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.

"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

• 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 Ito 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

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.

"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

• 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 Ito 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