Bayesian Stochastic Volatility Models And Applications:Based On Monte Carlo Simulation Algorithms

During the development of financial economics and mathematical finance, the instability always existed in the whole system. Therefore, the main ingredient in order to avoid risk capital theory and financial instruments has got a wide range of study and attention. To modeling the fluctuation process in the instability system, a lot of empirical studies indicate that the time series of the financial and economic areas have some new features against the classic assumptions in the econometric theory. The features include the high kurtosis and heavy tails, volatility clustering, the nonlinear dynamic structure and so on. Breakthrough limits of the computation, the modeling process of the time varying volatility provides effective analysis tools for the effectively risk management. There are two classes of time varying volatility specifications which are autoregressive conditional heteroskedasitic models(ARCH) and stochastic volatility models(SV). The SV model is an alternative to autoregressive conditional heteroscedastic model providing a more flexible structure for describing the time varying volatility. However, the variation of SV model is decided by an unobserved stochastic process which provides a more flexible structure for describing the time-varying volatility.As the SV model add an error term in the dynamics of volatility introducing another source of randomness, it has been found to fit asset returns better, have residuals closer to standard normal and have better statistical properties. However, they are typical nonlinear non-Gaussian state space models, with the exact likelihood function being a highly complex and high-dimensional integration in analytical, leading to parameters and latent log-volatilities very difficult to estimate. The extensions of the SV model are even more complicated and the studies on the estimation problem have important theoretical and practical significance. In recent years, with the development of the computer science, the Monte Carlo estimation algorithm show the more advantages in dealing with high-dimensional integration problems. The paper mainly carries out the Markov Chain Monte Carlo estimation and the Sequential Monte Carlo method of the SV class of models. These methods are kinds of the Bayesian analysis which regards the parameters of the model as stochastic variables. In the Bayesian analysis, the critical values of test statistics can be defined exactly which is one of the drawback in traditional statistics. Moreover, the Bayesian methodology is effective in estimating the parameters in the anticipation of variates’production caused by the development of financial economics.The MCMC algorithm is one of developed rapidly and applied widely Monte Carlo methods in dealing with the Bayesian estimation of time varying volatility models. However, due to the unobservable state variables in the SV class of models, the traditional MCMC algorithms converge rather slowly in estimating this kind of models and don’t fit for the empirical analysis. Therefore, a major concern in this paper is to studying the effective MCMC methods. The simulation study shows that the multiple steps sampling MCMC based on the mixture normal distributions outperform other three MCMC methods both in sampling efficiency and the estimation precision. Furthermore, we give the state space approximation of the long memory stochastic volatility models with limit rank and study the effective MCMC methods. In the application area, two extensions of SV model have been used to investigate the dynamic relationship of the inflation and inflation uncertainty in China and describe the dynamic behavior of credit spreads didn’t concentrate on the different time scales of the mean-reversion process. All the results provide useful advices for the financial risk management and economic policies.On the other hand, one major problem of MCMC algorithm is whenever a new observation has been obtained the posterior probability has to be assessed once more in the process of MCMC algorithm, which cause the inefficiency of the estimation. Moreover, the samples take a lot of storage space. However, the SMC technology first predicts the prior probability density based on the transformation model of the dynamic system status, and then updates the posterior probability density using the recent observation. Therefore, the algorithm is suitable for the on-line data set which is common in the financial economics analysis, especially providing a more general solution outline for the system identification and parameter estimation of the nonlinear non-Gauss state space models. First of all, we study the system identification process of state space models based on the SMC algorithm. The simulation experiments have been conducted to both dynamic linear model and standard SV model. The results show that the auxiliary particle filter method for the SV model gets higher sampling efficiency than the common particle filter method, especially for the higher quantile probabilities.In the case of unknown parameters, we develops a sequential Bayesian filtering parameter learning algorithm based on the parameter learning method of artificial noise process prevailing in the SMC analysis. The spirits of the Bayesian filtering parameter learning algorithm are the using of sequential Bayesian analysis and the introducing of sufficient statistics which reduce the dimension of the target distribution. Therefore, the Bayesian filtering algorithm can avoid the degeneration of the samples and improve the efficiency of the sampling procedure. Our algorithm has been examined by many simulation analysis and the results show that our method can get high accuracy estimations. The comparison of several main SMC filtering algorithms show that our method is more effective than the auxiliary particle filter learning method and the Storvik filter learning method.Moreover, we pays particular attention to the SV models with structure change character which will improve the forecast accuracy of the volatility in financial market, avoid the investment risk and provide the basic results for financial risk management and decision-making. In the Bayesian perspective the SMC algorithms are applied to filtering the hidden states and estimated the parameters. The models have been used to the time series of China’s stock market and show the ability to capture the different volatility regimes and to improve the estimation precise. In empirical study of the futures, the sequential Bayesian filter which can avoid the subjective prejudice of the prior information, as well as the noise infection of posterior information, is more effective than the MCMC method. Moreover, the co-movement between the tail characteristics of the two markets shows not only the price discovery ability but also the structure discovery capability.