  We develop a new stock market index that captures the chaos existing in the
market by measuring the mutual changes of asset prices. This new index relies
on a tensor-based embedding of the stock market information, which in turn
frees it from the restrictive value- or capitalization-weighting assumptions
that commonly underlie other various popular indexes. We show that our index is
a robust estimator of the market volatility which enables us to characterize
the market by performing the task of segmentation with a high degree of
reliability. In addition, we analyze the dynamics and kinematics of the
realized market volatility as compared to the implied volatility by introducing
a time-dependent dynamical system model. Our computational results which
pertain to the time period from January 1990 to December 2019 imply that there
exist a bidirectional causal relation between the processes underlying the
realized and implied volatility of the stock market within the given time
period, where it is shown that the later has a stronger causal effect on the
former as compared to the opposite. This result connotes that the implied
volatility of the market plays a key role in characterization of the market's
realized volatility.
