arXiv:cond-mat/0111257v2 [cond-mat.stat-mech] 3 Jun 2003
Power law relaxation in a complex system: Omori law after a financial market crash
F. Lillo
and R. N. Mantegna ,†
Istituto Nazionale per la Fisica della Materia, Unit`a di Palermo, Viale delle Scienze, I-90128, Palermo, Italia
†
Dipartimento di Fisica e Tecnologie Relative, Universit`a di Palermo, Viale delle Scienze, I-90128 Palermo, Italia.
We study the relaxation dynamics of a financial market just after the occurrence of a crash by
investigating the number of times the absolute value of an index return is exceeding a given threshold
value. We show that the empirical observation of a power law evolution of the number of events
exceeding the selected threshold (a behavior known as the Omori law in geophysics) is consistent with
the simultaneous occurrence of (i) a return probability density function characterized by a power
law asymptotic behavior and (ii) a power law relaxation decay of its typical scale. Our empirical
observation cannot be explained within the framework of simple and widespread stochastic volatility
models.
PACS numbers: 89.65.-s,89.75.-k
Several complex systems are statistically characterized
by power-law distributions. Examples are earthquakes,
financial markets, landslides, forest fires and scale free
networks. Power law distributions imply that rare events
are occurring with a finite non-negligible probability in
complex systems. It is therefore meaningful to ask the
following scientific question: how is the dynamics of a
complex system affected when the system undergoes to
an extreme event? An answer to this question concerning
earthquakes was provided by Omori more than a century
ago [1]. The Omori law describes the non stationary period
observed after a big earthquake. In his study, the
number of aftershocks per unit of time is described by
a power law and a time scale for the relaxation process
of the complex system to its typical state does not exist.
Non exponential relaxation to a typical state has
also been observed in several physical and social systems.
For example, power law relaxation has been theoretically
predicted and experimentally observed in spin glasses [2],
condensed matter systems [3], microfracturing phenomena
[4], physical systems described by a fractional Fokker-
Planck equation [5], in the kinetics of reversible bimolecular
reactions [6], in two-dimensional arrays of magnetic
dots interacting by long-range dipole-dipole interactions
[7], in the Internet dynamical response [8] and in the Internet
traffic [9].
In the present study we investigate the dynamics of a
model complex system when it is moved far away from its
typical state by the occurrence of an extreme event. This
is done by investigating the statistical properties of time
series of financial indices in the time period immediately
after a financial crash. These market phases are indeed
strongly non-stationary and we show that a time power
law relaxation is detected when the financial market is
moved far away from its typical behavior.
Financial time series of stock or index returns are modeled
in terms of random processes [10, 11]. Empirical investigations
show that the time series of stock or index
return is not strictly sense stationary. In fact the volatility
of the financial asset, i.e. the standard deviation of
asset returns describing the typical scale of the process,
is itself a stochastic process fluctuating in time [12, 13].
The non-stationary evolution of asset returns can sometimes
show relaxation time patterns. Specifically, decaying
patterns of volatility are observed in time periods
immediately after a financial crash. An illustrative example
of such non-stationary time pattern is given in Fig.
1 where we plot the one-minute logarithm changes of the
index
r(t) (a quantity essentially equivalent to return) for
the Standard and Poor’s 500 (S&P500) index during 100
trading days after the Black Monday (19 October 1987).
The pattern observed in Fig. 1 is not invariant under
time-reversal. Other examples of statistical properties of
market which are not time-reversal have been observed in
the investigation of cross-sectional quantities computed
for a set of stocks before and after financial crashes [14].
A direct characterization of the time evolution of the
scale of the random process of return is extremely difficult
in financial markets and in several other complex
systems due to the fact that the random variable is highly
fluctuating and that system is unavoidably monitored
by just recording a single random realization. We make
use of a different and statistically more robust method.
Specifically, we quantitatively characterize the time series
of index returns in the non-stationary time period
by investigating the number of times
|r(t)| is exceeding a
given threshold value. This investigation is analogous to
the investigation of the number
n(t) of aftershock earthquakes
measured at time
t after the main earthquake.
The Omori law
n(t) ∝ t−p says that the number of aftershock
earthquakes per unit time measured at time
t after
the main earthquake decays as a power law. In order to
avoid divergence at
t = 0 Omori law is often rewritten as
n
(t) = K(t + τ)−p (1)
where
K and τ are two positive constants. An equivalent
formulation of the Omori law more suitable for comparison
with real data can be obtained by integrating
equation (1) between 0 and
t. In this way the cumulative
number of aftershocks observed until time
t after the
main earthquake is
N
(t) = K[(t + τ)1−p − τ1−p]/(1 − p) (2)
2
when
p 6= 1 and N(t) = K ln(t/τ + 1) for p = 1. The
value of the exponent
p for earthquakes ranges between
0
.9 and 1.5. Because N(t) is related to n(t) by a summation,
the fluctuation in
N(t) is substantially reduced
compared with the fluctuation in
n(t). Hence customary
measurement of
N(t) leads to a more reliable characterization
of the aftershock period than measurement of
n
(t).
