Financial Stochastic Volatility Models And Applications: Based On State Space Models With Finite Mixture

In recent years, stochastic volatility (SV) class of models are widely used in areas offinancial economics and mathematical finance. They can capture time-varying volatilities offinancial assets, and have important impacts on financial decision-making. The developmentof SV models is highly interdisciplinary, covering relevant theories and contents of financialeconomics, probability and statistics theory, and econometrics. They provide us with help tounderstand methods and models of real option pricing, e?ective asset allocation and accuraterisk assessment. As e?ective methods for describing time-varying volatility of an financialassets, the studies on the SV models have important theoretical significance and practicalsignificance.The SV class of models is a basic alternative to autoregressive conditional heteroscedastic(GARCH) class of models, but they provide a more ?exible structure for describing the time-varying volatility. As the model adds an error term in the dynamics of volatility introducinganother source of randomness, it has been found to fit asset returns better, have residualscloser to standard normal and have better statistical properties. However, they are typicalnonlinear non-Gaussian state space models, with the exact likelihood function being a highlycomplex and high-dimensional integration in analytical, leading to parameters and latent log-volatilities very di?cult to estimate. Although several extended SV models can be furtherused to examine some practical problems, such as asymmetric e?ect or leverage e?ect betweenvolatilities and returns, volatility clustering, volatility persistence and structural break, theyare more complex and thus too di?cult to estimate. Thus, searching for a computationallysimple and algorithmically e?ective estimation method is still the subject to explore in financialeconometrics, but also a major concern in this article.This paper carries out a series of studies around SV models. Facing some problems fromthe basic SV model and their extension, we review and sum up past researches, carding thetheoretical relation between SV models and modern finance. We mainly discuss estimationmethods for the basic SV model, extended SV models and the Markov-switching SV model.Then these models are applied to return data of China's Shanghai and Shenzhen stock market,and short-term lending interest rate of bank credit market.In many parameter estimation methods for the basic SV model, some studies advocateto use the Markov chain Monte Carlo procedure or the simulated expected maximum pro- cedure, through taking the square and the logarithm of the SV model, and approximatingthe log-chi-square distribution by a mixture of normal distributions. However, the maximumlikelihood estimation procedure for this transformed SV model has not yet been put forward.Not only that, but approximate estimation of the coupled model for Gaussian mixture process,Markov switching process and state space process has also not been presented in time serieseconometrics. Based on the exact analysis of Gaussian mixture state space model, Markovswitching state space model, and Markov switching Gaussian mixture state space model, wedevelop three approximate filters, i.e. AMF(k), MSAMF(k), and MSMIXAMF(k) respectively,and the corresponding approximate maximum likelihood estimators and smoothing techniques.With these approximate filter, approximate maximum likelihood estimator and approximatesmoothing, this paper presents estimation methods for the basic SV model, the extended SVmodel, and the Markov switching SV model. The approximate estimation results are alsocompared to the results from the particle filter.The proposed approximate methods are examined by many simulation analysis usingsimulated data and repeated trials. The results show that our methods for Gaussian mixturestate space model, Markov switching state space model, and their coupled model work quitewell. The approximate filters have high accuracy, with the fitted results to the true stateprocess closing to the exact filters. Even with a smaller value of parameter k, they are also veryaccurate. Moreover, simulation results from the Gaussian mixture state space model also showthat parameter estimates are almost same under di?erent k using the approximate maximumlikelihood estimator. Similarly, simulation experiments for the basic SV model show that theapproximate filter (AMF) is very close to the particle filter, better than the Monte Carlolikelihood (MCL) filter (Koopman & Uspensky, 2002) when the number of simulation is smalland the Kalman filter. For the sample performance of parameter estimates, our approximatemaximum likelihood (AML) estimator performs competitively with the MCL estimtor when thenumber of simulation is large and the N-ML estimtor suggested by Fridman & Harris (1998),better than the QMl estimator based on the Kalman filter. At last, simulation experimentsfor the Markov switching SV model show that the approximate filter MSMIXAMF(k) notonly can better fit to the log-volatility process than MSAMF(k), but also can well describethe hidden Markov process under the true data generation process. Moreover, it is shownthat the approximate filter for the Markov switching Gaussian mixture state space model,MSMIXAMF(k), is obviously much better than that for the Markov-switching state spacemodel, MSAMF(k), while the latter will be the Kim approximate filter suggested by Kim(1994) when k = 1.Equipped with the SV models and the approximate methods introduced in this study, wecarried out empirical analysis on returns of China's Shanghai and Shenzhen stock market andshort-term bank lending rate. On the one hand, we compare the in-sample performance of time-varying volatilities with di?erent models and di?erent estimation methods. Firstly, the fittedresults from GARCH-type models and SV-type models with return data and short-term interestrate data show that the SV models have higher explanatory power in modeling time-varying volatility, with stronger tail fitting e?ect. Secondly, we implement Bayesian Markov chainMonte Carlo estimation (JPR estimator and KSC estimator), QML estimator, MCL estimatorand proposed AML estimator for the basic SV model. It is shown that our AML estimatorperforms competitively with the Bayesian estimation methods and the MCL estimator. Theresults from the particle filter further show that the AML estimator is slightly better thanthe Bayesian estimation methods and the MCL estimator. Finally, with introducing leveragee?ects and regime switching, we compare in-sample performance of the basic SV model, theasymmetric SV model and the Markov switching SV model to stock returns and short terminterest rate.On the other hand, this study e?ectively tests and describes dynamic characteristics ofChina's stock returns and short-term interest rate using di?erent SV models, so as to provideus with basic results for financial risk management and decision-making. First of all, althoughthe Shanghai stock returns and Shenzhen stock returns share a common trend at the majorityof periods, the di?erences between them can be described by the GARCH-type models and theSV-type models. For instance, Shanghai stock market has obvious leverage e?ect, while thereis no leverage e?ects in Shenzhen stock market. Both stock market volatilities of Shanghaiand Shenzhen have obvious regime-switching characteristics. In the majority of time periodsregime shifts are the same for them, but in 1995 and early 1996 Shanghai stock market takesa high-volatility regime, while Shenzhen stock market is in a low-volatility regime. Secondly,level e?ects are found between the time-varying volatility and the level of short-term interestrate, the estimated values of them are less than one using the SV-type models. Moreover,the results from the asymmetric SV model show the existence of leverage e?ect between theinterest rate volatility and interest rates, and the results from the Markov switching SV modelsshow obvious regime shifts in interest rate volatility. Finally, the results also show that interestrate volatilities not only have regime switching between high and low volatilities, but also haveobvious regime shifts between volatility consistence and volatility of volatility at the sametime. Thus our study reveals an internal mechanism that the short-term lending rate volatilitychange between a low volatility - low persistence - high volatility of volatility regime and a highvolatility - high persistence - low volatility of volatility regime. In addition, this study also findthat the regime switching characteristics have an obvious linkage with the GDP growth rate inChina's macroeconomy, which further demonstrates the importance of interest rate volatility.