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Key Takeaways
- In a random walk, each increment is independent of the last and drawn from the same distribution, so the past path leaves the odds of the next move unchanged.
- The idea is closely tied to the efficient market hypothesis, where prices already reflect available information and adjust as new information arrives.
- Drift is the mean increment per period and can be positive, zero or negative.
- With independent equal-variance increments, standard deviation grows with √n, so twenty days of 1-point daily shocks give a 4.47-point standard deviation.
- A strategy earns its keep only by beating the random walk at the same horizon, after costs and at matched exposure.
What is a random walk?
A random walk is a series built by adding successive random increments to a starting value. In its strict form, the increments are independent and identically distributed with finite mean and variance, so the past path leaves the distribution of the next increment unchanged. Applied to prices, no pattern in past changes improves the forecast of the next one.
Write P(t) = P(t − 1) + mu + epsilon(t). mu is drift in price points per period; epsilon has mean zero. Drift can be positive, negative or zero.
Formula
P_t = P_(t−1) + μ + e_t, where e_t has mean zero and is independent of past changes- P_t is the price at time t and P_(t−1) is the prior price.
- μ is the mean additive change per period, measured in the same units as P. It can have either sign or be zero.
- Shocks are independent and identically distributed, with mean zero and constant finite variance; that is what lets variances add.
- The independence of e_t is the core claim: no function of past shocks improves the forecast of the next one.
Random walk example: a 20-day index path
Start an index at 100 points. Each daily additive change is 0.03 points plus an independent normal shock with standard deviation 1 point. Calculate the next-day distribution and the 20-day distribution after five consecutive up sessions.
- Expected next level = current level + 0.03 points. The five-day streak leaves the next increment distribution unchanged.
- Probability of a positive next increment = Φ(0.03) = 51.2%.
- Day-20 mean from 100 = 100 + 20 × 0.03 = 100.60 points.
- Day-20 standard deviation = √20 = 4.47 points; the central one-sigma interval is 96.13 to 105.07.
Five up days in a row change nothing about the next day: the expected gain is still 0.03 points and the chance of an up close is still 51.2%. By day 20, the index centers on 100.60 with a 4.47-point standard deviation, so the day-20 level lands between 96.13 and 105.07 with 68.27% probability.
The next-day up probability is Phi(0.03) = 51.2%. Five preceding up sessions leave that next increment distribution unchanged.
The chart draws one 20-step sampled path, the expected level and the one-sigma interval for each day. Read the band one day at a time: on any single day the level sits inside it with 68.27% probability, and the odds that a path stays inside for all 20 days are lower.
Additive vs compounded random walks
For independent equal-variance additive increments, variances add and standard deviation scales with √n, so 20 days of 1-point shocks give 4.47 points. The square-root rule needs equal, finite variance, and a volatility spike breaks it.
A percentage-return model compounds prices. A random walk in log price adds log returns, then exponentiates to obtain price. Its dollar-price interval is different from adding a fixed number of points each day.
Random walk vs martingale
A martingale has conditional expected next value equal to its current value, together with the required integrability. Its increments only need a zero conditional mean, so their size and shape can change from step to step.
A zero conditional mean leaves the up probability open. An increment of +1 with probability 90% and −9 with probability 10% has mean zero and a 90% up probability.
In option pricing, the discounted stock price is a martingale under the risk-neutral measure, which is why an option’s price grows the stock at the risk-free rate, whatever return you expect.
Random walk vs the efficient market hypothesis
Market efficiency describes how much information prices reflect, while a random walk describes how prices step. Lo and MacKinlay’s weekly-index study rejected the random walk and gave no support to a stationary mean-reverting alternative, so the rejection hands a trader a hypothesis to test at the horizon traded.
How traders use the random walk as a benchmark
Set the strategy, exposure, costs and horizon, then measure the results against a random walk with the same drift and volatility. A rule earns a place in the book only by beating it after costs, so test direction, volatility timing and payoff shape one at a time to find where any edge comes from.
Limitations of the random walk model
- Resize the range when volatility changes. The strict model holds the shock at 1 point a day, so a volatility spike makes the 96.13 to 105.07 band too narrow and a calm stretch makes it too wide.
- Test at the horizon you trade. Lo and MacKinlay rejected the random walk for their 1962–1985 weekly US index sample, and their result gives no support to a mean-reverting alternative, so a weekly pattern needs its own test before it becomes a trade.
- Beating the random walk on paper is the start. The rule still has to clear costs at matched exposure, measured against the same benchmark, before it earns capital.
- Use log prices for long horizons. Over long horizons an additive walk in price points puts real probability below zero, while a walk in log price keeps modeled prices positive.
- Price a whole-path bet off the whole path. On any one day the level sits inside the one-sigma band with 68.27% probability, and the chance it stays inside on all 20 days is lower, which matters for barrier options and stop placement.
Bottom line
A random walk is the baseline model of price changes that past moves cannot predict, and the source of the square-root-of-time rule behind option ranges. Use it as the benchmark a strategy has to beat at the same horizon, exposure and cost before it earns capital.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute (Financial Analysts Journal): Random Walks in Stock Market Prices, Eugene F. Fama
- NBER: Stock Market Prices Do Not Follow Random Walks, Lo and MacKinlay
How entries are sourced, calculated and updated: editorial methodology.