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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.
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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 GARCH-family models to capture the tail properties using Monte Carlo simulation in framework of Conditional Extreme Value Theory. Analysis is carried out for three different GARCH-type models: GARCH, EGARCH, GJR-GARCH using Normal and Student's t-distributed innovations on four well-known 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 GARCH-type 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, GJR-GARCHT with t-distributed innovations is identified to be the best model, closely followed by other t-distributed GARCH-type models. Finally, a pattern in all Q-Q 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...
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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 best-practices 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 well-known caveats Value-at-Risk remains a predominant risk measure used in the context of operational risk management. We describe several serious drawbacks of Value-at-Risk 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...
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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 t-test reveals that potential one-year global losses due to data breach risk are larger than the GDP of the Czech Republic. Moreover, one-year 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.
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Value-at-Risk 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 well-known 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
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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...
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Stability of the Banking Sector: Dependence Beyond Correlation
Fiala, Tomáš ; Šopov, Boril (advisor) ; Zelený, Tomáš (referee)
We analyze systemic risk of banks in countries of the so-called Visegrad Group (V4). Particularly, we focus on the relationship between a foreign mother and its local subsidiary which we compare to the relationship between subsidiaries in a given V4 country. We find that the systemic risk between two subsidiaries is higher than that between a mother and the respective subsidiary. In our analysis, we employ a technique stemming from a nonparametric multivariate Extreme Value Theory which is distribution free. Thus, our results are robust to heavy tails. Keywords Extreme value theory, systemic risk, financial stability, Visegrad Group, mother-subsidiary re- altionship Author's e-mail tomas.fiala@gjn.cz Supervisor's e-mail boril.sopov@gmail.com
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Heavy Tails and Market Risk Measures: the Case of the Czech Stock Market
Bulva, Radek ; Zápal, Jan (advisor) ; Bubák, Vít (referee)
One of the stylized facts about the behaviour of financial returns is that they tend to exhibit more probability mass in the tails of the distribution than would be suggested by the normal distribution. This phenomenon is called heavy tails. The first part of this thesis focuses on examining the tails of a distribution of returns on Czech stock market index PX. Parametric and semi-parametric approaches to estimation of the tail index, a measure of heaviness of tails, are applied and compared. The results indicate that the tails behave in a way one would expect from an emerging market stock index. In the second part of the thesis, implications for two quantile-based market risk measures, Value at Risk and Expected Shortfall, are investigated. The main conclusion is that heavy-tailed alternatives should be preferred to the normal distribution in order to avoid serious underestimation of risks embedded in the underlying process. JEL classification: C13, C14, C16, G15; Keywords: Heavy Tails, Parametric and Semi-parametric Estimation, Statistics of Extremes, Extreme Value Theory, Market Risk, Value at Risk, Expected Shortfall.
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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 GARCH-family models to capture the tail properties using Monte Carlo simulation in framework of Conditional Extreme Value Theory. Analysis is carried out for three different GARCH-type models: GARCH, EGARCH, GJR-GARCH using Normal and Student's t-distributed innovations on four well-known 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 GARCH-type 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, GJR-GARCHT with t-distributed innovations is identified to be the best model, closely followed by other t-distributed GARCH-type models. Finally, a pattern in all Q-Q 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...
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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.
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