# [Stochastic Process] Geometric Brownian Motion

1. Motivation

Definition: Let be Brownian motion with drift coefficient and variance parameter . Let . Then is geometric Brownian motion.

Motivation: Let be the price of a stock at time (where n is distance); Let be the fractional increase/decrease in the price of the stock from time n-1 to time n.
We suppose that are i.i.d. Then

The process looks like a random walk.

2. Property of Geometric Brownian Motion

Proof:
=
=
• E[Y(t) | Y(s) = y_s] = y_s e^{mu(t-s) + sigma^2(t-s)/2}
Proof: If W ~ , then is lognormal with mean
Thus, {X(t) – X(s)} ~ , since is Brownian motion with drift

• Note: if , then . Thus is increasing even though the jump process with is symmetric
3. Example
Question: You invest 1000 dollars in the stock market. Suppose that the stock market can be modeled using geometric Brownian motion with an average daily return of 0.03% and a standard deviation of 1.02%. what is the probability that your money increases after 1 year (260 business days?) 10 years? 30 years?

Answer: Let , where is standard Brownian Motion.
Let where is standard Brownian motion, is Brownian motion with drift, and is geometric Brownian motion.

=
=
=
=
=