A volatilidade implícita é um importante tema no campo das Finanças. Do ponto de vista acadêmico, é crescente o número de pesquisas sobre o conteúdo informacional embutido no preço dos ativos. Na visão do mercado, a volatilidade implícita pode ser negociada diretamente no mercado de derivativos como um ativo, permitindo o seu emprego para diversificação de riscos em carteiras de investimentos. No entanto, o mercado brasileiro carece de um índice de volatilidade oficial e os estudos sobre o tema no Brasil são bastante limitados, sendo urgente a proposta de métodos de obtenção deste índice coerentes com o cenário de liquidez. Assim, essa pesquisa tem como objetivo analisar o conteúdo informacional da volatilidade implícita no Brasil. Para isso foi necessário estimar um índice de volatilidade implícita para o mercado brasileiro (“VIX Brasil”), investigar o impacto da liquidez na volatilidade implícita, analisar a capacidade preditiva da volatilidade implícita em relação à volatilidade realizada futura e verificar a sua eficiência na emissão de sinais de proximidade de eventos de stress. Foram utilizados dados diários sobre o mercado de opções sobre índice de 2002 a 2013. Os principais resultados sugerem que a liquidez das opções afeta a variabilidade da volatilidade implícita ao longo do tempo. Em relação ao conteúdo informacional da volatilidade implícita no Brasil, obteve-se indícios de que (i) há significativa relação com o retorno da bolsa, sendo esta uma relação assimétrica e concentrada nos extremos da distribuição; (ii) a volatilidade implícita brasileira possui informações sobre volatilidade futura realizada além daquela contida na volatilidade histórica, porém é um estimador viesado e ineficiente; (iii) o “VIX Brasil” possui capacidade sinalizadora da proximidade de eventos de stress, em especial quando utiliza-se o limiar de 10% sobre a sua média móvel de 90 dias como abordagem de emissão de sinal. / The Implied volatility is an important topic of research in Finance. From the academics point of view, there is a growing interest in the information embedded in asset prices. From the practitioners view, the implied volatility can be directly traded in the derivatives market as an asset, being a tool for risk diversification in investment portfolios. However, the Brazilian capital market lacks an official volatility index and studies on the subject in Brazil are very limited. Hence, models for volatilities indexes consistent with the liquidity scenario of the Brazilian market are an urgent issue. Thus, this research aims to analyse the information content of implied volatility in Brazil. For achieving this goal, it was necessary to estimate an implied volatility index for the Brazilian market ("VIX Brazil"), to investigate the impact of liquidity in implied volatility, to analyse the predictive power of implied volatility for the future realized volatility and to check its efficiency for issuing early warning signals (EWS) of stress events. Daily data on the options market index over 2002 to 2013 were used. The main results suggest that the liquidity of options affects the variability of implied volatility over time. Regarding the information content of implied volatility in Brazil, evidence was obtained that (i) there is significant relationship with the market return, which is an asymmetric relationship and concentrated at the tails of the probability distribution; (ii) the Brazilian implied volatility has information about the future realized volatility than that contained in the historical volatility, but it´s a biased and inefficient estimator; (iii) the "VIX Brazil" has signalling power concerning the proximity of stress events, especially when it is used the 10% threshold on its moving average 90 days as signal emission approach.

