High frequency volatility of volatility estimation free from spot volatility estimates

We define a new estimator of the volatility of volatility process based only on a pre-estimation of the Fourier coefficients of the volatility process. We prove the consistency and investigate the finite sample properties of the estimator in the presence of noise contamination by computing the bias of the estimator due to noise and showing that it tends to zero as the number of observations increases, under suitable assumptions. In a simulation study, the performance of the Fourier estimator 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.