In this work we define a multivariate semi-Markov process. We derive an
explicit expression for the transition probability of this multivariate
semi-Markov process in the discrete time case. We apply this multivariate model
to the study of the counterparty credit risk, with regard to correlation in a
CDS contract. The financial crisis has stressed the importance of the study of
the correlation in the financial market. In this regard, the study of the risk
of default of the counterparty in any financial contract has become crucial in
the credit risk. Many works has been done to trying to describe the
counterparty risk in a CDS contract, but all this work are based on the
Markovian approach to risk. In the our opinion this kind of model are too
restrictive, because they require that the distribuction function of the
waiting times has to be exponential or geometric, for discrete time. In the our
model, we describe the evolution of credit rating of the financial subjects
like a multivariate semi-Markov model, so we allow for arbitrarily distributed
sojourn time. The age state dependency, typical of the semi-Markov environment,
allow us to insert the correlation in a dynamical way. In particular, suppose
that A is a default-free bondholder and C is the relative firm. The bondholder
buy protection against C's default by another defaultable subject, say B the
protection seller. Our model describe the evolution of the credit rating of the
couple B and C. We admit for simultaneus default of C and B, the single default
of C or single default of B.