Nas últimas décadas a volatilidade transformou-se num conceito muito importante na área financeira, sendo utilizada para mensurar o risco de instrumentos financeiros. Neste trabalho, o foco de estudo é a modelagem da volatilidade, que faz referência à variabilidade dos retornos, sendo esta uma característica presente nas séries temporais financeiras. Como ferramenta fundamental da modelação usaremos o modelo GARCH (Generalized Autoregressive Conditional Heteroskedasticity), que usa a heterocedasticidade condicional como uma medida da volatilidade. Considerar-se-ão duas características principais a ser modeladas com o propósito de obter um melhor ajuste e previsão da volatilidade, estas são: a assimetria e as caudas pesadas presentes na distribuição incondicional da série dos retornos. A estimação dos parâmetros dos modelos propostos será feita utilizando a abordagem Bayesiana com a metodologia MCMC (Markov Chain Monte Carlo) especificamente o algoritmo de Metropolis-Hastings. / In the last decades volatility has become a very important concept in the financial area, being used to measure the risk of financial instruments. In this work, the focus of study is the modeling of volatility, that refers to the variability of returns, which is a characteristic present in the financial time series. As a fundamental modeling tool, we used the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model, which uses conditional heteroscedasticity as a measure of volatility. Two main characteristics will be considered to be modeled with the purpose of a better adjustment and prediction of the volatility, these are: heavy tails and an asymmetry present in the unconditional distribution of the return series. The estimation of the parameters of the proposed models is done by means of the Bayesian approach with an MCMC (Markov Chain Monte Carlo) methodology , specifically the Metropolis-Hastings algorithm.

nÃo hà / O presente artigo busca uma mÃtrica refinada e confiÃvel para mensurar riscos financeiros. RiskMetrics (1994) marcou o inÃcio dessa busca e desde entÃo vÃrios pesquisadores contribuÃram com inovaÃÃes e novos modelos para essa medida e aqui se apresenta mais um passo desse caminho, ao se agregar uma modelagem multivariada. Com essa modelagem à possÃvel capturar o efeito contÃgio e a interdependÃncia financeira global. O grupo de 10 paÃses presente no estudo representa 49,9% do PIB mundial e possuem representantes de 5 continentes. O modelo de volatilidade segue sugestÃo apresentada por Cappielo, Engle e Sheppard (2006) e modelos de Value-at-Risk (VaR) seguem Matos, Cruz, Macedo e Jucà (CAEN-UFC Workingpaper). AtravÃs desse procedimento à possÃvel calcular VaR levando em consideraÃÃo o efeito contÃgio e a interdependÃncia entre os mercados ao longo do tempo. Os resultados encontrados sÃo robustos contra problemas de variÃveis omitidas, heterocedasticidade e endogeneidade, alÃm de considerar quebras estruturais. De acordo com os resultados encontrados, a interdependÃncia apresenta um papel importante dentro do processo de mensuraÃÃo de risco de mercado, apesar de atà agora ter sido esquecida pelos pesquisadores. Isso se deve, principalmente, porque a integraÃÃo financeira a nÃvel global leva ao cenÃrio de dependÃncia crescente entre os mercados financeiros e, dessa forma, aumentando o contÃgio de um impacto que ocorre em um mercado nos outros. Convidamos outros pesquisadores a rever nossa metodologia, utilizando inclusive mais informaÃÃes e incluindo outros paÃses. Acredita-se que o mundo està ano a ano se tornando mais globalizado e suas economias por consequÃncia. Nesse artigo esse efeito està sendo considerado dentro da mensuraÃÃo do risco de mercado. Incorporar esse efeito leva a modelagem, legal e interna, mais acurada, que ajuda supervisores de mercado a garantirem estabilidade de longo prazo para os mercados e possuÃrem mÃtricas mais confiÃveis dentro das instituiÃÃes sob sua tutela. AlÃm disso, à de grande valia para Ãreas de GestÃo de Risco de bancos e instituiÃÃes financeiras ao ajuda-las a compreender melhor seu perfil de risco, melhorar a comunicaÃÃo com investidores institucionais internacionais e ranquear de maneira mais eficiente seus investimentos e aplicaÃÃes. Estudos anteriores possuem um aspecto comum: Apenas levam em consideraÃÃo mudanÃas de volatilidade nos mercados domÃsticos, nÃo levando em consideraÃÃo os efeitos que outros paÃses possuem neles. No presente estudo, esse efeito se provou como importante e representativo, os modelos univariados domÃsticos falharam