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statistical methods on risk management of extreme events

Abstract
The goal of the dissertation is the investigation of financial risk analysis methodologies, using the schemes for extreme value modeling as well as techniques from copula modeling. Extreme value theory is concerned with probabilistic and statistical questions re- lated to unusual behavior or rare events. The subject has a rich mathematical theory and also a long tradition of applications in a variety of areas. We are interested in its application in risk management, with a focus on estimating and forcasting the Value-at-Risk of financial time series data. Extremal data are inherently scarce, thus making inference challenging. In order to obtain good estimates for risk measures, we develop a two-stage approach: (1) fitting the GARCH-type models at the first stage to describe the volatility clustering and other stylized facts of financial time series; (2) using the extreme value theory based models to fit to the tails of the residuals. Additionally, the performance measures provide information in terms of the comparison of the two-stage semi-parametric approach with the parametric methodologies, through robust backtesting. Copula is a particular branch of probability theory, with which, given sufficient data, we can separate the marginal behavior of individual risks and their dependence structure from a multivariate random variable. Linear correlation is widely used to model dependence but has limitations as a measure of association and thus we opt to use copulas to analyze the dependence structure and build models for our different problems arising in risk management. For this part of the dissertation, we take a look at different copula families, highlight for some when they are most appropriate to use for a particular application, discuss some of their drawbacks as diverse scenarios occur in different risk management models, and explore the possibility of developing the copula modeling to reflect the complicated dependence structure of portfolios.
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