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 S_t is said to follow a GBM if it satisfies the following stochastic differential equation:

<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t </math>

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

The equation has an analytic solution:

<math> S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) </math>

for an arbitrary initial value S₀. The correctness of the solution can be verified using Itō's lemma. The random variable log(S_t/S₀) is normally distributed with mean <math> (\mu - \sigma^2/2)t </math> and variance <math> \sigma^2t </math>, 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'.

See Also

Black–Scholes