# Geometric Brownian motion

A

**Geometric Brownian motion** (occasionally,

**exponential Brownian motion** and, hereafter, GBM) is a continuous-time

stochastic process in which the

**logarithm** of the randomly varying quantity follows a

Brownian motion.
It is appropriate 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. This is precisely the nature of a stock price.

A stochastic process S_{t} is said to follow a GBM if it satisfies the following stochastic differential equation:

where {W

_{t}} is a

Wiener process or Brownian motion and u ('the percentage drift') and v ('the percentage volatility') are constants.

The equation has a analytic solution:

for an arbitrary initial value S

_{0}. The correctness of the solution can be verified using

Ito's Lemma. The

random variable log( S

_{t}/S

_{0}) is

Normally distributed with mean (u-v.v/2).t and variance (v.v).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'.