[pt] Esta tese investiga a estimação de curvas de juros sob o ponto de vista de métodos não-paramétricos. O texto está dividido em dois blocos. O primeiro investiga a questão do critério utilizado para selecionar o método de melhor desempenho na tarefa de interpolar a curva de juros brasileira em uma dada amostra. Foi proposto um critério de seleção de método baseado em estratégias de re-amostragem do tipo leave-k-out cross validation, onde K k £ £ 1 e K é função do número de contratos observados a cada curva da amostra. Especificidades do problema reduzem o esforço computacional requerido, tornando o critério factível. A amostra tem freqüência diária: janeiro de 1997 a fevereiro de 2001. O critério proposto apontou o spline cúbico natural -utilizado com método de ajuste perfeito aos dados - como o método de melhor desempenho. Considerando a precisão de negociação, este spline mostrou-se não viesado. A análise quantitativa de seu desempenho identificou, contudo, heterocedasticidades nos erros simulados. A partir da especificação da variância condicional destes erros e de algumas hipóteses, foi proposto um esquema de intervalo de segurança para a estimação de taxas de juros pelo spline cúbico natural, empregado como método de ajuste perfeito aos dados. O backtest sugere que o esquema proposto é consistente, acomodando bem as hipóteses e aproximações envolvidas. O segundo bloco investiga a estimação da curva de juros norte-americana construída a partir dos contratos de swaps de taxas de juros dólar-Libor pela Máquina de Vetores Suporte (MVS), parte do corpo da Teoria do Aprendizado Estatístico. A pesquisa em MVS tem obtido importantes avanços teóricos, embora ainda sejam escassas as implementações em problemas reais de regressão. A MVS possui características atrativas para a modelagem de curva de juros: é capaz de introduzir já na estimação informações a priori sobre o formato da curva e sobre aspectos da formação das taxas e liquidez de cada um dos contratos a partir dos quais ela é construída. Estas últimas são quantificadas pelo bid-ask spread (BAS) de cada contrato. A formulação básica da MVS é alterada para assimilar diferentes valores do BAS sem que as propriedades dela sejam perdidas. É dada especial atenção ao levantamento de informação a priori para seleção dos parâmetros da MVS a partir do formato típico da curva. A amostra tem freqüência diária: março de 1997 a abril de 2001. Os desempenhos fora da amostra de diversas especificações da MVS foram confrontados com aqueles de outros métodos de estimação. A MVS foi o método que melhor controlou o trade- off entre viés e variância dos erros. / [en] This thesis investigates interest rates curve estimation under non-parametric approach. The text is divided into two parts. The first one focus on which criterion to use to select the best performance method in the task of interpolating Brazilian interest rate curve. A selection criterion is proposed to measure out-of-sample performance by combining resample strategies leave-k-out cross validation applied upon the whole sample curves, where K k £ £ 1 and K is function of observed contract number in each curve. Some particularities reduce substantially the required computational effort, making the proposed criterion feasible. The data sample range is daily, from January 1997 to February 2001. The proposed criterion selected natural cubic spline, used as data perfect-fitting estimation method. Considering the trade rate precision, the spline is non-biased. However, quantitative analysis of performance determinant factors showed the existence of out-of-sample error heteroskedasticities. From a conditional variance specification of these errors, a security interval scheme is proposed for interest rate generated by perfect-fitting natural cubic spline. A backtest showed that the proposed security interval is consistent, accommodating the evolved assumptions and approximations. The second part estimate US free-for-floating interest rate swap contract curve by using Support Vector Machine (SVM), a method derived from Statistical Learning Theory. The SVM research has got important theoretical results, however the number of implementation on real regression problems is low. SVM has some attractive characteristics for interest rates curves modeling: it has the ability to introduce already in its estimation process a priori information about curve shape and about liquidity and price formation aspects of the contracts that generate the curve. The last information set is quantified by the bid-ask spread. The basic SVM formulation is changed in order to be able to incorporate the different values for bid-ask spreads, without losing its properties. Great attention is given to the question of how to extract a priori information from swap curve typical shape to be used in MVS parameter selection. The data sample range is daily, from March 1997 to April 2001. The out-of-sample performances of different SVM specifications are faced with others method performances. SVM got the better control of trade- off between bias and variance of out-of-sample errors.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.

This thesis deals with regularization parameter selection methods in the context of Tikhonov-type regularization with Poisson distributed data, in particular the reconstruction of images, as well as with the identification of the volatility surface from observed option prices. In Part I we examine the choice of the regularization parameter when reconstructing an image, which is disturbed by Poisson noise, with Tikhonov-type regularization. This type of regularization is a generalization of the classical Tikhonov regularization in the Banach space setting and often called variational regularization. After a general consideration of Tikhonov-type regularization for data corrupted by Poisson noise, we examine the methods for choosing the regularization parameter numerically on the basis of two test images and real PET data. In Part II we consider the estimation of the volatility function from observed call option prices with the explicit formula which has been derived by Dupire using the Black-Scholes partial differential equation. The option prices are only available as discrete noisy observations so that the main difficulty is the ill-posedness of the numerical differentiation. Finite difference schemes, as regularization by discretization of the inverse and ill-posed problem, do not overcome these difficulties when they are used to evaluate the partial derivatives. Therefore we construct an alternative algorithm based on the weak formulation of the dual Black-Scholes partial differential equation and evaluate the performance of the finite difference schemes and the new algorithm for synthetic and real option prices.