No / Based on a reduced-form model of credit risk, we explore statistical arbitrage opportunities in the CDS spreads of North American companies. Specifically, we develop a trading strategy using the model to trade market-neutral portfolios while controlling for realistic transaction costs. Empirical results show that our arbitrage strategy is of significant economic value, and also cast doubt on the efficiency of the CDS market. The aggregate returns of the trading strategy are positively related to the square of market-wide credit and liquidity risks, indicating that the market is less efficient when it is more volatile.

Yes / Based on a reduced-form model of credit risk, we explore mispricing in the CDS spreads of North American companies and its economic content. Specifically, we develop a trading strategy using the model to trade out of sample market-neutral portfolios across the term structure of CDS contracts. Our empirical results show that the trading strategy exhibits abnormally large returns, confirming the existence and persistence of a mispricing. The aggregate returns of the trading strategy are positively related to the square of market-wide credit and liquidity risks, indicating that the mispricing is more pronounced when the market is more volatile. When implemented on the Markit data, the strategy shows significant economic value even after controlling for realistic transaction costs.

Esta tese foca no problema de estimação de estado e de identificação de parâametros para modelos afins por partes. Modelos afins por partes são obtidos quando o domínio do estado ou da entrada do sistema e particionado em regiões e, para cada região, um submodelo linear ou afim e utilizado para descrever a dinâmica do sistema. Propomos um algoritmo para estimação recursiva de estados e um algoritmo de identificação de parâmetros para uma classe de modelos afins por partes. Propomos um estimador de estados Bayesiano que utiliza o filtro de Kalman em cada um dos submodelos. Neste estimador, a função distribuição cumulativa e utilizada para calcular a distribuição a posteriori do estado assim como a probabilidade de cada submodelo. Já o método de identificação proposto utiliza o algoritmo EM (Expectation Maximization algorithm) para identificar os parâmetros do modelo. A função distribuição cumulativa e utilizada para calcular a probabilidade de cada submodelo a partir da medida do sistema. Em seguida, utilizamos o filtro de Kalman suavizado para estimar o estado e calcular uma função substituta da função likelihood. Tal função e então utilizada para identificar os parâmetros do modelo. O estimador proposto foi utilizado para estimar o estado do modelo não linear para vibrações causadas por folgas. Foram realizadas simulações, onde comparamos o método proposto ao filtro de Kalman estendido e o filtro de partículas. O algoritmo de identificação foi utilizado para identificar os parâmetros do modelo do jato JAS 39 Gripen, assim como, o modelos não linear de vibrações causadas por folgas. / This thesis focuses on the state estimation and parameter identi cation problems of piecewise a ne models. Piecewise a ne models are obtained when the state domain or the input domain are partitioned into regions and, for each region, a linear or a ne submodel is used to describe the system dynamics. We propose a recursive state estimation algorithm and a parameter identi cation algorithm to a class of piecewise a ne models. We propose a Bayesian state estimate which uses the Kalman lter in each submodel. In the this estimator, the cumulative distribution is used to compute the posterior distribution of the state as well as the probability of each submodel. On the other hand, the proposed identi cation method uses the Expectation Maximization (EM) algorithm to identify the model parameters. We use the cumulative distribution to compute the probability of each submodel based on the system measurements. Subsequently, we use the Kalman smoother to estimate the state and compute a surrogate function for the likelihood function. This function is used to estimate the model parameters. The proposed estimator was used to estimate the state of the nonlinear model for vibrations caused by clearances. Numerical simulations were performed, where we have compared the proposed method to the extended Kalman lter and the particle lter. The identi cation algorithm was used to identify the model parameters of the JAS 39 Gripen aircraft as well as the nonlinear model for vibrations caused by clearances.

