6

## John C. Hull, Izzy Nelken and Alan D. White

p r o m i s e d d e b t p a y m e n t a n d l e t L = D * ⁄ A 0 b e a m e a s u r e o f l e v e r a g e . U s i n g t h e s e definitions the equity value is

# E_{0 }= A_{0}[N(d_{1}) – LN(d_{2})]

(1)

where

d_{1 }=

−ln(L) σ_{A }T

+

0.5σ

_{A }

T;

d_{2 }= d_{1 }− σ_{A }

T

As shown by Jones, Mason and Rosenfeld (1984), because the equity value is a function of the asset value, we can use Itô’s lemma to determine the instantaneous volatility of the equity from the asset volatility:^{4 }

∂E E A E A 0 ∂A 0 σ σ =

w h e r e σ E i s t h e i n s t a n t a n e o u s v o l a t i l i t y o f t h e c o m p a n y ’ s e q u i t y a t t i m e z e r o From Equation (1), this leads to .

σ_{E }=

σ_{A }N ( d_{1 }) (2) N(d_{1}) − L N(d_{2})

E q u a t i o n s ( 1 ) a n d ( 2 ) a l l o w A 0 a n d σ A t o b e o b t a i n e d f r o m E 0 , σ E , L a n d T The risk-neutral probability, P, that the company will default by time T is the probability that shareholders will not exercise their call option to buy the assets . 5

of the company for D at time T. It is given by P = N(– d_{2})

(3)

T h i s d e p e n d s o n l y o n t h e l e v e r a g e , L , t h e a s s e t v o l a t i l i t y , σ A , a n d t h e t i m e t repayment, T. o

# 2.2 Debt value and the implied credit spread of risky debt

Merton’s model can be used to explain risky debt yields. Define B_{0 }as the market price of the debt at time zero. The value of the assets at any time equals the total value of the two sources of financing, so that

B 0 = A 0 – E 0

4 Jones, Mason and Rosenfeld (1984) actually use Equations (1) and (2) in conjunction with some estimates of A and s.

5 The implementation of Merton’s model, based on Equations (1) and (2), has received considerable commercial attention in recent years. Moody’s KMV uses it to estimate relative probabilities of default. See Kealhofer (2003a, 2003b). CreditGrades (a venture supported by Deutsche Bank, Goldman Sachs, JP Morgan and the RiskMetrics Group) uses it to estimate credit default swap spreads. CreditGrades performs empirical tests similar to those we carry out for the traditional Merton model. See Finger (2002).

## Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05