累计报价收益率分析：反常扩散的证据
Analysis of aggregated tick returns: evidence for anomalous diffusion
摘要：
为了探讨股价大幅波动的根源，我们分析了2002年纳斯达克十大交易频繁的股票的股价变化。虽然交易频率在一定时间间隔内对总收益率的影响很重要，但它不能单独解释股票价格变化的厚尾分布。因此，我们分析了固定交易次数的区间，以消除交易频率的影响，并调查其他因素对总收益的相关性。我们发现，在报价时间的价格遵循一个离散的扩散过程与一个可变的步长，而在一个区间内的正负方向的步骤之间的差异是高斯分布。步长由单笔交易的回报给出，并与报价时间长期相关。因此，它的平均值可以很好地表征一个区间的许多行业，原来是一个重要的决定因素，为大的总回报。我们还提出了一个统计模型再现的累积分布的总收益。为了与经验分布的准确一致，我们还考虑了在不同方向上的步长的不对称性，以及这些不对称性与平均步长以及步长的符号之间的互相关。
英文摘要：
In order to investigate the origin of large price fluctuations, we analyze stock price changes of ten frequently traded NASDAQ stocks in the year 2002. Though the influence of the trading frequency on the aggregate return in a certain time interval is important, it cannot alone explain the heavy tailed distribution of stock price changes. For this reason, we analyze intervals with a fixed number of trades in order to eliminate the influence of the trading frequency and investigate the relevance of other factors for the aggregate return. We show that in tick time the price follows a discrete diffusion process with a variable step width while the difference between the number of steps in positive and negative direction in an interval is Gaussian distributed. The step width is given by the return due to a single trade and is long-term correlated in tick time. Hence, its mean value can well characterize an interval of many trades and turns out to be an important determinant for large aggregate returns. We also present a statistical model reproducing the cumulative distribution of aggregate returns. For an accurate agreement with the empirical distribution, we also take into account asymmetries of the step widths in different directions together with crosscorrelations between these asymmetries and the mean step width as well as the signs of the steps.
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