Neste trabalho realizamos um amplo estudo de simulação com o objetivo principal de avaliar o desempenho de carteiras de mínima variância global construídas com base em modelos de previsão da volatilidade que utilizam dados de alta frequência (em comparação a dados diários). O estudo é baseado em um abrangente conjunto de dados financeiros, compreendendo 41 ações listadas na BM&FBOVESPA entre 2009 e 2017. Nós avaliamos modelos de previsão de volatilidade que são inspirados na literatura ARCH, mas que também incluem medidas realizadas. Eles são os modelos GARCH-X, HEAVY e Realized GARCH. Seu desempenho é comparado com o de carteiras construídas com base na matriz de covariância amostral, métodos de encolhimento e DCC-GARCH, bem como com a carteira igualmente ponderada e o índice Ibovespa. Uma vez que a natureza do trabalho é multivariada, e a fim de possibilitar a estimação de matrizes de covariância de grandes dimensões, recorremos à especificação DCC. Utilizamos três frequências de rebalanceamento (diária, semanal e mensal) e quatro conjuntos diferentes de restrições sobre os pesos das carteiras. A avaliação de desempenho baseia-se em medidas econômicas tais como retornos anualizados, volatilidade anualizada, razão de Sharpe, máximo drawdown, Valor em Risco, Valor em Risco condicional e turnover. Como conclusão, para o nosso conjunto de dados o uso de retornos intradiários (amostrados a cada 5 e 10 minutos) não melhora o desempenho das carteiras de mínima variância global. / In this work we perform an extensive backtesting study targeting as a main goal to assess the performance of global minimum variance (GMV) portfolios built on volatility forecasting models that make use of high frequency (compared to daily) data. The study is based on a broad intradaily financial dataset comprising 41 assets listed on the BM&FBOVESPA from 2009 to 2017. We evaluate volatility forecasting models that are inspired by the ARCH literature, but also include realized measures. They are the GARCH-X, the High-Frequency Based Volatility (HEAVY) and the Realized GARCH models. Their perfomances are benchmarked against portfolios built on the sample covariance matrix, covariance matrix shrinkage methods, DCC-GARCH as well as the naive (equally weighted) portfolio and the Ibovespa index. Since the nature of this work is multivariate and in order to make possible the estimation of large covariance matrices, we resort to the Dynamic Conditional Correlation (DCC) specification. We use three different rebalancing schemes (daily, weekly and monthly) and four different sets of constraints on portfolio weights. The performance assessment relies on economic measures such as annualized portfolio returns, annualized volatility, Sharpe ratio, maximum drawdown, Value at Risk, Expected Shortfall and turnover. We also account for transaction costs. As a conclusion, for our dataset the use of intradaily returns (sampled every 5 and 10 minutes) does not enhance the performance of GMV portfolios.

A volatilidade implícita é um importante tema no campo das Finanças. Do ponto de vista acadêmico, é crescente o número de pesquisas sobre o conteúdo informacional embutido no preço dos ativos. Na visão do mercado, a volatilidade implícita pode ser negociada diretamente no mercado de derivativos como um ativo, permitindo o seu emprego para diversificação de riscos em carteiras de investimentos. No entanto, o mercado brasileiro carece de um índice de volatilidade oficial e os estudos sobre o tema no Brasil são bastante limitados, sendo urgente a proposta de métodos de obtenção deste índice coerentes com o cenário de liquidez. Assim, essa pesquisa tem como objetivo analisar o conteúdo informacional da volatilidade implícita no Brasil. Para isso foi necessário estimar um índice de volatilidade implícita para o mercado brasileiro (“VIX Brasil”), investigar o impacto da liquidez na volatilidade implícita, analisar a capacidade preditiva da volatilidade implícita em relação à volatilidade realizada futura e verificar a sua eficiência na emissão de sinais de proximidade de eventos de stress. Foram utilizados dados diários sobre o mercado de opções sobre índice de 2002 a 2013. Os principais resultados sugerem que a liquidez das opções afeta a variabilidade da volatilidade implícita ao longo do tempo. Em relação ao conteúdo informacional da volatilidade implícita no Brasil, obteve-se indícios de que (i) há significativa relação com o retorno da bolsa, sendo esta uma relação assimétrica e concentrada nos extremos da distribuição; (ii) a volatilidade implícita brasileira possui informações sobre volatilidade futura realizada além daquela contida na volatilidade histórica, porém é um estimador viesado e ineficiente; (iii) o “VIX Brasil” possui capacidade sinalizadora da proximidade de eventos de stress, em especial quando utiliza-se o limiar de 10% sobre a sua média móvel de 90 dias como abordagem de emissão de sinal. / The Implied volatility is an important topic of research in Finance. From the academics point of view, there is a growing interest in the information embedded in asset prices. From the practitioners view, the implied volatility can be directly traded in the derivatives market as an asset, being a tool for risk diversification in investment portfolios. However, the Brazilian capital market lacks an official volatility index and studies on the subject in Brazil are very limited. Hence, models for volatilities indexes consistent with the liquidity scenario of the Brazilian market are an urgent issue. Thus, this research aims to analyse the information content of implied volatility in Brazil. For achieving this goal, it was necessary to estimate an implied volatility index for the Brazilian market ("VIX Brazil"), to investigate the impact of liquidity in implied volatility, to analyse the predictive power of implied volatility for the future realized volatility and to check its efficiency for issuing early warning signals (EWS) of stress events. Daily data on the options market index over 2002 to 2013 were used. The main results suggest that the liquidity of options affects the variability of implied volatility over time. Regarding the information content of implied volatility in Brazil, evidence was obtained that (i) there is significant relationship with the market return, which is an asymmetric relationship and concentrated at the tails of the probability distribution; (ii) the Brazilian implied volatility has information about the future realized volatility than that contained in the historical volatility, but it´s a biased and inefficient estimator; (iii) the "VIX Brazil" has signalling power concerning the proximity of stress events, especially when it is used the 10% threshold on its moving average 90 days as signal emission approach.

Spectral tools in econometrics have lately experienced a renewed surge in interest. This dissertation contributes to this literature by providing conceptually different spectral-based methods and their applications to problems of modern economics. In the first part, we take a spectral decomposition of realized volatility and construct a multivariate GARCH style model that we fit by standard quasi-maximum likelihood and generalized autoregressive score procedures. We build our model on a belief that market agents obtain information in various time horizons and therefore form their expectations in various informational horizons. This behavior creates an overall volatility process that is a mixture of spectrum specific processes. We then apply the model to the currency markets, namely GBP, CHF, and EUR. With the help of the model confidence set test we show that the multi-scale model and the generalized autoregressive score based models produce forecasts that are in most cases superior to the competing models. Moreover, we find that most of the information for future volatility comes from the high frequency part of the spectra representing the very short investment horizons. In the second part, we provide a spectral decomposition of a system multivariate connectedness measure based on Diebold and Yilmaz...

Neste trabalho realizamos um amplo estudo de simulação com o objetivo principal de avaliar o desempenho de carteiras de mínima variância global construídas com base em modelos de previsão da volatilidade que utilizam dados de alta frequência (em comparação a dados diários). O estudo é baseado em um abrangente conjunto de dados financeiros, compreendendo 41 ações listadas na BM&FBOVESPA entre 2009 e 2017. Nós avaliamos modelos de previsão de volatilidade que são inspirados na literatura ARCH, mas que também incluem medidas realizadas. Eles são os modelos GARCH-X, HEAVY e Realized GARCH. Seu desempenho é comparado com o de carteiras construídas com base na matriz de covariância amostral, métodos de encolhimento e DCC-GARCH, bem como com a carteira igualmente ponderada e o índice Ibovespa. Uma vez que a natureza do trabalho é multivariada, e a fim de possibilitar a estimação de matrizes de covariância de grandes dimensões, recorremos à especificação DCC. Utilizamos três frequências de rebalanceamento (diária, semanal e mensal) e quatro conjuntos diferentes de restrições sobre os pesos das carteiras. A avaliação de desempenho baseia-se em medidas econômicas tais como retornos anualizados, volatilidade anualizada, razão de Sharpe, máximo drawdown, Valor em Risco, Valor em Risco condicional e turnover. Como conclusão, para o nosso conjunto de dados o uso de retornos intradiários (amostrados a cada 5 e 10 minutos) não melhora o desempenho das carteiras de mínima variância global. / In this work we perform an extensive backtesting study targeting as a main goal to assess the performance of global minimum variance (GMV) portfolios built on volatility forecasting models that make use of high frequency (compared to daily) data. The study is based on a broad intradaily financial dataset comprising 41 assets listed on the BM&FBOVESPA from 2009 to 2017. We evaluate volatility forecasting models that are inspired by the ARCH literature, but also include realized measures. They are the GARCH-X, the High-Frequency Based Volatility (HEAVY) and the Realized GARCH models. Their perfomances are benchmarked against portfolios built on the sample covariance matrix, covariance matrix shrinkage methods, DCC-GARCH as well as the naive (equally weighted) portfolio and the Ibovespa index. Since the nature of this work is multivariate and in order to make possible the estimation of large covariance matrices, we resort to the Dynamic Conditional Correlation (DCC) specification. We use three different rebalancing schemes (daily, weekly and monthly) and four different sets of constraints on portfolio weights. The performance assessment relies on economic measures such as annualized portfolio returns, annualized volatility, Sharpe ratio, maximum drawdown, Value at Risk, Expected Shortfall and turnover. We also account for transaction costs. As a conclusion, for our dataset the use of intradaily returns (sampled every 5 and 10 minutes) does not enhance the performance of GMV portfolios.

