
Analysis of extreme values
Vyhlídka, Jan ; Hendrych, Radek (advisor) ; Antoch, Jaromír (referee)
The goal of this thesis is to introduce basic concepts of the extreme value theory. The first chapter describes two fundamentally different approaches  block maxima and peaks over threshold models. Furthermore, it presents generalized extreme value distribution and generalized Pareto distribution. Moreover, relevant theorems and characteristics that are tied to these probabilistic distributions are discussed. The second chapter is a survey of various methods of parameter estimation of discussed distributions. The last chapter shows a simple application of how extreme value theory can be applied in finance on selected shares listed on the Prague Stock Exchange.


Stability of the Financial System: Systemic Dependencies between Bank and Insurance Sectors
Procházková, Jana ; Šopov, Boril (advisor) ; Janda, Karel (referee)
The central issue of this thesis is investigating the eventuality of systemic break downs in the international financial system through examining systemic depen dence between bank and insurance sectors. Standard models of systemic risk often use correlation of stock returns to evaluate the magnitude of intercon nectedness between financial institutions. One of the main drawbacks of this approach is that it is oriented towards observations occurring along the central part of the distribution and it does not capture the dependence structure of outlying observations. To account for that, we use methodology which builds on the Extreme Value Theory and is solely focused on capturing dependence in extremes. The analysis is performed using the data on stock prices of the EU largest banks and insurance companies. We study dependencies in the pre crisis and postcrisis period. The objective is to discover which sector poses a higher systemic threat to the international financial stability. Also, we try to find empirical evidence about an increase in interconnections in recent post crisis years. We find that in both examined periods systemic dependence in the banking sector is higher than in the insurance sector. Our results also in dicate that extremal interconnections in the respective sectors increased,...


Extreme value theory: Empirical analysis of tail behaviour of GARCH models
Šiml, Jan ; Šopov, Boril (advisor) ; Kocourek, David (referee)
This thesis investigates the capability of GARCHfamily models to capture the tail properties using Monte Carlo simulation in framework of Conditional Extreme Value Theory. Analysis is carried out for three different GARCHtype models: GARCH, EGARCH, GJRGARCH using Normal and Student's tdistributed innovations on four wellknown stock market indices: S&P 500, FTSE 100, DAX and Nikkei 225. After conducting 3000 simulations of every estimated model, the Hill estimate of shape parameter implied by the GARCHtype models will be calculated and the models' performance will be assessed based on histograms, descriptive statistics and Root Mean Squared Error of simulated Hill estimates. Interesting results and im plications for further research have been identified. Firstly, we highlight the Normal distribution's inappropriate nature in this case and its inability to capture the tail properties. Furthermore, GJRGARCHT with tdistributed innovations is identified to be the best model, closely followed by other tdistributed GARCHtype models. Finally, a pattern in all QQ plots forecasting the simulation study results is appar ent, with the exception of the DAX. This anomalous behaviour therefore necessitated further analysis and a significant right tail influence was recorded. Even though Hill estimates...


The use of coherent risk measures in operational risk modeling
Lebovič, Michal ; Teplý, Petr (advisor) ; Doležel, Pavel (referee)
The debate on quantitative operational risk modeling has only started at the beginning of the last decade and the bestpractices are still far from being established. Estimation of capital requirements for operational risk under Advanced Measurement Approaches of Basel II is critically dependent on the choice of risk measure, which quantifies the risk exposure based on the underlying simulated distribution of losses. Despite its wellknown caveats ValueatRisk remains a predominant risk measure used in the context of operational risk management. We describe several serious drawbacks of ValueatRisk and explain why it can possibly lead to misleading conclusions. As a remedy we suggest the use of coherent risk measures  and namely the statistic known as Expected Shortfall  as a suitable alternative or complement for quantification of operational risk exposure. We demonstrate that application of Expected Shortfall in operational loss modeling is feasible and produces reasonable and consistent results. We also consider a variety of statistical techniques for modeling of underlying loss distribution and evaluate extreme value theory framework as the most suitable for this purpose. Using stress tests we further compare the robustness and consistency of selected models and their implied risk capital estimates...


Cyber risk modelling using copulas
Spišiak, Michal ; Teplý, Petr (advisor) ; Baruník, Jozef (referee)
Cyber risk or data breach risk can be estimated similarly as other types of operational risk. First we identify problems of cyber risk models in existing literature. A large dataset consisting of 5,713 loss events enables us to apply extreme value theory. We adopt goodness of fit tests adjusted for distribution functions with estimated parameters. These tests are often overlooked in the literature even though they are essential for correct results. We model aggregate losses in three different industries separately and then we combine them using a copula. A ttest reveals that potential oneyear global losses due to data breach risk are larger than the GDP of the Czech Republic. Moreover, oneyear global cyber risk measured with a 99% CVaR amounts to 2.5% of the global GDP. Unlike others we compare risk measures with other quantities which allows wider audience to understand the magnitude of the cyber risk. An estimate of global data breach risk is a useful indicator not only for insurers, but also for any organization processing sensitive data.


