<p>We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model.</p><p>We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation.</p><p>In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.</p>

We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model. We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation. In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.

In this study, we show that realized bipower variation which is an extension of realized power variation is an alternative method that estimates integrated variance like realized variance. It is seen that realized bipower variation is robust to rare jumps. Robustness means that if we add rare jumps to a stochastic volatility process, realized bipower variation process continues to estimate integrated variance although realized variance estimates integrated variance plus the quadratic variation of the jump component. This robustness is crucial since it separates the discontinuous component of quadratic variation which comes from the jump part of the logarithmic price process. Thus, we demonstrate that if the logarithmic price process is in the class of stochastic volatility plus rare jumps processes then the difference between realized variance and realized bipower variation process estimates the discontinuous component of the quadratic variation. So, quadratic variation of the jump component can be estimated and jump detection can be achieved.

利用風險值作為投資組合的風險管理工具，必須考慮金融資產報酬率通常具有厚尾、高峰、波動叢聚以及資產間訊息與波動性的變化也會交互影響等現象；因此實證上通常以多變量GARCH模型作為估計投資組合變異數矩陣的方法。然而多變量GARCH模型卻存在有維度上的詛咒，當投資組合包含資產數增加時會加重參數估計上的困難度。另一種估計波動率的方法，稱為實現波動率，能比多變量GARCH模型更簡易地處理投資組合高維度的問題。本文即以實現波動率、BEKK多變量GARCH模型與CCC模型，並以中鋼、台積電、國泰金為研究對象，比較三種方法估計風險值的表現。而實證結果得到利用實現波動率確實適合應用在風險值的估計上，且在表現上有略勝一籌的現象。

Le prix efficient est latent, il est contaminé par les frictions microstructurelles ou bruit. On explore la mesure et la prévision de la volatilité fondamentale en utilisant les données à haute fréquence. Dans le premier papier, en maintenant le cadre standard du modèle additif du bruit et le prix efficient, on montre qu’en utilisant le volume de transaction, les volumes d’achat et de vente, l’indicateur de la direction de transaction et la différence entre prix d’achat et prix de vente pour absorber le bruit, on améliore la précision des estimateurs de volatilité. Si le bruit n’est que partiellement absorbé, le bruit résiduel est plus proche d’un bruit blanc que le bruit original, ce qui diminue la misspécification des caractéristiques du bruit. Dans le deuxième papier, on part d’un fait empirique qu’on modélise par une forme linéaire de la variance du bruit microstructure en la volatilité fondamentale. Grâce à la représentation de la classe générale des modèles de volatilité stochastique, on explore la performance de prévision de différentes mesures de volatilité sous les hypothèses de notre modèle. Dans le troisième papier, on dérive de nouvelles mesures réalizées en utilisant les prix et les volumes d’achat et de vente. Comme alternative au modèle additif standard pour les prix contaminés avec le bruit microstructure, on fait des hypothèses sur la distribution du prix sans frictions qui est supposé borné par les prix de vente et d’achat. / The high frequency observed price series is contaminated with market microstructure frictions or noise. We explore the measurement and forecasting of the fundamental volatility through novel approaches to the frictions’ problem. In the first paper, while maintaining the standard framework of a noise-frictionless price additive model, we use the trading volume, quoted depths, trade direction indicator and bid-ask spread to get rid of the noise. The econometric model is a price impact linear regression. We show that incorporating the cited liquidity costs variables delivers more precise volatility estimators. If the noise is only partially absorbed, the remaining noise is closer to a white noise than the original one, which lessens misspecification of the noise characteristics. Our approach is also robust to a specific form of endogeneity under which the common robust to noise measures are inconsistent. In the second paper, we model the variance of the market microstructure noise that contaminates the frictionless price as an affine function of the fundamental volatility. Under our model, the noise is time-varying intradaily. Using the eigenfunction representation of the general stochastic volatility class of models, we quantify the forecasting performance of several volatility measures under our model assumptions. In the third paper, instead of assuming the standard additive model for the observed price series, we specify the conditional distribution of the frictionless price given the available information which includes quotes and volumes. We come up with new volatility measures by characterizing the conditional mean of the integrated variance.

