Numerical Solution of the Time-Fractional Black-Scholes Equation and Its Application to European Option Pricing

Elza Rahma Dihna, Endang Rusyaman, Sukono Sukono

Abstract


The classical Black-Scholes model is widely used in option pricing but relies on idealized assumptions such as constant volatility and memoryless market dynamics, which limit its accuracy in capturing real-world financial behavior. To overcome these limitations, the time-fractional Black-Scholes model incorporates a fractional-order derivative—specifically the Caputo derivative—which introduces memory effects and accommodates time-varying volatility. This study focuses on numerically solving the time-fractional Black-Scholes equation using the finite difference method (FDM) and applying the results to the pricing of European call options. The model is discretized using an implicit finite difference scheme to ensure stability and accuracy over the time domain. Numerical simulations are conducted for various values of the fractional order α, illustrating that the option price is sensitive to the fractional parameter. Lower values of α tend to increase option prices, highlighting the influence of memory effects on pricing behavior. The results confirm that the finite difference method is an effective numerical tool for solving fractional partial differential equations and demonstrate that the fractional Black-Scholes model offers improved flexibility and realism in option  valuation, particularly in markets characterized by irregular volatility and non-Markovian features.

Keywords


Black-Scholes Fractional; European options; finite difference method; fractional calculus; option pricing

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References


[1] L. Miljkovic, “The role of financial derivatives in financial risks management,” MEST Journal, vol. 11, no. 1, 2023. DOI: 10.12709/mest.11.11.01.09.

[2] M. Farahani, S. Babaei, and A. Esfahani, ““black-scholes-artificial neural network": A novel op287 tion pricing model,” International Journal of Financial, Accounting, and Management, vol. 5, no. 4, pp. 475–509, 2024. DOI: 10.35912/ijfam.v5i4.1684.

[3] E. Zahran and A. Bekir, “The paul-painlevé approach of the black scholes model and its exact and numerical solutions,” Journal of Science and Arts, vol. 24, no. 1, pp. 111–122, 2024. DOI: 291 10.46939/j.sci.arts-24.1-a10.

[4] J. Mohapatra, S. Santra, and H. Ramos, “Analytical and numerical solution for the time fractional black-scholes model under jump-diffusion,” Computational Economics, vol. 63, no. 5, pp. 1853–1878, 2024. DOI: 10.1007/s10614-023-10386-3.

[5] P. Stinga, “Fractional derivatives: Fourier, elephants, memory effects, viscoelastic materials, and anomalous diffusions,” Notices of the American Mathematical Society, vol. 70, no. 4, pp. 576–587, 2023. DOI: 10.1090/noti2663.

[6] F. Emmanuel and B. Teniola, “On the analysis of black–scholes equation for european option involving a fractional order with generalized two dimensional differential transform method,” Fractional Differential Calculus, vol. 11, pp. 161–173, 2022. DOI: 10.7153/fdc-2021-11-11.

[7] R. Luca, “Advances in boundary value problems for fractional differential equations,” Fractal and Fractional, vol. 7, no. 5, p. 406, 2023. DOI: 10.3390/fractalfract7050406.

[8] A. Owoyemi, I. Sumiati, E. Rusyaman, and S. Sukono, “Laplace decomposition method for solving fractional black-scholes european option pricing equation,” International Journal of Quantitative Research and Modeling, vol. 1, no. 4, pp. 194–207, 2020. DOI: 10.46336/ijqrm.v1i4.91.

[9] Z. Dere, G. Sobamowo, and A. de Oliveira Siqueira, “Analytical solutions of black-scholes partial differential equation of pricing for valuations of financial options using hybrid transformation methods,” The Journal of Engineering and Exact Sciences, vol. 8, no. 1, pp. 15 223–01i, 2022. DOI: 10.18540/jcecvl8iss1pp15223-01i.

[10] S. Saratha, G. Krishnan, M. Bagyalakshmi, and C. Lim, “Solving black–scholes equations using fractional generalized homotopy analysis method,” Computational and Applied Mathematics, vol. 39, pp. 1–35, 2020. DOI: 10.1007/s40314-020-01306-4.

[11] M. Mohamed, M. Yousif, and A. Hamza, “Solving nonlinear fractional partial differential equations using the elzaki transform method and the homotopy perturbation method,” Abstract and Applied Analysis, vol. 2022, no. 1, p.4743 234, 2022. DOI: 10.1155/2022/4743234.

[12] A. Nirmala and S. Kumbinarasaiah, “A robust numerical technique based on the chromatic polynomials for the european options regulated by the time-fractional black–scholes equation,” Journal of Umm Al-Qura University for Applied Sciences, pp. 1–18, 2024. DOI: 10.1007/s43994- 024-00193-3.

[13] M. Al-Safi, “Numerical solutions for the time fractional black-scholes model governing european option by using double integral transform decomposition method,” Results in Nonlinear Analysis, vol. 7, no. 2, pp. 64–78, 2024. DOI: 10.31838/rna/2024.07.02.007.

[14] J. Gao, “Numerical methods and computation in financial mathematics: Solving pdes, monte carlo simulations, and machine learning applications,” Theoretical and Natural Science, vol. 55, pp. 17–23, 2024. DOI: 10.54254/2753-8818/55/20240139.

[15] C. Murwaningtyas, S. Kartiko, H. Gunardi, and H. Suryawan, “Finite difference method for pricing of indonesian option under a mixed fractional brownian motion,” Mathematics and Statistics, vol. 8, no. 5, pp. 610–619, 2020. DOI: 10.13189/ms.2020.080516.

[16] A. Palbeno and N. Putri, “Analysis of put and call option pricing on bca stock using the black scholes model: Financial and risk management perspective,” International Journal of Global Operations Research, vol. 5, no. 3, pp. 176–183, 2024. DOI: 10.47194/ijgor.v5i3.299.

[17] H. Zhang, M. Zhang, F. Liu, and M. Shen, “Review of the fractional black-scholes equations and their solution techniques,” Fractal and Fractional, vol. 8, no. 2, p. 101, 2024. DOI: 10.3390/fractalfract8020101.




DOI: https://doi.org/10.18860/cauchy.v10i2.35248

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