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Index tracking problem using risk measures
Polakovičová, Andrea ; Branda, Martin (advisor) ; Šmíd, Martin (referee)
In this thesis, we will introduce various methods for measuring risk known as Value at Risk (V aR) and Conditional Value at Risk (CV aR). We will use their properties and formulations in deriving a linear optimization problem. The linear programming problem will consist of minimizing the objective function representing the deviation between the portfolio and a chosen index. The calculation will be carried out based on multiple constraints, where one of them will use the aforementioned risk measures V aR and CV aR. The goal is to create a portfolio based on this program that replicates the S&P 500 index. We will perform the entire calculation using Python based on historical data. Subsequently, we will use the optimal solution found by the software and construct a replication portfolio that we will track in the following time periods. In conclusion, we will analyze and discuss the individual results for various input parameters. 1
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Sparsity and regularization in portfolio selection problems
Kaľatová, Monika ; Branda, Martin (advisor) ; Šmíd, Martin (referee)
This thesis focuses on a problem which decision vector has limited number of non- zero elements. This limitation is ensured by adding cardinality constraint, but solving the mixed-integer reformulation of the problem is difficult. This mixed-integer problem is relaxed and then regularized or the exact penalty function is added. These two apporaches are described and applied on the portfolio theory. For this special type of problems we show relations between these two approaches. Basic summary of the theory of risk measures is used in numerical study, in which we compare penalization functions for few types of problems. 1
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Modeling COVID Pandemics: Strengths and Weaknesses of Epidemic Models
Šmíd, Martin
We generally discuss modeling the present COVID pandemics. We argue that useful models have to be simple in the first case, yet their uncertainty has to be handled properly. In order to study circumstances of the upcoming wave of infection,\nwe construct a simple stochastic model and present predictions it gives. We conclude that the autumn wave is most likely unavoidable and suggest concentrating to mitigation.
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Value-at-Risk estimation - non standard approaches.
Picková, Radka ; Dupačová, Jitka (advisor) ; Šmíd, Martin (referee)
The topic of the presented work is Value-at-Risk (VaR) and its estimation. VaR is a financial risk measure and is defined as a quantile of the distribution of future returns, resp. losses. There exist various methods based on different assumptions how to estimate VaR. The most commonly used methods usually assume that the returns, resp. losses, are independently and identically distributed, especially that they are normally distributed. Since this assumption is not satisfied for most daily financial data, many alternative approaches have been suggested to estimate VaR. In the presented work two of them are discussed in detail, the CAViaR method and its asymptotic properties and the method of filtered historical simulation. One part of the work are numerical experiments with real data.
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Výběr volitelných parametrů částečného zapomínání
Votava, Adam ; Kárný, Miroslav (advisor) ; Šmíd, Martin (referee)
Presented work deals with the choice of optional parameters determining partial forgetting. The main objective is to design an algorithm for the development of the optional parameters in time in the optimal way, which would be better than usage of constant parameters. For this purpose, the Bayesian dynamic decision making, general principles of tracking the slowly varying parameters via forgetting and partial forgetting method are presented. To make computations feasible the exponential family of probability distribution functions is used. Applied algorithm is described mathematically using Bayesian learning. The stress is laid on the forgetting factor's choice, that is regarded as a Bayesian hypothesis testing. Moreover, the set of hypotheses on the forgetting factor varies in time. To hypotheses, forgetting is also applied. The presented methods are then applied to the normal regression model. However, the generality of the theoretical part allows the application to other models, e.g. Markov chain model, too. The algorithm is then programmed within the Python environment and tested on the real traffic data and on the simulated data as well.
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Bayesian modeling of market price using autoregression model
Šindelář, Jan ; Kárný, Miroslav (advisor) ; Pawlas, Zbyněk (referee) ; Šmíd, Martin (referee)
1 Bayesian modeling of market price using autoregression model 1Šindelář Jan Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Miroslav Kárný, DrSc. Abstract: In the thesis we present a novel solution of Bayesian filtering in autoregression model with Laplace distributed innovations. Estimation of regression models with lep- tokurtically distributed innovations has been studied before in a Bayesian framework [2], [1]. Compared to previously conducted studies, the method described in this article leads to an exact solution for density specifying the posterior distribution of parameters. Such a solution was previously known only for a very limited class of innovation distributions. In the text an algorithm leading to an effective solution of the problem is also proposed. The algorithm is slower than the one for the classical setup, but due to increasing com- putational power and stronger support of parallel computing, it can be executed in a reasonable time for models, where the number of parameters isn't very high. Keywords: Bayesian, Autoregression, Optimal Trading, Time Series References [1] P. Congdon. Bayesian statistical modelling. Wiley, 2006. [2] A. Zellner. Bayesian and Non-Bayesian analysis of the regression model with multivari- ate Student-t error term. Journal...
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Generating of Random Samples with Given Properties and Application to Banking
Voronin, Alexander ; Franěk, Petr (advisor) ; Šmíd, Martin (referee)
The work concerns the searching for the algorithm for generating of the random variables with the given properties. There are made analyses of comparisons of the algorithms, and the optimal algorithm was chosen based on it. Since we focus on generating of random variables of defaults and explanatory variables of defaults, it concentrates mainly on the conservation of the dependence of these variables. Further we are looking for the optimal sample size of the generated samples under conservation of the required properties. And in the last Chapter we have applied the surveyed techniques to the real data.
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A verification of an approximation of the continuous double auction by a sequence of call auctions.
Kubík, Petr ; Šmíd, Martin (advisor) ; Branda, Martin (referee)
The thesis deals with two kinds of double auction - with the continuous auction and a sequence of call auctions. We explain their rules and we define their models. We present results of simulations of the both kinds of double auction - the aim is to look for the call auction with such parameters that the prices and the traded volume of the continuous auction are approximated best. Finally, in a theoretical part, we characterize the dis- tribution of the order book in the continuous auction and then we specify the joint distribution of the price and the traded volume in the call auction (the distribution of bid, ask and the traded volume given by the continuous auction may be immediately devised from the distribution of the order book).
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