We show that a two-harmonic log-periodic formula fits the high-frequency data from the Dow Jones Industrial Average index, which encompass the recent episode known as the â€œflash crashâ€ of May 6, 2010.flash crash, crashes, log-periodicity

Raul Matsushita

Sergio Da Silva

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 2 
  
A log-periodic fit for the flash crash of May 6, 2010 
  
 
 
Raul Matsushita  
University of Brasilia 
Sergio Da Silva  
Federal University of Santa Catarina
Abstract 
We show that a two-harmonic log-periodic formula fits the high-frequency data from the Dow Jones Industrial 
Average index, which encompass the recent episode known as the “flash crash” of May 6, 2010.
 
Citation: Raul Matsushita and Sergio Da Silva, (2011) ''A log-periodic fit for the flash crash of May 6, 2010'', Economics Bulletin, Vol. 31 
no.2 pp. 1772-1779. 
Submitted: Jun 15 2011.   Published: June 16, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
1. Introduction 
 
The log-periodicity hypothesis proposes that financial crashes, like material rupture, can be 
governed by log-periodic formulas (Sornette and Vanneste 1992, Sornette and Johansen 
2001,  Sornette  and  Zhou  2002,  Sornette  2003).  One-harmonic  and  two-harmonic  log-
periodic equations have been employed to fit daily data (Sornette and Zhou 2002), and a 
three-harmonic formula has been used to fit anti-bubbles (bear markets) in the Nikkei (from 
1990 to 1998) and future gold prices (from 1980 to 1998) (Johansen and Sornette 1999). 
The three-harmonic formula has also been shown by us to fit intraday data for both bubble 
and anti-bubble episodes (Matsushita et al. 2006). There are other works using intraday 
data (Cajueiro et al. 2009), and Cajueiro et al. (2009) also offer a literature review. 
  One  explanation  of  the  physics  underlying  the  log-periodic  formulas  posits  that 
cooperative imitation between traders in a bull market is responsible for log-periodicity to 
emerge  because  crashes  could  possibly  result  from  the  buildup  of  correlations.  Self-
reinforcing imitation in a bubble episode builds up strains that push the market to a critical 
point, from which sooner or later many traders will place the same order (that is, “sell”) at 
the same time, thereby provoking a crash (Sornette 2003). It is imitation that makes the 
system periodic on the eve of a crash, and in that sense crashes are outliers with properties 
that are statistically distinct from the rest of the population. 
  Here, we show that a two-harmonic log-periodic formula fits well the data sampled 
at a one-minute frequency from the Dow Jones Industrial Average (DJIA) index ranging 
from  September  1,  2009  to  May  31,  2010.  The  data  purposely  encompass  the  recent 
episode known as the “flash crash” of May 6, 2010. On this day, the DJIA suffered its 
largest intraday decline, that is, 998.5 points. Most of the losses occurred between 2:40 pm 
and 3:00 pm, with a peak at 2:45 pm (Figure 1). The stocks of Accenture, for example, 
briefly traded for one cent. The crash, however, was followed by an almost immediate 
rebound. 
  What triggered the crash remains unknown, but some observers point to possible 
causes  like  computer-automated  trades  and  error  by  human  traders.  However,  if  stock 
markets are viewed as complex systems, there is no need for a trigger to explain a crash 
(see Bak and Paczuski 1995 for the general case, and Mazzeu et al. 2011 for the flash 
crash). In particular, under the log-periodic hypothesis, after the critical time a crash may 
suddenly occur without any early warning signs. An initial rumor that the trigger was a 
trader who had typed a sell order for 16 billion shares of Procter & Gamble instead of 16 
million  was  later  dismissed  by  regulators.  On  October  1
st,  2010  the  Securities  and 
Exchange  Commission  issued  a  report  blaming  a  sloppily  executed  sell  order  of  one 
mutual-fund group (Waddel & Reed), which started to sell $4.1 billion of “E-Mini” futures 
contracts through robot trading, taking account only of volume, not time or price. Some 
analysts  blame an intermarket  sweep order, anxiety over Greece‟s bailout  package, the 
British election‟s outcome, and simply two previous days‟ declines in the index. Even if not 
the  main  cause,  robot  trading  through  electronic  platforms  (such  as  Direct  Edge  and 
BATS), which executes trades in milliseconds, certainly played a role in magnifying the 
crash. Also, thanks to increasing high-frequency trading, correlations previously only seen 
across hours or days in trading time-series are now possibly showing up in timescales of 
seconds or minutes (Smith 2010). The buildup of correlations brought by high-frequency 
trading may thus explain the log-periodic nature of the flash crash. 
 
