What stock markets do in the weeks after a crash
In 2003 two physicists showed that the wild minutes after the 1987 crash died away the way earthquake aftershocks do. We set the earthquake law against the volatility models traders already use, on 225 shocks in nine stock indices since 1962.
Do stock market crashes have aftershocks that follow the earthquake law?
In 2003 the physicists Fabrizio Lillo and Rosario Mantegna counted the minutes in which the S&P 500 made an extreme move after the crashes of October 1987, October 1997 and August 1998. Those minutes thinned out over the next 60 trading days the way aftershocks thin out after an earthquake. Their rate fell as a power of the time since the crash, the law Fusakichi Omori wrote down for aftershocks in 1894 (Omori, 1894). The two called it “a new stylized statistical law” of markets (Lillo and Mantegna, 2003). Later studies found the same fade after shocks in US stocks minute by minute, and in ten emerging stock markets day by day (Petersen et al., 2010; Selçuk, 2004).
We tested the law on nine stock indices in North America, Europe, Asia and Australia, from 1962 to September 21, 2026. A shock is a day when an index moved at least four times as far as on a typical day of the three months before, which happens on about one trading day in 270. The test holds 225 shocks, and 176 of them were falls. A big move is any later day at least twice that typical size, and an ordinary day brings one 6.1% of the time.
The aftershocks are real, and the earthquake law forecasts them no better than the volatility model finance already uses. A month after a shock, in weeks three to six, an index still made a big move on 17% of days, almost three times the ordinary rate. Of the 762 big moves in those weeks, the volatility model expected 665, a miss small enough to sit inside its error band. The earthquake law expected 527, and a rule that carries the last 20 days forward expected 985 (Figs. 1 and 2). The law fits the years it is fitted on, and carried to another era it missed by up to 85%.
Big moves fade week by week after a shock, and the volatility model follows the fade more closely than the earthquake law.
Share of days with a big move, by trading days since the shock, across 225 shocks in nine stock indices: what happened (red dot, labelled), the volatility model's forecast (black dot) and the earthquake law's (ring), each forecast fitted on the other era.
How we tested it
The test runs on daily closes of the S&P 500 from 1962, the S&P/TSX Composite from 1979, the FTSE 100 from 1984, the Nikkei 225 from 1989, the Hang Seng from 1990, the Dow Jones Industrial Average from 1992, the DAX from 1994, the S&P/ASX 200 from 2001 and the Euro Stoxx 50 from 2007, all to September 21, 2026. A shock is a daily move of at least four times the standard deviation of the index’s previous 60 daily moves, and a big move is a later day at least twice that standard deviation. Shocks less than 31 trading days apart on the same index count once, as the first of them. The test scores trading days 3 to 30 after each shock: days 3 to 10 are the first two weeks, and days 11 to 30 are weeks three to six.
Three forecasts give each of those days a chance of a big move. The earthquake law sets it at a background rate plus a term that falls as a power of the days since the shock and rises with the shock’s size, the form seismologists call ETAS (Ogata, 1988). The volatility model is HAR, which forecasts coming moves from the average of the last day’s, week’s and month’s squared moves (Corsi, 2009). It was built to carry the long memory of market volatility, the way a burst of it lingers for weeks. The third, the 20-day rule, expects the coming weeks to look like the last 20 trading days, the shock included.
Each forecast is fitted on 1962 to 1994 and scored on 1995 to 2026, then fitted on the later era and scored on the earlier, so no forecast is scored on the years it was fitted to. Each count of big moves set against a forecast carries a 95% error band, from a bootstrap that resamples whole weeks of shocks.
Big moves keep coming for six weeks after a shock, and fade by the week
On the third trading day after a shock, the index made another big move more than a third of the time, almost six times as often as on an ordinary day. The rate fell to 22% in the second week and to 14% in the sixth, still more than twice the ordinary 6.1% (Fig. 1). That is the pattern the physicists described, here across 225 shocks in nine markets over 64 years.
Clustering of big moves is old news in finance: Mandelbrot (1963) observed that large price changes tend to follow large changes, up or down. The earthquake law adds two sharper claims. The first is a shape for the fade, a power of the time since the shock with an exponent near 1, under which the rate roughly halves each time the days since the shock double. The second is that bigger shocks bring more aftershocks. Those two claims decide what a trader should expect in the fourth week after a crash, and both can be tested against a forecast that makes neither.
The volatility model finance already uses forecasts the aftershocks best
Across all 225 shocks, the volatility model missed by the least: big moves ran 18% above its forecast in the first two weeks and 15% above it in weeks three to six (Fig. 2). Its error band for weeks three to six runs from 9% below the forecast to 44% above it, wide enough to include a forecast that was exactly right. Against the earthquake law, big moves ran 43% to 45% above the forecast in both windows. The law’s error band for weeks three to six, 17% to 79% above, rules out a correct forecast.
The volatility model missed the big moves after a shock by the least, and the earthquake law by the most.
How far the big moves that came ran above (right) or below (left) each forecast, in percent, in the first two weeks after a shock (red) and in weeks three to six (grey), across 225 shocks, with every forecast fitted on the other era.
A stricter test gives the same verdict. It follows the test Vuong (1989) built to choose between two models of different forms: it scores each day by how likely each forecast said the outcome was, and charges the earthquake law for its four extra parameters. On that score the law did not beat the volatility model in either era. To pass a 95% test the law needed an edge of 1.64 standard errors. It managed 0.60 since 1995, and before 1995 the volatility model came out ahead. Since 1995 even the 20-day rule beat the law, by 1.80 standard errors. The volatility model knows nothing about earthquakes and still forecasts the aftershocks as well as the law built for them.
The law fitted on one era misses the next
The exponent that sets how fast the aftershocks fade depends on the years the law is fitted on. Fitted since 1995 it is 1.01, inside the 0.9 to 1.5 range usually measured for earthquakes (Utsu et al., 1995), and close to the 0.70 to 0.99 that Lillo and Mantegna found after 1987, 1997 and 1998 (Lillo and Mantegna, 2003). Fitted on 1962 to 1994 it is 1.88, a law under which the aftershocks die out far faster.
So each version of the law misses the other era. The version fitted before 1995 expected too few big moves after it, and since 1995 they ran 85% above its forecast in weeks three to six. The version fitted since 1995 expected too many before it, where big moves ran 41% below its forecast (Fig. 3).
Since 1995, big moves in weeks three to six ran 85% above the earthquake law's forecast, twice the volatility model's miss.
How far the big moves in weeks three to six ran above (right) or below (left) the earthquake law's forecast (red) and the volatility model's (grey), in percent, for the 46 shocks before 1995 and the 179 since, each forecast fitted on the other era.
The volatility model swung with the eras too, because markets since 1995 have held on to their volatility longer. Its forecast for weeks three to six was 40% too low since 1995 and 48% too high before. The sum of its three slopes, which measures how long a burst of volatility lasts, rose from 0.48 fitted before 1995 to 0.76 fitted since. Every forecast here carries one era’s habits into another, and the earthquake law, offered as universal, missed the present era by the most.
A bigger crash does not bring more aftershocks a month later
Earthquakes obey a productivity law, under which a bigger mainshock sets off more aftershocks, and Petersen et al. (2010) found the same relation in US stocks. The 11 biggest shocks in the test, days when an index moved at least eight times as far as on a typical day, were followed in weeks three to six by big moves on 17.3% of days. The 137 smallest, at four to five times, were followed by big moves on 17.3% of days as well (Fig. 4). Across all four size groups the rate stayed near one day in six, so the size of the crash told nothing about how often the month that followed brought big moves.
Weeks three to six bring big moves on about one day in six whatever the size of the shock, while both forecasts expect more after the biggest ones.