Esta tese foca no problema de estimação de estado e de identificação de parâametros para modelos afins por partes. Modelos afins por partes são obtidos quando o domínio do estado ou da entrada do sistema e particionado em regiões e, para cada região, um submodelo linear ou afim e utilizado para descrever a dinâmica do sistema. Propomos um algoritmo para estimação recursiva de estados e um algoritmo de identificação de parâmetros para uma classe de modelos afins por partes. Propomos um estimador de estados Bayesiano que utiliza o filtro de Kalman em cada um dos submodelos. Neste estimador, a função distribuição cumulativa e utilizada para calcular a distribuição a posteriori do estado assim como a probabilidade de cada submodelo. Já o método de identificação proposto utiliza o algoritmo EM (Expectation Maximization algorithm) para identificar os parâmetros do modelo. A função distribuição cumulativa e utilizada para calcular a probabilidade de cada submodelo a partir da medida do sistema. Em seguida, utilizamos o filtro de Kalman suavizado para estimar o estado e calcular uma função substituta da função likelihood. Tal função e então utilizada para identificar os parâmetros do modelo. O estimador proposto foi utilizado para estimar o estado do modelo não linear para vibrações causadas por folgas. Foram realizadas simulações, onde comparamos o método proposto ao filtro de Kalman estendido e o filtro de partículas. O algoritmo de identificação foi utilizado para identificar os parâmetros do modelo do jato JAS 39 Gripen, assim como, o modelos não linear de vibrações causadas por folgas. / This thesis focuses on the state estimation and parameter identi cation problems of piecewise a ne models. Piecewise a ne models are obtained when the state domain or the input domain are partitioned into regions and, for each region, a linear or a ne submodel is used to describe the system dynamics. We propose a recursive state estimation algorithm and a parameter identi cation algorithm to a class of piecewise a ne models. We propose a Bayesian state estimate which uses the Kalman lter in each submodel. In the this estimator, the cumulative distribution is used to compute the posterior distribution of the state as well as the probability of each submodel. On the other hand, the proposed identi cation method uses the Expectation Maximization (EM) algorithm to identify the model parameters. We use the cumulative distribution to compute the probability of each submodel based on the system measurements. Subsequently, we use the Kalman smoother to estimate the state and compute a surrogate function for the likelihood function. This function is used to estimate the model parameters. The proposed estimator was used to estimate the state of the nonlinear model for vibrations caused by clearances. Numerical simulations were performed, where we have compared the proposed method to the extended Kalman lter and the particle lter. The identi cation algorithm was used to identify the model parameters of the JAS 39 Gripen aircraft as well as the nonlinear model for vibrations caused by clearances.

Esta tese foca no problema de estimação de estado e de identificação de parâametros para modelos afins por partes. Modelos afins por partes são obtidos quando o domínio do estado ou da entrada do sistema e particionado em regiões e, para cada região, um submodelo linear ou afim e utilizado para descrever a dinâmica do sistema. Propomos um algoritmo para estimação recursiva de estados e um algoritmo de identificação de parâmetros para uma classe de modelos afins por partes. Propomos um estimador de estados Bayesiano que utiliza o filtro de Kalman em cada um dos submodelos. Neste estimador, a função distribuição cumulativa e utilizada para calcular a distribuição a posteriori do estado assim como a probabilidade de cada submodelo. Já o método de identificação proposto utiliza o algoritmo EM (Expectation Maximization algorithm) para identificar os parâmetros do modelo. A função distribuição cumulativa e utilizada para calcular a probabilidade de cada submodelo a partir da medida do sistema. Em seguida, utilizamos o filtro de Kalman suavizado para estimar o estado e calcular uma função substituta da função likelihood. Tal função e então utilizada para identificar os parâmetros do modelo. O estimador proposto foi utilizado para estimar o estado do modelo não linear para vibrações causadas por folgas. Foram realizadas simulações, onde comparamos o método proposto ao filtro de Kalman estendido e o filtro de partículas. O algoritmo de identificação foi utilizado para identificar os parâmetros do modelo do jato JAS 39 Gripen, assim como, o modelos não linear de vibrações causadas por folgas. / This thesis focuses on the state estimation and parameter identi cation problems of piecewise a ne models. Piecewise a ne models are obtained when the state domain or the input domain are partitioned into regions and, for each region, a linear or a ne submodel is used to describe the system dynamics. We propose a recursive state estimation algorithm and a parameter identi cation algorithm to a class of piecewise a ne models. We propose a Bayesian state estimate which uses the Kalman lter in each submodel. In the this estimator, the cumulative distribution is used to compute the posterior distribution of the state as well as the probability of each submodel. On the other hand, the proposed identi cation method uses the Expectation Maximization (EM) algorithm to identify the model parameters. We use the cumulative distribution to compute the probability of each submodel based on the system measurements. Subsequently, we use the Kalman smoother to estimate the state and compute a surrogate function for the likelihood function. This function is used to estimate the model parameters. The proposed estimator was used to estimate the state of the nonlinear model for vibrations caused by clearances. Numerical simulations were performed, where we have compared the proposed method to the extended Kalman lter and the particle lter. The identi cation algorithm was used to identify the model parameters of the JAS 39 Gripen aircraft as well as the nonlinear model for vibrations caused by clearances.

