The forward Kolmogorov equation for two dimensional options

Antoine Conze; Nicolas Lantos; Olivier Pironneau

doi:10.3934/cpaa.2009.8.195

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2009, Volume 8, Issue 1: 195-208. Doi: 10.3934/cpaa.2009.8.195

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The forward Kolmogorov equation for two dimensional options

1.
Natixis Corporate Solutions bank, 30 av George V 75008 PARIS
2.
Laboratoire Jacques-Louis Lions (LJLL), UPMC Univ Paris 06, UMR 7598, LJLL, F-75005, Paris, CNRS, UMR 7598, LJLL, F-75005, Paris

Received: April 30, 2008

Revised: August 31, 2008

Published: December 2008

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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.

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Mathematics Subject Classification: Primary: 91B28, 65L60; Secondary: 82B31.

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