Statistisk analyse af ekstreme hændelser i finansiel data

Abstract

The aim of this thesis is to analyze extreme events in financial data by applying Extreme Value Theory (EVT). In the first major section, we explore the classical EVT framework, assuming that the sequence of random variables is independent and identically distributed (i.i.d.). We present key theo- rems that ensure the convergence of maxima to a non-degenerate limiting distribution. Additionally, we introduce the concept of maximum domains of attraction, which specifies the conditions that lead to convergence towards one of the three possible extreme value distributions: Gumbel, Fr´echet, or Weibull. The Fisher-Tippett theorem, which describes the limiting distribution for the centered and normalized maxima in case the limit exists, is also presented. Furthermore, we discuss the Block Max- ima Method and the Peaks Over Threshold (POT) approach, which are utilized to identify maxima and to fit the Generalized Extreme Value (GEV) distribution and the Generalized Pareto Distribution (GPD), respectively. We link these models to key financial risk measures—such as return period, re- turn level, Value at Risk (VaR), and Expected Shortfall (ES)—providing insight into the probability of extreme losses in financial assets. An empirical analysis follows, under the assumption that extreme events occur independently over time. We fit both the GEV distribution and the GPD to assess the probability of extreme losses. The data used in this section consists of daily closing prices for Novo Nordisk B and Danske Bank across all trading days between January 1, 2001, and December 31, 2023.

Financial data often exhibits time-varying volatility and clustering of extreme events, particularly around periods of economic and financial crises. This motivates the second major section of the the- sis. We relax on the assumption that extreme events occur independently over time and extend our analysis to dependent stationary sequences, incorporating appropriate mixing and anti-clustering con- ditions. We further introduce methods such as the blocks method and the runs method to estimate the extremal index. Finally, we estimate the extremal index for Danske Bank’s stock and reintroduce the VaR measure, demonstrating that dependence in the data reduces the likelihood of extreme losses in a financial asset.