# Statistical properties of the aftershocks of stock market crashes: evidence
based on the 1987 crash, 2008 financial crisis and COVID-19 pandemic

Anish Rai, Ajit Mahata and Md Nurujjaman<EMAIL_ADDRESS>Department of Physics, National Institute of Technology Sikkim, Sikkim,
India-737139.  Om Prakash  Department of Mathematics, National Institute of
Technology Sikkim, Sikkim, India-737139.

###### Abstract

Every unique crisis, a new and novel risk factor, leads to a rapid,
synchronous and panic sell-off by the investors that lead to a massive stock
market crash, termed as mainshock, which usually continues for more than one
day. Though most of the stocks start recovering from the crash within a short
period, the effect of the crash remains throughout the recovery phase. During
the recovery, as the market remains in stress, any small perturbation leads to
a relatively smaller aftershock, which may also occur for a few days.
Statistical analysis of the mainshock and the aftershocks for the crash of
1987, the financial crisis of 2008 and the COVID-19 pandemic shows that they
follow the Gutenberg-Richter (G-R) power law. The duration of the influence of
the mainshock, within which aftershocks are considered, has been calculated
using structural break analysis. The analysis shows that high magnitude
aftershocks comparable to the mainshock are rare but low magnitude aftershocks
can be seen frequently till the full recovery of the market. It is also
consistent with the psychology of the investors that when the unique crisis
becomes known, the market does not react too irrationally as it did initially,
and hence subsequent crashes becomes relatively smaller. The results indicate
that there is a possibility of occurrence of future low magnitude aftershocks
due to the ongoing COVID-19 pandemic. The analysis may help investors make
rational investment decisions during the stressed period after a major market
crash.

Market crash, aftershock, Gutenberg-Richter power law, Structural break,
Financial crisis of 2008, COVID-19 pandemic

##  I Introduction

Complex dynamical systems, such as stock market, earth crust, climate, ocean
show extreme events in the form of price crash, earthquake, storm and tsunami
under certain crisis sornette2017stock ; mcnamara2015earthquake ;
sarlis2018natural ; pisarenko2003characterization ; nguyen2017vietnam ;
piccoli2017stock ; zhang2009estimating ; sornette1998hierarchical ;
sornette1996stock . Some of these extreme events which can be termed as the
main-shock, leads to a series of smaller aftershocks sornette2017stock ;
sornette1996stock ; abe2004aging ; siokis2012stock . The dynamics of such
systems can be characterized statistically by power-law distributions
bak2002unified ; bai2010power ; gabaix2007unified ; siokis2012stock ;
selccuk2004financial ; kapopoulos2005stock . In the stock market, an extreme
event occurs in the form of a major crash due to the emergence of every new
and novel risk factor, for which investors are not prepared
albuquerque2020resiliency ; mazur2020covid . The new unique risk leads to a
wide fluctuation in price and volume because of the investors’ irrational
behavior in the volatile stock markets, and causes a massive crash
mazur2020covid ; lyocsa2020fear ; baker2020unprecedented ; wang2009stock ;
barlevy2003rational .

Crashes in stock market occurs mainly due to instability formed due to herding
behavior chiang2010empirical ; tan2008herding , speculation, overvaluation
like in the 1929 crash white1990stock , 1987 crash choudhry1996stock ;
carlson2007brief ; grant1990stock  and the financial crisis of 2008
farmer2012stock . The other reason for the stock market crash is due to the
emergence of novel risk factor like the COVID-19 pandemic of 2020
mazur2020covid ; lyocsa2020fear ; baker2020unprecedented ; wang2009stock . The
events like financial crisis and pandemic have had devastating effects in all
sectors of the economy that lasted for several months to years, and the stock
market remains in stress till the economy or the market fully recovered
sornette1997large ; mazur2020covid ; mahata2020modeling . During the stressed
period after the major crash, many aftershocks of comparatively smaller
magnitudes occur due to investors’ cautious approach in the investment
siokis2012stock ; selccuk2004financial ; kapopoulos2005stock ; netter1989stock
; hong2003differences .

