The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 700 Wednesday, February 4, 2004 1:12 PM

700 The Mathematics of Financial Modeling and Investment Management

The Calibration of Jarrow-Turnbull Model

Exhibit 22.2 best represents the Jarrow-Turnbull model.^27 The branches

that lead to default will terminate the contract and incur a recovery pay-

ment. The branches that lead to survival will continue the contract which

will then face future defaults. This is a very general framework to describe

how default occurs and contract terminates. Various models differ in how

the default probabilities are defined and the recovery is modeled.

Since a debt contract pays interest under survival and pays recovery

upon default, the expected payment is naturally the weighted average of

the two payoffs. For the ease of exposition, we shall denote the survival

probability from now to any future time as Q(0,t) where t is some

future time. As a consequence, the difference between two survival

times, Q(0,s) – Q(0,t) where s > t, by definition, is the default probabil-

ity between the two future time points t and s.

The above binomial structure can be applied to both structural

models and reduced form models. The default probabilities can be easily

computed by these models. The difference resides in how they specify

recovery assumptions. In the Geske model, the asset value at the time is

EXHIBIT 22.2 Tree-Based Diagram of Binomial Default Process for a

Debt Instrument

(^27) As recent articles by Ren-Raw Chen and Jinzhi Huang [“Credit Spread Bonds and
Their Implications for Credit Spread Modeling,” Rutgers University and Penn State
University (2001)] and Ren-Raw Chen [“Credit Risk Modeling: A General Frame-
work,” Rutgers University (2003)] show, the binomial process is also applicable to
structural models.