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Part Ⅰ The Term Structure of Interest Rates 3

1 Data and Instruments of the Term Structure of Interest Rates 3

1.1 Time Value of Money and Zero Coupon Bonds 3

1.1.1 Treasury Bills 4

1.1.2 Discount Factors and Interest Rates 5

1.2 Coupon Bearing Bonds 6

1.2.1 Treasury Notes and Treasury Bonds 7

1.2.2 The STRIPS Program 9

1.2.3 Clean Prices 10

1.3 Term Structure as Given by Curves 11

1.3.1 The Spot(Zero Coupon)Yield Curve 11

1.3.2 The Forward Rate Curve and Duration 13

1.3.3 Swap Rate Curves 14

1.4 Continuous Compounding and Market Conventions 17

1.4.1 Day Count Conventions 17

1.4.2 Compounding Conventions 19

1.4.3 Summary 20

1.5 Related Markets 21

1.5.1 Municipal Bonds 22

1.5.2 Index Linked Bonds 22

1.5.3 Corporate Bonds and Credit Markets 23

1.5.4 Tax Issues 25

1.5.5 Asset Backed Securities 25

1.6 Statistical Estimation of the Term Structure 25

1.6.1 Yield Curve Estimation 26

1.6.2 Parametric Estimation Procedures 27

1.6.3 Nonparametric Estimation Procedures 30

1.7 Principal Component Analysis 33

1.7.1 Principal Components of a Random Vector 33

1.7.2 Multivariate Data PCA 34

1.7.3 PCA of the Yield Curve 36

1.7.4 PCA of the Swap Rate Curve 39

Notes ＆ Complements 41

2 Term Structure Factor Models 43

2.1 Factor Models for the Term Structure 43

2.2 Affine Models 46

2.3 Short Rate Models as One-Factor Models 49

2.3.1 Incompleteness and Pricing 50

2.3.2 Specific Models 51

2.3.3 A PDE for Numerical Purposes 55

2.3.4 Explicit Pricing Formulae 57

2.3.5 Rigid Term Structures for Calibration 59

2.4 Term Structure Dynamics 60

2.4.1 The Heath-Jarrow-Morton Framework 60

2.4.2 Hedging Contingent Claims 62

2.4.3 A Shortcoming of the Finite-Rank Models 63

2.4.4 The Musiela Notation 64

2.4.5 Random Field Formulation 66

2.5 Appendices 67

Notes ＆ Complements 71

Part Ⅱ Infinite Dimensional Stochastic Analysis 75

3 Infinite Dimensional Integration Theory 75

3.1 Introduction 75

3.1.1 The Setting 77

3.1.2 Distributions of Gaussian Processes 78

3.2 Gaussian Measures in Banach Spaces and Examples 80

3.2.1 Integrability Properties 82

3.2.2 Isonormal Processes 83

3.3 Reproducing Kernel Hilbert Space 84

3.3.1 RKHS of Gaussian Processes 86

3.3.2 The RKHS of the Classical Wiener Measure 87

3.4 Topological Supports,Carriers,Equivalence and Singularity 88

3.4.1 Topological Supports of Gaussian Measures 88

3.4.2 Equivalence and Singularity of Gaussian Measures 89

3.5 Series Expansions 91

3.6 Cylindrical Measures 92

3.6.1 The Canonical(Gaussian)Cylindrical Measure of a Hilbert Space 93

3.6.2 Integration with Respect to a Cylindrical Measure 94

3.6.3 Characteristic Functions and Bochner's Theorem 94

3.6.4 Radonification of Cylindrical Measures 95

3.7 Appendices 96

Notes ＆ Complements 99

4 Stochastic Analysis in Infinite Dimensions 101

4.1 Infinite Dimensional Wiener Processes 101

4.1.1 Revisiting some Known Two-Parameter Processes 101

4.1.2 Banach Space Valued Wiener Process 103

4.1.3 Sample Path Regularity 103

4.1.4 Absolute Continuity Issues 104

4.1.5 Series Expansions 105

4.2 Stochastic Integral and It？ Processes 106

4.2.1 The Case of E＊-and H＊-Valued Integrands 108

4.2.2 The Case of Operator Valued Integrands 110

4.2.3 Stochastic Convolutions 112

4.3 Martingale Representation Theorems 114

4.4 Girsanov's Theorem and Changes of Measures 117

4.5 Infinite Dimensional Ornstein-Uhlenbeck Processes 119

4.5.1 Finite Dimensional OU Processes 119

4.5.2 Infinite Dimensional OU Processes 123

4.5.3 The SDE Approach in Infinite Dimensions 125

4.6 Stochastic Differential Equations 129

Notes ＆ Complements 132

5 The Malliavin Calculus 135

5.1 The Malliavin Derivative 135

5.1.1 Various Notions of Differentiability 135

5.1.2 The Definition of the Malliavin Derivative 138

5.2 The Chain Rule 141

5.3 The Skorohod Integral 142

5.4 The Clark-Ocone Formula 145

5.4.1 Sobolev and Logarithmic Sobolev Inequalities 146

5.5 Malliavin Derivatives and SDEs 149

5.5.1 Random Operators 150

5.5.2 A Useful Formula 152

5.6 Applications in Numerical Finance 153

5.6.1 Computation of the Delta 153

5.6.2 Computation of Conditional Expectations 155

Notes ＆ Complements 157

Part Ⅲ Generalized Models for the Term Structure 163

6 General Models 163

6.1 Existence of a Bond Market 163

6.2 The HJM Evolution Equation 164

6.2.1 Function Spaces for Forward Curves 164

6.3 The Abstract HJM Model 168

6.3.1 Drift Condition and Absence of Arbitrage 169

6.3.2 Long Rates Never Fall 171

6.3.3 A Concrete Example 173

6.4 Geometry of the Term Structure Dynamics 175

6.4.1 The Consistency Problem 176

6.4.2 Finite Dimensional Realizations 177

6.5 Generalized Bond Portfolios 182

6.5.1 Models of the Discounted Bond Price Curve 183

6.5.2 Trading Strategies 185

6.5.3 Uniqueness of Hedging Strategies 187

6.5.4 Approximate Completeness of the Bond Market 188

6.5.5 Hedging Strategies for Lipschitz Claims 189

Notes ＆ Complements 193

7 Specific Models 195

7.1 Markovian HJM Models 195

7.1. 1 Gaussian Markov Models 196

7.1.2 Assumptions on the State Space 197

7.1.3 Invariant Measures for Gauss-Markov HJM Models 198

7.1.4 Non-Uniqueness of the Invariant Measure 200

7.1.5 Asymptotic Behavior 201

7.1.6 The Short Rate is a Maximum on Average 201

7.2 SPDEs and Term Structure Models 203

7.2.1 The Deformation Process 204

7.2.2 A Model of the Deformation Process 205

7.2.3 Analysis of the SPDE 206

7.2.4 Regularity of the Solutions6 208

7.3 Market Models 210

7.3.1 The Forward Measure 210

7.3.2 LIBOR Rates Revisited 213

Notes ＆ Complements 214

References 217

Notation Index 225

Author Index 227

Subject Index 231