Geometric Brownian motion
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A geometric Brownian motion (GBM) (occasionally, exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, or, perhaps more precisely, a Wiener process. 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, and only the fractional changes of the random variate are significant. 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:
- <math>dS_t=u\,S\,dt+v\,S\,dW_t<math>
where {Wt} is a Wiener process or Brownian motion and u ('the percentage drift') and v ('the percentage volatility') are constants.
The equation has an analytic solution:
- <math>S_t=S_0\exp\left((u-v^2/2)t+vW_t\right)<math>
for an arbitrary initial value S0. The correctness of the solution can be verified using Itô'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'.de:Geometrische Brownsche Bewegung