Geometric Brownian motion

A geometric Brownian motion (GBM) (occasionally, exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, or a Wiener process. It is applicable to mathematical modelling of some phenomena in financial markets. It is used particularly in the field of option pricing because a quantity that follows a GBM may take any value strictly greater than zero, and only the fractional changes of the random variate are significant. This is a reasonable approximation of stock price dynamics.

A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation:


 * $$ dS_t = \mu S_t\,dt + \sigma S_t\,dW_t $$

where $$ W_t $$ is a Wiener process or Brownian motion and $$ \mu $$ ('the percentage drift') and $$ \sigma $$ ('the percentage volatility') are constants.

The equation has an analytic solution:


 * $$ S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) $$

for an arbitrary initial value S0. The correctness of the solution can be verified using Itō's lemma. The random variable log(St/S0) is normally distributed with mean $$ (\mu - \sigma^2/2)t $$ and variance $$ \sigma^2t $$, which reflects the fact that increments of a GBM are normal relative to the current price, which is why the process has the name 'geometric'.