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We first revisit the problem of estimating the spot volatility of an It\^osemimartingale using a kernel estimator. We prove a Central Limit Theorem withoptimal convergence rate for a general two-sided kernel. Next, we introduce anew pre-averaging/kernel estimator for spot volatility to handle themicrostructure noise of ultra high-frequency observations. We prove a CentralLimit Theorem for the estimation error with an optimal rate and study theoptimal selection of the bandwidth and kernel functions. We show that thepre-averaging/kernel estimator's asymptotic variance is minimal for exponentialkernels, hence, justifying the need of working with kernels of unboundedsupport as proposed in this work. We also develop a feasible implementation ofthe proposed estimators with optimal bandwidth. Monte Carlo experiments confirmthe superior performance of the devised method.

Kernel Estimation of Spot Volatility with Microstructure Noise Using Pre-Averaging