Random walk theory: the math, the history and the tests that challenged it

The red line in the chart below starts at 100 and ends ten years later at 176. It drifts down for two years, climbs through years three to five, gives back a chunk in year seven and finishes on a strong run. A chartist could draw trend lines on it, mark support levels and name a few patterns along the way. The line was produced by drawing one random number per trading day, with no trend built in, and the seven grey lines came from the same generator.

Eight of the 10,000 simulated price paths described further down. Each daily move is a random draw with a mean of zero and a standard deviation of 1%. Log scale, 2,520 trading days.

That is the claim at the center of random walk theory: real stock price changes behave like those random draws, so the history of prices holds no usable information about where they go next. The idea dates from 1900, was rejected in its strictest form in 1988, and still underlies index investing and the Black-Scholes option pricing formula.

What random walk theory says

Random walk theory holds that successive changes in a stock's price are independent of each other and cannot be forecast from past prices. A rise is as likely to be followed by a fall as by another rise, and apart from a small average drift, the best estimate of tomorrow's price is today's price.

The theory is closely tied to the efficient market hypothesis, and the two names are often used as if they meant the same thing. The efficient market hypothesis is about information, the idea that prices reflect what is known. Random walk theory is about the statistical behavior of price changes. The two meet in the weak form of market efficiency, under which it is impossible to systematically beat the market using historical asset prices. If prices follow a random walk, the weak form holds, and technical analysis built on price patterns alone has nothing to work with.

Random walk

A series in which each new value is the previous value plus a step, and the steps are independent draws from the same distribution. A counter that moves up one on heads and down one on tails is the simplest example.

Random walk with drift

The same series with a constant added at every step. For stocks, the drift stands for the average expected return, which is how a random walk and a market that rises over decades fit together.

Martingale

A series whose expected next value, given everything known today, equals its current value. It is a looser condition than a random walk: the size of the moves can change over time, as long as their direction stays unpredictable. Paul Samuelson showed in 1965 that prices in a well-functioning market should follow a generalized random walk of this kind.

The history of random walk theory, from Bachelier to Malkiel

Malkiel's best-known line is an image: a blindfolded monkey throwing darts at the financial pages could pick a portfolio as good as one chosen by experts. The Wall Street Journal later ran the experiment for 14 years, with staff throwing darts at the stock listings against picks from professionals. After 142 contests the professionals' picks had averaged 10.2% against 3.5% for the darts. Malkiel put part of the gap down to publicity, since a stock named by a professional in a national newspaper tends to jump when the column appears.

The mathematics of a random walk

Written for the logarithm of a price, the whole model fits on one line:

ln P(t) = ln P(t-1) + μ + ε(t)

Here μ is the drift, and ε(t) is a random shock with a mean of zero that is independent of every earlier shock. Using log prices keeps the price above zero and lets returns over consecutive periods add up. Three properties follow directly.

1. No predictable change. Beyond the drift, the expected change in the log price is zero, whatever happened before.
2. Uncertainty grows with the square root of time. After n periods the variance of the total change is n times the one-period variance, so the standard deviation is σ multiplied by the square root of n. A stock with 1% daily volatility has a one-year spread of about 1% × √252, or 15.9%. Bachelier had the square-root rule in 1900.
3. Zero autocorrelation. The correlation between one period's return and any earlier period's return is zero, which gives the theory its first and simplest test.

What 10,000 simulated random walks look like

To see what pure chance produces, we generated 10,000 price paths of 2,520 trading days each, about ten years, using NumPy's default random generator with seed 2026. Every daily return is drawn from a normal distribution with a mean of zero and a standard deviation of 1%, close to the Dow Jones average's long-run daily standard deviation of 1.07%. There is no drift, no memory and no news.

The spread tracks the square-root rule closely. The other results:

• Long one-sided stretches. 41.7% of the paths spent more than 90% of their trading days on the same side of their starting price. Only 13.3% spent between 40% and 60% of their days above it. Probability theory predicts 41.0% and 12.8% through a result called the arc sine law.
• Large gains and losses from nothing. The median path finished at 100.2, but 8.6% of paths more than doubled and 8.5% lost more than half their value.
• Deep drawdowns. The median path suffered a peak-to-trough fall of 43.3% at some point in the ten years, and 59.3% of paths had a fall of more than 40%.
• Streaks. Across the first 2,000 paths, the longest run of consecutive up days had a median of 10 days and a maximum of 20.
• A trend rule with nothing to find. A rule that holds the stock when its price is above its 200-day moving average and sits in cash otherwise earned an average cumulative log return of 0.23%, against 0.10% for buying and holding, before any trading costs. It finished positive on 47.0% of paths and switched position a median of 89 times.

How economists test for a random walk

The first tests, from the 1950s and 1960s, measured autocorrelation: whether one day's or one week's return helps predict the next. They found very little, and the early studies could not reject the random walk.

The variance ratio test uses the square-root rule. If returns are independent, the variance of two-week returns should be exactly twice the variance of one-week returns, and the variance of q-week returns should be q times the one-week figure. The variance ratio divides the first by the second:

• A ratio of 1 fits a random walk.
• A ratio above 1 means returns tend to continue in the same direction, positive autocorrelation.
• A ratio below 1 means moves tend to reverse, negative autocorrelation, or mean reversion.

For two-period returns the ratio is approximately 1 plus the first-order autocorrelation. In our simulation, the two-week ratio computed on each path's 504 weekly returns averaged 0.999, and 95% of paths fell between 0.912 and 1.086. None of the 10,000 reached 1.30.

Lo and MacKinlay's 1988 rejection

Andrew Lo and Craig MacKinlay ran the test on 1,216 weekly returns from September 6, 1962 to December 26, 1985, built from the CRSP daily file of NYSE and AMEX stocks, with each week running from one Wednesday close to the next. Their paper's title gave the verdict: Stock market prices do not follow random walks.

• The equal-weighted index had a two-week variance ratio of 1.30, a first-order autocorrelation of about 30% in weekly returns.
• The portfolio of the smallest fifth of stocks had a weekly autocorrelation of 42% over the full period, reaching 51% between May 1974 and December 1979.
• The largest fifth showed much weaker autocorrelation, still statistically significant.
• Infrequent trading of small stocks and changing volatility could not fully account for the rejection.
• At weekly horizons, prices kept moving in the same direction for a while, the opposite of mean reversion.

Short-term predictability in US stocks is statistically significant but small, stronger in small and thinly traded stocks, and so low that hardly any profit remains once transaction costs are counted. Lo and MacKinlay collected their work in a 1999 book whose title answered Malkiel's: A Non-Random Walk Down Wall Street.

Momentum and mean reversion at longer horizons

At horizons of months and years, two patterns point in opposite directions.

Momentum, over 3 to 12 months. Narasimhan Jegadeesh and Sheridan Titman documented momentum in 1993: buying stocks with the best returns over the previous 3 to 12 months and selling those with the worst generated significant positive returns over the following 3 to 12 months. Systematic risk did not explain the profits, and part of them disappeared over the next two years. Fama, the economist most associated with efficient markets, has called momentum the primary embarrassment to market efficiency. Our guide to factor investing covers how momentum is traded through funds today.

Mean reversion, over 3 to 5 years. Among NYSE stocks from 1926 to 1982, portfolios of the 35 worst performers over the previous three years went on to beat the market by 19.6% over the next 36 months, while the 35 best performers trailed it by about 5.0%, a gap of 24.6%. Much of the losers' excess return arrived in January, and the losers carried lower betas than the winners. Mean reversion accounts for about 40% of the variance of 3 to 5-year returns for small-firm portfolios and around 25% for large firms. Variance ratios on US returns from 1871 to 1986 and on 17 other markets show positive serial correlation over short horizons and negative serial correlation over long ones.

Over a one-year horizon, the dividend-to-price ratio explains about 15% of the variation in excess stock returns. Over five years it explains as much as 60%. The 2013 Nobel prize in economics, shared by Fama, Lars Peter Hansen and Robert Shiller, was announced with the same contrast: prices are close to unpredictable over days and weeks, and their broad course is partly foreseeable over three to five years.

Malkiel's reply, in his 2003 survey The efficient market hypothesis and its critics, is that statistical significance and economic significance differ. The dependencies behind momentum are small, trading costs absorb them, momentum strategies did well in some years of the late 1990s and lost heavily in 2000, and patterns such as the January effect faded soon after they were published.

From Brownian motion to Black-Scholes

The random walk has a continuous-time version, Brownian motion, in which the steps become infinitely small and infinitely frequent. Bachelier used it for prices in 1900. It has one flaw for stocks: a Brownian price can fall below zero.

