Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**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