No abstract

We derive the partial integro-differential equations (PIDEs) verified by the values of European and barrier options in exponential Lévy models. We discuss the conditions under which options prices are classical solutions of the PIDEs. Since these conditions may fail in general, we consider the notion of continuous viscosity solution. We give sufficient conditions on the Lévy triplet for the option price to be continuous; in this case we show that it is the unique viscosity solution of the PIDE.