arXiv:physics/0601126v1 [physics.data-an] 18 Jan 2006 re rm19 o20,tehigh-frequency the coun- 2005, seven to of 1991 indices from market tries international daily study Theiler by proposed and method al. [16] test data surrogate series the utilizes time of property long-term-memory iueidxfo 95t 04 n h high-frequency the and 2004, to 1995 from index minute DA,wihwsitoue yPeng by introduced was which (DFA), rbto,wietemtr akt olwapower-law a exponent follow an markets with mature distribution the dis- function exponential while an distribution tribution, follows markets the emerging in that However, returns revealed of has [12-15]. work property recent returns long-term-memory of volatility a the walks, shows random of process the lows exponent an fol- with ζ series distribution power-law time universal financial a of facts. lows stylized distribution of the stocks variety example, both a For have [2-15]. of rate returns exchange facts foreign the hy- stylized and that market called showed studies efficient are Previous fi- the which the or from (EMH), on different economic pothesis based features of model exhibits analysis random-walk series statistical complicated time and The nancial diverse complex of [1]. the up agents understand made to systems out economic carried been have series ‡ † ∗ lcrncades [email protected] address: Electronic [email protected] address: Electronic lcrncades [email protected] address: Electronic ≈ nti ae,w s erne lcuto Analysis Fluctuation Detrended use we paper, this In pt o,nmru tde nlzn nniltime financial analyzing studies numerous now, to Up omauetennlnaiyo iesre 1] We [17]. series time of non-linearity the measure to 37.Wietetmoa orlto frtrsfol- returns of correlation temporal the While [3-7]. 3 ttsia rpriso h eun fSokPie fInt of Prices Stock of Returns the of Properties Statistical nlsso h rgnlrtr,adismgiueadsg s sign and magnitude its and return, original the of analysis rpry hc sasn ntertrsadsg eis exi series, sign and returns the in 0 absent is which property, tc aktflosa xoeta itiuinwt nex an with distribution exponential an follows market stock ewrs cln,Ln-emMmr,Nnlnaiy Vol Non-linearity, Long-term-Memory, 05.45.TP. Scaling, 05.40.-a, Keywords: 89.90.+n, 02.50.-r, 5 numbers: S&P PACS the in as KOSDAQ such the market that mature reveal a to than helps volatility which higher shows series, return the of onre olwauieslpwrlwdsrbto iha interna of with returns distribution the power-law that universal found a have follow We countries test. surrogate the n high-frequency and . 7 eivsiaesaitclpoete fdiyinternatio daily of properties statistical investigate We ≤ .INTRODUCTION I. eateto elhaeMngmn,Ctoi University Catholic Management, Healthcare of Department H ≤ oagUiest fSineadTcnlg,Phn,Gyeon Pohang, Technology, and Science of University Pohang 0 . .Tesroaets hw httemgiuetm eisre series time magnitude the that shows test surrogate The 8. saPcfi etrfrTertclPyis ainlCr R Core National Physics, Theoretical for Center Pacific Asia etrfrSsesBoDnmc n eateto Physics, of Department and Bio-Dynamics Systems for Center ζ S ≈ & P [11]. 3 0 n ODQdt,b sn h erne utainmetho fluctuation detrended the using by data, KOSDAQ and 500 tal. et aJnOh GabJin S Rcie 6Spebr2005) September 16 (Received ofidthe find to & P 0 5- 500 ho-u Um Cheol-Jun ∗ n enha Kim Seunghwan and et aedvretm hrceitc ffiaca aktin- market financial of investi- characteristics to dices. 2004, time to diverse 1997 gate from index 1-minute KOSDAQ itiuinwt nepnn of exponent power-law an universal with a distribution follow countries seven of indices ket sasn ntertr n intm eis xssi the in exists series, 0 time with sign series and time return time magnitude the sign in and which absent property, magnitude is long-term-memory its the that analysis and reveal exponent return, series, original Hurst the The Hurst of mag- corresponding computed. into the are and divided series, exponents is time series sign time and nitude return original the ysis, tc aktflosa xoeta itiuinwt an with distribution of exponential exponent an follows market stock oaiiyta auemre uha h & 500 S&P the as such market higher shows mature index. markets, a KOS- emerging the than the that volatility of reveal to one helps find index, non-linearity which We DAQ the series, return reflects series. the series time of time all magnitude for the performed that is test surrogate r rsne.Fnly eedwt umr four of summary a with data market end the we of findings. Finally, statis- series the presented. time of various are results for the analysis the IV., fluctuation tical Section introduce detrended In and we (DFA). test III., analysis surrogate Section the of In methods investigations. our in tlt,DFA atility, efudta h eun fitrainlsokmar- stock international of returns the that found We nodrt ettenniert ftetm eis the series, time the of nonlinearity the test to order In ntenx eto,w xli h aktdt used data market the explain we section, next the In ‡ xoetof exponent n re,rva httelong-term-memory the that reveal eries, a aktidcso ee countries, seven of indices market nal oetof ponent t ntemgiuetm eiswith series time magnitude the in sts inlsokmre nie fseven of indices market stock tional e,oeo h mrigmarkets, emerging index. 00 the of one dex, β ≈ fPsn ua 609-757 Busan Pusan, of † 0 β . .Framr ealdsaitclanal- statistical detailed more a For 7. ≈ ζ 0 bk790-784 gbuk . .TeHrtexponent Hurst The 7. ≈ et h non-linearity the flects esearch rainlMarkets ernational ,wieteKorean the while 3, . 7 ≤ ζ H ≈ ≤ ,wieteKorean the while 3, and d 0 . 8. 2

