Technically the definition of volatility is the amount of movement either up or down, meaning a true volatility index would drop if the underlying bucket moves less than X up or down, and rises if the bucket moves more than X. We have VIX, but this seems to be considered a fear index, moving inversely with the market. There's more nuance to it, but am I getting that essentially correct?
If so, does a true volatility index exist, moving up on big underlying movement and down on calm days? This would make it a fear/exuberance index.