Comparing the Exact and Approximate Solutions of Financial-Type Stochastic Differential Equations

Abstract

In financial engineering, stock price prediction remains one of the most interesting problems of stock market analysis. One approach to solution consists of using historical data of the underlying asset to predict future prices while relying on stochastic differential equations for modeling stock price dynamics. The analytical solutions of financial-like stochastic differential equations are not always obvious; numerical methods for approximating the solutions are often considered. In this work, we propose the Euler-Maruyama method to compute the approximate solutions on both geometric Brownian motion and Heston-type models. We first perform the parameters estimation of the models and then compare the exact and approximate solutions of the models. We investigate the convergence of numerical solutions to analytical solutions for various situations and uncover interesting behaviors as we increase the time steps. Additionally, we discuss the impact of the volatility in price prediction with the specific case of stochastic volatility from the Heston model.