In this article, we have studied the Black-Scholes partial differential equation (BSPDE) which is a fundamental model in financial mathematics that describes the behaviour of option pricing. One important application of the Black-Scholes equation is in the pricing of European call and put options traded in the stock markets. Traders and financial analysts use this model to estimate the fair value of options, helping them to decide when to buy or sell these contracts. Under the theoretical framework of BS model supporting the efficient functioning of financial markets, we can have accurate risk assessment and execute perfect investment strategies. In this study, we have employed the Crank-Nicolson (CN) method to solve the Black-Scholes equation for real volatility. An accurate second-order solution in both asset space and time dimension are provided by the finite difference Crank-Nicolson method. We have applied a tri-diagonal matrix algorithm to solve the Black-Scholes PDE using the Crank-Nicolson method, in which a system of linear equations is constructed. In real-time situation, the accurate volatility estimation is crucial because it deeply affects the option pricing and the risk assessment. Volatility reflects the uncertainty or risk associated with the price movement of an underlying asset. Its precise estimate ensures that traders take well-informed decisions. Using accurate volatility estimates, investors can better manage their portfolios and hedge against potential market fluctuations, leading to more stable, predictable, and profitable outcomes. The Crank-Nicolson method provides an accurate and efficient solution to the Black-Scholes equation for real volatility. The method is stable and can easily be extended to more complex financial models. Our scheme is designed to describe the behaviour of financial options in the presence of real-world volatility, which is often characterized by a nonconstant volatility. Here, we have applied the CN method to solve BS PDE numerically and simulated the call option results through MATLAB. We have also presented the graphical and numerical solutions to the option pricing problem

Black-Scholes partial differential equation, European Option Pricing, Call option, Numerical scheme, CN Method, Indian financial market

The main objective of this paper is to investigate the European call option pricing problem by using differential equations. We presented the derivation of the numerical scheme and simulated it in MATLAB. The results demonstrated the effectiveness of the numerical scheme in approximating the option price. The findings suggest that differential equations provide a robust framework for understanding and solving complex financial problems. Using the data set of the State Bank of India security. We have also performed volatility estimation for the same data set. The estimated volatility turns out to be 0.016. It suggests that the underlying asset experiences relatively small price fluctuations over the time variable. This implies that the option traded is likely to have lower risk, which can affect its pricing and enhances its attractiveness for trading among investors. As a result, traders might adjust their strategies when considering to get into the option contracts offered by the SBI in view of its stable market condition. The parameter values on which option pricing depends are indicated in the section “results & discussion”. Results obtained here through Crank-Nicolson method are in line with the results obtained by the analytical BS formula. The volatility of the underlying is a critical factor in option pricing as it measures the extent to which the price of the underlying asset is expected to fluctuate over time. Generally, higher volatility increases the option premium because it increases the likelihood of significant price movements, which could lead to more profitable outcomes. Conversely, lower volatility, as seen in this case, typically results in a lower premium, making the option less costly but also suggesting that major price changes are less likely

R.K. Gangele and S. Asati (2025). A study of the Black Scholes Pricing model by applying estimated volatility in Indian securities. International Journal on Emerging Technologies, 16(1): 149–158