When Will N(d1) Be the Probability of Exercising an Option?
A proof by comparing change of measure with the Black-Scholes formula.
Consider the value of a vanilla European call option:
Assuming constant interest rate:
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).