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Volume 36, Issue 1
​A Fast and High Accuracy Numerical Simulation for a Fractional Black-Scholes Model on Two Assets

Hongmei Zhang, Fawang Liu, Shanzhen Chen & Ming Shen

Ann. Appl. Math., 36 (2020), pp. 91-110.

Published online: 2020-08

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  • Abstract

In this paper, a two dimensional (2D) fractional Black-Scholes (FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasi-differential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized (FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is showed in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.

  • Keywords

2D fractional Black-Scholes model, Lévy process, fractional derivative, numerical simulation, fast bi-conjugate gradient stabilized method.

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COPYRIGHT: © Global Science Press

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@Article{AAM-36-91, author = {Zhang , HongmeiLiu , FawangChen , Shanzhen and Shen , Ming}, title = {​A Fast and High Accuracy Numerical Simulation for a Fractional Black-Scholes Model on Two Assets}, journal = {Annals of Applied Mathematics}, year = {2020}, volume = {36}, number = {1}, pages = {91--110}, abstract = {

In this paper, a two dimensional (2D) fractional Black-Scholes (FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasi-differential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized (FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is showed in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/18094.html} }
TY - JOUR T1 - ​A Fast and High Accuracy Numerical Simulation for a Fractional Black-Scholes Model on Two Assets AU - Zhang , Hongmei AU - Liu , Fawang AU - Chen , Shanzhen AU - Shen , Ming JO - Annals of Applied Mathematics VL - 1 SP - 91 EP - 110 PY - 2020 DA - 2020/08 SN - 36 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aam/18094.html KW - 2D fractional Black-Scholes model, Lévy process, fractional derivative, numerical simulation, fast bi-conjugate gradient stabilized method. AB -

In this paper, a two dimensional (2D) fractional Black-Scholes (FBS) model on two assets following independent geometric Lévy processes is solved numerically. A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established. The fractional derivative is a quasi-differential operator, whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix. In order to speed up calculation and save storage space, a fast bi-conjugate gradient stabilized (FBi-CGSTAB) method is proposed to solve the resultant linear system. Finally, one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique. The pricing of a European Call-on-Min option is showed in the other example, in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.

HongmeiZhang, FawangLiu, ShanzhenChen & MingShen. (2020). ​A Fast and High Accuracy Numerical Simulation for a Fractional Black-Scholes Model on Two Assets. Annals of Applied Mathematics. 36 (1). 91-110. doi:
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