22

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

Equating this option price to the Black–Scholes price defines the implied volatil- ity v:

κe^{−λτ }N(−d_{2}) − N(−d_{1})+ κ(1− e^{−λτ }

κ = − ( ) − − ( ) N d N d ) * * 2 1

(11)

where, as before,

d 1 * =

−ln(κ) vτ

+

0.5v τ ;

d d v 2 1 * * = − τ

I f a v o l a t i l i t y , σ E , a n d a d e f a u l t i n t e n s i t y , λ , a r e c h o s e n a n d E q u a t i o n ( 1 1 ) i solved to determine the Black–Scholes implied volatility for various values of κ, a volatility skew results. Similarly to the case of the Merton model discussed in Section 2.3, the lower the value of κ, the higher is the implied volatility, v.^{17 }s

Merton’s (1976) model can also be used to determine the price of a zero- coupon bond issued by the firm that matures at time T. If no default occurs, the bond is assumed to pay $1 at maturity, and if a default occurs at or before matu-

rity, the bondholder is assumed to recover R ≤ $1 at maturity.^{18 }

Conditional on

the intensity of the Poisson process, the risk-neutral probability that no default will occur before maturity, π, is exp[–λT] and the probability that a default will occur is 1 – π. The bond price is the present value of the risk-neutral expected value discounted back to the present using the risk free rate of interest.

[π + (1− π)R]e^{−rT }= e^{−yT }

where y is the yield on the zero-coupon bond. The credit spread on the debt is then

y−r =

−ln[π + (1− π)R] T

or

−ln[e

^{−λT }+ (1− e_{−λT }T

)R]

(12)

In the event that the recovery rate, R, is zero, the credit spread equals the default intensity, λ, for all maturities.

17 A version of the jump–diffusion model in which the diffusion part of the process obeys a constant elasticity of variance process was also tested. The results from the CEV version were not materially different from the results given by the version described here and are not reported.

18 This is equivalent to the assumption that the claim in the event of default is proportional to the default-risk-free value of the debt.

## Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05