Asymptotics for the Fourier estimators of the volatility of volatility and the leverage

In this paper, we construct non parametric estimators of the volatility of volatility and the leverage component (covariance between the asset price and the volatility process) in the framework of one dimensional stochastic volatility model. The main feature of our estimator is that, given discrete observations of the price process, we are able to reconstruct the entire trajectory of the volatility. Thus, we handle the volatility as an observable variable and the Fourier coefficients of the volatility of volatility and the leverage processes can be computed. The estimators of the integrated quantities are easily obtained by means of the zero-Fourier coefficients. We prove consistency and feasible central limit theorems for the proposed estimators.