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An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options

Author

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  • Anshima Singh

    (Indian Institute of Technology (BHU))

  • Sunil Kumar

    (Indian Institute of Technology (BHU))

Abstract

In this paper a time-fractional Black–Scholes model (TFBSM) is considered to study the price change of the underlying fractal transmission system. We develop and analyze a numerical method to solve the TFBSM governing European options. The numerical method combines the exponential B-spline collocation to discretize in space and a finite difference method to discretize in time. The method is shown to be unconditionally stable using von-Neumann analysis. Also, the method is proved to be convergent of order two in space and $$2-\mu $$ 2 - μ is time, where $$\mu $$ μ is order of the fractional derivative. We implement the method on various numerical examples in order to illustrate the accuracy of the method, and validation of the theoretical findings. In addition, as an application, the method is used to price several different European options such as the European call option, European put option, and European double barrier knock-out call option. Moreover, the classical Black–Scholes model is also incorporated into our numerical study to validate the competence of our method in handling not only fractional problems, but also classical ones with favorable results.

Suggested Citation

  • Anshima Singh & Sunil Kumar, 2024. "An Efficient Numerical Method Based on Exponential B-splines for a Time-Fractional Black–Scholes Equation Governing European Options," Computational Economics, Springer;Society for Computational Economics, vol. 64(4), pages 1965-2002, October.
    • Handle: RePEc:kap:compec:v:64:y:2024:i:4:d:10.1007_s10614-023-10500-5
    DOI: 10.1007/s10614-023-10500-5
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    References listed on IDEAS

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    1. S. C. S. Rao & S. Kumar & M. Kumar, 2010. "A Parameter-Uniform B-Spline Collocation Method for Singularly Perturbed Semilinear Reaction-Diffusion Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 795-809, September.
    2. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    3. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. Fall, Aliou Niang & Ndiaye, Seydou Nourou & Sene, Ndolane, 2019. "Black–Scholes option pricing equations described by the Caputo generalized fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 108-118.
    6. A. Golbabai & E. Mohebianfar, 2017. "A New Stable Local Radial Basis Function Approach for Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 49(2), pages 271-288, February.
    7. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
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