Encyclopedia > GBM

  Article Content

Geometric Brownian motion

Redirected from GBM

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 {Wt} 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 log( St/S0) 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'.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article

... rhetoric as the counterpart of dialectic. By this, he means that, while dialectical methods are necessary to find truth, rhetorical methods are required to communicate ...