PhD Thesis Defense: "Kernel Estimation of Spot Volatility and Its Application in Volatility Functional Estimation"

Speaker: Bei Wu, Washington University in Saint Louis

Abstract: It\^o semi-martingale models for the dynamics of asset returns have been widely studied in financial econometrics. A key component of the model, spot volatility, plays a crucial role in option pricing, portfolio management, and financial risk assessment. In this dissertation, we consider three problems related to the estimation of spot volatility using high-frequency asset returns. 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 an optimal convergence rate for a general two-sided kernel under a quite broad class of models, which includes leverage effects and jumps of bounded and unbounded variations on both the return and volatility processes.

For our second project, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra-high-frequency observations of the asset returns under a continuous It\^o semimartingale model with additive microstructure noise. We establish a new Central Limit Theorem for the estimation error with an optimal rate and, as an application, we study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator's asymptotic variance is minimal for two-sided exponential kernels, hence, justifying the need of working with kernels of unbounded support as opposed to the most commonly used uniform kernel. We also develop a feasible implementation of the proposed estimators with optimal bandwidth. Monte Carlo experiments confirm the superior performance of the devised method.
Lastly, as an application of spot volatility estimation, we study the problem of estimating integrated volatility functionals; that is, integrals of a given function of the volatility over a specified period (typically, one day). We propose a Riemann sum bases estimator with kernel spot volatility estimator as plug-in. We prove a Central limit theorem for the estimator with optimal bandwidth. We show our estimator's bias is minimal for exponential kernels and provide an unbiased central limit theorem for the estimator with a proper de-biasing term.  Monte Carlo experiments confirm the advantage of using a general kernel estimator.


Host: Jose Figueroa-Lopez

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