Statistisk modellering af ekstremværdier: Statistical Modeling of Extreme Values

Abstrakt

The focus in this thesis is to study Extreme Value Theory (EVT) and statistical methods, to describe extreme observations, with regard to nancial risk management. These di erent statistical methods is used to model independent and identically distributed (i.i.d.) data and clusters of volatility. For the selection of the extreme values, the Method of Block Maxima and Peaks over Threshold (POT) are introduced, which are two di erent approaches used to select extreme values. The main focus of the rst part is the assumption of i.i.d. data, where the focus is on the POT method and the generalized pareto distribution (GPD). To model the GPD and estimate the parameters and risk measures, several di erent techniques are used. The estimation methods Method-of-Moments (MOM), Elemental-Percentile-Method (EPM), Probability-Weighted-Method (PWM), Maximum Likelihood Estimation (MLE) and L-Moments (LMOM) method are implemented in daily prices for the Vestas stock, wherein the risk measures Value-at-Risk (VaR) and Expected Shortfall (ES) are used. The achieved results of ve di erent estimation methods are: the MOM and EPM method is not a valid choise for modeling the extreme observations. PWM, MLE and LMOM are preferred. These methods provide approximately the same values for VaR and ES. The focus in the second part of the thesis is point processes and clusters of volatility, where the Poisson process and Hawkes Self-Exciting POT are introduced. The Poisson point process gives the same results as the i.i.d. POT model, and the Hawkes POT models, which take clustering into account, result in almost identically estimates, however smaller VaR and ES measures.