Estimation of geometric Brownian motion model with a t-distribution–based particle filter

Abstract

Orientation: Geometric Brownian motion (GBM) model basically suggests whether the distribution of asset returns is normal or lognormal. However, many empirical studies have revealed that return distributions are usually not normal. These studies, time and again, discover evidence of non-normality, such as heavy tails and excess kurtosis. Research purpose: This work was aimed at analysing the GBM with a sequential Monte Carlo (SMC) technique based on t-distribution and compares the distribution with normal distribution. Motivation for the study: The SMC or particle filter based on the t-distribution for the GBM model, which involves randomness, volatility and drift, can precisely capture the aforementioned statistical characteristics of return distributions and can predict the random changes or fluctuation in stock prices. Research approach/design and method: The particle filter based on the t-distribution is developed to estimate the random effects and parameters for the extended model; the mean absolute percentage error (MAPE) were calculated to compare distribution fit. Distribution performance was assessed through simulation study and real data. Main findings: Results show that the GBM model based on student’s t-distribution is empirically more successful than the normal distribution. Practical/managerial implications: The proposed model which is heavier tailed than the normal does not only provide an approximate solution to non-normal estimation problem. Contribution/value-add: The GBM model based on student’s t-distribution establishes an efficient structure for GBM and volatility modelling.

Keywords

• Geometric Brownian motion
• student-t distribution
• normal distribution
• drift
• volatility
• particle filter