mais e com mais severidade que os modelos multivariados. Portanto, no presente artigo, buscou-se o desafio de dar o passo de nÃo mais modelar modelos univariados domÃsticos, mas modelos 4 multivariados globais. Acredita-se que esse avanÃo metodolÃgico ajudarà a melhor mensurar e entender o comportamento do risco de mercado atravÃs do mundo. / This paper enter into the search of a refined and trustable metric for measuring financial risk. RiskMetrics (1994) marked the start of this search and since them many researches contributed with innovations and new models for that measure, and here we find a stepforward into the search, by aggregating multivariate models, with this itâs possible to capture the effect of a worldwide contagion and financial interdependence. The group of 10 countries presents in this study represents 49,9% of world GDP and has representation across 5 continents. We follow the model of volatilities suggested in Cappielo, Engle e Sheppard (2006) and Value-at-Risk follows Matos, Cruz, Macedo e Jucà (CAEN-UFC Working paper), though this procedure itâs possible to accurate VaR model, and take in count the contagion and interdependence between markets, in long term. Our results are robust to problems with omitted variable, heteroskedasticity and endogeneity. We also take into account for structural break. According to our results, the interdependence plays an important role into financial risk measure process, although its until now usually forbidden by modelers, mostly because worldâs financial integration leads the global economies to the scenario of increasing dependence among them and contagion effect that spreads the impacts that occur into one market to the others. We invite researchers to revisit this issue in order obtain evidences using larger data and other countries as well. We claim that the world is year by year more globalized, and so are the other economies, here we add this into account for measuring financial risks. This leads to model, legal and internal, more accurate that help supervisors to guarantee the long term stability across the markets, have trustable measure of the financial institutions under their responsibility. Besides, helps the Risk Management area of banks and other financial institutions to better understand their risk profile, improve communication with institutional investors worldwide and rank effiently their investments and applications into the markets. Previous studies have a common aspect: they only consider the volatilities change across the domestic market, not tanking in consider the effect of the other countries into the domestic volatility, and this effect here is proven to be important and representative, the univariate domestic risk measure fails more and harder than the multivariate model. That being said, here we take this step, the challenge of modeling no more univariate, domestic risk measures, but a worldwide multivariate. This is a methodological innovation that helps better measure and understands the financial risks behavior across the world.

O presente trabalho analisou as volatilidades dos mercados de renda fixa e variável de onze países, sendo eles: Brasil, Rússia, Índia, China, África do Sul (neste país apenas renda fixa), Argentina, Chile, México, Estados Unidos, Alemanha e Japão no período de janeiro de 2000 a dezembro de 2011. Os indicadores utilizados para representar cada mercado foram os índices dos mercados de ações e as taxas de juros interbancárias. Para tanto, o estudo se utilizou de modelos de heterocedasticidade condicional auto-regressiva: ARCH, GARCH, EGARCH, TGARCH e PGARCH, verificando quais destes processos eram mais eficientes para modelagem da volatilidade dos mercados dos países da amostra. Esta pesquisa também verificou qual dos modelos (ARIMA ou modelos GARCH e suas extensões) conseguiria prever melhor as séries de tempo analisadas. Além disso, por meio dos índices de correlação, covariância e causalidade Granger, foram comparados os retornos e a volatilidade do mercado de ações entre os países BRIC, entre os países latinos americanos e entre os países desenvolvidos e o Brasil. Os resultados sugerem que a volatilidade, tanto do mercado de renda fixa quanto do mercado de renda variável, é mais bem modelada por processos GARCH assimétricos (EGARCH e TGARCH), demonstrando efeitos de alavancagem nas séries estudadas. Quanto aos modelos de previsão, os modelos ARIMA, também para os dois mercados, mostrou-se mais eficiente que os modelos GARCH e suas extensões. Além disso, as volatilidades