Document Type : Research Paper

**Authors**

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 -136, Iran.

**Abstract**

The time-fractional Black-Scholes model (TFBSM) governing European options in which the temporal derivative is focused on the Caputo fractional derivative with 0 < β ≤ 1 is considered in this article. Approximating financial options with respect to their hereditary characteristics can be well understood and explained due to its outstanding memory effect current in fractional derivatives. Compelled by the stated cause, It is important to find reasonably accurate and successful numerical methods when approaching fractional differential equations. The simulation model given here is developed in two ways: one, the semi-discrete is produced in the time using a quadratic interpolation with the order of precision τ3−α in the case of a smooth solution, and subsequently, the unconditional stability and convergence order are investigated. The spatial derivative variables are simulated using the collocation approach based on a Legendre basis for the designed full-discrete scheme. Last, we employ various test problems to demonstrate the suggested design’s high precision. Moreover, the obtained results are compared to those obtained using other methodologies, demonstrating that the proposed technique is highly accurate and practicable.

**Keywords**

- [1] M. Aalaei and M. Manteqipour, An adaptive Monte Carlo algorithm for European and American options, Com- putational Methods for Diﬀerential Equations., (2021), 1–16.
- [2] A. A. Alikhanov, A new diﬀerence scheme for the time fractional diﬀusion equation, Journal of Computational Physics., 280 (2015), 424–438.
- [3] M. N. Anwar and L. S. Andallah, A study on numerical solution of Black-Scholes model, Journal of Mathematical Finance., 8(2) (2018), 372–381.
- [4] A. Bekir and O¨ . Gu¨ner, Analytical approach for the space-time nonlinear partial diﬀerential fractional equation, International Journal of Nonlinear Sciences and Numerical Simulation., 15(7-8) (2014), 463–470.
- [5] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of political economy., 81(3) (1973), 637–654.
- [6] R. H. De Staelen and A. S. Hendy, Numerically pricing double barrier options in a time-fractional Black-Scholes model, Computers & Mathematics with Applications., 74(6) (2017), 1166–1175.
- [7] Z. Ding, A. Xiao, and M. Li, Weighted finite diﬀerence methods for a class of space fractional partial diﬀerential equations with variable coeﬃcients, Journal of Computational and Applied Mathematics., 233(8) (2010), 1905– 1914.
- [8] R. Farnoosh, A. Sobhani, H. Rezazadeh, and M. H. Beheshti, Numerical method for discrete double barrier option pricing with time-dependent parameters Computers & Mathematics with Applications., 70(8) (2015), 2006–2013.
- [9] G. Gao, Z. Sun, and H. Zhang, A new fractional numerical diﬀerentiation formula to approximate the Caputo fractional derivative and its applications, Journal of Computational Physics., 259 (2014), 33–50.
- [10] A. Golbabai and O. Nikan, A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black-Scholes model, Computational Economics., (2019), 1–23.
- [11] D. Hackmann, Solving the Black Scholes equation using a finite diﬀerence method, Available online: math. yorku.ca/˜ dhackman/BlackScholes7., 2009.
- [12] R. Hejazi, E. Dastranj, N. Habibi, and A. Naderifard, Stochastic analysis and invariant subspace method for handling option pricing with numerical simulation, Computational Methods for Diﬀerential Equations., 1 (2021), 1–14.
- [13] B. Jin, R. Lazarov, and Z. Zhou, Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview, Computer Methods in Applied Mechanics and Engineering., 346 (2019), 332–358.
- [14] M. N. Koleva and G. L. Vulkov, Numerical solution of time-fractional Black-Scholes equation, Computational and Applied Mathematics., 36(4) (2017), 1699–1715.
- [15] K. Kumar, R. K. Pandey, and S. Sharma, Comparative study of three numerical schemes for fractional integro- diﬀerential equations, Journal of Computational and Applied Mathematics., 315 (2017), 287–302.
- [16] A. G. Lakoud, R. Khaldi, and A. Kılı¸cman, Existence of solutions for a mixed fractional boundary value problem, Advances in Diﬀerence Equations., 2017(1) (2017), 164.
- [17] N. N. Leonenko, M. M. Meerschaert, and A. Sikorskii, Fractional pearson diﬀusions, Journal of mathematical analysis and applications., 403(2) (2013), 532–546.
