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Stochastic Calculus of Variations in Mathematical Finance

Paul Malliavin and Anton Thalmaier
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
142
Format: 
Hardcover
Series: 
Springer Finance
Price: 
59.95
ISBN: 
3-540-43431-3
Category: 
Monograph
[Reviewed by
Ita Cirovic Donev
, on
02/25/2006
]

This book deals with applications of the Malliavin calculus to finance problems. It is rather technical and is written in a theorem-proof style, so it would be best suited for graduate students and researchers. It is a research book more than anything else. The main aim of the book is to give the results of Malliavin calculus which are applied in finance in form of theorems with detailed proofs. The first couple of chapters are considered as prerequisities, while the rest can be read independently.

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.

1 Gaussian Stochastic Calculus of Variations . . . . . . . . . . . . . . . . . 1

1.1 Finite-Dimensional Gaussian Spaces,

Hermite Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Wiener Space as Limit of its Dyadic Filtration . . . . . . . . . . . . . . 5

1.3 Stroock–Sobolev Spaces

of Functionals on Wiener Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Divergence of Vector Fields, Integration by Parts . . . . . . . . . . . . 10

1.5 Itˆo’s Theory of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Differential and Integral Calculus

in Chaos Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 Monte-Carlo Computation of Divergence . . . . . . . . . . . . . . . . . . . 21

2 Computation of Greeks

and Integration by Parts Formulae . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 PDE Option Pricing; PDEs Governing

the Evolution of Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Stochastic Flow of Diffeomorphisms;

Ocone-Karatzas Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Principle of Equivalence of Instantaneous Derivatives . . . . . . . . 33

2.4 Pathwise Smearing for European Options . . . . . . . . . . . . . . . . . . . 33

2.5 Examples of Computing Pathwise Weights . . . . . . . . . . . . . . . . . . 35

2.6 Pathwise Smearing for Barrier Option . . . . . . . . . . . . . . . . . . . . . . 37

3 Market Equilibrium and Price-Volatility Feedback Rate . . . 41

3.1 Natural Metric Associated to Pathwise Smearing . . . . . . . . . . . . 41

3.2 Price-Volatility Feedback Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Measurement of the Price-Volatility Feedback Rate . . . . . . . . . . 45

3.4 Market Ergodicity

and Price-Volatility Feedback Rate . . . . . . . . . . . . . . . . . . . . . . . . 46

X Contents

4 Multivariate Conditioning

and Regularity of Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Non-Degenerate Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Regularity of the Law of a Non-Degenerate Map. . . . . . . . . . . . . 53

4.4 Multivariate Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Riesz Transform and Multivariate Conditioning . . . . . . . . . . . . . 59

4.6 Example of the Univariate Conditioning . . . . . . . . . . . . . . . . . . . . 61

5 Non-Elliptic Markets and Instability

in HJM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 Notation for Diffusions on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 The Malliavin Covariance Matrix

of a Hypoelliptic Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Malliavin Covariance Matrix

and H¨ormander Bracket Conditions . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Regularity by Predictable Smearing . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Forward Regularity

by an Infinite-Dimensional Heat Equation . . . . . . . . . . . . . . . . . . 72

5.6 Instability of Hedging Digital Options

in HJM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.7 Econometric Observation of an Interest Rate Market . . . . . . . . . 75

6 Insider Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1 A Toy Model: the Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Information Drift and Stochastic Calculus

of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Integral Representation

of Measure-Valued Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.4 Insider Additional Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5 An Example of an Insider Getting Free Lunches . . . . . . . . . . . . . 84

7 Asymptotic Expansion and Weak Convergence . . . . . . . . . . . . 87

7.1 Asymptotic Expansion of SDEs Depending

on a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2 Watanabe Distributions and Descent Principle . . . . . . . . . . . . . . 89

7.3 Strong Functional Convergence of the Euler Scheme . . . . . . . . . 90

7.4 Weak Convergence of the Euler Scheme . . . . . . . . . . . . . . . . . . . . 93

8 Stochastic Calculus of Variations for Markets with Jumps . 97

8.1 Probability Spaces of Finite Type Jump Processes . . . . . . . . . . . 98

8.2 Stochastic Calculus of Variations

for Exponential Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.3 Stochastic Calculus of Variations

for Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Contents XI

8.4 Mean-Variance Minimal Hedging

and Clark–Ocone Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A Volatility Estimation by Fourier Expansion . . . . . . . . . . . . . . . . 107

A.1 Fourier Transform of the Volatility Functor . . . . . . . . . . . . . . . . . 109

A.2 Numerical Implementation of the Method . . . . . . . . . . . . . . . . . . 112

B Strong Monte-Carlo Approximation

of an Elliptic Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

B.1 Definition of the Scheme S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B.2 The Milstein Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B.3 Horizontal Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B.4 Reconstruction of the Scheme S . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C Numerical Implementation

of the Price-Volatility Feedback Rate . . . . . . . . . . . . . . . . . . . . . . 123

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Tags: 
Stochastic Calculus