dos mercados de ações entre os países analisados parecem ser mais correlacionadas e possuir maior causalidade Granger do que os retornos destes países. Entre os dois mercados, renda fixa e variável dentro de cada país, as correlações dos retornos e da volatilidade são muito baixas, em algumas vezes negativa, e há pouca relação de causalidade Granger. / This study analyzed the volatility of fixed income and stocks markets for eleven countries, namely: Brazil, Russia, India, China, South Africa (just fixed income), Argentina, Chile, Mexico, United States, Germany and Japan from January 2000 to December 2011, using interbank interest rate as a fixed income market indicator and stock index to each country, as a stock market indicator. Therefore, the study used models of autoregressive conditional heteroscedasticity: ARCH, GARCH, EGARCH, TGARCH e PGARCH to verify which of these processes were more effective for in volatility modeling in each country. This research also found that the models (ARIMA or GARCH models and their extensions) could be used as the best forecast models. Moreover, by means of correlation coefficients, covariance and Granger causality, were used to compare the returns and volatility of the stock market among the BRIC countries, among the Latin American countries and between developed countries and Brazil. The results suggest that the volatility of both the fixed income market as the stock market is best modeled by processes asymmetric GARCH (EGARCH and TGARCH) demonstrating leverage effects in the time series. Regarding prediction ARIMA models was more efficient for both markets than GARCH models and extensions. In addition, the volatility of stock markets across countries analyzed seem to be more correlated and have higher Granger causality than returns these countries. Between the two markets, for each country, the correlations of returns and volatility are very low, if not positive, and there is low Granger causality.

O objetivo deste trabalho foi estudar o comportamento da volatilidade do preço da soja negociada em contratos futuros na BM&FBOVESPA (série SFI). O estudo foi realizado por meio da comparação entre duas abordagens: na primeira, foi utilizada a série de retornos absolutos da série em questão para representar a volatilidade da mesma, que se mostrou persistente ao longo do tempo, comprovando o fato de que a série possui o comportamento de memória longa. Por ter apresentado tal comportamento, fez-se necessária a utilização de modelos ARFIMA (\"Autorregressivos Fracionários Integrados de Médias Móveis\") estes, que são capazes de capturar de maneira efetiva tal comportamento. Ainda dentro desta abordagem, os modelos foram estimados de duas maneiras distintas: a primeira, em que todos os parâmetros foram estimados simultaneamente e a segunda, em que primeiramente foi estimado o parâmetro de memória longa, diferenciada a série e, posteriormente, foram ajustados os modelos ARIMA nos dados diferenciados. Por fim, a segunda abordagem utilizada no trabalho é a mais comum em pesquisas acadêmicas: foi realizada a estimação dos modelos GARCH (\"Autorregressivos Generalizados de Heteroscedasticidade Condicional\") diretamente na série de retornos. Neste estudo, concluímos que a primeira abordagem se mostrou mais eficiente, dados os critérios de comparação utilizados. / The purpose of this article was to study the volatility of the soybean price traded in futures contracts on the BM&FBOVESPA (SFI series). The study was conduct by comparison between two approaches: first, was use the series of absolute returns of the respective series, to represent its volatility, which was persistent over time, proving the fact that the series has a long memory behavior. Because of such behavior, it was necessary to use ARFIMA models (\"Autoregressive Fractional Integrated Moving Average\"), which are able to capture effectively such behavior. Still using this approach, the models were estimate in two different ways: first, which all parameters were estimate simultaneously, and the second one, that was first estimated the long memory parameter, differentiated the series and, later, adjusted the ARIMA models in differentiated data. Finally, the second approach used in this work is the most common in academic research: the estimation of GARCH models (\"Generalized Autoregressive Conditional Heretoscskedasticity\") directly in the returns series of the studied series. In this study, we conclude that the first approach was more effective, given the comparison criteria used.

Made available in DSpace on 2016-12-23T14:00:40Z (GMT). No. of bitstreams: 1 Martinho de Freitas Salomao.pdf: 1262175 bytes, checksum: ef4dc9b7a603fc2332f25a6fb3d3bcae (MD5) Previous issue date: 2011-05-20 / Neste trabalho são analisadas as propriedades teóricas e empíricas de três modelos de precificação de opções financeiras sobre ações: Black Scholes (1973), ad-hoc Black Scholes (Dumas, Fleming e Whaley, 1998), e o modelo GARCH assimétrico proposto por Heston e Nandi (2000), ou HN-GARCH. Os modelos são testados em opções de compra sobre ações preferenciais da Petrobras. É mostrado que o modelo Black Scholes (1973), por supor que a variância do ativo subjacente seja constante, apresentou o pior desempenho de predição comparativamente aos outros dois modelos, que consideram a volatilidade uma variável. Enquanto o modelo ad-hoc Black Scholes precificou melhor as opções muito dentro do dinheiro, dentro do dinheiro e muito fora do dinheiro, o modelo HN-GARCH obteve desempenho superior em opções no dinheiro e fora do dinheiro / This study analyzes the theoretical and empirical properties of three models for pricing options on financial stocks: Black Scholes (1973), ad-hoc Black Scholes (Dumas, Fleming and Whaley, 1998), and the asymmetric GARCH model proposed by Heston and Nandi (2000), or HN-GARCH. The models are tested in call s options on shares of Petrobras. It is shown that the Black Scholes model (1973), by assuming that the variance of the underlying asset is constant, showed the worst performance prediction compared to the other two models that consider volatility a variable. While the model adhoc Black Scholes priced much better options deep in the money, in the money and deep out of the money, the HN-GARCH model had superior performance for at the money and out of the money options

The main objective of this thesis is to summarize and possibly interconnect the existing methodology on principal components analysis, hierarchical clustering and topological organization in the financial and economic networks, linear regression and GARCH modeling. In the thesis the clustering ability of PCA is compared with the more conventional approaches on a set of world stock market indices returns in different time periods where the time division is represented by The World Financial Crisis of 2007-2009. It is also observed whether the clustering of DJIA index components is underlied by the industry sector to which the individual stocks belong. Joining together PCA with classical linear regression creates principal components regression which is further in the thesis applied to the German DAX 30 index logarithmic returns forecasting using various macroeconomic and financial predictors. The correlation between two energy stocks returns - Chevron and ExxonMobil is forecasted using orthogonal (or PCA) GARCH. The constructed forecast is then compared with the predictions constructed by the conventional multivariate volatility models - EWMA and DCC GARCH.

L’objectif de cette thèse est d’étudier les divers aspects de la crise financière 2007-09. Dans l’ensemble, les deux types d’objectifs sont poursuivis dans cette thèse: le premier objectif est de déchiffrer les liaisons entre les différents marchés boursiers, immobiliers et pétroliers afin d’évaluer les retombées du rendement et de la volatilité. L’accent dans ce champ est mis sur le niveau d’intégration entre les marchés pendant différents périodes de temps y compris la crise. Ce domaine est examiné par le développement de trois essais distincts. Le premier essai examine la déclaration du gouvernement Russe affirmant que ce sont les chocs initiés par les marchés étrangers qui ont été essentiellement responsables de la panique sur leur marché boursier pendant la période Septembre - Octobre 2008. En utilisant l’approche de la contagion financière, les résultats indiquent que le marché boursier Russe est intégré faiblement avec les marchés Américain et Européen ce qui met à l’écart l’affirmation du gouvernement. Les résultats de la comparaison bivariée des marchés montrent que le marché Russe émet un niveau élevé des chocs en affectant la structure de corrélation entre la Russie et les marchés étrangers tandis que l’inverse est vrai dans le cas des retombées de la volatilité. Il est conclu que les gouvernements ne devraient pas utiliser la justification des chocs étrangers qui affectent les marchés locaux pendant la crise globale. Comme dans l’analyse précédente, nous examinons la transmission des chocs et de la volatilité sur les marchés des sociétés d’investissements immobiliers cotées (SIIC). Etant donné que la loi exige des SIIC de consacrer une grande partie de leurs investissements dans les actifs