Bivariate semi-Markov reward chain and credit spreads

In this paper, a bivariate semi-Markov reward chain (BVSMC) model is presented. Equations for the higher order moments of the reward process are presented for the first time and applied to the problem of modelling the credit spread evolution of an obligor by considering the dynamic of its own credit rating and that of a dependent obligor called the counterpart. The paper shows how to compute the expected value of the accumulated credit spread (expressed in basis points) that the obligor should expect to pay in addition to the risk free interest rate. Higher order moments of the accumulated credit spread process convey important financial information in terms of variance, skewness and kurtosis of the total basis points the obligor should with pay in a given time horizon. The paper contributes to the literature by extending on previous results on the semi-Markov reward chains, BVSMC and credit spread modelling by providing unifying approach to these problems. The model and the validity of the results are illustrated through a numerical example.