Thesis (Ph.D.)--Bowling Green State University, 2008. / Document formatted into pages; contains xii, 166 p. Includes bibliographical references.

Thesis (Ph. D.)--University of California, San Diego, 2009. / Title from first page of PDF file (viewed August 11, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 134-136).

Orientador: Luiz Koodi Hotta / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T03:56:46Z (GMT). No. of bitstreams: 1 Sakamoto_CarolinedeFreitas_M.pdf: 4538610 bytes, checksum: 977f3115a7248c40284073adc86889ac (MD5) Previous issue date: 2012 / Resumo: A construção de distribuições multivariadas com dependências assimétricas, especialmente com dependências complexas nas caudas, é um requisito necessário em muitas aplicações, particularmente em finanças. A teoria de cópulas pode ser bastante útil nesta tarefa. Neste sentido, algumas das propostas sugeridas na literatura são os modelos hierárquicos arquimedianos, os modelos pair-cópula e a cópula t-Student assimétrica. Esta dissertação está focada no estudo e aplicação de modelos de cópulas com dimensões maiores que três através dos modelos Pair-Cópula, que têm sido de fundamental importância para estender o conceito de dependência do caso bivariado para o caso multivariado. A metodologia de Pair-Cópula propõe a utilização de diagramas vine para a organização dos possíveis modelos. A ênfase é dada para o diagrama D-vine, que permite diversas permutações entre as séries. Por meio de simulação, é verificado o impacto dessas diferentes permutações do diagrama D-vine, e também do uso de diferentes funções de cópulas sob o cálculo do Valor em Risco (VaR). São realizadas comparações com cópulas multivariadas arquimedianas, normal e t-Student multivariadas. é apresentada uma aplicação de cópulas tetravariadas a dados reais de retornos financeiros / Abstract: The construction of multivariate distributions with asymmetric dependencies, especially with complex dependencies in the tails, is a necessary requirement in many applications, particularly in finance. The theory of copulas can be very useful in this task. In this sense, some of the proposals suggested in the literature are the Archimedean hierarchical models, Pair-Copula models and asymmetric t-Student copula. This dissertation is focused on the study and application of models of more than three dimensions through the Pair-Copula models, which have been essential to extend the concept of dependence of bivariate case to the multivariate case. The Pair-Copula methodology proposes the use of vine tree for the organization the possible models. Emphasis is given to the D-vine tree, which allows permutation among the variables. The influence and the importance of the order of the variables in the D-vine in the estimation of the Value at Risk (VaR) is investigated by simulation. The pair-copula model is compared with the t-Student multivariate distribution, the multivariate Archimedean copula, and paircopula models using different copula functions. The model is also applied to estimate the VaR of a portfolio with four assets / Mestrado / Estatistica / Mestre em Estatística

Orientador: Mauricio Enrique Zevallos Herencia / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T23:05:02Z (GMT). No. of bitstreams: 1 Abbara_OmarMuhieddineFranco_M.pdf: 3046257 bytes, checksum: d44908ec7942d0684a67c6de5242a86c (MD5) Previous issue date: 2009 / Resumo: A modelagem multivariada de séries financeiras se constitui em um dos mais importantes e desafiadores problemas na área de econometria financeira. Um dos modelos populares nesta área é o modelo de cópulas, dada sua flexibilidade para construir funções de distribuição multivariadas que reproduzam dependências não lineares. Este trabalho está focado no estudo e aplicação de modelos de cópulas com dimensão maior que três, em problemas de interdependência, contágio e gerenciamento de risco. Primeiramente é realizada a modelagem bivariada de retornos de índices considerando os mercados de Estados Unidos, os principais mercados financeiros latino-americanos e europeus, utilizando copulas variando no tempo segundo a metodologia proposta por Patton(2006). Em seguida é proposta a especificação de um modelo de cópulas trivariado com parâmetros variando no tempo combinando as propostas de Patton (2006) e Aas et. al. (2009). Em terceiro lugar a análise de dependência e contagio entre os retornos estudados é feita através do uso de copulas condicionais. Esta análise, conjuntamente com a proposta do modelo trivariado de cópulas com parâmetros variando no tempo, constituem as principais contribuições metodológicas deste trabalho. Finalmente, cópulas