Stochastic Calculus of Variations in Mathematical Finance / Edition 1

Stochastic Calculus of Variations in Mathematical Finance / Edition 1

by Paul Malliavin, Anton Thalmaier
     
 

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

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Overview

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.

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Product Details

ISBN-13:
9783540434313
Publisher:
Springer Berlin Heidelberg
Publication date:
12/19/2005
Series:
Springer Finance Series
Edition description:
2006
Pages:
142
Product dimensions:
0.44(w) x 6.14(h) x 9.21(d)

Table of Contents

1Gaussian Stochastic Calculus of Variations1
1.1Finite-Dimensional Gaussian Spaces, Hermite Expansion1
1.2Wiener Space as Limit of its Dyadic Filtration5
1.3Stroock-Sobolev Spaces of Functionals on Wiener Space7
1.4Divergence of Vector Fields, Integration by Parts10
1.5Ito's Theory of Stochastic Integrals15
1.6Differential and Integral Calculus in Chaos Expansion17
1.7Monte-Carlo Computation of Divergence21
2Computation of Greeks and Integration by Parts Formulae25
2.1PDE Option Pricing; PDEs Governing the Evolution of Greeks25
2.2Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedging30
2.3Principle of Equivalence of Instantaneous Derivatives33
2.4Pathwise Smearing for European Options33
2.5Examples of Computing Pathwise Weights35
2.6Pathwise Smearing for Barrier Option37
3Market Equilibrium and Price-Volatility Feedback Rate41
3.1Natural Metric Associated to Pathwise Smearing41
3.2Price-Volatility Feedback Rate42
3.3Measurement of the Price-Volatility Feedback Rate45
3.4Market Ergodicity and Price-Volatility Feedback Rate46
4Multivariate Conditioning and Regularity of Law49
4.1Non-Degenerate Maps49
4.2Divergences51
4.3Regularity of the Law of a Non-Degenerate Map53
4.4Multivariate Conditioning55
4.5Riesz Transform and Multivariate Conditioning59
4.6Example of the Univariate Conditioning61
5Non-Elliptic Markets and Instability in HJM Models65
5.1Notation for Diffusions on R[superscript N]66
5.2The Malliavin Covariance Matrix of a Hypoelliptic Diffusion67
5.3Malliavin Covariance Matrix and Hormander Bracket Conditions70
5.4Regularity by Predictable Smearing70
5.5Forward Regularity by an Infinite-Dimensional Heat Equation72
5.6Instability of Hedging Digital Options in HJM Models73
5.7Econometric Observation of an Interest Rate Market75
6Insider Trading77
6.1A Toy Model: the Brownian Bridge77
6.2Information Drift and Stochastic Calculus of Variations79
6.3Integral Representation of Measure-Valued Martingales81
6.4Insider Additional Utility83
6.5An Example of an Insider Getting Free Lunches84
7Asymptotic Expansion and Weak Convergence87
7.1Asymptotic Expansion of SDEs Depending on a Parameter88
7.2Watanabe Distributions and Descent Principle89
7.3Strong Functional Convergence of the Euler Scheme90
7.4Weak Convergence of the Euler Scheme93
8Stochastic Calculus of Variations for Markets with Jumps97
8.1Probability Spaces of Finite Type Jump Processes98
8.2Stochastic Calculus of Variations for Exponential Variables100
8.3Stochastic Calculus of Variations for Poisson Processes102
8.4Mean-Variance Minimal Hedging and Clark-Ocone Formula104
AVolatility Estimation by Fourier Expansion107
A.1Fourier Transform of the Volatility Functor109
A.2Numerical Implementation of the Method112
BStrong Monte-Carlo Approximation of an Elliptic Market115
B.1Definition of the Scheme [characters not reproducible]116
B.2The Milstein Scheme117
B.3Horizontal Parametrization118
B.4Reconstruction of the Scheme [characters not reproducible]120
CNumerical Implementation of the Price-Volatility Feedback Rate123
References127
Index139

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