The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 696 Wednesday, February 4, 2004 1:12 PM

696 The Mathematics of Financial Modeling and Investment Management

Advantages and Drawbacks of Structural Models

Structural models have many advantages. First, they model default on

the very reasonable assumption that it is a result of the value of the

firm’s assets falling below the value of its debt. In the case of the BSM

model, the outputs of the model show how the credit risk of a corporate

debt is a function of the leverage and the asset volatility of the issuer.

The term structure of spreads also appear realistic and empirical evi-

dence argues for and against their shape. Some of the more recent struc-

tural models have addressed many of the limitations and assumptions of

the original BSM model.

However structural models are difficult to calibrate and so are not

suitable for the frequent marking to market of credit contingent securi-

ties. Structural models are also computationally burdensome. For

instance, as we have seen, the pricing of a defaultable zero-coupon bond

is as difficult as pricing an option. Just adding coupons transforms the

problem into the equivalent of pricing a compound option. Pricing any

subordinated debt requires the simultaneous valuation of all of the more

senior debt. Consequently, structural models are not used where there is

a need for rapid and accurate pricing of many credit-related securities.

Instead, the main application of structural models is in the areas of

credit risk analysis and corporate structure analysis. As explained later

in this chapter, a structural model is more likely to be able to predict the

credit quality of a corporate security than a reduced form model. It is

therefore a useful tool in the analysis of counterparty risk for banks

when establishing credit lines with companies and a useful tool in the

risk analysis of portfolios of securities. Corporate analysts might also

use structural models as a tool for analyzing the best way to structure

the debt and equity of a company.

CREDIT RISK MODELING: REDUCED FORM MODELS

The name reduced form was first given by Darrell Duffie to differentiate

from the structural form models of the BSM type. Reduced form models

are mainly represented by the Jarrow-Turnbull^21 and Duffie-Singleton^22

models. Both types of models are arbitrage free and employ the risk-

neutral measure to price securities. The principal difference is that

(^21) Robert Jarrow and Stuart Turnbull, “Pricing Derivatives on Financial Securities
Subject to Default Risk,” Journal of Finance (March 1995), pp. 53–86.
(^22) Darrell Duffie and Kenneth Singleton, “Modeling the Term Structure of Default-
able Bonds” (1997), working paper, Stanford University.