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Correlation analysis and betting odds
Josefus, Pavel ; Pešta, Michal (advisor) ; Večeř, Jan (referee)
This bachelor's thesis focuses on a statistical method called correlation analysis. The aim of the thesis is to explain various correlation coefficients such as Pearson's correlation coefficient, point biserial correlation, Spearman's rank correlation coefficient and Kendall's rank order correlation coefficient. The thesis presents confidence intervals for each of them and also tests hypotheses about correlation coefficients. The practical part of the thesis applies established methods to real data concerning courses on women's tennis match results. 1
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Log-optimal approach in betting, compound events
Macek, Tomáš ; Kupsa, Michal (advisor) ; Večeř, Jan (referee)
In this Thesis we deal with the log-optimal betting approach. The goal is to maximize the gambler's wealth in the long term. In the course of the Thesis, we will work our way from the basic cases to a completely general problem, while the task is always to obtain a log-optimal betting strategy. For the simplest cases, we use the connection to information theory, and for others we will formulate and prove a version of the Karush-Kuhn-Tucker conditions suitable precisely for the log-optimal betting aproach. In this work, we focus primarily on the tree betting scheme and we will derive the algorithm for obtaining the log-optimal strategy of any betting opportunity from the tree betting scheme, which co- vers a large variety of betting opportunities. We will then use this algorithm to program an application in Python, which will print out the log-optimal strategy of a given betting opportunity to the user. Finally, we will verify that the obtained results correspond to the Kelly criterion and we will show several examples of the use of the Thesis. 1
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Kelly criterion and Bayesian statistics
Pardubický, Štěpán ; Večeř, Jan (advisor) ; Kalina, Jan (referee)
The classic problem of the investor is the search for profitable investment opportu- nities. But how should an investor behave if he finds such an opportunity? The Kelly criterion, named after the American scientist J.L. Kelly, answers this question. The crite- rion maximises the asymptotic exponential growth rate of capital in repeated bets, which it achieves by maximising the expected value of the logarithmic utility function. The criterion assumes a fixed investor's view of the true probability distribution. In practice, however, it is not clear how this opinion should be formed. In this paper, we combine the Kelly criterion with a Bayesian approach that allows to consider multiple opinions instead of a fixed opinion and let them be validated by the evolution of capital. Finally, we apply the findings to the investor's situation in the binomial market. 1
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Complex random variables
Kovalčíková, Emma ; Zichová, Jitka (advisor) ; Večeř, Jan (referee)
This bachelor thesis deals with complex random variables and complex random vec- tors. We introduce the complex normal distribution by deriving it from the multivariate normal distribution and we describe maximum likelihood estimators of mean and variance matrix of a vector with a complex normal distribution. We conclude the theoretical part by describing a test for the nullity of the mean of the complex normal distribution. The practical part of the thesis consists of a simulation study in which we generate realizations of random vectors with complex normal distribution. We investigate the behavior of the parameter estimates of the complex normal distribution and the properties of the test of nullity of the mean for different sample sizes and different numbers of samples. Finally, we compare the empirical distribution of the test statistics derived in the theoretical part of the paper with the corresponding theoretical distribution. 1
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Markov Chains and Martingales
Hlas, Adam ; Čoupek, Petr (advisor) ; Večeř, Jan (referee)
In this thesis, we examine connection between Markov chains and martingales. These two notions are defined at the beginning and then we formulate a theorem that connects these two notions. By using this result, we examine several properties of Markov chain. At the end, we state the Lyapunov theorem about recurrence and with its help we prove in which dimensions is the random walk on lattice recurrent and when transient. 1
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On Market Efficiency, Optimal Distributional Trading Gain, and Utility Maximization
Navrátil, Robert ; Večeř, Jan (advisor)
On Market Efficiency, Optimal Distributional Trading Gain, and Utility Maximization Robert Navr'atil The aim of this thesis is multifold. First, using results from the optimal distributional trading gain problem, we determine a utility-maximizing portfolio that optimizes the benefit an agent may receive by trading the difference between his perceived future distribution of a security price and the risk-neutral density provided by the corresponding option market. Moreover, we show how one can fit the risk-neutral density directly from option market data using the SVI parameterization. We use integer programming with kernel search heuristics to statically replicate the optimal payoff. Second, we show that the United States equity market was inefficient during the weeks following the initiation of the COVID-19 pandemic. This is demonstrated by showing that utility-maximizing agents over the period ranging from mid-February to late March 2020 could generate statistically significant profits by utilizing historical price and virus- related data to forecast future equity ETF returns. Finally, we focus on the passport option. We present a version of insurance of a traded account that symmetrically treats both of its underlying assets. In our approach, we impose a natural symmetric limit in which the agent can...
