Efficient solution of a partial integro-differential equation in finance

Abstract

Jump-diffusion models for the pricing of derivatives lead under certain assumptions to partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a non-local integral. We transform the PIDE to eliminate the convection term, discretize it implicitly, and use finite differences on a uniform grid. The resulting dense linear system exhibits so much structure that it can be solved very efficiently by a circulant preconditioned conjugate gradient method. Therefore, this fully implicit scheme requires only on the order of O ( n log n ) operations. Second order accuracy is obtained numerically on the whole computational domain for Merton's model.