Stochastic Calculus of Variations in Mathematical Finance

by Paul Malliavin (Author), Anton Thalmaier (Author)



Contents
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 Ito's Theory of Stochastic Integrals  15
1.6 Differential and Integr al 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

4 Multivariate Conditioning and Regularity of Law . 49
4.1 Non-Degenerate Maps 49
4.2 Divergences . 51
4.3 Regulari ty of the Law of a Non-Degenerate Map 53
4.4 Multivariate Con ditioning 55
4.5 Riesz Transform and Mult ivari ate Condit ioning 59
4.6 Example of the Univar iate Condit ioning 61

5 Non-Elliptic Markets and Instability in HJM Models  65
5.1 Notation for Diffusions on l~N 66
5.2 The Malliavin Covariance Matrix of a Hyp oelliptic Diffusion 67
5.3 Malliav in Covariance Matrix and Horrnander 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 Informat ion 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 J ump Processes . 98
8.2 Stochastic Calculus of Variations for Exponential Variables 100
8.3 Stochastic Calculus of Variations for Poisson Processes 102
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 Implementat ion of the Method 112

B Strong Monte-Carlo Approximation
of an Elliptic Market 115
B.1 Definition of the Scheme Y 116
B.2 The Milstein Scheme 117
B.3 Hori zont al Parametrization 118
B.4 Reconstruction of the Scheme Y 120

C Numerical Implementation of the Price-Volatility Feedback Rate 123
References 127
Index 139