We first investigate the index returns during the time
period after the Black Monday crash (19 October 1987)
occurred at New York Stock Exchange (NYSE). This
crash was one of the worst crashes occurred in the entire
history of NYSE. The S&P500 went down 20
.4%
that day. In our investigation, we select a 60 day after
crash time period ranging from 20 October 1987 to
14 January 1988. This time period is chosen to maximize
the time period investigated by simultaneously ensuring
that the relaxation process is still going on. The selected
value is not a critical one and time windows of 50 or
70 trading days provide similar results. For the selected
time period, we investigate the one-minute return time
series of the S&P500 Index. The first estimate concerns
the unconditional one-minute volatility which is equal to
σ
= 4.91 × 10−4. In Fig. 2 we show the cumulative
number of events
N(t) detected by considering all the occurrences
observed when the the absolute value of index
return exceeds a threshold value
ℓ chosen as 4σ, 5σ, 6σ
and 7
σ. For all the selected threshold values we observe
a nonlinear behavior. Nonlinear fits performed with the
functional form of equation (2) well describes the empirical
data for the entire time period. This paradigmatic
behavior is not specific of the Black Monday crash of the
S&P 500 index. In fact, we observe similar results also
for a stock price index weighted by market capitalization
for the time periods occurring after the 27 October
1997 and the 31 August 1998 stock market crashes. This
index has been computed selecting the 30 most capitalized
stocks traded in the NYSE and by using the highfrequency
data of the
Trade and Quote database issued
by the NYSE. In Fig. 3 we show
N(t) for ℓ = 4σ where
σ
is again the unconditional one-minute volatility in the
considered periods. We estimate
σ = 4.54 × 10−4 during
the period from 28 October 1997 to 23 January 1998
and
σ = 6.09 × 10−4 during the period from 1 September
1998 to 24 November 1998. In the left part of Table
I, we summarize the values of the
p exponents obtained
by best fitting with equation (2) the cumulative number
of events exceeding the selected threshold values for the
considered market crashes. The value of the exponent
p
varies in the interval between 0.70 and 0.99 . The estimate
of the exponent
p is slightly increasing when the
threshold value
ℓ is increasing. Below we will comment
the relation between this observation and the properties
of the index return probability density function (pdf).
The detected nonlinear behavior of
N(t) is specific to
aftercrash market period. In fact an approximately linear
behavior of
N(t) is observed when a market period of
roughly constant volatility such as, for example, the 1984
year is investigated. This is due to the fact that when
the process is stationary the frequency of aftershock
n(t)
is on average constant in time and therefore the cumulative
number
N(t) increases linearly in time. In terms
of equation (2) this implies that the exponent
p is equal
to zero. For independent identically distributed random
time series it is possible to characterize
n(t) in terms of
an homogeneous Poisson process [15]. The results summarized
in the left part of Table I imply that the time
period immediately after a big market crash has statistical
properties which are different from constant volatility
periods. In particular, index return cannot be modeled
in terms of independent identically distributed random
process after a big market crash.
The empirical evidence of the power-law decrease of
the frequency of aftershocks is consistent with a powerlaw
decay of volatility after a major crash. In order to
prove this claim, we describe the empirical behavior of
N
(t) by assuming that during the time period after a
big crash the stochastic variable
r(t) is the product of a
time dependent scale
γ(t) times a stationary stochastic
process
rs(t). For the sake of simplicity, we also assume
that the pdf of
r(t) is approximately symmetrical. Under
these assumptions, the frequency of events of
|r(t)| larger
than
ℓ observed at time t is
n
(t) ∝ 2 Z +1
ℓ
f
(r, t)dr (3)
where
f(r, t) is the pdf of r(t) at time t. One can rewrite
equation (3) in terms of the cumulative distribution function
F
s(rs) of the random variable rs(t) as
n
(t) ∝ 1 − Fs (ℓ/γ(t)) . (4)
In this description, the specific form of the time evolution
of
n(t) is, for large values of the threshold ℓ, controlled
by the properties of (i) the time evolution of the
scale
γ(t) and (ii) the asymptotic behavior of the pdf for
large values of
|rs(t)|.
By assuming that the stationary return pdf behaves
asymptotically as a power law
f
s(rs) ∼
1
r
α+1
s
,
(5)
the frequency of events
n(t) becomes for large values of ℓ
n
(t) ∼ (γ(t)/ℓ)α . (6)
By hypothesizing that
γ(t) ∼ exp (−kt), the frequency
of events above threshold is expected exponentially decreasing
n
(t) ∼ exp (−αkt). Conversely, when the scale
of the stochastic process decays as a power law
γ(t) ∼ t−β, the frequency of events above threshold is power
law decaying as
n(t) ∼ 1/tp. It is worth noting that the
exponent
p is given by
p
= α β. (7)
3
TABLE I: Exponents obtained from the empirical analyses of 60 day market periods occurring after the 19 October 1987, 27
October 1997 and 31 August 1998 market crashes.
p α β α β
4
σ 5 σ 6 σ 7 σ
1987 0.85 0.90 0.99 0.99 3
.18 ± 0.34 0.32 ± 0.02 1.02 ± 0.13
1997 0.70 0.73 0.73 0.76 3
.67 ± 0.40 0.22 ± 0.04 0.81 ± 0.17
1998 0.99 0.99 0.99 0.99 3
.49 ± 0.37 0.32 ± 0.05 1.12 ± 0.21
The previous relation links the exponent
p governing the
number of events exceeding a given threshold to the
α
exponent of the power law return cumulative distribution
and to the
β exponent of the power law decaying
scale. It is worth noting that a power law behavior of
the return pdf is observed only for large absolute values
of returns. Hence, the relation between exponents (eq.
(7)) is valid only for large values of the threshold
ℓ used
to determine the exponent
p. Our theoretical considerations
show that a number of events above threshold
decaying as a power law, i.e. the analogous of the Omori
law, is consistent with the simultaneous occurrence of:
(i) a return pdf characterized by a power law asymptotic
behavior and (ii) a non-stationary time evolution of the
return pdf whose scale is decaying in time as a power
law. These hypotheses are consistent with recent empirical
results. In fact, a return pdf characterized by a
power law asymptotic behavior has been observed in the
price dynamics of several stocks [16, 17]. To the best
of our knowledge the only investigation on the decay of
volatility after a crash has been performed in Ref. [18]
where a power law or power law log-periodic decay of
implied volatility has been observed in the S&P500 after
the 1987 financial crash. We would like to stress that
implied volatilty is different from our
γ(t) because implied
volatiltiy is obtained from index derivative prices
by using the Black and Scholes formula instead that directly
from data. Moreover the value of the exponent
governing the decay of volatility is different in our study
and in Ref. [18]. The analytical considerations developed
above indicate that stochastic volatility models of price
dynamics are able to describe the behavior of an index
after a crash when they predict the volatility power-law
decay in time after a crash. Therefore simple autoregressive
models, such as GARCH(1,1) [19] models, are
unable to describe the observed behavior. GARCH processes
in their most compact form cannot show a scale
of the stochastic process decaying as a power law after a
big event. By analytical calculation and by performing
numerical simulations we have shown that these models
are characterized by an exponential decay of the scale of
the process [20].
In order to show that empirical data are consistent
with our description of aftershock periods, we empirically
study the time evolution of the scale of the process. To
this end, by using the ordinary least square method, we
fit the absolute value of return with the functional form
f
(t) = c1t−β + c2 in the 60 days after each considered
market crash. We check that the relation
c1t−β >> c2 is
verified in the investigated period. The best estimation
of
c1 and c2 are 6.3 10−4 and 2.8 10−6 for the 1987 crash,
5
.1 10−4 and 4.3 10−5 for the 1997 crash and 4.4 10−4
and 1
.0 10−4 for the 1998 crash. The time t is expressed
in trading day. By using the relation,
r(t) = γ(t)rs(t),
the
β exponent obtained is also the exponent controlling
the scale
γ(t). In order to estimate the α exponent
governing the stationary part of the return evolution we
define a new variable
rp(t) obtained dividing r(t) by the
moving average of its absolute value. The averaging window
is set to 500 trading minutes. The quantity
rp(t)
is a proxy for the stationary return
rs(t). We investigate
the asymptotic properties for large absolute values
of the stochastic process
rp(t) by computing the Hill’s
estimator [21] of the process computed over the largest
1% values of
|rp(t)|. To assess the reliability of the α
estimate obtained with this method we also compute its
95% confidence interval. The 95% confidence interval is
obtained by computing
C95α/√m where C95 is the value
at which the normal distribution is equal to 0.95 and
m
is the number of records located in the distribution tail.
With our procedure, we obtain a value of the exponent
α
which is ranging from 3.18 to 3.67. These values are
consistent with the observations performed by different
authors on the power law behavior governing large absolute
returns in stocks and stock indices [16, 17].
The estimates of
α and β values are shown in the right
part of Table I for all the investigated market crashes.
The last column of the Table gives the value of the product
α β
that is to be compared with the values of p summarized
in the left part of the Table. The agreement is
increasingly good for values of
p obtained for large values
of the threshold. This is expected because only for
large threshold the relevant part of the return pdf is well
described by a power law behavior.
Finally, we investigate the properties of
N(t) computed
for the random variable
rp(t). This variable is our proxy
for
rs(t) and therefore a linear behavior of N(t) is expected
for each value of the threshold chosen. ¿From our
definition of
rp(t), it follows that the mean of the absolute
value of
rp(t) is equal to one. In Fig. 4, we show N(t)
for the market crash of 19 October 1987 when
ℓ is ranging
from 4 to 13. For all values of the threshold,
N(t) is
approximately linear showing that
rp(t) provides a good
proxy for
rs(t). Moreover, starting from equation (4), one
4
can show that the slope
η of N(t) is proportional to the
quantity 1
−Fs(ℓ). We determine η with a best linear fit
of
N(t) for each value of ℓ. The results are shown in the
inset of Fig. 4. Under the assumption of equation (5),
the expected relation between
η and α is η ∼ ℓ−α. The
inset also shows our best fit of
η with a power law relation
as a solid line. The best fitting exponent is
α = 3.14
when
ℓ ≥ 7. This value of α is consistent with the value
obtained with the Hill estimator (see Table I).
In conclusion our results show that time periods of
the order of 60 trading days (approximately 3 months
in calendar time) occurring after a major financial crash
can be modeled in terms of a new stylized statistical law.
Specifically, the number of index returns computed at a
given time horizon occurring above a large threshold is
well described by a power law function which is analogous
to the Omori law of geophysics.
The presence of a power law relaxation seems to be
a common behavior observed in a wide range of complex
systems. One possibility for this common occurrence
is that the Omori law is a phenomenological manifestation
of underlying common microscopic mechanisms
governing the dynamics of complex systems after an extreme
event. An example of such mechanisms has been
proposed to model the magnetization relaxation in spin
glasses where it has been shown that the presence of
many metastable states whose lifetime are distributed
according to a broad, power law distribution implies a
power law decay of the magnetization during aging [2].
Acknowledgments
Authors wish to thank INFM, ASI, MIUR and MIURFIRB
research projects for financial support. F.L. thanks
Vittorio Loreto for introducing him to the Omori law.
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0 20 40 60 80 100
t (trading day)
−0.01
−0.005
0
0.005
0.01
r(t)
FIG. 1: One-minute change of the natural logarithm of the
Standard and Poor’s 500 index during the 100 trading day
time period immediately after the Black Monday financial
crash (20 October 1987 - 11 March 1998). A decrease of
the typical scale of the stochastic process (volatility in the
financial literature) is manifest making the stochastic process
non-stationary.
5
0 20 40 60
t (trading day)
0
100
200
300
N(t)
FIG. 2: Cumulative number
N(t) of the number of times |r(t)|
is exceeding a threshold
ℓ during the 60 trading days immediately
after the Black Monday financial crash. ¿From top to
bottom we show the curves for values of
ℓ equal to 4σ, 5σ, 6σ
and 7
σ, respectively. The parameter σ is the standard deviation
of the process
r(t) computed over the entire investigated
period. The dashed lines are best fits of equation (2).
0 20 40
t (trading day)
0
50
100
N(t)
0 20 40 60
t (trading day)
a b
FIG. 3: Cumulative number
N(t) of the number of times |r(t)|
is exceeding the threshold
ℓ during the 60 trading days immediately
after (a) the 27 October 1997 and (b) the 31 August
1998 financial crashes. In both panels, from top to bottom
we show the curves for values of
ℓ equal to 4σ, 5σ, 6σ and 7σ,
respectively. The parameter
σ is the standard deviation of
the process
r(t) computed over the entire investigated period.
The dashed lines are best fits of equation (2).
6
0 20 40 60
t (trading day)
0
500
1000
N(t)
4 8 16
l
10
−2
10
−1
h
FIG. 4: Cumulative number
N(t) of the number of times
|
rp(t)| is exceeding a threshold ℓ. The data refer to the S&P
500 index just after the 1987 crash.
N(t) is computed for
different values of the threshold
ℓ ranging from 4 to 13. A
linear behavior of
N(t) is observed for all values of ℓ. In the
inset, we show the values of the slope
η as a function of ℓ in
a log-log plot. These values are computed by performing a
best linear fitting of
N(t). The continuous line is the best fit
of
η(ℓ) with a power law behavior. The best fitting exponent
is 3.14.
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