Lie symmetry analysis, and exact solutions to the time–fractional Black–Scholes equation of the Caputo–type

Najafi, Ramin; Bahrami, Fariba; Vafadar, Parisa

doi:10.22034/cmde.2024.59096.2510

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	Lie symmetry analysis, and exact solutions to the time–fractional Black–Scholes equation of the Caputo–type
Computational Methods for Differential Equations
مقاله 1، دوره 12، شماره 4، دی 2024، صفحه 638-650 اصل مقاله (625.84 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.59096.2510
نویسندگان
Ramin Najafi^* ¹؛ Fariba Bahrami²؛ Parisa Vafadar²
¹Department of Mathematics, Maku Branch, Islamic Azad University, Maku, Iran.
²Faculty of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran.
چکیده
In this study, the Lie symmetry analysis, and exact solutions are investigated to the fractional Black–Scholes(B-S) equations of the Caputo–type modeling of the pricing options under the absence of arbitrage and self-financing portfolio assumptions. A class of exact invariant and solitary solutions are given to B-S equations. Some examples are presented in which we use the obtained reductions to find their exact solutions.
کلیدواژه‌ها
Lie symmetry analysis؛ Time–fractional Black–Scholes equation؛ Caputo fractional derivative؛ Invariant solutions

مراجع
[1] T. Akram, M. Abbas, K. M. Abualnaja, A. Iqbal, and A. Majeed, An efficient numerical technique based on the extended cubic B–spline functions for solving time fractional Black–Scholes model, Eng. Comput., 38 (2022), 1705–1716. [2] M. S. Aktar, M. A. Akbar, and M. S. Osman, Spatio–temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator–prey system and the diffusion–reaction equations, Chaos Soliton. Fract., 160 (2022), 112212. [3] F. Bahrami, R. Najafi, and M. S. Hashemi, On the invariant solutions of space/time–fractional diffusion equations, Indian J. Phys., 91 (2017), 1571–1579. [4] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989. [5] Y. Chatibi, E. H. El Kinani, and A. Ouhadan, On the discrete symmetry analysis of some classical and fractional differential equations, Math. Meth. Appl. Sci., 44 (2021), 2868–2878. [6] Y. Chatibi, E. H. El Kinani, and A. Ouhadan, Lie symmetry analysis and conservation laws for the time fractional Black–Scholes equation, Int. J. Geometr. Methods Mod. Phys., 17 (2020), 2050010. [7] W. Chen, X. Xu, and S. P. Zhu, Analytically pricing double barrier options based on a time–fractional Black– Scholes equation, Comput. Math. Appl., 69 (2015), 1407–1419. [8] K. Y. Chong and J. G. O’Hara, Lie Symmetry Analysis of a Fractional Black–Scholes Equation, AIP Conference Proceedings 2153, 020007 (2019). [9] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. [10] S. E. Fadugba, Homotopy analysis method and its applications in the valuation of European call options with time–fractional Black–Scholes equation, Chaos Soliton. Fract., 141 (2020) 110351. [11] R. K. Gazizov, N. H. Ibragimov, and S. Yu. Lukashchuk, Nonlinear self–adjointness, conservation laws and exact solutions of time–fractional Kompaneets equations, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 153–163. [12] R. K. Gazizov, A. A. Kasatkin, and S. Yu. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr., T136 (2009), 014016. [13] A. Goswami, J. Singh, D. Kumar, S. Gupta, and Sushila, An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma, J. Ocean Eng. Sci., 4 (2019), 85–99. [14] M. S. Hashemi, F. Bahrami, and R. Najafi, Lie symmetry analysis of steady–state fractional reaction–convection– diffusion equation, Optik, 138 (2017), 240–249. [15] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000. [16] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007), 311–328. [17] M. Iqbal, A. R. Seadawy, D. Lu, and Z. Zhang, Physical structure and multiple solitary wave solutions for the nonlinear Jaulent–Miodek hierarchy equation, Mod. Phys. Lett. B, 38 (2024), 2341016. [18] G. Jumarie, Derivation and solutions of some fractional Black–Scholes equations in coarse–grained space and time. Application to Merton’s optimal portfolio, Comput. Math. Appl., 59 (2010), 1142–1164. [19] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, 2006. [20] S. Kumar, D. Kumar, and A. Kumar, Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher–dimensional Fokas equation, Chaos Soliton. Fract., 142 (2021), 110507. [21] V. Kumar and A. M. Wazwaz, Lie symmetry analysis and soliton solutions for complex short pulse equation, Wave. Random Complex Media, 32 (2022), 968–979. [22] S. Yu. Lukashchuk, Conservation laws for time–fractional subdiffusion and diffusion–wave equations, Nonlinear Dynam., 80 (2015), 791–802. [23] R. Najafi, Group-invariant solutions for time–fractional Fornberg–Whitham equation by Lie symmetry analysis, Computational Methods for Differential Equations, 8 (2020), 251–258. [24] R. Najafi, Approximate nonclassical symmetries for the time–fractional KdV equations with the small parameter, Computational Methods for Differential Equations, 8 (2020), 111–118.
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Lie symmetry analysis, and exact solutions to the time–fractional Black–Scholes equation of the Caputo–type