Integro-differential equations for option prices in exponential Lévy models

We explore the precise link between option prices in exponential Lévy models and the related partial integro-differential equations (PIDEs) in the case of European options and options with single or double barriers. We first discuss the conditions under which options prices are classical solutions of the PIDEs. We show that these conditions may fail in pure jump models and give examples of lack of smoothness of option prices with respect to the underlying. We give sufficient conditions on the Lévy triplet for the prices of barrier options to be continuous with respect to the underlying and show that, in a general setting, option prices in exp-Lévy models correspond to viscosity solutions of the pricing PIDE.

Authors and Affiliations

1. Ecole Polytechnique, Centre de Mathématiques Appliquées, 91128, Palaiseau, France

   Rama Cont & Ekaterina Voltchkova

Corresponding author

Correspondence to Rama Cont.

JEL Classification:

G13

Mathematics Subject Classification (2000):

45K05, 49L25, 60G51, 60J75

This work was presented at the Bachelier seminar (Paris 2003), Workshop on Lévy processes and partial integro-differential equations (Palaiseau 2003), IFIP 2003 (Nice), the Workshop on Semimartingale theory and applications in finance (Banff) and the Workshop on Computational Finance (Zürich 2003). We thank Yves Achdou, Mariko Arisawa, Daniel Gabay, Huyên Pham and Peter Tankov for helpful discussions.

Manuscript received: July 2004; final version received: November 2004

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