
Comparison of Models for Probabilities in Football Betting
Kožnar, František ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of the thesis is to compare different statistical models for football betting odds and determine the best performing once based on the historical performance of sport teams. There are at least three possible approaches for computing the odds, namely logistic regression, Poisson regression and methods based on statistical machine learning. The idea is that the historical performance of teams is a good predictor of the future performance. Thus we can take the past performances, say all matches in the full season of the English Premier League (380 matches), and use these data for predicting the odds for the following season. The resulting odds should be compared with the actual results using the scoring rules, which will identify the best performing model.


Profit Maximization of Car Manufacturers Facing EU CO2 Emission Penalties From 2021
Leamer, Anthony David ; Večeř, Jan (advisor) ; Antoch, Jaromír (referee)
Title: Profit maximization of car manufacturers facing EU CO2 emission penalties from 2021 Author: Anthony David Leamer Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Jan Večeř, Ph.D., Department of Probability and Mathematical Statistics Abstract: This paper sheds light on the newly coming emissions penalization sys tem imposed on passenger vehicles registered in the EU. We analyze the penalty based on how it influences profit of the car manufacturers. After optimizing the profit margins car manufacturers impose on different vehicles we discuss what this means for the consumer and the manufacturer. We seek to answer the ques tion 'Who is going to pay the penalty?'. In the last chapter we analyze real data to see if the penalty will motivate manufacturers to produce more ecofriendly passenger cars. The data shows that the manufacturers will lose profit until the fleets' average emissions fall within the limits. The maximization apparatus developed in this paper is indeed standard  in the sense that there are no new theories developed  although the problem is new to the extent that it requires new creative use of specific parts of optimization theory. Moreover the decision of the EU to implement drastic measures to bring down, 'on road CO2 emissions', leads...


Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to train a computer on Beatles' songs using the re search project Magenta from the Google Brain Team to produce its own music, to derive backpropagation formulas for recurrent neural networks with LSTM cells used in the Magenta music composing model, to overview machine learning techniques and discuss its similarities with methods of mathematical statistics. In order to explore the qualities of the artificially composed music more thor oughly, we restrict ourselves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and At tention) and compare generated results. Even though the artificially composed music is not as interesting as the original Beatles, it is quite likeable. According to our analysis based on musically informed metrics, artificial melodies differ from the original ones especially in lengths of notes and in pitch differences be tween consecutive notes. The artificially composed melodies tend to use shorter notes and higher pitch differences. 1


Asian Perpetuities
Svoboda, Miroslav ; Večeř, Jan (advisor) ; Čoupek, Petr (referee)
This Master thesis studies the Asian perpetuity, which is the European type option with the average asset as the underlying asset and the execution time of the option in infinity. Assuming the geometric Brownian motion model of an asset, the thesis studies the behavior of the average of the asset. Three different types of averaging are considered: arithmetic, geometric and harmonic average. The average values of the lognormals maintain the known distribution only for the geometric average but, as it is shown in the thesis, when the average is examined on infinite time horizon, the arithmetic and harmonic averages maintain the inverse gamma distribution or gamma distribution, respectively. This result enables the computation of the price of Asian perpetuity which is also examined in the thesis. 1


Logoptimal investment
Král, Stanislav ; Dostál, Petr (advisor) ; Večeř, Jan (referee)
1. Abstrakt Suppose we have a capital, which we will redistribute into investment op portunities. The financial valuation of these investments will be a sequence of independent, identically distributed random vectors that acquire finite amount of values. We will have full knowledge of the entire history of these valuations before each investment. It turns out that if our strategy is to always maximizes the mean value of the logarithm of the investment value, denoted by Λ∗ , then this strategy is asymptotically the best one possible. If strategy Λ is not asymptotically close to Λ∗ and if x goes to infinity, then the mean of the time we earn atleast x using Λ∗ is infinitely smaller than the time if we used Λ. We also earn infinitely times more money using the strategy Λ∗ . 1


