Merton’s model, credit risk and volatility skews

The rest of this paper is organized as follows. Section 2 develops the theory that underlies our implementation of Merton’s model. Section 3 describes the data we use. In Section 4 we compare the credit spreads implied by Merton’s model with CDS spreads for both our implementation of Merton’s model and the traditional implementation. In Section 5 we present some results on the theoretical relation- ships between implied volatilities and credit spreads under Merton’s model and test whether these relationships hold. In Section 6 we develop a relatively simple model, based on Merton’s (1976) jump–diffusion model, for relating credit spreads to implied volatilities and use it as a benchmark to test whether the more elaborate structure underlying our implementation of Merton (1974) provides a better explanation of observed credit spreads. Conclusions are presented in Section 7.

# 2 MERTON’S MODEL

Both Merton (1974) and Black and Scholes (1973) propose a simple model of the firm that provides a way of relating credit risk to the capital structure of the firm. In this model the value of the firm’s assets is assumed to obey a lognormal diffusion process with a constant volatility. The firm has issued two classes of securities: equity and debt. The equity receives no dividends. The debt is a pure discount bond where a payment of D is promised at time T.

If at time T the firm’s asset value exceeds the promised payment, D, the lenders are paid the promised amount and the shareholders receive the residual asset value. If the asset value is less than the promised payment, the firm defaults, the lenders receive a payment equal to the asset value, and the shareholders get nothing.

# 2.1 Equity value and the probability of default

Define E as the value of the firm’s equity and A as the value of its assets. Let E_{0 }and A_{0 }be the values of E and A today and let E_{T }and A_{T }be their values at time T. In the Merton framework the payment to the shareholders at time T is given by

E_{T }= max [A_{T }– D, 0]

This shows that the equity is a call option on the assets of the firm with strike price equal to the promised debt payment. The current equity price is therefore

E 0 = A 0 N ( d 1 ) – D e

–

rT

N ( d 2 )

where

d_{1 }=

ln(A_{0}e^{rT }σ_{A }T

D)

+

0.5σ

_{A }

T;

d_{2 }= d_{1 }− σ_{A }

T

σ A i s t h e v o l a t i l i t y o f t h e a s s e t v a l u e a n d r i s t h e r i s k - f r e e r a t e o f i n t e r e s t , b o t h o which are assumed to be constant. Define D^{* }= De^{–rT }as the present value of the f

### Research papers

www.journalofcreditrisk.com

5