Estimation of Extremal Events for Financial Data

Abstract

This thesis seeks to investigate the estimation procedure of extremal events for financial data. Specifically, it seeks to investigate the potential of methods within the area of Extreme Value Theory by comparing these with the more traditional models from Time Series Analysis. Firstly, the thesis describes the basis of financial data and the characteristics of this type of data. In this section, the relevant risk measures - their meaning and derivation - will be described too. Secondly, the theory of both Time Series Analysis and Extreme Value Theory is introduced. It is assumed that the reader is familiar with the concepts in Time Series Analysis, and these are therefore described briefly. Since the focus is on Extreme Value Theory, the Time Series models are also basic, as they are meant to solely represent a basis for comparison. Regarding Extreme Value Theory, a wide range of theory and modelling approaches are presented. This covers the basics of the weak convergence for maxima, The Generalized Extreme Value Distribution, and the Maximum Domain of Attraction. The modelling approach spans from the classic Hill Estimator and Peaks-over-Thresholds method to the more modern bias correction method presented by De Haan et al. (2016). Thirdly, the described methods are used in practice by analyzing the log-returns of Jyske Bank. The Value-at-Risk and Expected-Shortfall is estimated for all methods and the best model from Extreme Value theory is found. This will show that the bias correction method is by far the most robust. Finally, the bias correction method is compared to the classic ARMA-GARCH-model. This will show that each method has pros and cons which relate to their statistical background. Because of this, it is proposed to follow the procedure by McNeil and Frey in which ARMA-GARCH and Extreme Value Theory is combined. Further, it is proposed to potentially improve the procedure further, by applying the bias correction method by De Haan et al. (2016) instead of using Peaks-over-Thresholds as McNeil and Frey (2000).