The MCMC Method Research On Volatility Of CSI 300 Stock Index Base On The GARCH-Jump Model

The dynamics of financial asset price and its volatility model have been important research topics in the academe and business. Because of some disadvantages of the existing financial asset prices and its volatility model, they are difficult to fit the movement behavior of financial markets which have emergencies, and with the strengthening of the complexity of the model, parameter estimation brings a lot of restrictions to the practical application of the model. Therefore, it is highly significant to seek suitable models and suitable methods for studying the volatility of financial assets.When people estimated the model of financial time series, mostly based on classical statistical methods, and used the actual volatility which is calculated in the low-frequency data as a measure of the estimated model of volatility. Therefore, in this paper, we establish the GARCH-Jump model and try to research on volatility of CSI 300 stock index based on markov chain monte carlo(MCMC)simulation method. At the same time, we use the 5 minutes high-frequency data to build intraday volatility yields which called realized volatility as the real market volatility benchmark. Then we use the realized volatility as evaluation criteria of the volatility forecast series and evaluate prediction accuracy rate of the volatility model by the loss function metric indexes and DM test statistics, and select the optimal model to forecast the volatility.First of all, in order to examine the effectiveness of the model, we apply the ML (maximum likelihood) and MCMC method to GARCH-Jump model which showed in formula (2.15) in simulation experiment. Simulation experimental results of ML method show that in addition to a small number of parameters, the rest of the parameter estimates are close to the true values. Among the model parameters, except the p value of y is slightly larger than 0.1, the rest of the coefficients are significant. For using MCMC method, after reaching convergence, output parameter estimates of the model are very close to the true values, and all true values are within the 95% confidence interval. Using loss function metrics to evaluate the model fitting degree, we can see the MSE and MAE values are both very small of two different methods, which show these two methods both have a good fitting effect. But, in contrast, the MSE and MAE values which are produced by applying MCMC method estimate are smaller. And then we use the DM statistics to test, results show that MSE and MAE values produced by loss function metric indexes have significant differences. From what have been discussed above, we can know MCMC method is better than ML method in estimating GARCH-Jump model simulation effect.Secondly, in the empirical study of the volatility of CSI 300 stock index, we find the CSI 300 index logarithm yield sequence is a smooth and non-white noise sequence, which has left, rush fat-tailed features, heteroscedastic phenomena and obviously fluctuating changes. Thereby we build a ARMA (p, q) model for the conditional mean of the sequence, and according to the BIC worthy of the model to get the optimal sequence average equations for ARMA (0,0) model. Therefore, we directly establish a inhomogeneous and asymmetric GARCH-Jump model for the sequence established. By using the ML method to estimate parameters of the model and using the LR test statistics inspection we found that the model structure is unreasonable, the CSI 300 index is jumping, and leaping gathered, inhomogeneous, but not with asymmetrical characteristics. According to the result of likelihood ratio, we reconstruct a new model namely GARCH- Jump as in formula (4.1). Compared with GARCH (residual errors based on the standard normal distribution, t distribution and generalized error distribution) model, using the realized volatility as evaluation criteria of the volatility forecast series. According to the result of the loss function metric indexes and DM test statistics, the GARCH-Jump model as in formula (4.1) can depict the volatility characteristics of CSI 300 index, namely the GARCH-Jump model prediction ability is better than GARCH model.Finally, we use another method namely MCMC method to estimate the GARCH-Jump model as in formula (4.1), constructing two markov chains to Gibbs sampling. By increasing the number of iterations gradually and observing autocorrelation figure and variance ratio of each parameter. When the number of iterations is 15000 times, the model achieves convergence. Comparing with the ML method of estimation results, through loss function metric indexes and DM test statistics results show that using MCMC method of volatility forecast is better than ML method, which provides a certain reference value to financial investment and risk control.