Topics in Statistical Modeling and Estimation of Extremes and Their Dependence.

The main goal of this thesis is to use concepts and tools from extreme value theory (EVT) to model, make inference and develop prediction tools for the extremes of dependent data with a focus on the financial risk management applications. In the first part of the thesis, a new estimator of the extremal index based on the asymptotic scaling of block maxima and resampling is introduced. The extremal index is one of the main parameters that describes and quantifies the dependence of the extremes for many stationary time series. It is shown to be consistent and asymptotically normal for m-dependent time series. The estimator is applied to two real data sets of daily Crude Oil prices and extreme temperatures. In the second part, Value-at-Risk (VaR), one of the most widely used risk measures in finance, is estimated via point processes and tools from EVT. Our method is shown to be superior to the more widely accepted methods of VaR estimation. In the final part of the thesis, new risk measures are proposed to incorporate the clustering of the large losses as often exhibited by the asset returns. Such clustering can have grave financial consequences if not taken into account. The new risk measures are shown to be good predictors for the accumulation of the extreme losses.