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:
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'.