We model the evolution of the credit migration of a corporate bond as an inhomogeneous semi-Markov chain. The valuation of a defaultable bond is done with the use of the forward probability of no default up to maturity time. It is proved that, under the forward probability measure, the semi-Markov property is maintained. We find the functional relationships between the forward transition probability sequences and the real-world probability sequences. The stochastic monotonicity properties of the inhomogeneous semi-Markov model, which play a prominent role in these issues, are studied in detail. Finally, we study the term structure of credit spread, provide an algorithm for the estimation of the forward probabilities of transitions under the risk premium assumptions, and present an estimation method for the real-world probability sequences.
