Merton’s model, credit risk and volatility skews

and is significant at the 5% level for the high-volatility subset. This is consistent with the model, although less strongly so than the results in Section 5.1.

A general observation is that the volatility skews observed in practice are much higher than those that could reasonably be predicted by Merton’s model. The reason may be what Rubinstein (1994) has referred to as “crash-o-phobia”.

# 6 IS OUR VERSION OF MERTON’S MODEL OVER-STRUCTURED?

As shown by Equations (8) and (9), Merton’s model provides a fairly complex relationship between implied volatilities and credit spreads. Some banks have instead tried much simpler linear or log-linear regression models of credit spreads against implied volatilities and other variables. An important question is whether Merton’s model outperforms such simpler models. In this section we develop as a benchmark a model for relating credit spreads to implied volatilities with far less structure. We test whether Merton’s (1974) model outperforms this benchmark. In Merton’s (1976) jump–diffusion model, the stock price obeys the process:

dS = (µ −λk)dt + σ_{E }dz + dq S

where dq is a Poisson process with intensity λ and jump size k. The probability that a jump occurs in some small time interval, dt, is λdt. In the event of a jump the change in the stock price is dS = kS. The expected return on the stock is

E

(dS S) dt

= (µ −λk)+ λk = µ

We consider the particular case of this model where k = –1. In this case jumps always lead to a zero stock price. We assume that a zero stock price coincides with a default. Defaults are therefore generated by a Poisson process with intensity λ.^{16 }

Consider a put option with strike price K and time to maturity τ. As before, we define a moneyness variable κ by

K = κS_{0}e

rτ

where S_{0 }is today’s stock price. Merton showed that the option’s price is

# S_{0 }[κe^{−λτ }N(−d_{2}) − N(−d_{1}) +κ(1− e^{−λτ })]

(10)

where

λ − lnκ

d_{1 }=

+

0.5σ

_{E }

τ;

d_{2 }= d_{1 }− σ_{E }

τ

σ_{E }

τ

16 This is similar to reduced-form models such as those proposed by Duffie and Singleton (1999).

## Research papers

www.journalofcreditrisk.com

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