8

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

Define v as the implied volatility of the put at time zero based on the Black– Scholes model. Also define parameters α and κ by

A

* τ

= α A 0 e r τ ,

K = κ E 0 e

rτ

The parameter α is the ratio of the critical asset price to the forward asset price (both being observed at time zero). We will refer to it as the implied strike level. The parameter κ is the ratio of the option strike price to the forward equity price (observed at time zero). We will refer to it as the option’s moneyness.

Implied volatilities are the volatilities which, when substituted into the Black–Scholes model, give the market price. If we assume that market prices are given by Merton’s model, the implied volatility of an option can be determined by solving

D M a d * , ; − − 2 2

T A M a d , ; − − − 0 1 1 τ

T E N a + − ( ) 0 2 τ κ

E N d E N d * * = − ( ) − − ( ) 0 2 0 1 κ

(7)

where

d 1 * =

−ln(κ) vτ

+

0.5v τ ;

d d v 2 1 * * = − τ

a_{1 }=

−ln(α)

+

0.5σ

_{A }

τ;

a_{2 }= a_{1 }− σ_{A }

τ

σ_{A }τ Substituting Equation (1) into Equation (7) results in

L M a d − − 2 2 , ;

τ T

M a d − − − 1 1 , ;

τ T

N a N d L N d + − ( ) − [ ] 2 1 2 ( ) ( ) κ

N d N d N d L N d = − ( ) − − ( ) [ ] − [ ] 2 1 1 2 ( ) ( ) * * κ

(8)

# A variation of Equation (1) can be used to determine the implied strike level, α:

κ α τ τ τ τ E A N d L N d r 0 1 2 e = − ( ) [ ] * , , ( ) ( )

so that

κ=

αN(d_{1,τ }) − LN(d_{2,τ }N(d_{1}) − LN(d_{2})

)

(9)

where

d_{1,τ }

=

−ln(L α)

+

0.5σ

_{A }

T −τ ;

d_{2,τ }

= d_{1,τ }

− σ_{A }

T −τ

A σ

# T −τ

Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05