| Volatility is an index commonly used to measure the risk of financial market. In recent decades,the estimation and prediction about volatility is one of hot topics in the financial research field. Spot volatility plays an crucial role in hedge, option pricing, risk analysis and portfolio management and some other financial activities. Thanks to the popularity of electronic trading and the development of the information storage technology, it is much easier today to obtain the data saved by the scale of "minutes" and "seconds". This so called high-frequency data can capture the market information quickly and effectively. Moreover, compared to the low-frequency data, the high-frequency data can reflect the true circumstance of the financial markets, and make the estimation of the volatility possible.In recent years, financial theory and empirical study indicate the existence of jumps in the asset prices. Moreover, the jumps and their type have significant influence on volatility estimators. In this thesis, using intra-day high-frequency data, we consider the volatility estimations in two kinds of jump-diffusion processes, i.e. the finite activity jumps(compound Poisson jumps) and infinite activity jumps. The estimators for the volatility will be constructed and asymptotic properties will be established, which extend corresponding results of the continuous case.For the jump-diffusion process containing finite activity jumps(compound poisson jumps),applying the threshold method and kernel function technology, this paper present the volatility estimator, and then prove its asymptotic normality. Furthermore, by G?rtner-Ellis theorem and Delta method in large deviations, the moderate deviations with explicit rate functions are obtained. Finally,for the jump-diffusion process containing infinite activity jumps, under the condition that the Lévy measure is ?-stable, the law of large numbers and asymptotic normality for the volatility estimator are given. |
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