We investigate multifractality in the Korean stock-market index KOSPI. The generalized qth order height–height correlation function shows multiscaling properties. There are two scaling regimes with a crossover time around View the MathML sourcetc=40min. We consider the original data sets and the modified data sets obtained by removing the daily jumps, which occur due to the difference between the closing index and the opening index. To clarify the origin of the multifractality, we also smooth the data through convolution with a Gaussian function. After convolution we observe that the multifractality disappears in the short-time scaling regime t<tct<tc, but remains in the long-time scaling regime t>tct>tc, regardless of whether or not the daily jumps are removed. We suggest that multifractality in the short-time scaling regime is caused by the local fluctuations of the stock index. But the multifractality in the long-time scaling regime appears to be due to the intrinsic trading properties, such as herding behavior, information outside the market, the long memory of the volatility, and the nonlinear dynamics of the stock market.
In recent years, 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], [14], [15], [16], [17], [18] and [19], and 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 distribution functions [10], [11], [12] and [19], correlation functions [12], [13] and [14], multifractal analysis [19], [20] and [21], and network analysis of the market structure [15] and [16].
Multiscaling properties have been reported for many economic time series [22], [23], [24], [25], [26], [27], [28], [29], [30] and [31]. Multifractality has been observed in stock markets [20], [29], [32], [33] and [34], the price of crude oil [25], the price of commodities [34], and foreign exchange rates [24] and [35]. In daily stock indexes and foreign exchange rates, the generalized Hurst exponent HqHq decreases monotonically with q [24], [29], [32], [33], [34] and [35] (see formal definitions of HqHq and q in Eqs. (1) and (2) below). In the US NASDAQ index, two scaling regimes have been reported: one quasi-Brownian and the other multifractal [33]. Two scaling domains have also been reported in the fluctuations of the price of crude oil: HqHq increases with q in the short-time domain, but decreases with q in the long-time domain [25]. The existence of multiple scaling regimes seems to depend on the resolution of the data sets in the economic time series.
Although many economic time series display multifractality (in the theory of surface scaling analysis referred to as multiaffinity), the origins of multifractality in the stock market are not well understood. It has been suggested that herding behavior and nonlinear complex dynamics of the stock market induce multiscaling [10]. However, it is very difficult to quantify herding behavior and complex dynamics in the stock market. Buendía et al. observed multiaffinity in a frustrated spring-network model simulating the surface structure of cross-linked polymer gels [36]. Removing vertical discontinuities from the rough surface by convolution with a Gaussian, they observed that the multiaffine surface changed into a self-affine one [37]. They concluded that vertical discontinuities can be one cause of multiaffinity. Mitchell conversely introduced artificial vertical discontinuities into a self-affine surface and observed that the surface became multiaffine [38].
In the present paper we investigate multifractality in the Korean stock-market index (Korean Composite Stock Price Index (KOSPI)). We observe that local fluctuations, including jumps, are responsible for multifractality in the short-time scaling regime. However, multifractality in the long-time scaling regime is not removed by smoothing of the time series.
In summary, we have studied multifractality of the Korean KOSPI stock index. Multifractality is observed in two scaling regimes in the original time series of the stock index. However, multifractality in the short-time regime can be removed by convoluting the time series by a Gaussian. In the long-time regime the multifractality is unchanged by smoothing the time series. We propose that multifractality in the short-time regime is caused by the local fluctuations in the stock index. However, multifractality in the long-time regime appears to be a result of the complex behaviors in the stock market, such as herding behavior, information outside the market, the long memory of the volatility, and the intrinsic nonlinear dynamics of the market. Understanding the origins of multifractality in the long-time scaling regime remains an active research topic.