Maximum Return Portfolio
Palko, Maximilián ; Večeř, Jan (advisor) ; Šmíd, Martin (referee)
Classical method of portfolio selection is based on minimizing the variabi lity of the portfolio. The Law of Large Numbers tells us that in case of longer investment horizon it should be enough to invest in the asset with the highest expected return which will eventually outperform any other portfolio. In our thesis we will suggest some portfolio creation methods which will create Maxi mum Return Portfolios. These methods will be based on finding the asset with maximal expected return. That way we will avoid the problem of estimation errors of expected returns. Two of those methods will be selected based on the results of simulation analysis. Those two methods will be tested with the real stock data and compared with the S&P 500 index. Results of the testing suggest that our portfolios could have an application in the real world. Mainly because our portfolios showed to be significantly better than the index in the case of 10 year investment horizon. 1


The StiglerLuckock model for a limit order book
Fornůsková, Monika ; Swart, Jan (advisor) ; Večeř, Jan (referee)
THE STIGLERLUCKOCK MODEL FOR A LIMIT ORDER BOOK Abstract One of the types of modernday markets are socalled orderdriven markets whose core component is a database of all incoming buy and sell orders (order book). The main goal of this thesis is to extend the StiglerLuckock model for order books to give a better insight into the price forming process and behaviour of the market participants themselves. The model introduced in this thesis focuses on a comparison of behaviour and various strategies of market makers who are sophisticated market participants profiting from extensive trading. The market is described using Markov chains, and the strategies are compared using Monte Carlo simulations and game theory. The results showed that market makers' orders should have small spread and large volumes. The final model compares two strategies in which market makers monitor their portfolio. In case of having more cash than asset (or vice versa), they shift prices of their orders to equalise the portfolio. The model recommends checking the market quite often, but acting conservatively, which means not changing prices that frequently and not jumping to conclusions just from a small imbalance in the portfolio.


Statistical machine learning with applications in music
Janásková, Eliška ; Večeř, Jan (advisor) ; Hlávka, Zdeněk (referee)
The aim of this thesis is to review the current state of machine learning in music composition and to train a computer on Beatles' songs using research project Magenta from the Google Brain Team to produce its own music. In order to explore the qualities of the generated music more thoroughly, we restrict our selves to monophonic melodies only. We train three deep learning models with three different configurations (Basic, Lookback, and Attention) and compare generated results. Even though the generated music is not as interesting as the original Beatles, it is quite likable. According to our analysis based on musically informed metrics, generated melodies differ from the original ones especially in lengths of notes and in pitch differences between consecutive notes. Generated melodies tend to use shorter notes and higher pitch differences. In theoretical background, we cover the most commonly used machine learning algorithms, introduce neural networks and review related work of music generation. 1


Parameter choice in portfolio optimization problems based on outofsample performance
Vaňková, Kateřina ; Kopa, Miloš (advisor) ; Večeř, Jan (referee)
This thesis investigates three optimization models using the rolling window method. These models are based on maximizing profits and minimizing risk. Two statistics are considered in the models: expected value and a risk measure. Risk measures analyzed in this thesis are: the variance, the Conditional ValueatRisk at a specified confidence level, and the Mean Absolute Deviation. Models are tested on the real US stock data of ten companies in the time period of twenty years: from January 30th, 1999 to January 30th, 2019. The aim of this thesis is to analyze these models using the rolling window method and to investigate its sensitivity towards changes in the values of several parameters in order to identify the best parameter setting.


Volatility modeling
Jurka, Vojtěch ; Prášková, Zuzana (advisor) ; Večeř, Jan (referee)
In the thesis we deal with modelling volatility conditional on past shocks. Traditional ARCH and GARCH models proposed by Engle(1982) and Bollerslev(1986) are investigated as well as several generalizations of GARCH model that capture asymmetric reaction on positive and negative excess returns, namely GJRGARCH, TGARCH and EGARCH. Selected models are then applied to four commodities traded on Chicago Mercantile Exchange that represent various sectors of commodity market. Our first key finding is that in short horizon all considered models have similar performance, while in longer horizon, EGARCH and TGARCH give more precise results. The second is that, measured by an average percentage error, there is no significant difference in quality of predictions among selected assets across commodity sectors.
