Bootstrap Methods for Univariate and Multivariate Volatility

Abstract

This thesis focuses on developing bootstrap procedures for realized volatility estimators, which are often used to measure the financial market volatility based on high frequency intraday data. Unlike the commonly used continuous-time stochastic volatility (SV) models, discrete-time models for the logarithmic returns with underlying time varying volatility functions are investigated. In these models, the innovation term is not necessarily normally distributed and weak dependence is allowed. To describe this weak dependence, we make use of the geometric-moment contracting (GMC) property as an underlying assumption. The central limit results for our discrete-time models are given. For the univariate discrete-time models, we propose a kernel estimator to capture the time varying volatility structure, and thereafter estimate the underlying innovations. In chapter 2, the innovations are assumed to be independent. We propose a nonparametric i.i.d. bootstrap procedure by resampling the estimated noise innovations, and a nonparametric wild bootstrap procedure by generating pseudo-noise that imitates correctly the first and second order properties of the underlying noise. In chapter 3, the innovation term is assumed to be a time series with weak dependence. We combine the kernel volatility estimation with the linear process bootstrap of McMurry and Politis (2010). This proposal highly depends on the accuracy of the kernel estimation. For our proposed kernel estimator, the application is restricted to those time series, in which the autocovariance decays geometrically. In the multivariate discrete-time models, the varying volatility structure cannot be estimated anymore. However, the underlying volatility is assumed to be a smooth function, and the return process is therefore locally stationary. Based on this property, we propose to use the local bootstrap approach of Shi (1991) for the model with independent innovations, and the local block bootstrap of Paparoditis and Politis (2002) for the model with weak dependence. As an alternative to the local block bootstrap method, we propose a local dependent wild bootstrap procedure by application of the dependent wild bootstrap of Shao (2010) in nonoverlapping local windows. The validity of all bootstrap proposals is proved, and the finite sample properties of some proposals are investigated in simulation studies.