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Decision Problems and Empirical Data; Applications to New Types of Problems
Odintsov, Kirill ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
This thesis concentrates on different approaches of solving decision making problems with an aspect of randomness. The basic methodologies of converting stochastic optimization problems to deterministic optimization problems are described. The proximity of solution of a problem and its empirical counterpart is shown. The empirical counterpart is used when we don't know the distribution of the random elements of the former problem. The distribution with heavy tails, stable distribution and their relationship is described. The stochastic dominance and the possibility of defining problems with stochastic dominance is introduced. The proximity of solution of problem with second order stochastic dominance and the solution of its empirical counterpart is proven. A portfolio management problem with second order stochastic dominance is solved by solving the equivalent empirical problem. Powered by TCPDF (www.tcpdf.org)
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Modeling of risk aversion
Navrátil, František ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
of the master thesis Title: Modeling of risk aversion Author: František Navrátil Department: Department of Probability and Mathematical Statistics Supervisor: Doc. RNDr. Petr Lachout, CSc. Abstract: The thesis discusses various theories that are able to model investor's subjective attitude to risk. The goal of the thesis is to clearly recapitulate possible mathematical approaches and to apply them in a real situation. One of the ways to tackle the problem is to use expected utility theory and a specific shape of a utility function. Another way is to choose a suitable risk measure. Especially useful for the modelling of risk aversion is the class of spectral risk measures that enables investor to choose a risk spectrum that meets his perception of risk. The thesis contains basic definitions concerning stochastic programming - a theory essential to solve the related optimization problems. Keywords: Risk aversion, utility function, probability constraint.
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Selected risk parameters in IRB approach and their modeling
Malec, Jaromír ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
The determination of lending (credit) risk is one of the most important fields of bank activities. This thesis discusses the IRB approach under Basel II. This approach includes the LGD, EAD and PD parameters. All parameters are individually modelled by the bank using regulator approved models. Parameter PD is the most focused one in this thesis. Theory for this parameter is of interest in many papers. However, at present the need for modelling of PD parameter over more years is appearing. Parameter LGD is also discussed in this thesis. The parameter EAD is only briefly presented. The thesis begins with the IRB approach, regression models and evaluation indicators, and then it focuses on the above parameters.
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Optimization of flow in graph
Popovič, Viktor ; Lachout, Petr (advisor) ; Kozmík, Václav (referee)
When it comes to maximization of effectively or minimizing of cost, optimization represents the key activity. There is a number of practical examples that can be implemented into Theory of Graphs and subsequently optimized. This thesis includes the introduction to transportation problem where the consumer demand is met by the lowest price. Also there is maximum flow problem which is to transfer maximum of commodity (petroleum, gas...) through the network where each edge has a capacity restriction. We will also look into the alternative situations where we will maximize the flow along with minimizing of cost. To resolve these problems we will establish numeric algorithms like distribute method, labeling algorithm, shortest augmented path algorithm, and Preflow-Push algorithms. We will also illustrate functionality on example which confirm appropriate application of algorithms and differences among them.
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Robust portfolio selection problem
Zákutná, Tatiana ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
In this thesis, a portfolio optimization with integer variables which influ- ence optimal assets allocation, is studied. Measures of risk are defined and the cor- responding mean-risk models are derived. Two methods are used to develop robust models involving uncertainty in probability distribution: the worst-case analyses and contamination. The uncertainty in values of scenarios and in their probabili- ties of the discrete probability distribution is assumed separately followed by their combination. These models are applied to stock market data with using optimization software GAMS.
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Analysis of number lotteries
Jedličková, Veronika ; Pawlas, Zbyněk (advisor) ; Lachout, Petr (referee)
This bachelor thesis focuses on most well-known lotteries on the Czech market, in particular Sportka and Loto. Thesis observes many aspects influencing progress of these games. Winnning prices and lottery participant's expectations are examined. Total sum of these winnings is influenced by the amount of money in jackpot. Therefore, jackpot sum modelling and period between wins is taken into account. Moreover, expected period between two jackpot hits, distribution of drawn numbers and probability of drawing the same winning sequence is examined.
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Numerical study on simultanious equations
Šaroch, Vojtěch ; Lachout, Petr (advisor) ; Hendrych, Radek (referee)
Title: Numerical study on simultanious equations Author: Vojtěch Šaroch Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Petr Lachout, CSc. Abstract: In this thesis we deal with simultaneous equation model. In first chapter we introduce to theoretical aspect of this problem, especially estimation procedures and their properties. We mention issues of identification and an inconsistency of OLS-estimates for the simultaneous modeling. In second chapter we introduce theory of estimation, especially we will focus on interval estimation and precision. We mention empirical approach too. In the third chapter we perform a numerical study on simple macroeconomic model on generated dates. We are interested in properties interval estimations of parameters, the convergence rate, difference between empirical and theoretical extimation etc. Keywords: simultaneous equations model, interval estimation, empirical estimation 1
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Searching for optimal path in graphs
Znamenáčková, Gabriela ; Lachout, Petr (advisor) ; Kopa, Miloš (referee)
It's possible to simulate a lot of real decision-making situations by a weighted graph. Consequently it's important to find the optimal solution of a given situation based on this model. The subject of this Bachelor Thesis is to present the typical problems of combinatorial optimization, that deal with finding the optimal path in a graph considering the given conditions, and algorithms to find their optimal solution. It's focused on following problems: the shortest path problem, the minimum cost spanning-tree problem, the minimum cost Steiner tree problem, the travelling salesman problem and the optimal network flow. Working of some algorithms is shown on illustrative examples.
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Roulette and particular probabilities
Oberhauserová, Simona ; Lachout, Petr (advisor) ; Prokešová, Michaela (referee)
Title: Roulette and particular probabilities Author: Simona Oberhauserová Department: Department of probability and mathematical statistics Supervisor: Doc.RNDr. Petr Lachout, CSc., Department of probability and mathe- matical statistics Abstract: The thesis formulates roulette as a mathematical problem and examines the best roulette strategies in terms of probability of winning, gambler's ruin and probability distribution of profit. This game follows Kolmogor axiomatic probability model, therefore the calculations were counted by the basic formulas and axioms. In the calculations of the gambler's ruin differential equations were also used and built with random walk. In the longest expected run of red (black) were used sto- chastic processes and extreme value theory. In addition to interesting calculations, the conclusion also contains finding that there is no winning strategy in roulette. Even though one-time probabilities of winning are high, the finding indicates nega- tive mean value of profit. Keywords: Roulette, Kolmogorov axiomatic probability space
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