Submitted by Marcia Bacha (marcia.bacha@fgv.br) on 2012-03-07T12:45:08Z No. of bitstreams: 1 20120306084421880.pdf: 1716342 bytes, checksum: e7f9f7df4b67ff4e12f57770620942d8 (MD5) / Approved for entry into archive by Gisele Isaura Hannickel (gisele.hannickel@fgv.br) on 2012-03-07T12:50:42Z (GMT) No. of bitstreams: 1 20120306084421880.pdf: 1716342 bytes, checksum: e7f9f7df4b67ff4e12f57770620942d8 (MD5) / Made available in DSpace on 2012-03-07T12:51:26Z (GMT). No. of bitstreams: 1 20120306084421880.pdf: 1716342 bytes, checksum: e7f9f7df4b67ff4e12f57770620942d8 (MD5) / Com origem no setor imobiliário americano, a crise de crédito de 2008 gerou grandes perdas nos mercados ao redor do mundo. O mês de outubro do mesmo ano concentrou a maior parte da turbulência, apresentando também uma explosão na volatilidade. Em meados de 2006 e 2007, o VIX, um índice de volatilidade implícita das opções do S&P500, registrou uma elevação de patamar, sinalizando o possível desequilíbrio existente no mercado americano. Esta dissertação analisa se o consenso de que a volatilidade implícita é a melhor previsora da volatilidade futura permanece durante o período de crise. Os resultados indicam que o VIX perde poder explicativo ao se passar do período sem crise para o de crise, sendo ultrapassado pela volatilidade realizada. / Started in the U.S. housing sector, the credit crisis of 2008 caused great damage in markets around the world. The effects were concentrated in October of the same year, which also showed an explosion in volatility. In mid-2006 and mid-2007, the VIX, an index of implied volatility of options on the S&P500, recorded a rise in level signaling the possible imbalance in the U.S. market. This dissertation examines whether the consensus that implied volatility is the best predictor of future volatility remains during the crisis. The results indicate that the VIX loses explanatory power to move from a period of economic stability for a period of crisis, been surpassed by the realized volatility.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-07-18T14:26:58Z No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-07-22T15:03:54Z (GMT) No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) / Made available in DSpace on 2016-07-22T15:03:54Z (GMT). No. of bitstreams: 1 daniellealdepaulaestevesdosreis.pdf: 1239258 bytes, checksum: 75cc07cdf6eba15d62c43b78ac783fbc (MD5) Previous issue date: 2011-12-16 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Medidas de volatilidade se constituem numa preocupação por parte de estudiosos e profissionais do mercado financeiro. Modelos da família ARCH/GARCH a partir dos retornos diários produzem um indicador de volatilidade, mas, não conferem ao pesquisador uma medida observável do grau de variabilidade dos retornos em torno de seu valor esperado. A recente disponibilidade de dados de frequência inferior a um dia de negociação permitiu a elaboração de indicadores de volatilidade observáveis por meio de uma medida conhecida como volatilidade realizada. A partir de então, é possível elaborar um indicador observável de volatilidade diária com base em dados de natureza intradiária, de modo a representar uma medida mais apropriada do grau de risco de um ativo ou carteira de ativos, e, a partir de então, estimar a volatilidade por meio de processo da família ARIMA. De posse dos dados de alta-frequência de um papel preferencial da Petrobrás S.A., o presente trabalho se propõe, portanto, em construir a medida de volatilidade realizada por meio da soma dos quadrados dos retornos obtidos em intervalos regulares (5, 15 e 30 minutos) durante cada dia de negociação do papel PETR4 durante o período de 02/01/2007 à 29/10/2010. Posteriormente à criação do indicador de volatilidade realizada que se supõe como mais apropriado para se mensurar o grau de risco, pretende-se comparar a qualidade do ajustamento e a capacidade preditiva de cada um dos métodos de modelagem da volatilidade. A comparação dos modelos baseados em dados diários e intradiários dar-se-á por meio do cômputo do erro quadrático médio (EQM) e dos testes de Diebold e Mariano e de Harvey para avaliação da acurácia preditiva dos modelos. Os resultados mostraram que, em geral, os modelos da família ARIMA são mais apropriados para a