Black-Scholes Model, comparison between Analytical Solution and Numerical Analysis

TL;DR

This paper provides an overview of the Black-Scholes model, comparing its analytical solution with numerical methods, and discusses its historical context, applications, and practical implementation through code examples.

Contribution

It offers a comprehensive comparison between analytical and numerical solutions of the Black-Scholes model, including practical coding resources for implementation.

Findings

Analytical and numerical solutions are both viable for Black-Scholes.

Numerical methods can handle more complex variations of the model.

The paper clarifies the model's relevance and application in modern finance.

Abstract

The main purpose of this article is to give a general overview and understanding of the first widely used option-pricing model, the Black-Scholes model. The history and context are presented, with the usefulness and implications in the economics world. A brief review of fundamental calculus concepts is introduced to derive and solve the model. The equation is then resolved using both an analytical (variable separation) and a numerical method (finite differences). Conclusions are drawn in order to understand how Black-Scholes is employed nowadays. At the end a handy appendix (A) is written with some economics notions to ease the reader's comprehension of the paper; furthermore a second appendix (B) is given with some code scripts, to allow the reader to put in practice some concepts.