Random Walks

Here is a chart of something over 12 months. It rises in the first quarter. Dips sharply in the spring. Recovers, then trends upward through autumn before pulling back at year end.

Is it a stock price? A company's monthly revenue? Or is it the result of flipping a coin 250 times, adding +1 for heads and -1 for tails, and plotting the cumulative total?

Most people looking at such a chart cannot tell. The ones who are confident are usually wrong.

This is not a gotcha. It is one of the most important facts in statistics — and in finance.

Building a random walk from scratch

Start at zero. Every step, flip a coin. Heads: move up one unit. Tails: move down one unit. Write down your position after each flip and plot it.

That's a random walk.

Each individual step is independent — the coin has no memory. Heads on the last flip tells you nothing about the next flip. And yet the cumulative path drifts, trends, reverses, and forms structures that feel meaningful. Long runs appear. Momentum seems to build. Turning points emerge.

None of it means anything. It's all noise accumulated over time.

Chartist Fallacy

A

B

C

Drift per step (μ)0.000

0.0000.150

—

Your pick

μ = 0.000

Drift per step

It's AIt's BIt's CReset

Three financial charts. Which one shows a real upward trend?

Why cumulative randomness fools everyone

A single coin flip is obviously random. Nobody sees a trend in one flip.

But cumulative processes — where each outcome adds to the previous total — stretch randomness across time in a way that looks structured. After 100 steps, you expect to be roughly $\sqrt{100} = 10$ units away from where you started, but in some arbitrary direction. After 1,000 steps, roughly $\sqrt{1000} \approx 32$ units away.

This square-root scaling is a deep property of random walks. The displacement grows, but far slower than the number of steps. You're drifting, not flying. And the shape of that drift — wandering, reversing, trending, reversing again — is what we're hardwired to interpret as signal.

The efficient market hypothesis, briefly

The reason this matters for finance is precisely that well-functioning financial markets incorporate information quickly. If a stock's price reflects everything publicly known about a company's future, then only genuinely new information should move it. New information is unpredictable by definition.

Under this view, the next price change is independent of the previous price change — and the cumulative path of prices over time follows something close to a random walk.

This doesn't mean markets are perfectly efficient, or that no one can ever find an edge. It means the bar for claiming you've found a real pattern is much higher than it feels, because random walks naturally produce what looks like patterns.

How to tell the difference (and the limits)

Distinguishing genuine trends from random walks is not impossible — it just requires the right evidence.

Length matters: a random walk that drifts upward for three months is unremarkable. One that drifts consistently upward for ten years, while the baseline noise would predict frequent reversals, is more meaningful.

Variance matters: if the steps are large relative to the drift, you need many more observations before the drift becomes distinguishable from noise. A stock with 2% daily volatility needs a very long run of gains before you can rule out randomness.

Mechanism matters: a genuine trend usually has a causal story. Revenue grows because customer count grows. A price trends upward because earnings grow consistently. Pure randomness, by definition, has no mechanism.

Without all three — sufficient length, small noise relative to signal, and a plausible mechanism — you're probably looking at a random walk wearing a costume.

The formal structure

A simple random walk is defined by the recursion:

$S_n = S_{n-1} + \epsilon_n, \quad \epsilon_n \overset{\text{iid}}{\sim} \{-1, +1\}$

where each $\epsilon_n$ is drawn independently. The expected position after $n$ steps is zero ($\mathbb{E}[S_n] = 0$), but the expected distance from the origin scales as:

$\mathbb{E}[|S_n|] \approx \sqrt{n}$

In continuous time and with normally distributed steps, this becomes Brownian motion — the mathematical model underlying the Black-Scholes options pricing formula, models of particle diffusion in physics, and population genetics.

Key takeaways

• Random walks produce trends, reversals, and patterns entirely by chance — cumulative randomness looks structured
• The typical distance from the start grows as $\sqrt{n}$, not $n$ — drift is slow relative to the number of steps
• Claiming a financial chart shows a real trend requires length, low noise, and a causal mechanism
• This is one of the hardest concepts to internalize: the chart looking like a trend is not evidence that it is one
• The efficient market hypothesis is, at its core, an argument that stock prices should behave like random walks