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Stochastic Programming Problems via Economic Problems
Kučera, Tomáš ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis' topic is stochastic programming, in particular with regard to portfolio optimization and heavy tailed data. The first part of the thesis mentions the most common types of problems associated with stochastic programming. The second part focuses on solving the stochastic programming problems via the SAA method, especially on the condition of data with heavy tailed distributions. In the final part, the theory is applied to the portfolio optimization problem and the thesis concludes with a numerical study programmed in R based on data collected from Google Finance.
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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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Modelování portfolií s risk faktory s těžkými chvosty
Kyselá, Eva ; Málek, Jiří (advisor) ; Fičura, Milan (referee)
The thesis aims to investigate some of the approaches to modelling portfolio returns with heavy-tailed risk factors. It first elaborates on the univariate time series models, and compares the benchmark model (GARCH with Student t innovations or its GJR extension) predictive performance with its two competitors, the EVT-GARCH model and the Markov-Switching Multifractal (MSM) model. The motivation of EVT extension of GARCH specification is to use a more proper distribution of the innovations, based on the empirical distribution function. The MSM is one of the best performing models in the multifractal literature, a markov-switching model which is unique by its parsimonious specification and variability. The performance of these models is assessed with Mincer-Zarnowitz regressions as well as by comparison of quality of VaR and expected shortfall predictions, and the empirical analysis shows that for the risk management purposes the EVT-GARCH dominates the benchmark as well as the MSM. The second part addresses the dependence structure modelling, using the Gauss and t-copula to model the portfolio returns and compares the result with the classic variance-covariance approach, concluding that copulas offer a more realistic estimates of future extreme quantiles.
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Empirical Estimates in Stochastic Optimization: Special cases
Kaňková, Vlasta
Classical optimization problems depending on a probability measure belong mostly to nonlinear deterministic optimization problems that are relatively complicated. On the other hand, these problems fulfil very often "suitable" mathematical properties guaranteing the stability (w.r.t. probability measure) and, moreover, giving a possibility to replace the "underlying" probability measure by an empirical one to obtain "good" stochastic estimates of the optimal value and the optimal solution. Properties of thess estimates have been investigated mostly for standard types of probability measures with suitable (thin) tails and independent random samples. However distributions with heavy tails correspond to many economic problems and, moreover, many applications do not correspond to the "classical" problems. The aim of the paper is, first, to try to recall stability results including also heavy tails and more general problems.
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