Geometric Brownian Motion
Overview
A geometric brownian motion, {% S(t) %} is a stochastic process. As such, the random variable {% S(t) %}
has
Moments
like any other random variable.
Moments of a Geometric Brownian Motion
Given a geometric brownian motion defined as
{% dS(t) = \mu S(t) dt + \sigma S(t) dW %}
and define
{% \nu = \mu - \frac{1}{2} \sigma^2 %}
we have the following moments.
{% \mathbb{E}[log[S(t)/S(0)]] = \nu t %}
{% StdDev[log[S(t)/S(0)]] = \sigma \sqrt{t} %}
{% \mathbb{E} [S(t)/S(0)] = e^{\mu t} %}
{% StdDev[S(t)/S(0)] = e^{mu t} \sqrt{ (e^{\sigma^2 t} - 1) } %}
