Physica A 391 (2012) 1315–1322

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Physica A

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Stock market dynamics: Before and after stock market crashesFotios M. Siokis ∗

University of Macedonia, Thessaloniki, Greece

a r t i c l e i n f o

Article history:Received 21 June 2010Received in revised form 11 July 2011Available online 8 September 2011

Keywords:Financial crisisStock market crashesStock returnsPower law

a b s t r a c t

This paper presents a brief analysis on the distribution of magnitude of major stockmarketshocks. Based on the Gutenberg–Richter law in geophysics, we model the dynamics ofmarket index returns prior and after major crashes in search of statistical regularities. Fora large number of market crashes, our analysis suggests that the distribution of marketvolatility before and after the stock market crash is described well by the Gutenberg–Richter law, which reflects the scale-invariance and self-similarity of the underlyingdynamics by a robust power-law relation. In addition, the rate of the decay of the aftershocksequence is well described by another power law, which is known as the Omori law. Powerlaw relaxation seems to be a common behavior observed in complex systems such as thefinancial markets.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Aquestion central to research on financial crisis is howstockmarket crashes are generated andhowstoutly they influenceeconomic activity. Financial crises and crashes attracted enormous attention and have always been on the center of policymakers’ attention.With the recent 2007–2009 financial crash, policymakers aremore concerned than ever about the impactof the stock market crash and the distortions would have for investment, consumption and growth and on the financialsystem stability. In this respect, stock market movements are viewed as barometer of current and future economic growth.Upward stock price movements including high price volatility contain valuable information and send signals about thestage of the economy. Many believe that soaring prices reflect a new sustained andmore rapid economic growth. Others arevery skeptical when prices increase rapidly and talk about bubble’s rising and stock market crashes, distorting all economicdecisions and creating costly imbalances that can take years to dissipate.

New theoretical literature on stockmarket crashes is based on the rational panic model [1] which states that uninformedtraders can precipitate a price crash because as prices decline, they surmise that informed traders received negativeinformation, which leads them to reduce their demand for assets and drive the prices of stocks even lower and on thehedging model [2] where exogenous portfolio insurers precipitate price declines by selling stocks when prices fall, thusmagnifying small price changes into large discontinuous jumps. Othermodels consider financial crashes as episodes inwhichagents learn about some underlying fundamental they were previously uncertain about and react to this information [3–6].Madrigal and Scheinkman [7] generate a price crash that is due to strategic manipulation by a fully informed market makerwho finds it optimal to coarsen the information set for potential buyers in bad states of the world, causing a discrete jumpdown in prices when bad outcomes occur. Finally, market crashes are modeled as self-organizing processes [8–10] in adynamical out-of equilibrium system and [11,12].

On the empirical side [13] studied relaxation dynamics after the occurrence of three stock market crashes. By applyingthe Omori law to the number of index returns for a period of 60 trading days, after a major financial crash, and by setting

∗ Tel.: +30 2310 891 459.E-mail address: fsiokis@uom.gr.

0378-4371/$ – see front matter© 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2011.08.068

1316 F.M. Siokis / Physica A 391 (2012) 1315–1322

a large threshold, they found that the index return is well described by a power law function. Selcuk [14] investigateddaily stock market data for a number of emerging markets and concludes that all crashes tested follow the Omori law [15].Also, [16] tested Power-law relaxation of aftershocks for long periods following several other intermediate-size crashes.Omori processes after market crashes exist not only on very large scales, but on less significant shocks as well. Moreover,they show that such Omori processes on different scales can occur within the same time period. This leads to self-similarfeatures of the volatility time series, meaning that some of the aftershocks of a large crash can be considered as subcrashesthat themselves initiate Omori processes on a smaller scale. Mu and Zhou [17] on the other hand, studied four stock marketcrashes of the Shanghai Stock Exchange Composite index and concluded that although volatility decays as a power law itdeviates significantly from the Omori law. Lastly, Petersen et al. [18] investigated cascading dynamics immediately beforeand after 219 market crashes with the use of three different empirical laws, i.e. the Omori law, the productivity law and theBath law. Their findings quantitative relate themain shockmagnitude and the parameters quantifying the decay of volatilityaftershocks as well as the volatility preshocks.

