  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.
