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Practical finite difference method for solving multi-dimensional black-Scholes model in fractal market

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  • Wang, Jian
  • Wen, Shuai
  • Yang, Mengdie
  • Shao, Wei

Abstract

In this paper, we employ a practical finite difference method to research the multi-dimensional fractional Balck-Scholes model under one asset and three assets. In the case of one asset, we establish explicit scheme and Crank-Nicolson scheme to study the effect of different Hurst exponent (H) on numerical results and Greeks. With the increase of H, the numerical figures of the finite difference scheme also change. In addition, we also verify the effectiveness of Crank-Nicolson scheme in numerical solution of Greeks. We observe that when H=0.5, the results of Delta, Gamma and Theta are consistent with the accurate results. In the case of three assets, we use operator splitting method (OSM) and establish semi-implicit scheme. We hold that H will affect the numerical results and Greeks results in fractional Black-Scholes model. If the effect of H is not considered in option hedging, the result will deviate greatly from the actual result.

Suggested Citation

  • Wang, Jian & Wen, Shuai & Yang, Mengdie & Shao, Wei, 2022. "Practical finite difference method for solving multi-dimensional black-Scholes model in fractal market," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    • Handle: RePEc:eee:chsofr:v:157:y:2022:i:c:s0960077922001060
    DOI: 10.1016/j.chaos.2022.111895
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    References listed on IDEAS

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    1. Kim, Junseok & Kim, Taekkeun & Jo, Jaehyun & Choi, Yongho & Lee, Seunggyu & Hwang, Hyeongseok & Yoo, Minhyun & Jeong, Darae, 2016. "A practical finite difference method for the three-dimensional Black–Scholes equation," European Journal of Operational Research, Elsevier, vol. 252(1), pages 183-190.
    2. Wang, Jian & Yan, Yan & Chen, Wenbing & Shao, Wei & Wang, Jian & Tang, Weiwei, 2021. "Equity-linked securities option pricing by fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    3. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xi-Li & Wang, Ying-Luo, 2010. "Pricing currency options in a fractional Brownian motion with jumps," Economic Modelling, Elsevier, vol. 27(5), pages 935-942, September.
    4. Kim, Sangkwon & Kim, Junseok, 2021. "Robust and accurate construction of the local volatility surface using the Black–Scholes equation," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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