  Volatility forecasting is crucial to risk management and portfolio
construction. One particular challenge of assessing volatility forecasts is how
to construct a robust proxy for the unknown true volatility. In this work, we
show that the empirical loss comparison between two volatility predictors
hinges on the deviation of the volatility proxy from the true volatility. We
then establish non-asymptotic deviation bounds for three robust volatility
proxies, two of which are based on clipped data, and the third of which is
based on exponentially weighted Huber loss minimization. In particular, in
order for the Huber approach to adapt to non-stationary financial returns, we
propose to solve a tuning-free weighted Huber loss minimization problem to
jointly estimate the volatility and the optimal robustification parameter at
each time point. We then inflate this robustification parameter and use it to
update the volatility proxy to achieve optimal balance between the bias and
variance of the global empirical loss. We also extend this Huber method to
construct volatility predictors. Finally, we exploit the proposed robust
volatility proxy to compare different volatility predictors on the Bitcoin
market data. It turns out that when the sample size is limited, applying the
robust volatility proxy gives more consistent and stable evaluation of
volatility forecasts.
