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

External links