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Part Ⅰ The Term Structure of Interest Rates3

1 Data and Instruments of the Term Structure of Interest Rates3

1.1 Time Value of Money and Zero Coupon Bonds3

1.1.1 Treasury Bills4

1.1.2 Discount Factors and Interest Rates5

1.2 Coupon Bearing Bonds6

1.2.1 Treasury Notes and Treasury Bonds7

1.2.2 The STRIPS Program9

1.2.3 Clean Prices10

1.3 Term Structure as Given by Curves11

1.3.1 The Spot(Zero Coupon)Yield Curve11

1.3.2 The Forward Rate Curve and Duration13

1.3.3 Swap Rate Curves14

1.4 Continuous Compounding and Market Conventions17

1.4.1 Day Count Conventions17

1.4.2 Compounding Conventions19

1.4.3 Summary20

1.5 Related Markets21

1.5.1 Municipal Bonds22

1.5.2 Index Linked Bonds22

1.5.3 Corporate Bonds and Credit Markets23

1.5.4 Tax Issues25

1.5.5 Asset Backed Securities25

1.6 Statistical Estimation of the Term Structure25

1.6.1 Yield Curve Estimation26

1.6.2 Parametric Estimation Procedures27

1.6.3 Nonparametric Estimation Procedures30

1.7 Principal Component Analysis33

1.7.1 Principal Components of a Random Vector33

1.7.2 Multivariate Data PCA34

1.7.3 PCA of the Yield Curve36

1.7.4 PCA of the Swap Rate Curve39

Notes ＆ Complements41

2 Term Structure Factor Models43

2.1 Factor Models for the Term Structure43

2.2 Affine Models46

2.3 Short Rate Models as One-Factor Models49

2.3.1 Incompleteness and Pricing50

2.3.2 Specific Models51

2.3.3 A PDE for Numerical Purposes55

2.3.4 Explicit Pricing Formulae57

2.3.5 Rigid Term Structures for Calibration59

2.4 Term Structure Dynamics60

2.4.1 The Heath-Jarrow-Morton Framework60

2.4.2 Hedging Contingent Claims62

2.4.3 A Shortcoming of the Finite-Rank Models63

2.4.4 The Musiela Notation64

2.4.5 Random Field Formulation66

2.5 Appendices67

Notes ＆ Complements71

Part Ⅱ Infinite Dimensional Stochastic Analysis75

3 Infinite Dimensional Integration Theory75

3.1 Introduction75

3.1.1 The Setting77

3.1.2 Distributions of Gaussian Processes78

3.2 Gaussian Measures in Banach Spaces and Examples80

3.2.1 Integrability Properties82

3.2.2 Isonormal Processes83

3.3 Reproducing Kernel Hilbert Space84

3.3.1 RKHS of Gaussian Processes86

3.3.2 The RKHS of the Classical Wiener Measure87

3.4 Topological Supports,Carriers,Equivalence and Singularity88

3.4.1 Topological Supports of Gaussian Measures88

3.4.2 Equivalence and Singularity of Gaussian Measures89

3.5 Series Expansions91

3.6 Cylindrical Measures92

3.6.1 The Canonical(Gaussian)Cylindrical Measure of a Hilbert Space93

3.6.2 Integration with Respect to a Cylindrical Measure94

3.6.3 Characteristic Functions and Bochner's Theorem94

3.6.4 Radonification of Cylindrical Measures95

3.7 Appendices96

Notes ＆ Complements99

4 Stochastic Analysis in Infinite Dimensions101

4.1 Infinite Dimensional Wiener Processes101

4.1.1 Revisiting some Known Two-Parameter Processes101

4.1.2 Banach Space Valued Wiener Process103

4.1.3 Sample Path Regularity103

4.1.4 Absolute Continuity Issues104

4.1.5 Series Expansions105

4.2 Stochastic Integral and It？ Processes106

4.2.1 The Case of E＊-and H＊-Valued Integrands108

4.2.2 The Case of Operator Valued Integrands110

4.2.3 Stochastic Convolutions112

4.3 Martingale Representation Theorems114

4.4 Girsanov's Theorem and Changes of Measures117

4.5 Infinite Dimensional Ornstein-Uhlenbeck Processes119

4.5.1 Finite Dimensional OU Processes119

4.5.2 Infinite Dimensional OU Processes123

4.5.3 The SDE Approach in Infinite Dimensions125

4.6 Stochastic Differential Equations129

Notes ＆ Complements132

5 The Malliavin Calculus135

5.1 The Malliavin Derivative135

5.1.1 Various Notions of Differentiability135

5.1.2 The Definition of the Malliavin Derivative138

5.2 The Chain Rule141

5.3 The Skorohod Integral142

5.4 The Clark-Ocone Formula145

5.4.1 Sobolev and Logarithmic Sobolev Inequalities146

5.5 Malliavin Derivatives and SDEs149

5.5.1 Random Operators150

5.5.2 A Useful Formula152

5.6 Applications in Numerical Finance153

5.6.1 Computation of the Delta153

5.6.2 Computation of Conditional Expectations155

Notes ＆ Complements157

Part Ⅲ Generalized Models for the Term Structure163

6 General Models163

6.1 Existence of a Bond Market163

6.2 The HJM Evolution Equation164

6.2.1 Function Spaces for Forward Curves164

6.3 The Abstract HJM Model168

6.3.1 Drift Condition and Absence of Arbitrage169

6.3.2 Long Rates Never Fall171

6.3.3 A Concrete Example173

6.4 Geometry of the Term Structure Dynamics175

6.4.1 The Consistency Problem176

6.4.2 Finite Dimensional Realizations177

6.5 Generalized Bond Portfolios182

6.5.1 Models of the Discounted Bond Price Curve183

6.5.2 Trading Strategies185

6.5.3 Uniqueness of Hedging Strategies187

6.5.4 Approximate Completeness of the Bond Market188

6.5.5 Hedging Strategies for Lipschitz Claims189

Notes ＆ Complements193

7 Specific Models195

7.1 Markovian HJM Models195

7.1. 1 Gaussian Markov Models196

7.1.2 Assumptions on the State Space197

7.1.3 Invariant Measures for Gauss-Markov HJM Models198

7.1.4 Non-Uniqueness of the Invariant Measure200

7.1.5 Asymptotic Behavior201

7.1.6 The Short Rate is a Maximum on Average201

7.2 SPDEs and Term Structure Models203

7.2.1 The Deformation Process204

7.2.2 A Model of the Deformation Process205

7.2.3 Analysis of the SPDE206

7.2.4 Regularity of the Solutions6208

7.3 Market Models210

7.3.1 The Forward Measure210

7.3.2 LIBOR Rates Revisited213

Notes ＆ Complements214

References217

Notation Index225

Author Index227

Subject Index231