The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 705 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 705

To make the model mathematically tractable, Jarrow-Lando-Turn-

bull assume that the transition matrix follows a Markov chain; that is,

the n-period transition is the above matrix raised to the n-th power. The

main purpose to derive such a matrix is that we can calibrate it to the

historical transition matrix published by rating agencies. Note that the

historical transition matrix consists of real probabilities which are dif-

ferent from the risk-neutral probabilities in the tree. Hence, Jarrow-

Lando-Turnbull make a further assumption that the risk-neutral proba-

bilities are proportional to the actual ones. For a risk averse investor,

the risk-neutral default probabilities are larger than the actual ones

because of the risk premium.

Since historical default probabilities are observable, we can then

directly compute the prices of credit derivatives. For example, let the

transition probability matrix for a 1-year period be

Future state

2 1 0

2 0.80 0.15 0.05

Current state (^1) 0.15 0.70 0.15
0 0 0 1
Then, for a one-year, 0-recovery coupon bond, if the current state is
1, it has 85% to receive the coupon and 15% to go into default in the
next period. So the present value of the next coupon is
0.85 × $6
------------------------ = $4.81
1.06
In the second period, the bond could be upgraded with probability of
15% or remain the same with probability of 70%. If it is at the good rat-
ing, then the probability of survival is 95% and if it is at the bad rating, the
probability of survival is 85%. Hence, the total probability of survival is
0.15 × 0.95 + 0.7 × 0.85 = 0.7375 = 73.75%
Therefore, the present value of the maturity cash flow (coupon and face
value) is
0.7375 × 106
---------------------------------- = $69.58
1.06
2