Statistical Properties of the Returns of Stock Prices of International Markets

0.48 MB·English

by  GabJin Oh

Statistical Properties of the Returns of Stock Prices of International Markets
GabJin Oh∗ and Seunghwan Kim†
Asia Pacific Center for Theoretical Physics, National Core Research
Center for Systems Bio-Dynamics and Department of Physics,
Pohang University of Science and Technology, Pohang, Gyeongbuk 790-784
Cheol-Jun Um‡
6 Department of Healthcare Management, Catholic University of Pusan, Busan 609-757
0 (Received 16 September2005)
0
We investigate statistical properties of daily international market indices of seven countries,
2
and high-frequency S&P500 and KOSDAQ data, by using the detrended fluctuation method and

n the surrogate test. We have found that the returns of international stock market indices of seven
a countries follow a universal power-law distribution with an exponent of ζ ≈ 3, while the Korean
J stock marketfollows an exponentialdistributionwith anexponentof β ≈0.7. TheHurstexponent

8 analysisoftheoriginalreturn,anditsmagnitudeandsignseries,revealthatthelong-term-memory
1 property, which is absent in the returns and sign series, exists in the magnitude time series with
0.7 ≤ H ≤ 0.8. The surrogate test shows that the magnitude time series reflects the non-linearity

] of the return series, which helps to reveal that the KOSDAQ index, one of the emerging markets,
n
shows higher volatility than a mature market such as theS&P 500 index.
a
-
a
t PACSnumbers: 02.50.-r,89.90.+n,05.40.-a,05.45.TP.
a
Keywords: Scaling,Long-term-Memory,Non-linearity,Volatility,DFA
d
.
s
c I. INTRODUCTION KOSDAQ 1-minute index from 1997 to 2004, to investi-
i
s gate diverse time characteristics of financial market in-
y
dices.
h Up to now, numerous studies analyzing financial time
p series have been carried out to understand the complex
[ economic systems made up of diverse and complicated
We found that the returns of international stock mar-
agents [1]. The statistical analysis of economic or fi-
1 ketindicesofsevencountriesfollowauniversalpower-law
v nancial time series exhibits features different from the distributionwithanexponentofζ ≈3,whiletheKorean
6 random-walk model based on the efficient market hy- stockmarketfollowsanexponentialdistribution withan
2 pothesis (EMH), which are called stylized facts [2-15].
exponentofβ ≈0.7. Foramoredetailedstatisticalanal-
1 Previous studies showed that the returns of both stocks
ysis, the original return time series is divided into mag-
1
andforeignexchangeratehaveavarietyofstylizedfacts.
nitude andsigntime series,andthe correspondingHurst
0
For example, the distribution of financial time series fol-
6 exponents are computed. The Hurst exponent analysis
lowsauniversalpower-lawdistributionwithanexponent
0 of the original return, and its magnitude and sign time
/ ζ ≈3[3-7]. Whilethetemporalcorrelationofreturnsfol- series,revealthatthelong-term-memoryproperty,which
s
lowstheprocessofrandomwalks,thevolatilityofreturns
c is absent in the return and sign time series, exists in the
i shows a long-term-memory property [12-15]. However, magnitude time series with 0.7≤H ≤0.8.
s
recent work has revealed that the distribution function
y
h ofreturnsinemergingmarketsfollowsanexponentialdis-
p tribution, while the mature markets follow a power-law In orderto test the nonlinearity of the time series,the
: distribution with an exponent ζ ≈3 [11]. surrogate test is performed for all time series. We find
v
i In this paper, we use Detrended Fluctuation Analysis that the magnitude time series reflects the non-linearity
X (DFA), which was introduced by Peng et al. to find the of the return series, which helps to revealthat the KOS-
r long-term-memory property of time series data [16] and DAQ index, one of the emerging markets, shows higher
a
utilizesthesurrogatetestmethodproposedbyTheileret volatility than a mature market such as the S&P 500
al. to measure the non-linearity of time series [17]. We index.
study daily international market indices of seven coun-
tries from 1991 to 2005, the high-frequency S&P 500 5-
In the next section, we explain the market data used
minute index from 1995 to 2004,and the high-frequency
in our investigations. In Section III., we introduce the
methods of the surrogate test and detrended fluctuation
analysis (DFA). In Section IV., the results of the statis-
tical analysis for various time series of the market data
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] are presented. Finally, we end with a summary of our
‡Electronicaddress: [email protected] findings.
