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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...

V tomto článku zkoumáme model stochastické volatility s dlouhou pamětí (LSV), ve kterém je cena aktiva modelována skokově-difuzním stochastickým procesem a jeho stochastická volatilita je řízená spojitým frakcionálním procesem s dlouhou pamětí. LSV model by měl být schopen napodobit většinu z pozorovaných vlastností finančních trhů a na rozdíl od jiných přístupů popisuje explicitně shlukování volatility pomocí parametru dlouhé paměti. tato vlastnost byla navíc pozorována v časových řadách realizované volatility napříč různými aktivy s různou časovou periodou. V první části článku odvodíme alternativní oceňovací formuli pro Evropské opce. Formule nám umožní efektivně oceňovat deriváty a kalibrovat model na reálná data. V druhé části článku se zaměříme na empirické výsledky kalibrace modelu. K tomuto účelu použijeme sadu obchodovaných opcí na index FTSE 100 pro která je LSV model porovnán s oblíbeným oceňovacím přístupem - Hestonovým modelem. K prozkoumání stability zkalibrovaných parametrů a pro ověření kalibračních výsledků použijeme alternativní sady dat opcí na Apple Inc. akcie obchodované na NYSE. In this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.

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...

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,...

The aim of this thesis is to provide an empirical evidence of multifractality in financial time series and to discuss the relevance of this concept for the current financial theory. We have applied two methods, the Multifractal Detrended Fluctuation analysis and the Generalized Hurst exponent method, on components of the Dow Jones Industrial Average. We analyzed daily data of 30 companies traded on U.S. stock markets from 2002 to 2012. We present results supporting presence of multiscaling in open-close returns. Contrary to published literature, we were not able to find any significant multiscaling in volatility. Moreover based on our analysis, multiscaling is not present in standardized returns and as multifractality requires relatively complicated models, this is our most valuable result. 1

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...

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.

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.

1 Abstract There is a discrepancy between two important horizon for Value at Risk modelling in the Basel context. We take 10-day values for determining the regulatory capital but we consider 1-day models for backtesting. The main objective of this thesis is to examine the suitability of the currently used Square Root of Time rule for Value at Risk scaling. We compare its performance with the method utilizing Hurst exponent. Our analysis is performed for both the normal and stable distribution. We conclude that the normality assumption and the Square Root of Time rule prevail under the regulatory parameters. The results of the Hurst exponent method are not favourable under normality. On the other hand, the performance for the stable distribution is quite satisfactory under non-Basel parameters and the Hurst exponent complements this distribution very well. Therefore, the use of stable distribution and the Hurst exponent method is justified when dealing with complex non-linear instruments, during turbulent periods, or for general non-Basel setting. In general however, our results are strongly data-dependent and further evidence is needed for any conclusive implications. JEL Classification G21, G28, C58, G32, C14, G18 Keywords value at risk, backtesting, volatility scaling, Basel II, stable distribution, Hurst...

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

In this thesis, we analyze new possibilities in predicting daily ranges, i.e. the differences between daily high and low prices. The main focus of our work lies in investigating how models commonly used for daily ranges modeling can be enhanced to provide better forecasts. In this respect, we explore the added benefit of using more efficient volatility measures as predictors of daily ranges. Volatility measures considered in this work include realized measures of variance (realized range, realized variance) and range-based volatility measures (Parkinson, Garman & Klass, Rogers & Satchell, etc). As a subtask, we empirically assess efficiency gains in volatility estimation when using range-based estimators as opposed to simple daily ranges. As another venue of research in this work, we analyze the added benefit of slicing the trading day into different sessions based on trading activity (e.g. Asian, European and American session). In this setting we analyze whether whole-day volatility measures reliably aggregate information coming from all trading sessions. We are led by intuition that different sessions exhibit significantly different characteristics due to different order book thicknesses and trading activity in general. Thus these sessions are expected to provide valuable information concealed in...