Abstract

It\^o semimartingale 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 quite mild assumptions, 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. 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 any given function of the volatility over a specified period (typically, one day). We propose a Riemann sum based estimator with kernel spot volatility estimators as plug-ins. We prove a Central Limit Theorem for the estimator with optimal convergence rate. We show our estimator's bias can be reduced by adopting a general kernel, and we 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.

Committee Chair

Jose E Figueroa-Lopez

Committee Members

Jimin Ding

Degree

Doctor of Philosophy (PhD)

Author's Department

Statistics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Winter 12-15-2022

Language

English (en)

Author's ORCID

http://orcid.org/0000-0002-0551-6466

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