Recursive formulas for the default probability distribution with applications in Markov chain-based intensity models

Abstract

The probability distribution of the number of defaults plays an important role in pricing problems of multiple-name credit derivatives. When the group size gets large, it becomes increasingly difficult to obtain its whole distribution. We use a financial argument to prove that, for these default probabilities, there exists a recursive formula that is useful for the calculation of the whole distribution. A major advantage of the proposed formula is that it is model free and allows for a general correlation structure among group entities. For a fully heterogeneous group, the usefulness of the proposed formula is limited by its high computational complexity. To balance model heterogeneity with computational feasibility, the recursive formulas are further extended to new versions, allowing for multiple heterogeneous subgroups of homogeneous entities. The recursive algorithms developed from these results are applied to the calculation of default probability distribution under Markov chain-based default intensity models.We demonstrate the computational benefits using a number of numerical examples.