Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007, pp. 178∼181
Scaling and Multiscaling Properties in the Korean Stock Market Kyoung Eun Lee and Jae Woo Lee∗ Department of Physics, Inha University, Incheon 402-751 (Received 7 August 2006) We consider the scaling and the multiscaling behaviors of the Korean stock-market index. We consider the return and the absolute return of Korean composite stock price index (KOSPI). We observed scaling behaviors in the tail parts of the probability distribution of the return and in the autocorrelation function of the absolute return. At early time, the autocorrelation function of the return decays exponentially with a characteristic time of 5.9 min. However, the absolute return reveals a long-range correlation. The generalized q-th order height-height correlation functions show multiscaling properties. There are two scaling regions, with a crossover time of tc = 40 min. The multiscaling in the long-time region appears to be due to intrinsic trading properties of the stock market. PACS numbers: 05.40.-a, 05.45.Tp, 89.65.Gh Keywords: Stock market, Multifractal, Correlation function, Econophysics
tic time has been reported in the S & P 500 [7]. The autocorrelation function of the absolute return (volatility) shows a scaling behavior in the stock market [9]. The autocorrelation function of the squared logarithmic return also follows a power law in restricted scaling regions [22]. Multiscaling properties have been reported for many economic time series [22–29]. Multifractality has been observed in stock markets [18, 27, 30–32], the price of crude oil [24], the price of commodities [32], and foreign exchange rates [33]. In daily stock indices and foreign exchange rates, the generalized Hurst exponent Hq decreases monotonically with q [27,30–33]. In the present paper, we investigate the probability density function of the return, the autocorrelation functions, and the multifractality in the Korean stock-market index, the KOSPI (Korean Composite Stock Price Index). In Section II, we present the probability density function of the return. In Section III, we consider the autocorrelation function of the return and the absolute return. In Section IV, we report the multiscaling properties of the return. We give concluding remarks in Section V.
I. INTRODUCTION In recent years, concepts from statistical physics have been widely applied to economics [1–17]. Stock market indices around the world have been accurately recorded for many years and, therefore, represent a rich source of data for quantitative analysis. The statistical behaviors of stock markets have been studied by using various methods, such as probability density functions [7–9,13], correlation functions [9–11], multifractal analysis [13,18, 19], and network analysis of the market structure [12]. Bachelier proposed a financial model of a stochastic process of returns, which considers the variation of the share price as an independently, identically distributed (i.i.d) Gaussian random variable [20]. However, the distribution of returns in financial markets does not follow a Gaussian distribution. Mandelbrot analyzed a relatively short-time series of cotton prices and observed that returns had a L´evy stable symmetric distribution with a Pareto fat tail [21]. Gopikrishnan et al. reported departures from the L´evy stable distribution of returns by analyzing high-frequency data from the S & P 500 index [7–11]. They observed that large events were very frequent in the data, a fact largely underestimated by a Gaussian process. They also found a power-law behavior of the probability density function (pdf) of returns with a fat tail exceeding the L´evy stable distribution. A similar behavior was reported for the probability distribution of returns of other indices, including the DAX [18] and the Hang-Seng indices [19]. An autocorrelation function with a short characteris∗ E-mail:
II. PROBABILITY DENSITY FUNCTION OF RETURN We consider a set of data recorded every minute of trading from March 30, 1992, through November 30, 1999. We count the time during trading hours and remove closing hours, weekends, and holidays from the data. For a time series I(t) of stock market index, the logarithmic return RT (t) over a time lag or a return time T is defined as
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Fig. 1. Normalized probability density function of the return in the Korean stock market index, the KOSPI. 0
Fig. 3. Semilogarithmic plot of the autocorrelation function of the return versus the time τ for the return time ∆t = 1 min. Inset: Fitting with the exponential function exp(−τ /τc ) (solid line) with τc = 5.9 min.
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with an exponent α > 2. The cumulated pdf of the return is defined by Z ∞ P (GT > x) = p(y)dy. (4)
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Fig. 2. Cumulated probability density function of the return with a return time T = 30 min in the Korean stock market index, the KOSPI.
RT (t) = log I(t) − log I(t − T ).
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The normalized return is defined by GT (t) =
RT (t)− < RT (t) > , σ(GT (t))
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where σ(GT ) is the standard deviation and < · · · > denotes averaging over time. Let’s consider a probability density function (pdf) of the normalized return. Fig. 1 present the pdf of the return for the KOSPI. The central part of the pdf is fitted better by a Lorentzian function than a Gaussian function. However, the tail parts of the pdf deviate from the Gaussian and the Lorentzian. The tails of the pdf decay according to a power law such as p(x) ∼ x−(1+α) ,
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In Fig. 2, we present the cumulative pdf for the time lag T = 30 min. We observe an obvious power law. The exponent α is greater than 2, which means that the pdf of the return deviates from the stable L´evy distribution with 0 < α < 2. We obtain exponents such as α = 2.16 (T = 1 min), 2.46( T= 10 min), 2.71 (T = 30 min) and 2.87 (T = 60 min) and 2.87 (T = 600 min) for the positive tail. We also obtain exponents such as α = 2.29 (T = 1 min), 2.56 (T = 10 min), 2.73 (T = 30 min), 3.03 (T = 60 min), and 2.91 (T = 600 min) for the negative tail. The exponents for the negative tail are greater than those for the positive tail. The exponetns of the pdf depend on the return time. Nonuniversal behaviors of the pdf of the return have been reported in the KOSDAQ [13], the DAX [18], the Hang-Seng [19], and the S & P 500 [7].