The analysis methods that are applied to study the geophysical phenomena have
also been applied to understand the statistical properties of the price
crashes and subsequent aftershocks due to a crisis in the stock market
siokis2012stock ; selccuk2004financial ; kapopoulos2005stock . The Gutenberg-
Richter (G-R) power law and Omori law are frequently used to explain the
statistical properties of the mainshock and subsequent aftershocks
siokis2012stock ; selccuk2004financial ; kapopoulos2005stock ;
petersen2010market . In these studies, the mainshock and aftershocks were
identified for one-day data with the one-minute absolute log return
petersen2010market , and 60, 100 and 240 days data with the one-day absolute
log return selccuk2004financial ; lillo2003power ; siokis2012stock ;
kapopoulos2005stock ; selccuk2004financial . However, these analyses have the
following limitations.

  1. 1.

As the market crash can continue for more than a day sornette1997large ;
sornette1996stock ; sornette1998hierarchical , the mainshock and aftershock
can not be limited to a single day or minute return.

  2. 2.

After the implementation of the circuit breakers in the market, it takes more
than a day to complete an intense crash. Hence, taking one day fall as a
mainshock or aftershock may be erroneous.

  3. 3.

When one-day absolute log return is considered, both the crashes and
recoveries act as a crash.

  4. 4.

The duration of the aftershocks sequence has been taken arbitrarily though the
effect of major crash may stay for a longer/shorter period on the market and
economy. Hence, identifying the proper data length is also very essential to
analyze the statistics of mainshock and its aftershocks.

This paper aims to study the statistical properties of main and secondary
crashes due to a crisis, overcoming the above limitations. We have estimated
the mainshock and aftershocks as the cumulative consecutive fall of a stock
price. If the secondary crashes are above 7.0% of the main crash, they are
considered as an aftershock. Further, the structural break analysis method
bai1998estimating ; bai2003computation  is applied to estimate the duration of
influence of the mainshock to identify the number of aftershocks. We obtained
that the aftershocks sequence follows the G-R power law. We also intend to
extract some new information about the possible occurrence of aftershocks
during the latest stock market crash in 2020 due to the COVID-19 pandemic.

The paper is organised as follows: Sec. II discusses about the methods used
for the analysis. Sec. III provides the data description. Sec. IV discusses
about the results obtained from the analysis. Finally, Sec. V includes the
summary of the paper and our concluding remarks.

##  II Methodology

###  II.1 Gutenberg-Richter power law

The Gutenberg-Richter (G-R) power law is one of the important empirical law
that is applied to study the statistical properties of an earthquake in
geophysics sornette1999general ; gutenberg1944frequency . It shows the
relationship between the number of earthquakes (aftershocks) to the magnitude
of each earthquakes. It states that the cumulative number of earthquakes ($N$)
with magnitude $M$ larger than or equal to $M$ is proportional to $M$ and is
given by

$\log_{10}N(M)=\alpha-\beta M$ (1)

Where $\alpha$ and $\beta$ are two positive constants. The slope $\beta$ shows
the relationship between the convergence process after a shock to its
magnitude. Eqn. 1 states that the larger the magnitude, the fewer the
remaining aftershocks. Hence, the larger the magnitude of the aftershock, the
shorter will be the length of the aftershocks sequence. The large absolute
value of $\beta$ indicates the occurrence of more aftershocks with low
magnitude and the smaller absolute $\beta$ value indicates fewer aftershocks
with high magnitude. The constant parameter $\alpha$ represents the number of
remaining aftershocks, which does not depend on the magnitude of the
aftershocks siokis2012stock ; kapopoulos2005stock .

