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and artificial financial markets Minimal spanning tree ... · 0.35 0.55 0.75 0.95 1.15 1.35 d< i,j....
Transcript of and artificial financial markets Minimal spanning tree ... · 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 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)
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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
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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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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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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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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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Minimal spanning tree networks in real and artificial markets
Associated volatility MST
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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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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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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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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
AIT
UCM
GTE
BELBHI HAL
SLB
USB
JPM
WFCONE
BACMER
FLR
AXP
JNJ
MRK
BMY
CL
PG
MOB
ARCHM
CHVPEP
KO
GM
F
AIG
CI
HWP
IBM
UIS
BCC CHA
IP
WY
NSM TXN
INTC
ORCL
CSCO
SUNW
MSFT
GE
XON
∆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