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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...

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-linesapproach 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 speed-up 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.

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.