immobiliers, le rôle des SIIC dans la propagation de la crise hypothécaire des subprimes à travers le globe a été évalué. L’analyse préliminaire démontre que pendant la crise tous les marchés possèdent entre eux des liens de causalité dans le sens de Granger. Ce résultat est en accord avec le point de vue largement répandu que les marchés boursiers se comportent de la même manière pendant la crise globale. Ensuite l’intégration entre les SIIC américaines (USREITs) et les SIIC globales et le S&P500 a été examiné. Les résultats indiquent que les SIIC américaines sont faiblement intégrées avec les SIIC globales impliquant un niveau faible des retombées bidirectionnelles du choc et de la volatilité tandis que l’inverse est vrai dans le cas des SIIC américaines (USREITs) - S&P500. Enfin, l’intégration entre le S&P500 et les SIIC globales a été exploré. Les résultats suggèrent une faible intégration entre le S&P500 et les SIIC globales. Les chocs sont essentiellement transmis du S&P500 vers les SIIC globales. D’une manière générale, l’étude amène à la conclusion que ni les SIIC américaines ni le S&P500 ne peuvent pas créer une panique plus grande sur les marchés des SIIC globales pendant la crise. Ces liens faibles indiquent également les avantages de la diversification d’un portefeuille.En étudiant la crise au niveau suivant, nous analysons la relation à court ainsi qu’à long terme entre le prix du pétrole brut et les marchés boursiers pour le Brésil, la Russie, l’Inde et la Chine (BRIC) dans le cadre des modèles structurels contraints. Nos conclusions indiquent que les marchés boursiers du BRIC suivent dans certaine mesure l’hypothèse de l’efficience des marchés comme dans le cas d’un pays importateur du pétrole un choc positif de prix du pétrole entraîne une chute du marché boursier et l’inverse est vrai pour tous les pays exportateurs du pétrole. Les deux comportements importants ont été identifiés qui sont liés au taux d’intérêt à court terme et à la production industrielle. La montée des prix du pétrole engendre l’inflation qui est enrayée par une hausse du taux d’intérêt à court terme. En même temps, la production industrielle a tendance à s’accroître en termes réels au lieu de diminuer vu le choc des prix du pétrole (une hausse des prix du pétrole). Ce résultat peut être imputé à la couverture du risque d’une hausse des prix du pétrole avec la livraison physique. Dès que le contrat de couverture commence à expirer après 30, 90 ou 180 jours l’impact des prix du pétrole commence à réduire la production industrielle. Le deuxième objectif de la thèse est d’étudier l’intervention gouvernementale particulièrement sur les marchés boursiers et dans l’économie en général. D’un point de vue boursier, nous analysons le cas de l’intervention répétée du gouvernement Russe sur ses marchés boursiers nationaux pendant la fin d’année 2008. En utilisant la méthodologie des études d’événements, les résultats sont peu concluants sur l’efficacité de l’intervention gouvernementale pour protéger le marché boursier contre des chocs financiers extérieurs. Ainsi l’étude préconise aux gouvernements de ne pas intervenir pendant la crise des marchés boursiers.En étudiant le cas de l’économie en général, une nouvelle idée a été développée et lancée concernant l’intervention de la banque centrale pour contrecarrer une Bulle des Prix des Actifs (BPA). Nous avons détecté différents problèmes dans la théorie économique concernant l’intervention de la banque centrale sur le marché monétaire en cas d’apparition d’une BPA comme par exemple, - un décalage dans le temps ne peut pas avoir une incidence sur le secteur formant une bulle spéculative tout seul ainsi que l’inadéquation des canaux traditionnels des prêts bancaires. Pour faire face à ces problèmes l’étude fait avancer l’idée d’une intervention réglementaire basée sur certaines suppositions classiques. L’idée implique que contrairement à l’intervention traditionnelle de la politique monétaire la banque centrale devrait imposer aux institutions de crédit des limites d’exposition au risque de crédit pour chaque secteur. Ces limites devraient être imposées une fois que la banque centrale découvre une hausse anormale des prix dans un secteur économique donné. Nos résultats préliminaires suggèrent que l’idée d’une intervention réglementaire a du potentiel de contrecarrer la BPA. / The objective of the dissertation is to study various aspects of financial crisis 2007-09. Overall there are two kinds of objectives that are pursued in this dissertation: the first objective is to decipher the linkages between different stock markets, real estate markets and oil markets in order to assess the return and volatility spillover