Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. Weak convergence of numerical integration of SDE is interpreted as a functional belonging to a Sobolev space of negative order. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear. Contents......Page 9 1.1 Finite-Dimensional Gaussian Spaces, Hermite Expansion......Page 12 1.2 Wiener Space as Limit of its Dyadic Filtration......Page 16 1.3 Stroock–Sobolev Spaces of Functionals on Wiener Space......Page 18 1.4 Divergence of Vector Fields, Integration by Parts......Page 21 1.5 Itô’s Theory of Stochastic Integrals......Page 26 1.6 Differential and Integral Calculus in Chaos Expansion......Page 28 1.7 Monte-Carlo Computation of Divergence......Page 32 2.1 PDE Option Pricing; PDEs Governing the Evolution of Greeks......Page 35 2.2 Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedging......Page 40 2.4 Pathwise Smearing for European Options......Page 43 2.5 Examples of Computing Pathwise Weights......Page 45 2.6 Pathwise Smearing for Barrier Option......Page 47 3.1 Natural Metric Associated to Pathwise Smearing......Page 51 3.2 Price-Volatility Feedback Rate......Page 52 3.3 Measurement of the Price-Volatility Feedback Rate......Page 55 3.4 Market Ergodicity and Price-Volatility Feedback Rate......Page 56 4.1 Non-Degenerate Maps......Page 59 4.2 Divergences......Page 61 4.3 Regularity of the Law of a Non-Degenerate Map......Page 63 4.4 Multivariate Conditioning......Page 65 4.5 Riesz Transform and Multivariate Conditioning......Page 69 4.6 Example of the Univariate Conditioning......Page 71 5 Non-Elliptic Markets and Instability in HJM Models......Page 74 5.1 Notation for Diffusions on R[sup(N)]......Page 75 5.2 The Malliavin Covariance Matrix of a Hypoelliptic Diffusion......Page 76 5.4 Regularity by Predictable Smearing......Page 79 5.5 Forward Regularity by an Infinite-Dimensional Heat Equation......Page 81 5.6 Instability of Hedging Digital Options in HJM Models......Page 82 5.7 Econometric Observation of an Interest Rate Market......Page 84 6.1 A Toy Model: the Brownian Bridge......Page 86 6.2 Information Drift and Stochastic Calculus of Variations......Page 88 6.3 Integral Representation of Measure-Valued Martingales......Page 90 6.4 Insider Additional Utility......Page 92 6.5 An Example of an Insider Getting Free Lunches......Page 93 7 Asymptotic Expansion and Weak Convergence......Page 95 7.1 Asymptotic Expansion of SDEs Depending on a Parameter......Page 96 7.2 Watanabe Distributions and Descent Principle......Page 97 7.3 Strong Functional Convergence of the Euler Scheme......Page 98 7.4 Weak Convergence of the Euler Scheme......Page 101 8 Stochastic Calculus of Variations for Markets with Jumps......Page 105 8.1 Probability Spaces of Finite Type Jump Processes......Page 106 8.2 Stochastic Calculus of Variations for Exponential Variables......Page 108 8.3 Stochastic Calculus of Variations for Poisson Processes......Page 110 8.4 Mean-Variance Minimal Hedging and Clark–Ocone Formula......Page 112 A. Volatility Estimation by Fourier Expansion......Page 114 A.1 Fourier Transform of the Volatility Functor......Page 116 A.2 Numerical Implementation of the Method......Page 119 B. Strong Monte-Carlo Approximation of an Elliptic Market......Page 122 B.1 Definition of the Scheme S......Page 123 B.2 The Milstein Scheme......Page 124 B.3 Horizontal Parametrization......Page 125 B.4 Reconstruction of the Scheme S......Page 127 C. Numerical Implementation of the Price-Volatility Feedback Rate......Page 130 References......Page 133 D......Page 145 L......Page 146 S......Page 147 W......Page 148
Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.