tetravariadas são empregadas na análise de risco de ações negociadas no mercado à vista brasileiro / Abstract: Multivariate modelling of financial time series is one of the most important and challenging issue in financial econometrics. One of the most popular model in this subject is copula models, mainly because its flexible properties to construct multivariate distributions in which it is possible to reproduce nonlinear dependence. This work studies and applies copula models with dimension higher than two in issues of interdependence, contagion and risk management. At first it is fitted bivariate copula models with time-varying parameters proposed by Patton (2006) considering the north american stock markets and the most important markets of latin America and europe. After that it is proposed a trivariate copula model with time-varying parameter which combines the methodologies of Patton (2006) and Aas et. al. (2009). At third the analysis of dependence and contagion among returns under study is made through a conditional copula model. Both the analysis through a conditional copula and the trivariate copula model with time-varying parameters are the main methodological contributions of this work. Finally, 4-variate copula models are applied in risk management in brazilian stock market / Mestrado / Econometria Financeira / Mestre em Estatística

A copula is the representation of a multivariate distribution. Copulas are used to model multivariate data in many fields. Recent developments include copula models for spatial data and for discrete marginals. We will present a new methodological approach for modeling discrete spatial processes and for predicting the process at unobserved locations. We employ Bayesian methodology for both estimation and prediction. Comparisons between the new method and Generalized Additive Model (GAM) are done to test the performance of the prediction. Although there exists a large variety of copula functions, only a few are practically manageable and in certain problems one would like to choose the Gaussian copula to model the dependence. Furthermore, most copulas are exchangeable, thus implying symmetric dependence. However, none of them is flexible enough to catch the tailed (upper tailed or lower tailed) distribution as well as elliptical distributions. An elliptical copula is the copula corresponding to an elliptical distribution by Sklar's theorem, so it can be used appropriately and effectively only to fit elliptical distributions. While in reality, data may be better described by a "fat-tailed" or "tailed" copula than by an elliptical copula. This dissertation proposes a novel pseudo-copula (the modified Gaussian pseudo-copula) based on the Gaussian copula to model dependencies in multivariate data. Our modified Gaussian pseudo-copula differs from the standard Gaussian copula in that it can model the tail dependence. The modified Gaussian pseudo-copula captures properties from both elliptical copulas and Archimedean copulas. The modified Gaussian pseudo-copula and its properties are described. We focus on issues related to the dependence of extreme values. We give our pseudo-copula characteristics in the bivariate case, which can be extended to multivariate cases easily. The proposed pseudo-copula is assessed by estimating the measure of association from two real data sets, one from finance and one from insurance. A simulation study is done to test the goodness-of-fit of this new model. / Graduation date: 2012

El concepto de dependencia aparece por todas partes en nuestra tierra y sus habitantes de manera profunda. Son innumerables los ejemplos de fenómenos interdependientes en la naturaleza, así como en aspectos médicos, sociales, políticos, económicos, entre otros. Más aún, la dependencia es obviamente no determinística, sino de naturaleza estocástica. Es por lo anterior que resulta sorprendente que conceptos y medidas de dependencia no hayan recibido suficiente atención en la literatura estadística. Al menos hasta 1966, cuando el trabajo pionero de E.L. Lehmann probó el lema de Hoeffding. Desde entonces, se han publicado algunas generalizaciones de este. Nosotros hemos obtenido una generalización multivariante para funciones de variación acotada que agrupa a las planteadas anteriormente, al establecer la relación entre los planteamiento presentados por Quesada-Molina (1992) y Cuadras (2002b) y extendiendo este último al caso multivariante. Uno de los conceptos importante en la interpretación estadística esta relacionada con la dimensión. Es por eso que hemos definido la dimensionalidad geométrica de una distribución conjunta H en función del cardinal del conjunto de correlaciones canónicas de H, si H se puede representar mediante una expansión diagonal. La dimensionalidad geométrica ha sido obtenida para algunas de las familias de cópulas más conocidas. Para determinar la dimensionalidad de algunas de las copulas, se utilizaron métodos numéricos. De acuerdo con la dimensionalidad, hemos clasificado a las cópulas en cuatro grupos: las de dimensión cero, finita, numerable o continua. En la mayoría de las cópulas se encontro que poseen dimensión numerable. Con el uso de dos funciones que satisfacen ciertas condiciones de regularidad, se ha obtenido una extensión generalizada para la cópula Gumbel-Barnett, a la que hemos deducido sus principales propiedades y medidas de dependencia para algunas funciones en particular. La cópula FGM es una de las cópulas con más aplicabilidad en campos como el análisis financiero, y a la que se le han obtenido un gran número de generalizaciones para el caso simétrico. Nosotros hemos obtenido dos nuevas generalizaciones. La primera fue obtenida al adicionar dos distribuciones auxiliares y la segunda generalización es para el caso asimétrico. En está última caben algunas de las generalizaciones existentes. Para ambos casos se han deducido los rangos admisibles de los parámetros de asociación, las principales propiedades y las medidas de dependencia. Demostramos que si se conocen las funciones canónicas de una función de distribución, es posible aproximarla a otra función de distribución a través de combinaciones lineales de las funciones canónicas. Como ejemplo, consideramos la cópula FGM en dos dimensiones, en el sentido geométrico, debido a que se conocen sus funciones canónicas, y hemos comprobado numéricamente que su aproximación a otras cópulas con dimensión numerable es aceptablemente bueno. / Contributions to Dependence and Dimensionality in copulas The concept of dependency is everywhere in our land and its inhabitants in a profound way. There are countless examples of interdependent phenomena in nature, or related to medical, social, political and economic aspects. Moreover, dependence is obviously non deterministic, but stochastic in nature. For this reason, it is surprising that concepts and measures of dependence have not been paid enough attention in the statistical literature; at least until 1966 when the pioneering work of E.L. Lehmann proved Hoeffding’s lemma, some generalizations of this have been released since then. We have obtained a multivariate generalization for functions of bounded variation that groups the above mentioned generalizations, by ascertaining the relation between the approaches presented by Quesada-Molina (1992) and Cuadras (2002b) and extending the latter to the multivariate case. One of the important concepts in statistical interpretation deals with dimensionality, which is why we have defined the geometric dimensionality of a joint distribution H as a function of the cardinal of the set of canonical correlations of H, if H can be represented by a diagonal expansion. The geometrical dimensionality has been obtained for some of the best known families of copulas. To determine the dimensionality of some copulas, numerical methods were used. According to the dimensionality, we have classified the copulas into four groups: the zero-, finite-, countable- or continuous-dimensional. Most of the copulas were found to possess countable dimension. With the use of two functions that satisfy certain regularity conditions, we have obtained a generalized extension of the Gumbel-Barnett copula, for which we have derived its main properties and measures of dependence, particularly for some functions. The FGM copula is one of the copulas with more applicability in fields such as financial analysis, and for which a large number of generalizations for the symmetric case have been obtained. We have obtained two new generalizations: the first was obtained by adding two auxiliary distributions and the second generalization is to the asymmetric case, in the latter some existing generalizations do fit. For both cases, the allowable ranges of association parameters, as well as the main properties and dependence measures have been deducted. We show that if the canonical functions of a distribution function are known, it is possible to approximate it to another distribution function through linear combinations of canonical functions. As an example, consider the two-dimensional FGM copula, in the geometric sense, because their canonical functions are known and we have numerically found that their approximation to other copulas with countable dimension is acceptably good.