2
II. DATA Then, r is multiplied by random phases,
k
We use the return series in eight daily international
r˜ =r eiφk, (4)
market indices of seven countries from 1991 to 2005,the k k
S&P 500 index (5 minutes) from 1995 to 2004, and the
where φ is uniformly distributed in [0, 2π]. The inverse
KOSDAQindex(1minute)from1997to2004. Theseven k
FFTofr˜ givesthesurrogatedataretainingthelinearity
countries are France (CAC40), Germany (DAX), United k
in the original time series,
Kingdom (FTSE100), Hong Kong (HangSeng), KOREA
(KOSPI), America (NASDAQ), Japan (Nikkei225), and
America (S&P 500). We make use of the normalized
N
return often used in the financial time series analysis in- r′ = 1 r˜e−i2πnk/N. (5)
stead of the stock prices. Let y1,y2,....yn, be the daily n N Xk=1 k
stock prices. The normalizedreturnR ata giventime t
t
is defined by In the third step, non-linear measurements with the en-
tropy, the dimension, and Lyapunov exponents are per-
formed for the original data and the surrogate data, re-
rt =lnyt+1−lnyt, spectively. Finally,thedifferenceinmeasurementsofthe
R ≡ lnyt+1−lnyt, (1) original data and the surrogate data is tested for signifi-
t σ(rt) cance. If significant, the hypothesis will be rejected and
the original data are regarded as having non-linearity.
where σ(r ) is the standard deviation of the return. The
t
normalized returns R are divided into magnitude and
t
sign series by using the following relation:
B. Detrended Fluctuation Analysis
R =|R |×Sign , (2)
k,t k,t k,t Thetypicalmethodstoanalyzethelong-term-memory
where R is the return series of the k-th market index propertyinthetimeseriesdataarelargelyclassifiedinto
k,t
calculated by the log-difference,|R | the magnitude se- three types: the re-scaled range analysis (R/S) method
k,t
ries of the returns of the k-th market index, and Sign proposed by Mandelbrot and Wallis [19], the modified
k,t
the sign series with +1 for the upturn and −1 for the R/S analysis by Lo et al. [18], and the DFA (detrended
downturn. Notethatthemagnitudeseries|R |fromtak- fluctuation analysis) method by Peng et al. [20]. In this
t
ing the absolute value of the return measures the size of paper, the DFA method is used due to its effectiveness
thereturnchange,andthesignseriesSign measuresthe even for the absence of long-term memory. The Hurst
t
directionofthe change. The volatilityofthe returns can exponentcanbe calculatedby the DFA method through
be studied though the magnitude series |R |. the following process.
t
Step (1): The time series after the subtraction of the
mean are accumulated as follows:
III. METHODS
N
A. Surrogate Test
y(i)= [x(i)−x¯], (6)
X
i=1
The surrogate test method was first proposed by
Theiler et al. to prove the non-linearity contained in where x(i) are the i-th time series, and x¯ is the mean of
the time series [17]. The surrogate data test can be ex- the whole time series. This accumulation process is one
plained by the following four steps [16]. First, a null thatchangesthe originaldata into a self-similarprocess.
hypothesis is made and the features of the linear pro- Step(2): Theaccumulatedtimeseriesaredividedinto
cess following the hypothesis aredefined. In general,the boxes of the same length n. In each box of length n,
linearityuses the mean, the variance,and the autocorre- the trend is estimated through the ordinary least square
lation of the originaltime series. The surrogate data are method, called DFA(m), where m is the order of fitting.
randomly generatedbut retain the autocorrelationfunc- Ineachbox,theordinaryleastsquarelineisexpressedas
tion, the mean, and the variance of the original data. In y (i). Bysubtractingy (i)fromtheaccumulatedy(i)in
n n
the second step, the surrogate data are created through each box, the trend is removed. This process is applied
theFastFourierTransform(FFT)method. Letrn bethe to everyboxandthe fluctuationmagnitude is calculated
original time series. The Fourier Transform rk of rn is by using
given by
1 N v 1 N
rk = N Xrnei2πnk/N. (3) F(n)=uuN X[y(i)−yn(k)]2. (7)
n=1 t i=1
3
(a) 1.1
CAC40 r e t u r n s
DAX m a g n i t u d e
−1 FHNTaASnSgEDS1Ae0Qn0g 1 sr ei gt un r n s ( s u r r o )
Nikkei225 m a g n i t u d e ( s u r r o )
SP&owPe5r0 L0aw(3.3) 0.9 s i g n ( s u r r o )
−1.5 3.3
CDF 0.8
−2 0.7
0.6
−2.5
0.5
log10 R 0.5
(b)
Exponential ( 0.7 )
−1 kospi 0.41 2 3 Cou4ntry 5 6 7 8
−2
0.7
FIG. 2: Hurst exponents of international market indices [1:
CDF−3 France (CAC40), 2: Germany (DAX), 3: United Kindom
−4 (FTSE 100), 4: Hong Kong (HangSeng), 5: Korea (KOSPI),
6: America (Nasdaq), 7: Japan (Nikkei 225), 8: America
−5
(S&P 500)] from the return, magnitude and sign time se-
−6 ries. The notation (surro) denotes the corresponding surro-
0 1 2 3 4 5 6 7 8 9 gate data.
