Volatility plays a central role in modern finance especially in the pricing of derivative securities. Research on changing volatility can be categorized into two groups: the time-varying volatility models represented by ARCH type models and the Stochastic Volatility Models. Research on ARCH type models offers straightforward implementation and has been empirically successful but they generally lack economic intuition. Stochastic Volatility models are statistically elegant and have strong connection to continuous-time finance models. Yet estimation of Stochastic Volatility models has been very difficult which makes further development of the model and comparison of empirical results with ARCH/GARCH type models difficult.

We propose an efficient Bayesian Markov Chain Monte Carlo estimation procedure for a Log-AR(1) Stochastic Volatility Model. We develop new simulation-based model diagnostics methods for in-sample model adequacy check and compare with the popular EGARCH model. Our in-sample diagnostic check shows better kurtosis properties and different Smile effect generated by the Stochastic Volatility Model than the EGARCH model. We also discuss issues of the comparison of historical volatility and implied volatility and propose a new model which combines the historical volatility and implied volatility under one model framework. This new model can be used for both forecasting and testing of the hypothesis of the existence of stochastic volatility.

Two common methods exist for frequency estimation in cyclical time series: probability theory and Fourier transform. Recent work of Jaynes and Bretthorst has shown the connection of the two methods and the theoretical advantage of the probability method. We develop a unified approach for accurate frequency estimation under the Bayesian MCMC framework for the single-frequency and multi-frequency harmonic model which can be generalized to more complex models for the frequency. We apply the method to real cyclical data. Motivated by the study of Oxygen isotope data in geology study, we discuss timing issues in harmonic analysis, particularly the impact of uncertain timing to the estimation of frequencies. We develop a harmonic model with uncertain timing to investigate the impact of uncertain timing in frequency estimation and to illustrate the use of Bayesian MCMC simulation methods as a general method for complex models in Bayesian spectral analysis. We illustrate our idea using real Oxygen data in geology study and provide evidence of the impact of uncertain timing to frequency estimation.