Integro-differential equations for option prices in exponential Levy models

We explore the precise link between option prices in exponential Levy 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 Levy 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-Levy models correspond to viscosity solutions of the pricing PIDE.