Geometric Brownian motion

A Geometric Brownian motion (occasionally, exponential Brownian motion and, hereafter, GBM) iscontinuous-time stochastic processwhichlogarithm ofrandomly varying quantity followsBrownian motion. Itappropriatemathematical modellingsome phenomenafinancial markets. Itused particularly infieldoption pricing becausequantity that followsGBM may take any value strictly greater than zero. Thispreciselynature ofstock price.

A stochastic process S_tsaidfollowGBM ifsatisfiesfollowing stochastic differential equation:

where {W_t} isWiener process or Brownian motionu ('the percentage drift')v ('the percentage volatility')constants.

The equation hasanalytic solution:

for an arbitrary initial value S₀. The correctness ofsolution can be verified using Ito's Lemma. The random variable log( S_t/S₀)Normally distributedmean (u-v.v/2).tvariance (v.v).t, which reflectsfact that increments ofGBMNormal relative tocurrent price, whichwhyprocess hasname 'geometric'.