In this study we aim to better understand market shocks and to model the dynamics of a market’s index returns beforeand after of a major crash, by looking at the activity of the stock market index in order to arrive at a statistical regularitywith respect to the number of times the absolute value index return exceeds a given threshold value. Most of the previousstudies have focused on the applicability of the Omori law in order to document the relaxation dynamics of aftershocks. Inour studywemake use of Gutenberg–Richter law (G–R law) formulated in 1944 [19]. We analyze a large number of US stockmarket crashes, based on daily data of the Dow Jones Industrial Index (DJIA) and compare two distinct crashes, namely thegreat crash of 1929 and the crash of 1987. As a robustness test and in line with Lillo and Mantegna and Weber et al., wealso compare the results with two other shocks, the 1997 and 1998 crashes. We identify common regularities and derivevaluable results with policy making implications. To best of our knowledge this is the first attempt made to test the validityof G–R law in U.S. stock market, by using daily data and for a large number of stock market shocks. Finally, we test powerlaw relaxation of aftershocks by introducing another empirical power law, the Omori law.

2. The Gutenberg–Richter power law

The Gutenberg–Richter (G–R) law introduces a simple relationship between the number and the magnitude ofearthquakes (aftershocks, foreshocks). It is one of the longest observed empirical relationships in geophysics, and is a powerlaw relating the number of earthquakes to magnitude. The general form of G–R law is given by Eq. (1) where the constant αcharacterizes the general level of seismicity and the parameter b is commonly known as the b-value describing the ratio oflarge-to-small earthquakes. Lastly, N(M) is the cumulative number of earthquakes with magnitudes equal to or larger thanM . Because the RichtermagnitudeM is a logarithmic scale,M ∝ log[E], where E is the energy released,we haveN(E) ∝ E−w ,a power law.

The G–R takes the following form

N = 10a−bM

or it can be written as

Log10N(M) = a − bM. (1)

The b-value represents the relationship between the convergence process after a shock and its altitude: the larger the sizeof each aftershock the fewer the remaining aftershocks. Therefore, a large absolute b-value implies large aftershocks withhigh volatility and short duration. On the contrary, if b is small, a longer high volatility period is needed. Finally, the constantparameter α reflects the number of expected remaining aftershocks independently from the magnitude of aftershocks. Italso characterizes the fluctuation in a given stock market and is proportional to the volume productivity, i.e. the higher theα value the greater the value.

2.1. Stock market crash

The stress of the financial system from a crash should become visible in the time required to set to a new steady state.This is a key factor since long periods of high volatility and uncertainty in the financial markets tend to increase interestrates, and in general the credit terms. But how can we define a stock market crash? Although a precise definition is difficult,we define a crash as a large drawdown, or as precipitous declines in value for securities that represent a large proportion ofwealth [20].

We investigate a number of crashes and major shocks of the Dow Jones Industrial Index. We extend our analysis to the1929 and 1987 stock market crashes since they make an important pairing mainly for the following two reasons.

Firstly, the 1929 and the 1987 crashes, which, by the way were the largest crashes in the history of the Dow JonesIndustrial, occurred after extended bull markets that attracted many inexperienced investors into the stock market. Theevent of October 19, 1987 caused DJIA index to drop 508 points or 22.6%. From the close of trading on Tuesday October 13,1987 to the close on October 19 (four trading days), the DJIA fell 769 points or 31%, representing a decline in the value ofoutstanding US equity of some $1 trillion. Only the 13.5% drop in the Dow on October 28, 1929 and the fall of 11.7% on thefollowing day have approached the October 1987 decline in magnitude.

F.M. Siokis / Physica A 391 (2012) 1315–1322 1317

Table 1Descriptive statistics.

Description Total sample The 1929 event The 1987 eventForeshocks Aftershocks Foreshocks Aftershocks

n 20,405 100 100 100 100Mean 0.03% −0.01% 0.13% −0.02% 0.18%Ku 21.1 3.8 6.0 3.1 4.4Sk −0.2 −0.5 −0.3 −1.1 0.2Std 1.16% 1.64% 3.00% 1.16% 2.33%Min −6.30% −11.70% −4.60% −8.00%Max 6.30% 12.30% 3.00% 10.10

Note: The sample size covers the period from January 1928 to end of February 2008.

Secondly, according to Mishkin andWhite [21] the pattern of the two great crashes were very similar as was the FederalReserve’s successful lender-of-last-resort intervention to prevent the effects of a crash from spilling into the rest of thefinancial system. Therefore, based on Fed’s evaluation and reaction to the above stock market crashes, seems importantto compare these two crashes and observe their statistical dynamics. We define the dates of the major shocks during thesample period as the days of crashes / mainshocks. We also define aftershocks and foreshocks as the daily absolute returnsof the index, which are greater than one standard deviation (σ ) immediately (100 days) after (and prior to) the mainshock.

