Communications on Pure and Applied Analysis (CPAA)

The forward Kolmogorov equation for two dimensional options

Pages: 195 - 208, Volume 8, Issue 1, January 2009

doi:10.3934/cpaa.2009.8.195       Abstract        Full Text (580.2K)       Related Articles

Antoine Conze - Natixis Corporate Solutions bank, 30 av George V 75008 PARIS, France (email)
Nicolas Lantos - Natixis Corporate Solutions bank, 30 av George V 75008 PARIS, France (email)
Olivier Pironneau - Laboratoire Jacques-Louis Lions (LJLL), UPMC Univ Paris 06, UMR 7598, LJLL, F-75005, Paris, CNRS, UMR 7598, LJLL, F-75005, Paris, France (email)

Abstract: Pricing options on multiple underlyings or on an underlying modeled with stochastic volatility may involve solving multi-dimensional parabolic partial differential equations (PDE). Computing several such options at once for various moneyness levels can be a numerical challenge. We investigate here the Kolmogorov equation and Dupire or “pre-Dupire" equations to solve the problem faster and we validate the approach numerically. The heart of the method is to use the adjoint of the PDE of the option at the discrete level and to use discrete duality identities to obtain Dupire-like relations. The method works on every linear models. Numerical results are given for a European call option on a basket of two assets.

Keywords:  Kolmogorv equation, financial mathematics, option pricing, finite element methods.
Mathematics Subject Classification:  Primary: 91B28, 65L60; Secondary: 82B31.

Received: May 2008;      Revised: September 2008;      Published: October 2008.