###  II.2 Structural Break Technique

The structural break analysis method developed by Bai and Perron (1998, 2003)
bai1998estimating ; bai2003computation  is applied to capture the structural
changes in a time series data. As the structural breaks emerge because of a
sudden change in a time series due to a crisis, it helps us to identify the
duration of the influence of a crisis bai1998estimating ; bai2003computation ;
adamovic2017prediction ; MAHATA2020123612 . A detailed discussion on the
calculation of structural breaks is given below bai1998estimating ;
bai2003computation . Let us consider the following structural change model
with $m$ break points

$y_{t}=x_{t}^{{}^{\prime}}\zeta+z_{t}^{{}^{\prime}}\delta_{j}+u_{t},~{}~{}~{}t=T_{j-1}+1,.....,T_{j}$
(2)

where, $j=1,...,~{}m$. $x_{t}(p\times 1)$ and $z_{t}(q\times 1)$ are vectors
of covariates and $u_{t}$ is the disturbance at time t. The break dates
($T_{1},...,T_{m}$) are explicitly treated as unknown. The purpose is to
estimate the unknown regression coefficients $\zeta$, $\delta_{j}$ and the
break dates. For each $m$-partition ($T_{1},...,T_{m}$), denoted by $T_{j}$,
the associate least square estimates of $\zeta$, $\delta_{j}$ are obtained by
minimizing the following objective function

$\sum_{i=1}^{m+1}\sum_{t=T_{i-1}+1}^{T_{i}}(y_{t}-x_{t}^{{}^{\prime}}\zeta-
z_{t}^{{}^{\prime}}\delta_{i})^{2}$ (3)

Let $\hat{\zeta}(T_{j})$ and $\hat{\delta}(T_{j})$ denote the resulting
estimates that minimize the objective function. The sum of squared residual,
$SSR_{T}(T_{1},....,T_{m})$ is obtained by substituting $\hat{\zeta}(T_{j})$
and $\hat{\delta}(T_{j})$ in the above objective function. The resulting
estimated break dates $T_{1},....,T_{m}$ are obtained as

$\\{(\hat{T_{1}}),...,(\hat{T_{m}})\\}=arg\min_{T_{1},...T_{m}}SSR_{T}\\{T_{1},....,T_{m}\\}$
(4)

where the minimization is done over all partitions $(T_{1},...,T_{m})$. The
break date estimators are global minimizer of the objective function. Let
$v(i,j)$ be the recursive residual at time $j$ obtained using a sample that
starts at date $i$. $SSR(i,j)$ is the sum of squared residual obtained by
applying least-squares to a segment that starts at date $i$ and ends at date
$j$ with $SSR(i,j)=SSR(i,j-1)+v(i,j)^{2}$. In order to evaluate the global
optimal partition, a dynamical programming approach is applied on the sums of
squared residual of all the relevant combination of ($i,j$) segments. Finally,
$m$ break points have been obtained from the global optimal partition function
$SSR(T_{m,K})$ by using first $K$ observation as given in the following
recursive equation.

$SSR(T_{m,K})=\min_{mh\leq K\leq K-h}\\{SSR(T_{m-1,j})+SSR(j+1,K)\\}$ (5)

In our regression procedure, we have taken upto 3-break points or four
segments and $z_{t}=\\{1\\}$ as a constant regressor.

##  III Data Analysed

The proposed methodologies have been applied to identify the statistics of the
stock price crashes on the time series of the daily closing price of indices
and companies. The stock data were taken from Yahoo Finance Yahoofi . We have
analyzed three mainshocks: the 1987 crash, the financial crisis of 2008,
ongoing COVID-19 pandemic and their subsequent aftershocks. In the 1987 crash
case, we have taken DJIA index data as daily closing prices of other companies
were unavailable.

Fourteen dominant world stock indices have been taken, and they are Nasdaq
(USA), DJIA (USA), HSI (Hongkong), BEL20 (Belgium), IBOVESPA (Brazil), BSE
SENSEX (India), DAX (Germany), IBEX35 (Spain), IPC (Mexico), Nifty50 (India),
NIKKEI225 (Japan), S&P500 (USA), SSE (China), CAC40 (France). Further, we have
also taken thirty-one listed companies under these indices.

