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 . We prove Central Limit Theorems for the estimation error with an optimal rate and study the problems of optimal bandwidth and kernel selection. As in the case of a simple kernel estimator of spot volatility in the absence of microstructure noise, we show that the asymptotic variance of the pre-averaging/kernel estimator  is minimal for exponential or Laplace kernels, hence, justifying the need of working with unbounded kernels as proposed in this work. Feasible implementation of the proposed estimators with optimal bandwidth is also developed . Monte Carlo experiments confirm the superior performance of the devised method.
We first revisit the problem of estimating the spot volatility of an It\^o semimartingale using a kernel estimator. We prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel . Next, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra high-frequency observations . We prove a Central Limit Theorem for the estimation error with an optimal rate and study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator 's asymptotic variance is minimal for exponential kernels, hence, justifying the need of working with kernels of unbounded support as proposed in this work. We also develop a feasible implementation of the proposed estimators with optimal bandwidth . Monte Carlo experiments confirm the superior performance of the devised method.
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kernel estimation of spot volatility in a general continuous
estimating the spot volatility of an
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model in the absence of microstructure noise, and
using a kernel estimator. We
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, which is an extension of Figueroa and Li (2020) as we allow
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function
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to handle the microstructure noise of ultra high-frequency observations, we present a new type of
we introduce a new
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under the presence of additive microstructure noise
to handle the microstructure noise of ultra high-frequency observations
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Central Limit Theorems
a Central Limit Theorem
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problems of optimal
optimal selection of the
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selection. As in the case of a simple kernel estimator of spot volatility in the absence of microstructure noise, we
functions. We
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asymptotic variance of the
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's asymptotic variance
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or Laplace
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unbounded kernels
kernels of unbounded support
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Feasible
We also develop a feasible
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is also developed