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Statistical analysis and modeling of inflation
Baniar, Matúš ; Zichová, Jitka (advisor) ; Hurt, Jan (referee)
Title: Inflation modeling Author: Matúš Baniar Department: Department of probability and mathematical statistics Supervisor: RNDr. Jitka Zichová Dr., Department of probability and mathematical statistics Abstract: Inflation, the growth of the general price level, is a common economic phenomenon, which is a macroeconomic problem. The thesis deals with the me- thods by which it is possible to model inflation and therefore to understand its de- velopment. In the first case, the correlation and regression analysis, which deal with the relationship of two or more variables and the following selection of the appro- priate mathematical model. The model of linear regression is described also with methods by which we analyze its adequacy. Another described method is the analy- sis of one-dimensional time series, which we apply so called Box-Jenkins methodol- ogy. Both approaches are illustrated on real financial data using the software Wol- fram Mathematica 8. Keywords: inflation, correlation analysis, regression analysis, time series
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Interest Rates
Holotňáková, Dominika ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
This thesis is focused on the study of interest rates, It consists of four chapters. The first chapter provides introduction to this issue, presents basic terminology and different method of interest rate process. The second chapter re- presents theoretical one-factor and two-factor models of interest rates, it is mainly aimed at Vasicek, Dothan and Cox-Ingersoll-Ross model, which are used in the practical part. The third chapter is devoted to internal bank policy, describing the most important factors influencing amount of interest rate and credit limit. The last part of the paper is the practical application of one-factor models on real data. At the beginning of the chapter, we describe methods of parameters esti- mation, which are used for individual models. Numerically estimated parameters are inputs for simulations of yield curves by these models. 1
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Probability distributions in finance
Malec, Jaromír ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
This thesis presents a summary of distributions suitable for modelling returns and losses. First discusses the basic properties of returns and losses, and then on specific distributions. Particular emphasis is placed on the asymmetric distribution and distribution with heavy tails. These distributions are discussed in depth, and the basic properties concerning the behaviour of tails are summarized. It is also supplemented with numerical observations on real data. The motive for writing this work is the inadequacy of symmetric distribution, because they are not good for modelling extreme returns and losses. The work should help people, who are interested in studying asymmetric distribution with heavy tails, as a source of further investigation.
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Concentration Risk
Marchalínová, Zuzana ; Herman, Jiří (advisor) ; Hurt, Jan (referee)
The goal of this thesis is to measure the concentration risk of a portfolio as a part of a investment risk considered from the view of insurance companies by various methods and also to compare achieved results. Concentration risk in credit portfolios originates in uneven distribution of invested funds to individual obligors and it is important to manage it. In the theoretical part there are two methods presented - one is being used in practice CreditMetrics), the other one, the EU Directive, will be put into effect in the near future (Solvency II). In the practical part the methods are applied on model portfolios and the results are compared in order to decide how the methods reflect the concentration risk.
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Selected methods for multivariate financial data analysis
Andráš, Adrián ; Zichová, Jitka (advisor) ; Hurt, Jan (referee)
In practice, we often meet data in the form of observations of several variables at various points in time. These data are called time series. We present various approaches in time series analysis; graphical models, vector autoregres- sive models and vector moving-average models. We try to get information about mutual relationship of the variables and then to model their behavior. The used techniques are illustrated on log returns of monthly average exchange rates. The programs are processed in the software Mathematica 7 and can be found on the CD. 1
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Hypothesis Testing of interest rates models
Petrík, Daniel ; Myška, Petr (advisor) ; Hurt, Jan (referee)
V předložené práci se zabýváme problematikou stochastického modelování úro- kových sazeb. Jedním z nejobvyklejších postup· je modelovat dynamiku úroko- vých sazeb pomocí stochastické diferenciální rovnice difúze, jejímiž základními kameny jsou funkce driftu a funkce difúze. Od 70. let 20. století byla navržena celá řada model· tohoto typu, a ačkoli se tyto modely neustále zdokonalují, vyvstává přirozená otázka, zda se historicky pozorované úrokové sazby skutečně takovými difúzními rovnicemi řídily. V této práci budeme právě uvedenou hypo- tézu testovat pro několik nejběžnějších jednofaktorových model· úrokové sazby první generace. Z historických dat odhadneme obecnou momentovou metodou a metodou maximální věrohodnosti parametry jednotlivých difúzních rovnic a následně provedeme statistické testy dobré shody proložení těchto rovnic pozo- rovanými daty. 1
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Control of Company Risk
Nagy, Gergely ; Hurt, Jan (advisor) ; Zichová, Jitka (referee)
In the present thesis we study the most common risks a company faces every day. We deal with risk valuation from different points of view, but mostly the most modern risk measure, the Value at Risk is discussed. We study different ways of VaR esti- mation based on historical data. Further, we study the most common types of risks, market risk, liquidity and credit risk, and operational risks. This work is accompanied with sample examples and a procedure, programmed in the system Mathematica, for illustration and computational purposes.
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Moddeling of interest rates at the financial markets
Myška, Petr ; Hurt, Jan (advisor) ; Vejmělek, Jan (referee) ; Keprta, Stanislav (referee)
Traditional Monte Carlo methods for a calculation of risk quantities (mainly VaR and TVaR) use for modeling of individual risk factors very simplified models of stochastic differential equations, where the drift and diffusion functions contain usually only one or two parameters. Such approach naturally reduces the accuracy of the final result and may significantly underestimate the risk of the portfolio. In this paper we focus on the construction of a portfolio risk model that uses nonparametric statistics theory. We shall assume the development of risk factors (specifically interest rate curve) is described by stochastic differential equation, but set minimum requirements for the drift and diffusion functions and thus better reflect the information contained in historical observations. Keywords: stochastic process, nonparametric estimation, diffusion, drift, local time, VaR, TVaR
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Option Pricing
Moravec, Radek ; Hurt, Jan (advisor)
Title: Option Pricing Author: Radek Moravec Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Jan Hurt, CSc., Department of Probability and Mathematical Statistics In the present thesis we deal with European call option pricing using lattice approaches. We introduce a discrete market model and show a way how to find an arbitrage price of financial instruments on complete markets. It's equal to the discounted value of future expected cash flow. We present the binomial option pricing model and generalize it into multinomial model. We test the resulting formula on real market data obtained from NYSE and NASDAQ. We suggest a parameter estimate method which is based on time series of historical observations of daily close price. We compare calculated option prices with their real market value and try to explain the reasons of the differences. 1
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Credit risk
Srbová, Eliška ; Herman, Jiří (advisor) ; Hurt, Jan (referee)
This thesis deals with credit risk and selected methods of its evalua- tion. It is focused on assumptions, calculation methods, results and specifics of the CreditMetrics and the CreditRisk+ models. The CreditRisk+ model analytically determines the portfolio credit losses distribution that is caused by defaults of counterparties. In the CreditMetrics model, the credit migration risk is addition- ally considered and the future portfolio value distribution is calculated using the Monte Carlo simulation. The third approach covered in this thesis is the Solvency II, the set of requirements proposed by the European Union for determination of regulatory capital for insurance companies. In the practical part the three ap- proaches are applied on a set of three portfolios of different credit quality. Their results, particularly the determined level of capital required to cover the risk of unexpected credit losses, are analyzed and compared.
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