White Noise and Random Walks

If price changes were completely random, no pattern or indicator could predict them. That idea is captured by the random walk model, in which each step is a random draw unrelated to the past. White noise is the randomness itself: a sequence of independent values with a constant mean and variance. These models are the benchmark against which every trading strategy is judged. If a strategy cannot beat a random walk after costs, it has no edge. Understanding them helps traders recognise how much of what they see on charts is noise.

White noise#

A white noise series has:

• Constant mean, usually zero.
• Constant variance.
• No autocorrelation: each value is unrelated to past values.

White noise is unpredictable by definition. Its best forecast is always its mean.

Random walk#

A random walk adds up white noise steps:

P_t = P_(t-1) + ε_t

With drift (a steady average move):

P_t = P_(t-1) + μ + ε_t

The changes (P_t minus P_(t-1)) are white noise; the level wanders without returning to any fixed value. A random walk is non stationary. See Stationarity, Differencing and Unit Roots.

Random walks look like real charts#

The random walk hypothesis#

The idea that stock prices follow a random walk was popularised by Burton Malkiel's 1973 book "A Random Walk Down Wall Street", building on earlier work by Louis Bachelier (1900), Maurice Kendall (1953) and Eugene Fama's efficient market hypothesis. If markets quickly incorporate all available information, future price changes should depend only on new, unpredictable information.

Evidence against a pure random walk#

Researchers have found departures from a pure random walk:

These patterns are generally small, can weaken after publication and may not survive costs, but they suggest prices are not perfectly random.

Testing for randomness#

Why it matters for traders#

• Benchmark: compare strategies to random entries with the same risk management. See Bootstrap and Permutation Tests.
• Humility: much short term movement is noise; avoid overinterpreting.
• Risk: under a random walk, the spread of outcomes grows with the square root of time, which underpins volatility scaling. See Variance and Standard Deviation.
• Edge hunting: look for the small, persistent departures from randomness that research has documented, and test them carefully.

Frequently asked questions#

What is a random walk in finance?#

A model in which each price change is random and unrelated to past changes, so future prices cannot be predicted from past prices.

What is white noise?#

A sequence of independent random values with a constant mean and variance and no correlation over time.

Do stock prices follow a random walk?#

Approximately, in the short term, but research has found small departures such as momentum, reversal and volatility clustering.

Next, learn to measure links between past and present values in Autocorrelation and Partial Autocorrelation.