Le premier volet de cette thèse traite de l’évaluation du risque de crédit. Après un chapitre introductif offrant une synthèse technique des modèles de risque, nous nous intéressons à la modélisation de la dépendance entre les risques de défaut par les copules qui permettent de mieux fonder les mesures du risque de crédit. Ces dernières assurent une description intégrale de la structure de dépendance et ont l’avantage d’exprimer la distribution jointe en termes des distributions marginales. Nous les appréhendons en termes probabilistes telles qu’elles sont désormais familières, mais également selon des perspectives algébriques, démarche à certains égards plus englobante que l’approche probabiliste. Ensuite, nous proposons un modèle général de pricing des dérivés de crédit inspiré des travaux de Cherubini et Luciano (2003) et de Luciano (2003). Nous évaluons un Credit Default Swap « vulnérable », comprenant un risque de contrepartie. Nous y intégrons la Credit Valuation Adjustment (CVA)préconisée par Bâle III pour optimiser l’allocation du capital économique. Nous reprenons la représentation générale de pricing établie par Sorensen et Bollier (1994) et contrairement aux travaux cités ci-dessus, le paiement de protection ne survient pas forcément à l’échéance du contrat. La dépendance entre le risque de contrepartie et celui de l’entité de référence est approchée par les copules. Nous examinons la vulnérabilité du CDS pour des cas de dépendance extrêmes grâce à un choix de copule mixte combinant des copules usuelles « extrêmes ». En variant le rho de Spearman, la copule mixte balaie un large spectre de dépendances, tout en assurant des closed form prices. Le modèle qui en résulte est adapté aux pratiques du marché et facile à calibrer.Nous en fournissons une application numérique. Nous mettons ensuite en évidence le rôle des dérivés de crédit en tant qu’instruments de couvertures mais aussi comme facteurs de risque, accusés d’être à l’origine de la crise des subprime. Enfin, nous analysons cette dernière ainsi que celle des dettes souveraines, héritant également de l’effondrement du marché immobilier américain. Nous proposons à la suite une étude de soutenabilité de la dette publique des pays périphériques surendettés de la zone euro à l’horizon 2016. / The first part of this thesis deals with the valuation of credit risk. After an introductory chapter providing a technical synthesis of risk models, we model the dependence between default risks with the copula that helps enhancing credit risk measures. This technical tool provides a full description of the dependence structure; one could exploit the possibility of writing any joint distribution function as a copula, taking as arguments the marginal distributions. We approach copulas in probabilistic terms as they are familiar nowadays, then with an algebraic approach which is more inclusive than the probabilistic one. Afterwards, we present a general credit derivative pricing model based on Cherubini and Luciano (2003) and Luciano (2003). We price a “vulnerable”Credit Default Swap, taking into account a counterparty risk. We consider theCredit Valuation Adjustment (CVA) advocated by Basel III to optimize theeconomic capital allocation. We recover the general representation of aproduct with counterparty risk which goes back to Sorensen and Bollier (1994)and differently from the papers mentioned above, the payment of protectiondoes not occur necessarily at the end of the contract. We approach the dependence between counterparty risk and the reference credit’s one with the copula. We study the sensitivity of the CDS in extreme dependence cases with a mixture copula defined in terms of the “extreme” ones. By varying the Spearman’s rho, one can explore the whole range of positive and negative association. Furthermore, the mixture copula provides closed form prices. Our model is then closer to the market practice and easy to implement. Later on, we provide an application on credit market data. Then, we highlight the role of credit derivatives as hedging instruments and as risk factors as well since they are accused to be responsible for the subprime crisis. Finally, we analyze the subprime crisis and the sovereign debt crisis which arose from the U.S. mortgage market collapse as well. We then study the public debt sustainability of the heavily indebted peripheral countries of the eurozone by 2016.