Summary: Geometric Brownian Motion is the predominant way to model stock prices that is utilized by option pricing models, such as the Black-Scholes model. In part one of this two-part series, we explore the foundation of this model and examine some of the notable properties that are inherent within it. Some people love Geometric Brownian Motion and believe it to be a good representation of actual stock price behavior, while others feel that it is severely lacking in its realistic application.
Proponents of Geometric Brownian Motion point to the following as pros to real-world application: expected returns are independent of stock prices, stock prices can take positive values only, and these prices follow an irregular, or random, path. While opponents of Geometric Brownian Motion suggest that both heteroskedasticity (changing volatility over time) and the presence of jumps in the distribution (from earnings announcements, news events, etc.) renders Geometric Brownian Motion difficult to use with real stock prices.
Some properties of the model are:
1) Regardless of where you are in the process, the distribution is normally distributed with mean μ (positive drift) and variance t
2) Incremental steps in the process are also distributed with mean μ and variance t
3) Geometric Brownian Motion is a Markov Process, which simply means that it has no memory – the past is irrelevant to the future
4) It is also a Martingale Process, which means that our expectation of some future value is nothing more than the current value itself (plus the positive drift…making this a ‘Sub-Martingale’)
5) The process is fractal in nature, which is a mathematical term used to describe some infinitely complex pattern that continually repeats itself
6) All paths of Geometric Brownian Motion that originate from some point will be random in nature