Eq. (2) calculates the daily absolute returns of the Index which are greater than 1 standard deviation.ri,t = ln(pi,t/pi,t−1) = ln pi,t − ln pi,t−1 > 1σr (2)

where p is the price of the Dow Jones Industrial Average Index and r is the Index return.

3. Data description and results

Data are taken fromDow Jones Indexes for theDown Jones Industrial Average Index and the sample size covers the periodfrom January 1928 to February 2008. Table 1 presents the descriptive statistics of our sample and for the two main crashes,such as the sample standard deviation, kurtosis, skewness, aswell as theminimumandmaximumof the daily rate of returns.Based on the sample kurtosis and skewness, the daily rate of returns seems not normally distributed. The kurtosis is 21.1while the skewness is −0.16. The distribution is leptokurtic and skewed to the left. In addition, we depict the statistics forthe pre and post 100 day period of the twomain crashes. The kurtosis for the 1929 sample is 3.8 for the foreshock period and6 for the aftershock, while for the 1987 the kurtosis is 3.1 and 4.4 respectively. Obviously, the kurtosis for the 1929 sampleis more leptokurtic than the 1987. Furthermore, the data shows that in both cases skewness is negative with the 1987 eventto exhibit more lack of symmetry. Lastly, in terms of comparison, foreshocks have lower kurtosis, than the aftershocksand are skewed more to the left. In terms of minimum/maximum both extreme values are recorded in the aftershockperiods.

As a second stepwe apply the least-squaresmethod in estimatingα and b-value of Eq. (1) and test the hypothesis that theG–R law provides a good description of market movements following a crash. By setting threshold (M) equal to 1σ , Table 2reports the results of all aftershock and foreshock sequences.1 First, all aftershock and foreshocks sequences reproducethe G–R law. All b-values are significantly different from zero, while the R2 indicates that the data fits well to the powerlaw. However the b-values are statistically different from each other. Secondly, almost in all foreshock sequences b-valueis greater than the aftershocks b-value. This is a key finding since in geophysics the opposite has been observed. Lastly,aftershock and foreshock sequences later than the 1987 event have greater b-value than earlier aftershock and foreshocktime sequences and their value is almost equal to or greater than 1.

Apart for estimating the magnitude–frequency distribution for a large number of stock market crashes we investigatethe crashes of 1929 and 1987. For robustness purposes we run the regression based on two different threshold values ofthe returnmagnitudeM .2 Specifically, two definitions of aftershocks were employed, the benchmark threshold as discussedearlier (with absolute return greater than 1.0 standard deviation) and the threshold with 1.5σ for the 100 days after theshock. Based on the results reported in Table 3, we find that both distributions, i.e. the 1929 and 1987 aftershock sequencesreproduce the G–R law. The 1929 aftershock sequence is characterized by a smaller absolute exponent, b ∼ 0.33 than the1987 aftershock sequence, b ∼ 0.49. The results imply a different behavior of aftershocks dynamics among the two crashesin question. Based on estimated b-valueswe can say that the 1929 aftershock periodwas characterized by a longer turbulentperiodwhile, on the other hand, for the 1987 event, In otherwords, the 1987 event depicts a greater sensitivity of the numberof remaining aftershocks to an increase in the relative magnitude of the aftershock. The result is fairly plausible since the1929 crisis was longer in duration compared to the 1987. Lastly we can say that the b-value fluctuates in time and dependson the shock magnitude. Fig. 1 depicts the Gutenberg–Richter law for 1929 and 1987 aftershock sequences.

1 We test more crashes and higher volatility periods, as well as the volume traded and we derived at the same results that all crashes are well describedby the G–R law.2 We also run various other regressions based on different threshold values and we have arrived at indifferent results.

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Table 2Description of the main shocks for the Dow Jones industrial index.

