Realized kernel estimation of integrated volatility using high frequency data with random trading times

Estimating integrated volatility is one of the most important and challenging tasks in quantitative finance. As the ultra-high frequency data becomes available, nowadays new methods for integrated volatility estimation which can take use of as much high frequency data as provided are needed. Because of the complexity of the ultra-high frequency data, these new methods mostly nonparametric in nature. However, most of these nonparametric methods require the price data to be equally spaced in order to obtain asymptotic result. Examples include the class of realized kernel method, the two-time scale method, the bi-power method, and so on. On the other hand, the ultra-high frequency data available in the market is generally non-equally spaced. Applying these nonparametric methods means that we throw away a lot of the data available to make the data equally spaced. In this thesis, we redefine the classical realized kernel estimator in order to accommodate random trading times. In order to characterize its asymptotic behavior, we prove several law of large numbers (LLN) and central limit theorems (CLT) for functionals of unequally-spaced data sampled from a continuous semi-martingale. We show that, based on appropriate constrains, the effect of randomness in sampling time or trading time can be controlled well enough to obtain useful asymptotic results for functionals of the didifferences of price data. Moreover, the CLTs provide us important tools to construct nonparametric methods to estimate integrated volatility using ultra-high frequency data. Based on the main CLT we further develop the asymptotic theory for our redefined realized kernel estimator of the integrated volatility. We show explicitly how the randomness of trading times and the micro-structural noise have influenced the asymptotic properties of our estimator