Numerical Methods and Option Pricing

This dissertation will describe the fundamental concepts of the theory of options valuation. Since the behavior of derivative financial instruments is described by partial differential equations, there is often no analytical solution or universal formula. Therefore, various numerical methods are used to estimate their values. The basic concepts of the theory of random processes will be introduced to describe the Black-Scholes equation, which is a mathematical model of the dynamics of a financial market containing derivative investment instruments that give option prices under certain assumptions about the dynamics of stock prices. The paper will also describe one of the main methods for estimating the values of derivatives, namely the method of finite differences or difference schemes. In this paper, we will use the following difference schemes: explicit, implicit, mixed, and explicit for vulnerable options. Calculations of european, american and vulnerable options will also be provided. Computer calculations were performed to study the effectiveness of these methods for solving the Black-Scholes equation in option pricing.