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.
Setup
Given a geometric brownian motion
{% dS(t) = \mu S(t) dt + \sigma S(t) dW %}
then {% \nu %} is defined as
{% \nu = \mu - \frac{1}{2} \sigma^2 %}
Moments of a Geometric Brownian Motion
The geometric brownian motion has the following moments.
- {% \mathbb{E}[log[\frac{S(t)}{S(0)}]] = \nu t %}
- {% StdDev[log[\frac{S(t)}{S(0)}]] = \sigma \sqrt{t} %}
- {% \mathbb{E} [\frac{S(t)}{S(0)}] = e^{\mu t} %}
- {% StdDev[\frac{S(t)}{S(0)}] = e^{\mu t} \sqrt{ (e^{\sigma^2 t} - 1) } %}
(see Luenberger pg 310)