Topological Properties of the Minimal Spanning Tree in Korean and American Stock Markets

0.22 MB·English

by  Cheoljun Eom

Topological Properties of the Minimal Spanning Tree in the Korean and American
Stock Markets
Cheoljun Eom
Division of Business Administration, Pusan National University, Busan 609-735, Korea
∗
Gabjin Oh
NCSL, Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk, 790-784, Korea
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0
Seunghwan Kim
0
NCSL, Department of Physics, Pohang University of Science and Technology, Pohang, Gyeongbuk, 790-784, Korea and
2
Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk, 790-784, Korea

n
a Weinvestigateafactorthatcanaffectthenumberoflinksofaspecificstockinanetworkbetween
J stocks created by the minimal spanning tree (MST) method, by using individual stock data listed

1 on the S&P500 and KOSPI. Among the common factors mentioned in the arbitrage pricing model
1 (APM), widely acknowledged in the financial field, a representative market index is established as
a possible factor. We found that the correlation distribution, ρij, of 400 stocks taken from the
] S&P500 index shows a very similar with that of the Korean stock market and those deviate from
n
the correlation distribution of time series removed a nonlinearity by the surrogate method. We
a
also shows that the degree distribution of the MSTs for both stock markets follows a power-law
-
a distribution with the exponent ζ ∼ 2.1, while the degree distribution of the time series eliminated
t a nonlinearity follows an exponential distribution with the exponent, δ ∼ 0.77. Furthermore the
a
d correlation, ρiM, between the degree k of individual stock, i, and the market index, M, follows
. a power-law distribution, hρiM(k)i ∼ kγ, with the exponent γS&P500 ≈ 0.16 and γKOSPI ≈ 0.14,
s
respectively. Thus, regardless of the markets, the indivisual stocks closely related to the common
c
i factor in the market, the market index, are likely to be located around the center of the network
s betweenstocks,whilethoseweaklyrelatedtothemarketindexarelikelytobeplacedintheoutside.
y
h
PACSnumbers: 89.75.Fb,89.65.Gh,89.75.Hc
p
Keywords: econophysics, stocknetwork,minimalspanningtree
[

2
v I. INTRODUCTION foundthatthisgeneratednetworkbetweenstockshasan
8 economically significant grouping process [6, 7].
6
The studies of the past several years showed that the
0 Recently,researchersfromdiversedisciplinesareshow-
degree distribution of the network created by the MST
2 ing great interest in the topological properties of net-
1 method follows a power-law distribution with the expo-
works. In particular, the network observed in natural
6 nent ξ ≈2 [8, 9]. That is, most individual companies in
and social science shows a variety of properties different
0 thestockmarkethaveasmallnumberoflinkswithother
fromthoseofrandomgraphs[1,2]. Theeconomicworld,
/ stocks,whileafewstockshaveagreatnumberofconnec-
s known as having the most complex structure among
c tions. However the KOSPI200 companies of the Korean
i them, evolves through the nonlinear-interaction of the
s stock market, one of the emerging market, does not fol-
diverse heterogeneous agents. The stock market is a
y low a power-lawdistribution andfor the Americanstock
representative example. The stock prices of individual
h market, S&P500, the relation between market capital-
p companies are formed by a complex evolution process of
izationand|q|,theinfluencestrength(IS),hasapositive
: diverse information generated in the market and these
v correlation, while the KOSPI200 has no correlation [10].
have strong correlationswith eachother by the common
i Moreover,previousresultsshowedthatthenetworkofin-
X factors in the market [3, 4, 5]. In other words, indi-
dividual stocks tends to gather around the companies of
r vidual stocks are connected with each other and compa-
a ahomogeneousindustrialgroup[11,12,13,14]. Further-
nies with the same properties tend to be grouped into a
more, the stocks forming a Markowitz efficient portfolio
community. Toinvestigatetheseproperties,Mantegna.et
in the financial field are almost located at the outside of
al. proposed the minimal spanning tree (MST) method,
the network [14, 15]. However, until recently, studies on
which can be used to observe the grouping process of
the possible factors important in determining the num-
individual stocks transacted in the market, on the basis
ber of linkages with other stocks in the network between
of the correlation of stocks [6]. Mantegna constructed
stocks were insufficient.