Date ofmainshock

Magnitude of shock in absolutevalue Ln(pt ) − Ln(pt−1) (%)

Standard deviation (σ )(%)

Relativeshock

Aftershocks Foreshocks

α β Rsq α β Rsq

10/28/1929 14.5 1.16 12.5 4.07 −0.33 0.95 4.26 −0.68 0.956/16/1930 8.2 1.16 7.1 4.60 −0.41 0.98 4.50 −1.00 0.9712/17/1930 5.0 1.16 4.3 4.87 −0.73 0.99 5.18 −0.88 0.9910/6/1931 13.9 1.16 11.9 4.67 −0.29 0.99 4.48 −0.42 0.969/21/1932 10.8 1.16 9.3 4.56 −0.43 0.98 5.00 −0.47 0.973/15/1933 14.3 1.16 12.3 4.71 −0.47 0.99 4.82 −0.53 0.967/26/1934 6.9 1.16 5.9 4.75 −1.07 0.94 5.11 −1.22 0.9810/18/1937 7.5 1.16 6.5 4.80 −0.75 0.98 4.15 −0.69 0.979/5/1939 9.1 1.16 7.8 3.58 −0.90 0.91 5.22 −1.51 0.949/3/1946 5.7 1.16 4.9 4.37 −0.83 0.94 3.76 −1.19 0.9510/19/1987 25.6 1.16 22.1 4.32 −0.49 0.95 4.20 −0.97 0.9410/13/1989 7.2 1.16 6.2 4.30 −1.28 0.97 4.95 −2.17 0.9610/27/1997 7.5 1.16 6.4 4.23 −1.04 0.90 4.87 −0.95 0.998/31/1998 6.6 1.16 5.7 4.74 −0.97 0.99 4.59 −1.26 0.904/14/2000 5.8 1.16 5.0 5.40 −1.80 0.98 4.68 −1.02 0.989/17/2001 7.4 1.16 6.4 5.00 −1.32 0.98 5.31 −1.80 0.987/24/2002 6.2 1.16 5.3 4.66 −0.58 0.98 4.95 −1.18 0.97

Fig. 1. Cumulative number of events versus magnitude. The 1929 and 1987 aftershock sequences.

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Table 3The Gutenberg–Richter relation: Ln N(M) = a − bM .

Description Magnitude of shock (%) σ (%) Relative shock α b modal α/b R square

1929 14.51 std 1.16 12.50 4.07 −0.33 −12.32 0.951.5 std 1.74 8.33 3.85 −0.30 −12.93 0.971987 25.61 std 1.16 22.07 4.32 −0.49 −8.79 0.951.5 std 1.74 14.71 4.18 −0.46 −9.04 0.95

Table 4The G–R Relation. Comparison of fore–aftershock sequences.

Event Number of shocks over 1σ α b modal α/b R square

1929Foreshocks 28 4.26 0.68 6.26 0.95Aftershocks 39 4.07 0.33 12.33 0.95

1987Foreshocks 19 4.2 0.97 4.34 0.94Aftershocks 42 4.32 0.49 8.82 0.95

Note: Threshold is set to at 1 and 1.5 standard deviation respectively. Event magnitude is calculated asM = abs(rt ) where (rt ) = ln pt − lnpt−1 .

As a robustness test and following [13]we report results of twomore stockmarket drawdown, the time periods occurringafter the 27 October 1997 and the 31 August 1998. The estimates of the b-value are much higher and close to 1, with 0.97for the 1998 crisis and 1.04 for the 1997, meaning that there is a greater sensitivity of the number of remaining aftershocksto an increase in the relative magnitude of the aftershocks. Also, the higher b-value (in absolute terms) implies a shorteraftershock period for the index to settle into a new course. Furthermore, it is interesting to report the value of the modal,which is calculated by dividing the productivity α, by the b value. The modal for the last two crashes have values around 4while for the 1929 and 1987 crashes the number is 8.8 and 12.3 respectively.

3.1. Foreshocks

In this sectionwe investigate the pre-shock period and the relationshipwith themain shock and the aftershock sequence.Although aftershocks and foreshocks in general can be taken to be causally interactive with the main shock, precisedefinitions of these events have not been made consistently. Since there is no widely accepted definition of foreshocks,some earthquakes were preceded by a very remarkable short-term increase in the number of small events. These eventsmay be causally and/or physically related to the main shock, and may be called foreshocks or preshocks. By defining theforeshocks events it is very important to recognize and to determine if a bubble or energy has actually accumulated beforethe main shock.