##  IV Results and Discussion

In this section we have analyzed the mainshock and its subsequent aftershocks
of 1987 crash, the financial crisis of 2008 and COVID-19 pandemic using our
proposed definition of market crashes. We have calculated the influence time
length of these crashes using 3-points structural break analysis. All the
stocks mentioned in Sec. III have been analyzed and their results are
discussed herewith.

Figure 1:  represents the daily closing price of DJIA index from January 1987
to December 1989. The vertical dotted lines represent the structural break
points.

We first investigate the DJIA index from 1987 to 1990 during and after the
Black Monday crash occurred at New York Stock Exchange (NYSE) that was
considered one of the worst crashes in the history of the NYSE. We found that
the crash continued for eleven consecutive days. Fig. 1 represents the daily
closing price of DJIA index during that period. In order to identify the
number of aftershocks due to the mainshock of the 1987 crash, the duration of
the influence of it is identified using the structural break analysis as given
in Subsec. II.2. The closest breakpoints is shown by the vertical dotted lines
in Fig. 1. The first vertical dotted line represents the starting point of the
mainshock, and the second dotted line represents the end of the effect of the
mainshock. Hence, the time of influence of the 1987 crash is from the time of
the mian crash till the end of 1989. The identified period is consistent with
the recovery of the market.

Figure 2: The figure represents the plot between cumulative number of events
(N) and the corresponding magnitude (M) of the 1987 crash. Figure 3:  Solid
line represents the daily closing price of S&P500 index from 2006 to December
2012. The dotted lines represent the quarterly GDP growth rate of USA from
2006 to 2012. The vertical lines represents the structural break points.

Fig. 2 shows the cumulative number ($N$) of aftershocks of magnitude ($M$)
larger than or equal to 7% of the mainshock. We performed a fit with the
functional form of Eq 1. Fig. 2 shows the typical plot of $M$ versus $\log(N)$
that describes the empirical data well for the study period, i.e., the 1987
crash and its aftershocks follow the G-R power law. The straight line
represents the best fit, and we obtained the value of $\alpha$ and $\beta$
mentioned in Eqn. 1 are 1.6 and 25.0, respectively. The figure clearly shows
that the aftershocks with high magnitude are significantly less and
aftershocks with low magnitude occur more frequently after the mainshock. We
have also obtained similar paradigmatic behavior of the leading stock price
indices and companies due to the financial crisis of 2008 and COVID-19
pandemic, 2020, and the detailed analyses are given below.

In Fig. 3 the solid line and dashed line represent the daily closing price of
S&P 500 index from 2006 to 2012 and the quarterly GDP growth rate of USA from
2006 to 2012, respectively. The vertical solid line are the structural breaks
obtained for the financial crisis of 2008. The financial crisis of 2008 can
been clearly captured from these solid vertical line. The closest breakpoints
of the mainshock and recovery are mid of 2008 and end of 2011 respectively.
Hence, we have taken the data from the time of mainshock till the end of 2011
to analyze the influence of the mainshock. Similar analysis has been carried
out for the rest of the 44 companies and indices. Detailed analysis of all the
indices and companies are given in Table 1.