the stock network visually using the MST method and
Inordertoinvestigatethetopologicalpropertiesinthe
network of the stock market, we used the MST method
introducedby Mantegnaet al. We alsoconsider the mar-
∗Electronicaddress: [email protected] ketindex intermsofapossiblefactorthatcanaffectthe
2
number of links of a specific stock with other stocks in III. METHODOLOGY
the network created by the MST method. We used the
data of 400 individual companies listed on the S&P500 We make the network by using stocks listed on the
index from January 1993 to May 2005, and 468 individ- S&P500 and KOSPI, respectively, through the MST
ual companies listed on the KOSPI from January 1991 method proposed by Mantegna et al. As the MST
to May 2003. method makes the network based on the correlation of
We found that the correlation distribution, ρij, of the stocks, the cross-correlation of stocks listed on the S&P
400 stocks in S&P500 index shows a very similar with 500andKOSPIstockmarkets,respectively,iscalculated
that of the KOSPI and those deviate from the correla- as follows
tion distribution of time series removed a nonlinearity
by the surrogate method introduced by J. Theiler et al.
hR R i−hR ihR i
[16] for both stock markets. We also found that the de- ρ = i j i j , (2)
ij
gree distribution of the network, like those from previ- (hR2i−hR i2)(hR2i−hR i2)
q i i j j
ous research, follows a power-law distribution with the
exponent ζS&P500,KOSPI ≈ 2.1 for both the Korean and where h.i means the mean value of the whole period and
Americanstockmarkets. Inordertoobservethepossible thecorrelationlieswithintherangeof−1≤ρ ≤+1. If
ij
factorindetermingthedegreekontheMSTnetwork,we ρ is 1, two time series have a complete correlation and
ij
calculatethecross-correlation,ρiM,betweenaindividual if ρij is -1, they have a complete anti-correlation. In the
stock and market index for both stock markets. We also case where ρ is 0, the correlation of two time series is
ij
found that the cross-correlation, ρiM between the mar- 0. On the basis of ρij calculated by Eq. 2, the distance
ket index and the companies with the degree k follows between nodes is calculated as follows
a power-law distribution, hρ (k)i ∼ kγ, where the ex-
iM
ponents are calculated to be γ ∼ 0.16, γ ∼
S&P500 KOSPI
0.14. Inotherwords,individualstockshavingmanycon- d = 2(1−ρ ). (3)
ij q i,j
nections with other stocks in the network obtained by
the MST method are more highly related to the market In order to find out the correlation between the num-
index than those having a comparatively small number berofconnectionsofaindividualstockwithotherstocks
of links. inthenetworkandthemarketindex,weinvestigatedthe
Inthenextsection,wedescribethefinancialdataused correlation, ρiM, between the market index and individ-
in this paper. In Section 3, we introduce the methodol- ual stocks. Using the Eq. 4, we calculated the correla-
ogy. In Section 4, we present the results in this investi- tion, ρiM, between returns of individual stocks, Ri, and
gation. Finally, we end with the summary. the market index, RM, and defined as follow
hR R i−hR ihR i
ρ = i M i M . (4)
iM
II. DATA q(hRi2i−hRii2)(hRM2 i−hRMi2)
Weused400individualdailystocksdatafromJanuary
1993 to May 2005 taken from individual stocks listed on IV. RESULTS
the S&P500 index of the American stock market (from
the Yahoo website) and 468 individual daily stocks data
Inthissection,usingtheMSTmethodweinvestigated
from January 1991 to May 2003 taken from individual
the network properties of 400 individual stocks listed on
stocks listed on the KOSPI of the Korean stock mar-
the S&P500 index from January 1993 to May 2005, and
ket (from the Korean Stock Exchange). In order to in-
468 individual stocks listed on the KOSPI from January
vestigate the possible factors determined the number of
1991 to May 2003, respectively.
