Finance and Stochastics

, Volume 3, Issue 4, pp 391–412

Applications of Malliavin calculus to Monte Carlo methods in finance

  • Eric Fournié
  • Jean-Michel Lasry
  • Jérôme Lebuchoux
  • Pierre-Louis Lions
  • Nizar Touzi

DOI: 10.1007/s007800050068

Cite this article as:
Fournié, E., Lasry, JM., Lebuchoux, J. et al. Finance Stochast (1999) 3: 391. doi:10.1007/s007800050068

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 {\it 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.

Key words:Monte Carlo methods, Malliavin calculus, hedge ratios and Greeks
JEL classification :G13
Mathematics Subject Classification (1991):60H07, 60J60, 65C05

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Eric Fournié
    • 1
  • Jean-Michel Lasry
    • 1
  • Jérôme Lebuchoux
    • 3
  • Pierre-Louis Lions
    • 3
  • Nizar Touzi
    • 3
  1. 1.PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom (e-mail: eric_fournie@paribas.com; lasry@paribas.com; lebuchoux@paribas.com) GB
  2. 2.Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 5, France FR
  3. 3.CEREMADE, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France (e-mail: lions@ceremade.dauphine.fr; touzi@ceremade.dauphine.fr) FR
  4. 4.Consultant Caisse Autonome de Refinancement (CDC group), 33, rue de Mogador, F-75009 Paris, France FR