The Random Walk

Picture someone standing on a number line who flips a coin each second: heads they step one unit right, tails one unit left. Their position is a random walk, the running sum of independent $\pm 1$ steps. It's the plainest model of something that drifts nowhere on purpose but wanders anyway, and it's the discrete ancestor of how quants model prices.

Write the steps as $X_1, X_2, \dots$, each equal to $+1$ or $-1$ with probability $\tfrac12$. The position after $n$ steps is their cumulative sum:

$$S_n = X_1 + X_2 + \cdots + X_n .$$

Each symbol is small: $X_i$ is a single coin-flip step, and $S_n$ is where you stand after $n$ of them. The path is jagged and unpredictable step to step, yet its statistics are strikingly regular.

The two numbers that describe it

Because each step averages to zero, the walk's expected position is always zero, no matter how long it runs:

$$E[S_n] = E[X_1] + \cdots + E[X_n] = 0 .$$

The spread is the interesting part. Each step has variance $1$, and independent variances add (from variance), so

$$\mathrm{Var}(S_n) = n, \qquad \sigma_{S_n} = \sqrt{n} .$$

That square root is the headline result. Typical distance from the start grows like $\sqrt{n}$, not $n$. The walk keeps wandering, but ever more slowly relative to the clock.

Worked example: how far after many steps?

After $n = 100$ steps, the expected position is still $0$, but the standard deviation is $\sqrt{100} = 10$. So a typical endpoint is about ten units from the origin, and by the The Central Limit Theorem the position is approximately normal, meaning you'd land beyond $\pm 20$ (two sigma) only about $5\%$ of the time.

Now push to $n = 10{,}000$ steps, a hundred times longer. The spread is only $\sqrt{10{,}000} = 100$, ten times wider, not a hundred. To double how far you typically roam, you must run the walk four times as long. This $\sqrt{\text{time}}$ scaling is exactly the diffusion behavior that carries over to Brownian Motion, the continuous-time limit of the random walk.

Common pitfalls

• Zero drift is not zero risk. The expected position never moves, yet the range of where you might be keeps widening. "Fair on average" and "safe" are different claims.
• Distance grows like $\sqrt{n}$, not $n$. Assuming the walk drifts proportionally to time is the single most common error; it overstates long-run movement badly.
• No memory, no reversion. A pure random walk does not get "pulled back" after a run of heads. Believing losses must reverse is the gambler's fallacy.
• Independence is doing the work. The clean $\mathrm{Var}(S_n)=n$ relies on steps being independent. If steps are correlated, the variance no longer adds so simply.

The random walk is the bridge from coin flips to continuous price models: it's a martingale (a fair game with no predictable drift), its endpoint is normal by the The Central Limit Theorem, and its scaling limit is Brownian Motion, the engine under most of quantitative finance.