Share of days in weeks three to six with a big move (red dot, labelled), against the volatility model's forecast (black dot) and the earthquake law's (ring), by the shock's size as a multiple of the typical daily move over the 60 days before it.
The volatility model builds the shock’s own size into its forecast, so after the 11 biggest shocks it expected big moves on 31.0% of days, almost twice the 17.3% that came. Those 11 include October 19, 1987 in Toronto and London, the S&P 500’s 6.1% fall on October 13, 1989 and the Hang Seng’s 13.2% fall on April 7, 2025. In weeks three to six it expected 68 big moves after them and 38 came, little more than half. The error band on that ratio, 0.24 to 0.84, rules out a correct forecast.
Pricing the month after a crash off the crash itself overstates the danger
The simplest way to size the risk after a crash is the 20-day rule, which expects the coming weeks to look like the last 20 trading days, crash included, and it gets the timing wrong. In the first two weeks big moves ran 18% above its forecast, and in weeks three to six they ran 23% below it (Fig. 2). Its error band for those later weeks, 6% to 36% below, rules out a correct forecast. The danger after a shock comes early, and a forecast built on the shock’s own volatility spreads it evenly over the weeks.
The April 2025 sell-off shows the size of that error. The United States announced tariffs on imports from all its trading partners on April 2, 2025, and between April 3 and April 7 eight of the nine indices had a shock, the Hang Seng’s 13.2% fall the largest (Executive Office of the President, 2025a). On April 9 the higher rates were suspended for 90 days for every country except China (Executive Office of the President, 2025b). In weeks three to six after the eight shocks, the indices made 8 big moves. The volatility model expected 15 and the earthquake law 16, about twice as many. The 20-day rule expected 40, five times what came.
The case for the earthquake law is in the first hours
The strongest case for the law comes from finer data than daily closes. Lillo and Mantegna fitted it to minute-by-minute returns, and Petersen et al. (2010) found the Omori, productivity and Båth laws of earthquakes in 219 shocks across the 531 most-traded US stocks in 2001 and 2002, also by the minute. A daily test sees nothing of the first hours after a crash, and the law may describe those hours well.
For anyone who holds a position for days or weeks, the daily test is the one that counts, and there the law adds nothing a volatility model does not already carry. Weber et al. (2007) found the same Omori-type fade after bursts of volatility in artificial series built only to have long memory, and concluded that the memory in volatility is related to the Omori processes. Long memory is the property HAR approximates with its daily, weekly and monthly averages (Corsi, 2009). Lillo and Mantegna’s objection was to GARCH(1,1), whose memory fades by a fixed fraction each day and so cannot produce a power-law decay. A model with long memory produces the aftershocks without being told about earthquakes.
What to watch
The next shock is the next test. At the September 21, 2026 close, a one-day fall of 2.8% would count as a shock in the S&P 500, and a fall of 2.3% in the FTSE 100. The DAX would need 3.3% and the Hang Seng 4.0%, and the Nikkei 225, after a volatile summer, 7.3%. The last shock in the test came on March 3, 2026, when the FTSE 100, the DAX and the Euro Stoxx 50 fell 2.8% to 3.6% on the same day.
Two results would bring the earthquake law back. Big moves in weeks three to six that run well above the volatility model’s forecast and fade as a power of time with an exponent near 1 would support the version fitted since 1995. A biggest shock followed by more big moves than the smaller ones would revive the productivity law, which the last 64 years do not support.
References
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- Executive Office of the President. (2025a). Executive Order 14257: Regulating imports with a reciprocal tariff to rectify trade practices that contribute to large and persistent annual United States goods trade deficits. Federal Register, 90, 15041, April 7.
- Executive Office of the President. (2025b). Executive Order 14266: Modifying reciprocal tariff rates to reflect trading partner retaliation and alignment. Federal Register, 90, 15625, April 15.
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