The geometry of the market

The geometry of the
market
Contents
1 The Market Geometry
Market metric
Principal directions
Effective dimensions
A measure of structure
New structure after 1997
2 Market networks
Networks in and outside crises
Connection intensities
3 Market shape: precursor of endogeneous crisis and return to
normality
Geometrical volume
Skewness and kurtosis
1 - The Market Geometry
r: return
r ( k ) = log( p t ( k )) − log( p t −1 ( k ))
p: price
The metric
(From the Euclidean distance of the time series)
dij = 2(1 − cij )
(Mantegna & Stanley, 1999)
Market space
1.
2.
3.
4.
5.
6.
The N stock coordinates are computed from the
distances. Embedding space dimension = N-1
Find the center of mass R=Σ m(k) x(k) / Σ m(k)
Find coordinates in the center of mass system
y(k) = x(k) - R
Construct the inertial tensor Tij =Σ m(k) yi(k) yj(k)
Find the eigenvectors and eigenvalues of T : (λi,ei)
Compare with the same construction for permuted
data
IBM
d
GE
Geometria
Effective
do Mercado
dimensions
Financeiro
What is the number of dimensions of the manifold
containing the systematic components of all stocks ?
S&P500 and Dow Jones, daily data
z 35 stocks, 10 years
z 70 stocks, 10 years
z 249 stocks, 33 years
z 253 stocks, 35 years
z 253 stocks, 22 years
z 424 stocks, 10 years
In all cases: 6 dimensions are enough !
IBM
The systematic component
is contained in f dimensions << N
d
Dimensions are not necessarily sectors.
Impact on the choice of portfolios
GE
Geometria
Effective
do Mercado
dimensions
Financeiro
Ordered and normalized eigenvalues
1988-2008
λ + (1- λ*)
λ : actual
λ* : random
253 Stocks
1998-2008 424 Stocks
1
1
0.9
0.9
0.8
0.8
0.7
0.7
λ + (1- λ*)
λ+(1-λ*)
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0
10
20
30
0.2
0
10
20
30
Redes
Shape de
and
Mercado
crises
Shape changes in the market space
Jul.1998 to Jul.2000
“Spherical” market
Typical in all projections
for “surrogate data” or in
“business-as-usual”
periods
0.5
0
During crises there is
distortion and reduction
of volume
-0.5
0.5
0.5
0
0
-0.5
Uti
IT
Fin
-0.5
Heal
Cons
Ind
Ene
Redes
Shape de
and
Mercado
crises
Surrogate Data 253 stocks
1992 253 stocks
0.5
e2
e2
0.5
0
0
-0.5
-0.5
0.5
0.5
0
0.5
0
-0.5
e1
e3 -0.5
2001 253 stocks
0
e1
0.5
0.5
e2
e3
-0.5
2007 253 stocks
0.5
0
Mar.1998 to Mar.2008
0
-0.5
-0.5
0.5
0
-0.5
e3
0
e2
-0.5
-0.5
0
0.5
0.5
0 -0.5
e3
0.5
-0.5
e1
0
0.5
0
-0.5
-0.5
0.5
0
Uti
IT
0
Fin
0.5 Heal
-0.5
Cons
Ind
Ene
Geometria
A structure
do Mercado
index
Financeiro
230 stocks
50
40
Structure index
30
S
λ : actual
λ’: random
20
10
⎞
⎛ λt ' ( i )
⎟
⎜
St =
⎜ λ (i ) − 1 ⎟
⎠
i =1 ⎝ t
6
∑
0
17/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/09
253 stocks
A new regime after 1997
20
What happened ?
S
15
10
5
0
S>5
After 1997 deviations
from sphericity become
more frequent.
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08
Redes
Marketde
networks
Mercado
Market networks: From a fully connected
network to a sparse one
1. minimal spanning tree
Aug2000
28
2. LD6 = smallest distance in
R6 that guarantees the
network connectivity
3. hierarchical clustering
Industrial
73
Health
44
d
6
i, j
≤
d 6i, j >
LD 6
2
LD 6
2
⇒ bi , j = 1
Energy
Technology
56
Consumer
114
Financial
70
Utilities
39
⇒ bi , j = 0
Few connections in normal periods
Redes
Marketdenetworks
Mercado
Market networks
Jan2008
Energy
28
The economic agents
are much more
correlated during
market crises
Industrial
73
Health
44
Technology
56
Consumer
114
Financial
70
Utilities
39
An increased number of connections, mostly
inside sectors
Redes
Marketdenetworks
Mercado
Sep2001
Energy
28
Industrial
73
Health
44
Technology
56
Consumer
114
Financial
70
Utilities
39
Redes
Marketde
networks
Mercado
Another structure index
424 stocks
From Jan.1998 to Mar.2008
1.5
Ratio of weak and
strong links
R
1
Rt =
d t6 ( i , j ) ≤ L
D6
∑ d t6 (i , j )
d t6 ( i ,
j)> L
0.5
0
1997
1998
2000
2001
2002
2004
2005
2006
2008
1998
2000
2001
2002
2004
2005
2006
2008
D6
R>0.5
∑
d t6 ( i , j )
1997
Redes
Synchronization
de Mercado
Synchronization
si = 1 ⇔ ∃
d t6 ( i ,
j )≤
si = 0 otherwise
LD 6
2
Mar1998
The state
depends on
the existence
of a strong
link inside or
Sep2001
outside the
sector
Uti
IT
Fin
Heal
Cons
Ind
Ene
Redes
Synchronization
Topologia
de Mercado
Aug2000
Dec2007
Sep2000
Jan2008
Feb2008
Uti
IT
Fin
Heal
Cons
Ind
Ene
Market shape as a precursor
of crises
and indicator of return to normality
Redes
de Mercado
-
Characterizing shape:
Volume, asymmetry and tails
Mardia’s Skewness K3 and Kurtosis K4
K3 =
1
n
2
∑∑ {(X
n
n
i =1 j =1
n
{(
) (X
'
i
−X S
) (
−1
j
' −1
1
K4 = ∑ X i − X S X i − X
n i =1
-
−X
)}
3
)}
2
The volume is the product of the first 6 eigenvaues
K3 and K4 are computed after the dimensions are
renormalized to have the same volume
Market shape as a precursor
of crises
and indicator of return to normality
Redes
de Mercado
2
w=8 2nd Black Monday: 27/10/97
10
w=8 2nd Black Monday: 27/10/97
K3
-3
1
10
V
10
07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97
0
10
28/08/97 07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97 16/11/97
1000
950
900
2
10
850
07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97
K4
PS&P500
w=8 2nd Black Monday: 27/10/97
07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97
Market shape as a precursor
of crises
and indicator of return to normality
Redes
de Mercado
2
w=8 Black Monday: 19/10/1987
10
−2
w=8 Black Monday: 19/10/1987
K
3
10
−3
1
10
V
10
−4
10
−5
10
−6
10
30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/87
0
10
30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/87
350
3
w=8 Black Monday: 19/10/1987
10
PS&P500
300
250
4
30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/87
K
200
2
10
1
10
30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/87
Market shape as a precursor
of crises
and indicator of return to normality
Redes
de Mercado
- From some of the cases studied so far it looks as if the
shape of the market might provide some precursors of
impending crises and would also be an indicator of when
normality has returned.
- However this is just work in progress. More to come.
Redes
References
de Mercado
- RVM., T. Araújo and F. Louçã; Reconstructing an Economic Space
from a Market Metric, Physica A 323 (2003) 625-648
- T. Araújo and F. Louçã; The Geometry of Crashes: A Measure of
the Dynamics of Stock Market Crises, Quantitative Finance 7 (2007)
63-74
- T. Araújo and F. Louçã; The Seismography of Crashes in Financial
Markets, Physics Letters A 372 (2008) 429-434
- T. Araújo and F. Louçã; Tribes under Threat - The Collective
Behavior of Firms During Stock Market Crises, Working paper ISEG
2008/28, to appear in Int. Journ. of Industrial and Corporate Change
- T. Araújo and F. Louçã; Trouble Ahead - The Subprime Crisis as
Evidence of a New Regime in the Stock Market, Working paper
ISEG 2008/26, Economics Letters