International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 11, No. 4, 8 Dec. 2025
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Solutions of Black-Scholes Equation by Some Numerical Approaches
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Author(s)
Index Terms
European Options, Black-Scholes Equation, Partial Differential Equation, Financial Mathematics, Numerical Techniques
Abstract
The Black-Scholes equation plays an important role in financial mathematics for the evaluation of European options. It is a fundamental PDE in financial mathematics, models the price dynamics of options and derivatives. While a closed-form of analytical solution exists for European options, numerical methods remain essential for validating computational approaches and extending solutions to more complex derivatives. This study explores and compares various numerical techniques for solving the Black-Scholes partial differential equation, including the finite difference method (explicit, implicit, and Crank-Nicolson schemes), and Monte Carlo simulation. Each method is implemented and tested against the analytical Black-Scholes formula to assess accuracy, convergence, and computational efficiency. The results demonstrate the strengths and limitations of each numerical approach, providing insights into their suitability for different option pricing scenarios. This comparative analysis highlights the importance of method selection in practical financial modeling applications.
Cite This Paper
Md. Mehedi Hasan, Md. Biplob Hossain, "Solutions of Black-Scholes Equation by Some Numerical Approaches", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.11, No.4, pp. 24-38, 2025. DOI: 10.5815/ijmsc.2025.04.03
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