Index/Company | The financial Crisis of 2008 | COVID-19 Crisis
---|---|---
% fall | Magnitude(M) | $\alpha$ | $\beta$ | %fall | Magnitude(M) | $\alpha$ | $\beta$
Nasdaq Composite Index | 22 | 0.1043 | 1.7 | 14.0 | 14 | 0.0639 | 1.4 | 21.0
Hang Seng Index | 28 | 0.1433 | 1.3 | 13.0 | 19 | 0.0910 | 1.5 | 34.0
BEL20 Index | 23 | 0.1134 | 1.7 | 13.0 | 30 | 0.1531 | 1.5 | 26.0
Ibovespa Index | 29 | 0.1456 | 1.6 | 9.9 | 21 | 0.1040 | 1.4 | 17.0
BSE SENSEX Index | 25 | 0.1251 | 1.7 | 12.0 | 20 | 0.0990 | 1.2 | 12.0
DAX Index | 27 | 0.1342 | 1.6 | 11.0 | 28 | 0.1422 | 1.5 | 34.0
DJIA Index | 22 | 0.1085 | 1.9 | 21.0 | 15 | 0.0719 | 1.3 | 16.0
IBEX35 Index | 25 | 0.1251 | 1.7 | 13.0 | 20 | 0.1006 | 1.2 | 12.0
IPC Index | 26 | 0.1311 | 1.5 | 11.0 | 13 | 0.0627 | 1.3 | 26.0
Nifty50 Index | 22 | 0.1077 | 1.7 | 14.0 | 17 | 0.0809 | 1.3 | 24.0
Nikkei225 Index | 32 | 0.1655 | 1.7 | 12.0 | 29 | 0.1590 | 1.3 | 29.0
S&P500 Index | 23 | 0.1130 | 1.7 | 13.0 | 14 | 0.0652 | 1.3 | 15.0
SSE Index | 26 | 0.1291 | 1.5 | 10.0 | 12 | 0.0547 | 1.3 | 27.0
CAC40 Index | 27 | 0.1340 | 1.8 | 13.0 | 29 | 0.1486 | 1.5 | 28.0
Adidas AG | 29 | 0.1504 | 1.8 | 14.0 | 32 | 0.1667 | 1.4 | 19.0
Amazon Inc. | 29 | 0.1481 | 1.9 | 15.0 | 13 | 0.0615 | 1.5 | 21.0
3M Co. | 26 | 0.1331 | 1.7 | 14.0 | 14 | 0.0669 | 1.4 | 23.0
BASF SE | 31 | 0.1640 | 1.8 | 13.0 | 28 | 0.1424 | 1.5 | 19.0
Torrent Pharmaceuticals Ltd. | 26 | 0.1316 | 1.8 | 16.0 | 17 | 0.0817 | 1.2 | 12.0
TCS Ltd. | 23 | 0.1159 | 2.0 | 18.0 | 17 | 0.0794 | 1.5 | 25.0
Reliance Industries Ltd. | 27 | 0.1362 | 1.9 | 12.0 | 17 | 0.0811 | 1.3 | 15.0
Allianz SE | 43 | 0.2472 | 1.8 | 11.0 | 39 | 0.2145 | 1.3 | 13.0
Daikin Industries | 43 | 0.2455 | 1.9 | 10.0 | 24 | 0.1218 | 1.4 | 26.0
Bajaj Finance Ltd. | 48 | 0.2855 | 1.9 | 13.0 | 23 | 0.1148 | 1.5 | 13.0
Braskem | 43 | 0.2445 | 1.7 | 8.7 | 49 | 0.2930 | 1.5 | 13.0
Bharat Petroleum Corp. Ltd. | 26 | 0.1292 | 1.8 | 11.0 | 20 | 0.0978 | 1.3 | 12.0
Canon Inc. | 26 | 0.1298 | 1.9 | 17.0 | 25 | 0.1250 | 1.5 | 22.0
Casio Computer Co. Ltd. | 33 | 0.1757 | 1.9 | 11.0 | 24 | 0.1208 | 1.5 | 19.0
HDFC Bank Ltd. | 26 | 0.1319 | 1.8 | 16.0 | 29 | 0.0864 | 1.4 | 17.0
Infosys Ltd. | 25 | 0.1240 | 1.8 | 14.0 | 30 | 0.1523 | 1.4 | 26.0
Intel Corporation | 27 | 0.1354 | 2.0 | 18.0 | 10 | 0.0864 | 1.5 | 25.0
Microsoft Corporation | 26 | 0.1278 | 1.7 | 13.0 | 16 | 0.0733 | 1.5 | 16.0
Tencent Holdings Ltd. | 29 | 0.1514 | 1.9 | 14.0 | 16 | 0.0750 | 1.6 | 31.0
British American Tobacoo plc | 25 | 0.1246 | 1.7 | 18.0 | 20 | 0.0987 | 1.3 | 17.0
Advantest Corporation | 47 | 0.2826 | 1.9 | 9.5 | 18 | 0.0848 | 1.4 | 13.0
BMW AG | 37 | 0.1973 | 2.0 | 14.0 | 17 | 0.1874 | 1.6 | 24.0
Honeywell International Inc. | 37 | 0.2009 | 1.6 | 8.6 | 30 | 0.1579 | 1.5 | 22.0
Home Depot Inc. | 25 | 0.1270 | 1.8 | 14.0 | 20 | 0.0958 | 1.4 | 21.0
Daiichi Sankyo Company | 45 | 0.2649 | 1.6 | 9.3 | 19 | 0.0924 | 1.1 | 13.0
Apple Inc. | 28 | 0.1408 | 1.8 | 13.0 | 17 | 0.0781 | 1.4 | 14.0
Mastercard Inc. | 27 | 0.1391 | 1.8 | 12.0 | 25 | 0.1232 | 1.5 | 24.0
Glaxosmithkline plc | 23 | 0.1144 | 1.6 | 14.0 | 16 | 0.0745 | 1.3 | 21.0
Ultratech cement Ltd. | 34 | 0.1798 | 1.7 | 12.0 | 16 | 0.0780 | 1.3 | 15.0
Cocacola Co. | 23 | 0.1156 | 1.7 | 20.0 | 20 | 0.0990 | 1.5 | 29.0
Pepsi Co. | 21 | 0.1028 | 1.6 | 16.0 | 19 | 0.0895 | 1.2 | 25.0
Table 1:  The above table consists of 45 indices and companies and its
properties during the financial crisis of 2008 and the ongoing pandemic
COVID-19. The % fall represents the fall during the mainshock and the
magnitude(M) shows strength of the mainshock. The $\alpha$ and $\beta$ are the
variables in the G-R power law i.e., $\log_{10}N(M)=\alpha-\beta M$.  Figure
4: (a) and (b) represent the plot between cumulative number of events (N) and
the corresponding magnitude (M) of S&P500 index and Amazon Inc., respectively
during the 2008 financial crisis.