Osborne's 1959 paper applied Brownian motion to the logarithm of the price. The result is geometric Brownian motion, in which percentage changes are random, the price stays positive, and prices at any future date follow a lognormal distribution. Samuelson proposed it as a model for stock prices in 1965, and it became the assumption behind the Black-Scholes model.

Fischer Black and Myron Scholes published their option pricing formula in 1973 in the Journal of Political Economy, deriving it from the principle that options and their underlying stocks should leave no risk-free profit when combined. Scholes and Robert Merton, who generalized the method, received the 1997 Nobel prize in economics. Black had died in 1995. The same announcement credited Bachelier's thesis as one of the earliest attempts to value derivatives, with a formula flawed in several ways. The volatility term in Black-Scholes scales with the square root of the time to expiry, the random walk property from Bachelier's thesis carried into continuous time. Our guide to how options work covers the contracts themselves.

Fat tails and volatility clustering

The simulation above draws every return from a normal distribution, and its kurtosis, a measure of how often extreme values appear, came out at 3.00, the textbook value for a bell curve.

Over 29,190 trading days of Dow Jones history, a bell curve implies about 79 days with moves beyond three standard deviations. There were 461. Benoit Mandelbrot documented the same excess of large moves in cotton prices in 1963, and it became one of the starting points of econophysics.

The second departure is volatility clustering: turbulent periods with large moves are followed by more turbulence, and calm periods by calm. Robert Engle's ARCH models of time-varying volatility, which describe this pattern, earned him a share of the 2003 Nobel prize.

Neither departure makes the direction of prices easier to forecast. A series can have unpredictable returns and predictable volatility at the same time, which is the martingale case defined earlier. Fat tails change the measurement of risk: a model built on Brownian motion understates how often crashes occur. Our article on fat tails counts them in S&P 500 data since 1928, and our beginner's guide to econophysics explains fat tails and power laws with worked examples.

What random walk theory means for investors

The practical conclusion Malkiel drew was to stop picking stocks and buy a broad-based index fund that holds the whole market at low cost. In 2025, 79% of active US large-cap funds trailed the S&P 500, and in the first half of 2026 the figure was 67%. The full evidence, including why the minority that outperforms rarely keeps doing so, is in our comparison of active and passive investing.

Three other consequences follow from the mathematics:

• Time widens the range of outcomes. Under the square-root rule, ten years of 1% daily volatility produces a spread about 3.2 times the one-year spread, while the drift grows in proportion to time. Over long horizons the drift accounts for more of the result, and the range around it stays wide.
• Deep falls need no cause in the model. In the simulation, most paths lost more than 40% from a peak at some point in ten years with no news at all. Real returns have fatter tails than the simulation, so it understates how often extreme days occur.
• Trading on price patterns costs money. Weekly autocorrelation, momentum and long-run reversal are all measurable in the data, and trading costs absorb much of them. In the simulation, the 200-day trend rule switched position a median of 89 times for an average gain of 0.13 percentage points over buying and holding, before costs.

This article is general information, not investment advice.

Questions about random walk theory

Is random walk theory true?

In its strict form, no. Weekly returns on US stock indexes from 1962 to 1985 showed positive serial correlation of about 30%, strong enough to reject a random walk, and stocks also show momentum over 3 to 12 months and partial reversals over 3 to 5 years. The patterns are small at short horizons and strongest in small, thinly traded stocks, and trading costs absorb much of them. Day-to-day price changes remain close to unpredictable.

What is the difference between random walk theory and the efficient market hypothesis?

The efficient market hypothesis is a statement about information: prices reflect what is known. Random walk theory is a statement about the statistics of price changes: they are independent and cannot be forecast from past prices. A random walk implies weak-form efficiency, but an efficient market does not have to produce a strict random walk, because expected returns and volatility can change over time.

Who came up with random walk theory?

Louis Bachelier modeled prices on the Paris exchange as random in his 1900 thesis. Maurice Kendall found random-looking price changes in British data in 1953, Paul Samuelson gave the economic argument in 1965, Eugene Fama tested it on US stocks and named the efficient markets hypothesis in 1965, and Burton Malkiel popularized it in A Random Walk Down Wall Street in 1973.

Does random walk theory mean you cannot make money in stocks?

No. A random walk can include drift, a steady average gain, and the stock market has risen over long periods while its daily moves stayed close to unpredictable. The theory says it is hard to earn more than the market from past price patterns. Owning the market itself fits the theory, which is why its best-known advocate recommends broad, low-cost index funds.