Highly efficient parallel algorithms for solving the Bates PIDE for pricing options on a GPU

Abstract

In this paper we investigate faster and memory efficient parallel techniques to numerically solve the Bates model for European options. We have followed method-of-lines approach and implemented the numerical algorithms on a graphics processing unit (GPU). Two second order finite difference (FD) schemes are taken into account that yield discretization matrices with tridiagonal and pentadiagonal block structures. Three recent adaptations of an alternating direction implicit scheme are employed for time-stepping. Spatial and temporal errors corresponding to our chosen FD and time-stepping schemes are numerically studied. For parallel computation of solutions we have applied the well-know parallel cyclic reduction (PCR) algorithm for tridiagonal systems and our novel PCR algorithm for pentadiagonal systems. Ample numerical experiments are performed to study speed and accuracy on three platforms: single GPU using CUDA, multi-core CPU using OpenMP and an efficient sequential algorithm on a single core using MATLAB, where substantial speedup is observed on the GPU. Sensitivities of computational times of the sequential algorithm in MATLAB with respect to certain parameters in the Bates model are also analysed.