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.

翻译：本文的主要目的是对首个被广泛使用的期权定价模型——Black-Scholes模型——提供一个全面的概述与理解。文中介绍了该模型的历史背景及其在经济学领域中的应用价值与意义。为推导和求解该模型，文章简要回顾了基础微积分概念。随后，分别采用解析方法（变量分离法）和数值方法（有限差分法）对方程进行求解。通过结论分析，阐述了Black-Scholes模型在当今的实际应用方式。文末附有两个附录：附录（A）提供了一些经济学基础概念，以帮助读者理解全文；附录（B）则给出部分代码脚本，便于读者实践相关概念。