Extreme value theory
Pelinka, Adam ; Čabla, Adam (advisor) ; Gerthofer, Michal (referee)
Extreme value theory is a modern statistical method for modelling events with a very low probability. During the analysis, we deal with convergence of distribution of these extremal events to their limit distributions. These distributions are generalized extreme value distribution and generalized Pareto distribution, which estimate tails of empirical probability distribution, where extremal events occur. In the last years, extreme value theory is used in many fields of study, e.g. in estimating financial risk or in estimating size of floods. In this work, two methods of modelling extremal events are presented  block maxima method and peaks over threshold method. Both methods are used during the real data analysis of daily discharge of river Vltava and estimated models are summarized. Although both used methods give slightly different results, choice of the appropriate model is not clear.


ValueatRisk Calculation Using Extreme Value Theory
Lipták, Patrik ; Hendrych, Radek (advisor) ; Mazurová, Lucie (referee)
This diploma thesis studies extreme value theory and its application in finan cial risk management, when focusing on computation of wellknown risk measure  Value at Risk (VaR). The first part of the thesis reviews theoretical background. In particular, it rigorously discusses the extreme value theory when emphasi zing fundamentals theorems and their consequences followed by the summary of methods based on this theory, specifically, Block Maxima method, Hill met hod and Peaks over Threshold method. Moreover, specific issues that may arise in such applications and ways how to deal with these problems are described. The second part of the thesis contains extensive empirical study, which together with theoretical foundings applies each of the examined method to real market data of the closing prices of Dow Jones Industrial Average stock index, stocks of JPMorgan and stock index Russell 2000 in order to compare methods based on extreme value theory together with the classic methodology RiskMetrics. 1


Estimation and Application of the Tail Index
Pokorná, Markéta ; Šopov, Boril (advisor) ; Zelený, Tomáš (referee)
Examining the nature of extreme values plays an important role in financial risk management. This thesis investigates tail behaviour of distribution of re turns using the framework of univariate Extreme Value Theory. The empirical research was conducted on the S&P 500 index and its seven constituents. The goal of this thesis was to use the Hill method to estimate the tail index of the series which characterizes the tail behaviour, especially the speed of the tail decay. To select the tail threshold several graphical methods were performed as they represent empirical measures of model stability. Classical Hill plots as well as alternative Hill plots and smoothing procedure were presented. The threshold choice based on stable regions in the graphs was found to be highly subjective. Hill method modified by Huisman was used instead and the results confirmed that the classical Hill method yields estimates which overestimate the tail thickness. All the examined series were found to have heavy tails with polynomial tail decay. This thesis stressed the need to model the left and the right tail separately as both extreme losses and profits are important depending on whether an investor takes a long or a short position on portfolio. Finally, the tail index was used to demonstrate the need to compute the...


Stability of the Financial System: Systemic Dependencies between Bank and Insurance Sectors
Procházková, Jana ; Šopov, Boril (advisor) ; Janda, Karel (referee)
The central issue of this thesis is investigating the eventuality of systemic break downs in the international financial system through examining systemic depen dence between bank and insurance sectors. Standard models of systemic risk often use correlation of stock returns to evaluate the magnitude of intercon nectedness between financial institutions. One of the main drawbacks of this approach is that it is oriented towards observations occurring along the central part of the distribution and it does not capture the dependence structure of outlying observations. To account for that, we use methodology which builds on the Extreme Value Theory and is solely focused on capturing dependence in extremes. The analysis is performed using the data on stock prices of the EU largest banks and insurance companies. We study dependencies in the pre crisis and postcrisis period. The objective is to discover which sector poses a higher systemic threat to the international financial stability. Also, we try to find empirical evidence about an increase in interconnections in recent post crisis years. We find that in both examined periods systemic dependence in the banking sector is higher than in the insurance sector. Our results also in dicate that extremal interconnections in the respective sectors increased,...


Multivariate extreme value models and their application in hydrology
Drápal, Lukáš ; Jarušková, Daniela (advisor) ; Hušková, Marie (referee)
Present thesis deals with the multivariate extreme value theory. First, concepts of modelling block maxima and threshold excesses in the univariate case are reviewed. In the multivariate setting the point process approach is chosen to model dependence. The dependence structure of multivariate extremes is provided by a spectral measure or an exponent function. Models for asymptotically dependent variables are provided. A construction principle from Ballani and Schlather (2011) is discussed. Based on this discussion the pairwise beta model introduced by Cooley et al. (2010) is modified to provide higher flexibility. Models are applied to data from nine hydrological stations from northern Moravia previously analysed by Jarušková (2009). Usage of the new pairwise beta model is justified as it brought a substantial improvement of loglikelihood. Models are also compared with Bayesian model selection introduced by Sabourin et al. (2013). Powered by TCPDF (www.tcpdf.org)
