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The impact of stochastic dominance generator on a portfolio efficiency
Dungl, Martin ; Dupačová, Jitka (referee) ; Kopa, Miloš (advisor)
This contribution focuses on the sets of efficient portfolios and their properties, given the class of utility functions. Firstly the basic concepts of stochastic dominance are recalled, then we limit our attention to the scenario approach and the concept of portfolio eeffciency is introduced. Di erent de nitions of efficient portfolios are taken into account - we call them admissibility, strict admissibility and optimality. We summarize the most important results concerning the shapes of sets composed by portfolios efficient according to the above mentioned approaches. The problem of path connectedness of the sets of optimal portfolios with limited short sales is analyzed. We prove that the set of optimal portfolios is path connected under the assumption that the generator of the stochastic dominance is a path connected set of strictly concave utility functions. Then the convexity of the set of optimal portfolios with respect to the set of exponential utility functions, in case of allowed short sales, is analyzed. We conclude that the sets of optimal portfolios generally need not be convex and we prove the necessary and sufficient condition of convexity.
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The transportation problem, its generalizations and applications in probability and statistics
Doležel, Pavel ; Kopa, Miloš (referee) ; Dupačová, Jitka (advisor)
Author describes a specific optimization problem-the transportation problem and analyzes relevant solution methods. Several methods of solving the transportation problem are listed, applied or introduced and applications of the transportation poblem in the theory of probability and mathematical statistics are presented, namely the statistical sorting in L1 norm and re-construction of contingency tables. Special interest is devoted to several modifications of ordinary transportation problem, mainly the multiindex transportation problem. The crucial part of the work are selected applications of the transportation problem to particular problems and showing some algorithms used for finding solutions.
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Portfolio efficiency with continuous probability distribution of returns
Kozmík, Václav ; Dupačová, Jitka (referee) ; Kopa, Miloš (advisor)
Present work deals with the portfolio selection problem using mean-risk models. The main goal of this work is to investigate the convergence of approxi mate solutions using generated scenarios to the analytic solution and its sensitivity to chosen risk measure and probability distribution. The considered risk measures are: variance, VaR, cVaR, absolute deviation and semivariance. We present analytical solutions for all risk measures under the assumption of normal or Student distribution. For log-normal distribution, we use the approximate assumption that the sum of log-normal random variables has log-normal distribution. Optimization models for discrete scenarios are derived for all risk measures and compared with analytical solution. In case of approximate solution with scenarios, we repeat the procedure multiple times and present our own approach to nding the optimal solution using the cluster analysis. All optimization models are written in GAMS language. Testing and estimating are realized using an application developed in C++ language.
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Credit Derivatives Valuation
Promer, Marek ; Franěk, Petr (advisor) ; Dupačová, Jitka (referee)
Credit risk plays an important role in the pricing of financial instruments. In effort to avoid the dangers resulting from this risk were developed new financial instruments called credit derivatives. In this work, the main features of three types of credit derivatives are discussed: credit default swap, total return swap and credit linked note. Regarding to the major portion of credit default swap on the credit derivatives market, the work deals with the valuation of this exact instrument with three models for valuing a credit default swap. These models estimate the value of credit default swap, under which the premium required from one participant of the contract is meant.
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Expected value of information in stochastic programming
Čížková, Jitka ; Lachout, Petr (referee) ; Dupačová, Jitka (advisor)
Stochastic problems (both two-stage and multistage) can be formulated in several di erent ways which utilize to various extent available information on a future realization of incorporated random parameters. When comparing optimal objective function values resulting from di erent formulations of the given problem with the same available information, we obtain a value of using one of these formulations rather than the other one (e.g., VSS). Level of the available information can be changed by a partial or full relaxation of nonanticipativity constraints, which assure that a present decision is independent of future (unknown) realizations of random parameters. By comparing optimal objective function values gained when solving the given problem with distinct levels of available information we obtain (expected) value of partial or perfect information. In this work we present de nitions of various information value types and related values connected with the problem formulation and we derive their properties (nonnegativity, bounds). In the last part we introduce their summary classi cation.
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Multicriterial Optimization Problems with a Random Element and Stochastic Programming
Líkař, Jan ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
In practice we often have to solve optimization problems with several criteria. These problems are called multicriteria optimization problems. Such problems are presented in this thesis. It is important, whether parameters take unknown values at the moment of making decision. If these parameters are random variables, resulting problem is called stochastic multiobjective problem, otherwise it is called deterministic multiobjective problem. We describe how to choose some "good" solutions of deterministic problem. We investigate their relations as well. In the stochastic case we have to convert such problem to deterministic one. We introduce some possibilities how to do it. Then we are able to solve the problem. These concepts are demonstrated using examples. We present a numerical illustration as well (the Portfolio Selection problem).
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