The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 694 Wednesday, February 4, 2004 1:12 PM

694 The Mathematics of Financial Modeling and Investment Management

Et()= At()– DtT( , 1 )– DtT( , 2 )

= At()Md( + ,d+ ( –

22 )– e

• r(T 1 – t)
  12 K 1 Nd 12 )


• r(T 2 – t) – –


• e K 2 Md( 12 ,d 22 )

which is precisely the compound option formula derived by Geske. The

two debt values in the example are $95.12 and $81.27, respectively. The

equity is $23.61.

Using the information given in our earlier example, we solve for the

“internal strike price”—the asset price at time 1 for E(1) = K 11 to be

$195.12. In other words, if the asset price at time 1, A(1), exceeds this

value, the company survives; otherwise the company defaults. As a

result, we can calculate the default probability of the first year to be

Pr(AT( 1 ) < K 12 )= 1 – Nd( 12 ) = 1 – 0.6078 = 0.3922

The two-year total default probability is the one whereby the com-

pany defaults in year 1 or it survives the first year but defaults the sec-

ond year:

Pr[AT K ∪AT – –

( 1 ) < 12 ( 2 ) < K 22 ]= 1 – Md( 12 ,d 22 )

= 1 – 0.6077 = 0.3923

The default probability therefore between the first year and the second

year is only 0.0001. In other words, the Geske model indicates that the

majority default probability is in the first year, and then the company

can survive with almost certainty.

In general, structural models are not easy to calibrate since informa-

tion regarding the size and priority of claimants on a company’s assets is

not readily available. Typically companies only publish details of their

balance sheets at most quarterly, and some companies, particularly

those facing severe financial difficulties, do not disclose the full picture.

Instead, practitioners tend to take equity volatility as a proxy for the

asset value volatility.^12

Barrier Structural Models

In addition to the Geske (compound option) model, another series of

models have also evolved to extend the BSM model to multiple periods.

(^12) For example, KMV uses σ = (AE⁄ )Nd()σ, where is the volatility of eq-
E 1 A σE
uity and σA is the volatility of the asset.