Bayesian Inference of Stochastic Volatility Models and Applications in Risk Management

The recent financial crisis triggered by the mortgage-backed securities motivates researchers to develop better models to monitor the risk of financial institutions. In this paper, we discuss the Bayesian Inference of univariate and multivariate stochastic volatility models and their applications in risk management.;We first review the most basic volatility modeling methodology of univariate stochastic model. The conditional distribution of return shocks is standard Normal distribution. The stylized features of financial time series, such as fat tails and asymmetry, have made it clear that the need to go beyond the normality assumption. To accommodate these features, we adopt Generalized Error Distribution and Skewed Generalized Error Distribution. To explore the higher persistence in the volatility, we also study Integrated Stochastic Volatility models with all three distribution assumptions and test the goodness-of-fit through Bayes factor. We compare the performances of these models in risk management, i.e., the accuracy of Value-at-Risk forecasting.;During stressful time, the expected capital shortage of a firm is used to capture its systemic risk. We develop a multivariate stochastic volatility model to estimate the capital shortage of a firm and its correlation with the whole financial system. We name this capital shortage SRISK. SRISK is a function of Marginal Expected Shortfall (MES), which is a tail expectation of firm loss when a financial crisis occurs. The model is built upon the Dynamic Correlation Multivariate Stochastic Volatility Model (DCMSV). To adjust the model for fat tails, we extend the DCMSV model and assume the vector of return shocks follows a Multivariate Generalized Error Distribution.;There are two problems with DCMSV. First, we find that a parameter matrix in the model is partially identified. We proved that the scale of this matrix is unidentified and show that the trace of the parameter matrix could be an unidentified scale. Another problem is about the model estimation. Because of the penalty of high dimensional integration, the model estimation is computationally expensive. A Two-stage approach was developed for DCMSV model which were claimed to be accurate and efficient. But it is not a Gibbs sampler or any other Bayesian methods. We develop a Two-stage algorithm and a Unified algorithm for our extended DCMSV model. The Unified approach is a Gibbs sampler. We compare these two approaches through a simulation study and find that the Unified approach outperforms the Two-stage approach.;At the end, we conduct an empirical study using our SRISK model with the stock prices of Bank of America, Lehman Brothers and the S&P 500 index. Our model successfully provides alerts of the excessive leverage taken by Lehman right before it went bankruptcy. It also indicates the increase in risk of Bank of America after the purchase of Countrywide Financial.