  Using a large dataset on major FX rates, we test the robustness of the rough
fractional volatility model over different time scales, by including smoothing
and measurement errors into the analysis. Our findings lead to new stylized
facts in the log-log plots of the second moments of realized variance
increments against lag which exhibit some convexity in addition to the
roughness and stationarity of the volatility. The very low perceived Hurst
exponents at small scales is consistent with the rough framework, while the
higher perceived Hurst exponents for larger scales leads to a nonlinear
behavior of the log-log plot that has not been described by models introduced
so far.
