Analytical and Numerical Approaches to Solving Black-Scholes Equation for European Options

Abstract

This thesis investigates option pricing through both analytical and numerical approaches. On the analytical side, the Black–Scholes equation was derived and its closed-form solutions for European options were presented. To address more complex structures, numerical methods were explored, with a focus on finite difference schemes. A comparative study of the explicit, implicit, and Crank–Nicolson methods was conducted, highlighting their respective advantages and limitations in terms of stability and accuracy. The explicit scheme, while simple, was shown to be only conditionally stable. The implicit scheme achieved unconditional stability but at the cost of reduced accuracy. The Crank–Nicolson method, by contrast, combined unconditional stability with second-order accuracy in time, making it the most effective balance between robustness and precision. Overall, the thesis demonstrates how analytical and numerical frameworks complement one another, providing both theoretical insight and .practical tools for option pricing