- [18] H. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction-subdiﬀusion equations, SIAM Journal on Numerical Analysis, 56(2) (2018), 1112–1133.
- [19] F. Liu, P. Zhuang, I. Turner, K. Burrage, and V. Anh, A new fractional finite volume method for solving the fractional diﬀusion equation, Applied Mathematical Modelling., 38(15-16) (2014), 3871–3878.
- [20] W. H. Luo, T. Z. Huang, G. C. Wu, and X. M. Gu, Quadratic spline collocation method for the time fractional subdiﬀusion equation, Applied Mathematics and Computation., 276 (2016), 252–265.
- [21] W. H. Luo,C. Li, T. Z. Huang, X. M. Gu, and XG. C. Wu, A high-order accurate numerical scheme for the Caputo derivative with applications to fractional diﬀusion problems, Numerical functional analysis and optimization., 39(5) (2018), 600–622.
- [22] F. Mehrdoust, A. H. R. Sheikhani, M. Mashoof, and S. Hasanzadeh,Block-pulse operational matrix method for solving fractional Black-Scholes equation, Journal of Economic Studies., (2017).
- [23] R. C. Merton, Theory of rational option pricing, The Bell Journal of economics and management science., (1973), 141–183.
- [24] H. Mesgarani, A. Adl, and Y. Esmaeelzade Aghdam, Approximate price of the option under discretization by applying quadratic interpolation and Legendre polynomials, Mathematical Sciences., (2021), 1–8.
- [25] H. Mesgarani, S. Ahanj, and Y. Esmaeelzade Aghdam, Numerical investigation of the time-fractional Black- Scholes equation with barrier choice of regulating European option, Journal of Mathematical Modeling., (2021), 1–10.
- [26] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional diﬀerential equations, Wiley., (1993).
- [27] B. P. Moghaddam and Z. S. Mostaghim, Modified finite diﬀerence method for solving fractional delay diﬀerential equations, Boletim da Sociedade Paranaense de Matem´atica., 35(2) (2017), 49–58.
- [28] S. Momani and Z. Odibat, Analytical approach to linear fractional partial diﬀerential equations arising in fluid mechanics, Physics Letters A., 355(4-5) (2006), 271–279.
- [29] Z. M. Odibat, Analytic study on linear systems of fractional diﬀerential equations, Computers & Mathematics with Applications., 59(3) (2010), 1171–1183.
- [30] P. Phaochoo, A. Luadsong, and N. Aschariyaphotha, The meshless local Petrov-Galerkin based on moving kriging interpolation for solving fractional Black-Scholes model, Journal of King Saud University-Science., 28(1) (2016), 111–117.
- [31] M. Rezaei Mirarkolaei, A. Yazdanian, S. M. Mahmoudi, and A. Ashrafi, A compact diﬀerence scheme for time- fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options, Com- putational Methods for Diﬀerential Equations., 9(2) (2021), 523-552.
- [32] S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, and P. Agarwal, On analytical solutions of the fractional diﬀerential equation with uncertainty: application to the basset problem, Entropy., 17(2) (2015), 885–902.
- [33] L. Song and W. Wang, Solution of the fractional Black-Scholes option pricing model by finite diﬀerence method, Abstract and applied analysis., (2013), 1-10.
- [34] M. Stynes, E. OR´iordan, and J. L. Gracia, Error analysis of a finite diﬀerence method on graded meshes for a time-fractional diﬀusion equation, SIAM Journal on Numerical Analysis., 55(2) (2017), 1057–1079
- [35] Z. Tian, S. Zhai, H. Ji, and Z. Weng, A compact quadratic spline collocation method for the time-fractional Black-Scholes model, Journal of Applied Mathematics and Computing., (2020), 1–24.
- [36] M. K. S. Uddin,M. Ahmed, and S. K. Bhowmilk, A note on numerical solution of a linear Black-Scholes model, GANIT: Journal of Bangladesh Mathematical Society., 33 (2013), 103–115.
- [37] J. Wang and G. Liu, On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer methods in applied mechanics and engineering., 191(23-24) (2002), 2611–2630.
- [38] C. Xie, X. Xie, Y. Esmaeelzade Aghdam, B. Farnam, and H. Jafari, The Numerical Strategy of Tempered Fractional Derivative in European Double Barrier Option, Fractals., (2021), 1–9.
- [39] H. Zhang, F. Liu, I. Turner, and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Computers & Mathematics with Applications., 71(9) (2016), 1772–1783.
- [40] M. Zheng, F. Liu, V. Anh, and I. Turner, A high-order spectral method for the multi-term time-fractional diﬀusion equations, Applied mathematical modelling., 40(7-8) (2016), 4970–4985.

October 2022

Pages 1075-1085

**Receive Date:**24 April 2021**Revise Date:**19 October 2021**Accept Date:**22 November 2021