Large-volatility dynamics in financial markets

We investigate the large-volatility dynamics in financial markets, based on the minutely and daily data of the Chinese Indices and German DAX. The dynamic relaxation both before and after large volatilities is characterized by a power law, and the exponents p± usually vary with the strength of the large volatilities. The large-volatility dynamics is time-reversal symmetric at the minutely time scale, while asymmetric at the daily time scale. Careful analysis reveals that the time-reversal asymmetry is mainly induced by exogenous events. It is also the exogenous events which drive the financial dynamics to a non-stationary state. In general, the Chinese Indices and German DAX are in different universality classes. An interacting herding model without and with exogenous driving forces could qualitatively describe the large-volatility dynamics. Financial markets are complex systems which share common features with those in traditional physics. In recent years, it has been piled up large amount of financial data. This allows an analysis of the fine structure and interaction of the financial dynamics, and many empirical results have been documented [1–10]. Although the price return of a financial index is short-range correlated in time, the volatility exhibits a long-range temporal correlation [2, 3]. The dynamic behavior of volatilities is an important topic in econophysics [2, 3, 11, 12]. In usual cases, one assumes that the financial market is in the stationary state, and analyzes the statistical properties of the financial dynamics. For a comprehensive understanding of the financial markets, however, it is also important to investigate the non-stationary dynamic properties. A typical example is the so-called financial crash [6,13]. Lillo and Mantegna study three huge crashes of the stock market, and find that the rate of volatilities larger than a given threshold after such market crashes decreases by a power law with certain corrections in shorter times [14]. This dynamic behavior is analogous to the classical Omori law, which describes the aftershocks following a large earthquake [15]. Selcuk analyzes the daily data of the financial indices from 10 emerging stock markets and also observed the Omori law after the two largest crashes [16]. Recently, Weber et al. demonstrate that the Omori law holds also after ”intermediate shocks”, and the memory of (a)corresponding author; email: [email protected] volatilities is mainly related to such relaxation processes [17]. Stimulated by these works, we systematically analyze the large-volatility dynamics in financial markets, based on the minutely and daily data of the Chinese Indices and German DAX. In our study, a large volatility is so selected that it is sufficiently large compared with the average volatility, but maybe not yet a real financial crash or rally. The purpose of this paper is multi-folds. We investigate the dynamic relaxation both before and after large volatilities. We focus on the time-reversal symmetry or asymmetry at the minutely and daily time scales. To achieve more reliable results, we introduce the remanent and anti-remanet volatilities to describe the largevolatility dynamics. In particular, we examine the dynamic behavior of different categories of large volatilities, and search for the origin of the time-reversal asymmetry at the daily time scale. We reveal how the dynamic system is driven to a non-stationary state by exogenous events. We compare the results of the mature German market and the emerging Chinese market. Finally we present a multiagent model to simulate the large-volatility dynamics. In this paper, we have collected the daily data of the German DAX from 1959 to 2009 with 12407 data points, and the minutely data from 1993 to 1997 with 360000 data points. The daily data of the Shanghai Index are from 1990 to 2009 with 4482 data points, and the minutely data are from 1998 to 2006 with 95856 data points. The daily data of the Shenzhen Index are from 1991 to 2009