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Dynamic Interest-Rate Modelling in Incomplete Markets

In the first section, 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 cadlag" sample paths. The proposed process is specifically designed to derive interest-rates modelling because it allows us to introduce a jump-term structure as an increasing sequence of Levy 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 Levy-Khintchine characteristic function and Levy-Ito 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 section. 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 semidefinite programming (convex) problem and the calibration of the Levy measure is proposed using a non-parametric (non linear least-square with a regularization term) calibration.