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Applications of Malliavin calculus to Monte Carlo methods in finance

Author

Listed:
  • Eric Fournié

    (PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom)

  • Jean-Michel Lasry

    (PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom)

  • Pierre-Louis Lions

    (CEREMADE, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France)

  • Jérôme Lebuchoux

    (CEREMADE, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France)

  • Nizar Touzi

    (CEREMADE, Université Paris IX Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris Cedex 16, France)

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.

Suggested Citation

  • Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    • Handle: RePEc:spr:finsto:v:3:y:1999:i:4:p:391-412
    Note: received: July 1997; final version received: September 1998
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    More about this item

    Keywords

    Monte Carlo methods; Malliavin calculus; hedge ratios and Greeks;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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