II. DATA Then, rk is multiplied by random phases,

We use the return series in eight daily international iφk market indices of seven countries from 1991 to 2005, the r˜k = rke , (4) S&P 500 index (5 minutes) from 1995 to 2004, and the φ π KOSDAQ index (1 minute) from 1997 to 2004. The seven where k is uniformly distributed in [0, 2 ]. The inverse r countries are France (CAC40), Germany (DAX), United FFT of ˜k gives the surrogate data retaining the linearity Kingdom (FTSE100), Hong Kong (HangSeng), KOREA in the original time series, (KOSPI), America (NASDAQ), Japan (Nikkei225), and America (S&P 500). We make use of the normalized N ′ 1 return often used in the financial time series analysis in- r = r˜ e−i2πnk/N . (5) n N X k stead of the stock prices. Let y1,y2, ....yn, be the daily k=1 stock prices. The normalized return Rt at a given time 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- r y y , t = ln t+1 − ln t spectively. Finally, the difference in measurements of the ln yt+1 − ln yt original data and the surrogate data is tested for signifi- Rt ≡ , (1) σ(rt) cance. If significant, the hypothesis will be rejected and the original data are regarded as having non-linearity. where σ(rt) is the standard deviation of the return. The normalized returns Rt are divided into magnitude and sign series by using the following relation: B. Detrended Fluctuation Analysis

R R Sign , k,t = | k,t|× k,t (2) The typical methods to analyze the long-term-memory where Rk,t is the return series of the k-th market index property in the time series data are largely classified into calculated by the log-difference, |Rk,t| the magnitude se- three types: the re-scaled range analysis (R/S) method ries of the returns of the k-th market index, and Signk,t proposed by Mandelbrot and Wallis [19], the modified the sign series with +1 for the upturn and −1 for the R/S analysis by Lo et al. [18], and the DFA (detrended et al. downturn. Note that the magnitude series |Rt| from tak- fluctuation analysis) method by Peng [20]. In this ing the absolute value of the return measures the size of paper, the DFA method is used due to its effectiveness the return change, and the sign series Signt measures the even for the absence of long-term memory. The Hurst direction of the change. The volatility of the returns can exponent can be calculated by the DFA method through be studied though the magnitude series |Rt|. the following process. 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, andx ¯ 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 that changes the original data into a self-similar process. hypothesis is made and the features of the linear pro- Step (2): The accumulated time series are divided into cess following the hypothesis are defined. In general, the boxes of the same length n. In each box of length n, linearity uses the mean, the variance, and the autocorre- the trend is estimated through the ordinary least square lation of the original time series. The surrogate data are method, called DFA(m), where m is the order of fitting. randomly generated but retain the autocorrelation func- In each box, the ordinary least square line is expressed as tion, the mean, and the variance of the original data. In yn(i). By subtracting yn(i) from the accumulated y(i) in the second step, the surrogate data are created through each box, the trend is removed. This process is applied the Fast Fourier Transform(FFT) method. Let rn be the to every box and the fluctuation magnitude is calculated original time series. The Fourier Transform rk of rn is by using given by

N 1 N 1 i2πnk/N v 2 r = r e . (3) F (n)= u [y(i) − yn(k)] . (7) k N X n uN X 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 FTSE100 HangSeng 1 s i g n NASDAQ r e t u r n s ( s u r r o ) Nikkei225 m a g n i t u d e ( s u r r o ) S&P500 s i g n ( s u r r o ) Power Law(3.3) 0.9 −1.5 3.3

CDF 0.8

−2 0.7

0.6 −2.5 0.5 log R 10 0.5 (b) Exponential ( 0.7 ) kospi 0.4 −1 1 2 3 4 5 6 7 8 Country

−2 0.7

−3 FIG. 2: Hurst exponents of international market indices [1:

CDF France (CAC40), 2: Germany (DAX), 3: United Kindom −4 (FTSE 100), 4: Hong Kong (HangSeng), 5: Korea (KOSPI),

−5 6: America (Nasdaq), 7: Japan (Nikkei 225), 8: America (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 R gate data.

