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

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, X_t, is a geometric Brownian motion (GBM) if for all t \ge 0
X_t = e^{(\mu - \frac{\sigma^2}{2})t} + \sigma W_t
where W_t is a Standard Brownian Motion
Here \mu is the drift and \sigma is the volatility. We write X_t \sim GBM (\mu, \sigma)

Also note that
X_{t+s}
= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t+s}}
= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t} + (\mu - \frac{\sigma^2}{2})s + \sigma (W_{t+s} - W_t)}; This is a common technique for solving expectations.
= X_t e^{(\mu - \frac{\sigma^2}{2})(s) + \sigma (W_{t+s} - W_t)}. This is very useful for simulating security prices.

Consider \mathbb{E}_t [X_{t+s}]

\mathbb{E}_t [X_{t+s}]
= \mathbb{E}_t [X_{t}e^{(\mu - \frac{\sigma^2}{2})s + \sigma(W_{t+s} - W_t)}]; Notice this expansion is similar to before.
Latex = X_t e^{(\mu - \frac{\sigma^2}{2})s} \mathbb{E}^t [e^{\sigma (W_{t+s} - W_t)}]
= X_t e^{(\mu - \frac{\sigma^2}{2})s} e^{\frac{\sigma^2}{2}s}
= e^{\mu s} X_t
This result tells us that the expected growth rate of X_t is \mu.

From the definitions of Brownian Motion introduced earlier, we extend them to Geometric Brownian motion.
1. Fix t_1, t_2, \ldots , t_n. Then \frac{X_{t_2}}{X_{t_1}}, \frac{X_{t_3}}{X_{t_2}}, \ldots \frac{X_{t_n}}{X_{t_{n-1}}} are mutually independent.
2. Paths of X_t are continuous as function of t, meaning they do not jump.
3. For s > 0, \mathrm{log}(\frac{X_{t+s}}{X_t}) \sim \mathrm{N}((\mu - \frac{\sigma^2}{2})s, \sigma^2 s)

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

Suppose X_t \sim GBM(\mu, \sigma). Clearly
1. X_t > 0 \Rightarrow X_{t+s} > 0 for any s > 0
This tells us that the limited liability of stock price is not violated.
2. The distribution of \frac{X_{t+s}}{X_t} 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.

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