Research On Stochastic Volatility Models And Their Modelling Methods

There are two classes of financial time varying volatility specifications---autoregressive conditional heteroskedasitic (ARCH) models and stochastic volatility(SV) models. The lack of estimation procedures for SV models made them for a longtime an unattractive class of models in comparison to ARCH, because it is impossibleto obtain explicit expressions for the likelihood function, the traditional estimationmethods are difficult to implement. This dissertation mainly studies and discusses SVmodels and their new estimation methods. Aiming at the leverage effects in stock markets, SV model with leverage effect isused to study the relationship between return and volatility of Shanghai and Shenzhenstock markets; a special MCMC algorithm—Gibbs sampling is used to estimatemodel via BUGS in this dissertation. The empirical results show obvious leverageeffects in these two markets, and this phenomenon is more remarkable in Shanghaimarket. The empirical characteristic function (ECF) method is proposed for SV modelestimation in this dissertation. The basic idea of the empirical characteristic functionmethod is to match the theoretical characteristic function from model with empiricalcharacteristic function from sample observation, minimizing certain distancemeasurement to obtain the estimates of parameters. The ECF method is extended toestimate the SV model with leverage effect, comparing the estimation result usingECF with the result using MCMC method shows ECF method is a simple method,and the estimator is consistent and asymptotically normal. A class of non-linear SV model with leverage effect is developed in thisdissertation. This class SV model encompasses many traditional SV models, includinglogarithm normal SV model. An advantage of our proposed class is the ease withwhich different specifications on stochastic volatility can be tested. In fact, thespecification test is based on a single parameter. MCMC method is used to estimatethe new class SV model. We empirically test logarithm normal SV model against ourproposed one using daily index return data in Shanghai and Shenzhen. The empiricaltest rejects logarithm normal SV models and favors a nonlinear SV model. Box-Coxtransformation is introuduced to SV-M model with leverage effect, a class of moregeneral SV model is achieved.The characteristic functions of affine diffusion and affine jump diffusion processare calculated in this dissertation. Based on the results, the ECFs of continuous timesquare-root stochastic volatility model and asset return process are proved existent.ECF method is extended to estimate continuous time stochastic volatility model. Thismethod requires neither discretization nor simulation. We illustrate ECF approachwith a detailed examination of the continuous time square root stochastic volatility(SV) model, along with an empirical application using Shanghai and Shenzhen indexreturns. The result suggesting there exists mean reversion in the volatility processes ofthese two markets.