Dupire-like identities for complex options

Abstract

Dupire's identity is very useful to compute all financial options based on a single asset at once and also for the calibration of models. We show that it is not limited to European options based on a single Brownian driven asset. By using the adjoint equations of the financial models we extend the concept to barrier options, Lévy driven options, basket options and partially to stochastic volatility models. The technique does not work for American and Asian options. The analytic derivations of these Dupire-like formulae is tested numerically and excellent agreement is found proving henceforth that the method is also numerically feasible. To cite this article: O. Pironneau, C. R. Acad. Sci. Paris, Ser. I 344 (2007).