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

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	45
تعداد شماره‌ها	1,532
تعداد مقالات	18,594
تعداد مشاهده مقاله	60,953,812
تعداد دریافت فایل اصل مقاله	21,761,854

	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.
آمار تعداد مشاهده مقاله: 656 تعداد دریافت فایل اصل مقاله: 1,035

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

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