The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 716 Wednesday, February 4, 2004 1:12 PM

716 The Mathematics of Financial Modeling and Investment Management

No Need For Stochastic Hazard Rate or Interest Rate

The analysis above demonstrates that to price a default swap, we only need

a recovery rate, the risk-free yield curve (the P-curve), and the survival

probability curve (the Q-curve). This implies that regardless of which model

is used to justify the P-curve or the Q-curve, default swaps should be priced

exactly the same. This further implies that there is no need to be concerned

if the risk-free rate and the hazard rate are stochastic or not, because they

do not enter into the valuation of the default swap. In other words, random

interest rates and hazard rates are “calibrated out” of the valuation.^39

Delivery Option in Default Swaps

As explained earlier in this chapter, a credit default swap trade can spec-

ify a reference entity or a reference obligation. In the former case, the

protection buyer has the option to deliver one of severable deliverable

obligations of the reference entity. This effectively creates a similar situ-

ation to the well-known quality option for Treasury note and bond

futures contracts where more than one bond can be delivered. In this

case, the value of the credit default swap is

n

V = (^) ∑P t T( , j )[Qt T( , j – 1 )– Qt T( , j )][ 1 – minRT()j]
j = 1
The difference between the above equation and equation (22.11) is the
recovery. The delivery of the lowest recovery bond, min{R(Tj)}, for all j
bonds is what the payoff is.
It is natural that the worst quality bond should be delivered upon
default. For a credit default swap, the one with the lowest recovery
should be delivered. Unlike Treasury bond and note futures where the
cheapest-to-deliver issue can change due to interest rate changes, recovery
is mostly determined contractually and usually the lowest priority bond
will remain the lowest priority for the life of the contract. The only uncer-
tainty in determining the cheapest-to-deliver issue is the future introduc-
tion of new bonds. This is largely related to the capital structure of the
company and beyond the scope of risk-neutral pricing. The model that
can incorporate capital structure issues (i.e., using debt to optimize capi-
tal structure) needs to be a structural model with wealth maximization.^40
(^39) For the stochastic hazard rate model, see Daniel Lando, “On Cox Processes and
Credit Risky Securities,” Review of Derivatives Research (1998), pp. 99–120.
(^40) Issues about optimal capital structure and default risk are discussed in Hayne E. Le-
land and Klaus Bjerre Toft, “Optimal Capital Structure, Endogenous Bankruptcy, and
the Term Structure of Credit Spreads,” Journal of Finance (July 1996), pp. 987–1019.