Finance and Stochastics

, Volume 5, Issue 2, pp 201–236

Applications of Malliavin calculus to Monte-Carlo methods in finance. II

  • Eric Fournié
  • Jean-Michel Lasry
  • Jérôme Lebuchoux
  • Pierre-Louis Lions
Original Paper

DOI: 10.1007/PL00013529

Cite this article as:
Fournié, E., Lasry, J., Lebuchoux, J. et al. Finance Stochast (2001) 5: 201. doi:10.1007/PL00013529

Abstract.

This paper is the sequel of Part I [1], where we showed how to use the so-called Malliavin calculus in order to devise efficient Monte-Carlo (numerical) methods for Finance. First, we return to the formulas developed in [1] concerning the “greeks” used in European options, and we answer to the question of optimal weight functional in the sense of minimal variance. Then, we investigate the use of Malliavin calculus to compute conditional expectations. The integration by part formula provides a powerful tool when used in the framework of Monte Carlo simulation. It allows to compute everywhere, on a single set of trajectories starting at one point, solution of general options related PDEs.

Our final application of Malliavin calculus concerns the use of Girsanov transforms involving anticipating drifts. We give an example in numerical Finance of such a transform which gives reduction of variance via importance sampling.

Finally, we include two appendices that are concerned with the PDE interpretation of the formulas presented in [1] for the delta of a European option and with the connections between the functional dependence of some random variables and their Malliavin derivatives.

Key words: Monte Carlo methods, Malliavin calculus, hedge ratios and greeks, conditional expectations, PDE, anticipative Girsanov transform, functional dependence
JEL Classification: G13
Mathematics Subject Classification (1991): 60H07, 60J60, 65C05

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Eric Fournié
    • 1
  • Jean-Michel Lasry
    • 1
  • Jérôme Lebuchoux
    • 1
  • Pierre-Louis Lions
    • 2
  1. 1.PARIBAS Capital Markets, 10, Harewood Avenue, NW1 6AA London, England GB
  2. 2.Ceremade, UMR 9534, Université Paris-Dauphine, Place Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France (e-mail: mazzella@ceremade.dauphine.fr) FR
  3. 3.Laboratoire de Prohabilités, Université Pierre er Marie Curie, 4, Place Jussieu, 75252 Paris Céedex 05, France FR