Fig. 4 (a) and Fig. 4 (b) show the typical plot of $M$ versus $\log(N)$ of S&P
500 index and Amazon Inc., respectively during the 2008 financial crisis. The
points show the aftershocks with its magnitude and the straight line
represents the best fit of the aftershocks sequence. We clearly see that the
number of aftershocks with high magnitude is very low, and the number of
aftershocks with low magnitude is very high that is observed from the cluster
of points formed around low magnitude region. This shows that the aftershocks
with high magnitude are rare and aftershocks with low magnitude occurs
frequently until the market fully recovers from the shock. We obtained the
value of $\alpha$ and $\beta$ of Eqn.1 during the financial crisis for S&P 500
are 1.7 and 13.0 and for Amazon Inc. are 1.9 and 15.0 respectively. The
absolute value of $\beta$ in Fig. 4 (a) is less than Fig. 4 (b) that indicates
the later has more aftershocks than the former. The values of $\alpha$ and
$\beta$ is calculated for all the 45 stocks which is shown in Table 1. From
the figure it is clear that aftershocks sequence of S&P 500 and Amazon Inc.
follow the G-R power law during the financial crisis of 2008. Similar results
have been found for the rest of the 43 companies for financial crisis of 2008.
Detailed analysis of all the indices and companies are given in Table 1.

Figure 5: (a) and (b) represent the plot between cumulative number of events
(N) and the corresponding magnitude (M) of S&P500 index and Amazon Inc.
respectively during COVID-19.

