The integrated covariance is a multivariate and stochastic generalization of the univariate constant volatility parameter in continuous time finance models. Its estimation is indispensable for a broad range of financial risk management. The availability of intra-daily financial data renders a non-parametric estimation of this variable possible, yet the measurement error likely to contaminate the intra-daily data poses another challenge for currently available estimators.

In chapter one I propose a bias-corrected non-parametric estimator of this variable. It is constructed using a linear combination of two realized kernels. In addition to its simplicity, my estimator has desirable statistical properties including consistency, asymptotic normality and the best parametric rate of convergence n^1/4 for the intra-daily sample size n, given serially dependent measurement error. Simulations show that my estimator has much smaller bias than currently available estimators with only a mild increase in variance, so that the overall mean squared error is also smaller. The proposed method is applied to the estimation of the dynamic hedge ratio between spot and futures prices of the SP 500 index.

In chapter two I study the inferential performance of my estimator proposed in chapter one. Simulations confirm that the confidence intervals of my estimator have smaller sizes and better coverage rates than those of the realized kernel estimator, given the common proxy for the integrated quarticity in constructing the intervals. In an application to the estimation of integrated variances of two representative futures returns, I obtained tighter confidence intervals of the estimator than those of the realized kernel estimator.

In chapter three I propose a novel non-parametric estimator of the instantaneous volatility of security returns within a daytime trading period. It is constructed using two different kernel windows, the one for the estimation of the integrated variance and the other for its localization at a particular time within a sample period. I prove its consistency, asymptotic normality and the supremal rate of convergence as a function of the smoothness of the instantaneous volatility path. Based on noisy observations with volatility driven by a Brownian diffusion process, the rate is n^1/8.