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Minimal spanning tree networks in real and artificial markets

Minimal spanning tree networks in realand artificial financial markets

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Minimal spanning tree networks in real and artificial markets

Overview

• Hierarchical structures can be detected in financial portfolios;

• One filtering procedure is a correlation-based clustering procedure;

• I discuss examples observed in the equity market and for financial markets located around the world;

• Artificial market models;

• Topology of MSTs in empirical data and market models;

• Conclusion.

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Minimal spanning tree networks in real and artificial markets

The Collaboration

• Rosario N. Mantegna

• Fabrizio Lillo

• Giovanni Bonanno

• Guido Caldarelli

• Salvatore Miccichè

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Minimal spanning tree networks in real and artificial markets

Hierarchical treeA meaningful taxonomy can be extracted from returns time series by using a correlation-based clustering procedure†:

A hierarchical tree is obtained

†R.N. Mantegna, Eur. Phys. J. B 11, 193-197 (1999)

0.35

0.55

0.75

0.95

1.15

1.35

d<

i,j

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Minimal spanning tree networks in real and artificial markets

Minimal spanning tree

Together with a minimal spanning tree

This is the set of the 100 most capitalized US stocks in 1998.

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Minimal spanning tree networks in real and artificial markets

Cross-correlation

Cross-correlation between stock returns are well-known

2222jjii

jijiij

SSSS

SSSS

−−

−=ρ

)()1(ln)(ln tRtYtYS iiii ≅−−≡

They may be quantified by the correlationcoefficient ρij

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Minimal spanning tree networks in real and artificial markets

A metric similarity measure

10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

10.00

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90.00

100.00

0.60

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0.90

1.00

1.10

1.20

1.30

1.40

The function

( )ijijd ρ−= 12

verifies the axioms of a metric distance.

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Minimal spanning tree networks in real and artificial markets

Filtering procedure

The use of the single linkage clustering method using the distance dij as a similarity measure is one possible filtering procedure selectinginformation about the hierarchical structure of the portfolio

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Minimal spanning tree networks in real and artificial markets

The ultrametric distance matrix

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0.600.700.800.901.001.101.201.301.40

This filtering israther severe byretaining only(n-1) correlationcoefficients fromthe original n (n-1)/2

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Minimal spanning tree networks in real and artificial markets

The (ordered) distance matrix

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0.600.700.800.901.001.101.201.301.40

10 20 30 40 50 60 70 80 90 100

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0.600.700.800.901.001.101.201.301.40

A direct comparison shows the amount of informationnot considered after the filtering procedure

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Minimal spanning tree networks in real and artificial markets

The global financial market

1 G. Bonanno, N. Vandewalle and R. N. Mantegna, Physical Review E62, R7615 (2000)

We investigate51 MSCI stockindices both inlocal currency and in US dollars

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Minimal spanning tree networks in real and artificial markets

The global financial market

A hierarchical tree showing regionalrelations among stock indices of major stock exchanges has be detected1

1 G. Bonanno, N. Vandewalle and R. N. Mantegna, Physical Review E62, R7615 (2000)

1 US, Canada and Europe; 2 South-America;3 Japan; and 4 Asia

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Minimal spanning tree networks in real and artificial markets

The global financial market

1996

MSTs show a clear regional clustering

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Minimal spanning tree networks in real and artificial markets

VolatilityThe same approachcan of course be applied also at volatilitytime series1

1 Salvatore Miccichè Giovanni Bonanno, Fabrizio Lillo, Rosario N. Mantegna, Physica A 324, 66-73 (2003)

The set investigated isthe set of 93 highly capitalized stocks tradedat NYSE during thetime period 87-98

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Associated volatility MST

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Minimal spanning tree networks in real and artificial markets

High frequency data† (stocks)† B

onan

no, L

illo,

Man

tegn

a, Q

uant

itativ

e Fi

nanc

e 1,

96-

104

(200

1)

∆t=d/2 ∆t=d

∆t=d/5∆t=d/20

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Minimal spanning tree networks in real and artificial markets

MSTs at different time horizons†

†Bonanno, Lillo, Mantegna, Quantitative Finance 1, 96-104 (2001)

∆t=19 min 30 sec ∆t=6 h 30 min

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Minimal spanning tree networks in real and artificial markets

Topology of MSTs

Topology† of MSTs in

• empirical data;• in an (unrealistic) random marketand• in a (realistic) one-factor model

†Bonanno, Caldarelli, Lillo and Mantegna, cond-mat/0211546, PRE (in press 2003)

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Minimal spanning tree networks in real and artificial markets

Empirical data

The color code refers tothe SIC index:• finance • manufacturing• construction • utilities• wholesale trade • mining• retail trade • services• public administration

1071 stocks continuouslytraded at the NYSE during the period 1987-1998 (3030 tradingdays)

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Empirical data: degree distribution

It is approximatelypower-law decayingfor low k values and a few nodes with very high degreeare present.

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A random (unrealistic) model

( ) ( )ttR ii ε=

εi(t) is a Gaussian zero-mean i.i.d. random variable

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Minimal spanning tree networks in real and artificial markets

MST of a random artificial market

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Topological properties of the random model

Degree distribution

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Theoretical prediction of the random model

Theory developedin M.D.Penrose, The Annals of Probability 24,1903 (1996)

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Eigenvalue spectrum

It may be worth mentioning that the random modeldescribes a significant part of the spectrum ofcorrelation matrix eigenvalues

From Laloux et al, PRL 83, 1467 (1999)

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One-factor model We compare our empirical results with numerical simulations based on the one-factor model

where αi and βi are two real parameters, εi is a zero-mean noise term characterized by a variance equal to σi

2

RM(t) is the market factor. We choose it as the SP 500 index.

In our investigation, we estimate the model parameters and generate an artificial market.

( ) ( ) ( )ttRtR iMiii εβα ++=

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Correlation in the one-factor model

The one-factor model explains more than 85% of the elements of correlation matrix

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MST of a one-factor model

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Topological properties of the one-factor model

Degree distribution

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Topological properties ofMSTs are different in the degree pdf f(k) and areenough to falsifya widespread financial model

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Topological properties are able to falsify the model

Other topologicalquantities, as forexample, thein-degreecomponentf(a) allow us toconclude thesame way

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Can we use our results for predictive purposes?

• Are empirical results suggesting that one-factor model better describes high-frequency returns rather than daily returns?

SOETR

AEP

BNI

NSC

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UCM

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BELBHI HAL

SLB

USB

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WFCONE

BACMER

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∆t19 min30 sec

A preliminaryanswer is

YES!

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Conclusion

•The topology of MSTs obtained by filtering the information present in the time dynamics of returns of an asset portfolio traded in a financial market allows to falsify simple models of markets dynamics.

•Topological properties of MSTs obtained at very short time horizons suggest that one-factor model is more appropriate for intraday time horizons rather than for daily time horizons of stock prices, weekly index returns of stock exchanges and volatility .

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The OCS website

http://lagash.dft.unipa.it