The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 688 Wednesday, February 4, 2004 1:12 PM

688 The Mathematics of Financial Modeling and Investment Management

lnAt()– lnK+ (r– σ^2 ⁄ 2 )(Tt– )

d 2 = -----------------------------------------------------------------------------------

σ Tt–

Equation (22.4) implies that the risk neutral probability of in the

money N(d 2 ) is also the survival probability. To find the current value of

the debt, D(t,T) (maturing at time T), we need to first use the BSM

result to find the current value of the equity. As shown above, this is

equal to the value of a call option:

Et()= At()Nd() 1 – e –rT t ( – )KN d() 2 (22.5)

where d 1 =d 2 + σ Tt–. The current value of the debt is a covered call

value:

DtT ( , ) = At()– Et() (22.6)

= At ()Nd

( –

()– [At ()– e ()]

• rT t )
  1 KN d 2
  = At()[ 1 – Nd() 1 ]+ e –rT t ( – )KN d() 2

Note that the second term in the last equation is the present value of

probability-weighted face value of the debt. It means that if default does

not occur (with probability N(d 2 )), the debt owner receives the face

value K. Since the probability is risk neutral, the probability-weighted

value is discounted by the risk-free rate. The first term represents the

recovery value. The two values together make up the value of debt.

The yield of the debt is calculated by solving D(t,T) = Ke–y(T–t) for yto

give

lnK– lnDtT ( , )

y= ---------------------------------------- (22.7)

Tt–

Consider the case of a company which currently has net assets

worth $140 million and has issued $100 million in debt in the form of a

zero-coupon bond which matures in one year. By looking at the equity

markets, we estimate that the volatility of the asset value is 30%. The

risk-free interest rate is at 5%. We therefore have

A(t) = $140 million

K = $100 million

σ = 30%