Random Walks, Brownian Motion, and Why Everything Scales with √t

If you retain one quantitative fact from this entire library, make it this: randomness accumulates with the square root of time. Nearly every scaling rule in finance is this one fact wearing different clothes.

The simple random walk

Let $S_n = X_1 + \cdots + X_n$ where each $X_i$ is $\pm 1$ with probability $\tfrac12$. Then

$$E[S_n] = 0, \qquad \operatorname{Var}(S_n) = n, \qquad \sigma_{S_n} = \sqrt{n}.$$

The variance calculation is one line — variances of independent steps add — but the consequence is deep: after 100 steps, the walk is typically about $10$ from the origin, not $0$ and not $100$. The walk does wander (contrary to the intuition "it averages out"), but slowly (contrary to the intuition "randomness compounds like growth").

Interview staples that are just this fact:

• Daily vol to annual vol: multiply by $\sqrt{252} \approx 16$. A 1% daily-vol stock has $\approx 16\%$ annual vol. (And the reverse: annual 32% ⇒ daily 2%.)
• "How far will the price wander in 4 hours vs 1 hour?" — twice as far, not four times.
• Why doubling your holding period does not double your risk-adjusted return: mean P&L scales with $t$, noise with $\sqrt{t}$, so Sharpe scales with $\sqrt{t}$.

Brownian motion

Shrink the step size and time interval together (steps of size $\sqrt{\Delta t}$ every $\Delta t$) and the walk converges to Brownian motion $W_t$: continuous paths with independent, normal increments,

$$W_t - W_s \sim N(0,\, t - s).$$

The defining weirdness: over a tiny interval $dt$, the move $dW$ has magnitude $\sqrt{dt}$, which is enormous relative to $dt$. Consequently $(dW)^2 \approx dt$ is not negligible — the source of the extra term in Itô's lemma and ultimately of option time-decay. You don't need stochastic calculus fluency for trading interviews, but you should be able to say: "squared Brownian increments are order $dt$, so second derivatives enter the drift — that's why gamma and theta are linked."

Asset prices are modeled as geometric Brownian motion,

$$\frac{dS}{S} = \mu\, dt + \sigma\, dW,$$

meaning returns, not prices, are the random walk — prices stay positive and the distribution of $S_t$ is lognormal.

Two classic results you should know cold

Reflection / first passage. For a driftless walk, the probability of hitting a level before another follows from the martingale argument (see the gambler's-ruin pattern): starting at $x$ between barriers at $0$ and $a$, $P(\text{hit } a \text{ first}) = x/a$. Clean, and constantly reused.

Expected time to exit. From the same setup, the expected time to hit either barrier is $x(a-x)$ steps — maximized in the middle, quadratic in the distance. The $\sqrt{t}$ law in reverse: to wander a distance $d$ takes order $d^2$ time.

The interview version

"A stock is at 100 with 2% daily vol. Make me a market on where it'll be in a month." The $\sqrt{t}$ machinery: monthly vol $\approx 2\% \times \sqrt{21} \approx 9\%$, so a 1-standard-deviation band is roughly $100 \pm 9$. Quote around that, widen for fat tails, and you've demonstrated the entire toolkit — vol scaling, normal quantiles, and market-making in one answer.

The failure mode to avoid: scaling risk linearly with time. It's the most common quantitative error in interviews, and the $\sqrt{t}$ correction is usually the entire point of the question.