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 St is said to follow a GBM if it satisfies the following stochastic differential equation:
} 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 S0
. The correctness of the solution can be verified using Ito's Lemma
. The random variable
) 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'.