The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 693 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 693

If the company survives the first period, it has to pay off the first

debt, which clearly causes the asset price to be discontinuous. The dis-

continuity of the asset value makes the valuation of the second debt

more difficult. Geske suggests that the if the firm issues equity to pay for

the first debt, then the asset value should remain continuous and a

closed-form solution can be achieved. Here, we simply show the result:

( , 1 ) = e

• r(T 1 – t) –
  DtT K 1 Nd( 11 )+ At()[ 1 – Nd( 11 +)]

( , +

2 )= At ( 11 )– Md

+ ,d

DtT ()[Nd 22 )]

+ (

12

• r(T 2 – t) – –

• e K 2 Md( 12 ,d 22 )

• –

• e

• r(T 1 – t)
  K 1 [Nd( 12 )– Nd( 11 )]

where

± ln A()^0 ln – Kij+ (r± σ^2 ⁄^2 )

dij = ------------------------------------------------------------------------

σ Tij

K 12 is the internal solution to E(T 1 ) = K 11 which is given as the face

value of the first debt (maturing at t= 1 year) and K 22 is the face value

of the second debt (maturing at t= 2). This formulation can be extended

to include any number of debts, T 11 = T 12 = T 1 = 1 and T 22 = 2. The

correlation in the bivariate normal probability functions is the square

root of the ratio of two maturity times. In this case, it is ¹⁄₂.

Note that the total debt values add to

DtT( , 1 )+ DtT( , 2 )

= At()[ 1 – Md( 12 + ,d+ 22 )]+ e ( –

• r(T 1 – t)
  K 1 Nd 12 )


• r(T 2 – t) – –

• e K 2 Md( 12 ,d 22 )

which implies that the one-year survival probability is Nd( – ) and

• –^12
  two-year is Md( 12 ,d 22 ) which is a bivariate normal probability func-
  tion with correlation T 1 ⁄T 2. The equity value, which is the residual
  value