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

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.