Keliami uždaviniai: GARCH modelių klasės taikymas ilgo periodo finansiniams duomenims: modelių parametrų paieška, jų vertinimas, testavimas ir taikymas. Ilga atmintis sąlyginiame variantiškume yra viena iš empirinių savybių, kurią turi daugelis finansinių laiko eilučių. Viena modelių klasė, kuri atvaizduoja šį elgesį yra vadinama Dalinai Integruotu GARCH (Baillie, Bollerslev ir Mikkelsen 1996). Dalinės integracijos idėją pateikė ir ją pritaikė GARCH struktūrai Granger (1980) ir Hosking (1981). Šiame darbe bus surastos analitinės FIGARCH proceso antros eilės logaritminės tikėtinumo funkcijos išvestinės. Ilgo diapazono priklausomybė bus apskaičiuota parametriniu dalinai integruotu GARCH modeliu. Finansinių laiko eilučių duomenys bus įvertinti GARCH (CGARCH(1), CGARCH(2)) ir FIGARCH(1,d,1)) modeliais maksimalaus tikėtinumo metodu. Taip pat bus sukurtas NASDAQ- NYSE santykinio stiprumo indikatorius bei patikrintos jo panaudojimo sąlygos. Iiustracija yra pateikta 5 akcijų indeksais, 2 valiutų santykiais, aukso bei NNSS duomenims. / The paper deals with the problems of applying GARCH model/framework to a long term financial data, the search of the models, their evaluation, testing/validation and application. Long memory in conditional variance is one of the empirical features exhibited by many financial time series. One class of models that was suggested to capture this behavior is the so-called Fractionally Integrated GARCH (Baillie, Bollerslev and Mikkelsen 1996) in which the ideas of fractional integration originally introduced by Granger (1980) and Hosking (1981) for processes of the mean are applied to GARCH framework. In this paper we derive analytic expressions for the second-order derivatives of the log-likelihood function of FIGARCH processes. Long-range dependence is assessed through the parametric fractionally integrated GARCH model. Financial time series data will be estimated Component GARCH (CGARCH(1), CGARCH(2)) and FIGARCH models maximum likelihood method. Also we built NASDAQ- NYSE relative strength indicator and tested its usage conditions. An illustration is provided on 2 exchange rate, 5 stock index, gold and NNSS data.

The purpose of this paper is to select the best GARCH-type model for modelling the volatility of Bitcoin, Bitcoin Cash, Litecoin, Dogecoin and Ethereum. GARCH (1,1), IGARCH(1,1), EGARCH(1,1), TGARCH(1,1) and CGARCH(1,1) are used on the cryptocurrencies closing day return. We select the model with the highest Maximum Likelihood and run an OLS regression on the conditional volatility to measure the day-of-the-week effect. The findings show that EGARCH(1,1) model best suits Bitcoin, Litecoin, Dogecoin and Ethereum data and that the GARCH(1,1) model suits best Bitcoin data. The results show a significant presence of day-of-the-week effects on the conditional volatility of some days for Bitcoin, Bitcoin Cash and Ethereum. Wednesday has a significant negative effect on Bitcoin conditional volatility. Friday, Saturday and Sunday are found to be significant and positive on Bitcoin Cash conditional volatility. Finally, Saturday is found to be significant and positive on Ethereum conditional volatility.

[pt] Esta dissertação tem como objetivo buscar evidências de prêmio de risco a partir do mercado de opções e de futuro de dólar no Brasil. Para isso dois ensaios foram realizados: um que mede o prêmio de risco por volatilidade no mercado de opções e outro que mede o prêmio de risco cambial no mercado futuro. No primeiro caso, o prêmio é estimado como o excesso de retorno de um portfolio protegido. No segundo caso, o prêmio é estimado com base na Teoria da Paridade de Juros ajustada a risco pelo modelo CGARCH-M. Verificou-se evidências de forward bias puzzle e de prêmio de risco por volatilidade e cambial ambos negativos e variantes no tempo. O primeiro é responsável por aumento nos preços das opções de moeda enquanto o segundo é consistente com a teoria de média-variância, ou seja, o investidor avesso ao risco requer mais retorno com o aumento do risco. Além disso, choques não antecipados possuem influência na determinação do componente de longo prazo da volatilidade do prêmio de risco cambial. Em momentos de incerteza global no mercado e aumento nas restrições de liquidez a volatilidade de curto prazo se eleva. Entretanto somente com o prêmio de risco não é possível explicar os preços viesados. Portanto, são necessários estudos futuros que envolvam tanto custo de transação, quanto o desenvolvimento de modelo econômico mais tratável para determinação da taxa de câmbio. / [en] This work aims to seek evidence of risk premium in the option and future foreign exchange markets of dollar in Brazil. For that we used two essays: one that measures the premium for volatility risk in the option market and other which measures the currency risk premium in the future market. In the first case, the premium is estimated as excess return of hedge portfolio. In the second case, the premium is estimated based on risk-adjusted Interest Rate Parity Theory from a CGARCH-M model. There was evidence of forward bias puzzle and premium for volatility and for currency risk both negative and time-varying. The first is responsible for increasing currency option price, while the second is consistent with the mean-variance theory, so risk averse investors required more return when they face higher risk. In addition, unanticipated shocks have an influence in determining the long-term volatility component of currency risk premium. In times of global market uncertainty and increasing liquidity constraints the short-term volatility raises. But only the risk premium can not explain the price biased. So transaction cost and a more effective economic model must be including in futher studies about exchange rate discovering.