Random walks and Brownian motion

From coin flips to Brownian motion

Take a random walk that steps $\pm\sqrt{\Delta t}$ every $\Delta t$ units of time. As $\Delta t \to 0$ it converges to Brownian motion $W_t$ (also called a Wiener process), with three defining properties:

1. $W_0 = 0$.
2. Increments over non-overlapping intervals are independent.
3. $W_t - W_s \sim \mathcal{N}(0,\, t - s)$.

The variance grows linearly with time, so the typical size of a move grows with $\sqrt{t}$. That is where every $\sqrt{T}$ in finance comes from.

Geometric Brownian motion

Prices are modelled as growing proportionally, with proportional noise:

$$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$$

$\mu$ is the drift (expected return) and $\sigma$ the volatility. The solution is

$$S_t = S_0 \exp\!\Big(\big(\mu - \tfrac{\sigma^2}{2}\big)t + \sigma W_t\Big)$$

so $\ln S_t$ is normal and $S_t$ is lognormal. The hero chart on the home page is exactly this formula; drag its sliders to feel how $\sigma$ widens the fan.

Where the minus sigma squared over two comes from

In ordinary calculus, $d(\ln S) = dS/S$. In stochastic calculus there is an extra term, because $(dW)^2$ is not negligible: it behaves like $dt$. Itô’s lemma says that for a smooth function $f(S, t)$,

$$df = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial S}dS + \frac{1}{2}\frac{\partial^2 f}{\partial S^2}(dS)^2$$

with $(dS)^2 = \sigma^2 S^2\,dt$. Applying it to $f = \ln S$ gives

$$d(\ln S) = \big(\mu - \tfrac{\sigma^2}{2}\big)dt + \sigma\,dW$$

The $-\sigma^2/2$ is the same volatility drag you met in returns: the expected price grows at $\mu$, but the median price — the typical path — grows at only $\mu - \sigma^2/2$.

Hitting probabilities

A useful closed form: for Brownian motion with drift $\nu$ and volatility $\sigma$, starting at 0, the probability of ever touching a level $-a$ below (with $\nu > 0$) is $e^{-2\nu a/\sigma^2}$. Applied to log-prices, this gives the odds that a position ever hits its stop before drifting away — the continuous cousin of gambler’s ruin.

Limits of the model

GBM assumes constant volatility, no jumps and independent increments. Real markets violate all three: volatility clusters, gaps happen, and microstructure creates short-term reversal. Extensions include stochastic volatility (Heston), jump-diffusion (Merton), and mean-reverting processes (Ornstein–Uhlenbeck). GBM remains the baseline because it is simple, tractable and right about the most important thing: risk scales with $\sqrt{t}$.