Volume 4, Issue 1
Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

Hong-Kui Pang & Hai-Wei Sun

East Asian J. Appl. Math., 4 (2014), pp. 52-68.

Published online: 2018-02

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  • Abstract

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

  • Keywords

Stochastic volatility jump diffusion, European option, barrier option, partial integro-differential equation, matrix exponential, shift-invert Arnoldi, matrix splitting, multigrid method.

  • AMS Subject Headings

91B28, 62P05, 35K15, 65F10, 65M06, 91B70, 47B35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-4-52, author = {}, title = {Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {4}, number = {1}, pages = {52--68}, abstract = {

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.280313.061013a}, url = {http://global-sci.org/intro/article_detail/eajam/10820.html} }
TY - JOUR T1 - Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models JO - East Asian Journal on Applied Mathematics VL - 1 SP - 52 EP - 68 PY - 2018 DA - 2018/02 SN - 4 DO - http://doi.org/10.4208/eajam.280313.061013a UR - https://global-sci.org/intro/article_detail/eajam/10820.html KW - Stochastic volatility jump diffusion, European option, barrier option, partial integro-differential equation, matrix exponential, shift-invert Arnoldi, matrix splitting, multigrid method. AB -

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Hong-Kui Pang & Hai-Wei Sun. (1970). Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models. East Asian Journal on Applied Mathematics. 4 (1). 52-68. doi:10.4208/eajam.280313.061013a
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