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Cette thèse traite de plusieurs sujets en mathématiques financières: risque de crédit, optimisation de portefeuille et modélisation des taux d’intérêts. Le chapitre 1 consiste en trois études dans le domaine du risque de crédit. La plus innovante est la première dans laquel nous construisons un modèle tel que la propriété d’immersion n’est vérifiée sous aucune mesure martingale équivalente. Le chapitre 2 étudie le problème de maximisation de la somme d’une utilité de la richesse terminale et d’une utilité de la consommation. Le chapitre 3 étudie l’évaluation des produits dérivés de taux d’intérêt dans un cadre multicourbe, qui prend en compte la différence entre une courbe de taux sans risque et des courbes de taux Libor de différents tenors. / This thesis deals with several topics in mathematical finance: credit risk, portfolio optimization and interest rate modeling. Chapter 1 consists of three studies in the field of credit risk. The most innovative is the first one, where we construct a model such that the immersion property does not hold under any equivalent martingale measure. Chapter 2 studies the problem of maximization of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where a sudden jump in the risk-free interest rate induces market incompleteness. Chapter 3 studies the valuation of Libor interest rate derivatives in a multiple-curve setup, which accounts for the spreads between a risk-free discount curve and Libor curves of different tenors.

This thesis consists of two related parts. In the first part we conduct an empirical examination of the dynamics of Australian interest rates of six different maturities, covering the whole yield curve. This direct study of the long rates is quite novel. We use maximum likelihood estimation on a variety of models and find some results that are in stark contrast to previous studies. We estimate Poisson-jump diffusion (PJD) models and find very strong evidence for the existence of jumps in all daily interest rate series. We find that the PJD model fits short-rate data significantly better than a Bernoulli-jump diffusion model. We also estimate the CKLS model for our data and find that the only model not rejected for all six maturities is the CEV model in stark contrast to previous findings. Also, we find that the elasticity of variance estimate in the CKLS model is much higher for the short-rates than for the longer rates where the estimate is only about 0.25, indicating that different dynamics seem to be at work for different maturities. We also found that adding jumps to the simple diffusion model gives a larger improvement than comes from going from the simple diffusion to the CKLS model. In the second part of the thesis we examine the Flesaker and Hughston (FH) term structure model. We derive the dynamics of the short rate under both the original measure and the risk-neutral measure, and show that some criticisms of the bounds for the short rate may not be significant in actual applications. We also derive the dynamics of bond prices in the FH model and compare them to the HJM model. We also extend the FH model by allowing the martingale to follow a jump-diffusion process, rather than just a diffusion process. We derive the unique change of measure that guarantees the family of bond prices is arbitrage-free. We derive prices for caps and swaptions, and extend the results to include Bermudan swaptions and show how to price options with the jump-diffusion version of the FH model.

This thesis consists of two related parts. In the first part we conduct an empirical examination of the dynamics of Australian interest rates of six different maturities, covering the whole yield curve. This direct study of the long rates is quite novel. We use maximum likelihood estimation on a variety of models and find some results that are in stark contrast to previous studies. We estimate Poisson-jump diffusion (PJD) models and find very strong evidence for the existence of jumps in all daily interest rate series. We find that the PJD model fits short-rate data significantly better than a Bernoulli-jump diffusion model. We also estimate the CKLS model for our data and find that the only model not rejected for all six maturities is the CEV model in stark contrast to previous findings. Also, we find that the elasticity of variance estimate in the CKLS model is much higher for the short-rates than for the longer rates where the estimate is only about 0.25, indicating that different dynamics seem to be at work for different maturities. We also found that adding jumps to the simple diffusion model gives a larger improvement than comes from going from the simple diffusion to the CKLS model. In the second part of the thesis we examine the Flesaker and Hughston (FH) term structure model. We derive the dynamics of the short rate under both the original measure and the risk-neutral measure, and show that some criticisms of the bounds for the short rate may not be significant in actual applications. We also derive the dynamics of bond prices in the FH model and compare them to the HJM model. We also extend the FH model by allowing the martingale to follow a jump-diffusion process, rather than just a diffusion process. We derive the unique change of measure that guarantees the family of bond prices is arbitrage-free. We derive prices for caps and swaptions, and extend the results to include Bermudan swaptions and show how to price options with the jump-diffusion version of the FH model.

