
Statistical analysis of interval data
Troshkov, Kirill ; Antoch, Jaromír (advisor) ; Branda, Martin (referee)
Traditional statistical analysis starts with computing the basic statisti cal characteristics such as the population mean E, population variance V , cova riance and correlation. In computing these characteristics, it is usually assumed that the corresponding data values are known exactly. In real life there are many situations in which a more complete information can be achieved by describing a set of statistical units in terms of interval data. For example, daily tempera tures registered as minimum and maximum values offer a more realistic view on the weather conditions variations with respect to the simple average values. In environmental analysis, we observe a pollution level x(t) in a lake at different mo ments of time t, and we would like to estimate standard statistical characteristics such as mean, variance and correlation with other measurements. Another exam ple can be given by financial series. The minimum and the maximum transaction prices recorded daily for a set of stocks represent a more relevant information for experts in order to evaluate the stocks tendency and volatility in the same day. We must therefore modify the existing statistical algorithms to process such interval data. In this work we will analyze algorithms and their modifications for computing various statistics under...


Methods of modelling and statistical analysis of an extremal value process
Jelenová, Klára ; Volf, Petr (advisor) ; Branda, Martin (referee)
In the present work we deal with the problem of etremal value of time series, especially of maxima. We study times and values of maximum by an approach of point process and we model distribution of extremal values by statistical methods. We estimate parameters of distribution using different methods, namely graphical methods of data analysis and subsequently we test the estimated distribution by tests of goodness of fit. We study the stationary case and also the cases with a trend. In connection with distribution of excesess and exceedances over a threshold we deal with generalized Pareto distribution.


Logistic regression with applications in financial sector
Bílková, Kristýna ; Branda, Martin (advisor) ; Pešta, Michal (referee)
In this bachelor thesis binary logistic regression model is described. Its parameters are estimated by maximum likelihood method. NewtonRaphson's algorithm is used for enumeration of these estimates. There are defined some statistics for testing the significance of the coefficients. Then stepwise regression is desribed. For assessing the quality of the model Pearson's Chi Square Test and HosmerLemeshow's Test of the goodness of fit are defined. Diversification abilitz of the model is illustrated bz the Loreny curve and is quantificated by Gini coefficient, KolmogorovSmirnov statistics and generalized coefficient of determination. The theoretical knowledge is applied to insurance area data.


Pricing with interest rate trees
Novotová, Simona ; Starinská, Katarína (advisor) ; Branda, Martin (referee)
This thesis deals with interest rate trees, their construction and use in pricing. At the beginning, work deals with various types of interest and selected financial derivatives, which need to be properly priced. For pricing, it is necessary to know the evolution of interest rates, which is simulated by various stochastic models. Next section offers summary of the models. For two models (RendelmanBarter and HullWhite model) the construction of a binomial and trinomial tree is studied. The work describes a twophase algorithm, which is used for generation of trinomial tree. Interest rates obtained from the constructed tree are used for the valuation of bond options.


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).


Almost stochastic dominance
Štefánik, Adam ; Kopa, Miloš (advisor) ; Branda, Martin (referee)
Title: Almost stochastic dominance Author: Adam Štefánik Department: Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Department of Probability and Mathematical Statistics, MFF UK Abstract: In the presented work we study the almost stochastic dominance and it's properties. Almost stochastic dominance is a relaxation of stochastic dominance. Almost stochastic dominance also deals with paradox situations occurring in case of stochastic dominance. This is a situation when stochastic dominance determines indifferent relation ship between two portfolios, but in fact almost all investors can choose the better one. The original almost stochastic dominance presented by Leshno and Levy (2002) is compu tationally expensive. Lizyayev and Ruszczy'nski (2012) suggested an alternative approach. This work introduces both approaches. The most interesting part of this work is a search for efficient portfolio with respect to the almost stochastic dominance by the simple linear programming. Lizyayev and Ruszczy'nski (2012) approach is applied to Kopa and Chovanec (2008) quantile approach for portfolio efficiency testing with respect to second order stochastic dominance. Keywords: almost stochastic dominance, efficiency, CVaR


Data Envelopment Analysis with financial application
Marcinek, Daniel ; Branda, Martin (advisor) ; Zichová, Jitka (referee)
This thesis deals with various methods of Data Envelopment Analysis and their use in finance. Efficiency is measured by a ratio of weighted outputs to weighted inputs. From this model, a fractional programming problem is formed, which is then transformed into a linear programming problem. We derive a dual problem for that one. We also introduce another methods of Data Envelopment Analysis. We explain difference between a constant return to scale and a variable return to scale. We deal with a risk measures, which are considered as the inputs together with the management fees. We use gross returns as the single input. We apply these models to 15 mutual funds, determine efficiency of these mutual funds and compare these methods with another one. At the end we determine how the efficiency changes if we use only the risk measures as the inputs.


Interest rate models
Radič, Pavol ; Branda, Martin (advisor) ; Hurt, Jan (referee)
This thesis deals with modeling the development of interest rates. It discusses the most popular models of short interest rate. It focuses in detail on widely known shortrate models, i.e. Vašíček, CIR and Ho & Lee model where the dynamics of the system is described by stochastic diferential equation. The next section offers dealing with the calibration and the creation of a binomial interest rate tree. The main purpose of this work is to describe different stochastic models of interest rates and compare them with each other.

 

Economic scenarios in insurance
Krýcha, Daniel ; Branda, Martin (advisor) ; Hurt, Jan (referee)
In this thesis we will focus on interest rate modelling and related practical aspects. We will explain the significance of generated scenarios of interest rate's movement for economic results of both life and nonlife insurance companies. We will analyse presently known ways of approaching this matter and describe the selected models in detail. Taking into consideration the practical focus of this thesis, we will address the applied methods of model's calibration. Furthermore, we will employ these methods in an extensive numerical study, that will aim to reveal the weaknesses and strengths of particular calibration methods while implementing a specific model and to evaluate their potential application in actuarial practice. Central model of this work is CIR (CoxIngersollRoss) model.