effects. The focus in this area is on the level of integration among the markets during different periods of time including crisis. This area is investigated through developing three separate essays. The first essay tests the Russian government claim that shocks originating in foreign markets were primarily responsible for its stock market panic during September-October 2008. Using financial contagion framework, the results indicate that the Russian stock market is weakly integrated with the US and European market in turn discarding the government claim. In bivariate market comparison, the results indicate that Russian market emits high level of shocks affecting the correlation structure between Russia and foreign markets while the reverse is true in case of volatility spillover effects. It is concluded that the governments should not use the justification of foreign shocks affecting the local markets during global crisis. Akin to foregoing analysis, we look at the transmission of shock and volatility in the Real Estate Investment Trust (REIT) markets. Since by law REITs are required to invest a large portion of their investments in real estate, the role of REITs in spreading the subprime mortgage crisis across the globe has been assessed. The initial analysis indicates that during crisis all markets are granger causing each other. The result is in compliance with the widely held view that the stock markets behave alike during global crisis. Next the integration between USREITs and global REITs and S&P500 has been examined. The results indicate USREITs is weakly integrated with the global REITs implying low level of bidirectional shock and volatility spillover while the reverse is true in case of USREITs- S&P500. Finally the integration between S&P500 and global REITs has been explored. The results suggest weak integration between S&P500 and global REITs. The shocks are mainly transmitted from S&P500 to global REITs. Over all the study concludes that neither USREITs nor S&P500 can create a wider panic in the global REIT markets during crisis. These weak linkages points towards portfolio diversification benefits as well.Studying the crisis at the next level, we analyze short-run as well as long-run relationship between crude oil price and stock markets for Brazil, Russia, India and China (BRIC) within a constrained structural modeling framework. Our findings indicate that BRIC stock markets to certain extent follow the efficient market hypothesis such that in case of oil importing country a positive oil price shock cause the stock market to fall and the reverse is true for an oil exporting country. Two important behaviors have been identified related to short-run interest rate and industrial production. The rise in oil prices generate inflation which is countered by increase in short-run interest rate. At the same time, industrial production tends to increase in real terms instead of decreasing in view of oil price shock (increase in oil price). The result can be attributed to hedging oil price risk with physical delivery. Once the hedge contract starts expiring after 30, 90 or 180 days the impact of oil price starts reducing the industrial production. The second objective of the dissertation is to study the government intervention specifically in the stock markets and generally in the economy. From stock market perspective, we analyze the case of Russian government repeated intervention in its national stock markets during late 2008. Using event-study methodology the findings indicate weak evidence that government intervention can in fact prevent stock market from external financial shocks. The study strongly recommends that the governments should not intervene during stock market crisis.Studying the case of general economy, a new idea has been developed and floated regarding central bank’s intervention directed to preempt an Asset Price Bubble (APB). The economic theory regarding central bank monetary policy intervention has been found to suffer from various problems in the event an APB occurs, such as, -time lag, -cannot affect bubbled sector alone as well as –irrelevance of traditional bank-lending channel. To deal with these issues the study brings forward the idea of regulatory intervention based on certain text book assumptions. The idea entails that contrary to traditional monetary policy intervention, the central bank should impose credit exposure limits for a particular sector on credit institutions. These limits should be imposed once the central bank finds out the abnormal increase in prices in a given sector of the economy. Our preliminary findings suggest that idea of regulatory intervention has the potential to preempt the APB.

