Finance and Stochastics

, Volume 3, Issue 4, pp 391–412

Applications of Malliavin calculus to Monte Carlo methods in finance

Authors

  • Eric Fournié
    • PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom (e-mail: eric_fournie@paribas.com; lasry@paribas.com; lebuchoux@paribas.com)
  • Jean-Michel Lasry
    • PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom (e-mail: eric_fournie@paribas.com; lasry@paribas.com; lebuchoux@paribas.com)
  • Jérôme Lebuchoux
    • 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)
  • Pierre-Louis Lions
    • 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)
  • Nizar Touzi
    • 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)

DOI: 10.1007/s007800050068

Cite this article as:
Fournié, E., Lasry, J., 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