Advanced Applications of Generalized Hyperbolic Distributions in Portfolio Allocation and Measuring Diversificatio

This thesis consists of two parts. The first part addresses the parameter estimation and calibration of the Generalized Hyperbolic (GH) distributions. In this part we review the classical expectation maximization (EM) algorithm and factor analysis for the GH distribution. We also propose a simple shrinkage estimator driven from the penalized maximum likelihood. In addition an on-line EM algorithm is implemented to the GH distribution; and its regret for general exponential family can be represented as a mixture of Kullback-Leibler divergence. We compute the Hellinger distance of the joint GH distribution to measure the performances of all the estimators numerically. Empirical studies for long-term and short-term predictions are also performed to evaluate the algorithms.;In the second part we applied the GH distribution to portfolio optimization and risk allocation. We show that the mean-risk portfolio optimization problem of a certain type of normal mixture distributions including the GH distribution can be reduced to a two dimensional problem by fixing the location parameter and the skewness parameter. In addition, we show that the efficient frontier of the mean-risk optimization problem can be extended to the three dimensional space. We also proposed a simple algorithm to deal with the transaction costs. The first and second derivatives of the CVaR are computed analytically when the underlying distribution is GH. With these results we are able to extend the effective number of bets (ENB) to general risk measures with the GH distribution. By diagonalizing the Hessian matrix of a risk measure we are able to extract locally independent marginal contributions to the risk. The minimal torsion approach can still be applied to get the local coordinators of the marginal contributions.