Book Description

This dissertation consists of three essays on modeling financial risk under Bayesian framework. The first essay compares the performances of Maximum Likelihood Estimation (MLE), Probability-Weighted Moments (PWM), Maximum Product of Spacings (MPS) and Bayesian estimation by using the Monte Carlo Experiments on simulated data from GEV distribution. I compare not only how close the estimates are to the true parameters, but also how close the combination of the three parameters in terms of estimated Value-at-Risk (VaR) to the true VaR. The Block Maxima Method based on student-t distribution is used for analysis to mimic the real world situation. The Monte Carlo Experiments show that the Bayesian estimation provides the smallest standard deviations of estimates for all cases. VaR estimates of the MLE and the PWM are closer to the true VaR, but we need to choose the initial values carefully for MLE. MPS gives the worst approximation in general. The second essay analyzes the movement of implied volatility surface from 2005 to 2014. The study period is divided into four sub-periods: Pre-Crisis, Crisis, Adjustment period and Post-Crisis. The Black-Scholes model based daily implied volatility (IV) is constructed and the time series of IV given different moneyness and time to maturity is fitted into a stochastic differential equation with mean-reverting drift and constant elasticity of variance. After estimating the parameters using a Bayesian Metropolis Hastings algorithm, the comparison across different time periods is conducted. As it is natural to expect abnormality in Crisis and Adjustment period, it is interesting to see the difference between Post-Crisis movement and the Pre-Crisis's. The results reveal that if the catastrophe does not permanently change the investment behavior, the effect from Crisis may last longer than expected. It is unwise to assume the market movement or investment behavior would be identical in Pre-Crisis and Post-Crisis periods. Market participants learn from Crisis and behave differently in Post-Crisis comparing to Pre-Crisis. The third essay attempts to predict financial stress by identifying leading indicators under a Bayesian variable selection framework. Stochastic search variable selection (SSVS) formulation of George and McCulloch (1993) is used to select more informative variables as leading indicators among a number of financial variables. Both linear model and Probit model under normal error assumption and fat tail assumption are used for analysis. Financial stress indexes issued by Federal Reserve Banks combined with Bloom(2009) and Ng(2015)'s paper are used to identify financial stress. An ex-post approach based on historical perspective and ex ante approach combined with rolling window are used for analysis. The results show promising predictive power and the selection of variables can be used to signal financial crisis period.