COMPACT FINITE DIFFERENCE SCHEMES OF THE TIME FRACTIONAL BLACK-SCHOLES MODEL

Zhaowei Tian; Shuying Zhai; Zhifeng Weng

doi:10.11948/20190148

2020 Volume 10 Issue 3

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Zhaowei Tian, Shuying Zhai, Zhifeng Weng. COMPACT FINITE DIFFERENCE SCHEMES OF THE TIME FRACTIONAL BLACK-SCHOLES MODEL[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 904-919. doi: 10.11948/20190148

Citation:

Zhaowei Tian, Shuying Zhai, Zhifeng Weng. COMPACT FINITE DIFFERENCE SCHEMES OF THE TIME FRACTIONAL BLACK-SCHOLES MODEL[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 904-919. doi: 10.11948/20190148

COMPACT FINITE DIFFERENCE SCHEMES OF THE TIME FRACTIONAL BLACK-SCHOLES MODEL

1.
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian, China

Corresponding author: Email address:zfwmath@163.com(Z. Weng)
Fund Project: The authors were in part supported by the National Natural Science Foundation of China (Nos. 11701197 and 11701196), the Fundamental Research Funds for the Central Universities (No. ZQN-702), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(No. ZQN-YX502), and the Project of Education and Scientific Research for Young and Middle-aged Teachers in Fujian Province(No. JAT160024)

Abstract

Abstract

In this paper, three compact difference schemes for the time-fractio-nal Black-Scholes model governing European option pricing are presented. Firstly, in order to obtain the fourth-order accuracy in space by applying the Padé approximation, we eliminate the convection term of the B-S equation by an exponential transformation. Then the time fractional derivative is approximated by $ L1 $ formula, $ L2 - 1_\sigma $ formula and $ L1 - 2 $ formula respectively, and three compact difference schemes with oders $ O(\Delta t^{2-\alpha}+h ^4) $, $ O(\Delta t^{2}+h ^4) $ and $ O(\Delta t^{3-\alpha}+h ^4) $ are constructed. Finally, numerical example is carried out to verify the accuracy and effectiveness of proposed methods, and the comparisons of various schemes are given. The paper also provides numerical studies including the effect of fractional orders and the effect of different parameters on option price in time-fractional B-S model.
- Time-fractional Black-Scholes equation /
- European option /
- exponential transformation /
- compact difference scheme

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References

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