We present a symmetry analysis of the distribution of variations of different financial indices, by means of a statistical procedure developed by the authors based on a symmetry statistic by Einmahl and Mckeague. We applied this statistical methodology to the here introduced financial uninterrupted daily trends returns and to other novel multi-scale observable defined as trend returns divided by their durations. In our opinion, to study distributional symmetry, trends returns offer more advantages than the commonly used daily financial returns; the two most important being: (1) Trends returns involve sampling over different time scales and (2) By construction, this variable time series contains practically the same number of non-negative and negative entry values.

In addition, we show that the two studied time multi-scale returns display distributional bimodality and the mechanism of its emergence is explained. Daily financial indices analyzed in this work, are the Mexican IPC, the American DJIA, DAX from Germany and the Japanese Market index Nikkei, covering a time period from 11-08-1991 to 06-30-2017. Finally, it is shown that, at the time scale resolution and significance considered in this paper, it is almost always feasible to find an interval of possible symmetry points containing one most plausible symmetry point denoted by $C$. Finally, we study the temporal evolution of $C$ showing that this point is seldom zero and responds with sensitivity to extreme market events.

通过作者根据基于Einmahl和Mckeague的对称统计量开发的统计程序，我们对不同财务指标的变动分布进行对称分析。我们将此统计方法应用于此处介绍的财务不间断的每日趋势收益以及其他新颖的多尺度可观测值，定义为趋势收益除以持续时间。我们认为，研究分布对称性，趋势收益比常用的每日财务收益更具优势。最重要的两个因素是：（1）趋势收益涉及在不同时间范围内进行采样；（2）通过构造，此可变时间序列实际上包含相同数量的非负和负输入值。

此外，我们表明，这两个研究的时间多尺度收益显示出分布双峰态，并解释了其出现的机理。这项工作中分析的每日财务指数是墨西哥IPC，美国DJIA，德国DAX和日本市场指数日经指数，涵盖时间范围为1991年11月8日至2017年6月30日。最后表明，在本文考虑的时间尺度分辨率和重要性下，找到一个可能的对称点的间隔几乎总是可行的，其中可能的对称点包含一个用符号表示的最合理的对称点$C$。最后，我们研究了时间的演变$C$ 表明这一点很少为零，并且对极端的市场事件敏感。