Date of Award
8-8-2024
Degree Name
Doctor of Philosophy (PhD)
Degree Type
Dissertation
Abstract
In financial economics, semimartingales hold great importance among all the stochastic processes, and the Itô framework, driven by Brownian motion and Poisson random measure, provides a robust way to model a wide range of phenomena, particularly in finance, and therefore, Itô semimartingales are almost universally preferred in practical applications. Based on high-frequency return data, the accurate estimation of the spot volatility of Itô semimartingales is crucial for addressing numerous issues such as hedging, option pricing, risk analysis and portfolio management. In this dissertation, we investigate kernel type estimators and examine two issues concerning the estimation of spot volatility. Firstly, for a multidimensional Itô semimartingale, we revisit the problem of estimating integrated volatility functionals. [27] studied a plug-in type estimator based on a Riemann sum approximation of the integrated functional and a spot volatility estimator with a forward uniform kernel. Motivated by recent results that show that spot volatility estimators with general two-side kernels are more accurate, in this project, an estimator using a general kernel spot volatility estimator as plug-in is considered. As for the discontinuous Itô semimartingale with jumps, we introduce the corresponding truncated version of the spot volatility estimator as in [28]. A central limit theorem for estimating the integrated functional with a general kernel estimator is established with an optimal convergence rate. An unbiased central limit theorem for the estimator with a proper de-biasing term is also obtained. In the case of the right-sided uniform kernel, the stable convergence in law for the estimator of the integrated functional in [28] can be recovered from the proposed theorem. Our results show that one can significantly reduce the estimator’s bias by adopting a general kernel instead of the standard uniform kernel. In the sensitivity test among different bandwidths, the proposed bias-corrected estimator maintains a remarkable record of robustness in a variety of sampling frequency and functions, compared with the multi-scaled Jackknife estimators in [31]. For our second project, we focus on the selection of the optimal bandwidth for the kernel estimators of spot volatility. Based on three different measures of the estimation errors, namely mean squared error (MSE), average squared error (ASE), and cross-validation function, a data-driven selection method is proposed, with the selected bandwidths being asymptotically optimal. Furthermore, under certain rate of convergence, we show the asymptotic behavior of both the bandwidth selector and estimation error. In the simulation study, Monte Carlo experiments confirm that the difference between the bandwidth minimizing MSE and the one minimizing ASE converges to normal distribution, while the limit distribution of difference between estimation errors is χ2 distribution.
Language
English (en)
Chair and Committee
José Figueroa-López
Committee Members
Carlos Misael Madrid Padilla; Jimin Ding; Nan Lin; Werner Ploberger
Recommended Citation
Pang, Jincheng, "Estimation of Integrated Volatility Functionals & Behavior of the Bandwidth Selector with Kernel Spot volatility Estimators" (2024). Arts & Sciences Electronic Theses and Dissertations. 3316.
https://openscholarship.wustl.edu/art_sci_etds/3316