I was wondering why the asymmetry between S and K — that is, why is the first term S*N(d1) and not S*N(d2), since N(d2) is the probability that S > K. But then I realized the S term is much more complicated than the K term. K is pre-determined. S is a stochastic variable. So real surprise here is that in the expression S*N(blah), that blah is so closely related to d2. They’re identical up to an additive constant. That’s amazing.

So that got me thinking — maybe the answer to my first question is that there’s a lot going on in that first term, and actual calculation of E[S] already contains the discount factor pre-built into it.

Which begs the question… can N(d1) be expressed as e^{-rT} * f(d2)?

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