Implicit--Explicit Numerical Schemes for Integro-differential Parabolic Problems arising in Financial Theory

Maya Briani

Abstract: We study the numerical approximation of viscosity solutions for integro-differential parabolic problems (PIDE). Similar models arise in option pricing, to generalize the Black--Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term an implicit differencing scheme, unconditionally stable, is not pratically feasible.

We present Implicit-Explicit (IMEX) Runge-Kutta methods for the time integration to solve the integral term explicitly, giving higher order accuracy schemes under weak stability time-step restrictions. Numerical tests are presented to show the computational efficiency of the proposed approximation.