The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 706 Wednesday, February 4, 2004 1:12 PM

706 The Mathematics of Financial Modeling and Investment Management

The bond price today is

$4.81 + $69.58 = $74.39

Similar analysis can be applied to the case where the current state is 2. In

the above example, it is quite easy to include various recovery assumptions.

It is costly to include the ratings migration risk in the Jarrow-Turn-

bull model. It is very difficult to calibrate the model to the historical

transition matrix. First of all, the historical probabilities computed by

the rating agencies are actual probabilities while the probabilities that

are used for computing prices must be risk neutral probabilities that we

introduced in Chapter 14. The assumption by Jarrow, Lando, and Turn-

bull that there is a linear transformation does not necessarily provide a

good fit to the data. Second, there are more variables to solve for than

the available bonds. In other words, the calibration is an underidentifi-

cation problem. Hence, more restrictive assumptions about the proba-

bilities need to be made. In general, migration risk is still modeled by

the traditional portfolio theory (non-option methodology). But the

model by Jarrow, Lando, and Turnbull is a first attempt at using the

option approach to model the rating migration risk.

The Duffie-Singleton Model

Obviously, the Jarrow-Turnbull assumption that recovery payment can

occur only at maturity is too far from reality. Although it generates a

closed-form solution for the bond price, it suffers from two major draw-

backs in reality: recovery actually occurs upon (or soon after) default

and the recovery amount can fluctuate randomly over time.^30

Duffie and Singleton take a different approach.^31 They allow the

payment of recovery to occur at any time but the amount of recovery is

restricted to be the proportion of the bond price at default time as if it

did not default. That is

Rt()= δDt T ( , )

where R is the recovery ratio, δis a fixed ratio, and D(t,T) represents the

debt value if default did not occur. For this reason the Duffie-Singleton

model is known as a fractional recovery model. The rationale behind this

approach is that as the credit quality of a bond deteriorates, the price

falls. At default the recovery price will be some fraction of the final price

(^30) Recovery fluctuates because it depends on the liquidation value of the firm at the
time of default.
(^31) Duffie and Singleton, “Modeling the Term Structure of Defaultable Bonds.”