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‡ Department of Healthcare Management, Catholic University of Pusan, Busan 609-757 (Received 16 September 2005) We investigate statistical properties of daily international market indices of seven countries, and high-frequency S&P 500 and KOSDAQ data, by using the detrended fluctuation method and the surrogate test. We have found that the returns of international stock market indices of seven countries follow a universal power-law distribution with an exponent of ζ ≈ 3, while the Korean stock market follows an exponential distribution with an exponent of β ≈ 0.7. The Hurst exponent analysis of the original return, and its magnitude and sign series, reveal that the long-term-memory 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, shows higher volatility than a mature market such as the S&P 500 index. PACS numbers: 02.50.-r, 89.90.+n, 05.40.-a, 05.45.TP. Keywords: Scaling, Long-term-Memory, Non-linearity, Volatility, DFA I. INTRODUCTION KOSDAQ 1-minute index from 1997 to 2004, to investi- gate diverse time characteristics of financial market in- dices. Up to now, numerous studies analyzing financial time 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- ket indices of seven countries follow a universal power-law nancial time series exhibits features different from the distribution with an exponent of ζ ≈ 3, while the Korean random-walk model based on the efficient market hy- stock market follows an exponential distribution with an pothesis (EMH), which are called stylized facts [2-15]. exponent of β ≈ 0.7. For a more detailed statistical anal- Previous studies showed that the returns of both stocks ysis, the original return time series is divided into mag- and foreign exchange rate have a variety of stylized facts. nitude and sign time series, and the corresponding Hurst For example, the distribution of financial time series fol- exponents are computed. The Hurst exponent analysis lows a universal power-law distribution with an exponent of the original return, and its magnitude and sign time ζ ≈ 3 [3-7]. While the temporal correlation of returns fol- series, reveal that the long-term-memory property, which lows the process of random walks, the volatility of returns is absent in the return and sign time series, exists in the shows a long-term-memory property [12-15]. However, magnitude time series with 0.7 ≤ H ≤ 0.8. recent work has revealed that the distribution function of returns in emerging markets follows an exponential dis- tribution, while the mature markets follow a power-law In order to 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 In this paper, we use Detrended Fluctuation Analysis that the magnitude time series reflects the non-linearity (DFA), which was introduced by Peng et al. to find the of the return series, which helps to reveal that the KOS- long-term-memory property of time series data [16] and DAQ index, one of the emerging markets, shows higher arXiv:physics/0601126v1 [physics.data-an] 18 Jan 2006 utilizes the surrogate test method proposed by Theiler et 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- minute index from 1995 to 2004, and the high-frequency In the next section, we explain the market data used 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- ∗Electronic address: [email protected] tical analysis for various time series of the market data †Electronic address: [email protected] are presented. Finally, we end with a summary of our ‡Electronic address: [email protected] findings. 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.