Nonlinear GARCH Models And Applications To The Volatilities Of Stock Returns

Nowadays, China opens her finance market to the world, and then the hot money from inside and outside will come into the capital market. It is somehow good news to the market, for it will excite the market. But in another hand, the investors should take more risk because of more uncertain factors. Especially, the financial institutions in China are experiencing the change that they lose the support form the country. So the monetary market and the capital will appear the unusual volatility. In this special period, it is a real challenge that we never faced before. To improve our competing ability in the finance market and the ability of enduring the risk, we must learn how to find the risk immediately and how to measure the risk and forecast it. By this, we could keep the country safe in finance.The purpose of this paper is giving a good method to describe the volatility and measure the risk by analyzing the GARCH models.The structure of this paper is as following:At first, we present the background of the models, and the problems we want to solve are given. And we also report the related researches published before. Secondly, the paper gives the specification and the characteristics of class of GARCH model. Especially, we pay more attention on the SWARCH. Thirdly, the paper gives the process of estimating the GARCH and SWARCH. Finally, we compare the different models from an empirical result of stock market.In chapter 1, the paper present the background and the development process of the model. As above, we learn that the fields of the research and the application are broad, from the gross and the business cycle to measuring the market risk and to pricing the financial asset. GARCH has become a popular model on the area of measuring violability. In chapter 2, the paper describes the specification of the GARCH and the SWARCH. We propose the GARCH model that was set by Bollerslev, Chou and Kroner(1992). And then, based on this model we discuss its characteristics.The GARCH model has generated some other specifications which is driven from the original. This paper we mainly introduced the TGARCH and EGARCH which could reflect the asymmetrical.After that, we discuss the SWARCH which is given by Hamilton and Susmel(1994). In this part, we give more explanation on the Markov state switching. And then, we also make some research on the forecast and the persistence. We found that the persistence factor is significantly smaller than the other GARCH, so its forecast is well.In chapter 3, we give the deduction of the estimation for the parameters. As to SWARCH, because of the addition of the Markov state switching in the model, it becomes more complicated. Here, we give the iteration of the calculation. Let:The iteration was started withξ_1|0, and consider the parameters are known. the iteration process, we got the maximum log-likelihood value.In chapter 4, we make a empirical test by using the stock index data, and by this, we compare the different models on forecast and the persistence. The result indicates that SWARCH model is better than the traditional models. We could learn the distribution of the smooth probabilities. From the result, we learn that, the returns have no relationship with the lag period.Now, more of the empirical test is about the GARCH, but little to the SWARCH. One of the reasons is that the effect on the fit between GARCH and SWARCH has no much difference, and the good character of SWARCH on forecast doesn't appear. The other is that we can't get the estimation, owing to the complicated calculation.The meaning of this paper is that, in one hand, we research the class of GARCH models on strict mathematical deduction. In another hand, we use the iteration calculation to estimate the SWARCH, and make a empirical research.