Option Valuation Using Finite Difference Methods

The finite difference method is a mathematical construct that can be used to solve partial differential equations. In this study, we used the finite difference method to solve the Black-Scholes-Merton partial differential equation to calculate options prices. Three methods were used: the Implicit Method, the Explicit Method, and the Crank-Nicolson Method. Using some code and the help of MATLAB I was able to calculate for each of the three methods listed above the values of both call and put options using the Black-Scholes-Merton partial differential equation. Furthermore, the Binomial Cox-Ross-Rubinstein Model was introduced briefly to conduct a comparative study using this model and the finite difference methods. An analysis was carried out to ascertain which of the above methods would agree with the Black-Scholes value of an option. It was found that only the Explicit Method, the Implicit Method, and the Binomial CRR Model produced similar values. A second analysis was done to compare which of the models would converge to the Black-Scholes value of an option given that the number of time intervals L and the intervals of the stock prices were increased. It was found that the Crank-Nicolson method converged faster than the Binomial Model. Hence, we conclude that the finite difference model is more appropriate than the Binomial CRR model.