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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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Granger's causality in financial time series
Marčiny, Jakub ; Voříšek, Jan (advisor) ; Lachout, Petr (referee)
The bachelor thesis discusses causality in multiple time series. Granger causality, along with its more general counterparts instantaneous causality and multistep causality, are utilized to study the mutual influence of the individual components of a multiple time series. These concepts are investigated within the framework of vector autoregressive models VAR. After the introduction of basic definitions and facts, the construction of VAR model is described including methods for order selection and verification. Subsequently, causal relations within the model are examined. Finally, empirical analysis of real financial market data is performed using tests procedures programmed with computational software Mathematica.
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Markowitz model with constraints
Němec, Jan ; Lachout, Petr (advisor) ; Hurt, Jan (referee)
Title: Markowitz model with constrains Author: Jan Němec Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Petr Lachout, CSc. Abstract: Composition of an optimal portfolio from available tradable comodities is very frequntly a discussed issue. One model, which considers not only the yield of the portfolio, but also its risk, is Markowitz model. Bachelor thesis will consider this ap- proach in cases when the searched portfolio is bounded with additional restrictions. This thesis will primarily address the constraints that are determined by legislation to conduct various banking entities investing in the stock market. Keywords: Markowitz model, portfolio constraints, banking regulation 1
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Design of dynamic decision-making strategies for futures trading
Vosáhlo, Jaroslav ; Guy, Tatiana Valentine (advisor) ; Lachout, Petr (referee)
This thesis deals with an issue of futures derivative trading from a perspective of a minor speculator. The aim of this work is to find and design an optimal trading strategy using dynamic programming and approximate dynamic programming. We use means of Bayesian statistics to obtain predictions of variate's behavior and risk indicators to form a rate of carefulness. Effectivity of algorithm is afterwards tested in Matlab program. Available data for testing the success of the method offer more then 15.000 trading days.
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Volatility bursts and order book dynamics
Plačková, Jana ; Swart, Jan (advisor) ; Lachout, Petr (referee)
Title: Volatility bursts and order book dynamics Author: Jana Plačková Department: Department of Probability and Mathematical Statistics Supervisor: Dr. Jan M. Swart Supervisor's e-mail address: swart@utia.cas.cz Abstract: The presented paper studies the dynamics of supply and demand through the electronic order book. We describe and define the basic rules of the order book and its dynamics. We also define limit and market orders and describe the differences between them and how they influenced the evolution of ask, bid price and spread. Next part of the paper is dedicated to the de- scription and definition of volatility and its basic models. The brief overview about volatility clustering and its modeling by economists and physicists can be found in the following part. In the last part we introduce a simple model of order book in which we observe ask, bid price and spread. Then we study the empirical distribution of spread and try to find its probability distribu- tion. The volatility clustering is then observed through the relative returns of spread. In the last part we introduce some possible improvement of the model. Keywords: volatility clustering, order book, limit orders, market orders 1
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General skew-probability distributions
Václavík, Jiří ; Lachout, Petr (advisor) ; Dostál, Petr (referee)
In the present work we study families of skew-probability distributions. We will gradually build concept of families of more and more general distributions. For us the most important ones are skew normal distribution, elliptical distri- bution and skew elliptical distribution. On the each of them we will present basic properties and visualize particular examples. At the end we will generate realizations of variates and propose how to estimate the original distribution.
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