avaliação do grau de ajustamento, e produz previsões mais satisfatórias que os modelos da família ARCH/GARCH. / Volatility measures constitute a concern among scholars and professionals of the financial market. Models of the ARCH/GARCH class from the daily returns produce an indicator of volatility, but do not give the researcher an observable measure of the degree of variability of returns around their expected value. The recent availability of data at frequencies below a trading day allowed the development of indicators of volatility observable through a measurement known as realized volatility. Since then, they can build an observable indicator of daily volatility based on intraday data, so as to represent a more appropriate measure of the riskiness of an asset, and from then estimate volatility through a process of ARIMA family. Provided with the data of a high frequency preferential role of Petrobrás S. A., the present paper therefore proposes to construct a measure of realized volatility by the sum of the squares of the returns obtained at regular intervals (5, 15 and 30 minutes ) during each trading day for the paper PETR4 during 02/01/2007 to 29/10/2010. After the creation of the realized volatility indicator that is supposed to be more appropriate to measure the degree of risk, the intent is to compare the goodness of fit and predictive ability of each of the methods of volatility’s models. The comparison of models based on daily data and intraday give will be through the calculation of the mean square error (MSE) and tests of Diebold and Mariano and Harvey to evaluate the predictive accuracy of models. The results in general showed that the models of the ARIMA class are more suitable for assessing the degree of adjustment and produces predictions more satisfactory than the models of the ARCH/GARCH class.

Operações em alta frequência demonstraram crescimento nos últimos anos; em decorrência disso, surgiu a necessidade de estudar o mercado de ações brasileiro no contexto dos dados em alta frequência. Os estimadores da volatilidade dos preços de ações utilizando dados de negociações em alta frequência são os principais objetos de estudo. Conforme Aldridge (2010) e Vuorenmaa (2013), o HFT foi definido como a rápida realocação de capital feita de modo que as transações possam ocorrer em milésimos de segundos por uso de algoritmos complexos que gerenciam envio de ordens, análise dos dados obtidos e tomada das melhores decisões de compra e venda. A principal fonte de informações para análise do HFT são os dados tick by tick, conhecidos como dados em alta frequência. Uma métrica oriunda da análise de dados em alta frequência e utilizada para gestão de riscos é a Volatilidade Percebida. Conforme Andersen et al. (2003), Pong et al. (2004), Koopman et al. (2005) e Corsi (2009) há um consenso na área de finanças de que as projeções da volatilidade utilizando essa métrica de risco são mais eficientes de que a estimativa da volatilidade por meio de modelos GARCH. Na gestão financeira, a projeção da volatilidade é uma ferramenta fundamental para provisionar reservas para possíveis perdas;, devido à existência de vários métodos de projeção da volatilidade e em decorrência desta necessidade torna-se necessário selecionar um modelo ou combinar diversas projeções. O principal desafio para combinar projeções é a escolha dos pesos: as diversas pesquisas da área têm foco no desenvolvimento de métodos para escolhê-los visando minimizar os erros de previsão. A literatura existente carece, no entanto, de uma proposição de método que considere o problema de eventual projeção de volatilidade abaixo do esperado. Buscando preencher essa lacuna, o objetivo principal desta tese é propor uma combinação dos estimadores da volatilidade dos preços de ações utilizando dados de negociações em alta frequência para o mercado brasileiro. Como principal ponto de inovação, propõe-se aqui de forma inédita a utilização da função baseada no Lower Partial Moment (LPM) para estimativa dos pesos para combinação das projeções. Ainda que a métrica LPM seja bastante conhecida na literatura, sua utilização para combinação de projeções ainda não foi analisada. Este trabalho apresenta contribuições ao estudo de combinações de projeções realizadas pelos