We investigate the probability distribution of the volatility return intervals ττ for the Chinese stock market. We rescale both the probability distribution Pq(τ)Pq(τ) and the volatility return intervals ττ as View the MathML sourcePq(τ)=1/τ¯f(τ/τ¯) to obtain a uniform scaling curve for different threshold value qq. The scaling curve can be well fitted by the stretched exponential function View the MathML sourcef(x)∼e−αxγ, which suggests memory exists in ττ. To demonstrate the memory effect, we investigate the conditional probability distribution Pq(τ|τ0)Pq(τ|τ0), the mean conditional interval 〈τ|τ0〉〈τ|τ0〉 and the cumulative probability distribution of the cluster size of ττ. The results show clear clustering effect. We further investigate the persistence probability distribution P±(t)P±(t) and find that P−(t)P−(t) decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in ττ. The scaling and long memory effect of ττ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.
In recent years, physicists have paid much attention on the dynamics of financial markets. Scaling behavior is discovered in the financial system by analyzing the indices and the stock prices y(t′)y(t′), such as the ‘fat tail’ of the probability distribution P(Z,t)P(Z,t) of the two point price change return View the MathML sourceZ(t′)=lny(t′)−lny(t′−1) [1] and [2]. The physical origin of the scaling behavior is often related to the long range correlation. It is interested to find, in spite of the absence of the return correlation, the volatility |Z(t′)||Z(t′)| is long range correlated [3] and [4].
Recently, the volatility return intervals ττ, which is defined as the return intervals that the volatility is above a certain threshold qq, is investigated for the United States and the Japanese financial markets [5], [6], [7], [8], [9] and [10]. Scaling behavior of the probability distribution in the volatility return intervals ττ is discovered, and long-range autocorrelation is demonstrated for ττ. The scaling and the long-range autocorrelation are rather robust independent of the stock markets and the foreign exchange markets for the developed countries. However, it is known that the emerging markets may behave differently [11], [12], [13], [14], [15], [16] and [17]. Especially, the Chinese stock market is newly set up in 1990 and shares a transiting social and political system. Due to the special background of the Chinese stock market, it may exhibit special features far different from the mature financial markets in some aspects [11], [12] and [13]. In Refs. [11] and [12], the prominent anti-leverage effect of the Chinese indices is reported, in contrast with the leverage effect of the mature markets. It suggests different investment propensity between the mature markets and the emerging markets since it just experiences the first stage of capitalism. It is important to investigate the financial dynamics of the Chinese stock market to achieve more comprehensive understanding of the financial markets.
In this paper, to broaden the understanding of the scaling and memory effect of the volatility return intervals ττ for the emerging markets, we investigate the probability distribution and the memory effect of ττ for the Chinese stock market. In the next section, we present the data set we analyzed. In Section 3, we show the way to remove the intraday pattern. In Section 4, we show the probability distribution of ττ. In Section 5, we investigate the clustering phenomena by analyzing the conditional probability distribution Pq(τ|τ0)Pq(τ|τ0), the mean conditional interval 〈τ|τ0〉〈τ|τ0〉 and the cumulative distribution of the cluster size of ττ. In Section 6, we investigate the persistence probability distribution P±(t)P±(t). Finally comes the conclusion.
In summary, we have investigated the probability distribution function Pq(τ)Pq(τ) of the volatility return intervals ττ for the Chinese stock market. Scaling behavior is observed after Pq(τ)Pq(τ) and ττ are rescaled as View the MathML sourcePq(τ)=1/τ¯f(τ/τ¯). The scaling curve can be fitted by a stretched exponential function View the MathML sourcef(x)∼e−αxγ with α=5.1α=5.1 and γ=0.30γ=0.30, which is far different from the Poisson distribution. It suggests there exists memory in ττ. We then study the conditional probability distribution Pq(τ|τ0)Pq(τ|τ0) and the mean conditional return interval 〈τ|τ0〉〈τ|τ0〉. The results show that both the Pq(τ|τ0)Pq(τ|τ0) and the 〈τ|τ0〉〈τ|τ0〉 depend on the previous volatility return intervals τ0τ0. To obtain the clustering phenomena in a more direct way, we investigate the cumulative probability distribution of the cluster size of ττ. Clear clustering effect is observed for the relative small value of ττ. We further investigate the persistence probability distribution of ττ. It is found that P−(t)P−(t) decays by a power law, with the exponent value far different from the value 0.5 for the random walk. The Pq(τ|τ0)Pq(τ|τ0), the 〈τ|τ0〉〈τ|τ0〉 and the cumulative probability distribution of the cluster size of ττ for the Chinese stock market are similar to those obtained from the United States and the Japanese stock markets. The persistence probability further confirms the long memory of ττ for the Chinese stock market. Compared with the mature financial markets, we find that, as a emerging market, the Chinese stock market may have some unique features, however, it shares the similar scaling and long memory properties for the volatility return intervals as the United States and the Japanese stock markets.
