The main purpose of this thesis is to propose the valuation method of PX index options. PX index consists of blue chip stocks traded on Prague Stock Exchange. There are traded a few futures contracts on PX index on Prague Stock Exchange. However, the options on PX index are traded neither on Prague Stock Exchange nor on the OTC market. It is reasonable to think that it is only question of time when the trading of these options will emerge and thus, it is highly relevant subject of research to propose the method for valuation of these options. The traditional Merton's approach for valuation of equity index options assumes constant volatility and constant risk free rate. This results in serious mispricing which can be easily seen when we compare market prices and Merton formula derived prices. Instead, this thesis releases the assumptions of constant risk free rate and constant volatility. Firstly, it is assumed that that the risk free rate is time dependent function based on current market expectations and secondly it is assumed that the volatility of underlying asset follows NGARCH-mean process. For the purpose of former, the validity of pure expectation theory assumption is made. This enables to employ the instantaneous forward rate curve estimation procedure. For the purpose of the latter, the locally risk-neutral valuation relationship is applied. The assumption of NGARCH-mean process is essential in an effort to capture usually observed patterns of volatility (volatility skews) whereas the assumption of time dependent risk free rate still moves the valuation option model closer to the reality. The author derives the expected path of risk free rate and estimates the parameters of NGARCH process. Subsequently, the empirical martingale Monte Carlo simulation is used to price the PX options with different moneyness and with different times to maturity. It is shown that this proposed model results in volatility pattern which is usually observed on developed markets and the author's results are in line with similar empirical studies testing the GARCH Option Pricing Theory. The author concludes that proposed valuation method superiors original Merton's model and thus is more appropriate for primary valuation of PX options.

The three-factor model of Fama and French has proved to be a seminal contribution to asset pricing theory, and was recently extended to include two more factors, yielding the Fama-French five-factor model. Other proposed augmentations of the three-factor model includes the introduction of a momentum factor by Carthart. The extensive use of such factors in asset pricing theory and investing motivates the study of the distributional properties of the returns of these factors. However, previous studies have focused on subsets of these six factors on the U.S. market. In this thesis, the distributional properties of daily log-returns of the five Fama-French factors and the Carthart momentum factor in European data from 2009 to 2019 are examined. The univariate distributional dynamics of the factor log-returns are modelled as ARMA-NGARCH processes with skewed t distributed driving noise sequences. The Gaussian and t copula are then used to model the joint distributions of these factor log-returns. The models developed are applied to estimate the one-day ahead Value-at-Risk (VaR) in testing data. The estimations of the VaR are backtested to check for correct unconditional coverage and exponentially distributed durations between exceedances. The results suggest that the ARMA-NGARCH processes are a valid approximation of the factor log-returns, and lead to good estimations of the VaR. The results of the multivariate analysis suggest that constant Gaussian and t copulas might be insufficient to model the dependence structure of the factors, and that there might be a need for more flexible copula models with dynamic correlations between factor log-returns. / Fama och Frenchs trefaktormodell har blivit en populär modell för aktieavkastning, och utvidgades nyligen av Fama och French genom att två ytterligare faktorer lades till för att skapa en femfaktormodell. Carthart föreslår en annan modell där trefaktormodellen kompletteras med en momentumfaktor. Då dessa faktorer används inom både akademiska sammanhang och kapitalförvaltning finns det ett tydligt behov av att undersöka vilka egenskaper fördelningen av faktorernas avkastning har. Dock har tidigare sådan forskning inte undersökt detta för alla sex faktorer, och endast använt data från USA:s marknad. I detta examensarbete undersökt därför sannolikhetsfördelningen för den logaritmiska dagliga avkastningen av de fem Fama-French-faktorerna och Cartharts momentumfaktor i europeisk data från åren 2009 till 2019. De endimensionella sannolikhetsfördelningarna modelleras som dynamiska med hjälp av ARMA-NGARCH-processer med feltermer som är fördelade enligt en generaliserad t-fördelning som tillåter skevhet. För att modellera multivariata fördelningar används en Gaussisk copula och en t-copula. De erhållna modellerna används sedan för att uppskatta daglig Value-at-Risk (VaR) i testdata. Dessa uppskattningar av VaR genomgår sedan statistiska test för att undersöka om antalet överträdelser är korrekt och tiderna mellan varje överträdelse är exponentialfördelade. Resultaten i detta examensarbete tyder på att ARMA-NGARCH-processer är en bra approximation av faktorernas logaritmiska dagliga avkastning, och ger bra uppskattningar av VaR. Resultaten för den multivariata analysen tyder på att en konstant copula kan vara en otillräcklig modell för beroendestrukturen mellan faktorerna, och att det möjligen finns ett behov av att använda mer flexibla copula-modeller med en dynamisk korrelation mellan faktorernas logaritmiska avkastning.

This thesis concerns with a comparison of two advanced option-pricing techniques applied on European-style DAX index options. Specifically, the study examines the performance of both the stochastic volatility model based on asymmetric nonlinear GARCH, which was proposed by Heston and Nandi (2000), and the artificial neural network, where the conventional Black-Scholes-Merton model serves as a benchmark. These option-pricing models are tested with the use of the dataset covering the period 3rd July 2006 - 30th October 2009 as well as of its two subsets labelled as "before crisis" and "in crisis" data where the breakthrough day is the 17th March 2008. Finding the most appropriate option-pricing method for the whole periods as well as for both the "before crisis" and the "in crisis" datasets is the main focus of this work. The first two chapters introduce core issues involved in option pricing, while the subsequent third section provides a theoretical background related to all of above-mentioned pricing methods. At the same time, the reader is provided with an overview of the theoretical frameworks of various nonlinear optimization techniques, i.e. descent gradient, quassi-Newton method, Backpropagation and Levenberg-Marquardt algorithm. The empirical part of the thesis then shows that none of the...

<p>The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.</p>

The work is focused on the Value at Risk and the Expected Shortfallcalculation. We assume the returns to be based on two pillars - the white noise and the stochastic volatility. We assume that the white noise follows the NIG distribution and the volatility is modeled using the nGARCH, NIG-GARCH, tGARCH and the non-parametric method. We apply the models into the stocks of three Banks of the Nordic market. We consider the daily and the intraday returns with the frequencies 5, 10, 20 and 30 minutes. We calculate the one step ahead VaR and ES for the daily and the intraday data. We use the Kupiec test and the Markov test to assess the correctness of the models. We also provide a new concept of improving the daily VaR calculation by using the high frequency returns. The results show that the intraday data can be used to the one step ahead VaR and the ES calculation. The comparison of the VaR for the end of the following trading day calculated on the basis of the daily returns and the one computed using the high frequency returns shows that using the intraday data can improve the VaR outcomes.