Intraday Periodic Volatility Curves

Torben G. Andersen, Tao Su, Viktor S Todorov*, Zhiyuan Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
JournalJournal of the American Statistical Association
StateAccepted/In press - 2023


  • Calendar effect
  • Functional central limit theorem
  • High-frequency data
  • Nonparametric estimation
  • Volatility

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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