Elementary Introduction To Stochastic Interest Rate Modeling, An (2nd Edition)

by Nicolas Privault

Overview

Interest rate modeling and the pricing of related derivatives remain subjects of increasing importance in financial mathematics and risk management. This book provides an accessible introduction to these topics by a step-by-step presentation of concepts with a focus on explicit calculations. Each chapter is accompanied with exercises and their complete solutions, making the book suitable for advanced undergraduate and graduate level students.This second edition retains the main features of the first edition while incorporating a complete revision of the text as well as additional exercises with their solutions, and a new introductory chapter on credit risk. The stochastic interest rate models considered range from standard short rate to forward rate models, with a treatment of the pricing of related derivatives such as caps and swaptions under forward measures. Some more advanced topics including the BGM model and an approach to its calibration are also covered.

Product Details

ISBN-13:	9789814390859
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/03/2012
Series:	Advanced Series On Statistical Science And Applied Probability , #16
Pages:	244
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface vii

1 A Review of Stochastic Calculus 1

1.1 Brownian Motion 1

1.2 Stochastic Integration 2

1.3 Quadratic Variation 8

1.4 Itô's Formula 10

1.5 Exercises 12

2 A Review of Black-Scholes Pricing and Hedging 15

2.1 Call and Put Options 15

2.2 Market Model and Portfolio 17

2.3 PDE Method 18

2.4 The Girsanov Theorem 20

2.5 Martingale Method 23

2.6 Exercises 30

3 Short Term Interest Rate Models 33

3.1 Mean-Reverting Models 33

3.2 Constant Elasticity of Variance (CEV) Models 34

3.3 Time-Dependent Models 35

3.4 Exercises 35

4 Pricing of Zero-Coupon Bonds 39

4.1 Definition and Basic Properties 39

4.2 Absence of Arbitrage and the Markov Property 40

4.3 Absence of Arbitrage and the Martingale Property 42

4.4 PDE Solution: Probabilistic Method 44

4.5 PDE Solution: Analytical Method 46

4.6 Numerical Simulations 47

4.7 Exercises 50

5 Foreward Rate Modeling 55

5.1 Forward Contracts 55

5.2 Instantaneous Forward Rate 58

5.3 Short Rates 60

5.4 Parametrization of Forward Rates 61

5.5 Curve Estimation 62

5.6 Exercises 63

6 The Heath-Jarrow-Morton (HJM) Model 65

6.1 Restatement of Objectives 65

6.2 Forward Vasicek Rates 67

6.3 Spot Forward Rate Dynamics 72

6.4 The HJM Condition 73

6.5 Markov Property of Short Rates 76

6.6 The Hull-White Model 78

6.7 Exercises 79

7 The Forward Measure and Derivative Pricing 81

7.1 Forward Measure 81

7.2 Dynamics under the Forward Measure 85

7.3 Derivative Pricing 88

7.4 Inverse Change of Measure 92

7.5 Exercises 93

8 Curve Fitting and a Two-Factor Model 97

8.1 Curve Fitting 97

8.2 Deterministic Shifts 100

8.3 The Correlation Problem 101

8.4 Two-Factor Model 104

8.5 Exercises 111

9 A Credit Default Model 115

9.1 Survival Probabilities 115

9.2 Stochastic Default 117

9.3 Defaultable Bonds 119

9.4 Credit Default Swaps 120

9.5 Exercises 122

10 Pricing of Caps and Swaptions on the LIBOR 125

10.1 Pricing of Caplets and Caps 125

10.2 Forward Rate Measure and Tenor Structure 127

10.3 Swaps and Swaptions 131

10.4 The London InterBank Offered Rates (LIBOR) Model 133

10.5 Swap Rates on the LIBOR Market 134

10.6 Forward Swap Measures 137

10.7 Swaption Pricing on the LIBOR Market 142

10.8 Exercises 143

11 The Brace-Gatarek-Musiela (BGM) Model 149

11.1 The BGM Model 149

11.2 Cap Pricing 152

11.3 Swaption Pricing 153

11.4 Calibration of the BGM Model 157

11.5 Exercises 160

12 Appendix A: Mathematical Tools 163

13 Appendix B: Some Recent Developments 171

14 Solutions to the Exercises 175

Bibliography 221

Index 225

Author Index 227