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Stochastic Models in Financial Mathematics
Waczulík, Oliver ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
Title: Stochastic Models in Financial Mathematics Author: Bc. Oliver Waczulík Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathe- matical Statistics Abstract: This thesis looks into the problems of ordinary stochastic models used in financial mathematics, which are often influenced by unrealistic assumptions of Brownian motion. The thesis deals with and suggests more sophisticated alternatives to Brownian motion models. By applying the fractional Brownian motion we derive a modification of the Black-Scholes pricing formula for a mixed fractional Bro- wnian motion. We use Lévy processes to introduce subordinated stable process of Ornstein-Uhlenbeck type serving for modeling interest rates. We present the calibration procedures for these models along with a simulation study for estima- tion of Hurst parameter. To illustrate the practical use of the models introduced in the paper we have used real financial data and custom procedures program- med in the system Wolfram Mathematica. We have achieved almost 90% decline in the value of Kolmogorov-Smirnov statistics by the application of subordinated stable process of Ornstein-Uhlenbeck type for the historical values of the monthly PRIBOR (Prague Interbank Offered Rate) rates in...
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Analysis and prediction of league games results
Šimsa, Filip ; Hanzák, Tomáš (advisor) ; Večeř, Jan (referee)
The thesis is devoted to an analysis of ice hockey matches results in the highest Czech league competition in seasons 1999/2000 to 2014/2015 and to prediction of the following matches. We describe and apply Kalman filter theory where forms of teams represent an unobservable state vector and results of matches serve as measurements. Goal differences are identified as a suitable transformation of a match result. They are used as a dependent variable in a linear regression to find significant predictors. For a prediction of a match result we construct an ordinal model with those predictors. By using generalized Gini coefficient, we compare a diversifica- tion power of this model with betting odds, which are offered by betting companies. At the end, we combine knowledge of odds before a match with other predictors to make a prediction model. This model is used to identify profitable bets. 1
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Robust methods in portfolio theory
Petrušová, Lucia ; Branda, Martin (advisor) ; Večeř, Jan (referee)
01 Abstract: This thesis is concerned with the robust methods in portfolio theory. Different risk measures used in portfolio management are introduced and the corresponding robust portfolio optimization problems are formulated. The analytical solutions of the robust portfolio optimization problem with the lower partial moments (LPM), value-at-risk (VaR) or conditional value-at-risk (CVaR), as a risk measure, are presented. The application of the worst-case conditional value-at-risk (WCVaR) to robust portfolio management is proposed. This thesis considers WCVaR in the situation where only partial information on the underlying probability distribution is available. The minimization of WCVaR under mixture distribution uncertainty, box uncertainty, and ellipsoidal uncertainty are investigated. Several numerical examples based on real market data are presented to illustrate the proposed approaches and advantage of the robust formulation over the corresponding nominal approach.
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Investment strategies
Mašát, Filip ; Hurt, Jan (advisor) ; Večeř, Jan (referee)
The goal of the thesis is to explain the portfolio diversification with graph theory. It introduces needed terminology from financial mathematics and from graph theory. The described methods is used on real data and are compared with classic method where risk is measured by the standard deviation and mean absolute deviation and used criteria are based on the Markowitz approach and Sharpe ratio. The software Mathematica is used for computation and graph rendering. 1
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Random walks on networks, hitting times and cover times
Havránek, Jiří ; Prokešová, Michaela (advisor) ; Večeř, Jan (referee)
This thesis studies the cover time of random walks on finite connected graphs. Work contains the derivation of upper and lower estimates of cover time which allows us to focus on hitting time instead of cover time. We show that in some cases the problem of searching for the hitting time can be further simplified with the usage of the electrical networks, which can provide a different model for considered random walks and can help finding the hitting time through the effective resistance between some vertices. This procedure is used to find the upper and lower bounds for specific families of structures, which ilustrates that in some cases the bounds are asymptotically very tight and in other cases they give poor results. 1
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Applications of Bayesian Model Selection
Macek, Tomáš ; Večeř, Jan (advisor) ; Komárek, Arnošt (referee)
The Thesis deals with Bayesian model selection. In the theoretical part, readers will get to know with the priciple of Bayesian approach and the Thesis also states Bayes theorem, which has a key role in the given problematics. Next, it elucidates possibilities of choosing prior distribution and introduces Bayesian regression model, especially Zellner's method, which can be used to choose the most suitable model. In the practical part, this method is then implemented using R on real data from English Premier League football matches. From several statistics considered, the method will select the most suitable model, which means that it will select those statistics which are the most important for the match outcome. 1
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Multivariate random walk model for multiple players games
Pavlech, Ján ; Hlubinka, Daniel (advisor) ; Večeř, Jan (referee)
The goal of this bachelor's thesis is to analyse a game of three players, as a multiva- riate random walk. Specifically, its probability distribution from a purely combinatoric approach, but also through generating functions and the inverse formula. We will exa- mine in detail the basic properties in a few simpler models: regular rotation of players who are equally skilled, regular rotation of players who are not equally skilled, and irregular rotation of players who are not equally skilled. We will also focus on the fairness of the game, return to its origin, and distribution of maximum achieved during the game. In the last chapter, we will inspect more closely some basic simulations of progress of the game. 1
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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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