A modelagem da volatilidade desempenha um papel fundamental em Econometria. Nesta dissertação são estudados a generalização dos modelos autorregressivos condicionalmente heterocedásticos conhecidos como GARCH e sua principal generalização multivariada, os modelos DCC-GARCH (Dynamic Condicional Correlation GARCH). Para os erros desses modelos são consideradas distribuições de probabilidade possivelmente assimétricas e leptocúrticas, sendo essas parametrizadas em função da assimetria e do peso nas caudas, necessitando assim de estimar esses parâmetros adicionais aos modelos. A estimação dos parâmetros dos modelos é feita sob a abordagem Bayesiana e devido às complexidades destes modelos, métodos computacionais baseados em simulações de Monte Carlo via Cadeias de Markov (MCMC) são utilizados. Para obter maior eficiência computacional os algoritmos de simulação da distribuição a posteriori dos parâmetros são implementados em linguagem de baixo nível. Por fim, a proposta de modelagem e estimação é exemplificada com dois conjuntos de dados reais / The modeling of volatility plays a fundamental role in Econometrics. In this dissertation are studied the generalization of known autoregressive conditionally heteroscedastic (GARCH) models and its main principal multivariate generalization, the DCCGARCH (Dynamic Conditional Correlation GARCH) models. For the errors of these models are considered distribution of probability possibility asymmetric and leptokurtic, these being parameterized as a function of asymmetry and the weight on the tails, thus requiring estimate the models additional parameters. The estimation of parameters is made under the Bayesian approach and due to the complexities of these models, methods computer-based simulations Monte Carlo Markov Chain (MCMC) are used. For more computational efficiency of simulation algorithms of posterior distribution of the parameters are implemented in low-level language. Finally, the proposed modeling and estimation is illustrated with two real data sets

O objetivo deste trabalho é apresentar e comparar diferentes métodos de modelagem da volatilidade (variância condicional) de séries temporais financeiras. O modelo ARFIMA é empregado para capturar o comportamento de memória longa observado na volatilidade de séries financeiras. Por sua vez, o modelo GARCH é utilizado para modelar a volatilidade variando no tempo destas séries. Finalmente, o modelo FIGARCH é utilizado para modelar a dinâmica dos retornos de séries temporais financeiras juntamente com sua volatilidade. Serão apresentados alguns estimadores para os parâmetros dos modelos estudados. Foram realizadas simulações dos três tipos de modelos com o objetivo de comparar o comportamento dos estimadores para diferentes valores dos parâmetros. Por fim, serão apresentadas aplicações em séries reais. / The goal of this project is to present and compare differents methods of modeling volatility (conditional variance) in financial time series. ARFIMA model is applied to capture long memory behavior of volatility in financial time series. GARCH model is used to model the temporal variation in financial volatility. Finally, FIGARCH model is used to model dynamic of financial time series returns as well as its volatility behavior. We present some estimators for the studied models. Estimators behavior of the three types of models for different parameters is assessed through a simulation study. At last, applications to real data are presented.

When the financial markets start to shake investors start looking for a safe asset for protection. When people talk about a safe asset, they for the most part refer to gold. But is that really the case? There are few studies about gold as a safe haven however they do not concern the Swedish financial market. That is the purpose of this research; to examine if gold could act as a safe haven in the financial market in Sweden. This is of interest since there is an economic crisis caused by covid19 at this particular moment. The result could help us understand how investors could use gold in their portfolio of investments. To do this we have gathered daily returns from OMXS30, gold, and a 10-year Swedish government bond. With a statistical model we answered the question. When the financial markets start to shake investors start looking for a safe asset for protection. When people talk about a safe asset, they for the most part refer to gold. But is that really the case? There are few studies about gold as a safe haven however they do not concern the Swedish financial market. That is the purpose of this research; to examine if gold could act as a safe haven in the financial market in Sweden. This is of interest since there is an economic crisis caused by covid19 at this particular moment. The result could help us understand how investors could use gold in their portfolio of investments. To do this we have gathered daily returns from OMXS30, gold, and a 10-year Swedish government bond. With a statistical model we answered the question.