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Volume 6, Issue 1
A Computational Scheme for Options Under Jump Diffusion Processes

K. Zhang & S. Wang

Int. J. Numer. Anal. Mod., 6 (2009), pp. 110-123.

Published online: 2009-06

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  • Abstract

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the discretization methods are unconditionally stable in time and the system matrices of the resulting linear systems are M-matrices. The resulting linear systems involve products of a dense matrix and vectors and an Fast Fourier Transformation (FFT) technique is used for the evaluation of these products. Furthermore, a splitting technique is proposed for the solution of the discretized system arising from the Crank-Nicolson scheme. Numerical results are presented to show the rates of convergence and the robustness of the numerical method.

  • Keywords

Jump diffusion processes, option pricing, finite volume method, integral partial differential equation, FFT.

  • AMS Subject Headings

65M12, 65M60, 91B28

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-6-110, author = {K. Zhang and S. Wang}, title = {A Computational Scheme for Options Under Jump Diffusion Processes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2009}, volume = {6}, number = {1}, pages = {110--123}, abstract = {

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the discretization methods are unconditionally stable in time and the system matrices of the resulting linear systems are M-matrices. The resulting linear systems involve products of a dense matrix and vectors and an Fast Fourier Transformation (FFT) technique is used for the evaluation of these products. Furthermore, a splitting technique is proposed for the solution of the discretized system arising from the Crank-Nicolson scheme. Numerical results are presented to show the rates of convergence and the robustness of the numerical method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/758.html} }
TY - JOUR T1 - A Computational Scheme for Options Under Jump Diffusion Processes AU - K. Zhang & S. Wang JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 110 EP - 123 PY - 2009 DA - 2009/06 SN - 6 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/758.html KW - Jump diffusion processes, option pricing, finite volume method, integral partial differential equation, FFT. AB -

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the discretization methods are unconditionally stable in time and the system matrices of the resulting linear systems are M-matrices. The resulting linear systems involve products of a dense matrix and vectors and an Fast Fourier Transformation (FFT) technique is used for the evaluation of these products. Furthermore, a splitting technique is proposed for the solution of the discretized system arising from the Crank-Nicolson scheme. Numerical results are presented to show the rates of convergence and the robustness of the numerical method.

K. Zhang and S. Wang. (2009). A Computational Scheme for Options Under Jump Diffusion Processes. International Journal of Numerical Analysis and Modeling. 6 (1). 110-123. doi:
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