Real World Economic Scenario Generator

Doutoramento em Matemática Aplicada à Economia e GestãoNeste trabalho apresentamos uma metodologia para simular a evolução das taxas de juros sob medida de probabilidade real. Mais precisamente, usando o modelo de mercado Shifted Lognormal LIBOR multidimensional e uma especificação do vetor do preço de mercado do risco, explicamos como realizar simulações das taxas de juro futuras, usando o método de Euler-Maruyama com preditor-corretor. A metodologia proposta permite acomodar a presença de taxas de juro negativas, tal como é observado atualmente em vários mercados. Após definir a estrutura livre de default, generalizamos os resultados para incorporar a existência de risco de crédito nos mercados financeiros e desenvolvemos um modelo LIBOR para obrigações com risco de crédito classificadas por ratings. Neste trabalho modelamos diretamente os spreads entre as classificações de ratings de acordo com uma dinâmica estocástica que garante a monotonicidade dos preços dos títulos relativamente às classificações por ratings.In this work, we present a methodology to simulate the evolution of interest rates under real world probability measure. More precisely, using the multidimensional Shifted Lognormal LIBOR market model and a specification of the market price of risk vector process, we explain how to perform simulations of the real world forward rates in the future, using the Euler-Maruyama scheme with a predictor-corrector strategy. The proposed methodology allows for the presence of negative interest rates as currently observed in many markets. After setting the default-free framework we generalize the results to incorporate the existence of credit risk to our model and develop a LIBOR model for defaultable bonds with credit ratings. We model directly the inter-rating spreads according to a stochastic dynamic that guarantees the monotonicity of bond prices with respect to the credit ratings.info:eu-repo/semantics/publishedVersio

Dynamic interest-rate modelling in incomplete markets

In the first chapter, a new kind of additive process is proposed. Our main goal is to define, characterize and prove the existence of the LIBOR additive process as a new stochastic process. This process will be defined as a piecewise stationary process with independent increments, continuous in probability but with discontinuous trajectories, and having "càdlàg" sample paths. The proposed process is specically designed to derive interest-rates modelling because it allows us to introduce a jump-term structure as an increasing sequence of Lévy measures. In this paper we characterize this process as a Markovian process with an infinitely divisible, selfsimilar, stable and self-decomposable distribution. Also, we prove that the Lévy-Khintchine characteristic function and Lévy-Itô decomposition apply to this process. Additionally we develop a basic framework for density transformations. Finally, we show some examples of LIBOR additive processes. A no-arbitrage framework to model interest rates with credit risk, based on the LIBOR additive process, and an approach to price corporate bonds in incomplete markets, is presented in the second chapter. We derive the no-arbitrage conditions under different conditions of recovery, and we obtain new expressions in order to estimate the probabilities of default under risk-neutral measure. Additionally, we study both the approximation of a continuous-time model by a sequence of discrete-time models with credit risk, and the convergence of price processes (in terms of the triplets) under a framework that allows the practitioner a multiple set of models (semimartingale) and credit conditions (migration and default). Finally, in the third chapter, we introduce a d-dimensional LIBOR additive process to model the forward LIBOR market model with different credit ratings. Additionally, we expose the risk-neutral and forward-neutral dynamic of forward LIBOR rates. Additionally, we introduce a new calibration methodology for the LIBOR market model driven by LIBOR additive processes. The calibration of the continuous part is based on a semide nite programming (convex) problem and the calibration of the Lévy measure is proposed using a non-parametric (non linear least-square with a regularization term) calibratio

The asymptotic behavior of the term structure of interest rates

In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity going to infinity, in the post-crisis interest rate market. Three different concepts of long-term interest rates are considered for this purpose: the long-term yield, the long-term simple rate, and the long-term swap rate. We analyze the properties as well as the interrelations of these long-term interest rates. In particular, we study the asymptotic behavior of the term structure of interest rates in some specific models. First, we compute the three long-term interest rates in the HJM framework with different stochastic drivers, namely Brownian motions, Lévy processes, and affine processes on the state space of positive semidefinite symmetric matrices. The HJM setting presents the advantage that the entire yield curve can be modeled directly. Furthermore, by considering increasingly more general classes of drivers, we were able to take into account the impact of different risk factors and their dependence structure on the long end of the yield curve. Finally, we study the long-term interest rates and especially the long-term swap rate in the Flesaker-Hughston model and the linear-rational methodology

An extended generalized Markov model for the spread risk and its calibration by using filtering techniques in Solvency II framework

The Solvency II regulatory regime requires the calculation of a capital requirement, the Solvency Capital Requirement (SCR), for the insurance and reinsurance companies, that is based on a market-consistent evaluation of the Basic Own Funds probability distribution forecast over a one-year time horizon. This work proposes an extended generalized Markov model for rating-based pricing of risky securities for spread risk assessment and management within the Solvency II framework, under an internal model or partial internal model. This model is based on Jarrow, Lando and Turnbull (1997), Lando (1998) and Gambaro et al. (2018) and models the credit rating transitions and the default process using an extension of a time-homogeneous Markov chain and two subordinator processes. This approach allows simultaneous modeling of credit spreads for different rating classes and credit spreads to fluctuate randomly even when the rating does not change. The estimation methodologies used in this work are consistent with the scope of the work and the scope of the proposed model, i.e., pricing of defaultable bonds and calculation of SCR for the spread risk sub-module, and with the market-consistency principle required by Solvency II. For this purpose, estimation techniques on time series known as filtering techniques are used, which allow the model parameters to be jointly estimated under both the real-world probability measure (necessary for risk assessment) and the risk-neutral probability measure (necessary for pricing). Specifically, an appropriate set of time series of credit spread term structures, differentiated by economic sector and rating class, is used. The proposed model, in its final version, returns excellent results in terms of goodness of fit to historical data, and the projected data are consistent with historical data and the Solvency II framework. The filtering techniques, in the different configurations used in this work (particle filtering with Gauss-Legendre quadrature techniques, particle filtering with Sequential Importance Resampling algorithm, Kalman filter), were found to be an effective and flexible tool for estimating the models proposed, able to handle the high computational complexity of the problem addressed

The asymptotic behavior of the term structure of interest rates

In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity going to infinity, in the post-crisis interest rate market. Three different concepts of long-term interest rates are considered for this purpose: the long-term yield, the long-term simple rate, and the long-term swap rate. We analyze the properties as well as the interrelations of these long-term interest rates. In particular, we study the asymptotic behavior of the term structure of interest rates in some specific models. First, we compute the three long-term interest rates in the HJM framework with different stochastic drivers, namely Brownian motions, Lévy processes, and affine processes on the state space of positive semidefinite symmetric matrices. The HJM setting presents the advantage that the entire yield curve can be modeled directly. Furthermore, by considering increasingly more general classes of drivers, we were able to take into account the impact of different risk factors and their dependence structure on the long end of the yield curve. Finally, we study the long-term interest rates and especially the long-term swap rate in the Flesaker-Hughston model and the linear-rational methodology

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal