We find 18,520 crashes and spikes with durations less than 1500 ms in our dataset, with examples of each given in Fig. 1A (crash) and 1B (spike). We define a crash (or spike) as an occurrence of the stock price ticking down (or up) at least ten times before ticking up (or down) and the price change exceeding 0.8% of the initial price, i.e. a fractional change of 0.008. We have checked that our main conclusions are robust to variations of these definitions. In order to have a standardized measure of the size of a UEE across stocks, we take the UEE size to be the fractional change between the price at the start of the UEE and the price at the last tick in the sequence of price jumps in a given direction. Since both crashes and spikes are typically more than 30 standard deviations larger than the average price movement either side of an event (see Figs. 1A and 1B), they are unlikely to have arisen by chance since, in that case, their expected number would be essentially zero whereas we observe 18,520.

Figure 1 Ultrafast extreme events (UEEs). (A) Crash. Stock symbol is ABK. Date is 11/04/2009. Number of sequential down ticks is 20. Price change is −0.22. Duration is 25 ms (i.e. 0.025 seconds). The UEE duration is the time difference between the first and last tick in the sequence of jumps in a given direction. Percentage price change downwards is 14% (i.e. crash size is 0.14 expressed as a fraction). (B) Spike. Stock symbol is SMCI. Date is 10/01/2010. Number of sequential up ticks is 31. Price change is + 2.75. Duration is 25 ms (i.e. 0.025 seconds). Percentage price change upwards is 26% (i.e. spike size is 0.26 expressed as a fraction). Dots in price chart are sized according to volume of trade. (C) Cumulative number of crashes (red) and spikes (blue) compared to overall stock market index (Standard & Poor's 500) in black, showing daily close data from 3 Jan 2006 until 3 Feb 2011. Green horizontal lines show periods of escalation of UEEs. Non-financials are dashed green horizontal lines, financials are solid green. 20 most susceptible stock (i.e. most UEEs) are shown in ranked order from bottom to top, with Morgan Stanley (MS) having the most UEEs. Full size image

Figure 2 shows that as the UEE duration falls below human response times26,27, the number of both crash and spike UEEs increases very rapidly. The fact that the occurrence of spikes and crashes is similar (i.e. blue and red curves almost identical in Fig. 1C and in Fig. 2) suggests UEEs are unlikely to originate from any regulatory rule that is designed to control market movements in one direction, e.g. the uptick regulatory rule for crashes16,17. Their rapid subsecond speed and recovery shown in Figs. 1A and 1B suggests they are also unlikely to be driven by exogenous news arrival. We have also checked that using ‘volume time’ instead of clock time, does not simplify or unify their dynamics. The extensive charts at www.nanex.net, of which Figs. 1A and 1B are examples, show that the total volume traded within each UEE does not differ significantly from trading volumes during typical few-second market intervals, nor do the UEEs originate from one large but possibly mistaken trade.

Figure 2 Number of UEEs as a function of UEE duration. The UEE duration is the time difference between the first and last tick in the sequence of jumps in a given direction. UEE crashes are shown as red curve, UEE spikes as blue curve. Since the clock time between ticks varies, two UEEs having the same number of ticks do not generally have the same durations. Full size image

The horizontal green lines in Fig. 1C show that the UEEs started appearing at different times in the past for individual stock, but then escalated in the build-up to the 2008 global financial collapse (black curve). Moreover, these escalation periods tend to culminate around the 15 September bankruptcy filing of Lehman Brothers. Indeed, the ten stock with the most UEEs (solid green horizontal lines) are all major banks with Morgan Stanley (MS) first, followed by Goldman Sachs (GS). Figure 2 in the SI shows explicitly the escalation of UEEs in the case of Bank of America (BAC) stock. For each stock shown in Fig. 1C, the start and end times of the escalation period (i.e. horizontal green line) are determined by examining the local trend in the arrival rate of the UEEs. In determining these start and end times, we checked various statistical methods such as LOWESS and found them all to give very similar escalation periods to those shown in Fig. 1C. We also find that the occurrence of UEEs is not simply related to the daily volatility, price or volume (see SI Fig. 2 for the explicit case of BAC). Figure 1C therefore suggests that there may indeed be a degree of causality between propagating cascades of UEEs and subsequent global instability, despite the huge difference in their respective timescales. Although access to confidential trade and exchange information is needed to fully test this hypothesis, at the very least Fig. 1C demonstrates a coupling between extreme market behaviours below the human response time and slower global instabilities2,5 above it and shows how machine and human worlds can become entwined across timescales from milliseconds to months. We have also found that UEEs build up around smaller global instabilities such as the 5/6/10 Flash Crash: although fast on the daily scale, Flash Crashes are fundamentally different to UEEs in that Flash Crashes typically last many minutes ( ) and hence allow ample time for human involvement. Future work will explore the connection to existing studies such as Ref. 28 of market dynamics immediately before and after financial shocks.

