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