Geometric Brownian motion

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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:

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

where $W t$ is a Wiener process or Brownian motion and $μ$ ('the percentage drift') and $σ$ ('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(S_t/S₀) is normally distributed with mean $(μ - σ 2 / 2) t$ and variance $σ 2 t$ , 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'.