On Solution of Scalar First Order Stochastic Differential Equations Via Semi-Implicit Milstein Method

Abstract

Scalar first order stochastic differential equation (SDEs) is a mathematical equation that describes the evolution of a single random variable over time, subject to random fluctuations. It consists of deterministic part and stochastic or diffusion part. The deterministic part contains the drift function while the diffusion part contains the diffusion function. This paper examined the solution of scalar first order SDEs with regard to Balck-Scholes option pricing model (BSOPM) commonly used in financial setting. Two forms of this model have been considered, they include the form where the drift function is greater than diffusion function and the form where the diffusion function is greater than drift function. The solutions of the SDEs were calculated using semi-implicit Milstein method. Having calculated the exact and numerical solutions of the models, the absolute errors were determined. The performance of the method was compared using mean absolute error (MAE) criteria. The strong order of convergence (SOC) for the method was also calculated to determine its accuracy. To know how best the method approximates the SDEs, the SOC obtained was compared with that of two existing methods such as explicit Euler-Maruyama and Milstein methods.  Graphical solutions were obtained for the method using two step-sizes.