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.
Highlights
Most of the models utilised in the description of financial time series are written in terms of a continuous time diffusion St that satisfies the stochastic differential equation (SDE): dSt = μStdt + σSt dBt [Eqn 1]where dBt ~ N(0,dt) is the increment to Brownian motion process, and σSt and μSt denote the volatility and drift function, respectively
We extended our investigations by introducing a geometric Brownian motion (GBM) model based on the t-distribution–based particle filter to approximate the return distributions of assets and compared the distribution with normal distribution
Geometric Brownian motion is the stochastic process used in the Black–Scholes methodology to model the evolution of prices in time
Summary
Where dBt ~ N(0,dt) is the increment to Brownian motion process, and σSt and μSt denote the volatility and drift function, respectively. This class of parametric models has been extensively used to portray the dynamics of financial variables, including stock prices, interest rates and exchange rates. The GBM is one of the most popular stochastic processes and undoubtedly an effective instrument in modelling and predicting the random changes in stock prices that evolve over time It is essentially useful for index price study because the process in question assumes that percentage changes are independent and identically distributed over equal and non-overlapping time length (Luenberger 1995; Ross 2000). The sequential Monte Carlo (SMC) methods or the particle filter have allowed for the accurate evaluation of likelihoods at fixed parameter values (Nemeth, Feamhead & Mihaylova 2014)
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