Numerical approximation of Black Scholes stochastic differential equation using Euler-Maruyama and Milstein methods

Abstract

This paper will introduce the Ito’s lemma used in the stochastic calculus to obtain the Ito-Taylor expansion of a stochastic differential equations. The Euler-Maruyama and Milstein’s methods of solving stochastic differential equations will be discussed and derived. We will apply these two numerical methods to the Black-Scholes model to obtain the values of a European call option of a stock at discretized time intervals. We will use a computer simulation to approximate while using the Ito’s formula to obtain the exact solution. The numerical approximations to the exact solution to infer on the effectiveness of the two methods.