# White Noise and Random Walks

> A random walk is a path built from random steps; white noise is pure randomness. Learn how they model prices, the random walk hypothesis and the evidence against it.

Source: https://learn.tradelabsai.com/math/white-noise-and-random-walks/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "White Noise and Random Walks", https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/stationarity/).

## Random walks look like real charts

**Example: Fake charts**
Generate 500 random daily returns with a mean of zero and a standard deviation of 1.5%, and plot the cumulative price path. The chart will usually show apparent trends, support and resistance levels, double tops and breakouts, even though every move was random. Many traders shown random charts mixed with real ones cannot reliably tell them apart. This is a warning against reading meaning into every pattern. See [Confirmation Bias](https://learn.tradelabsai.com/psychology/confirmation-bias/).

## 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:

| Finding | Source |
|---|---|
| Short term autocorrelation in weekly index returns | Lo and MacKinlay (1988), "Stock Market Prices Do Not Follow Random Walks" |
| Momentum over 3 to 12 months | Jegadeesh and Titman (1993). See [Momentum Trading](https://learn.tradelabsai.com/strategies/momentum-trading/) |
| Short term and long term reversal | De Bondt and Thaler (1985). See [Short and Long-Term Reversal](https://learn.tradelabsai.com/research/short-and-long-term-reversal/) |
| Volatility clustering | Engle (1982). See [GARCH](https://learn.tradelabsai.com/math/garch/) |
| Time series momentum in futures | Moskowitz, Ooi and Pedersen (2012). See [Trend Following](https://learn.tradelabsai.com/strategies/trend-following/) |

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

| Test | Checks |
|---|---|
| Autocorrelation and Ljung Box test | Whether returns are correlated with past returns. See [Autocorrelation and Partial Autocorrelation](https://learn.tradelabsai.com/math/autocorrelation/) |
| Variance ratio test | Whether variance grows in proportion to time, as a random walk implies |
| Runs test | Whether sequences of ups and downs are longer or shorter than random |
| Unit root tests (ADF) | Whether a series behaves like a random walk. See [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/) |

## Why it matters for traders

- **Benchmark:** compare strategies to random entries with the same risk management. See [Bootstrap and Permutation Tests](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/autocorrelation/).

## Continue learning

- Next lesson: [Autocorrelation and Partial Autocorrelation](https://learn.tradelabsai.com/math/autocorrelation/)
- Previous lesson: [Time Series Basics](https://learn.tradelabsai.com/math/time-series-basics/)
- Related: [Time Series Basics](https://learn.tradelabsai.com/math/time-series-basics/): Market data is a time series: values ordered in time. Learn the components of a time series, why order matters, returns vs prices and the core tools for analysis.
- Related: [Autocorrelation and Partial Autocorrelation](https://learn.tradelabsai.com/math/autocorrelation/): Autocorrelation measures how a series relates to its own past values. Learn the formula, the ACF, what positive and negative autocorrelation mean and why it matters.
- Related: [Stationarity, Differencing and Unit Roots](https://learn.tradelabsai.com/math/stationarity/): A stationary series has stable statistical properties over time. Learn why prices are non stationary, how to test with ADF and KPSS and how to make data stationary.
- Related: [Momentum Trading](https://learn.tradelabsai.com/strategies/momentum-trading/): Momentum trading buys assets that are rising fastest and sells those falling fastest. Learn the research, intraday and multi month methods and momentum crashes.
- Related: [Mean Reversion](https://learn.tradelabsai.com/strategies/mean-reversion/): Mean reversion trades bet that prices stretched far from their average will come back. Learn the signals, z scores, examples and the risk of fading strong trends.