links of an individual stock in the network, we used the
First, we analyze the distribution of the correlation
S&P500 and KOSPI index with the same period as in- matrix, ρ , for individual stocks in the S&P500 index
ij
dividual stocks, respectively. We used the normalized
and KOSPI. In Fig. 1, we shows the distribution of
returns, R , by the standard deviation, σ(r ), after cal-
t t the correlation matrix for both stock markets. The blue
culating the returns from the stock price, P , by the log-
t (square),red(circle),green(diamond),andblack(trian-
difference,rt ≡lnPt−lnPt−1,asinpreviousstudiesand gle) indicates the S&P500,KOSPI,S&P500 (surrogate),
defined as follow
KOSPI (surrogate) stock markets, respectively. We find
that the correlation distribution, ρ between stocks in
ij
the S&P500 index shows a very similar with that be-
R ≡ lnPt−lnPt−1, (1) tween stocks in the KOSPI and those deviate from the
t
σ(rt) correlationdistributionoftime seriescreatedby the sur-
rogate method. In order to estimate the topology struc-
where σ(r ) is the standard deviation of the return. ture of both stock markets,we employ the MST method
t
3
0.06
KOSPI
S&P500
0.05 KOSPI(surrogate)
S&P500(surrogate)
0.04
)
Cij0.03
(
P
0.02
(c) (d)
0.01 102 KSKS&&OOPPSS55PP00II(00s(usrurrorgoagtaet)e) 102 KSKS&&OOPPSS55PP00II(00s(usrurrorgoagtaet)e)
δ = 0.77
ζ = 2.1
−00.4 −0.2 0 0.2 C 0.4 0.6 0.8 1 P(k)101 P(k)101
ij
FIG. 1: The PDF of the cross-correlation ρij between the
stocks listed on for the S&P500 and KOSPI stock markets. 100 100
The red (circle), blue (square), green (diamond), and black
(triangle) denotes the KOSPI, S&P500, KOSPI (surrogate), Degree k 101 5 10 Deg15ree k 20 25
and S&P500 (surrogate) stock market, respectively.
FIG. 2: (a) MST structure composed of daily returns of 400
individualcompanieslisedontheS&P500indexfrom1993to
proposed by the Mantegna et al. Using the whole pe- 2005,and(b)MSTstructurecomposedofthedailyreturnsof
riod data of the Korean and American stock markets, 468individualcorporationslistedontheKOSPIfrom1991to
we presentthe networkstructure calculatedby the MST 2003. (c)showsthedegreedistributionoftheMSTstructure
methodanditsthedegreedistributionforboththestock composed of individual companies on the S&P500 index and
markets. Fig. 2 shows the network structures between KOSPI. The degree of both the S&P500 index and KOSPI
followsapowerlawdistributionwiththeexponentofζ ∼2.1.
stocks generated by the MST method, using individual
The red (circle), blue (square), green (diamond), and black
stockdata lisedonthe AmericanandKoreanstockmar-
(triangle) denotes the KOSPI, S&P500, KOSPI (surrogate),
ket, respectively and plot the degree distribution, P(k),
S&P500 (surrogate), respectively.
of the MST networks for both stock markets. In Fig. 2,
(a)and(b)displaytheMSTstructurecomposedofdaily
returns of the individual companies lised onthe S&P500
between an individual stock with the degree k and the
index andKOSPI,respectivelyand (c) and (d) show the
stock market index for both stock markets.
degree distribution of the MST structure for both stock
markets in the log-log and linear-log plot. The red (cir- InFig. 3, we showsthe distribution ofthe correlation,
cle), blue (square), green (diamond), and black (trian- ρiM, between stocks in the S&P500 index and KOSPI,
gle) indicates the KOSPI, S&P500, KOSPI (surrogate), respectively. The blue (square)andred(circle)indicates
S&P500 (surrogate), respectively and the notation (sur- the S&P500 and KOSPI.
rogate) denotes the corresponding surrogate data. Wefindthatthecorrelation,ρ ,betweenastockwith
iM
We find that the degree distribution for both stock thedegreek,the numberofconnectedwithotherstocks,
markets follows a power law distribution with the expo- and the market index for both stock markets follows a
nent ζ ∼ 2.1, while the degree distribution of MST net- power-law distribution, hρ (k)i ∼ kγ, with the expo-
iM
work of the time series created by the surrogate method nent γ ≈ 0.16 and γ ≈ 0.14, respectively.
