Pricing Basket Option in Using the Two-Dimensional Black Scholes Equation and the Monte Carlo Approach

The purpose of this article is to investigate and use the two-dimensional partial differential Black-Scholes equation using the Monte Carlo method, including Cholesky decomposition to generate correlated Brownian movements, to analyze options on two underlying assets. This study focuses on the performance evaluation and risk management of investment portfolios comprised of these two assets over a given time period. Using a numerical simulation method, this study intends to show how the two-dimensional Black-Scholes model improves the ability to reflect the intricacies of market dynamics with linked assets. The two-dimensional fractional differential equation could be expressed in the form of: \begin{gather*}\frac{\partial\mu}{\partial t}+\frac{1}{2}\sigma_{1}^{2}x^{2}\frac{\partial^{2}\mu}{\partial x^{2}}+\frac{1}{2}\sigma_{2w}^{2}y^{2}\frac{\partial^{2}\mu}{\partial y^{2}}+\rho\sigma_{1}\sigma_{2}xy\frac{\partial^{2}\mu}{\partial x\partial y}\\ +r.\ [x\frac{\partial\mu}{\partial x}+y\frac{\partial\mu}{\partial y}]-r.\mu=0\end{gather*}