Dans cette thèse, nous étudions plusieurs problèmes de mathématiques financières liés à la valorisation des produits dérivés. Par différentes approches asymptotiques, nous développons des méthodes pour calculer des approximations précises du prix de certains types d’options dans des cas où il n’existe pas de formule explicite.Dans le premier chapitre, nous nous intéressons à la valorisation des options dont le payoff dépend de la trajectoire du sous-jacent par méthodes de Monte-Carlo, lorsque le sous-jacent est modélisé par un processus affine à volatilité stochastique. Nous prouvons un principe de grandes déviations trajectoriel en temps long, que nous utilisons pour calculer, en utilisant le lemme de Varadhan, un changement de mesure asymptotiquement optimal, permettant de réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options.Le second chapitre considère la valorisation par méthodes de Monte-Carlo des options dépendant de plusieurs sous-jacents, telles que les options sur panier, dans le modèle à volatilité stochastique de Wishart, qui généralise le modèle Heston. En suivant la même approche que dans le précédent chapitre, nous prouvons que le processus vérifie un principe de grandes déviations en temps long, que nous utilisons pour réduire significativement la variance de l’estimateur de Monte-Carlo des prix d’options, à travers un changement de mesure asymptotiquement optimal. En parallèle, nous utilisons le principe de grandes déviations pour caractériser le comportement en temps long de la volatilité implicite Black-Scholes des options sur panier.Dans le troisième chapitre, nous étudions la valorisation des options sur variance réalisée, lorsque la volatilité spot est modélisée par un processus de diffusion à volatilité constante. Nous utilisons de récents résultats asymptotiques sur les densités des diffusions hypo-elliptiques pour calculer une expansion de la densité de la variance réalisée, que nous intégrons pour obtenir l’expansion du prix des options, puis de leur volatilité implicite Black-Scholes.Le dernier chapitre est consacré à la valorisation des dérivés de taux d’intérêt dans le modèle Lévy de marché Libor qui généralise le modèle de marché Libor classique (log-normal) par l’ajout de sauts. En écrivant le premier comme une perturbation du second et en utilisant la représentation de Feynman-Kac, nous calculons explicitement l’expansion asymptotique du prix des dérivés de taux, en particulier, des caplets et des swaptions. / In this thesis, we study several mathematical finance problems, related to the pricing of derivatives. Using different asymptotic approaches, we develop methods to calculate accurate approximations of the prices of certain types of options in cases where no explicit formulas are available.In the first chapter, we are interested in the pricing of path-dependent options, with Monte-Carlo methods, when the underlying is modelled as an affine stochastic volatility model. We prove a long-time trajectorial large deviations principle. We then combine it with Varadhan’s Lemma to calculate an asymptotically optimal measure change, that allows to reduce significantly the variance of the Monte-Carlo estimator of option prices.The second chapter considers the pricing with Monte-Carlo methods of options that depend on several underlying assets, such as basket options, in the Wishart stochastic volatility model, that generalizes the Heston model. Following the approach of the first chapter, we prove that the process verifies a long-time large deviations principle, that we use to reduce significantly the variance of the Monte-Carlo estimator of option prices, through an asymptotically optimal measure change. In parallel, we use the large deviations property to characterize the long-time behaviour of the Black-Scholes implied volatility of basket options.In the third chapter, we study the pricing of options on realized variance, when the spot volatility is modelled as a diffusion process with constant volatility. We use recent asymptotic results on densities of hypo-elliptic diffusions to calculate an expansion of the density of realized variance, that we integrate to obtain an expansion of option prices and their Black-Scholes implied volatility.The last chapter is dedicated to the pricing of interest rate derivatives in the Levy Libor market model, that generaliszes the classical (log-normal) Libor market model by introducing jumps. Writing the first model as a perturbation of the second and using the Feynman-Kac representation, we calculate explicit expansions of the prices of interest rate derivatives and, in particular, caplets and swaptions