Fig. 5 (a) and Fig. 5 (b) shows the plot of $M$ versus $\log(N)$ of S&P 500
index and Amazon Inc., respectively during the ongoing COVID-19 pandemic. The
best fit straight line of aftershocks gives us the value of $\alpha$ and
$\beta$ for S&P 500 are 1.3 and 15.0 and for Amazon Inc. are 1.5 and 21.0
respectively. It is observed that the absolute value of $\beta$ in Fig. 5 (a)
is less than Fig. 5 (b). This implies that the Amazon Inc. has more
aftershocks than the S&P 500. It is also clear from Fig. 5(a) and (b) that the
aftershocks sequence of S&P 500 and Amazon Inc. follow the G-R power law
during this ongoing pandemic. We have carried the analysis for all the 45
companies and indices and we found that these indices and companies follow the
G-R power law. Detailed analysis of all the indices and companies are given in
Table 1.

It is interesting to note that for COVID-19 pandemic, there is no cluster of
points in the low magnitude region as observed in the Fig. 4 (a) and (b), and
a gap between the highest magnitude aftershock and its next aftershock is
observed in Fig. 5 (a). These observations give us a strong indication that
more aftershocks with low magnitude can be anticipated in the near future due
to the ongoing COVID-19 pandemic. The gap seen in high magnitude region of
Fig. 5 (a) shows the possibility of high-magnitude aftershocks in the near
future.

Figure 6: Plot represents the mainshock and the aftershocks during the
financial Crisis of 2008. The data has been taken from 2007 to 2012. Figure 7:
Plot represents the mainshock and the aftershocks during the COVID-19.The data
has been taken from January 2012 to First week of October 2020.

The temporal variance of the aftershocks sequence is presented in Fig. 6 and
Fig. 7 of S&P 500 during the financial crisis of 2008 and the COVID-19
pandemic, respectively. They show that the number of aftershocks per unit time
does not decay with time. Sometimes the number of aftershocks per unit time
increases and hence does not follow the Omori law. This finding is in
contradiction to the previous studies siokis2012stock ; selccuk2004financial ;
petersen2010market . Our finding is consistent with the statistics of the
mainshock and aftershocks in stock price as explained below.

After a major crash, the market recovers for a certain period, but due to some
negative perturbation, stressed condition of economy and fear of investors
lead to smaller crashes. During the stressed period of the market recovery, it
is observed that the number of aftershocks per unit time increases
significantly at the later period of market recovery. Hence, it is unlikely to
follow Omori law during the market recovery after a major crash. The studies
showed in Ref. siokis2012stock ; selccuk2004financial ; petersen2010market  on
the decay of aftershocks with time that holds Omori law may be incomplete due
to the erroneous definition of the mainshock, aftershocks and duration of
influence that is discussed in Sec.I.

##  V Conclusion

This paper analyzes the statistical nature of the aftershocks sequence after
major stock market crashes for the 1987 crash, the financial crisis of 2008
and the ongoing COVID-19 pandemic. The analysis is carried out for 45 stocks
and indices, considering that the mainshock or aftershock is a cumulative
consecutive fall of the stock price. The duration of influence of the
mainshock is obtained by the 3-point structural break method. The mainshock
and aftershocks follow the Gutenberg-Richter power law. We found that in 1987
crash and 2008 financial crisis there is a dense cluster of aftershocks in the
low magnitude region. This indicates that aftershocks of low magnitude occurs
more frequently rather than high magnitude aftershocks.

The analysis shows that in the ongoing COVID-19 event, the number of
aftershocks in the low-magnitude region is less, and there is a gap in the
high-magnitude region. These results indicate that few more aftershocks with
smaller amplitudes may be anticipated in the near future during this ongoing
pandemic till the pandemic is completely eliminated. However, the probability
of an aftershock with a large magnitude during the ongoing COVID-19 pandemic
is very less as it is no longer a novel risk to the market. Further, the
analysis shows that the number of aftershocks does not necessarily decay with
time, and does not obey Omori’s power law. The analysis may help investors
make rational investment decisions during the stressed period after a major
market crash.

##  VI Acknowledgment

NIT Sikkim is appreciated for allocating doctoral research fellowships to A.R.
and A.M. We also appreciate the help of Sandeep Parajuli in this work.

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