Comparative Error Analysis of the Black-Scholes Equation

Finance is a rapidly growing area in our banking world today. With this ever-increasing development come more complex derivative products than simple buy-and-sell trades. Financial derivatives such as futures and options have been developed stemming from the traditional stock, bond, currency, and commodity markets. Consequently, the need for more sophisticated mathematical modeling is also rising. The Black-Scholes equation is a partial differential equation that determines the price of a financial option under the Black-Scholes model. The idea behind the equation is that there is a perfect and risk-free way for one to hedge the options by buying and selling the underlying asset in just the right way. This hedge implies that there is a unique and right price for the option, as returned by the Black-Scholes formula. Traditionally, the Black- Scholes equation is solved by first being reduced to a simple heat diffusion equation through exponentially scaling and changing the variables. This conversion ensures a simpler, faster, and more practical numerical scheme. However, there are several drawbacks with this method. Accuracy is often compromised and could be very unevenly distributed across the domain due to the variables being exponentially scaled. Transformation is also very limited and inflexible. For my research project, my adviser and I have proposed to solve the Black-Scholes equation directly using finite-difference schemes. By doing so, we can ensure robustness and accuracy. Error analysis of these two approaches was conducted. An assessment of the accuracy, efficiency, and robustness of each method is reported