When Will N(d1) Be the Probability of Exercising an Option?

A proof by comparing change of measure with the Black-Scholes formula.

Written on September 26, 2018


Consider the value of a vanilla European call option:

Assuming constant interest rate:

we have:

Compare this with the Black-Scholes formula:

Since both formulas hold true for any choice of the parameters, we have:

In other words, $N(d_1)$ is the probability of exercising the option under the stock measure, while $N(d_2)$ is the probability of exercising the option under the T-forward measure (and of course, the risk-neutral measure as well).