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