1773Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
2. Results 
 
As observed, we collected data from the DJIA index sampled at a one-minute frequency 
from September 1, 2009 to May 31, 2010, totaling 65,534 observations. In line with the 
previous literature (Sornette and Johansen 2001, Sornette and Zhou 2002, Johansen and 
Sornette 1999) we considered in the analysis the price index itself rather than the returns. 
Recent  developments  of  the  log-periodic  power  law  model  consider  returns  (Lin  et  al. 
2009). Indeed, in Lin et al. (2009) stochastic conditional expectations of returns describe 
continuous updates of the investors‟ beliefs and sentiments. 
  Let  () Zt  be the time series of the DJIA index, where  1, ,65,534 t  minutes. 
Log-periodic cycles with a smooth trend component are described by a sum of log-periodic 
harmonics, that is, 
 
  ln ( ) cos( ln( ) )
j
j j j Z A B C j ,                                                   (1) 
 
where  30,000 t  is a reparameterized starting time on the onset of the high-frequency 
bubble. Because the sample is large (1 65,534 t ), we reparameterize it and divide t  by 
30,000  in  order  to  stabilize  the  estimation  numerical  method.  (Taking  60,000  instead 
rendered the estimation unstable.) Term  AB is the trend across time, and  A,  B, and 
 give its shape. Parameters  j ,  j C , and  j  are, respectively, the angular log-frequency, 
amplitude, and phase of the  j
th harmonic. We set  ( ) 30,000 0 c tt , where  c t  is the 
critical time. Unlike in our previous work (Matsushita et al. 2006), here  j ,  j , and  j  
need not be equal, and  2 j  allows more flexibility for the adjustment. 
  From the starting date used for the fitting procedure  start 1 t=  (3:09 pm of September 
18, 2009), the estimated critical time  3,552 c t  corresponds to the lower price observed at 
9:32  am  on  October  2,  2009.  (It  is  worth  remarking  that,  on  this  day,  there  was  the 
announcement of the US employment indicators.) The gaps between the trading days were 
ignored. From  24,335 t  onwards, the series closely followed the log-periodic path given 
by the adjusted model. This data point corresponds to 11:17 am of December 18, 2009. 
Figure 2 shows that the two-harmonic fits well the natural logs of the DJIA index. 
  Table  1  shows  the  results  for  the  parameters  adjusted  by  nonlinear  regression 
(performed using SAS 9.2). The mean square error of the fit in Figure 2 is 0.000124. For 
comparison, the  case where  1 j  rendered a mean  square  error of 0.000255, which is 
almost two times greater than the mean square error for  2 j . That the two-harmonic log-
periodic power law model (LPPL2) fits the data better than that of one harmonic (LPPL1) 
can be further justified using the information criterion of Akaike (AIC) and the Schwarz 
Bayesian information criterion (SBIC). Because we have a large sample ( 61,983 n ) the 
cost of increasing the number of parameters is outweighed by the benefit of a better fit.  
Indeed, in the AIC 
 
  AIC(LPPL1) 512,849 AIC(LPPL2) 557,529 
 
1774Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
and in the SBIC 
 
  SBIC(LPPL1) 512,785 SBIC(LPPL2) 557,430 . 
 
