22-Credit Risk Model Derivs Page 710 Wednesday, February 4, 2004 1:12 PM
710 The Mathematics of Financial Modeling and Investment Managementdefault probability curve can be calibrated to only if a particular recov-
ery assumption is adopted. Hence the default probability is a function
of the assumed recovery rate.General Observations on Reduced Form Models
While the reduced form models lay a solid theoretical foundation, as
they attempt to model the underlying risk-neutral probability of default
which is not a market observable, they are not as intuitive as one might
like. They also suffer from the constraint that default is always a sur-
prise. While this is true under some rare circumstances, Both Moody’s
and Standard & Poor’s data show that there are very few defaults
straight out of investment-grade quality bonds. Default is usually the
end of a series of downgrades and spread widenings and so can be antic-
ipated to a large extent. Hence, although more and more financial insti-
tutions are starting to implement the Jarrow-Turnbull and Duffie-
Singleton models, spread-based diffusion models remain very popular.
The Jarrow-Turnbull and Duffie-Singleton models assume that defaults
occur unexpectedly and follow the Poisson process. This assumption
greatly reduces the complexity since the Poisson process has very nice
mathematical properties. In order to further simplify the model, Jarrow-
Turnbull and Duffie-Singleton respectively make other assumptions so
that there exist closed-form solutions to the basic underlying asset.PRICING SINGLE-NAME CREDIT DEFAULT SWAPS
There are two approaches to pricing default swaps—static replication
and modeling. The former approach is based on the assumption that if
one can replicate the cash flows of the structure which one is trying to
price using a portfolio of tradable securities, then the price of the struc-
ture should equal the value of the replicating portfolio. This is accom-
plished through what is known as an asset swap; however, there are
limitations of using of asset swaps for pricing.^35 In situations where
either the nature of the instrument we are trying to price cannot be rep-
licated or that we do not have access to prices for the instruments we
would use in the replicating portfolio, it becomes necessary to use a
modeling approach. That is the approach explained below for pricing
credit default swaps.(^35) See Chapter 4 in Anson, Fabozzi, Choudhry, and Chen, Credit Derivatives: Instru-
ments, Applications, and Pricing.