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Abstract
Geometric Brownian motion (GBM), a stochastic differential equation, can be used to model phenomena that are subject to fluctuation and exhibit long-term trends, such as stock prices and the market value of goods. The model uses two parameters, the rate of drift from previous values and volatility, to describe and predict how the continuous-time stochastic process evolves over time. Accurate estimates of the drift rate and volatility are necessary for these models to be useful within quantitative economic decision-making models. Multiple estimation methods have been proposed in previous research. We show how well these methods perform using a GBM with known parameters using different sample sizes. Using a GBM model, we estimated the parameters for historical oil prices and performed statistical analyses to determine how well the oil prices fit a GBM model.
Keywords
Geometric Brownian motion, data analytics, simulation, maximum likelihood
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1. Introduction
Many observable phenomena exhibit stochastic, or non-deterministic, behavior over time. Geometric Brownian motion (GBM) is a stochastic differential equation that may be used to model phenomena that are subject to fluctuation and exhibit long-term trends. In particular, [3] has referred to it as "the model for stock prices". [7] indicated that the accuracy of a GBM model for oil prices is yet to be determined. In this paper, we examine estimation methods for GBM model parameters in the context of modelling fossil fuel prices. This is a first step in developing a quantitative decision making model based on an underlying GBM.
GBM differs from the generalized Brownian motion by removing the assumption of a constant drift rate. Instead, the expected rate of return у is assumed constant [3]. Additionally, for GBM, the drift rate is equal to the current value of x multiplied by expected rate of return y, or yx. The volatility rate of this function is expressed as the parameter a. "The stochastic variable x(t) follows a geometric Brownian motion if it satisfies the stochastic differential equation,
... (1)
where, dz is the increment of a Wiener process and y and a are the parameters to be estimated" [4].
The ratios of x(t) to x(t-1) have a lognormal distribution [6]; therefore, the log values are IID with a normal distribution. In the...