Asymptotic results for the Fourier estimator of the integrated quarticity

In this paper, we prove a central limit theorem for an estimator of the integrated quarticity based on Fourier analysis, strictly related to the one proposed in Mancino and Sanfelici (Quant Finance 12: 607–622, 2012). Also, a consistency result is derived. We show that the estimator reaches the parametric rate $$\rho (n)^{1/2}$$ρ(n)1/2, where $$\rho (n)$$ρ(n) is the discretization mesh and n the number of points of such discretization. The optimal variance is obtained, with a suitable choice of the number of frequencies employed to compute the Fourier coefficients of the volatility, while the limiting distribution has a bias. As a by-product, thanks to the Fourier methodology, we obtain consistent estimators of any even power of the volatility function as well as an estimator of the spot quarticity. We assess the finite-sample performance of the Fourier quarticity estimator in a numerical simulation.