Random Walk

A random walk is a mathematical process in which each new value equals the previous value plus a random step. The steps are independent of one another, so the path has no memory: where it goes next does not depend on how it reached its current position. The term was introduced by the statistician Karl Pearson in 1905.

In finance the random walk is the simplest model of how prices move. The random walk hypothesis states that successive price changes are independent, so past prices cannot be used to predict future ones. The idea is closely tied to the efficient market hypothesis and is the reference point against which trading strategies based on price history are judged.

A random walk with drift is written as:

• $P_t$ is the value at time $t$. In finance this is normally the logarithm of the price, so that steps are returns.
• $P_{t-1}$ is the value one period earlier.
• $\mu$ is the drift, the average step per period. With $\mu = 0$ the process is a pure random walk.
• $\varepsilon_t$ is a random shock with mean zero and variance $\sigma^2$, independent of all earlier shocks.

Because the shocks accumulate and never fade, the uncertainty about the future value grows with the horizon. After $T$ periods the variance of the change is $T\sigma^2$, so the standard deviation grows with the square root of time:

For example, if the daily shock has a standard deviation of 1 percent, the standard deviation of the change over 25 trading days is $1 \times \sqrt{25} = 5$ percent, and over 252 trading days it is $1 \times \sqrt{252} \approx 15.9$ percent. This square-root-of-time rule is the standard way of converting daily volatility into an annual figure.

A random walk is the basic example of a non-stationary time series. Its variance has no fixed level, and it does not return to any long-run average. It is said to contain a unit root, and tests such as the Dickey-Fuller test are designed to detect one. Although the level is not stationary, the steps are: differencing a random walk once gives a series that satisfies stationarity.

A random walk is a special case of autoregression in which the coefficient on the previous value is exactly one. When the time step is made vanishingly small, the random walk becomes Brownian motion, the continuous-time process used in option pricing. Random walks can also deceive the eye. Simulated paths show apparent trends, cycles and chart patterns that arise purely by chance, and two unrelated random walks will often appear strongly correlated.

Louis Bachelier modelled prices on the Paris exchange as a random process in 1900. In 1953 the statistician Maurice Kendall examined British share and commodity price series and found little pattern in their changes. Eugene Fama's 1965 article "Random Walks in Stock Market Prices" brought the idea to a wide audience, and Burton Malkiel's 1973 book A Random Walk Down Wall Street made it popular with the general public.

The economic argument is that if prices could be predicted from public information, traders would act on the prediction until the opportunity disappeared. Prices therefore move only on new information, which is by nature unpredictable. On this view technical analysis cannot earn excess returns, and a low-cost fund tracking a market index is hard to beat.

Later research showed that prices do not follow a random walk exactly. In 1988 Andrew Lo and Craig MacKinlay used a variance ratio test, which checks whether variance grows in proportion to the horizon, and rejected the random walk for weekly United States share returns. Other studies documented momentum over periods of several months and short-term reversal over days and weeks. Volatility is also clearly predictable: large moves tend to follow large moves, a feature captured by models such as GARCH.

These departures are small relative to the noise in returns, and trading costs absorb much of the apparent profit. For most assets a random walk remains a close first approximation and a difficult benchmark to beat.

Practitioners use the random walk as a null model. A strategy backtest is compared with what would be expected if prices were a random walk, and Monte Carlo simulation of random paths shows how often a given result would appear by luck. Strategies based on mean reversion depend on finding series that are not random walks, such as the spread between two related shares tested for cointegration. In risk management the square-root-of-time rule scales a one-day risk estimate to longer horizons, an approximation that fails when returns are serially correlated or volatility is changing.