Abstract:
This paper is concerned with fast, parallel and numerically accurate pricing of two-asset American options under the Merton jump-diffusion model, which gives rise to a two-dimensional partial integro-differential complementarity problem (PIDCP) with a nonlocal two-dimensional integral term. Following method-of-lines approach, the solution to the PIDCP can be computed quite accurately by a robust numerical technique that combines Ikonen–Toivanen splitting with an alternating direction implicit scheme. However, we observed that computing the numerical solution with this technique becomes extremely time consuming, mainly due to the handling of the integral term. In this paper we parallelize this technique by applying a parallel fast Fourier transformation algorithm to all matrix-vector multiplications involving the huge and dense integral approximation matrix by exploiting its block Toeplitz with Toeplitz block structure. We also parallelize other computationally intensive steps of this technique by applying a recently developed parallel cyclic reduction algorithm for pentadiagonal systems. Our solutions computed on a graphics processing unit (GPU) using CUDA® platform are compared for accuracy with those available in the literature. It is observed that by solving the PIDCP parallelly we could bring down the computational times from several hours to a few seconds in certain cases in our experiments.
