This paper investigates the dynamics of stocks in the S&P500 for the last 33 years, considering the population of all companies present in the index for the whole period. Using a stochastic geometry tech- nique and defining a robust index of the dynamics of the market struc- ture, which is able to provide information about the intensity of the crises, the paper proposes a seismographic classification of the crashes that occurred during the period. The index is used in order to inves- tigate and to classify the impact of the twelve crashes between July 1973 and March 2006 and to discuss the available evidence of change of structure after the fin de sicle.Keywords: financial markets; stochastic geometry; complexity; market spaces; market structures.

Francisco Louçã

Tanya Araujo

English

Research Papers in Economics

 
 
 
  
 
 
Tanya Araújo & Francisco Louçã 
 
The Seismography of Crashes in Financial Markets 
 
 
 
 
 
 
WP 005/2007/DE/UECE 
_________________________________________________________ 
Departament of Economics 
WORKING PAPERS 
 
ISSN Nº 0874-4548 
School of Economics and Management 
TECHNICAL UNIVERSITY OF LISBON The Seismography of Crashes in Financial
Markets
Tanya Ara¶ ujo ¤ and Francisco Lou» c~ a y
Research Unit on Complexity in Economics (UECE)
ISEG, Universidade T¶ ecnica de Lisboa
January 19, 2007
Abstract
This paper investigates the dynamics of stocks in the S&P500 for
the last 33 years, considering the population of all companies present
in the index for the whole period. Using a stochastic geometry tech-
nique and de¯ning a robust index of the dynamics of the market struc-
ture, which is able to provide information about the intensity of the
crises, the paper proposes a seismographic classi¯cation of the crashes
that occurred during the period. The index is used in order to inves-
tigate and to classify the impact of the twelve crashes between July
1973 and March 2006 and to discuss the available evidence of change
of structure after the ¯n de sicle.
Keywords: ¯nancial markets, stochastic geometry, complexity, market
spaces, market structures.
JEL C0, G1
¤tanya@iseg.utl.pt
y°ouc@iseg.utl.pt
11 Introduction
The nature of ¯nancial crashes has been intensely discussed for long. Re-
cently some new methods are being used to analyze and describe the dy-
namics of changes in complex markets, based on di®erent contributions from
econophysics. This paper recurs to such methods and the available infor-
mation on the trajectories of returns of dominating ¯rms for three decades
in order to propose new measures and a classi¯cation of the intensity of the
crises. As a measure for these perturbations is de¯ned, our investigation
follows previous papers comparing them to "economic earthquakes" [1, 2, 3,
4] and, in particular, the suggestion that the histogram of price changes for
any stock is "the analog of the Gutenberg-Richter histogram of earthquake
magnitude" [1], developing a new strategy for the quanti¯cation and quali¯-
cation of these extreme events. Inquiries into the statistical properties of this
distribution suggest the existence of a hierarchical organization expressed as
scale invariance over the history of the values of a control parameter [1]. Sec-
tion 2 presents the method and measures, whereas Section 3 summarize the
results.
2 Method and Measures
The stochastic geometry strategy is simply stated in the following terms [5]:
1. Pick a representative set of N stocks and their historical data of re-
turns over some time interval and, from the returns data, using an
appropriate metric, compute the matrix of distances between the N
stocks.
The problem is reduced to an embedding problem in which, given a set
of distances between points, one asks what is the smallest manifold that
contains the set. Given a graph G and an allowed distortion there are
algorithmic techniques to map the graph vertices to a normed space in
such a way that distances between the vertices of G match the distances
between their geometric images, up to the allowed distortion. However,
these techniques are not directly applicable to our problem because in
the distances between assets, computed from their return °uctuations,
there are systematic and unsystematic contributions. Therefore, to
2extract relevant information from the market, we have somehow to
separate these two e®ects. The following stochastic geometry technique
is used:
2. From the matrix of distances compute the coordinates for the N stocks
in an Euclidean space of dimension smaller than N and then apply the
standard analysis of reduction of the coordinates to the center of mass
and compute the eigenvectors of the inertial tensor.
3. Apply the same technique to surrogate data, namely to data obtained
by independent time permutation for each stock and compare these
eigenvalues with those obtained in (2), in order to identify the directions
for which the eigenvalues are signi¯cantly di®erent as being the market
characteristic dimensions.
In so doing, we are attempting to identify the empirically constructed
variables that drive the market and the number of surviving eigenvalues
is the e®ective dimension of this economic space.
4. From the eigenvalues of order smaller than the number of characteristic
dimensions, compute the di®erence between eigenvalues in (2) with
those in (3). The normalized sum of those di®erences is the index S,
which measures the evolution of the distortion e®ect in the shape of
the market space.
For both surrogate and actual data, the sorted eigenvalues, from large to
small, decrease with their order. In the surrogate case, the amount of decrease
is linear in the order number, proving that the directions are being extracted
from a spherical con¯guration. The display of a uniform and smooth decrease
in the values of the sorted eigenvalues is characteristic of random cases and is
also experimentally observed when the market space is built from historical
data corresponding to a period of business as usual.
Considering the lack of uniformity among the market e®ective dimensions
we are able to characterize the extent to which crashes act di®erently on
speci¯c directions, causing changes in the shape of the market space. Looking
for relevant distortions in the shape of the S&P500 market space through
the last 33 years, we found that amongst the highest values of the index
are those computed for some important dates, as 19th October 1987, 27th
October 1997 and 11th September 2001.
From the returns for each stock
3r(k) = log(pt(k)) ¡ log(pt¡1(k)) (1)
a normalized vector
¡ ! ½ (k) =
¡ ! r (k)¡h¡ ! r (k)i
q
n(hr2(k)i¡hr(k)i2)
(2)
is de¯ned, where n is the number of components (number of time labels)
in the vector ¡ ! ½ . With this vector one de¯nes the distance between the stocks
k and l by the Euclidian distance of the normalized vectors.
dij =
p
2(1 ¡ Cij) = k¡ ! ½ (k) ¡ ¡ ! ½ (l)k (3)
as proposed in [6], with Cijbeing the correlation coe±cient of the returns
r(i),r(j) .
The fact that is a properly de¯ned distance gives a meaning to geometric
notions and geometric tools in the study of the market. Given that set
of distances between points, the question now is reduced to an embedding
problem: one asks what is the smallest manifold that contains the set. If the
proportion of systematic information present in correlations between stocks
is small, then the corresponding manifold will be a low-dimensional entity.
The following stochastic geometry technique was used for this purpose.
2.1 The stochastic geometry technique
After the distances ( dij) are calculated for the set of N stocks, they are
embedded in RD, where D < n, with coordinates ¡ ! x (k). The center of mass
¡ !
R is computed and coordinates reduced to the center of mass.
¡ !
R =
§¡ !
kx(k)
k (4)
¡ ! y (k) = ¡ ! x (k) ¡
¡ !
R (5)
and the inertial tensor
Tij = §k¡ ! yi(k)¡ ! y j(k) (6)
is diagonalized to obtain the set of normalized eigenvectors f ¸i;¡ ! e ig.
The eigenvectors ¡ ! e i de¯ne the characteristic directions of the set of stocks.
4The characteristic directions correspond to the eigenvalues ( ¸i) that are
clearly di®erent from those obtained from surrogate data. They de¯ne a
reduced subspace of dimension e, which carries the systematic information
related to the market correlation structure. In order to improve the decision
criterion on how many eigenvalues are clearly di®erent from those obtained
from surrogate data, a normalized di®erence ¿ is computed:
¿(i) = ¸(i) + 1 ¡ ¸0(i) (7)
and the signi¯cantly di®erent eigenvalues are those to which ¿(i) > 1=2
2.2 Index of the market structure
As market spaces can be described as low dimension objects, the geometric
analysis is able to provide crucial information about their dynamics. In
previous papers, we developed di®erent applications of this technique, namely
for the identi¯cation of periods of stasis and of mutation or crashes. Indeed,
market spaces tend to contract during crises along their e®ective dimensions,
but each crisis may act di®erently on speci¯c dimensions. It is in order
to capture that distortion, namely the lack of uniformity along the market
e®ective dimensions, that we de¯ned the market structure index, S [5]. Since
the largest f eigenvalues de¯ne the e®ective dimensionality of the economic
space, at time t, we compute S as:
St = §
f
i=1
¸t(i)¡¸0
t(i)
¸0
t(i) = §
f
i=1
¸t(i)
¸0
t(i) ¡ 1 (8)
where ¸t(1);¸t(2);:::;¸t(f) are the largest f eigenvalues of the market
space and ¸0
t(1);¸0
t(2);:::;¸0
t(f) are the largest f eigenvalues obtained from
surrogate data, namely from data obtained by independent time permutation
of each stock. In computing S, at a given time t, both ¸t and ¸0
t are obtained
over the same time window and for the same set of stocks.
3 Results and Discussion
Results were computed using actual daily returns data and comparing them
to surrogate data that are generated by permuting each stock (one-day return
data) randomly in time. As each stock is independently permuted, time cor-
relations among stocks disappear while the resulting surrogate data preserve
the mean and the variance that characterize actual data.
5The set of actual data consists in 231 stocks present in S&P500 from
July 1973 to March 2006, considering all the surviving ¯rms for the whole
period. Although we acknowledge that this population does not necessarily
represent the behavior of the whole economy, we consider the information
useful enough to provide information on trends of the dynamics of market,
as it includes a large part of the winners after a long period of competition.
The next section proposes a classi¯cation of these crashes according to the
value of S and using an inspiration from seismography.
3.1 A seismographic classi¯cation
An approximate value of the index S for the 1987 Crash, the Black Monday
(BM), is taken as the higher value of a scale from 8BM down to 1BM, or
S from 40 to 5. Market perturbations measuring less than 1BM are not
considered to qualify as market crashes. Although this does not suggest
any comparability or similarity of causes between earthquakes and ¯nancial
crises, this procedure for classi¯cation and quanti¯cation in seismography
[7] suggests a way of describing perturbations in speculative markets, since
both kinds of shocks tend to occur in all magnitudes and may be described
according to power laws.
3.2 The dynamics of crashes
The results are presented in Fig.1, where the plot shows the daily values of
S for the 33 years period. We used a time moving window of 16 days on a
market space including the 231 stocks. The highest values of S are marked
on the plot.
The highest peaks are identi¯ed and correspond to the following crashes
(chronological order):
1. October 1978, Iranian crisis
2. September 1979, Iranian crisis and second oil shock
3. August 1982, general recession
4. October 1987, the Black Monday
5. October 1989, a US crash
617/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/09
0
5
10
15
20
25
30
35
40
230 stocks
S
Figure 1: The evolution of the index S measuring the evolution of the S&P500
structure for the surviving ¯rms for 1973-2006
6. October 1997, the Asian crash
7. October 1998, the Russian crash
8. April 1999, the Japan crash
9. April 2000, Nasdaq
10. Dec.2000/Jan.2001, crashes in Argentina and Turkey
11. April 2001, the start of a world recession
12. September 2001, the terrorist attack against New York
13. July 2003, general recession
Table 1 shows the same 13 peaks and the corresponding values of Smax
and BM. The chronological order was replaced and the events were ordered
according to the values of Smax and BM.
7Table 1: Ranking of the crises according to Smax and BM
|||||||||||||||||||{
Ranking Date Smax BM
1 October 1987 38.6 8
2 Dec.2000/Jan.2001 18.3 3.8
3 October 1989 10.6 2.2
4 Mar/April 2001 10.3 2.13
5 July 2003 9.53 2
6 October 1998 9.2 1.95
7 April 1999 9 1.9
8 October 1997 8.36 1.73
9 September 2001 7.95 1.65
10 April 2000 7.55 1.6
11 August 1982 7.5 1.55
12 October 1979 7.22 1.50
13 October 1978 6.60 1.37
|||||||||||||||||||{
Along the whole period (33 years), the accumulated value of S (AS) is
computed and the successive cumulative histories are presented in Table 2,
the curve being drawn in Figure 2. In that ¯gure, we measure the slope of
the curve that better ¯ts AS for each year period, the slope quantifying each
successively longer history. The following table shows the values of the slope
for the histories obtained for each year.
Table 2: Slopes of the AS yearly ¯t
|||||||||||||||||||{
Year Slope Year Slope Year Slope
1973 1.5 1974 2.1 1975 1.5
1976 1.2 1977 1.3 1978 2
1979 2 1980 1.5 1981 1.3
1982 1.7 1983 0.87 1884 1.4
1985 1.1 1986 1.3 1987 3.2
1988 1.4 1989 1.8 1990 1.7
1991 1.2 1992 1.1 1993 1.2
1994 1.1 1995 1.7 1996 1.4
1997 1.8 1998 2.5 1999 2.5
2000 4 2001 4.3 2002 3.2
2003 2.5 2004 2.6 2005 2.5
|||||||||||||||||||{
817/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/09
0
2000
4000
6000
8000
10000
12000
14000
16000
230 stocks
S
Figure 2: Accumulated values of S and their yearly ¯t
3.3 Empirical evidence for the change of structure
Studying the cumulative history of the index of market structure (Figure 2),
it is intuitive that a major change is occurring since around 1997, imposing
a new dynamic structure. This intuition is now investigated by empirical
means.
Figure 3 presents the empirical evidence for the concentration of seisms
in the period after 1997.
Any of these crashes ever compares to the one of 1987. Indeed, there are
some sound reasons to suspect the existence of di®erent dynamics through
time in the evolution of ¯nancial markets. The 1987 crash singles out as the
deepest general crisis, incomparable to the following ones. The subsequent
years witnessed the response to that shock through the construction of new
methods of regulation. For a period, not only no large seisms occurred, but
also only smaller °uctuations are detectable. Yet, since the second half of the
nineties, we obtain higher average values of the index S and a concentration
of a number of crashes and their replica. This di®erence in the empirically
917/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/09
−100
0
100
200
300
400
230 stocks
Σ
S
>
1
B
M
17/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/09
0
0.2
0.4
0.6
0.8
1
S
>
1
B
M
Figure 3: Yearly ¯t of the slope of S (above) and Concentration of Crashes
through the period (below)
described evolution suggests that the Clinton period of the "Internet boom"
corresponded to a new structure of the market or to emergence of a new
phase of turbulence in the ¯nancial markets.
Complementary evidence is provided by the computation of the Hurst
exponent for the period under consideration: whereas for 1973-1997 the ex-
ponent is H = 0:7, for the time interval 1997-2006 it is H = 0:87, indicating
stronger evidence for long term memory. Figure 4 shows the value of the
Hurst exponents for both periods, compared to the values obtained from time
permuted data, suggesting the presence and the evolution of some structure.
For the recent period, this structure is not under the impact of larger seisms
(such as the Black Monday) although it is de¯ned by frequent and important
seismic activity.
1010
0
10
1
10
2 10
−1
10
0
10
1
10
2
∆
t
S
(
t
+
∆
t
)
−
S
(
t
)
Hurst exponents:  Time permuted(o)  1973−1997(.)  1997−2006(+)
Figure 4: Hurst exponents
References
[1] Stanley, EH, Amaral, LAN, Gopikrishnan, P, Ivanov, P, Keitt, TH, Pel-
rou, V, 2000, Scale Invariance and Universality: Organizing Principles
in Complex Systems, Physica A 281, 60-8
[2] Stanley, EH, Amaral, LAN, Gopikrishnan, P, Plerou, V, 2000, Scale
Invariance and Universality of Economic Fluctuations, Physica A 283,
31-41
[3] Stanley, EH, Amaral, LAN, Gabaix, X, Gopikrishnan, P, Pelrou, V,
2001, Quantifying Economic Fluctuations, Physica A 302, 126-37
[4] Stanley, EH, 2003, Statistical Physics and Economic Fluctuations: Do
Outliers Exist?, 2003, Physica A 318 279-92
[5] Araujo T, Lou» c~ a, F, 2007, The Geometry of Crashes - A Measure of the
Dynamics of Stock Market Crises, Quantitative Finance, forthcoming.
11[6] Mantegna, RN, Stanley, HE, 2000, An Introduction to Econophysics:
Correlations and Complexity in Finance, Cambridge, Cambridge Uni-
versity Press
[7] Gutenberg B, Richter, CF, 1954, Seismicity of the Earth and Associated
Phenomenon, Princeton, Princeton university Press
[8] Mandelbrot, B, 1983, The Fractal Geometry of Nature, New York, Free-
man
[9] Sornette, D, Knopo®, L, Kagan, Y, Vanneste, C, 1996, Rank-ordering
Statistics of Extreme Events: Applications to the Distribution of Large
Earthquakes, Journal of Geophysical Research Solid Earth 101 (B6),
13833-93
[10] Pacheco, JF, Scholz, CH, Sukes, LR, 1992, Changes in Frequency-Size
Relationship from Small to Large Earthquakes, Nature 355, 71-3
[11] Mandelbrot, B, 1963, The Variation of Certain Speculative Prices, Jour-
nal of Business 36, 394-419
12

Changes in Frequency-Size Relationship from Small to Large Earthquakes,

Rank-ordering Statistics of Extreme Events: Applications to the Distribution of Large Earthquakes,

Scale Invariance and Universality: Organizing Principles in

Statistical Physics and Economic Fluctuations: Do Outliers Exist?,

The Fractal Geometry of Nature,

The Geometry of Crashes - A Measure of the Dynamics of Stock Market Crises, Quantitative Finance,

The Variation of Certain Speculative Prices,

The Seismography of Crashes in Financial Markets

http://pascal.iseg.utl.pt/~depeco/wp/wp052007.pdf

The Seismography of Crashes in Financial Markets

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics