Applications of Malliavin calculus to Monte Carlo methods in finance
type: Item
instance of: scholarly article
title: Applications of Malliavin calculus to Monte Carlo methods in finance
zbMATH Open document ID: 0947.60066
author: Eric Fournié
author: J.-M. Lasry
author: Jérôme Lebuchous
author: P.-L. Lions
published in: Finance and Stochastics
publication date: 2000-05-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 91G60
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 1452407
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: integration-by-parts
zbMATH Keywords: volatility
zbMATH Keywords: hedge ratios
zbMATH Keywords: Greeks
zbMATH Keywords: Monte-Carlo methods
zbMATH Keywords: Black-Scholes model
MaRDI profile type: Publication
full work available at URL: s007800050068
OpenAlex ID: W2151832703
DOI: 10.1007/S007800050068
description: scientific article; zbMATH DE number 1452407