R
FIG. 1: Cumulative distribution function (CDF) P(Rt >R)
of normalized returns time series Rt. (a) Normalized return power-law distribution. These results indicate that the
distribution of international market indices of six countries,
distributionof returns in the KOSPIindex, that belongs
excludingKorea,fromJanuary1991toMay2005inalog-log
to the emerging markets, does not follow a power-law
plot. (b) Linear-log plot for theKOSPI index.
distribution with the exponent ζ ≈3.
Figure 2 shows the Hurst exponents for the returns
The process of Step (2) is repeated for every scale, from of each international market index, calculated from the
which we obtain a scaling relation return, magnitude and sign time series. The long-term-
memory property is not found for the return and sign
F(n)≈cnH, (8) series with H ≈ 0.5. However, we find that the magni-
tude time series has a long-term-memory property with
where H is the Hurst exponent. The Hurst exponent H ≈ 0.8. The surrogate test plots denoted as (surro) in
characterizesthecorrelationoftimeserieswiththreedif- Figure2showthatthemagnitudetimeseriesreflectsthe
ferent properties. If 0≤H <0.5, the time series is anti- non-linearity of the original returns, while the sign time
persistent. If 0.5<H ≤1, it is persistent. In the caseof series shows the linearity of the original returns.
H = 0.5, the time series correspond to random walks. In order to investigate the scaling in high-frequency
data, we use the S&P 500 5-minute index from 1995
to 2004 and the KOSDAQ 1-minute index from 1997
IV. RESULTS to 2004. Figure 3(a) shows the Hurst exponents of the
return, magnitude and sign series for the S&P 500 5-
In this section, we analyze the statistical features of minute index and the KOSDAQ 1-minute index. As for
dailyinternationalmarketindicesofsevencountriesfrom international stock market indices, the sign series corre-
January1993toMay31,2005,theS&P 5005-minutein- sponds to random walks (H ≈ 0.5), but the magnitude
dexfrom1995to2004,andtheKOSDAQ1-minuteindex serieshasalong-term-memoryproperty(0.7≤H ≤0.8).
from1997to2004. We presenttheresultsofthe statisti- Figure 3(b) shows that all Hurst exponents of the corre-
cal features such as the cumulated distribution function sponding surrogate data follow random walks.
(CDF) and the time correlation of the various financial In order to find the time evolution of the Hurst ex-
time series. Figure 1(a) is a log-log plot of the cumu- ponent, we also investigated the time series by shifting
lative distribution function of the market indices of six theS&P 5005-minuteindexandKOSDAQ1-minutein-
countries, excluding Korea, from January 1991 to May dex by 500 minutes and 100 minutes, respectively. Fig-
2005. Figure 1(b) is a linear-log plot of the distribution ure 4 shows the Hurst exponent calculated with 6,000
function of the KOSPI index. data points by shifting approximately 500 minutes for
In Figure 1(a), we find that the tail exponents of the the S&P 500 5-minute index, from 1995 to 2004. The
indicesofallthecountriesexceptKoreafollowauniversal average Hurst exponent H ≈0.5 for the S&P 500 index
power-lawdistributionwithanexponentζ ≈3. However, signseriesofthereturns,andH ≈0.7forthemagnitude
in Figure 1(b), we find that the Korean stock market time series. In addition, the surrogate test shows that
follows an exponential distribution with β ≈ 0.7, not a the non-linearity of the original time series is reflected
4
(a)
4.4
S & P 5 0 0 ( returns )
S & P 5 0 0 ( magnitude )
4.2
S & P 5 0 0 ( sign )
K O S D A Q ( return )
4 K O S D A Q ( magnitude )
K O S D A Q ( sign )
3.8
H=0.8
n ) ) 3.6
F ( 3.4 H=0.5
og ( 103.2
l
3
2.8
2.6
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
log n
10
(b)
4.4
S & P 5 0 0 ( returns )
S & P 5 0 0 ( magnitude )
4.2
S & P 5 0 0 ( sign )
K O S D A Q ( return )
4 K O S D A Q ( magnitude )
K O S D A Q ( sign )
3.8
3.6 H=0.5
F ( n ) ) 3.4 FIG. 4: Hurst exponent of S&P 500 5-minute index returns
g ( 103.2 dthiveidperdiceintoof mS&agPnit5u0d0efaronmd s1ig9n9:5tthoe2b0l0a4ck. sTolhide lointheedrelninoetes
o
l 3 denotes the Hurst exponents corresponding to the returns,
sign and magnitude time series and the Hurst exponents of
2.8 the returns, sign and magnitude time series of the surrogate
data. The notation (surro) denotes the corresponding surro-
2.6
gate data.
