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Modeling of Long Memory in Volatility Using Wavelets
Kraicová, Lucie ; Baruník, Jozef (advisor) ; Adam, Tomáš (referee)
ii Abstract This thesis focuses on one of the attractive topics of current financial literature, the application of wavelet-based methods in volatility modeling. It introduces a new, wavelet-based estimator (wavelet Whittle estimator) of a FIEGARCH model, ARCH- family model capturing long-memory and asymmetry in volatility, and studies its properties. Based on an extensive Monte Carlo experiment, both the behavior of the new estimator in various situations and its relative performance with respect to two more traditional estimators (maximum likelihood estimator and Fourier-based Whittle estimator) are assessed, along with practical aspects of its application. Possible solutions are proposed for most of the issues detected, including suggestion of a new specification of the estimator. This uses maximal overlap discrete wavelet transform instead of the traditionally used discrete wavelet transform, which should improve the estimator performance in all its applications, not only in the case of FIEGARCH model estimation. The thesis concludes that, after optimization of the estimation setup, the wavelet-based estimator may become an attractive robust alternative to the traditional methods.
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Ising model in finance: from microscopic rules to macroscopic phenomena
Dvořák, Pavel ; Krištoufek, Ladislav (advisor) ; Kukačka, Jiří (referee)
The main objective of this thesis is to inspect the abilities of the Ising model to exhibit selected statistical properties, or stylized facts, that are common to a wide range of financial assets. The investigated properties are heteroskedasticity of returns, rapidly decaying linear autocorrelation, volatility clustering, heavy tails, negative skewness and non-Gaussianity of the return distribution. In the first part of the thesis, we test the presence of these stylized facts in S&P 500 daily returns over the last 30 years. The main part of the thesis is dedicated to the Ising model-based simulations and to discussion of the results. New features such as Poisson process governed lag or magnetisation dependent trading activity are incorporated in the model. We conclude that the Ising model is able to convincingly replicate most of the examined statistical properties while even more satisfactory results can be obtained with appropriate tuning. 1
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Black-Scholes models of option pricing
Čekal, Martin
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Three Essays on Electricity Markets
Luňáčková, Petra ; Janda, Karel (advisor) ; Tashpulatov, Sherzod (referee) ; Knápek, Jaroslav (referee) ; Möst, Dominik (referee)
DISSERTATION - Abstract in English Three Essays on Electricity Markets Author: PhDr. Petra Luňáčková Academic Year: 2017/2018 This thesis consists of three papers that share the main theme - energy. The articles introduce characteristics and behavior of electricity focusing on its unique properties. The dissertation aims at the Czech electricity market and analyzes also highly discussed solar power plants. The first article studies long term memory properties of electricity spot prices through the detrended fluctuation analysis, as electricity prices are dominated by cycles. We conclude that Czech electricity prices are strongly mean reverting yet non-stationary. The second part of the dissertation investigates possible asymmetry in the gas - oil prices adjustment. Oil prices determine the price of electricity during the times of peak demand, as the reaction of power plants fueled by oil is quick but marginal costs are high. We chose the gasoline - crude oil relationship known as "rockets and feathers" effect and offer two new tests to analyze such type of relationship as we believe that error correction model is not the most suitable tool. Analyzing international dataset we do not find statistically significant asymmetry. The third study assesses the impact of renewable energy sources, solar plants in...
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Forecasting in futures markets: Front, back and rolling contracts
Badáňová, Martina ; Krištoufek, Ladislav (advisor) ; Adam, Tomáš (referee)
In the thesis we analyze sixteen commodity futures markets belonging to four families (energy type, grains, metals and other agricultural commodities) utilizing futures prices of front, back and roll futures contracts. As the tests for cointegration between front and back futures prices give us contradictory results we concentrate on roll contracts defined as the difference between front and back commodity futures contracts. We found that all commodity roll futures except natural gas and wheat futures exhibit long memory, which is usually connected with the fractal market hypothesis. Further, we employ specific ARMA and ARFIMA models and rolling window one-day-ahead technique to predict roll futures contract prices. Based on analysis of relation between resulting predictability and liquidity of roll futures contracts we concluded that lowest predictability is linked with the lowest liquidity among all commodities except metals and found evidence that predictability is positively dependent on liquidity among all commodities except metals, lumber, soybean oil and soybeans. The revealed dependence is strongest for energy type commodities. The relations and dependencies on the commodity futures markets are of high importance for all market participants such as hedge managers, investors, speculators and also for...
