Are prices a random walk? 120 years of answers, and their limits

“Share prices follow a random walk” is one of the first ideas any finance textbook introduces. Being clear about where the hypothesis came from, what it claims and where it parts company with observation makes it easier to see what the many models that followed — including the quantum random walks these notes go on to discuss — are trying to fix.

The starting point: Bachelier, 1900

Modelling prices as a random walk goes back to Louis Bachelier’s doctoral thesis, “Théorie de la spéculation”, submitted in 1900 (1). Bachelier studied the prices of French government bonds traded on the Paris Bourse and described price changes as the accumulation of small, mutually independent fluctuations — five years before Brownian motion was given its theoretical explanation in physics (2).

In its standard modern form, the assumption is made not about prices themselves but about changes in the logarithm of price (log returns): these are taken to be independent and drawn from the same normal distribution. This is geometric Brownian motion, which also underpins the Black–Scholes theory of option pricing (3).

Two properties of the model matter for what follows.

• Direction cannot be predicted. In the model’s assumptions, today’s return is independent of the returns that came before it.
• Spread grows with √t. The dispersion (standard deviation) of the log price after t days grows in proportion to the square root of time, and the shape of the distribution is normal at every time scale.

What holds: direction is hard to predict

In his 1970 survey, Eugene Fama brought together the idea of an “efficient” market — one in which prices already reflect the available information — and the empirical work testing it. Drawing on tests of serial correlation and of filter rules (rules that trade when prices move by a set amount), its central conclusion was that profiting from predicting the direction of future prices from past price movements is difficult (4).

Later research has broadly supported this. In a 2001 paper, Rama Cont gathered eleven statistical properties observed again and again across many markets and periods, known as “stylised facts” (5). The first is that linear autocorrelations of returns are largely absent, except at very short intraday time scales.

The part of the random walk hypothesis that concerns direction remains a good first approximation today. But it is a first approximation. Fama himself distinguished the general form of an efficient market from the random walk, treating the latter as a stronger assumption. And Lo and MacKinlay statistically rejected the random walk for weekly returns on US stock indices and size-sorted portfolios, the rejection being driven mainly by small stocks (6).

What fails: the distribution of size

Where the hypothesis fails is the distribution of the magnitude of returns.

Heavy tails

In 1963 Benoit Mandelbrot examined cotton price changes and showed that large moves occur far more often than a normal distribution would predict (7). Under a normal distribution, a value five or more standard deviations from the mean occurs only about once in 1.7 million draws — about once in 7,000 years of daily observations, at 250 trading days a year. In real markets such moves are observed far more frequently. On 19 October 1987 the Dow Jones Industrial Average fell 22.6% in a single day (8), a move that, measured against the typical size of daily changes, is all but impossible under a normal distribution.

Heavy tails are one of the central items among Cont’s stylised facts. The tails of the return distribution decay roughly like a power law, much more slowly than the tails of the normal distribution, which fall off like exp(−x²/2), that is, exponentially at a rate proportional to x².

Volatility comes in clusters

The other major discrepancy is volatility clustering. Large moves tend to be followed by large moves, and small moves by small ones. Although returns themselves are almost uncorrelated, the autocorrelation of their absolute values (or squares) is positive and decays only slowly over time (5).

In other words, returns are nearly uncorrelated but not independent. Today’s return says almost nothing, linearly, about the direction of tomorrow’s; but a large move today makes a large move tomorrow more likely. Of the hypothesis’s assumption of “independent and identically distributed”, the data support only the weaker property of no correlation; independence does not hold. The distribution given the recent size of moves (the conditional distribution) changes from day to day.

Rises and falls, and time scales

Cont also lists gain/loss asymmetry: in share prices and share indices one sees large drawdowns but not equally large upward movements. This is an observation about equities: it has been reported for share indices (9) but, according to some studies, not confirmed for individual stocks. A recent re-examination using intraday data on the Dow 30 stocks (about six months in 2018–19) also did not find it for individual stocks (10).

Cont further lists aggregational Gaussianity: the longer the time interval, the closer the distribution comes to normal. In other words, the departure from the normal distribution looks largest at short time scales.

As long as returns are nearly uncorrelated and their variance is finite, the property that dispersion grows with √t is itself preserved as a first approximation. What fails is the shape of the distribution.

Does tail heaviness change over time?

Some research reports that tail heaviness changes over time. Ahn et al. estimated a tail exponent year by year from the daily returns of 137 US companies that were continuously in the S&P 500 between 1992 and 2021, and reported that the exponent is smaller — the tails heavier — in recessions (11). The authors interpret the exponent as a proxy for investors’ herding behaviour and discuss the result as a relation between herding and the business cycle. But there are only 30 annual estimates, four of them in recession years, and reading the exponent as herding is the authors’ interpretation.

If the finding holds, the return distribution cannot simply be fixed once and for all.

Taking stock

Taken together, the random walk hypothesis can be assessed as follows.

Property	Random walk hypothesis	Observation
Autocorrelation of returns (linear)	None	Almost none (holds well)
Tails of the distribution	Normal (thin)	Close to a power law (heavy)
Persistence of size	None	Strong and long-lasting
Symmetry of rises and falls	Symmetric	Large drawdowns reported for share indices (not confirmed for individual stocks)
How the spread grows	√t, always normal	Dispersion roughly √t; shape changes with time scale

Many models have been proposed to close these gaps. The earliest was the stable distribution Mandelbrot himself put forward (7). Later empirical work, however, found tail exponents generally above two, implying finite variance, so stable distributions with infinite variance have been set aside in mainstream assessments (5) (though this is disputed (12)). The observation that distributions approach normal over longer intervals also sits badly with stable distributions, whose shape does not change under aggregation. Since then, GARCH-type models that build in changing volatility explicitly (13), stochastic volatility models such as Heston’s (14), and the various models of econophysics that bring in methods from physics have been proposed.

Some research examines the quantum random walk as an alternative to the classical random walk as a model of distributions (15). This is not a claim that markets are quantum-mechanical systems, nor is it about speeding up computation on quantum computers; it borrows the mathematics of quantum mechanics as a tool for generating probability distributions. In a quantum random walk, probability amplitudes rather than probabilities are added together, so interference occurs, and the way the distribution spreads and the shape it takes differ sharply from a classical random walk. Its distribution, however, stays within a finite range at each time, and as t grows its spread approaches a fixed shape on a finite interval (16); it arises from a mechanism different from the power-law tails discussed here. The next note looks at what a quantum random walk is and how it differs from its classical counterpart.

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This series explains research in quantum finance and econophysics from the published literature. It offers no view on any particular market or asset.