Positive default intensities

The problem of developing tractable stochastic default intensity models that allow one to 1) reproduce the term-structure of credit default swaps, 2) calibrate to options on credit default swaps and 3) guarantee the positivity of default intensities (or at least limit the probability of the latter becoming negative) is, to the best of our knowledge, not yet fully satisfactorily solved. In this presentation we develop the Markov Functional approach as a means to achieve these three goals. Markov Functional models were first introduced by Hunt, Kennedy and Pelsser and have since been used to model e.g. interest rates, equity prices, foreign exchange rates and credit. We believe the application to default intensities presented here to be new. We then compare our Markov Functional model to the Shifted Square-Root Diffusion (SSRD or CIR++) model proposed by Brigo and Alfonsi. The advantage of the Markov Functional approach over the SSRD model is its greater flexibility combined with a better control of the (possibly) negative default intensities generated by the models. Its main disadvantage is that it is slightly less tractable in the sense that less closed-form analytical results are available. However, this is not a major drawback as Markov Functional models allow a relatively efficient numerical implementation without having to resort to e.g. Monte Carlo simulation. In addition to giving an overview of the SSRD model and presenting the theory and implementation of the Markov Functional model in quite some detail, we compare the relative performance of both models in terms of achieving the goals listed above. For the sake of completeness, we include the Hull-White model in our comparison, despite its inability to control the negativity of default intensities. Finally, we apply the three models to the pricing of callable credit-linked notes