N(d1) and N(d2)

No, N(d1) is not the probability of exercise.

In one of my classes I derived the formulas for the values of standard options and various digital options in the Black-Scholes models, the point being to illustrate various concepts – the state price process, risk-adjusted probabilities, and the use of different numeraires.

We got into an argument about the meaning of N(d1) and N(d2).

I published a paper about this a number of years ago in Revue Finance, the journal of the French finance association: “Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model,” Revue Finance (Journal of the French Finance Association) 14 (1993), 95-106. [Abstract][Abstract on the journal’s website][Paper (pdf)]

The paper explains N(d1) and N(d2) and relates them to the single-period and multi-period binomial models.

After so many years, people are still puzzling over this.

What is N(d1)? Well, N(d2) is the risk-adjusted probability of exercise. N(d1) is something else: It is the factor by which the present value of contingent receipt of the stock (contingent upon exercise) falls short of the current stock price.

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