High-frequency volatility of volatility estimation free from spot volatility estimates

We define a new consistent estimator of the integrated volatility of volatility based only on a pre-estimation of the Fourier coefficients of the volatility process. We investigate the finite sample properties of the estimator in the presence of noise contaminations by computing the bias of the estimator due to noise and showing that it vanishes as the number of observations increases, under suitable assumptions. In both simulated and empirical studies, the performance of the Fourier estimator with high-frequency data is investigated and it is shown that the proposed estimator of volatility of volatility is easily implementable, computationally stable and even robust to market microstructure noise.