Wir modellieren einen Finanzmarkt mit unendlich vielen Wertpapieren als stochastischen Prozeß X in stetiger Zeit mit Werten in einem separablen Hilbertraum H. In diesem Rahmen zeigen wir die Äquivalenz von Vollständigkeit des Marktes und der Eindeutigkeit des äquivalenten Martingalmaßes unter der Bedingung, daß X stetige Pfade besitzt. Weiter zeigen wir, daß (unter gewissen technischen Bedingungen) für X die Abwesenheit von asymptotischer Arbitrage der ersten/zweiten Art (im Sinne von Kabanov/Kramkov) äquivalent zur Absolutstetigkeit des Referenzmaßes zu einem eindeutigen, lokal äquivalenten Martingalmaß ist. Hat X stetige Pfade, so ist die Abwesenheit von allgemeiner asymptotischer Arbitrage äquivalent zur Existenz eines äquivalenten lokalen Martingalmaßes. Außerdem geben wir ein Kriterium für die Existenz einer optionalen Zerlegung von X an. Dies wenden wir auf das Problem der Risikominimierung bei vorgegebener Investitionsobergrenze (effizientes Hedgen (Föllmer/Leukert)) an, um dieses im unendlichdimensionalen Kontext zu behandeln. Außerdem stellen wir eine unendlichdimensionale Erweiterung des Heath-Jarrow-Morton-Modells vor und nutzen den Potentialansatz nach Rodgers, um zwei weitere Zinsstrukturmodelle zu konstruieren. Als Beitrag zur allgemeinen stochastischen Analysis in Hilberträumen beweisen wir eine pfadweise Version der Itoformel für stochastische Prozesse mit stetigen Pfaden in einem separablen Hilbertraum. Daraus läßt sich eine pfadweise Version des Satzes über die Vertauschbarkeit von stochastischem und Lebesgue-Integral ableiten. Außerdem zeigen wir eine Version der Clark-Formel für eine Brownsche Bewegung mit Werten in einem Hilbertraum. / We model a financial market with infinitely many assets as a stochastic process X with values in a separable Hilbert space H. In this setting we show the equivalence of market completeness and the uniqueness of the equivalent martingale measure, if X has continuous paths. Another result for our model is, that under some technical conditions, the absence of asymptotic arbitrage of the first/second kind (in the sense of Kabanov/Kramkov) is equivalent to the absolute continuity of the reference measure to a unique, locally equivalent, martingale measure. If X has continuous paths, the absence of general asymptotic arbitrage is equivalent to the existence of an equivalent local martingale measure. Furthermore, we give a sufficient condition for the existence of the optional decomposition of X. We apply this result to the problem of risk minimization with given upper limit for investion (efficient hedging (Föllmer/Leukert)). This allows us to solve this optimization problem in our infinite dimensional context. Another result is an infinite dimensional extension of the Heath-Jarrow-Morton term structure model. Two further term structure models are constructed, using the Markov potential approach developed by Rodgers. As a contribution to the theory of stochastic analysis in Hilbert spaces, we proof a pathwise version of the Ito formula for stochastic processes with continuous paths in a separable Hilbert space. This leads to a pathwise version of the interchangability theorem for stochastic and Lebesgue integrals. We also show a version of the Clark formula for Hilbert space valued Brownian motion.

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

Submitted by Adriano Faria (afaria@fgvmail.br) on 2017-12-13T19:49:29Z No. of bitstreams: 1 Tese_deFaria.pdf: 3657553 bytes, checksum: 11ec67914c866ca46d83c67c1592e093 (MD5) / Approved for entry into archive by GILSON ROCHA MIRANDA (gilson.miranda@fgv.br) on 2017-12-21T11:41:13Z (GMT) No. of bitstreams: 1 Tese_deFaria.pdf: 3657553 bytes, checksum: 11ec67914c866ca46d83c67c1592e093 (MD5) / Made available in DSpace on 2017-12-27T12:18:22Z (GMT). No. of bitstreams: 1 Tese_deFaria.pdf: 3657553 bytes, checksum: 11ec67914c866ca46d83c67c1592e093 (MD5) Previous issue date: 2017-03-16 / This thesis is a collection of essays in empirical finance mainly focused on term structure models. In the first three chapters, we developed methods to extract the yield curve from government and corporate bonds. We measure the performance of such methods in pricing, Value at Risk and forecasting exercises. In its turn, the last chapter brings a discussion about the effects of different metrics of the optimal portfolio on the estimation of a CCAPM model.In the first chapter, we propose a segmented model to deal with the seasonalities appearing in real yield curves. In different markets, the short end of the real yield curve is influenced by seasonalities of the price index that imply a lack of smoothness in this segment. Borrowing from the flexibility of spline models, a B-spline function is used to fit the short end of the yield curve, while the medium and the long end are captured by a parsimonious parametric four-factor exponential model. We illustrate the benefits of the proposed term structure model by estimating real yield curves in one of the biggest government index-linked bond markets in the world. Our model is simultaneously able to fit the yield curve and to provide unbiased Value at Risk estimates for different portfolios of bonds negotiated in this market.Chapter 2 introduces a novel framework for the estimation of corporate bond spreads based on mixture models. The modeling methodology allows us to enhance the informational content used to estimate the firm level term structure by clustering firms together using observable firm characteristics. Our model builds on the previous literature linking firm level characteristics to credit spreads. Specifically, we show that by clustering firms using their observable variables, instead of the traditional matrix pricing (cluster by rating/sector), it is possible to achieve gains of several orders of magnitude in terms of bond pricing. Empirically, we construct a large panel of firm level explanatory variables based on results from a handful of previous research and evaluate their performance in explaining credit spread differences. Relying on panel data regressions we identify the most significant factors driving the credit spreads to include in our term structure model. Using this selected sample, we show that our methodology significantly improves in sample fitting as well as produces reliable out of sample price estimations when compared to the traditional models.Chapter 3 brings the paper “Forecasting the Brazilian Term Structure Using Macroeconomic Factors”, published in Brazilian Review of Econometrics (BRE). This paper studies the forecasting of the Brazilian interest rate term structure using common factors from a wide database of macroeconomic series, from the period of January 2000 to May 2012. Firstly the model proposed by Moench (2008) is implemented, in which the dynamic of the short term interest rate is modeled using a Factor Augmented VAR and the term structure is derived using the restrictions implied by no-arbitrage. Similarly to the original study, this model resulted in better predictive performance when compared to the usual benchmarks, but presented deterioration of the results with increased maturity. To avoid this problem, we proposed that the dynamic of each rate be modeled in conjunction with the macroeconomic factors, thus eliminating the no-arbitrage restrictions. This attempt produced superior forecasting results. Finally, the macro factors were inserted in a parsimonious parametric three-factor exponential model.The last chapter presents the paper “Empirical Selection of Optimal Portfolios and its Influence in the Estimation of Kreps-Porteus Utility Function Parameters”, also published in BRE. This paper investigates the effects on the estimation of parameters related to the elasticity of intertemporal substitution and risk aversion, of the selection of different portfolios to represent the optimal aggregate wealth endogenously derived in equilibrium models with Kreps-Porteus recursive utility. We argue that the usual stock market wide index is not a good portfolio to represent optimal wealth of the representative agent, and we propose as an alternative the portfolio from the Investment Fund Industry. Especially for Brazil, where that industry invests most of its resources in fixed income, the aforementioned substitution of the optimal proxy portfolio caused a significant increase in the risk aversion coefficient and the elasticity of the intertemporal substitution in consumption.

Cette thèse présente plusieurs extensions relatives aux modèles affines positifs de taux d'intérêt. Un premier chapitre introduit les concepts reliés aux modélisations employées dans les chapitres suivants. Il détaille la définition de processus dits affines, et la construction de modèles de prix d'actifs obtenus par non-arbitrage. Le chapitre 2 propose une nouvelle méthode d’estimation et de filtrage pour les modèles espace-état linéaire-quadratiques. Le chapitre suivant applique cette méthode d’estimation à la modélisation d’écarts de taux interbancaires de la zone Euro, afin d’en décomposer les fluctuations liées au risque de défaut et de liquidité. Le chapitre 4 développe une nouvelle technique de création de processus affines multivariés à partir leurs contreparties univariées, sans imposer l’indépendance conditionnelle entre leurs composantes. Le dernier chapitre applique cette méthode et dérive un processus affine multivarié dont certaines composantes peuvent rester à zéro pendant des périodes prolongées. Incorporé dans un modèle de taux d’intérêt, ce processus permet de rendre compte efficacement des taux plancher à zéro. / This thesis presents new developments in the literature of non-negative affine interest rate models. The first chapter is devoted to the introduction of the main mathematical tools used in the following chapters. In particular, it presents the so-called affine processes which are extensively employed in no-arbitrage interest rate models. Chapter 2 provides a new filtering and estimation method for linear-quadratic state-space models. This technique is exploited in the 3rd chapter to estimate a positive asset pricing model on the term structure of Euro area interbank spreads. This allows us to decompose the interbank risk into a default risk and a liquidity risk components. Chapter 4 proposes a new recursive method for building general multivariate affine processes from their univariate counterparts. In particular, our method does not impose the conditional independence between the different vector elements. We apply this technique in Chapter 5 to produce multivariate non-negative affine processes where some components can stay at zero for several periods. This process is exploited to build a term structure model consistent with the zero lower bound features.

