Milstein-Based Maximum Likelihood Estimators for Discretely Observed Geometric Brownian Motion Stochastic Model with Application to Financial Risk Volatility

Generally, the geometric Brownian motion (GBM) efficiently models financial instruments with constant drift and volatility, while the Milstein scheme refines the simulation, particularly in the case of high volatility or large time steps. The present research seeks to determine the parameter estimators for the discretely observed GBM process. This methodology commences with the Milstein discretization scheme to approximate the underlying diffusion process, from which the maximum likelihood-based estimation method is utilized to derive straightforward expressions of the parameter estimators. Subsequently, using relative absolute errors and R software, it is established that, under reasonable condition, the parameter estimators of the process converge to their real values. More importantly, the paper further illustrates the practical use of the GBM process through modeling Morocco’s real interest rates (IRs), considering the absolute errors of the estimators and reinforced by numerical simulations. The findings offer significant insights for predicting the future dynamics of Morocco’s real IRs over the next three years.