modelos HAR, MIDAS, ARFIMA e Nearest Neighbor, além de propor dois novos métodos de combinação -- estes denominados por LPMFE (Lower Partial Moment Forecast Error) e DLPMFE (Discounted LPMFE). Os métodos demonstraram resultados promissores pretendem casos cuja pretensão seja evitar perdas acima do esperado e evitar provisionamento excessivo do ponto de vista orçamentário. / The High Frequency Trading (HFT) has grown significantly in the last years, in this way, this raises the need for research of the high frequency data on the Brazilian stock market.The volatility estimators of the asset prices using high frequency data are the main objects of study. According to Aldridge (2010) and Vuorenmaa (2013), the HFT was defined as the fast reallocation of trading capital that the negotiations may occur on milliseconds by complex algorithms scheduled for optimize the process of sending orders, data analysis and to make the best decisions of buy or sell. The principal information source for HFT analysis is the tick by tick data, called as high frequency data. The Realized Volatility is a risk measure from the high frequency data analysis, this metric is used for risk management.According to Andersen et al. (2003), Pong et al. (2004), Koopman et al.(2005) and Corsi (2009) there is a consensus in the finance field that the volatility forecast using this risk measure produce better results than estimating the volatility by GARCH models. The volatility forecasting is a key issue in the financial management to provision capital resources to possible losses. However, because there are several volatility forecast methods, this problem raises the need to choice a specific model or combines the projections. The main challenge to combine forecasts is the choice of the weights, with the aim of minimizingthe forecast errors, several research in the field have been focusing on development of methods to choice the weights.Nevertheless, it is missing in the literature the proposition of amethod which consider the minimization of the risk of an inefficient forecast for the losses protection. Aiming to fill the gap, the main goal of the thesis is to propose a combination of the asset prices volatility forecasts using high frequency data for Brazilian stock market. As the main focus of innovation, the thesis proposes, in an unprecedented way, the use of the function based on the Lower Partial Moment (LPM) to estimate the weights for the combination of volatility forecasts. Although the LPM measure is well known in the literature, the use of this metric for forecast combination has not been yet studied.The thesis contributes to the literature when studying the forecasts combination made by the models HAR, MIDAS, ARFIMA and Nearest Neighbor. The thesis also contributes when proposing two new methods of combinations, these methodologies are referred to as LPMFE (Lower Partial Moment Forecast Error) and DLPMFE (Discounted LPMFE). The methods have shown promising results when it is intended to avoid losses above the expected it is not intended to cause provisioning excess in the budget.

Submitted by Eduardo Augusto Aun (auneduardo@gmail.com) on 2013-01-16T19:48:06Z No. of bitstreams: 1 DISSERTAÇÃO EDUARDO A AUN 2012 MPFE FGV.pdf: 1487732 bytes, checksum: 0fa68b5193a17357238c3ff083146e3d (MD5) / Approved for entry into archive by Eliene Soares da Silva (eliene.silva@fgv.br) on 2013-01-16T19:49:13Z (GMT) No. of bitstreams: 1 DISSERTAÇÃO EDUARDO A AUN 2012 MPFE FGV.pdf: 1487732 bytes, checksum: 0fa68b5193a17357238c3ff083146e3d (MD5) / Made available in DSpace on 2013-01-16T19:55:33Z (GMT). No. of bitstreams: 1 DISSERTAÇÃO EDUARDO A AUN 2012 MPFE FGV.pdf: 1487732 bytes, checksum: 0fa68b5193a17357238c3ff083146e3d (MD5) Previous issue date: 2012-12-18 / Este estudo compara previsões de volatilidade de sete ações negociadas na Bovespa usando 02 diferentes modelos de volatilidade realizada e 03 de volatilidade condicional. A intenção é encontrar evidências empíricas quanto à diferença de resultados que são alcançados quando se usa modelos de volatilidade realizada e de volatilidade condicional para prever a volatilidade de ações no Brasil. O período analisado vai de 01 de Novembro de 2007 a 30 de Março de 2011. A amostra inclui dados intradiários