8.1.1 Black–Karasinski as a Credit Model
We introduced the Black–Karasinski short‐rate model in Chapter 5 as an interest rate model. Its main advantage over the Hull–White model in that respect is that it preserves positive rates, the price we must pay for that convenience being that the resultant pricing kernel is not in closed form but is available only as an asymptotic series. Since, for interest rates, the convenience of being able to ensure positive rates is not, particularly under current market conditions, considered to be a necessary feature of a model we shall tend to use Hull–White as our default interest rate model.
However, our interest in the present context is not only in modelling interest rate dynamics but particularly in capturing credit default risk. Our approach to this will be through the reduced‐form modelling approach pioneered by Merton , wherein the assets and liabilities of a borrower are not considered explicitly, so allowing an insolvency event to be modelled and a default consequently predicted (the so‐called structural modelling approach). Rather, we look to model the occurrence as a Poisson process1 with a stochastic process representing the intensity (probability per unit time) of a default event occurring at time . Notably this probability is not conditioned on survival of the debt‐issuing ...