Foreshock activity is present in our observed data. Fig. 2 exhibits the foreshock activities of the 1929 and 1987 eventsthat are greater than 1σ and for a period of 100 days. There is a moderate but clearly visible increase of the earthquakerate as we are approaching to the main crash.3 By applying the G–R law to foreshock activity we can show that the periodbefore the major crash is characterized by increasing volatility, but the foreshocks are smaller in magnitude compared withthe magnitude exhibited in the aftershock sequence. The foreshock periods (Table 4) are characterized with higher b-valuecompared with the aftershock’s b-value, but they are shorter in duration. In other words, based on the above result, theperiod before the major crash is characterized by a bubble rising arrangement, but the shocks are smaller in magnitudecompared with the aftershock magnitude. In addition, the number of foreshocks per main shock is quite smaller than thenumber of aftershocks.

4. The Omori law and the aftershock sequence

Another empirical law that relates the time after the main shock to the number of aftershocks per unit time, n(t) is theOmori law (Fig. 1). It states that the number of aftershocks per unit time decays with power law n(t) ∝ t−p. In order to avoiddivergence at t = 0, the Omori law is rewritten as

n(t) = K(t + τ)−p, (3)

3 Although it is documented that the increased activity prior to the crash (main shock) follows the G–R law, the overall smaller number of foreshocks incomparison with the aftershock sequence, requires very long data sets.

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Fig. 2. Temporal variation of the two crises.

whereK and τ are twopositive constants. By Integrating equation (3) between 0 and t , the cumulative number of aftershocksbetween the main shock and the time t can be described as

N(t) = K [(t + τ)1−p− τ 1−p

]/(1 − p), (4)

where p = 1 and N(t) = K ln(τ/1 + τ) for p = 1.We investigate the index returns during the timeperiod after the 1929 and1987 crashes that tookplace atNewYork Stock

Exchange. In comparison with Lillo and Mantegna and Weber et al., we also investigate the October 1997 and August 1998crash. In our investigation we select 100 days after the crash and the aftershock is defined again as the daily absolute returngreater than a threshold value (1σ , 2σ and 3σ respectively) immediately after the major shock. The number of cumulativeaftershocksN(t) for t = 100 is calculated and an estimate of p in Eq. (4) is obtained for all aftershock periods. For all selectedthreshold values a nonlinear behavior is detected. The estimated exponent p differs from threshold to threshold and fromone aftershock to another aftershock period. Fig. 3 plots the cumulative number of aftershocks, N , and the best fit of Eq. (4)for λ = 1σ , 2σ and 3σ . As seen, the aftershock sequence is well described by the power law. Please note that the p valueswhen threshold λ is set to 3σ are close to 1.4 This means that the daily volatility of the DJIA index-measured as the absolutereturns—after a mainshock relaxes faster than the results derived by Weber et al., and Lillo and Mantegna. Furthermorethe relaxation exponent p increases with the threshold λ, implying that larger aftershocks decay faster. This behavior isanalogous with the findings of the G–R law.

4 The large p value > 1 could be due to the possibility of local optimization. We perform the same analysis and restricting p to be between 0.5 and 1 andvalues appear to be 0.99. We also follow Weber et al., by fitting Eq. N(t) = K( 1

1−p ) t(1−p) . All p-values derived are smaller than the ones derived by fittingEq. (4). P-values range from 0.10 to 0.70 while the sum square residuals (SSR) are much higher than the SSR derived by fitting Eq. (4).

F.M. Siokis / Physica A 391 (2012) 1315–1322 1321

Fig. 3. Cumulative number of aftershocks and the Omori law. Dotted lines are the best fit of Eq. (4). For 1929, 1987 and 1998 we depict the curves foraftershocks greater than 1σ , 2σ and 3σ , while for 1997, 1σ , 1.5σ and 2σ respectively. Plots show 100 days after the crash.

5. Conclusion

Firstly, in this present work we attempt to analyze the scaling properties of a number of sequences generated by crashes/mainshocks in the NY Stock Exchange. We study the frequency–magnitude distribution of aftershocks and foreshockscaused by mainshocks. In our attempt to find any statistical regularity we conclude that all sequences generated by thecrashes obey a power law and can be accurately depicted by the G–R law. The frequency–magnitude distribution exhibitsscale invariability and appears to be self-similar.

Furthermore, we find that the b-values of the magnitude–frequency distribution are different among earlier aftershocksand later ones and between the 1929 and 1987 crashes. The 1987 aftershock sequence is characterized by a shorter buthigher volatility period.

In addition, a comparison between foreshocks and aftershocks reveals different magnitude–frequency distributions. Ifthis is valid, it should have important implications for financial crash prediction. But in order to test this proposition aconstruction of a complete financial crash catalog is required.

