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, $X_t$ , 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.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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