A multi-scale symmetry analysis of uninterrupted trends returns in daily financial indices

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 financial uninterrupted daily trends returns and to other derived observable. 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. We also show that these time multi-scale returns display distributional bi-modality. 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. We show 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.