An Alternative Method for Analytical Solutions of Two-Dimensional Black-Scholes-Merton Equation

Abstract

We present a method of deriving analytical solutions for a two-dimensional Black-Scholes-Merton equation. The method consists of three changes of variables in order to reduce the original partial differential equation (PDE) to a normal form and then solve it. Analytical solutions for two cases of option pricing on the minimum and maximum of two assets are derived using our method and are shown to agree with previously published results. The advantage of our solution procedure is the ability of splitting the original problem into several components in order to demonstrate some solution properties. The solutions of the two cases have a total of five components; each is a particular solution of the PDE itself. Due to the linearity of the two-dimensional Black-Scholes-Merton equation, any linear combination of these components constitutes another solution. Some other possible solutions as well as the solution properties are discussed.