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We study the volatility functional inference by Fourier transforms. Thisspectral framework is advantageous in that it harnesses the power of harmonicanalysis to handle missing data and asynchronous observations without anyartificial time alignment nor data imputation. Under conditions, this spectralapproach is consistent and we provide limit distributions using irregular andasynchronous observations. When observations are synchronous, the Fouriertransform method for volatility functionals attains both the optimalconvergence rate and the efficient bound in the sense of Le Cam and H\'ajek.Another finding is asynchronicity or missing data as a form of noise produces"interference" in the spectrum estimation and impacts on the convergence rateof volatility functional estimators. This new methodology extends previousapplications of volatility functionals, including principal component analysis,generalized method of moments, continuous-time linear regression models etcetera, to high-frequency datasets of which asynchronicity is a prevailingfeature.

The Fourier Transform Method for Volatility Functional Inference by Asynchronous Observations