White noise, random walks, and why returns ≠ prices

The white-noise / random-walk distinction is the keystone of empirical finance. Prices are (approximately) random walks. Returns are (approximately) white noise. Conflating the two is the most common conceptual error among newcomers to time-series finance.

White noise

ε_t ~ WN(0, σ²)
E[ε_t] = 0,  Var(ε_t) = σ²,  Cov(ε_t, ε_s) = 0 for t ≠ s

Weak white noise requires only uncorrelatedness. Strong white noise (= i.i.d. with zero mean) is a stronger condition. Gaussian white noise = i.i.d. N(0, σ²). The error term in most time-series models is assumed white noise of one of these flavours.

Random walks

X_t = X_{t-1} + ε_t,    ε_t ~ WN(0, σ²)
X_t = X_0 + Σ_{s=1}^t ε_s    (cumulative sum)

• Mean: E[X_t] = X_0 (constant, conditional on the starting value).
• Variance: Var(X_t) = t σ² — grows linearly with t. Non-stationary.
• Increment: X_t - X_{t-1} = ε_t. The first difference is white noise.

Random walk with drift

X_t = μ + X_{t-1} + ε_t
E[X_t] = X_0 + μt    (linear trend in mean)
Var(X_t) = t σ²       (still grows linearly)

This is the standard model for log prices: log P_t = log P_{t-1} + μ + ε_t, equivalently log P_t = log P_0 + μ t + (sum of ε_s). The drift μ corresponds to the expected log return per period. Returns r_t = log P_t - log P_{t-1} = μ + ε_t are white noise around μ.

Why prices look random

Efficient market hypothesis (Fama, 1965-70): if prices reflect all available information, future price changes must be unpredictable given current information. Therefore returns are (approximately) white noise. This is a testable implication, not a metaphysical claim — and a vast literature documents deviations (momentum, mean reversion, microstructure), though most are small in magnitude or expensive to exploit.

Testing for randomness

• Ljung-Box test: joint test of zero autocorrelation up to lag h. Q-statistic ~ χ²_h under H₀.
• Variance ratio test (Lo-MacKinlay 1988): if returns are uncorrelated, the variance over k periods should be k times the one-period variance. Powerful against mean-reversion alternatives.
• Runs test, BDS test: non-parametric checks for independence vs serial dependence.

From white noise to GBM

Aggregating a random walk with drift in the continuous-time limit gives geometric Brownian motion (Stochastic Calculus Module 1). The bridge is the central limit theorem applied to compounded white-noise innovations. This is the connection between discrete-time empirical finance and the Black-Scholes continuous-time world.