22-Credit Risk Model Derivs Page 709 Wednesday, February 4, 2004 1:12 PM
Credit Risk Modeling and Credit Default Swaps 7091 1
-------------------- =------------[pδ +( 1 – p)]
1 ++rs 1 +rthis implies an effective discounting rate of r + s = 9.63% over the time
step from the 7% node. In this way we can proceed to value the other
nodes and roll back to calculate an initial price for the bond equal to
$84.79. On each node in Exhibit 22.5 is also shown the effective dis-
counting rate. Knowing these we can equally price the bond as though it
were default free but discounted at r + s rather than at the risk-free rate.
The Duffie-Singleton model has one very important advantage. The
above result implies that it can be made compatible with arbitrage-free
term structure models such as Cox-Ingersoll-Ross^33 and Heath-Jarrow-
Morton.^34 The difference is that now the discounting is spread adjusted.
Just like the yield curve for the risk-free term structure, the spread curve
is added to the risk-free yield curve and we arrive at a risky yield curve.
The spread curve is clearly based upon the probability curve (pt for all t)
and the recovery rate (δ).
Although the Duffie-Singleton model seems to be superior to the
Jarrow-Turnbull model, it is not generic enough to be applied to all
credit derivative contracts. The problem with the Duffie-Singleton
model is that if a contract that has no payoff at maturity such as a credit
default swap, their model implies zero value today, which is of course
not true. Recall that credit default swaps pay nothing if default does not
occur. If recovery is proportional to the no-default payment, then it is
obvious that the contract today has no value. It is quite unfortunate that
the Duffie-Singleton model is not suitable for the most popular credit
derivative contracts. Hence, the proportionality recovery assumption is
not very general.
The calibration of the Duffie-Singleton model is as easy as the Jarrow-
Turnbull model. The two calibrations are comparable. However, there
are significant differences. Note that in the Jarrow-Turnbull model, the
recovery assumption is separate from the default probability. But this is
not the case in the Duffie-Singleton model—the recovery and the default
probability together become an instantaneous spread. While we can cal-
ibrate the spreads, we cannot separate the recovery from the default
probability. On the other hand, in the Jarrow-Turnbull model, the(^33) John Cox, Jonathan Ingersoll, and Stephen Ross, “A Theory of the Term Structure
of Interest Rates,” Econometrica 53 (1985), pp. 385–407.
(^34) David Heath, Robert Jarrow, and Andrew Morton, “Bond Pricing and the Term
Structure of Interest Rates: A New Methodology,” Econometrica 59 (February
1992), pp. 77–105.