S&P500 KOSPI
[16] does follows an exponential distribution with δ ∼ Thus, through these results, we found that the stocks
0.77. Thus,astheresultsfindinginthecomplexnetwork havingmore links with other stocksare morehighly cor-
suchasthe internet,WWW, protein-proteininteraction, related with the market index than those with relatively
and so on, there is a scale-free network property. There- fewer connections. In other words, the stocks closely re-
fore, the hubs existence in the financial markets means lated to the market index have a larger number of links
that there are a dominant company which gives many withotherstocksandarelikelytobe locatedaroundthe
influencetotheotherstocks. Inordertoobserveapossi- center of the network of the stocks. On the other hand,
blefactordeterminationthedegreeofanindividualstock the stocks poorly relatedto the marketindex have fewer
onthe MST network,wecalculatedthe correlation,ρ , links with other stocks andtend to be placedat the out-
iM
4
tive among multiple common factors mentioned in the
arbitrage pricing model (APM). We used 400 individual
KOSPI stocks listed on the S&P 500 index and 463 stocks listed
−0.2 on the KOSPI.
10 S&P500
We found that the correlation distribution, ρ , be-
ij
tween stocks in the S&P500 index shows a very simi-
lar with that between stocks lised on the KOSPI and
those deviate from the correlation distribution of time
γ = 0.16
series removed a nonlinearity by the surrogate method.
S&P500
> 10−0.3 We shows that the degree distribution in the network
)
k between stocks obtained by the MST method for both
(
M stock markets follows a power-law distribution with the
ρi exponent ζ ∼ 2.1, while the degree distri-
< S&P500,KOSPI
bution from the time series eliminated a nonlinearity
follows an exponential distribution with the exponent,
10−0.4 δS&P500(surrogate),KOSPI(surrogate) ∼0.77. In order to in-
vestigate a factor determining the degree k on the MST
γ = 0.14
KOSPI network, we used the market index for both stock mar-
kets. We found that in the degree distribution, the cor-
relation,ρ , betweenthe degreek,the number oflinks,
iM
and the market index for both stock markets follows a
10−0.5 power-law distribution, hρiM(k)i ∼ kγ, with the expo-
nent γ ≈0.16 and γ ≈0.14, respectively. In
0 1 S&P500 KOSPI
10 Degree k 10 otherwords,thestockshavingtheintimaterelationwith
the market index have a larger number of links, while
the stocks poorly relatedto the marketindex have fewer
FIG. 3: Distribution of the correlation, ρiM, between an in- links. According to above finding results, we imply that
dividual companies and market index for both the S&P500
the degree k as most important quantity to describe the
index and KOSPI,respectively.
networktopology,hasacloselyrelationwiththecommon
factors such as market index.
side of the network of the stocks. Our results suggest
that the common factor such as the market index play
a important rules in terms of determing the networks in Acknowledgments
the financial markets.
ThisworkwassupportedbytheKoreaResearchFoun-
dation funded by the Korean Government (MOEHRD)
V. CONCLUSIONS (KRF-2006-332-B00152), and by a grant from the
MOST/KOSEFtotheNationalCoreResearchCenterfor
In this paper, in order to examine possible factors Systems Bio-Dynamics (R15-2004-033), and by the Ko-
capable of affecting the number of links that a specific rea Research Foundation (KRF-2005-042-B00075), and
stock has in relation to other stocks in the network be- bytheMinistryofScience&TechnologythroughtheNa-
tween stocks created using the MST method, we car- tionalResearchLaboratoryProject,andbytheMinistry
ried out research using the market index, a representa- of Education through the program BK 21.
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