| Volatility is central to many applied issues in financial economics and financial engineering, ranging from asset allocation and derivative pricing to risk management. In Markowitz's portfolio selection model (1959), volatility is one of the two dimensions. Inthe other backbone of modem financial economics ------ option pricing, volatility is themost important factor. And, in the dominating risk measure of applied finance ------ valueat risk, estimating volatility and covariance matrix is the first important step. Moreover, volatility is widely used in many other fields of financial economics, such as asset pricing and performance evaluation.For the last 20 years, volatility has been the hotspot of the financial economics. Since the first conditional volatility model by Engle (1982), thousands of papers concerning conditional heteroskedasticity have been published. Most recently, Anderson, Bollerslev, Diebold, Labys and other economists developed a new estimator of volatility, and, because of its high precision, we can regard it as observed volatility, or "realized volatility".Although some economists have developed and investigated the measuring and forecasting methods of realized volatility, there remain some deficiencies.First of all, asset price series does not follow normal diffusion process exactly. So the frequency of data used to estimate volatility should not be too high. Data of very high frequency would bring more errors due to microstructure friction. Torben G. Andersen and his coworker chose a data interval of 5 minute when studying the volatility of DAIJ 30 stocks. Later, they developed a method called "signature plot" to select proper data interval. However, the optimum data interval to balance the usual measuring error and the microstructure error would not be unique along all the sample periods. It seems to vary in different periods and different markets.Secondly, Andersen and his coworker developed FIVAR (fractional integrated vector auto regression) model to forecast exchange volatility. This model would not be proper when considering stock volatility because it does not take asymmetric effect into account.Thirdly, Ebens (1999) developed ARFIMAX model for forecasting stock volatility. He used the conditional sum-of-squares maximum likelihood method to estimate the parameters. This estimating method would cause large error when considering small datasample. Moreover, his ARFIMAX model deals with fractional differencing before asymmetric regression. As compared to the improved model developed in this thesis, it has disadvantages both in parameter estimation and explanatory adequacy.Lastly, previous models either consider only one time series, or consider more time series on the assumption that correlations do not change over time. This is unrealistic assumption.This paper develops a more accurate volatility measuring method, based on the previous ARFIMAX model, by systematically investigating the volatility of China stock indexes, covering high-frequency estimate, volatility characteristics, simulating and forecasting.First of all, we find that the microstructure bias of single stock is opposite to that of stock index. Using very high-frequency data, the realized volatility of single stock would be much higher, whereas that of stock index would be much lower. Based on the volatilities estimated with different data frequency, we develop a more accurate estimating method, which can effectively balance the microstructure error and usual estimate error.Secondly, we investigate some characteristics of the volatility for China stock indexes, such as the distribution of return, the distribution of volatility, the asymmetric effect of volatility, and the long memory effect of volatility.Thirdly, we develop a new mode based on ARFIMAX model to simulate and forecaste the volatility of China stock indexes and compare it with early models such as GARCH, EGARCH, FIGARCH and FIEGARCH. It has been shown that our model is better than the previous ones according to the data...
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