Numerical Approximation of Black-Scholes Equation

Abstract

This study deals with well-known Black-Scholes model in a complete financial market. We obtain numerical methods for european and exotic options, for one asset and for two assets models. Introduction. Options are used on markets and exchanges. The Black- Scholes model is a convenient way to calculate the price of an option. In this study numerical methods will be processed to solve that equation. The numerical methods are based on finite differences. We test fully implicit, semi-implicit and explicit methods. For the discretized problem, a linear algebraic system, we test direct and iterative methods. This way we intend to create a general numerical scheme for different types of options. Section 1 deals with the Black-Scholes model. In this part we show some options used, European call and put option, and the mathematical properties of these options. Section 2 shows our numerical setup for the equation presented in Sec- tion 1. This section deals with the space discretization of the parabolic differential equation with one underlying asset, using finite difference meth- ods. We obtain an explicit and two implicit methods (fully implicit and semi-implicit method) to solve the discretized system. Numerical experi- ments are related for each method.