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 S_t is said to follow a GBM if it satisfies the following stochastic differential equation:

[dS_t=u\,S_t\,dt+v\,S_t\,dW_t]

where 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:

[S_t=S_0\exp\left((u-v^2/2)t+vW_t\right)]

for an arbitrary initial value S₀. The correctness of the solution can be verified using Itō's lemma. The random variable log(S_t/S₀) is normally distributed with mean [(u-v^2/2)t] and variance [v^2t], 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'.

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