Cross- and Autocorrelation in multi-period Credit Portfolio Models

We investigate different well-known multi-step credit portfolio models, all endowed with cross dependency, in their time-discretized versions. Refining then the time discretization we show that the correlation structure of only one model is invariant under this refinement whereas the others are not. In particular, the correlation structure of Markov-Chain migration model converges to the limit of no cross correlation, whereas the discrete barrier model converges to an unknown continuous limit. This has important implication on the risk assessment and pricing of portfolios of structured credit products when using these models, as it means that for consistent valuation the correlation structure of these models has to be calibrated to a given time horizon and time discretization of the respective implementation.