  For robustness, we tested the stability of the fitting parameters by varying the size of 
the fit intervals through different time windows (Jiang et al. 2010). The windows were both 
squeezed and stretched. Figures 3 and 4 show overlaid fits with very similar patterns. In 
Figure 3 the size of the fit intervals were changed in steps of 592 minutes by squeezing the 
windows  using  a  fixed  end  time  at  10:09  am  of  May  25,  2010,  and  a  starting  time 
increasing from 3:09 pm of September 18, 2009 to 1:48 pm of October 26, 2009. In Figure 
4 the same was done but now by stretching the windows considering a fixed starting time at 
3:09 pm of September 18, 2009, and the end point increasing from 10:06 am of April 19, 
2010 to 10:09 am of May 25, 2010. The values at the end of the range were extrapolated 
until 10:09 am of May 25, 2010 on the basis of the adjusted parameters. 
 
3. Conclusion 
 
The flash crash of May 6, 2010 in the DJIA index was a crash in high-frequency trading. 
Log-periodic  crashes  are  the  result  of  the  buildup  of  correlations  thanks  to  imitation 
between  traders  in  bubble  episodes.  Although  robot  trading  may  not  be  the  culprit  in 
triggering  the  flash  crash,  high-frequency  trading  may  have  generated  correlations  in 
timescales of seconds. This may explain the fact that we cannot dismiss the hypothesis of a 
log-periodic nature for the flash crash. In particular, this study showed that a two-harmonic 
formula fits well the data. 
 
 
1775Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
Table 1. Log-periodic fit for the flash crash of May 6, 2010 
coefficient  estimate  approx. 
standard error 
approx. 95% confidence limits 
lower  upper 
 
A  9.289  0.001  9.288  9.290 
B  0.037  0.001  0.038  0.036 
1 C   0.0081  0.0001  0.0079  0.0083 
2 C   0.0178  0.0001  0.0176  0.0178 
  0.60  0.01  0.61  0.58 
1  1.42  0.03  1.36  1.47 
2  2.10  0.01  2.07  2.13 
1  17.20  0.03  17.15  17.26 
2  9.98  0.01  9.96  10.00 
1  12.81  0.01  12.78  12.83 
2  19.55  0.01  19.53  19.56 
 
 
 
Figure 1. Daily chart of the Dow Jones Industrial Average index during the May 6, 2010 
flash crash. 
 
 
 
 
 
 
1776Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
 
Figure 2.  Critical time interval of the natural log of the Dow Jones Industrial Average 
index sampled at a one-minute frequency (October 2, 2009 to May 31, 2010) and a two-
harmonic log-periodic fit. See Table 1. 
 
Figure 3. Overlaid fits. The size of the fit intervals were changed in steps of 592 minutes by 
squeezing the windows using a fixed end time at 10:09 am of May 25, 2010, and a starting 
time increasing from 3:09 pm of September 18, 2009 to 1:48 pm of October 26, 2009. 
Twelve different  time  windows were  considered, beginning  with  the  longest  of 61,984 
minutes, then the subsequent one of 61,392, and so on, following steps of 592 minutes 
each. Horizontal solid lines represent the range of variation of the time windows whereas 
vertical dotted lines represent the starting time of each fit. The original fit seems stable. 
 
bottom of the flash crash 
at 2:45 pm of May 6, 
2010 
critical time: 9:32 am of 
October 2, 2009 
1777Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
 
 
 
 
 
Figure 4. Overlaid fits. The size of the fit intervals were changed in steps of 592 minutes by 
stretching the windows considering a fixed starting time at 3:09 pm of September 18, 2009, 
and the end point increasing from 10:06 am of April 19, 2010 to 10:09 am of May 25, 
2010.  Twelve  different  time  windows  were  considered,  beginning  with  the  shortest  of 
54,880 minutes, then the subsequent one of 55,472, and so one, following steps of 592 
minutes each. The values at the end of the range were extrapolated until 10:09 am of May 
25, 2010 on the basis of the adjusted parameters. Horizontal solid lines represent the range 
of variation of the time windows whereas vertical dotted lines represent the ending time of 
each fit. The original fit seems stable. 
1778Economics Bulletin, 2011, Vol. 31 no.2 pp. 1772-1779
References 
 
Bak, P. and M. Paczuski (1995) “Complexity, contingency, and criticality” Proceedings of 
the National Academy of Sciences of the United States of America 92, 6689 6696. 
 
