On return-volatility correlation in financial dynamics

With the daily and minutely data of the German DAX and Chinese indices, we investigate how the return-volatility correlation originates in financial dynamics. Based on a retarded volatility model, we may eliminate or generate the return-volatility correlation of the time series, while other characteristics, such as the probability distribution of returns and longrange time-correlation of volatilities etc., remain essentially unchanged. This suggests that the leverage effect or anti-leverage effect in financial markets arises from a kind of feedback returnvolatility interactions, rather than the long-range time-correlation of volatilities and asymmetric probability distribution of returns. Further, we show that large volatilities dominate the returnvolatility correlation in financial dynamics. Financial markets are complex systems with many-body interactions. The possibility of accessing large amounts of historical financial data have spurred the interest of physicists, to analyze the financial dynamics with physical concepts and methods. Some ”stylized facts” of the financial markets are revealed [1–8]. Different models and theoretical approaches have been proposed to describe and reproduce the features of the financial dynamics [9–22]. A complex system is often characterized by time correlations and spatial correlations. A famous stylized fact of the financial dynamics is the ”volatility clustering”, i.e., the long-range time-correlation of volatilities, though the price return itself is short-range correlated in time [2,3,15]. Meanwhile recent researches are concerned with the crosscorrelations between different stocks and their statistical properties in different stock markets [8, 23–29]. To further understand the financial dynamics, one may consider the higher-order time-correlations. It was first observed by Black [30, 31] that past negative returns increase future volatilities, i.e., the return-volatility correlation is negative. This is the leverage effect in financial markets. In the past years many literatures have been devoted to the leverage effect, and various relevant correlation coefficients have been measured within GARCH-like models [32–35]. Recently Bouchaud et al quantitatively computed the return-volatility correlation function with the daily data of several financial markets, and observed that it decays by an exponential law [4, 36]. More recently, (a)corresponding author; email: [email protected] Zheng et al discovered a positive return-volatility correlation in Chinese financial markets [7,37], i.e., the so-called anti-leverage effect. Further, it is shown that both the leverage effect in German markets and the anti-leverage effect in Chinese markets can be detected on both daily and minutely time scales [7, 37]. How does the return-volatility correlation originate in financial dynamics? The economic interpretation of this phenomenon is still controversial [32, 35]. According to Black, a price drop increases the risk of a company to go bankrupt, and its stock therefore becomes more volatile. This induces the leverage effect. Different models have been proposed to explain the leverage effect with certain success [4,38–43]. The retarded volatility model is a good example [4]. The core thought of this model is that the reference price used to set the scale for price updates is not the instantaneous price but rather a moving average of the price over a past period of time. In fact, the retarded volatility model may generate both the leverage and antileverage effects by selecting appropriate coupling parameters K(t) [4,7,37]. More recently, there have been discussions whether the long-range time-correlation of volatilities may play an important role in the origination of the leverage effect [40,44,45]. Especially, it is argued that both the long-range time-correlation of volatilities and asymmetric probability distribution of returns are necessary in order to have a leverage effect [45]. By the definition of the return-volatility correlation function, the long-range time-correlation of volatilities and