In the past years, dynamics of financial markets has drawn much attention of physicists [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and [20]. Based on large amounts of historical data, some stylized facts have been revealed, such as the “fat tail” distribution of the price return, and the so-called “volatility clustering” where the magnitude of returns is observed to be long-range correlated [1], [2], [3] and [4]. On the other hand, different models and theoretical approaches have been developed to describe financial markets [3], [4], [21], [7], [22], [23], [24] and [25].
From the view of many-body systems, long-range correlation of the volatilities should originate from the strong interactions among individual stocks. To investigate however individual stocks interact with each other, the cross-correlation function is widely adopted as a common mathematical tool. Random and nonrandom properties of the cross-correlations and the relevant economic sectors are revealed [18], [19], [20], [26], [27], [28], [29], [30], [31], [32], [33] and [34]. Correlation-based hierarchical or network structures are studied with the graph or complexity theory [35], [36], [37], [38], [39], [40] and [41]. The so-called pull effect is found with a time-dependent cross-correlation function [42].
Within this framework, dynamics of the cross-correlations has attracted an increasing interest of physicists. Dynamics of the cross-correlation functions and the eigenvalues of the cross-correlation matrices are widely investigated. Recently, a detrended cross-correlation analysis is proposed to investigate the memory effect of the cross-correlations between two time series [43], where the long-range memory is characterized by a power law scaling. The relevant extensive studies and applications are implemented [44], [45], [46] and [47].
In this paper, we introduce an instantaneous cross-correlation and an average instantaneous cross-correlation by considering correlations of pairs of stocks at a single time step, and therefore can quantify the temporal correlation between individual stocks with the local information. Based on the daily data of the United States and the Chinese stock markets, we study the memory effect of the instantaneous cross-correlations. More importantly, we examine the memory effect of the average instantaneous cross-correlations with different timescales of returns, and reveal the relevant multifractal nature.
The remainder of this paper is organized as follows. In the next section, we give the details of the datasets. In Section 3, we investigate the cross-correlations of pairs of individual stocks from the common sense and introduce the instantaneous cross-correlation, with both the price return and the volatility. Possible relation between the cross-correlation from the common sense and the memory effect of the instantaneous cross-correlation is discussed. In Section 4, we introduce the average instantaneous cross-correlation, and study the corresponding memory effect with different timescales of returns. Section 5 reveals the multifractal properties of the average instantaneous cross-correlations. Finally, Section 6 contains the conclusion.
We have investigated the memory effect of the instantaneous cross-correlations and the average instantaneous cross-correlations defined with both the price return and the volatility, based on the daily data of the NYSE and the CSM. It is interesting to find that, even though the instantaneous cross-correlations for some pairs of stocks might be short-range correlated or anti-persistent, the average instantaneous cross-correlation of a set of stocks is found to be robustly long-range correlated, with the timescale View the MathML sourceΔt′=1 day. Two-stage power law scalings are observed for the longer timescales, with the long-range memory of the average instantaneous cross-correlations persisting up to the month magnitude of timescale for the NYSE and half a month magnitude of timescale for the CSM for the large time regime.
Multifractal properties are investigated for the average instantaneous cross-correlations by the MF-DFA. By examining the MF-DFA function Fq(t)Fq(t), the scaling exponent function τ(q)τ(q), and the extracted multifractal spectrum f(α)f(α), multifractal features are revealed for both the NYSE and the CSM.
