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Type of Document Dissertation Author Hu, Wenbo Author's Email Address whu@math.fsu.edu URN etd-10312005-131627 Title Calibration Of Multivariate Generalized Hyperbolic Distributions Using The EM Algorithm, With Applications In Risk Management, Portfolio Optimization And Portfolio Credit Risk. Degree Doctor of Philosophy Department Mathematics, Department of Advisory Committee
Advisor Name Title Alec Kercheval Committee Chair Bettye Case Committee Member Craig Nolder Committee Member Fred Huffer Committee Member Warren Nichols Committee Member Keywords
- Basket Credit Default Swaps
- Skewed T Distribution
- Generalized Hyperbolic Distributions
- Portfolio Credit Risk
- Portfolio Optimization
- Risk Management
- EM Algorithm
Date of Defense 2005-10-28 Availability unrestricted Abstract The distributions of many financial quantities arewell-known to have heavy tails, exhibit skewness, and have other non-Gaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized
hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the so-called skewed t distributions, among many others.
The primary obstacle to the applications of such distributions is the numerical difficulty of calibrating the distributional parameters to the data. In this dissertation we describe a way to
stably calibrate GH distributions for a wider range of parameters than has previously been reported. In particular, we develop a version of the EM algorithm for calibrating GH istributions. This is a modification of methods proposed in McNeil, Frey, and Embrechts (2005), and generalizes the algorithm of Protassov
(2004). Our algorithm extends the stability of the calibration procedure to a wide range of parameters, now including parameter values that maximize log-likelihood for our real market data sets. This allows for the first time certain GH distributions to be used in modeling contexts when previously they have been numerically
intractable.
Our algorithm enables us to make new uses of GH distributions in three financial applications. First, we forecast univariate Value-at-Risk (VaR) for stock index returns, and we show in out-of-sample backtesting that the GH distributions outperform the Gaussian distribution. Second, we calculate an efficient frontier for equity portfolio optimization under the skewed-t distribution and using Expected Shortfall as the risk measure. Here, we show that the Gaussian efficient frontier is actually unreachable if
returns are skewed t distributed. Third, we build an intensity-based model to price Basket Credit Default Swaps by calibrating the skewed t distribution directly, without the need to separately calibrate the skewed t copula. To our knowledge this is the first use of the skewed t distribution in portfolio optimization and in portfolio credit risk.
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