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must01 system relaxation market

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) tp 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)1p τ1p]/(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 × 104. 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

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 = 4σ where

σ

is again the unconditional one-minute volatility in the

considered periods. We estimate

σ = 4.54 × 104 during

the period from 28 October 1997 to 23 January 1998

and

σ = 6.09 × 104 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

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

observed at time t is

n

(t) 2 Z +1



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 , 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

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) tβ, 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

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) = c1tβ + c2 in the 60 days after each considered

market crash. We check that the relation

c1tβ >> c2 is

verified in the investigated period. The best estimation

of

c1 and c2 are 6.3 104 and 2.8 106 for the 1987 crash,

5

.1 104 and 4.3 105 for the 1997 crash and 4.4 104

and 1

.0 104 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

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(). We determine η with a best linear fit

of

N(t) for each value of . 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 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

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

during the 60 trading days immediately

after the Black Monday financial crash. ¿From top to

bottom we show the curves for values of

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

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

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 . The data refer to the S&P

500 index just after the 1987 crash.

N(t) is computed for

different values of the threshold

ranging from 4 to 13. A

linear behavior of

N(t) is observed for all values of . In the

inset, we show the values of the slope

η as a function of 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() with a power law behavior. The best fitting exponent

is 3.14.

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