# Geometric Brownian Motion

This is really important for anyone interested in Finance Modelling. As what the movie Wolf on Wall Street says:

Fugazi. Source: AZ Quotes

They are referring to a geometric brownian motion.

Firstly, we will begin with the definitions.

We say that a random process, , is a geometric Brownian motion (GBM) if for all

where is a Standard Brownian Motion
Here is the drift and is the volatility. We write

Also note that

; This is a common technique for solving expectations.
. This is very useful for simulating security prices.

Consider

; Notice this expansion is similar to before.

This result tells us that the expected growth rate of is .

From the definitions of Brownian Motion introduced earlier, we extend them to Geometric Brownian motion.
1. Fix . Then are mutually independent.
2. Paths of are continuous as function of , meaning they do not jump.
3. For ,

So now lets try to do some modelling of stock prices as a geometric brownian motion.

Suppose . Clearly
1. for any
This tells us that the limited liability of stock price is not violated.
2. The distribution of only depends on s and not on \$latex X_t.
We will look at the Black-Scholes option formula next time and will come back to review the geometric brownian motion for the underlying model.