Simulation of a Second-Order Fractional Differential Equation: The Black-Scholes Equation for Call and Put Options as a Model

The aim of this paper is to investigate and apply the one-dimensional partial differential Black Scholes equation to the MASI Index in order to reduce market risk during the three months preceding the COVID 19 crisis. This research would be immensely useful in appraising equity investments in the context of the Moroccan equity market during stress scenarios, as well as testing the usefulness of the Black-Scholes equation in risk minimization. The one-dimensional partial differential equation can be written as follows:\begin{equation*}\frac{\partial \varphi}{\partial t}+\frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} \varphi}{\partial x^{2}}+r \cdot x \frac{\partial \varphi}{\partial x}-r \cdot \varphi=0\end{equation*}