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Optimal FInancial Payoffs Maximizing Utility Function
Kožnar, František ; Večeř, Jan (advisor) ; Kříž, Pavel (referee)
The goal of this thesis is to characterize payoffs that maximize expected utility function in different market setups. One can solve this problem in its generality in terms of a function of a likelihood ratio between the subjective measure of an agent P and a risk neutral measure Q. Such payoffs should be transformed to the function of the terminal stock price. The question is what measure P should be chosen, the natural candidates would correspond to either the frequentist or the Bayesian choice of the parameters. The thesis should provide a link to the Kelly Criterion in the binomial evolution of the stock price and to the Merton's Portfolio Problem in the geometric Brownian motion exam- ple showing the possible extensions of these well known problems in the novel Bayesian setup. The thesis should discuss pricing and hedging of these contracts together with their asymptotic behavior. 1
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On Market Efficiency, Optimal Distributional Trading Gain, and Utility Maximization
Navrátil, Robert ; Večeř, Jan (advisor) ; Witzany, Jiří (referee) ; Pospíšil, Jan (referee)
On Market Efficiency, Optimal Distributional Trading Gain, and Utility Maximization Robert Navr'atil The aim of this thesis is multifold. First, using results from the optimal distributional trading gain problem, we determine a utility-maximizing portfolio that optimizes the benefit an agent may receive by trading the difference between his perceived future distribution of a security price and the risk-neutral density provided by the corresponding option market. Moreover, we show how one can fit the risk-neutral density directly from option market data using the SVI parameterization. We use integer programming with kernel search heuristics to statically replicate the optimal payoff. Second, we show that the United States equity market was inefficient during the weeks following the initiation of the COVID-19 pandemic. This is demonstrated by showing that utility-maximizing agents over the period ranging from mid-February to late March 2020 could generate statistically significant profits by utilizing historical price and virus- related data to forecast future equity ETF returns. Finally, we focus on the passport option. We present a version of insurance of a traded account that symmetrically treats both of its underlying assets. In our approach, we impose a natural symmetric limit in which the agent can...
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Spectral and distortion risk measures
Kočandrle, Erik ; Kopa, Miloš (advisor) ; Večeř, Jan (referee)
In this thesis we define risk measures as a way of quantifying the risk of an invest- ment and we formulate their essential properties, focusing mainly on coherency. Then we define the notions of admissible spectrum and spectral risk measures. Next we define the distortion function and distortion risk measures. We examine their core properties, relati- onships to coherency and formulate theorems describing their mutual equivalence with respect to the task of portfolio optimization. Lastly we tackle the problem of portfolio optimization on numerical data with respect to the MINVAR distortion function and its different values of the risk aversion parameter. 1
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Models for Forecasting Interest Rates with Application to Bond Portfolio Immunisation
Vaňková, Kateřina ; Kopa, Miloš (advisor) ; Večeř, Jan (referee)
Title: Models for Forecasting Interest Rates with Application to Bond Portfolio Immunisation Author: Kateřina Vaňková Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: The development and behaviour of interest rates play a crucial role in many financial fields. Interest rates can be forecasted using several models with different assumptions. In reality, these assumptions are not usually met. It leads to situations when a sophisticated and theoretically well-established model is not significantly better than simple methods, such as random walk. This thesis aims to study several approaches to interest rate forecasting, apply these approaches to European interest rate data, and find the best model for these real data. We will model European interest rates using several models. We will consider the Nelson- Siegel model (with two different approaches on how to estimate the shape parameter λ), the vector autoregression model with lag one (VAR(1)) and the Vasicek model. We will evaluate these models based on in-sample and also out-of-sample fit. We will use the Diebold-Mariano test to evaluate the statistical significance of models' forecast error differences. We select random walk as a benchmark...
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