22-Credit Risk Model Derivs Page 695 Wednesday, February 4, 2004 1:12 PM
Credit Risk Modeling and Credit Default Swaps 695Pioneered by Black and Cox,^13 these models view default as a knockout
(down-and-out barrier) option^14 where default occurred the moment the
firm value crossed a certain threshold.
More recently Longstaff and Schwartz^15 examined the effect of sto-
chastic interest rates as did Briys and de Varenne^16 who modeled the
default as being triggered when the forward price of the firm value hits a
barrier. Few studies within the structural approach of credit risk valua-
tion have incorporated jumps in the firm value process, because of lack
of analytic tractability. Zhou^17 incorporates jumps into a setting used in
Longstaff and Schwartz.^18 However, this model is very computation
intensive.
Huang and Huang propose a jump-diffusion structural model which
allows for analytically tractable solutions for both bond prices and
default probabilities and is easy to implement.^19 The presence of jumps
overcomes two related limitations of the BSM approach. First, it makes
it possible for default to be a surprise since the jump cannot be antici-
pated as the asset value process is no longer continuous. Jumps also
make it more likely that firms with low leverage can suddenly default in
the short term and so enable them to have wider spreads at the short
end than previously possible.^20(^13) Fischer Black and John Cox, “Valuing Corporate Securities: Some Effects of Bond
Indenture Provisions,” Journal of Finance 31, no. 2 (1976), pp. 351–367.
(^14) A barrier option is a path dependent option. For such options both the payoff of
the option and the survival of the option to the stated expiration date depends on
whether the price of the underlying or the underlying reference rate reaches a speci-
fied level over the life of the option. Barrier options are also called down-and-out
barrier options. Knockout options are used to describe two types of barrier options:
knock-out options and knock-in options. The former is an option that is terminated
once a specified price or rate level is realized by the underlying. A knock-in option is
an option that is activated once a specified price or rate level is realized by the un-
derlying.
(^15) Francis Longstaff and Eduardo Schwartz, “A Simple Approach to Valuing Risky
Fixed and Floating Rate Debt,” Journal of Finance 50, no. 3 (1995), pp. 789–819.
(^16) Eric Briys and Francois de Varenne, “Valuing Risky Fixed Rate Debt: An Exten-
sion,” Journal of Financial and Quantitative Analysis 32, no. 2 (1997), pp. 239–248.
(^17) Chunsheng Zhou, “An Analysis of Default Correlations and Multiple Defaults,”
Review of Financial Studies (2001), pp. 555–576.
(^18) Longstaff and Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating
Rate Debt.”
(^19) Ming Huang and Jay Huang, “How Much of the Corporate-Treasury Yield
Spread is Due to Credit Risk?” working paper, Stanford University (2002).
(^20) For a discussion of barrier-based models, see Chapter 8 in Anson, Fabozzi,
Choudhry, and Chen, Credit Derivatives: Instruments, Applications, and Pricing.