Intraday Periodic Volatility Curves

The volatility of financial asset returns displays pronounced variation over the trading day. Our goal is nonparametric inference for the average intraday volatility pattern, viewed as a function of time-of-day. The functional inference is based on a long span of high-frequency return data. Our setup allows for general forms of volatility dynamics, including time-variation in the intraday pattern. The estimation is based on forming local volatility estimates from the high-frequency returns over overlapping blocks of asymptotically shrinking size, and then averaging these estimates across days in the sample. The block-based estimation of volatility renders the error in the estimation due to the martingale return innovation asymptotically negligible. As a result, the centered and scaled calendar volatility effect estimator converges to a Gaussian process determined by the empirical process error associated with estimating average volatility across the trading day. Feasible inference is obtained by consistently estimating the limiting covariance operator. Simulation results corroborate our theoretical findings. In an application to S&P 500 futures data, we find evidence for a shift in the intraday volatility pattern over time, including a more pronounced role for volatility outside U.S. trading hours in the latter part of the sample. Supplementary materials for this article are available online.

金融资产回报率的波动性在交易日内表现出显著的日间变化。本研究的目的是对平均日内波动模式进行非参数推断，该模式被视为一天中不同时间点的函数。这种函数推断基于长期高频回报数据。我们的设置允许波动动态具有一般形式，包括日内模式的时变特性。估计方法基于从高频回报中形成局部波动估计，这些估计是在重叠的、随着样本规模趋于无穷大而递减的块中进行的，然后对样本中的每日估计值进行平均。基于块的波动估计使由于鞅回报创新引起的估计误差趋于可忽略。因此，中心化和缩放的日历波动效应估计量收敛于一个高斯过程，该过程由与日内平均波动估计相关的经验过程误差所决定。通过一致估计极限协方差算子，我们获得了可行的推断。仿真结果证实了我们的理论发现。在将该方法应用于标准普尔500期货数据的应用中，我们发现随着时间的推移，日内波动模式发生了变化，包括在样本后期，美国交易时间之外的波动性作用更为显著。本文的补充材料可在网上获取。