Document Type : Research Article
Authors
School of Mathematics and Computer Science, Damghan University, P.O. Box 36715-364, Damghan, Iran
Abstract
The Black-Scholes model is one of the most widely used frameworks for pricing options in financial markets. However, its analytical solutions are often limited to idealized conditions, necessitating the use of numerical methods for more complex scenarios. This study proposes a combined numerical approach to solve the Black-Scholes equation, specifically focusing on call option pricing in the context of Iran's financial market. The proposed method integrates fully implicit and explicit methods to enhance accuracy and computational efficiency. By applying this approach to historical data from the Iranian options market, we demonstrate its effectiveness in capturing market dynamics and pricing call options under local conditions. The results indicate that the combined numerical method not only provides reliable pricing estimates but also offers insights into the unique characteristics of option trading in emerging markets like Iran. This research contributes to the growing body of literature on numerical methods in financial engineering and provides practical tools for traders and analysts in developing economies.
Keywords
81 (1973), no. 3, 637–654.
[2] Hull, J. C., Options, Futures, and Other Derivatives, 10th ed., Pearson, New York, 2018.
[3] Wilmott, P., Dewynne, J., and Howison, S., The Mathematics of Financial Derivatives: A
Student Introduction, Cambridge Univ. Press, 1995.
[4] Wang, R. R., Wang, Y. Y., and Dai, C. Q., Influence of higher-order nonlinear effects on
optical solitons of the complex Swift-Hohenberg model in the mode-locked fiber laser, Opt.
Laser Tech. 152 (2022), 108103.
[5] Cao, Q. H. and Dai, C. Q., Symmetric and anti-symmetric solitons of the fractional second
and third-order nonlinear Schr¨odinger equation, Chin. Phys. Lett. 38 (2021), 090501.
[6] Dai, C. Q. and Wang, Y. Y., Coupled spatial periodic waves and solitons in the photovoltaic
photorefractive crystals, Nonlinear Dyn. 102 (2020), 1733–1741.
[7] Yuan, L., Ni, Y. Q., Deng, X. Y., and Hao, S., A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations, J.
Comput. Phys. 462 (2022), 111260.
[8] Aliyari Boroujeni, A., Pourgholi, R., and Tabasi, S. H., Solving inverse partial differential
equations problems by using teaching learning based optimization algorithm, TWMS J. Appl.
Engrg. Math. (2024).
[9] Aliyari Boroujeni, A., Pourgholi, R., and Tabasi, S. H., A new improved teaching-learningbased optimization (ITLBO) algorithm for solving nonlinear inverse partial differential equation problems, Comput. Appl. Math. 42 (2023), no. 99.
[10] Pourgholi, R., Dana, H., and Tabasi, H., Solving an inverse heat conduction problem using
genetic algorithm: Sequential and multi-core parallelization approach, Appl. Math. Model.
38 (2014), 1948–1958.
[11] Aliyari Boroujeni, A., Pourgholi, R., and Tabasi, S. H., Numerical solutions of KDV and
mKDV equations: Using sequence and multi-core parallelization implementation, J. Comput.
Appl. Math. 454 (2025), 116184.
[12] Tavella, D. and Randall, C., Pricing Financial Instruments: The Finite Difference Method,
Wiley, 2000.
[13] Boyle, P., Options: A monte carlo approach, J. Financial Econ. 4 (1977), no. 3, 323–338.
[14] Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer, 2004.
[15] Zhang, P. G., Exotic options: a guide to second generation options, World Scientific, 1997.
[16] Gracianti, G., Computing Greeks by finite difference using Monte Carlo simulation and
variance reduction techniques, J. Math. Nat. Sci. 25 (2018), no. 1, 80–93.
[17] Anderson, D. F., An efficient finite difference method for parameter sensitivities of continuous time Markov chains, SIAM J. Numer. Anal. 50 (2018), no. 5, 2237–2258.
[18] Muroi, Y. and Suda, S., Computation of Greeks using binomial tree, J. Math. Finance 7
(2017), no. 3, 597–623.
[19] Jeong, D., Yoo, M., and Kim, J., Finite difference method for the Black-Scholes equation
without boundary conditions, Comput. Econ. 51 (2017), no. 4, 961–972.
[20] Bertram, D. and Fourni´e, M., High-order compact finite difference scheme for option pricing
in stochastic volatility models, J. Comput. Appl. Math. 236 (2012), no. 17, 4462–4473.
[21] Golbabai, A., Nikan, O., and Nikazad, T., Numerical analysis of time fractional BlackScholes European option pricing model arising in financial market, Comput. Appl. Math.
38 (2019), no. 173.
[22] Jeong, D., Yoo, M., Yoo, C., and Kim, J., A hybrid Monte Carlo and finite difference method
for option pricing, Comput. Econ. 53 (2019), 111–124.
[23] Peymany, M., Amiri, M., and Sokout, S. M., Option pricing using stochastic interest rate in
Tehran Stock Exchange, J. Financ. Manag. Perspect. 13 (2023), no. 41, 91–115.
[24] Peymany, M., Mathematical modeling of stock price behavior and option valuation, J. Math.
Model. Finance 1 (2021), no. 1, 113–129.
[25] Safdari-Vaighani, A., Ahmadian, D., and Javid-Jahromi, R., An approximation scheme for
option pricing under two-state continuous CAPM, Comput. Econ. 57 (2021), 1373–1385.
[26] Safdari-Vaighani, A., Heryudono, A., and Larsson, E., A radial basis function partition of
unity collocation method for convection-diffusion equations arising in financial applications,
J. Sci. Comput. 64 (2015), 341–367.
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