Aug 07

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

by David Harper, CFA, FRM, CIPM


FRM |

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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:


Comments

  1. Thank you for this perfectly understandable explanation about the Merton model for credit risk. As a Dutch student following my master in Financial Economics, this explanation really helped me a lot in understanding how to calculate and understand credit risk according to Merton’s model, which I need to know for the course Financial Risk Management!

    Great work!

    Robert

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