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Black-Scholes models of option pricing
Čekal, Martin ; Maslowski, Bohdan (advisor) ; Beneš, Viktor (referee)
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Black-Scholes models of option pricing
Čekal, Martin
Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.
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Multifractal analysis of petrol and diesel prices in the Czech Republic
Baletka, Martin ; Krištoufek, Ladislav (advisor) ; Gregor, Martin (referee)
This thesis examines scaling properties of petrol and diesel prices in the Czech Republic and a crude oil price over the period from January 2004 to February 2013. Using generalised Hurst exponent and multifractal detrended fluctuation analysis techniques we find out that crude oil market is efficient, do not contain long memory and the returns exhibit monofractal behaviour. On the other hand, petrol and diesel markets in the Czech Republic are not efficient, because their returns contain long-range dependence in autocorrelations and exhibit multifractal behaviour caused mostly by fat-tailed distribution. Thus, fuels can be modelled by complex methods like Markov switching multifractal model. JEL Classification C15, C16, C46 Keywords petrol, diesel, crude oil, long memory, multifrac- tality, GHE, MF-DFA Author's e-mail martin.baletka@ies-prague.org Supervisor's e-mail kristoufek@ies-prague.org Abstrakt Tato práce zkoumá škálování cen benzínu a motorové nafty v České repub- lice a ceny ropy na datech v období od ledna 2004 do února 2013. Použitím metod zobecněného Hurstova exponentu a multifraktální detrendované fluk- tuační analýzy jsme zjistili, že trh s ropou je efektivní, bez přítomnosti dlouhé paměti v autokorelacích a výnosy na trhu s ropou vykazují monofraktální...
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Modeling of Long Memory in Volatility Using Wavelets
Kraicová, Lucie ; Baruník, Jozef (advisor) ; Adam, Tomáš (referee)
ii Abstract This thesis focuses on one of the attractive topics of current financial literature, the application of wavelet-based methods in volatility modeling. It introduces a new, wavelet-based estimator (wavelet Whittle estimator) of a FIEGARCH model, ARCH- family model capturing long-memory and asymmetry in volatility, and studies its properties. Based on an extensive Monte Carlo experiment, both the behavior of the new estimator in various situations and its relative performance with respect to two more traditional estimators (maximum likelihood estimator and Fourier-based Whittle estimator) are assessed, along with practical aspects of its application. Possible solutions are proposed for most of the issues detected, including suggestion of a new specification of the estimator. This uses maximal overlap discrete wavelet transform instead of the traditionally used discrete wavelet transform, which should improve the estimator performance in all its applications, not only in the case of FIEGARCH model estimation. The thesis concludes that, after optimization of the estimation setup, the wavelet-based estimator may become an attractive robust alternative to the traditional methods.
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Long-range cross-correlations: Tests, estimators and applications
Krištoufek, Ladislav ; Vácha, Lukáš (advisor) ; Di Matteo, Tiziana (referee) ; Peng Liu, Rui (referee) ; Onali, Enrico (referee)
The motivation of this thesis is to provide a basic framework for treating long-range cross-correlated processes while keeping the methodology and as- sumptions as general as possible. Starting from the definition of long-range cross-correlated processes as jointly stationary processes with asymptotically power-law decaying cross-correlation function, we show that such definition implies a divergent at origin cross-power spectrum and power-law scaling of covariances of partial sums of the long-range cross-correlated processes. Chap- ter 2 describes these and other basic definitions and propositions together with necessary proofs. Chapter 3 then introduces several processes which possess long-range cross-correlated series properties. Apart from cases when the mem- ory parameter of the bivariate memory is a simple average of the parameters of the separate processes, we also introduce a new kind of process, which we call the mixed-correlated ARFIMA, which allows to control for both the bi- variate and univariate memory parameters. Chapter 4 deals with tests for a presence of long-range cross-correlations. We develop three new tests, and Monte-Carlo-simulation-based statistical power and size of the tests are com- pared. The newly introduced tests strongly surpass the already existing one. In Chapter 5,...
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