Lastly, in line with Lillo and Mantegna and Weber et al., we test the Omori law for 1929, 1987, 1997 and 1998 crashes.Instead of minute-to-minute scale, we test the Omori law on a daily scale and we find that the power law holds for at least100 days after the mainshock. These findings reinforce the results derived by Lillo and Mantegna andWeber et al., showingthat the Omori law for earthquakes holds after crashes of very large magnitude in the NYSE. The stock market volatilityabove a certain threshold shows a power law decay well described by the Omori law.

References

[1] G. Barlevy, P. Veronesi, Rational panics and stock market crashes, Journal of Economic Theory 110 (2003) 234–263.[2] G. Gennotte, H. Leland, Market liquidity, hedging and crashes, American Economic Review 80 (1990) 221–999.[3] A. Kraus, M. Smith, Market created risk, Journal of Finance 44 (1989) 557–569.[4] I. Lee, Market crashes and informational avalanches, Review of Economic Studies 65 (1998) 741–759.[5] J. Zeira, Informational overshooting, booms and crashes, Journal of Monetary Economics (1999) 237–257.[6] H. Hong, J. Stein, Differences of opinion, short sales constraints, and market crashes, Review of Financial Studies 16 (2003) 487–525.[7] V. Madrigal, J. Scheinkman, Price crashes, information aggregation and market making, Journal of Economic Theory 75 (1997) 16–63.[8] S. Focardi, S. Cincotti, M. Marchesi, Self-organization and market crashes, Journal of Economic Behavior and Organization 49 (2002) 241–267.[9] R. Cont, J. Bouchaud, Herd behavior and aggregate fluctuations of financial markets, Macroeconomic Dynamics 4 (2000) 170–196.

[10] D. Stauffer, D. Sornette, Self organized percolation model for stock market fluctuations, Physica A 271 (1999) 496–506.[11] A. Johansen, D. Sornette, A hierarchical model of financial crashes, Physica A 261 (1998) 581–598.

1322 F.M. Siokis / Physica A 391 (2012) 1315–1322

[12] M. Levy, H. Levy, S. Solomon, A microscopic model of the stock market-cycles, booms and crashes, Economics Letters 45 (1995) 103–111.[13] F. Lillo, N.R. Mantegna, Power-law relaxation in a complex system: Omori law after a financial crash, Physical Review E 68 (2003) 016119.[14] F. Selcuk, Financial earthquakes, aftershocks and scaling in emerging stock markets, Physica A 333 (204) (2004) 306–316.[15] F. Omori, On the aftershocks of earthquakes, J. Coll. Sci., Imp. Univ. Tokyo 7 (1894) 111–200.[16] P. Weber, F. Wang, I. Vodenska-Chitkushev, S. Havlin, E., H. Stanley, Relation between volatility correlations in financial markets and Omori processes

occurring on all scales, Physical Review E 76 (2007) 016109.[17] G.H. Mu, W.X. Zhou, Relaxation dynamics of aftershocks after large volatility shocks in teh SSEC index, Physica A 387 (2008) 5211–5218.[18] M.A. Petersen, F. Wang, S. Havlin, E.H. Stanley, Market dynamics immediately before and after financial shocks: quantifying the Omori, productivity

and Bath laws, Physical Review E 82 (2010) 036114.[19] B. Gutenberg, C. Richter, Earthquake magnitude, intensity, energy and acceleration, Seismological Society American Bulletin 46 (1956) 105–145.[20] Crashes E.M. Garber, in: E Newman, M. Milgate, J. Eatwell (Eds.), The New Palgrave Dictionary of Money and Finance,1, Macmillan Press Limited,

London, 1992, 511–513.[21] F.S. Mishkin, E.N.White, 2002. US stockmarket crashes and their aftermath: implications for monetary policy. National Bureau of Economic Research,

working Paper 8992.

Further reading

[1] M.A. Petersen, F. Wang, S. Havlin, E.H. Stanley, Quantitative law describing market dynamics before and after interest rates change, Physical Review E81 (2010) 066121.

[2] A. Johansen, D. Sornette, Stock market crashes are outliers, European Physical Journal. B, Condensed Matter Physics 1 (1998) 141.[3] R. Mantegna, H. Stanley, Modelling of financial data: comparison of the truncated Levy flight and the ARCH(1) and GARCH(1,1) process, Physica A 254

(1998) 77–84.[4] E. Stanley, V. Plerou, X. Gabaix, A statistical physics viewof financial fluctuations: evidence for scaling anduniversality, Physica A 387 (2008) 3967–3981.