Bayesian inference in stable distributions and its applications in stable portfolio analysis

Stable distributions have been widely used as a model for many types of financial and economic systems to capture heavy tail and skewness. But the lack of closed formulas for density and distribution functions for all but a few stable distributions (Gaussian, Cauchy and Levy) has been a major drawback to the use of stable distributions by practitioners. The objectives of the research are to develop a more straightforward and easier programming method to do Bayesian inference for general univariate stable distributions and symmetric multivariate stable distributions, and then to apply Bayesian inference for estimating the parameters of stable distributions and to apply such estimating procedures to carry out stable portfolio analysis. Approaches that combine Markov Chain Monte Carlo methods and fast Fourier transform are developed to perform posterior analysis for the parameters of univariate and multivariate stable distributions. The stable probability densities are computed by applying fast Fourier transform to invert the characteristic functions which have closed analytical form. The posterior moments and marginal posterior densities are computed by combining Gibbs sampler and Metropolis-Hastings algorithm. Both vague and proper priors are studied. The new methods are shown to perform satisfactorily in simulation examples. The methods are applied to empirical stock market data as part of examination of the stable portfolio analysis problem. Bayesian inference for the expected return and risk matrix of an illustrative stock portfolio is studied.