Aug 07

Importance of d2 in Black-Scholes to Merton Model in Credit Risk – 10 min screencast

by David Harper, CFA, FRM, CIPM

Based on a forum conversation yesterday, I explain N(d2) in the Black-Scholes and its relation to the Merton Model in credit risk. Highlights:

In Black-Scholes, N(d2) is the probability that the option will be struck (S > K) in the risk-neutral world.
The Merton model for credit risk uses the Black-Scholes by treating equity as a call option on firm assets. In Merton, d2 becomes the “distance to default.” The d2 is the same as in the Black-Scholes given the correspondences (i.e., firm asset value = stock price, default threshold = strike price), except the riskless rate is replaced by the expected growth (return) in firm’s assets.
By this translation, d2 becomes the distance to default (DD), N(d2) becomes the probability that the firm’s asset value will exceed the default threshold at period’s end, such that N(-d2) becomes the probability of default.
KMV uses the Merton model that I review here, basically, with two key differences: first, the development of the option pricing inputs (i.e., firm asset value and asset volatility) are non-trivial. Second, as I note at the end, KMV does not get EDF by assuming normality; i.e., Moody’s KMV EDF is not N(-d2). Rather, they use d2 (i.e., distance to default) and map it to an EDF based on their proprietary database.

Screencast: