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 Spac es
of Functionals on Wiener Space ........................... 7
1.4 Divergence of Vector Fields , Integration by Parts ............ 10
1.5 It o'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 a n d 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 Inst abili ty of Hedging Digit al Opt ions
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 Calc ulus
of Variations ............................................ 79
6.3 Integral Representation
of Meas ure-Valued Martingales............................ 81
6.4 Insider Additional Ut ility ................................. 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 Wat anab e Dist ribut ions an d Descent Prin ciple 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