We consider the probability distribution function (pdf) and the multiscaling properties of the index and the traded volume in the Korean stock market. We observed the power law of the pdf at the fat tail region for the return, volatility, the traded volume, and changes of the traded volume. We also investigate the multifractality in the Korean stock market. We consider the multifractality by the detrended fluctuation analysis (MFDFA). We observed the multiscaling behaviors for index, return, traded volume, and the changes of the traded volume. We apply MFDFA method for the randomly shuffled time series to observe the effects of the autocorrelations. The multifractality is strongly originated from the long time correlations of the time series.
Recently, concepts and techniques from statistical physics have been widely applied to economics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and [14]. The complex behaviors of economic systems have been found to be very similar to those of complex systems customarily studied in statistical physics. Stock-market indexes around the world have been accurately recorded for many years and therefore represent a rich source of data for quantitative analysis, and the statistical behaviors of stock markets have been studied by various methods, such as probability distribution functions [7], [8], [9] and [13], correlation functions [9], [10] and [11], multifractal analysis [13], [15], [16] and [17], and network analysis of the market structure [12].
Scaling behaviors in economics have been reported in many time series, such as stock indexes [5], [7], [13] and [17], foreign exchange markets [18] and [19], individual wealth [20], [21] and [22], and business sizes [23] and [24]. In the stock market the probability distribution function (pdf) of the return shows a power law at tail parts [7], [25], [26], [27] and [28]. The pdf of the volatility also follows the power law. The exponents of the power law for the volatility are close to 3 in US stock market [9]. The pdf of the traded volume for individual stock price shows a power law, P(Q)∼Q-(1+λ)P(Q)∼Q-(1+λ) at large fluctuation with the exponent λ=1.7λ=1.7 where Q is the traded volume [29]. Kaizoji and Nuki reported the scaling law for the distribution of fluctuations in share volume. The cumulative distributions have an exponent close to unity [30].
Multiscaling properties have been reported for many economic time series [31], [32], [33], [34], [35], [36], [37], [38], [39] and [40]. Multifractality has been observed in stock markets [15], [36], [41], [42], [43] and [44], the price of crude oil [32], the price of commodities [43], and foreign exchange rates [45]. In daily stock indexes and foreign exchange rates, the generalized Hurst exponent HqHq decreases monotonically with q [36], [41], [42], [43] and [45]. The method of the multifractal detrended fluctuation analysis (MFDFA) has been applied to the high-frequency tick-by-tick data in the stock market [43] and [46].
In the present work, we consider the pdf of the index, the return, the traded volume and the change of traded volume in the Korean stock market. We report the power law at tail parts of the pdf. We also consider the multiscaling behavior of the original time series and the randomly shuffled time series for the index and the traded volume in the stock market.
We consider the probability distribution function (pdf) for the return of the index, the volatility, the traded volume, and the change of the traded volume. The fat tails of the pdf for the return deviate from the Gaussian function and show the power law. The exponents of the power law for the pdf of the return depend on the return time. The central part of the pdf for the volatility is well fitted by the log-normal distribution function, but the tail parts show the fat tails. The exponents of the power law for the volatility are close to 2.3 in the Korean stock market. The pdf of the traded volume shows the power law at the large volume. The power-law exponents of the pdf for the traded volume is close to 2.5. The pdf of the change of the traded volume shows the power law with the exponent λ=1.0λ=1.0. We investigate the multiscaling properties for the index, the return of the index, the traded volume, and the change of the traded volume. We observe the multiscaling behaviors for all cases. The randomly shuffling processes for the time series reduce the multifractality. The multiscaling behaviors are strongly influenced by the long-range autocorrelation of the time series.
