## Merton’s model, credit risk and volatility skews

# Using Equation (1), this becomes

B_{0 }= A_{0}[N(– d_{1}) + LN(d_{2})]

(4)

The yield to maturity on the debt is defined implicitly by

B_{0 }= De

–

yT

= D * e

(r–y)T

S u b s t i t u t i n g t h i s i n t o E q u a t i o n ( 4 ) a n d u s i n g A 0 = D * ⁄ L g i v e s t h e y i e l d t o m a t u r i t y a s

[N(d_{2}) + N(−d_{1}) L] y = r − ln T

The credit spread implied by the Merton model is therefore^{6 }

s = y − r = − ln

[N(d_{2}) + N(−d_{1}) L] T

(5)

Like the expression for the risk-neutral probability of default in Equation (3), the i m p l i e d c r e d i t s p r e a d 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 the time to repayment, T. d

# 2.3 Equity volatility and volatility skews

One point about Merton’s model that has not received much exploration is the role it plays in explaining the equity option implied volatilities and the volatility skews that are observed in the equity options market. Within the framework of the Merton model, an option on the firm’s equity that expires before the debt matures is a compound option, an option on a European call option. We can therefore use the model proposed by Geske (1979). Using the notation developed above, the value at time zero of a European put with strike price K and expiry time t < T on the equity is

p = De

M a d r T − − − 2 2 , ;

T − A M a d τ − 0 1 1 , ;

−

T K r τ + − e τ

N(−a_{2})

(6)

where

a_{1 }=

ln(A_{0 }r − A e * τ τ σ_{A }τ

)

+

0.5σ

_{A }

τ;

a_{2 }= a_{1 }− 0.5σ_{A }

τ

M i s t h e c u m u l a t i v e b i v a r i a t e n o r m a l d i s t r i b u t i o n f u n c t i o n a n d A τ * i s t h e c r i t i c a l asset value at time τ, the value for which the equity value at that time equals K.

That is, A^{* }is the asset value below which the put on the equity will be exercised. τ

6 The relationship between credit spreads and default risk is discussed by Duffie and Singleton (1999), Litterman and Iben (1991) and Rodriguez (1988) among others.

## Research papers

www.journalofcreditrisk.com

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