The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 704 Wednesday, February 4, 2004 1:12 PM

704 The Mathematics of Financial Modeling and Investment Management

For a flavor of how a rating transition model can be obtained, con-

sider a simple three-state model. At each time interval an issuer can be

upgraded, downgraded or even jump to default. This process is shown

in Exhibit 22.4. This time, the tree is more complex. From a “live”

state, the issuer can be upgraded or downgraded, or even jump to

default. The default state, on the other hand, is an absorbing barrier

which cannot become live again. In terms of Exhibit 22.4, a movement

from “good rating” to “middle rating” is downgrade, and vice versa.

To best describe the situation, we can establish the following transi-

tion matrix:

Future state

2 1 0

2 p 22 p 21 p 20

Current state 1 p 12 p 11 p 10

0 0 0 1

where 0 is the default state, 1 is the middle credit rating state, and 2 is

good credit rating state. pij is the transition probability to move from

the current state i to future state j. The sum of the probabilities of each

current state should be 1, that is

2

∑ pij =^1

j = 0

The last row of the matrix is all 0’s except for the last column. This

means that once the asset is in default, it cannot become live again and

it will remain in default forever.

EXHIBIT 22.4 Multistate Default Process