A volatilidade implícita é um importante tema no campo das Finanças. Do ponto de vista acadêmico, é crescente o número de pesquisas sobre o conteúdo informacional embutido no preço dos ativos. Na visão do mercado, a volatilidade implícita pode ser negociada diretamente no mercado de derivativos como um ativo, permitindo o seu emprego para diversificação de riscos em carteiras de investimentos. No entanto, o mercado brasileiro carece de um índice de volatilidade oficial e os estudos sobre o tema no Brasil são bastante limitados, sendo urgente a proposta de métodos de obtenção deste índice coerentes com o cenário de liquidez. Assim, essa pesquisa tem como objetivo analisar o conteúdo informacional da volatilidade implícita no Brasil. Para isso foi necessário estimar um índice de volatilidade implícita para o mercado brasileiro (“VIX Brasil”), investigar o impacto da liquidez na volatilidade implícita, analisar a capacidade preditiva da volatilidade implícita em relação à volatilidade realizada futura e verificar a sua eficiência na emissão de sinais de proximidade de eventos de stress. Foram utilizados dados diários sobre o mercado de opções sobre índice de 2002 a 2013. Os principais resultados sugerem que a liquidez das opções afeta a variabilidade da volatilidade implícita ao longo do tempo. Em relação ao conteúdo informacional da volatilidade implícita no Brasil, obteve-se indícios de que (i) há significativa relação com o retorno da bolsa, sendo esta uma relação assimétrica e concentrada nos extremos da distribuição; (ii) a volatilidade implícita brasileira possui informações sobre volatilidade futura realizada além daquela contida na volatilidade histórica, porém é um estimador viesado e ineficiente; (iii) o “VIX Brasil” possui capacidade sinalizadora da proximidade de eventos de stress, em especial quando utiliza-se o limiar de 10% sobre a sua média móvel de 90 dias como abordagem de emissão de sinal. / The Implied volatility is an important topic of research in Finance. From the academics point of view, there is a growing interest in the information embedded in asset prices. From the practitioners view, the implied volatility can be directly traded in the derivatives market as an asset, being a tool for risk diversification in investment portfolios. However, the Brazilian capital market lacks an official volatility index and studies on the subject in Brazil are very limited. Hence, models for volatilities indexes consistent with the liquidity scenario of the Brazilian market are an urgent issue. Thus, this research aims to analyse the information content of implied volatility in Brazil. For achieving this goal, it was necessary to estimate an implied volatility index for the Brazilian market ("VIX Brazil"), to investigate the impact of liquidity in implied volatility, to analyse the predictive power of implied volatility for the future realized volatility and to check its efficiency for issuing early warning signals (EWS) of stress events. Daily data on the options market index over 2002 to 2013 were used. The main results suggest that the liquidity of options affects the variability of implied volatility over time. Regarding the information content of implied volatility in Brazil, evidence was obtained that (i) there is significant relationship with the market return, which is an asymmetric relationship and concentrated at the tails of the probability distribution; (ii) the Brazilian implied volatility has information about the future realized volatility than that contained in the historical volatility, but it´s a biased and inefficient estimator; (iii) the "VIX Brazil" has signalling power concerning the proximity of stress events, especially when it is used the 10% threshold on its moving average 90 days as signal emission approach.

This study examines whether the implied volatility index can provide further information in forecasting volatility than historical volatility using GARCHfamily models. For this purpose, this researchhas been conducted to forecast volatility in two main markets the United States of America through its wildly used Standard and Poor’s 500 index and its correspondingvolatility index VIX and in Europe through its Euro Stoxx 50 and its correspondingvolatility index VSTOXX. To evaluate the in-sample content of information, the conditional variance equations of GARCH(1,1) and EGARCH (1,1) are supplemented by integrating implied volatility as an explanatory variable. The realized volatility has been generated from daily squared returns and was employed as a proxy for true volatility. To examine the out-of-sample forecast performance, one-day-ahead rolling forecasts have been generated, and Mincer–Zarnowitz regression and encompassing regression has been utilized. The predictive power of implied volatility has been assessed based on Mean Square Error (MSE). Findings suggest that the integration of implied volatility as an exogenous variable in the conditional variance of GARCHmodels enhancesthe fitness of modelsand decreasesvolatility persistency. Furthermore, the significance of the implied volatility coefficient suggests that implied volatility includes pertinent information in illuminating the variation of the conditional variance. Implied volatility is found to be a biased forecast of realized volatility. Empirical findings of encompassingregression testsimply that the implied volatility index does not surpass historical volatility in terms of forecasting future realized volatility.

