Forward evolution equations for knock-out options

Abstract

We derive forward partial integrodifferential equations (PIDEs) for pricing up-and-out and down-and-out call options when the underlying is a jump diffusion. We assume that the jump part of the returns process is an additive process. This framework includes the Variance-Gamma, finite moment logstable, Merton jump diffusion, Kou jump diffusion, Dupire, CEV, arcsinh normal, displaced diffusion, and Black–Scholes models as special cases.