Temporal and Spatial Analysis of the Black-Scholes Equation for Option Pricing

Abstract

Options are very important nancial instruments and, like any other its kind, require very accurate quantitative models in order to serve its usefulness. The Black-Scholes model provides a good option pricing formula in a given market setting. In this work, we quantify the e ects of the assumptions of the Black-Schole's nancial market on their pricing formula. We expand their model by relaxing these assumptions and derive an option pricing formula in a more realistic nancial environment. This requires the introduction of a new hedging strategy and the remodeling of stock prices and therefore the market volatility in the Black-Scholes pricing formula to re ect some activities and consequences of modern exchanges or trading markets. This work is organized such that the numerical treatment of the Black-Scholes' classical linear option pricing model is considered, followed by the derivation of an option pricing equation based on the relaxation of Black-Scholes market assumptions and then, the presentation of a numerical solution of the resulting nonlinear and generalized Black-Scholes model.