Stochastic Calculus of Variations in Mathematical Finance [Paperback]

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.

Gaussian Stochastic Calculus of Variations.- Computation of Greeks and Integration by Parts Formulae.- Market Equilibrium and Price-Volatility Feedback Rate.- Multivariate Conditioning and Regularity of Law.- Non-Elliptic Markets and Instability in HJM Models.- Insider Trading.- Asymptotic Expansion and Weak Convergence.- Stochastic Calculus of Variations for Markets with Jumps.

From the reviews:

This short book introduces Malliavin calculus and illustrates important applications in finance. & For readers with the necessary mathematical skills, this is a valuable introduction to the mathematics and financial applications of Malliavin calculus. & it provides a direct gateway to the relevant literature. (www.riskbook.com, November, 2006)

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