The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 690 Wednesday, February 4, 2004 1:12 PM

690 The Mathematics of Financial Modeling and Investment Management

Implications of BSM Model

As we can see from this example, the BSM model captures some impor-

tant properties of risky debt; namely, the risky yield increases with the

debt-to-asset leverage of the firm and its asset value volatility. Using the

above equations, one can also plot the maturity dependency of the

credit spread, defined as the difference between the risky yield and the

risk-free rate.

What is appealing about this model is that the shapes of the credit

spread term structures resemble those observed in the market. The highly

leveraged firm has a credit spread which starts high, indicating that if the

debt were to mature in the short term, it would almost certainly default

with almost no recovery. However as the maturity increases, the likeli-

hood of the firm asset value increasing to the point that default does not

occur increases and the credit spread falls accordingly. For the medium

leveraged firm, the credit spread is small at the short end—there are just

sufficient assets to cover the debt repayment. As the maturity increases,

there is a rapid increase in credit spread as the likelihood of the assets

falling below the debt value rises. For the low leveraged company, the

initial spread is close to zero and so can only increase as the maturity

increases and more time is allowed for the asset value to drop. The gen-

eral downward trend of these spread curves at the long end is due to the

fact that on average the asset value grows at the riskless rate and so given

enough time, will always grow to cover the fixed debt.

Empirical evidence in favor of these term structure shapes has been

reported by Fons who observed similar relationships between spread term

structure shapes and credit quality.^9 Contrary evidence was reported by

Helwege and Turner who observed that the term structure of some low-

quality firms is upward sloping rather than downward sloping.^10

Geske Compound Option Model

If the company has a series of debts (zero coupon), then it is quite easy

for the BSM model to characterize default at different times. The trick is

to use the compound option model by Geske.^11 A compound option is

(^9) Jerome Fons, “Using Default Rates to Model the Term Structure of Credit Risk,”
Financial Analysts Journal (September/October 1994), pp. 25–32.
(^10) Jean Helwege and Christopher Turner, “The Slope of the Credit Yield Curve for
Speculative-Grade Issuers,” Federal Reserve Bank of New York Working Paper
no.97-25 (1997).
(^11) See Geske, “The Valuation of Debt as Compound Options,” and Robert Geske
and Herbert Johnson, “The Valuation of Corporate Liabilities as Compound Op-
tions: A Correction,” Journal of Financial and Quantitative Analysis 19, no. 2
(1984), pp. 231–232.