Before revision: We first revisit the problem of kernel estimation of spot volatility in a general continuous It\^o semimartingale model in the absence of microstructure noise, and prove a Central Limit Theorem with optimal convergence rate , which is an extension of Figueroa and Li (2020) as we allow for a general two-sided kernel function . Next, to handle the microstructure noise of ultra high-frequency observations, we present a new type of pre-averaging/kernel estimator for spot volatility under the presence of additive microstructure noise .
After revision: We first revisit the problem of kernel estimation of spot volatility in a general continuous It\^o semimartingale model in the absence of microstructure noise, and prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel function . Next, to handle the microstructure noise of ultra high-frequency observations, we present a new type of pre-averaging/kernel estimator for spot volatility under the presence of additive microstructure noise .