III. CORRELATIONS IN THE KOSPI The autocorrelation function of the return (AFR) is defined by C∆t (τ ) =< GT (t)GT (t + τ ) > − < GT (t) >2 .
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In Fig. 3, we present the AFR as a function of time. The AFR decays according to an exponential function
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Fig. 4. Log-log plot of the autocorrelation function of the absolute return against time for the Korean stock index, the KOSPI.
with a characteristic time τc = 5.9 min. Note that the characteristic time of the S&P 500 is around τc = 4 min [7]. The characteristic time of the AFR is very short in the return of the stock index. Although the returns correlate with a very short time that does not mean there are no any correlations in the stock index. Actually, there are long-range correlations in the volatility of the stock index [7–10]. The autocorrelation function of the absolute return (AFAR) is defined by C|G| (τ ) =< |GT (t)GT (t + τ )| > − < |GT (t)| >2 . (7) In Fig. 4, we present a log-log plot of the AFAR against time. We observe two scaling regions. In the short-time scaling region, AFAR follows a power law such as C|r| (τ ) ∼ τ −β1 with β1 = 0.52(2) at τ < τ1 = 5.9 min. In the long-time scaling region we observed a power law such as C|r| (τ ) ∼ τ −β2 with β2 = 0.25(2) at τ1 < τ < τ2 = 100 min. At τ > τ2 , the AFAR fluctuates and shows many peaks. The crossover time, τ1 = 5.9 min, is consistent with the characteristic time τc = 5.9 min of the AFR. We observed a similar scaling behaviors of the autocorrelation function of the absolute logarithmic returns with the scaling exponents β1 = 0.53(2) at τ < τ1 = 5.9 min and β2 = 0.27(2) at τ1 < τ < τ2 = 100 min. Liu et al reported the scaling exponent of the AFAR as β2 = 0.30(8) for the S & P 500 [9]. IV. MULTIFRACTALITY The generalized q-th order height-height correlation function (GHCF), Fq (t), is defined by Fq (t) = h|I(t0 + t) − I(t0 )|q i1/q ,
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Fig. 5. (a) Log-log plot of the generalized q-th order height-height correlation function Fq (t) as a funtion of the time for the Korean stock market index, the KOSPI, with q = 1, 3, 5, 7, and 9 from bottom to top. There are two scaling regions separated by a crossover time tc = 40 min. (b) The generalized q-th Hurst exponent as a function of q in the scaling regions t < tc (¤) and t > tc (4).
where the angular brackets denote a time average over the time series. The GHCF, Fq (t), characterizes the correlation properties of the time series I(t), and for a multiaffine series, a power-law behavior like Fq (t) ∼ tHq .
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is expected, where Hq is the generalized qth-order Hurst exponent [34]. If Hq is independent of q, the time series is monofractal. If Hq depends on q, the time series is multifractal. Multifractality is a distinctive property of the stock market index observed. In Fig. 5(a), we present the GHCF as a function of the time interval t. We observe a clear multifractal behavior in the time series of the stock index. There are two different scaling regions separated by a crossover time of about tc = 40 min. The slope of the log-log plot depends on q in each scaling region. Fig. 5(b) shows, Hq ∼ 1/q for large q in both scaling regions while it saturates for small q. The 1/q behavior is consistent with numerical observations presented in Refs. [35] and [36] with the analytical results for functions that include discontinuities. These behaviors are different from the multifractality in the price of crude oil [24]. AlvarezRamirez et al. reported the existence of two scaling
Scaling and Multiscaling Properties in the Korean Stock Market – Kyoung Eun Lee and Jae Woo Lee
regions for crude-oil prices [24], but Hq increased with q in the long-time scaling region.
V. CONCLUSION We have studied the probability density function of the return, the autocorrelation function of the return, and the multifractality of the Korean KOSPI stock index. We observe that the probability density function has a fat tail. The power-law exponents of the fat tail depend on the return time. In the KOSPI, the powerlaw exponents are in the range 2 < α < 3. The autocorrelation function of the return shows a short-range correlation. At early time the autocorrelation function decays exponentially with the characteristic time τ = 5.9 min. The autocorrelation function of the absolute return (or volatility) shows a long-range correlation and follows power-law. We observe two scaling regions. The powerlaw exponent is close to that of the S & P 5000. Multifractality is observed in two scaling regions in the time series of the stock index. We propose that multifractality in the short-time region is caused by local fluctuations in the stock index where multifractality in the long-time region appears to be a result of complex behaviors in the stock market, such as herding behavior, information outside the market, a long memory of volatility, and the intrinsic nonlinear dynamics of the market [37–41]. Understanding the origins of multifractality in the long-time scaling region remains an active research topic.
ACKNOWLEDGMENTS This work has been supported by the Research Fund of Inha University.
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