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Extensions of Kalman Filtering
Tlustý, Pavel ; Hlávka, Zdeněk (advisor) ; Cipra, Tomáš (referee)
Title: Extension of Kalman filter Author: Pavel Tlustý Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Zdeněk Hlávka, Ph.D. Supervisor's e-mail address: hlavka@karlin.mff.cuni.cz Abstract: In this thesis we try to explain a theory of the Kalman filter and apply it on European CALL and PUT options. In the first chapter, we sum- marize a general theory of the Kalman filter for a linear state space and an extension of the Kalman filter for a non-linear state space. In the second chapter, we estimate the state price density from prices of European CALL options and European PUT options. In the third chapter, we mention ap- plications of the previous theory on the real data sets. Chapter 4 concerns smoothing techniques that are used to obtain a better looking (smoother) state price density estimates. Keywords: Kalman filter, extended Kalman filter, state price density, CALL option, PUT option 1
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Statistical Properties of the Estimate of Non-Life Insurance Technical Reserves
Pechanec, Jan ; Cipra, Tomáš (referee) ; Jedlička, Petr (advisor)
In the presented work we study two different statistical methods for estimating IBNR reserve that is a part of the reserve for outstanding claims reserves. The first method is stochastic version of the Chain ladder method and the second one is a PTF model. We describe theories of the methods and show their different properties. The Chain ladder method does not assume any distribution of claims amount, on the other hand PTF models assume normal distribution of logarithms of incremental data. In practical part of this work we apply both methods on illustrative data and then we compare the results. We inspect especially probabilistic distribution of estimate of reserves and their statistical characteristics. An important part of this work is statistical testing of assumptions of both methods.
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Analysis of interest rate markets
Kvasničková, Eva ; Cipra, Tomáš (referee) ; Janeček, Karel (advisor)
The aim of this thesis is to introduce probabilistic stochastic interest rate models in continuous time. Presented models are It^o processes defined by parameters, which are trying to describe interest rate behavior in the real world. For selected models we will discuss the di®erence between the forward and futures interest rates, called convexity adjustment. At the end of the thesis the analysis of arbitrage existence between interest rates and currency exchange rates, applied to the simplest Ho-Lee model, is presented.
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Exponential smoothing
Mikulka, Jakub ; Cipra, Tomáš (referee) ; Hanzák, Tomáš (advisor)
Nazev prace: Exponencialnivyrovnavani Autor: Jakub Mikulka Katedra: Katedra pravdepodobnosti a matematicke statistiky Vedouci bakalarske prace: Mgr. Tomas Hanzak e-mail vedouciho:hanzak@karlin.mff.cuni.cz Abstrakt: Prace se zabyva dvema metodami exponencialniho vyrovnavani pro nesezonni casove rady s lokalne linearnim trendem: Holtove metode a dvojitemu exponencialmmu vyrovnani (Brownove metode). Je ukazano, ze Brownova metoda je specialnim pnpadem Holtovy metody. Dale je uveden vztah procesu ARIMA(0, 2, 2) a Holtovy metody. Hlavni casti prace je teoreticke odvozeni hodnoty MSE a autokorelacniho koeficientu pfedpovedmch chyb Q pri pouziti Holtovy metody pro vsechny kombinace jejfch vyrovnavacich konstant za predpokladu generovani rady procesem ARIMA(0, 2, 2} pro vsechny hodnoty jeho parametru. Odvozene teoreticke vzorce jsou aplikovany tez na Brownovu metodu. Odvozene vzorce jsou pomoci simulaci overeny a vyzkouseny na realnych casovych radach. Jsou formulovany prakticke zavery tykajici se obou metod. Klicova slova: autokorelacni koeficient predpovednich chyb, Holtova metoda, dvojite exponencialni vyrovnavani,MSE, vyrovnavaci konstanty Abstract Title: Exponential smoothing Author: Jakub Mikulka Department: Department of Probability and Mathematical Statistics Supervisor: Mgr. Tomas Hanzak...
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Vector Autoregressive Models
Jonáš, Petr ; Lachout, Petr (referee) ; Cipra, Tomáš (advisor)
In the presented work vector autoregression (VAR) models of finite order are examined. The main part is concerned with stationary VAR processes, whose basic characteristics, various methods of coefficient matrices estimation including consistency conditions are derived. We discuss the point and interval forecasts based on VAR models as well. We also describe integrated processes, principle of cointegration and VEC models which are appropriate modifications of VAR models for cointegration processes. The work also pays attention to Granger's and multi-step causality in the context of VAR models. In the final chapter impulse response analysis and forecast error variance decomposition are presented. Everything is supplemented by illustrative examples on real data.
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Cointegrated Time Series Models
Mikoška, Marek ; Cipra, Tomáš (referee) ; Lachout, Petr (advisor)
The thesis deals with the concept of cointegration which represents appropriate tool in the analysis of nonstationary processes. First we summarized most commonly used test for the presence of the unit root in individual time series. Next we concentrate on the models which are commonly used in the cointegration analysis of the time series. We are extensively described error-correction (EC) model which could be used in the analysis of few cointegrating relations. We also pay attention to testing of the linear restrictions on cointegrating relations and testing the hypothesis of weekly exogeneity of examined series by employing likelihood ratio. For the single equation cointegration analysis we described autoregressive distributed lags model (ADL) in detail. We illustrated straight connection between EC and ADL models. Next we introduce the models VAR, VMA, Phillips triangular representation and cointegrating regression.We were concerned with description of relationships between models and we summarized their advantages and disadvantages. Final we illustrated theoretical results in the analysis of the real time series. In the final choice of model we could reduced vector error-correction model to single equation ADL model without the loss of e±ciency and we could verified the relationship between them. By...
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Long-Term Insurance and Permanent Health Insurance
Kočová, Karolína ; Šváb, Jan (referee) ; Cipra, Tomáš (advisor)
This thesis deals with long-term care insurance and disability insurance. Both products are classified as products of private health insurance which is described in the first chapter. We can find characteristics of longterm care and its providers in Czech republic in the second chapter. Further, some numerical methods used in long-term care insurance are stated. The third chapter describes the British model of length of stay in institutional long-term care, the results of the study in the Great Britain are summarized in brief. The fourth chapter deals with characteristics and numerical methods of disability insurance. The fifth chapter displays the use of state model in disability insurance. The emerging costs method is illustrated on simple example. The life tables for disabled people were designed to be used in the calculation. In the conclusion we can find reasoning for the future of products mentioned above on the Czech insurance market.
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