Es sei ein Heath-Jarrow-Morton Zinsstrukturmodell df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t gegeben, angetrieben von einem mehrdimensionalen Semimartingal X. Das Ziel dieser Arbeit besteht darin, die Existenz endlich dimensionaler Realisierungen für solche Modelle zu untersuchen, wobei wir als treibende Prozesse die Klasse der Grigelionis Prozesse wählen, die insbesondere Levy Prozesse enthält. Zur Bearbeitung der Fragestellung werden zwei veschiedene Ansätze verfolgt. Wir dehnen die Ideen aus der Differenzialgeometrie von Björk und Svensson (2001) auf die vorliegende Situation aus und zeigen, dass das in der zitierten Arbeit bewiesene Kriterium für die Existenz endlich dimensionaler Realisierungen in unserem Fall als notwendiges Kriterium dienlich ist. Dieses Resultat wird auf konkrete Volatilitätsstrukturen angewandt. Im Kontext von sogenannten Benchmark Realisierungen, die eine natürliche Verallgemeinerung von Short Rate Realisierungen darstellen, leiten wir Integro-Differenzialgleichungen her, die für die Untersuchung der Existenz endlich dimensionaler Realisierungen hilfreich sind. Als Verallgemeinerung eines Resultats von Jeffrey (1995) beweisen wir außerdem, dass Zinsstrukturmodelle, die eine generische Benchmark Realisierung besitzen, notwendigerweise eine singuläre Hessesche Matrix haben. Beide Ansätze zeigen, dass neue Phänomene auftreten, sobald der treibende Prozess X Sprünge macht. Es gibt dann auf einmal nur noch sehr wenige Zinsstrukturmodelle, die endlich dimensionale Realisierungen zulassen, was ein beträchtlicher Unterschied zu solchen Modellen ist, die von einer Brownschen Bewegung angetrieben werden. Aus diesem Grund zeigen wir, dass für die in der Literatur oft behandelten Modelle mit deterministischer Richtungsvolatilität eine Folge von endlich dimensionalen Systemen existiert, die gegen das Zinsmodell konvergieren. / Let f(t,T) be a term structure model of Heath-Jarrow-Morton type df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t, driven by a multidimensional semimartingale X. Our objective is to study the existence of finite dimensional realizations for equations of this kind. Choosing the class of Grigelionis processes (including in particular Levy processes) as driving processes, we approach this problem from two different directions. Extending the ideas from differential geometry in Björk and Svensson (2001), we show that the criterion for the existence of finite dimensional realizations, proven in the aforementioned paper, still serves as a necessary condition in our setup. This result is applied to concrete volatility structures. In the context of benchmark realizations, which are a natural generalization of short rate realizations, we derive integro-differential equations, suitable for the analysis of the realization problem. Generalizing Jeffrey (1995), we also prove a result stating that forward rate models, which generically possess a benchmark realization, must have a singular Hessian matrix. Both approaches reveal that, with regard to the results known for driving Wiener processes, new phenomena emerge, as soon as the driving process X has jumps. In particular, the occurrence of jumps severely limits the range of models that admit finite dimensional realizations. For this reason we prove, for the often considered case of deterministic direction volatility structures, the existence of finite dimensional systems converging to the forward rate model.

Insurance contracts will be reported at fair values on insurers’ balance sheets from 2010. In this thesis, we will review the conceptual and theoretical backbone of the insurance fair valuation project while providing a summary of the key features of the fair valuation project. Then, we will conduct a case study aimed at finding, under the fair valuation regime, the best asset allocation strategy for a particular business unit that carries a hypothetical annuity portfolio using a single modelling framework for valuation, risk calculation and business appraisal.