Applications of Malliavin calculus to Monte Carlo methods in finance
The authors propose an original probabilistic method for numerical computation of Greeks (i.e. price sensitivities) in finance. Suppose that the assets are described by \(n\)-dimensional diffusion processes \(X(t)\) and \(u(x)=E[\varphi(X(t_1),\dots,X(t_1))\mid X(0)=x]\) describes the price of a contingent claim defined by the payoff function \(\varphi\). Financial applications require not only the calculation of the function \(u(x)\) but also its differentials with respect to the initial condition, drift and volatility coefficients. Malliavin calculus, mainly integration-by-parts formula, is used to show that all the differentials of interest can be expressed as \(E[\pi \varphi(X(t_1),\dots,X(t_1))\mid X(0)]\), where the weight \(\pi\) is a random variable which does not depend on the payoff function \(\varphi\). The required differential can be computed numerically by Monte Carlo simulation and estimator achieves the \(n^{-1/2}\) convergence rate. A numerical experiment illustrates the application of this method to exotic European options in the framework of the Black and Scholes model and is compared to the Monte Carlo finite difference approach. The proposed procedure turns out to be rather efficient in the case of a discontinuous payoff function.
scientific article; zbMATH DE number 1452407