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Computational algorithm for financial mathematical model based on European option

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Abstract

In this manuscript, a computational approach based on the combination of finite difference with the operational matrix approach is constructed for the time-fractional Black–Scholes model (TFBSM) arising in the financial market. We have applied the L1-2 scheme to approximate the Caputo derivative and the operational matrix method (OMM) by using shifted Legendre polynomials (SLP) and shifted Chebyshev polynomials (SCP) to approximate the space derivatives. The proposed algorithm easily transforms the TFBSM into a system of algebraic equations, which can be solved easily to get the numerical solution. Furthermore, theoretical unconditional stability and convergence of the numerical scheme are established for \(\alpha \in (0,{\bar{\alpha }}]\) and the stability of the proposed algorithm is also verified numerically. Finally, the scheme is tested on five numerical problems, including the European double barrier call option and the European call and put option. It is observed that this computational approach gives almost the same accuracy with both the basis functions, but the CPU time taken by scheme with SLP basis is less than the SCP basis function. The effect of different parameters like volatility, interest rate, fractional order, etc., on the option pricing, is also investigated. A comparative study of the numerical results by the proposed algorithm with the schemes given in Zhang et al. (Comput Math Appl 71(9):1772–1783, 2016) and De Staelen and Hendy (Comput Math Applic 74(6):1166–1175, 2017), is also provided to show its effectiveness and accuracy.

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  • 12 June 2022

    The original version of this article was revised to update the email id of the authors.

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Acknowledgements

N. Srivastava acknowledges the financial support from the Ministry of Education, Govt. of India, under Senior Research Fellowship (SRF) scheme. A. Singh acknowledges the financial support from Council of Scientific & Industrial Research (CSIR), India, under Senior Research Fellow (SRF) scheme.

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  1. Nikhil Srivastava & Aman Singh

    Present address: Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India

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  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India

    Vineet Kumar Singh

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Correspondence to Vineet Kumar Singh.

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Srivastava, N., Singh, A. & Singh, V.K. Computational algorithm for financial mathematical model based on European option. Math Sci 17, 467–490 (2023). https://doi.org/10.1007/s40096-022-00474-0

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