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Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective

René Carmona and Michael Tehranchi
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
235
Format: 
Hardcover
Series: 
Springer Finance
Price: 
89.95
ISBN: 
3540270655
Category: 
Monograph
[Reviewed by
Ita Cirovic Donev
, on
10/21/2006
]

Interest Rate Models is not your typical study book; rather, it is a research monograph on the theory of interest rate models in infinite dimension. The fixed-income market represents a large portion of the world financial market and therefore efficient and accurate methods to model interest rates are neccessary. This text deviates from the usual texts on modeling term structure, as it deals with infinite dimensions. It is yet to be proved (or even convincingly argued) that this approach could be of potential use in the real problems when it comes to fixed income markets.

The book is divided into three parts:

Part I - The Term Structure of Interest Rates

Part II - Infinite Dimensional Stochastic Analysis

Part III - Generalized Models for the Term Structure

Part I is a very basic introduction to the mechanics of fixed income markets and statiscal analysis. Very few pre-requisites are needed for this part. Concepts are presented in detail with appropriate examples. Some general term structure models are given.

Part II is more advanced. Here the authors present methods of stochastic calculus for infinite dimension. Readers with appropriate background in stochastic and functional analysis will benefit from the great discussion, others can just skim the pages to get the overview and maybe later go back and try to follow the proofs given. These are all very technical proofs, but with the aid of the context, which is written in an informative manner, even the proofs can be conquered.

Part III takes us back to the initial problem of modeling term structure. It shows us how to implement new techniques developed in Part II.

As I mentioned before this is a research monograph, and that is the way it is written — very technical. It is most suitable for reasearchers with good background in stochastic and functional analysis and some knowledge of the term structure models, even though the last is not formally a requirement.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

Part I The Term Structure of Interest Rates

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

XII Contents

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

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

Contents XIII

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 Itˆo 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

XIV Contents

Part III Generalized Models for the Term Structure

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 Solutions . . . . . . . . . . . . . . . . . . . . . . . . 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

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Stochastic Processes