The uncertainty attached to future movements of interest rates is an essential part of the Financial Decision Theory and requires an awareness of the stochastic movement of these rates. Several approaches have been proposed for modeling the one-factor short rate models where some lead to arbitrage-free term structures. However, no definite consensus has been reached with regard to the best approach for interest rate modeling. In this work, we briefly examine the existing one-factor interest rate models and calibrate Vasicek and Hull-White (Extended Vasicek) Models by using Turkey&#039 / s term structure. Moreover, a trinomial interest rate tree is constructed to represent the evolution of Turkey&rsquo / s zero coupon rates.

Random field models have provided a flexible environment in which the properties of the term structure of interest rates are captured almost as observed. In this study we provide an overview of the forward rate random fiield models and propose an extension in which the forward rates fluctuate along with a two parameter process represented by a random field. We then provide a mathematical expression of the yield curve under this model and sketch the prospective utilities and applications of this model for interest rate management.

This diploma thesis deals with short-term interest rate models. Many interest models have been developed in the last decades. They focus on accuracy of prediction. The pioneering one was developed by Vasicek in 1977 followed by the work of others. Nowadays these vary in their level of comprehensiveness and technical difficulty. The main aim of the thesis is to introduce not only a basic Vasicek's work but also more sophisticated models such as Brennan-Schwartz or Longstaff-Schwartz.

An in depth study of the pricing of Game contingent claims under a general diffusion market model, in which interest rate is non constant, is presented. With the idea of providing a few numerical examples of the valuation of such claims, we present a detailed description of a Bootstrapping procedure to obtain interest rate information from Swaps rates. We also present a Stripping procedure that can be used to obtain initial spot (caplet) volatility from Market quotes on Caps/FLoors. These methods are of general application and could be used in the calibration of diffusion models of interest rate. Then we show several examples of calibration of the Hull--White model of interest rates. Our calibration examples are later used in the numerical approximation of the value of a particular form of Game option.

Este trabalho tem por objetivo principal aplicar o modelo de precificação de opções de taxas de juros proposto por Barbachan e Ornelas (2003), com base nos modelos de taxa de juro e avaliação de opções de Cox, Ingerssol e Ross (1985), para avaliação de opções de compra sobre o Índice de Taxa Média de Depósitos Interfinanceiros de Um Dia (IDI), negociadas na BM&FBovespa. Para estimação dos parâmetros deste modelo, foi empregado o método de Máxima Verossimilhança. Neste contexto, também fez-se uso da fórmula de precificação de opções proposta por Black (1976), adaptada para o mercado de derivativos brasileiros, conforme implementação verificada no trabalho de Gluckstern et al. (2002). Tal aplicação torna-se interessante, pois este modelo é amplamente utilizado pelo mercado brasileiro para avaliação de opções sobre o IDI. De forma a verificar a aderência dos preços teóricos gerados pelos modelos, em comparação aos preços de mercado, métricas de erro foram empregadas. De forma geral, nossos resultados mostraram que ambos os modelos apresentam erros sistemáticos de precificação, onde o modelo CIR subavalia os prêmios das opções e o modelo de Black superprecifica. No entanto, bons resultados foram encontrados ao avaliarmos opções in-the-money e out-of-money com o modelo de Black. / This work aims to apply the interest rate option pricing model proposed by Barbachan and Ornelas (2003), based on the interest rate model and option pricing model developed by Cox, Ingersoll and Ross (1985), to evaluate call options on the 1 day Brazilian Interfinancial Deposits Index - IDI, traded at BM&FBovespa. The Maximum Likelihood method was applied to estimate the model parameters. In this context, the option pricing formula proposed by Black (1976), adapted for the Brazilian derivative Market, was also used, according implementation verified in Gluckstern et al. (2002). This application becomes interesting because this model is widely used by the Brazilian Market to evaluate options on IDI. In order to verify the adherence of theoretical prices generated by the models, in comparison to the Market prices, error metrics were applied. In general, our results pointed out that both models presented systematic pricing errors, in which the CIR model underestimates the option prices and Black’s model overestimates. However, good results were found on the evaluation of options in-the-money and out-of-money with the Black’s Model.