The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 699 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 699

yield spread. This is so because in any one-period interval in the bino-

mial model, we have

DtT ( , )= PtT ( , )e –λ (Tt– )

= PtT ( , )QtT ( , )

This is known as the risky discount factor, which is the present value of

$1 if there is no recovery (i.e., the recovery ratio is zero, R= 0).

The Jarrow-Turnbull model is usually modified when it is used in

practice. One modification is to allow the Poisson intensity λto be a

function of time and the other is to allow recovery to be paid upon

default. As a result the bond equation is modified as follows:

T (^) n

Bt()= ∫ Ptu ( , )Ru()(–d Qu())+ ∑PtT( , j)cjQtT( , j)

t j=^1

T – u^ λw ()dw

∫ ()dw

n –

∫

Tj

tλw

= ∫ Ptu ( , )Ru()λ()ue t + ∑PtT( , j)cje

t j=^1

To actually implement this equation, it is usually assumed that λfol-

lows a step function. That is between any two adjacent time points, λis a

constant. Furthermore, it is also, as a matter of mathematical tractabil-

ity, assumed that default can occur only at coupon times.^26 As a result of

this further assumption, the above equation can be simplified as

j n

• ∑λ (Tk) – ∑λ (Tk)
  ()e

n n

Bt()= ( , ()λT k=^1 ( , k=^1

∑PtTj)RTj j + ∑PtTj)cje^

j= 1 j= 1

The major advantage of the Jarrow-Turnbull model is calibration.

Since default probabilities and recovery are exogenously specified, one

can use a series of risky zero-coupon bonds to calibrate out a default

probability curve and hence a spread curve.

Calibration has become a necessary first step in fixed-income trad-

ing recently for it allows traders to clearly see relative pricesand hence

be able to construct arbitrage trading strategies. The ability to quickly

calibrate is the major reason why reduced form models are strongly

favored by real-world practitioners in the credit derivatives markets.

(^26) This assumption is not unreasonable because between two coupon times, if the
company is not audited, the company should not have any reason to default.