Geometric Brownian motion
- Geometric Brownian motion
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 a Wiener process. It is applicable 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 positive value, and only the fractional changes of the random variate are significant. This is a reasonable approximation of stock price dynamics.
A stochastic process "S""t" is said to follow a GBM if it satisfies the following stochastic differential equation:
:dS_t = mu S_t,dt + sigma S_t,dW_t
where W_t is a Wiener process or Brownian motion and mu ('the percentage drift') and sigma ('the percentage volatility') are constants.
For an arbitrary initial value "S"0 the equation has the analytic solution
:S_t = S_0expleft( left(mu - frac{sigma^2}{2}
ight)t + sigma W_t
ight),
which is a log-normally distributed random variable with expected value mathbb{E}(S_t)= e^{mu t}S_0 and variance operatorname{Var}(S_t)= e^{2mu t}S_0^2 left( e^{sigma^2 t}-1
ight).
The correctness of the solution can be verified using Itō's lemma. The random variable log("S""t"/"S"0) is normally distributed with mean mu - sigma^2/2)t and variance sigma^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'.
ee also
*Black–Scholes
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