In this study, we implicitly solve the generalized distributed-order time-fractional Black-Scholes equation. We employ finite differences to approximate the time derivatives and cubic B-spline quasi-interpolation for the spatial derivatives. The error analysis of the presented method is investigated. The algorithm of this method is also presented, which shows the simplicity of implementing the method to solve the generalized distributed-order time-fractional Black-Scholes equation. Numerical results demonstrate the method’s convergence rate and accuracy.
[1] H. Aminikhah and J. Alavi, Applying cubic B-Spline quasi-interpolation to solve 1D wave equations in polar coordinates, International Scholarly Research Notices, 2013 (2013), Article ID 710529.
[2] F. Black and M. Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81(3) (1973), 637–654.
[3] F. Broucke and L. Oparnica, Distributed-order time-fractional wave equations, Z. Angew. Math. Phys., 74(1) (2023), 1.
[4] C. De Boor, A practical guide to splines, Revised edition, Springer-Verlag New York, Inc., (2001).
[5] W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti, Applications of Distributed-Order Fractional Operators: A Review, Entropy, 23 (2021), 110.
[6] J. C. Hull, Options, Futures and Other Derivatives, Prentice Hall, (2005).
[7] M. N. Koleva and L. G. Vulkov, Numerical solution of time-fractional Black-Scholes equation, Comp. Appl. Math. 36 (2017), 1699–1715.
[8] Z. Li, K. Fujishiro, and G. Li, An inverse problem for distributed order time-fractional diffusion equations, arXiv:1707.02556.
[9] R. Merton, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science, 4(1) (1973), 141–183.
[10] J. E. Nàpoles, P. M. Guzmàn, L. M. Lugo, and A. Kashuri, The local generalized derivative and Mittag Leffler function, Sigma J Eng & Nat Sci, 38(2) (2020), 1007–1017.
[11] F. Ndäırou and D. F. M. Torres, Distributed-Order Non-Local Optimal Control, arXiv: 2010.11648.
[12] M. She, L. Li, R. Tang, et al, A novel numerical scheme for a time fractional Black-Scholes equation, J. Appl. Math. Comput., 66 (2021), 853–870.
[13] W. Wyss, The fractional Black-Scholes equation, Fract. Cal. Appl. Anal., 3(1) (2000), 51–61.
[14] M. Zhang, J. Jia, and X. Zheng, Numerical approximation and fast implementation to a generalized distributedorder time-fractional option pricing model, Chaos, Solitons and Fractals, 170 (2023), 113353.
Montazeri, R. (2026). Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation. Computational Methods for Differential Equations, 14(2), 947-957. doi: 10.22034/cmde.2025.64886.2956
MLA
Montazeri, R. . "Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation", Computational Methods for Differential Equations, 14, 2, 2026, 947-957. doi: 10.22034/cmde.2025.64886.2956
HARVARD
Montazeri, R. (2026). 'Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation', Computational Methods for Differential Equations, 14(2), pp. 947-957. doi: 10.22034/cmde.2025.64886.2956
CHICAGO
R. Montazeri, "Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation," Computational Methods for Differential Equations, 14 2 (2026): 947-957, doi: 10.22034/cmde.2025.64886.2956
VANCOUVER
Montazeri, R. Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation. Computational Methods for Differential Equations, 2026; 14(2): 947-957. doi: 10.22034/cmde.2025.64886.2956
)