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| A FAST AND HIGH ACCURACY NUMERICAL SIMULATION FOR A FRACTIONAL BLACK-SCHOLES MODEL ON TWO ASSETS |
| Hongmei Zhang,Fawang Liu,Shanzhen Chen,Ming Shen |
| (College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, Fujian, PR China;School of Mathematical Sciences, Queensland University of Technology, Qld. 4001, Australia;School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, Sichuan, PR China) |
| DOI： |
| Abstract: |
| In this paper, a two dimensional (2D) fractional Black-Scholes (FBS) model on two assets following independent geometric L\'{e}vy 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. |
| Key words: 2D fractional Black-Scholes model; L\'{e}vy process; fractional derivative; numerical simulation; fast bi-conjugrate gradient stabilized method |