MATHEMATICAL AND STATISTICAL COMPUTATION OF OPTION PRICING USING THE BLACK-SCHOLES EQUATION

  • Jacob Emmanuel Kogi State Polytechnic, Lokoja
  • Zemira Aliyu National Aids and STI Control, Federal Ministry of Health
  • Paul Ogba Kogi State Polytechnic
Keywords: Black-Scholes Model, Numerical Solution, Explicit Scheme, Call Option, Stability Criterion, Error Estimation

Abstract

This study focuses on the numerical solution of the Black-Scholes model, a key framework in financial mathematics for pricing European-style options. The model describes the behavior of option prices in relation to asset price, volatility, interest rate, and time to maturity. While exact analytical solutions exist for simple cases, numerical methods offer greater adaptability for real-world applications. In this work, we implement an explicit finite difference scheme to approximate the solution of the Black-Scholes partial differential equation. A stability criterion is derived to ensure numerical reliability, and accuracy is measured using the L1-norm by comparing results with the analytical solution. MATLAB simulations are used to compute the price of a European call option with a strike price of $100, a 12% risk-free interest rate, 10% volatility, and a one-year maturity. The generated graph (Figure 1) illustrates how the option value increases as the stock price moves from $70 to $130, notably rising when it exceeds the strike price. A comparative study with a semi-implicit scheme from existing literature confirms the enhanced precision of our explicit approach. These findings demonstrate the accuracy, efficiency, and practical utility of the explicit finite difference method for solving the Black-Scholes model.

References

Anwar, M.N. and Andallah, L.S. (2018) A Study on Numerical Solution of Black-Scholes Model. Journal of Mathematical Finance, 8, 372-381. https://doi.org/10.4236/jmf.2018.82024

Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. https://doi.org/10.1086/260062

Merton, R.C. (1973) Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4, 141-183. https://doi.org/10.2307/3003143

Jodar, L., Sevilla-Peris, P., Cortes, J.C. and Sala, R. (2005) A New Direct Method for Solving the Black-Scholes Equation. Applied Mathematics Letters, 18, 29-32. https://doi.org/10.1016/j.aml.2002.12.016

Company, R., Gonzalez, A.L. and Jodar, L. (2006) Numerical Solution of Modified

Black-Scholes Equation Pricing Stock Options with Discrete Dividend. Mathematical and Computer Modeling, 44, 1058-1068. https://doi.org/10.1016/j.mcm.2006.03.009

Dura, G. and Mosneagu, A.-M. (2010) Numerical Approximation of Black-Scholes Equation. An. Stiint. Univ.Al. I. Cuza Iasi. Mat, 56, 39-64.

Durojaye M.O. and Kazeem J.A (2020) A semi-Analytical Solution of the black -Scholes Pricing Model for European Call Option. Direct Research Journal of Engineering and Information Technology Vol.7(2),pp.55 https://doi.org/10.26765/DRJEIT21060549

Jacob Emmanuel, Olumi T.T, Ibrahim Abdulqudus and Suberu I.K ( 2022) Numercal Simulation of Coronavirus Disease Epidemic Based on Established Susceptible- Exposed- Infectious Recovered-Undetectable Susceptible Model Vol 6 N0.5, October 2022, pp 224-230. https://doi.org/10.33003/fjs-2022-0605-1731

Shinde, A.S. and Takale, K.C. (2012) Study of Black-Scholes Model and its Applications. Procedia Engineering, 38, 270-279. https://doi.org/10.1016/j.proeng.2012.06.035

Smith, G.D. (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon Press, Oxford.

Duffy, D.J. (2013) Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach. John Wiley & Sons, Hoboken.

Published
2025-07-10
How to Cite
Emmanuel, J., Aliyu, Z., & Ogba, P. (2025). MATHEMATICAL AND STATISTICAL COMPUTATION OF OPTION PRICING USING THE BLACK-SCHOLES EQUATION. FUDMA JOURNAL OF SCIENCES, 9(7), 8 - 12. https://doi.org/10.33003/fjs-2025-0907-3667