Having established that the number of UEEs increases dramatically as the timescale drops below one second and hence drops below the human reaction time, we now seek to investigate how the character of the UEEs might also change as the timescale drops – and in particular, whether the distribution may become more or less akin to a power-law distribution. Power-law distributions are ubiquitous in real-world complex systems and are known to provide a reasonable description for the distribution of stock returns for a given time increment, from minutes up to weeks13,14,15. Our statistical procedure to test a power-law hypothesis for the distribution of UEE sizes and hence obtain best-fit power-law parameter values, follows Clauset et al.'s29 state-of-the-art methodology for obtaining best-fit parameters for power-law distributions and for testing the power-law distribution hypothesis on a given dataset. Following this procedure, we obtain a best estimate of the power-law exponent α and a p-value for the goodness-of-fit, for the distribution of UEE sizes. Specific details of the implementation, including a step-by-step recipe and documented programs in a variety of computer languages, are given in Ref. 29.

Figure 3 shows a plot of the goodness-of-fit p-value and the corresponding power-law exponent α, for the distribution of sizes of UEEs having durations above a particular threshold. As this duration threshold decreases, the character of the UEE size distribution exhibits a transition from a power-law above the limit of the human response time to a non-power-law below it -- specifically, the goodness-of-fit p falls from near unity to below 0.1 within a small timescale range in Figs. 3B and 3C. This loss of power-law character at subsecond timescales suggests that a lower limit needs to be placed on the validity of Mandelbrot's claim that price-changes exhibit approximate self-similarity (i.e. approximate fractal behavior and hence power-law distribution) across all timescales30. It can be seen that the transition for crashes is smoother than for spikes: this may be because many market participants are typically ‘long’ the market16 and hence respond to damaging crashes differently from profitable spikes. Not only is the crash transition onset (650 ms) earlier in Fig. 3B than for spikes in Fig. 3C, it surprisingly is the same as the thinking time of a chess grandmaster, even though individual traders are not likely to be as attentive or quick as a chess grandmaster on a daily basis26,27. This may be a global online manifestation of the ‘many eyes’ principle from ecology6 whereby larger groups of animals or fish may detect imminent danger more rapidly than individuals.

Figure 3 Empirical transition in size distribution for UEEs with duration above threshold τ, as function of τ. (A) Scale of times. 650 ms is the time for chess grandmaster to discern King is in checkmate. Plots show results of the best-fit power-law exponent (black) and goodness-of-fit (blue) to the distributions for size of (B) crashes and (C) spikes, as shown in the inset schematic. Full size image

Figures 4 and 5 show further evidence for this transition in UEE size character as timescales drop below human response times. Figure 4 shows that the cumulative distribution of UEE sizes for the example of spikes, exhibits a qualitative difference between UEEs of duration greater than 1 second, where p = 0.91 and hence there is strong support for a power-law distribution and those less than 1 second where p < 0.05 and hence a power-law can be rejected. A similar conclusion holds for crashes. Figure 5 shows the cumulative distribution of sizes for UEEs in different duration windows, with the distribution for the duration window 1200–1500 ms showing a marked change from the trend at lower window values. The following quantities that we investigated, also confirm a change in UEE character in this same transition regime: (1) a Kolmogorov-Smirnov two-sample test to check the similarity of the different UEE size distributions within different duration time-windows (see SI Fig. 5); (2) the standard deviation of the size of UEEs in a given window of duration (see SI Fig. 6); (3) the average and standard deviation in the number of price ticks making up the individual UEEs which lie in a given duration window (see SI Fig. 7); (4) a test for a lognormal distribution for UEE durations (see SI Fig. 8). Figure 9 of the SI confirms that using different binnings for the UEE durations does not change our main conclusions.

Figure 4 Extent to which the cumulative distribution for UEE spikes follows a power-law, for the subset having durations greater than 1 second (upper panel) and less than 1 second (lower panel). For durations more than 1 second, there is strong evidence for a power-law (p-value is 0.912). For durations less than 1 second, a power-law can be rejected. Black line shows best-fit power-law. Full size image