The mathematical analysis of dynamical systems covers a wide range of challenging problems related to the time evolution, transient and asymptotic behavior, or regulation and control of physical systems. A large part of my work has been motivated by new mathematical questions arising from biological systems, especially signaling and genetic regulatory networks, where the classical methods usually don't directly apply. Problems include parameter estimation, robustness of the system, model reduction, or model assembly from smaller modules, or control of a system towards a desired state. Although many different formalisms and methodologies can be used to study these problems, in the past decade my work has focused on discrete and hybrid modeling frameworks with the goal of developing intuitive, computationally amenable, and mathematically rigorous, methods of analysis. Discrete (and, in particular, Boolean) models involve a high degree of abstraction and provide a qualitative description of the systems' dynamics. Such models are often suitable to represent the known interactions in gene regulatory networks and their advantage is that a large range of theoretical analysis tools are available using, for instance, graph theoretical concepts. Hybrid (piecewise affine) models have discontinuous vector fields but provide a continuous and more quantitative description of the dynamics. These systems can be analytically studied in each region of an appropriate partition of the state space, and the full solution given as a concatenation of the solutions in each region. Here, I will introduce the two formalisms and then, using several examples, illustrate how a combination of different formalisms permits comparison of results, as well as gaining quantitative knowledge and predictive power on a biological system, through the use of complementary mathematical methods.

Submitted by Livio Cuzzi Maya (liviomaya@gmail.com) on 2015-04-14T14:31:39Z No. of bitstreams: 1 dis_ref.pdf: 631490 bytes, checksum: d730ea4e26e9e8795547f24ea6da9284 (MD5) / Approved for entry into archive by BRUNA BARROS (bruna.barros@fgv.br) on 2015-04-17T14:05:44Z (GMT) No. of bitstreams: 1 dis_ref.pdf: 631490 bytes, checksum: d730ea4e26e9e8795547f24ea6da9284 (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2015-05-04T12:20:07Z (GMT) No. of bitstreams: 1 dis_ref.pdf: 631490 bytes, checksum: d730ea4e26e9e8795547f24ea6da9284 (MD5) / Made available in DSpace on 2015-05-04T12:20:38Z (GMT). No. of bitstreams: 1 dis_ref.pdf: 631490 bytes, checksum: d730ea4e26e9e8795547f24ea6da9284 (MD5) Previous issue date: 2015-04-07 / We develop an affine jump diffusion (AJD) model with the jump-risk premium being determined by both idiosyncratic and systematic sources of risk. While we maintain the classical affine setting of the model, we add a finite set of new state variables that affect the paths of the primitive, under both the actual and the risk-neutral measure, by being related to the primitive's jump process. Those new variables are assumed to be commom to all the primitives. We present simulations to ensure that the model generates the volatility smile and compute the 'discounted conditional characteristic function'' transform that permits the pricing of a wide range of derivatives. / Desenvolvemos um model afim com saltos com o prêmio pelo risco dos saltos determinado tanto por variáveis idiossincráticas quanto por variáveis sistêmicas. Mantemos a clássica estrutura linear do modelo, mas adicionamos um conjunto finito de novas variáveis de estado que afetam o caminho percorrido pelo primitivo, tanto no distribuição real quanto na distribuição neutra ao risco, por afetar o processo de saltos do primitivo. Assumimos que essas novas variáveis de estado são comuns a todos os primitivos. Apresentamos simulações que garantem que o modelo gere o sorriso da volatilidade e computamos a transformação da 'função característica descontada condicional' que permite a precificação de uma ampla gama de derivativos.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal.

Thèse numérisée par la Division de la gestion de documents et des archives de l'Université de Montréal