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Applications of Malliavin calculus to Monte Carlo methods in finance

Eric Fournié (), Jean-Michel Lasry (), Pierre-Louis Lions (), Jérôme Lebuchoux () and Nizar Touzi ()
Additional contact information
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

Finance and Stochastics, 1999, vol. 3, issue 4, 391-412

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.

Keywords: Monte Carlo methods; Malliavin calculus; hedge ratios and Greeks (search for similar items in EconPapers)
JEL-codes: G13 (search for similar items in EconPapers)
Date: 1999-08-20
Note: received: July 1997; final version received: September 1998
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