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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.

Based on the parallel cyclic reduction technique, a promising new parallel algorithm is designed for pentadiagonal systems. Subject to fulfilling stability conditions, this highly parallelizable al...

One of the prominent alternating direction implicit (ADI) schemes for numerically pricing financial options, the modified Craig–Sneyd scheme, is put to test for its reliability and efficiency for s...

The Modified Craig-Sneyd scheme is an alternating direction implicit(ADI) type scheme that was introduced by In 't Hout and Welfert (2009) 12 in order to numerically solve multidimensional convection-diffusion equations with mixed-derivative terms. It is one of the most prominent ADI schemes currently known for their efficiency in solving above type of problems. This paper deals with a useful stability result for the Modified Craig-Sneyd scheme when applied to two-dimensional convection-diffusion equations with mixed derivative term. The stability of the scheme is analyzed in the von Neumann framework, effectively taking into account the actual size of the mixed derivative term. This study is relevant to an observation of apparent discrepancy in a real world application of the scheme, i.e., in computational finance. The obtained results not only generalize some of the existing stability results, but also clearly justify this surprising observation theoretically.

The modified Craig-Sneyd (MCS) scheme is a promising splitting scheme of the ADI type for multi-dimensional pure diffusion equations having mixed spatial-derivative terms. In this paper we investigate the extension of the MCS scheme to two-dimensional convection-diffusion equations with a mixed derivative. Both necessary and sufficient conditions on the parameter @q of the scheme are derived concerning unconditional stability in the von Neumann sense.

In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter θ of each scheme are obtained that take into account the actual size of the mixed derivative coefficients. Our results generalize results obtained previously by Craig & Sneyd (1988) and In ʼt Hout & Welfert (2009). Numerical experiments are presented illustrating our main theorems.

This paper discusses the stability of the Modified Craig-Sneyd scheme when applied to two-dimensional convection-diffusion equations with mixed derivatives. A preliminary result is summarized that specially considers the size of the mixed derivative coefficients, which is very useful in practice.

The stability of ADI type schemes is considered when applied to two- and three-dimensional pure diffusion equations with mixed derivative terms. We present a first stability result that takes the actual size of the mixed derivative coefficients into account.