Applications of Malliavin Calculus to Monte Carlo Methods in Finance

Posted: 21 Oct 1999

See all articles by Eric Fournie

Eric Fournie

BNP Paribas; Université Paris VI Pierre et Marie Curie

Jean-Michel Lasry

BNP Paribas; Université Paris Dauphine - CEREMADE

Jerome Lebuchoux

Université Paris Dauphine - CEREMADE; Consultant Caisse Autonome de Refinancement (CDC group)

Pierre-Louis Lions

Université Paris Dauphine - CEREMADE; Consultant Caisse Autonome de Refinancement (CDC group)

Nizar Touzi

Université Paris Dauphine - CEREMADE; Consultant Caisse Autonome de Refinancement (CDC group)

Abstract

This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the integration-by-parts formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

JEL Classification: G13, C63

Suggested Citation

Fournie, Eric and Lasry, Jean-Michel and Lebuchoux, Jerome and Lions, Pierre-Louis and Touzi, Nizar, Applications of Malliavin Calculus to Monte Carlo Methods in Finance. Available at SSRN: https://ssrn.com/abstract=179553

Eric Fournie (Contact Author)

BNP Paribas ( email )

London NW1 6AA
United Kingdom

Université Paris VI Pierre et Marie Curie

4, Place Jussieu, B.P. 169
Laboratoire de Probabilites
F-75252-Paris Cedex 05
France

Jean-Michel Lasry

BNP Paribas ( email )

London NW1 6AA
United Kingdom

Université Paris Dauphine - CEREMADE

Place du Marechal de Lattre de Tassigny
Paris Cedex 16, 75775
France

Jerome Lebuchoux

Université Paris Dauphine - CEREMADE ( email )

Place du Marechal de Lattre de Tassigny
Paris Cedex 16, 75775
France

Consultant Caisse Autonome de Refinancement (CDC group)

33 rue de Mogador
F-75009 Paris
France

Pierre-Louis Lions

Université Paris Dauphine - CEREMADE ( email )

Place du Marechal de Lattre de Tassigny
Paris Cedex 16, 75775
France

Consultant Caisse Autonome de Refinancement (CDC group)

33 rue de Mogador
F-75009 Paris
France

Nizar Touzi

Université Paris Dauphine - CEREMADE ( email )

Place du Marechal de Lattre de Tassigny
Paris Cedex 16, 75775
France

Consultant Caisse Autonome de Refinancement (CDC group)

33 rue de Mogador
F-75009 Paris
France

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