Cajueiro, D.O., B.M. Tabak and F.K. Werneck (2009) “Can we predict crashes? The case of 
the Brazilian stock market” Physica A 388, 1603 1609. 
 
Jiang, Z.Q., W.X. Zhou, D. Sornette, R. Woodard, K. Bastiaensen and P. Cauwels (2010) 
“Bubble diagnosis and prediction of the 2005–2007 and 2008–2009 Chinese stock market 
bubbles” Journal of Economic Behavior & Organization 74, 149 162. 
 
Johansen, A. and D. Sornette (1999) “Financial „anti-bubbles‟: log-periodicity in gold and 
Nikkei collapses” International Journal of Modern Physics C 10, 563 575. 
 
Lin,  L., R.E. Ren and D. Sornette (2009) “A consistent model of „explosive‟ financial 
bubbles with mean-reversing residuals” arXiv quantitative finance paper number 0905.0128 
(arXiv:0905.0128v1 [q-fin.RM]). 
 
Matsushita,  R.,  S.  Da  Silva,  A.  Figueiredo  and  I.  Gleria  (2006)  “Log-periodic  crashes 
revisited” Physica A 364, 331 335. 
 
Mazzeu, J., T. Otuki and S. Da Silva (2011) “The canonical econophysics approach to the 
flash crash of May 6, 2010” Applied Mathematical Sciences 5, 1373 1389. 
 
Smith, R.D. (2010) “Is high-frequency trading inducing changes in market microstructure 
and dynamics?” arXiv quantitative finance paper number 1006.5490 (arXiv:1006.5490v2 
[q-fin.TR]). 
 
Sornette,  D.  (2003)  Why  Stock  Markets  Crash:  Critical  Events  in  Complex  Financial 
Systems, Princeton University Press: Princeton. 
 
Sornette, D. and A. Johansen (2001) “Significance of log-periodic precursors to financial 
crashes” Quantitative Finance 1, 452 471. 
 
Sornette, D. and C. Vanneste (1992) “Dynamics and memory effects in rupture of thermal 
fuse networks” Physical Review Letters 68, 612 615. 
 
Sornette, D. and W.X. Zhou (2002) “The US 2000-2002 market descent: how much longer 
and deeper?” Quantitative Finance 2, 468 481. 
1779

A consistent model of „explosive‟ financial bubbles with mean-reversing residuals” arXiv quantitative finance paper number 0905.0128

Bubble diagnosis and prediction of the 2005–2007 and 2008–2009 Chinese stock market bubbles”

Can we predict crashes? The case of the Brazilian stock market” Physica A 388,

Complexity, contingency, and criticality”

Dynamics and memory effects in rupture of thermal fuse networks” Physical Review Letters 68,

Financial „anti-bubbles‟: log-periodicity in gold and Nikkei collapses”

Is high-frequency trading inducing changes in market microstructure and dynamics?” arXiv quantitative finance paper number 1006.5490

Log-periodic crashes revisited” Physica A 364,

Significance of log-periodic precursors to financial crashes”

The canonical econophysics approach to the flash crash of May 6,

The US 2000-2002 market descent: how much longer and deeper?”

Why Stock Markets Crash: Critical Events in Complex Financial Systems,

A log-periodic fit for the flash crash of May 6, 2010

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A log-periodic fit for the flash crash of May 6, 2010

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