<p>Due to the advancements in computing power and the availability of high-frequency data, the analyses of the high frequency stock data and market microstructure has become more and more important in econometrics. In the high frequency data setting, volatility is a very important indicator on the movement of stock prices and measure of risk. It is a key input in pricing of assets, portfolio reallocation, and risk management. In this thesis, we use the Heterogeneous Autoregressive model of realized volatility, combined with Bayesian inference as well as Markov chain Monte Carlo method’s to estimate the innovation density of the daily realized volatility. A Dirichlet process is used as the prior in a countably infinite mixture model. The semi-parametric model provides a robust alternative to the models used in the literature. I find evidence of thick tails in the density of innovations to log-realized volatility.</p> / Master of Science (MSc)

This thesis studies realized volatility (RV), implied volatility (IV) and their applications in stochastic volatility models. The first essay uses both daytime and overnight high-frequency price data for equity index futures to estimate the RV of the S\&P500 and NASDAQ 100 indexes. Empirical results reveal strong inter-correlation between the regular-trading-time and after-hour RVs, as well as a significant predictive power of overnight RV on daytime RV and vice versa. We propose a new day-night realized stochastic volatility (DN-SV-RV) model, where the daytime and overnight returns are jointly modeled with their RVs, and their latent volatilities are correlated. The newly proposed DN-SV-RV model has the best out-of-sample return distribution forecasts among the models considered. The second essay extends the realized stochastic volatility model by jointly estimating return, RV and IV. We examine how RV and IV enhance the estimation of the latent volatility process for both the S\&P500 index and individual stocks. The third essay re-examines asymmetric stochastic volatility (ASV) models with different return-volatility correlation structures given RV and IV. We show by simulation that estimating the ASV models with return series alone may infer erroneous estimations of the correlation coefficients. The incorporation of volatility measures helps identify the true return-volatility correlation within the ASV framework. Empirical evidence on global equity market indices verifies that ASV models with additional volatility measures not only obtain significantly different estimations of the correlations compared to the benchmark ASV models, but also improve out-of-sample return forecasts. / Thesis / Doctor of Philosophy (PhD)

<p>This doctoral thesis is comprised of four papers that all relate to the subject of Time Series Analysis.</p><p>The first paper of the thesis considers point estimation in a nonnegative, hence non-Gaussian, AR(1) model. The parameter estimation is carried out using a type of extreme value estimators (EVEs). A novel estimation method based on the EVEs is presented. The theoretical analysis is complemented with Monte Carlo simulation results and the paper is concluded by an empirical example.</p><p>The second paper extends the model of the first paper of the thesis and considers semiparametric, robust point estimation in a nonlinear nonnegative autoregression. The nonnegative AR(1) model of the first paper is extended in three important ways: First, we allow the errors to be serially correlated. Second, we allow for heteroskedasticity of unknown form. Third, we allow for a multi-variable mapping of previous observations. Once more, the EVEs used for parameter estimation are shown to be strongly consistent under very general conditions. The theoretical analysis is complemented with extensive Monte Carlo simulation studies that illustrate the asymptotic theory and indicate reasonable small sample properties of the proposed estimators.</p><p>In the third paper we construct a simple nonnegative time series model for realized volatility, use the results of the second paper to estimate the proposed model on S&P 500 monthly realized volatilities, and then use the estimated model to make one-month-ahead forecasts. The out-of-sample performance of the proposed model is evaluated against a number of standard models. Various tests and accuracy measures are utilized to evaluate the forecast performances. It is found that forecasts from the nonnegative model perform exceptionally well under the mean absolute error and the mean absolute percentage error forecast accuracy measures.</p><p>In the fourth and last paper of the thesis we construct a multivariate extension of the popular Diebold-Mariano test. Under the null hypothesis of equal predictive accuracy of three or more forecasting models, the proposed test statistic has an asymptotic Chi-squared distribution. To explore whether the behavior of the test in moderate-sized samples can be improved, we also provide a finite-sample correction. A small-scale Monte Carlo study indicates that the proposed test has reasonable size properties in large samples and that it benefits noticeably from the finite-sample correction, even in quite large samples. The paper is concluded by an empirical example that illustrates the practical use of the two tests.</p>