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
log n
10
and the linearity in the sign time series.
FIG. 3: (a) Hurst exponent of the S&P 500 5-minute index
andtheKOSDAQ1-minuteindexwith thetimeseries of the
returnsdividedintomagnitudeandsigntimeseries. (b)Hurst
exponentofthesurrogatedataoftheS&P 500andKOSDAQ V. CONCLUSION
indicies.
In this paper, we have investigated the statistical fea-
turesofinternationalstockmarketindicesofsevencoun-
by the magnitude time series. tries, high-frequency S&P 500 and KOSDAQ data. For
Figure 5 shows the Hurst exponent calculated with this purpose, the tail index was studied through a linear
6,000data points by shifting approximately100 minutes fitting method by using the Hurst exponent by the DFA
for the KOSDAQ 1-minute index from 1997 to 2004 . method. Also, the non-linearity was measured through
Though on average H ≈ 0.5, the Hurst exponent of the the surrogate test method. We find that the absolute
returns changes considerably over time, unlike the S&P value distribution of the returns of international stock
500 index with a more or less uniform Hurst exponent. marketindicesfollowsauniversalpower-lawdistribution,
In particular, in the KOSDAQ index during its bubble having a tail index ζ ≈ 3 . However, the Korean stock
period from the second half of 1999 to mid-2000,a large marketfollows an exponential distribution with β ≈0.7,
long-term-memoryproperty is observedin the returnse- not a power-law distribution.
ries. After the market bubble burst, we found that the We also found that in the time series of international
Hurstexponentofthereturnsdroppedto0.5. Thisresult market indices, the S&P 500 index and the KOSDAQ
indicatesthattheKOSDAQindexmayhaveimprovedits index, the returns and sign series follow random walks
marketefficiencyafterthe bubble. Asinthepreviousre- (H ≈ 0.5), but the magnitude series does not. On the
sults, the non-linearity of the original time series of the otherhand,wefoundthatinallthetimeseries,theHurst
KOSDAQ data is reflected in the magnitude time series, exponent of the magnitude time series has a long-term-
5
memory property (0.7 ≤ H ≤ 0.8). Furthermore, we
found that in high-frequency data, the KOSDAQ index,
oneoftheemergingmarkets,showshighervolatilitythan
amaturemarketsuchastheS&P500index,whichispos-
sibly caused by the abnormally generated bubble. We
found a long-term-memory property in the magnitude
time series of all data, regardlessof nation or time scale.
Non-linear features of the returns are generally observed
inthemagnitudetimeseries. However,thedegreeofdis-
tribution and correlation in the returns of all data differ
in emerging and mature markets. Our results may be
usefulinanalyzingglobalfinancialmarkets,forexample,
differentiating the mature and emerging markets.
Acknowledgments
FIG. 5: Hurst exponent of the KOSDAQ 1-minute index
returns divided into magnitude and sign series. The solid This work was supported by a grant from the
blacklineshowstheKOSDAQindexfrom 1997to2004. The MOST/KOSEF to the National Core Research Center
other lines denote the Hurst exponents for the returns, sign for Systems Bio-Dynamics (R15-2004-033), and by the
and magnitude time series and the corresponding surrogate Ministry of Science & Technology through the National
data. The notation (surro) denotes the corresponding surro- ResearchLaboratoryProject,andbytheMinistryofEd-
gate data.
ucation through the programBK 21.
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