Numerical Analysis
Dupire-like identities for complex options
Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 127-133.

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.

L'identité de Dupire est très utile pour calculer tous les prix d'options sur un seul sous-jacent par une unique résolution des équations aux dérivées partielles et aussi pour la calibration des modèles en mathématiques financières. Nous montrons ici comment obtenir de telles identités dans quelques cas plus complexes que le cadre traité par Dupire lui-même : options barrières, options paniers et options modélisées par des modèles à volatilité stochastique ou par des processus de Lévy. Les formules sont aussi testées sur des exemples numériques et une très bonne précision est obtenue.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.032
Pironneau, Olivier 1, 2

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France
2 Institut Universitaire de France, France
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Pironneau, Olivier. Dupire-like identities for complex options. Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 127-133. doi : 10.1016/j.crma.2006.11.032. http://www.numdam.org/articles/10.1016/j.crma.2006.11.032/

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