Comments on the geometric Brownian motion

• General properties of the geometric Brownian motion;
• Test for stationary time-series;

Hello, all! 

One of the main assumptions of the Black-Scholes model is that the underlying assets follows a geometric Brownian motion (GBM). In this post, we will walk through some important features and consequences of assuming a GBM, getting different insights always guided by the idea of the Occam´s razor.

In a differential form, GBM can be written as:

$\begin{gather*} dX=\mu Xdt + \sigma XdW \end{gather*}$ or $ \begin{gather*} d(lnX)=\mu dt + \sigma dW \end{gather*} $

where $W$ denotes a Brownian motion in determined measure. The analytical form of GBM, assuming $\mu$ and $\sigma$ constants, is given by:

$ X=X_0 exp\{(\mu - \frac {\sigma^2}{2})t + \sigma W  \} $

The analytical form of the GBM allows one to price different derivatives by means of an integral of a Gaussian-like function.

In the following, we discuss the main points about GBM:

(A) One could say that the most important feature of GBM is that this model prevents negative values for X. This is essential in the modelling of stock prices, for instance. In this sense, one could not directly model an stock, for instance, using an expression like this:

$ dX=\mu dt + \sigma dW $

since this model admit negative values.

(B) Another very important feature of this model is that it assumes the log of the returns between any times i and i+1 ($log(\frac{S_{i+1}}{S_i})$) presents an normal distribution. Put in other words, effects of extreme events (large scale ups and downs) are largely minimized in this model, since GBM assumes a continuous change of market, neglecting jumps that cause discontinuity in the price levels. The use of this assumption should never be forgotten, mainly in market stress situations.

An important role of a quant analyst of derivatives is to create models taking into account all relevant scenarios of the market.

(C) Models based on GBM assumes that no stationary pattern is followed, such as a long-term average described by Ornstein-Uhlenbeck processes. However, it doesn´t mean that a GBM may not follow some stationary pattern by chance. In order to test the time-series for stationary patterns one can use an ADF test. In the following, I provide a simple code in Python to test and exhibit stationarity by ADF:
which provides the following results:

In the example above, the value of the ADF test -1.92 is bigger than the critical values, in addition one observes a higher p-value. This result exemplifies the low probability of stationary patterns for GBM.

See you! May the Force be with you.

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Diogo de Moura Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso