The geometry of the The geometry of the marketmarket

ContentsContents1 The Market Geometry

Market metricPrincipal directionsEffective dimensionsA measure of structureNew structure after 1997

2 Market networksNetworks in and outside crises

Connection intensities3 Market shape: precursor of endogeneous crisis and return to

normalityGeometrical volumeSkewness and kurtosis

The metric

((FromFrom thethe EuclideanEuclidean distancedistance ofof thethe timetime seriesseries))

dd

IBMIBM

GEGE

)1(2 ijij cd −= (Mantegna & Stanley, 1999)

))(log())(log()( 1 kpkpkr tt −−= p: pricer: return

Market space1. The N stock coordinates are computed from the

distances. Embedding space dimension = N-12. Find the center of mass R=Σ m(k) x(k) / Σ m(k)3. Find coordinates in the center of mass system

y(k) = x(k) - R4.4. ConstructConstruct thethe inertialinertial tensor tensor TTijij ==ΣΣ m(km(k) ) yyii(k(k) ) yyjj(k(k))5.5. FindFind thethe eigenvectorseigenvectors andand eigenvalueseigenvalues ofof T : (T : (λλii,e,eii))6.6. Compare Compare withwith thethe samesame constructionconstruction for for permutedpermuted

datadata

1 - The Market Geometry

What is the number of dimensions of the manifoldcontaining the systematic components of all stocks ?

S&P500 S&P500 andand DowDow JonesJones, , dailydaily datadata

35 stocks, 10 years 70 stocks, 10 years249 stocks, 33 years 253 stocks, 35 years253 stocks, 22 years 424 stocks, 10 years

InIn allall cases: cases: 6 dimensions are enough !

Geometria do Mercado Financeiro

TheThe systematicsystematic componentcomponentisis containedcontained inin f f dimensionsdimensions

OrderedOrdered andand normalizednormalized eigenvalueseigenvalues

λλ+(1+(1--λλ*)*)

λ : actualλ* : random

Geometria do Mercado FinanceiroEffective dimensions

0 10 20 300.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11998-2008 424 Stocks

λ +

(1- λ

*)

0 10 20 300.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ +

(1- λ

*)

1988-2008 253 Stocks

Redes de Mercado

Shape changes in the market space

Shape and crises

-0.5

0

0.5

-0.5

0

0.5-0.5

0

0.5

Jul.1998 to Jul.2000

Uti IT Fin Heal Cons Ind Ene

“Spherical” marketTypical in all projections

for “surrogate data” or in “business-as-usual”

periods

During crises there isdistortion and reduction

of volume

Redes de MercadoShape and crises

-0.5

0

0.5-0.5

0

0.5

-0.5

0

0.5

Mar.1998 to Mar.2008

Uti IT Fin Heal Cons Ind Ene

-0.500.5

-0.500.5

-0.5

0

0.5

2001 253 stocks

e2

e3

-0.5 0 0.5-0.500.5

-0.5

0

0.5

e1

2007 253 stocks

e3

e2

-0.5 00.5

-0.500.5

-0.5

0

0.5

e1

1992 253 stocks

e3

e2

-0.50

0.5

-0.500.5

-0.5

0

0.5

e1

Surrogate Data 253 stocks

e3

e2

Geometria do Mercado FinanceiroA structure index

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

6

11

)()('

i t

tt i

iSλλ

Structure index

After 1997 deviationsfrom sphericity become

more frequent.

What happened ?

λ : actualλ’: random

17/04/71 07/10/76 30/03/82 20/09/87 12/03/93 02/09/98 23/02/04 15/08/090

10

20

30

40

50230 stocks

S

88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 080

5

10

15

20253 stocks A new regime after 1997

SS

>5

Redes de MercadoMarket networks

MarketMarket networksnetworks: : FromFrom a a fullyfully connectedconnectednetworknetwork to a to a sparsesparse oneone

1. minimal spanning tree2. LD6 = smallest distance in

R6 that guarantees the network connectivity

3. hierarchical clustering

02

12

,,6

,,6

6

6

=⇒>

=⇒≤

jiDji

jiDji

bL

d

bL

d

Aug2000

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

Few connections in normal periods

MarketMarket networksnetworks

The economic agentsare much more correlated duringmarket crises

Redes de MercadoMarket networks

Jan2008

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

An increased number of connections, mostly inside sectors

Redes de MercadoMarket networks

Sep2001

28

Energy

73

Industrial

114

Consumer

44

Health

70

Financial

56

Technology

39

Utilities

Redes de MercadoMarket networks

Another structure indexAnother structure index

Ratio of weak and

strong links

1997 1998 2000 2001 2002 2004 2005 2006 20080

0.5

1

1.5424 stocks From Jan.1998 to Mar.2008

R

1997 1998 2000 2001 2002 2004 2005 2006 2008

R>0

.5

∑

∑

>

≤=

66

66

),(

6

),(

6

),(

),(

Dt

Dt

Ljidt

Ljidt

tjid

jid

R

SynchronizationSynchronizationRedes de MercadoSynchronization

Sep2001

Uti IT Fin Heal Cons Ind Ene

Mar1998otherwises

Ljids

i

Dti

02

),(166

=

∃⇔= ≤

The state depends on the existence of a strong link inside or outside the sector

Redes de MercadoTopologiaSynchronization

Dec2007

Jan2008

Feb2008

Uti IT Fin Heal Cons Ind Ene

Aug2000

Sep2000

Redes de MercadoMarket shape as a precursor of crises and indicator of return to normality

-- CharacterizingCharacterizing shapeshape::-- Volume, Volume, asymmetryasymmetry andand tailstails-- MardiaMardia’’ss SkewnessSkewness KK33 andand KurtosisKurtosis KK44

-- TheThe volume volume isis thethe productproduct ofof thethe firstfirst 6 6 eigenvaueseigenvaues-- KK33 andand KK44 are are computedcomputed afterafter thethe dimensionsdimensions are are

renormalizedrenormalized to to havehave thethe samesame volumevolume

( ) ( ){ }∑∑= =

− −−=n

i

n

jji XXSXX

nK

1 1

31'

231

( ) ( ){ }∑=

− −−=n

iii XXSXXn

K1

21'

41

Redes de MercadoMarket shape as a precursor of crises and indicator of return to normality

07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97

102

K4

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

10-3

V

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 06/11/97850

900

950

1000

PS&

P500

28/08/97 07/09/97 17/09/97 27/09/97 07/10/97 17/10/97 27/10/97 06/11/97 16/11/97100

101

102

K3

w=8 2nd Black Monday: 27/10/97

Redes de MercadoMarket shape as a precursor of crises and indicator of return to normality

30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/8710

−6

10−5

10−4

10−3

10−2

V

w=8 Black Monday: 19/10/1987

30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/87200

250

300

350

PS

&P

500

30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/8710

0

101

102

K3

w=8 Black Monday: 19/10/1987

30/09/87 10/10/87 20/10/87 30/10/87 09/11/87 19/11/87 29/11/87 09/12/87 19/12/8710

1

102

103

K4

w=8 Black Monday: 19/10/1987

- 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 de MercadoMarket shape as a precursor of crises and indicator of return to normality

- 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

Redes de MercadoReferences