On the Valuation of Options in Jump-Diffusion Models by Variational Methods∗

We consider the valuation of European and American-style options under jump-diffusion processes by variational methods. In particular, the value function is seen to satisfy a parabolic partial (spatial) integro-differential variational inequality. A theoretical framework is developed and an analysis of a Þnite element implementation presented. A key feature is the introduction of separate approximation domains for both the state space and jump process variables. When coupled with any semi-implicit time integrator, this procedure presents a full discretization which is of optimal efficiency; the additional computational cost of evaluating the value function associated with a jump-diffusion as compared to pure diffusion process is, as a practical matter, negligible. Multi-dimensional computations are presented that validate the applicability and efficiency of the method.