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Option Pricing
Moravec, Radek ; Hurt, Jan (advisor) ; Cipra, Tomáš (referee)
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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Multidimensional risk measures
Chromíková, Dana ; Kopa, Miloš (advisor) ; Cipra, Tomáš (referee)
This thesis deals with multiperiod risk measures and multiperiod models with these risk measures in the objective are formulated. Multiperiod models consider the possibility of an intermediate actions within the investment horizont and represent the real situation in a better way than one-period models. First the basic properties for one-period risk measures are summarized. Then multiperiod risk measures are de ned and several ways of construction concrete risk measures are discussed as extension of one-period risk measures. Multiperiod portfolio selection mean-risk models with di erent risk measures are formulated, transaction costs are included and short sales are not allowed. Using scenario approach the analysis on real data is performed and optimal strategies for one-period and multiperiod models are compared. A transaction costs e ect on optimal strategy is examined.
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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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