The Bayesian Estimation Of GARCH-STABLE Model And Financial Application

The data from the financial market always have very fat tail, however the traditional distributions can not catch this characteristic. The stable distribution can not only describe the heavy tail, but also the skewness. It has a series of excellent properties, but it doesn't have a closed form of probability density function, so it is somewhat difficult to estimate its parameters. This article estimates its four parameters with MCMC method from the Bayesian view, then we use this method to complete the modeling of GARCH based on stable distribution. The structure of this paper is as follows,Firstly we introduce the return of financial market, the features of the financial market behavior and the distributions used to capture the characteristics. Then we define the stable distribution with four equivalent ways and analyze its properties carefully. To study the stable distribution, we must know how to simulate its random variables. We can learn it intuitively by watching its plot of density function.By adding an auxiliary variable in the Chapter 4, we estimate the four parameters using Gibbs sampler. The plot of auxiliary variable's posterior density function is uni-mode, so we use adaptive rejection sampling to generate auxiliary variables. However the posterior density functions of character parameter, skew parameter and location parameter are multi-mode, so the slice sampling is used to make the algorithm more efficient. In the Chapter 5 we define the GARCH model based on the stable distribution and show how to complete the estimates.At last we confirm the Bayesian estimation is appropriate by simulating stable random numbers. Then we use this method to analyze the Chinese financial market and find out that it is more appropriate than the normal distribution to fit the stock return of Shanghai and Shenzhen using stable distribution. The GARCH-STABLE model can evaluate the risk of financial market more clearly.