Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### First order schemes in the numerical quantization method

Auteur(s):

Code(s) de Classification MSC:

• 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60}
• 65C05 Monte Carlo methods
• 65C20 Models, numerical methods
• 65N50 Mesh generation and refinement
• 90A09 Finance, portfolios, investment

Résumé: The numerical quantization method is a grid method which relies on the approximation of the solution of a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first order scheme based on piecewise {\em linear} approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and {\em a priori} estimates of the errors are proposed.

Mots Clés: Numerical quantization ; american options ; Malliavin calculus

Date: 2002-05-30

Prépublication numéro: PMA-735

Pdf file : PMA-735.pdf