*On the computation of short-term credit transition matrices with scenario-dependence.*[Laurea magistrale], Università di Bologna, Corso di Studio in Matematica [LM-DM270]

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## Abstract

The computation of short-term credit transition matrices is a problem for which different solution approaches have been proposed over the years. In this thesis, carried out with the support of Prometeia S.p.A., we give a full overview of these methods; moreover, we include a study on the scenario-dependence. A credit transition matrix (CTM) is a stochastic matrix that represents the probabilities of migration from a rating class to another; following the path already traced by literature, we assume that the credit rating migrations can be described through a discrete time-homogeneous Markov model. The smallest time interval in which a CTM is available is usually one year, but for some financial applications a short-term CTM (e.g., monthly/quarterly CTM) is needed: thanks to the time-homogeneity assumption, we have that short-term CTMs can be obtained as roots of the annual CTM. However, we want the short-term CTM to be stochastic, and for this reason we present some optimization algorithms for finding the stochastic matrix that best approximates the exact root of a CTM: the QOM method that finds the nearest stochastic matrix to the exact root in Frobenius norm; the QOG method that finds the nearest generator to the exact logarithm and computes the root using the continuous-time Markov chain generated by such generator; the nonlinear optimization approach that works directly on the annual matrix to minimize the error with the sequential quadratic programming technique. We study the mathematical foundation of these methods and present a comparison of the numerical results of the application. Finally, since a CTM is obtained from historical data, to use it for future financial forecasts we need to include information about the macroeconomic scenario. For this reason, we study a classical strategy for the inclusion of the scenario-dependence in annual CTMs and propose two strategies for doing the same for short-time matrices, with an application to quarterly matrices.