Mean reversion

The idea

Measure what normal looks like for a price, a spread, or a rate. When it stretches far from normal, sell what has run up and buy what has fallen. The trade is paid when the stretch closes.

Who pays for the snap-back

Reversal is a fee for immediacy. In 1990 Bruce Lehmann showed that US stocks that won or lost in one week tended to reverse the next, and Narasimhan Jegadeesh found strong negative serial correlation in monthly stock returns. Grossman and Miller had described the mechanism in 1988. A trader who must deal now moves the price, and the market maker on the other side holds the inventory until a natural counterparty arrives. The snap-back is the payment for carrying it.

Only part of a move comes back. Price impact from informed trading is permanent, so only the part caused by someone's need to trade reverts. Campbell, Grossman and Wang found that index moves on heavy volume are more likely to reverse than moves on light volume, and explain it by market makers absorbing trades made for reasons other than information. Wesley Chan found that large monthly moves with no public news tended to reverse while stocks with bad news kept drifting down, and both patterns showed up mainly in smaller, less liquid stocks. The first question about any stretch is why the price moved.

The fee rises when providers retreat. Stefan Nagel's Evaporating Liquidity treats short-term reversal returns as the return to providing liquidity. He finds they track the VIX closely, ran high in the 1998 LTCM crisis and the 2000 to 2001 Nasdaq fall, declined steadily until 2007 and then rose above their 1998 levels in the 2007 to 2009 crisis. Hameed, Kang and Viswanathan find that liquidity dries up after large market falls, most of all when funding is tight, and that the return to supplying liquidity after those falls was economically significant. In Brunnermeier and Pedersen's model, margins can rise when markets turn illiquid, so the capital meant to absorb the flow shrinks as the flow peaks.

The edge and the danger can arrive together. Khandani and Lo show that in the week of 6 August 2007 a simple contrarian strategy took heavy losses over a few days, then won back a large part of them on 10 August. They trace the losses to the forced unwinding of similar quantitative portfolios and, in later work, to a temporary withdrawal of market-making capital from 8 August. Nagel calls that week exceptional within his 1998 to 2010 sample. Only books with room to hold through the unwind were still there for the rebound, and a book sized for calm markets can become a forced seller in the days before the snap-back it was built to catch.

Why regime matters

In a range, stretches tend to come back. In a trend, or after a break, normal itself moves.

The trade then keeps losing, and it looks cheaper every day it loses.

Measuring normal, and noticing when it moves

Fitted half-lives run short. The textbook model for a stretched spread is the Ornstein-Uhlenbeck process, in which the expected gap shrinks by the same fraction in each unit of time and the half-life is ln 2 divided by the speed of reversion. Econometric work summarised by Bao, Ullah and Wang shows that maximum likelihood overstates that speed in finite samples, so fitted half-lives come out shorter than the truth. In simulations they cite, the other parameters came out almost unbiased while the speed could be badly biased. Sampling more often does not shrink this bias, which depends on the span of history the data cover and is worst when reversion is slow.

Liquidity stretches are brief. Hendershott and Menkveld estimate that price pressure from NYSE intermediaries' inventories has a half-life of 0.92 days. A stretch that lasts weeks in a liquid market is more likely to carry information than inventory. When a gap has outlived several fitted half-lives, the fit is the first suspect.

Breaks hide in the statistics. Pierre Perron showed in 1989 that a series that moves around a fixed trend, apart from one break, can look like a random walk to standard tests. His breaks were the 1929 crash, a shift in level, and the 1973 oil shock, a shift in the growth rate. Traders meet the same problem from the other side. A mean estimated across a break sits between two regimes, and every price after the break looks like a stretch from a level that no longer exists.

Long-horizon reversion was a period piece. In 1988 Poterba and Summers reported evidence that stock prices mean-revert over horizons of several years. Kim, Nelson and Startz split the history and found the effect in 1926 to 1946 but not after 1946, when returns showed persistence, and the evidence before 1926 was mixed. Forecasts of post-war three-year returns built on the reversion pattern were negatively correlated with what happened, in every version they tried. A pattern that rests on the Depression and war years may belong to them.

Volatility reverts, at an uncertain speed

Options already price the pull. Engle and Patton list mean reversion among the core stylised facts of volatility, and note that long-dated implied volatilities move less than short-dated ones and sit closer to the long-run average. A trade in options or volatility futures that bets on high volatility falling back needs it to fall faster than those prices already assume.

The speed depends on the sampling. Engle and Patton's GARCH fit to daily Dow Jones data from 1988 to 2000 implied a volatility half-life of about 73 days. The same model fitted to the same index sampled every two, three or four days, or weekly, gave half-lives between 168 and 508 days. Every fit still implied a pull back to the mean, while the estimated speed varied up to sevenfold. A book sized on the fastest estimate can run out of room before the pull arrives.

The path can end the trade. On 5 February 2018 the VIX rose 20 points, its largest one-day rise since the 1987 crash, according to the BIS. The BIS found the rise far exceeded the change in expected volatility and judged that the trading strategies of issuers of leveraged and inverse volatility products appear to have been a key factor. XIV, one of the inverse volatility products, lost most of its value and was terminated. Reversion in the level of volatility says nothing about how far it travels first.

The level can move too. Kaiko data show bitcoin's realised volatility above 100% at points in 2021 and 2022, and its 60-day measure below 50% from the start of 2023 into mid-2024. A model anchored on the earlier years could have read the calmer market as a stretch due to snap back. Engle and Patton note that practitioners who agree volatility reverts may still differ on the normal level and on whether it holds through institutional change. Mean reversion in volatility needs the same regime test as mean reversion in price.

What is different in crypto

Liquidity sets the sign. Using daily prices for more than 3,600 coins, Zaremba and co-authors find that the previous day's losers outperform the previous day's winners, and attribute the reversal to the illiquidity of most coins. The handful of largest and most traded coins show daily momentum instead. Liu, Tsyvinski and Wu find that from 2014 to 2018, coins that rose over the past one to four weeks kept outperforming, most strongly among the larger coins. That is the opposite of the weekly reversal Lehmann found in US stocks, and a reversal rule calibrated on the long tail of small tokens can point the wrong way on the largest coins.

Normal has a clock. Hansen, Kim and Kimbrough find that bitcoin and ether volatility and volume are lowest at weekends, especially on Saturdays. Both spike at 00:00 and 16:00 UTC, with a smaller spike at 08:00, in line with futures funding times, and volatility bursts in the first minute of each hour. Kaiko reports that the weekend share of bitcoin volume fell from a peak of 28% in 2019 to 16% in the first half of 2024, and that spot ETFs sped up the decline. A z-score that ignores the clock flags ordinary funding-hour moves as stretches and treats a Saturday night move as if a weekday order book stood behind it.

Stress stretches are local. On 10 October 2025 more than $19 billion of crypto leverage was liquidated in roughly a day, according to FTI Consulting. Bitcoin top-of-book depth fell by more than 90% on key venues, and the USDe stablecoin traded in the mid-$0.60s on Binance while holding much closer to $1 on other venues. Where margin systems marked collateral to the venue's own price, accounts were liquidated at a price the wider market did not share. Auto-deleveraging also closed some profitable short hedges, leaving hedged books suddenly exposed.

This is Nagel's evaporating liquidity in a market that never closes and liquidates by algorithm. In that episode the widest stretches appeared on single venues, when free collateral was scarcest. A crypto reversal book has to survive the stretch on margin, venue and hedge before its signal matters.

Where it breaks

Trends. Selling what has run up and buying what has fallen is a bet against the trend.

Breaks. A level that was normal last year may not be normal now.

Adding. Buying more because it is cheaper is how these books get hurt. Size is set before the trade, not during it.

Why books keep adding

Doubling looks like skill until it fails. Brown and Steenbeek studied Nick Leeson's trading at Barings and say the evidence suggests he followed a doubling strategy, enlarging positions as prices fell. Such a strategy shows high returns and low volatility until the loss arrives, so the record reads as skill. In the final two months, which included the Kobe earthquake of 17 January 1995, he expanded his long Nikkei futures position to 49% of open interest in the March 1995 contract, and on 23 February Barings could not meet its SIMEX margin calls. Total losses were about $1.4 billion, and the authors note that stopping him at the end of January would have left about a quarter of that, which Barings could probably have absorbed.

Smooth records attract leverage. Khandani and Lo report that leverage of four to ten times capital was common among quantitative equity market-neutral managers in 2007, with the highest ratios in high-frequency mean-reversion books because those showed the lowest volatility. They also find that profits from the simple contrarian strategy had declined substantially over the previous decade, which they link to growing competition and changes in market structure. They argue that managers added leverage to make up for the thinner profits, and more leverage turns a few days of widening into forced selling.

The trade improves as the funding worsens. Royal Dutch and Shell shares were claims on the same cash flows in a fixed 60/40 split, yet Lamont and Thaler record the price ratio ranging from 30% below parity in 1981 to more than 15% above it in 1996. Long-Term Capital Management held a reported $2.3 billion pairs trade on the gap, and when its spreads widened in 1998 it tried to raise new money and found no takers, although its convergence trades had presumably become more attractive. Shleifer and Vishny had modelled this a year earlier. Arbitrage capital is concentrated and run with other people's money, and those backers pull money after losses, so it shrinks just as the mispricing peaks.

Some gaps have a cause. Lamont and Thaler point to one prop under the Royal Dutch premium, its place in the S&P 500, which forced index funds to own the dearer twin. The premium had been near zero for much of the previous year, but it stood at 6% the day before S&P announced on 10 July 2002 that it would drop all foreign stocks, and it fell to 1% on the announcement. Most of that gap went when the prop was removed, on a date no half-life model could have given.

Adding to losers is a documented habit. Odean's study of 10,000 accounts at a large discount brokerage found that investors held losers too long and sold winners too soon, and that later returns did not justify it. The same investors bought more of a stock they already held more often when it stood at a loss than at a gain, which Odean links to risk-seeking after losses or a belief that prices revert. Setting size before the trade is the control for a documented bias. For an allocator, one test of a reversal book is whether its position can grow past the limit set before the trade as mark-to-market losses build.