Random Walks: Why Stock Prices Follow a Drunkard's Path

A man leaves a bar in a state of complete disorientation. At each step, he moves forward or backward with equal probability. Where does he end up after n steps? On average, at the bar. But his expected distance from the bar — the root-mean-square displacement — is not zero. It is √n steps. This is the simplest random walk, and its mathematics describes phenomena across physics, biology, economics, and finance.

The mathematics of random walks

Position after n steps: Sₙ = X₁ + X₂ + ... + Xₙ
Where Xᵢ = +1 (probability p) or −1 (probability 1−p)

E[Sₙ] = n(2p − 1)       (zero for fair walk, p = 0.5)
Var[Sₙ] = 4p(1−p)n
RMS displacement = √(4p(1−p)n) = √n  for p = 0.5

Key insight: expected displacement grows as √n, not n.
To double your expected displacement, you need 4× as many steps.

The reflection principle

One of the most elegant results in random walk theory is the ballot problem: in a fair election where candidate A gets a votes and B gets b votes (a > b), what is the probability that A is ahead throughout the entire counting? The answer, derived from the reflection principle:

P(A leads throughout) = (a − b) / (a + b)

For a = 60, b = 40 (candidate A wins 60-40):
P(A always ahead) = (60 − 40) / (60 + 40) = 20/100 = 20%

Despite winning, there is an 80% chance B led at some point.
This is deeply counterintuitive.

Brownian motion and finance

As step size shrinks to zero and frequency increases to infinity, the discrete random walk converges to Brownian motion — a continuous-time process with independent, normally distributed increments. Louis Bachelier applied this to stock prices in his 1900 PhD thesis, five years before Einstein used the same mathematics to describe particle diffusion.

The Geometric Brownian Motion model for stock prices:

dS = μS dt + σS dW

Where:
  S  = stock price
  μ  = drift (expected return)
  σ  = volatility (standard deviation of returns)
  dW = increment of Brownian motion (random shock)

Solution: S(t) = S(0) × exp[(μ − σ²/2)t + σW(t)]

The log of the stock price follows a normal distribution.
This is the model underlying the Black-Scholes option pricing formula.

The limits of the random walk model

Real financial markets exhibit fat tails (extreme moves occur far more often than Gaussian models predict), volatility clustering (large moves tend to follow large moves), and mean reversion over long time horizons — none of which a pure random walk captures. The 2008 financial crisis was, in part, a crisis of models that assumed Gaussian randomness in mortgage default correlations.

The random walk is not wrong — it is a first-order approximation. It correctly captures the essential unpredictability of price movements while failing at the tails, where the most consequential events happen.