de 5 minutos. Os estimadores de volatilidade realizada que serão considerados neste estudo são o Bi-Power Variation (BPVar), desenvolvido por Barndorff-Nielsen e Shephard (2004b), e o Realized Outlyingness Weighted Variation (ROWVar), proposto por Boudt, Croux e Laurent (2008a). Ambos são estimadores não paramétricos, e são robustos a jumps. As previsões de volatilidade realizada foram feitas através de modelos autoregressivos estimados para cada ação sobre as séries de volatilidade estimadas. Os modelos de variância condicional considerados aqui serão o GARCH(1,1), o GJR (1,1), que tem assimetrias em sua construção, e o FIGARCH-CHUNG (1,d,1), que tem memória longa. A amostra foi divida em duas; uma para o período de estimação de 01 de Novembro de 2007 a 30 de Dezembro de 2010 (779 dias de negociação) e uma para o período de validação de 03 de Janeiro de 2011 a 31 de Março de 2011 (61 dias de negociação). As previsões fora da amostra foram feitas para 1 dia a frente, e os modelos foram reestimados a cada passo, incluindo uma variável a mais na amostra depois de cada previsão. As previsões serão comparadas através do teste Diebold-Mariano e através de regressões da variância ex-post contra uma constante e a previsão. Além disto, o estudo também apresentará algumas estatísticas descritivas sobre as séries de volatilidade estimadas e sobre os erros de previsão. / This study compares volatility forecasts of seven publicly traded companies using 2 different models of realized volatility and 3 models of conditional volatility. The intention is to find empirical evidence as to the difference in results that are achieved when using models of realized volatility and conditional volatility to predict the volatility of shares in Brazil. The sample period runs from 1 November 2007 to 30 March 2011. The sample includes 5 minutes intraday data. The realized volatility estimators that are considered in this study are the Bi-Power Variation (BPVar) developed by Barndorff-Nielsen and Shephard (2004b), and Weighted Realized Outlyingness Variation (ROWVar) proposed by Boudt, Croux and Laurent (2008a) . Both estimators are non-parametric, and are robust to jumps. The realized volatility forecasts were made by autoregressive models estimated for each share on the estimated volatility series. The conditional variance models considered here are the GARCH (1,1), the GJR (1,1), having asymmetries in its construction, and FIGARCH-CHUNG (1, d 1), having long memory. The sample was divided into two, one for the estimation period from 01 November 2007 to 30 December 2010 (779 trading days) and one for the validation period of 03 January 2011 to 31 March 2011 (61 trading days). The out of sample forecasts were made to 1 day ahead, and the models were reestimated at each step, including one more variable in the sample after each prediction. The predictions will be compared using the Diebold-Mariano test and through regressions of the variance ex-post against a constant and the prediction. Moreover, the study also shows some descriptive statistics on the estimated volatility series and on the forecasting errors.

In this masterthesis I have rewied basic approaches to volatility estimating. These approaches are based on classical and Bayesian statistics. I have applied the volatility models for the purpose of volatility forecasting of a different foreign exchange (EURUSD, GBPUSD and CZKEUR) in the different period (from a second period to a day period). I formulate the models EWMA, GARCH, EGARCH, IGARCH, GJRGARCH, jump diffuison with constant volatility and jump diffusion model with stochastic volatility. I also proposed an MCMC algorithm in order to estimate the Bayesian models. All the models we estimated as univariate models. I compared the models according to Mincer Zarnowitz regression. The most successfull model is the jump diffusion model with a stochastic volatility. On the second place they were the GJR- GARCH model and the jump diffusion model with a constant volatility. But the jump diffusion model with a constat volatilit provided much more overvalued results.The rest of the models were even worse. From the rest the IGARCH model is the best but provided undervalued results. All these findings correspond with R squared coefficient.