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.
翻译:挥发性预测对于风险管理和组合构建至关重要。 评估波动预测的一个特殊挑战是如何为未知的真实波动构建一个强有力的替代物。 在这项工作中,我们显示两个波动预测器之间的实证损失比较取决于波动替代物与真实波动的偏差。 然后我们为三个稳健的波动代理物建立非零散偏差界限, 其中两个基于剪裁数据, 第三个基于指数加权赫伯损失最小化。 特别是, 为了让赫伯方法适应非静止金融回报, 我们提议解决一个无调适的加权赫伯损失最小化问题, 以联合估计每个时间点的波动和最佳稳健化参数。 我们随后将这一稳健化参数扩大并用来更新波动替代物, 以便在全球经验损失的偏差和差异之间实现最佳平衡。 我们还推广了Huber方法来构建波动预测器。 最后, 我们利用拟议的稳健的波动替代物来比较Bitcoin市场数据上的不同波动预测器。 它指出, 当抽样规模有限时, 应用稳健的波动性代理物会提供更稳定且更稳定的预测。