that cover this topic, it was
felt that none were entirely appropriate for the present course.
The book is aimed at people who are just starting to learn the subject of interest rate
modelling. Thus, the primary readership is intended to include students on advanced
taught courses, doctoral students and financial market practitioners learning about
bond pricing and bond-derivative pricing for the first time. Other readers who are
familiar with the basics of interest rate modelling will hopefully also find much of
interest in the second half of the book, where I move on to more advanced and more
recent topics. Finally, there are other practitioners in areas such as insurance who
are not involved with the day-to-day running of a bond portfolio or a derivatives
operation but who nevertheless need good interest rate models. I am primarily
thinking of my original field of actuarial science, where having a good model for
interest rates is becoming increasingly important. These practitioners should also
find the book useful.
The level at which I have written the book is intended to make it accessible and
helpful to masters-level students in financial mathematics. Students typically will
have a good first degree in the mathematical sciences and will already be com­
fortable with probability, stochastic processes, stochastic differential equations and
arbitrage-free pricing of equity derivatives (including the Fundamental Theorem