The forward Kolmogorov equation for two dimensional options

Pricing options on multiple underlying or on an underlying modeled with stochastic volatility may involve solving multi-dimensional Black- Scholes like partial dierential 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 most linear models. Numerical results are given for a European call option on a basket of two assets.