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Log-optimal investment
Král, Stanislav ; Dostál, Petr (advisor) ; Večeř, Jan (referee)
Suppose we have capital, which we will redistribute into investment oppor- tunities. The financial valuation of these investments will form a sequence of independent, identically distributed random vectors taking values in some clo- sed, positive interval. 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 maximize the mean value of the logarithm return on these investments, then this strategy is in a sense asymptotically optimal. 1
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Stochastic Differential Equations with Gaussian Noise and Their Applications
Camfrlová, Monika ; Čoupek, Petr (advisor) ; Večeř, Jan (referee)
In the thesis, multivariate fractional Brownian motions with possibly different Hurst indices in different coordinates are considered and a Girsanov-type theo- rem for these processes is shown. Two applications of this theorem to stochastic differential equations driven by multivariate fractional Brownian motions (SDEs) are given. Firstly, the existence of a weak solution to an SDE with a drift coeffi- cient that can be written as a sum of a regular and a singular part and a diffusion coefficient that is dependent on time and satisfies suitable conditions is shown. The results are applied for the proof of existence of a weak solution of an equation describing stochastic harmonic oscillator. Secondly, the Girsanov-type theorem is used to find the maximum likelihood scalar estimator that appears in the drift of an SDE with additive noise. 1
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Optimal age for retirement
Eichler, Dominik ; Večeř, Jan (advisor) ; Cipra, Tomáš (referee)
This thesis studies the Czech retirement system and determines the best time of the retirement age in order to receive the highest amount of benefits in expectation. Firstly, we describe the retirement system in general. We summa- rize the upsides and downsides of the early and the late retirement. We study life tables and how to use the mortality rates in order to forecast the life ex- pectancy. In the practical part, we determine the optimal age for retirement that maximizes the expected value of future cash flows. We also define the worst retirement age and the difference between the expected best and worst pension benefits. The last section compares the Czech pension system with the US and UK pension systems. 1
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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 two possible approaches for computing the odds, namely 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 Bundesliga (306 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. 1
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Asian Perpetuities
Svoboda, Miroslav ; Večeř, Jan (advisor) ; Čoupek, Petr (referee)
This Master thesis studies Asian perpetuities, which is a term standing for European type of options with an average asset as the underlying asset and the execution time of the option in infinity. Assuming Geometric Brownian motion model of price of an asset, the goal of this thesis is to study behavior of the average of the asset price. Three different types of averaging are considered: arithmetic, geometric and harmonic average. The average values of the log-normals maintain the known distribution only for the geometric average. As it is shown in the thesis; however, 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
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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.
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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 eco-friendly 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...
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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
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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 log-normals 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
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Log-optimal 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
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