22-Credit Risk Model Derivs Page 703 Wednesday, February 4, 2004 1:12 PM
Credit Risk Modeling and Credit Default Swaps 703EXHIBIT 22.3 Immediate Recoveryrisk-neutral probabilities. This is very similar to the bootstrapping
method in calibrating the yield curve. The probabilities are solved recur-
sively.
No matter which model is used, the model has to match the default
probabilities implied by the bond prices observed in the market. It can
be seen in the above section that there is no closed-form solution. The
reason is that the recovery amount is the liquidation value of the com-
pany and can change as time changes (so called “stochastic recovery”).Transition Matrix
The binomial structure can be extended to multinomial to incorporate
various credit classes. It is as easy to specify n states (different credit rat-
ings) instead of just two states (default and survival). The probabilities
can always be given exogenously. Hence, instead of a single default for
default (and survival), there can be a number of probabilities, each for
the probability of moving from one credit rating to another credit rat-
ing. Based upon this idea, Jarrow, Lando, and Turnbull,^29 extend the
Jarrow-Turnbull model to incorporate the so-called migration risk.
Migration risk is different from default risk in that a downgrade in
credit ratings only widens the credit spread of the debt issuer and does
not cause default. No default means no recovery to worry about. This
way, the Jarrow-Turnbull model can be more closely related to spread
products, whereas as a model of default it can only be useful in default
products. One advantage of ratings transition models is the ability to
use the data published by the credit rating agencies.(^29) Robert Jarrow, David Lando, and Stuart Turnbull, “A Markov Model for the
Term Structure of Credit Spreads,” Review of Financial Studies 10 (1997), pp. 481–
532.