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, distribution of returns in the KOSPI index, that belongs excluding Korea, from January 1991 to May 2005 in a log-log plot. (b) Linear-log plot for the KOSPI index. to the emerging markets, does not follow a power-law 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 characterizes the correlation of time series with three dif- Figure 2 show that the magnitude time series reflects the 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 stock market indices, the sign series corre- January 1993 to May 31, 2005, the S&P 500 5-minute in- sponds to random walks (H ≈ 0.5), but the magnitude dex from 1995 to 2004, and the KOSDAQ 1-minute index series has a long-term-memory property (0.7 ≤ H ≤ 0.8). from 1997 to 2004. We present the results of the 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 the S&P 500 5-minute index and KOSDAQ 1-minute in- 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 indices of all the countries except Korea follow a universal average Hurst exponent H ≈ 0.5 for the S&P 500 index power-law distribution with an exponent ζ ≈ 3. However, sign series of the returns, and H ≈ 0.7 for the magnitude 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 3.6 H=0.5 3.4 ( F n ) 10

log 3.2

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 ) 4.2 S & P 5 0 0 ( magnitude ) 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

3.4 FIG. 4: Hurst exponent of S&P 500 5-minute index returns

( F n ) 3.2 divided into magnitude and sign: the black solid line denote 10 the price of S&P 500 from 1995 to 2004. The other lines log 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 2.6 data. The notation (surro) denotes the corresponding surro- 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 and the KOSDAQ 1-minute index with the time series of the returns divided into magnitude and sign time series. (b) Hurst exponent of the surrogate data of the S&P 500 and KOSDAQ V. CONCLUSION indicies. In this paper, we have investigated the statistical fea- tures of international stock market indices of seven coun- 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 6,000 data points by shifting approximately 100 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. market indices follows a universal power-law distribution, 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 market follows an exponential distribution with β ≈ 0.7, long-term-memory property is observed in the return se- 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 Hurst exponent of the returns dropped to 0.5. This result market indices, the S&P 500 index and the KOSDAQ indicates that the KOSDAQ index may have improved its index, the returns and sign series follow random walks market efficiency after the bubble. As in the previous re- (H ≈ 0.5), but the magnitude series does not. On the sults, the non-linearity of the original time series of the other hand, we found that in all the time series, the Hurst 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, one of the emerging markets, shows higher volatility than a mature market such as the S&P 500 index, which is pos- sibly caused by the abnormally generated bubble. We found a long-term-memory property in the magnitude time series of all data, regardless of nation or time scale. Non-linear features of the returns are generally observed in the magnitude time series. However, the degree of dis- tribution and correlation in the returns of all data differ in emerging and mature markets. Our results may be useful in analyzing global financial markets, for example, 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 black line shows the KOSDAQ index from 1997 to 2004. 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- Research Laboratory Project, and by the Ministry of Ed- gate data. ucation through the program BK 21.

[1] J. P. Bouchaud and M. Potters, Theory of Financial [11] K. Matia, M. Pal, H. Salunkay and H. E. Stanley, Euro- Risks: from Statistical Physics to Risk Managements, phys. Lett. 66, 909 (2004). Cambridge University Press, Cambridge, 2000; R. N. [12] Rogerio L. Costa and G. L. Vasconcelos, Physica A 329, Mantegna and H. E. Stanley, An Introduction to Econo- 231 (2003). physics : Correlations and Complexity in Finance, Cam- [13] D. O. Cajuerio and B. M. Tabak, Chaos, Solitons and bridge University Press, Cambridge, 1999. Fractals 22, 349 (2004). [2] E. F. Fama, Journal of Finance 25, 421 (1970). [14] D. O. Cajuerio and B. M. Tabak, Chaos, Solitons and [3] H. Ghashghaie, W. Breymann, J. Peinke, P. Talkner and Fractals 23, 671 (2005). Y. Dodge, Nature 381, 767 (1996). [15] T. D. Matteo, T. Aste and M. M. Dacorogna, Journal of [4] R. N. Mantegna, and H. Eugene Stanley, Nature 376, 46 Banking and Finance 29, 827 (2005). (1995). [16] Rogerio L. Costa and G. L. Vasconcelos, Physica A 32, [5] R. N. Mantegna, and H. Eugene Stanley, Nature 383, 231 (2003). 587 (1996). [17] J. Theiler, S. Eubank, A. Lontin, B. Galdrikian and J. [6] Xavier Gabaix, Parameswaran Gopikrishnan, Vasiliki Doyne, Physica D 58, 77 (1992). Plerou and H. Eugene Stanley, Nature 423, 267 (2003). [18] A. W. Lo, Econometrica 59, 1279 (1991). [7] B. B. Mandelbrot, Quantitative Finance 1, 113 (2001). [19] B. B. Mandelbrot and J. W. Van Ness, SIAM. Rev. 10, [8] Kyungsik Kim, Seong-Min Yoon and J. S. Col, J. Korean 422 (1968). Phys. Soc. 46, 6 (2005). [20] C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. [9] Jae Woo Lee and Kyoung Eun Lee, J. Korean Phys. Soc. E. Stanley and A. L. Goldberger, Phys. Rev. E 49, 1685 46, 3 (2005). (1994). [10] Tomas Lux, Quantitative Finance 1 , 560 (2001).