The field of financial
mathematics has developed tremendously over the past thirty years, and the underlying
models that have taken shape in interest rate markets and bond markets, being much richer
in structure than equity-derivative models, are particularly fascinating and complex. This
book introduces the tools required for the arbitrage-free modelling of the dynamics of
these markets. Andrew Cairns addresses not only seminal works but also modern
developments. Refreshingly broad in scope, covering numerical methods, credit risk, and
descriptive models, and with an approachable sequence of opening chapters, Interest Rate
Models will make readers--be they graduate students, academics, or
practitioners--confident enough to develop their own interest rate models or to price
nonstandard derivatives using existing models.
The mathematical chapters
begin with the simple binomial model that introduces many core ideas. But the main
chapters work their way systematically through all of the main developments in
continuous-time interest rate modelling. The book describes fully the broad range of
approaches to interest rate modelling: short-rate models, no-arbitrage models, the
Heath-Jarrow-Morton framework, multifactor models, forward measures, positive-interest
models, and market models. Later chapters cover some related topics, including numerical
methods, credit risk, and model calibration. Significantly, the book develops the
martingale approach to bond pricing in detail, concentrating on risk-neutral pricing,
before later exploring recent advances in interest rate modelling where different pricing
measures are important.
Andrew J. G. Cairns is
Professor of Financial Mathematics at Heriot-Watt University in the United Kingdom. After
completing his Ph.D. in statistics he worked as an actuary with a major life insurer, and
since rejoining academia he has specialized in interest rate modelling and financial risk
management for pension plans.
Endorsements:
"This book provides an
excellent introduction to the field of interest-rate modeling for readers at the graduate
level with a background in mathematics. It covers all key models and topics in the field
and provides first glances at practical issues (calibration) and important related fields
(credit risk). The mathematics is structured very well."--Rüdiger Kiesel, University
of Ulm, coauthor of Risk-Neutral Valuation
"A very useful book
that provides clear and comprehensive discussions of the topic that are not easily
available elsewhere."--Edwin J. Elton, New York University, author of Modern
Portfolio Theory and Investment Analysis
TABLE OF 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
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
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.1 The Multivariate Normal Distribution 241
A.2 Brownian Motion 241
A.3 Itô Integrals 242
A.4 One-Dimensional Itô 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
274 pages
