It’s possible to get an intuitive feel for the binary call valuation formula.
For a vanilla European call, C = … – K exp(-Rdisc T)*N(d2)
N(d2) = Risk-Neutral Pr(S_T > K). Therefore,
N(d2) = RN-expected payoff of a binary call
N(d2) exp(-Rdisc T) — If we discount that RN-expected payoff to Present Value, we get the current price of the binary call. Note all prices are measure-independent.
Based on GBM assumption, we can *easily* prove Pr(S_T > K) = N(d2) .
First, notice Pr(S_T > K) = Pr (log S_T > log K).
Now, given S_T is GBM, the random variable (N@T)
log S_T ~ N ( mean = log S + T(Rgrow – σ^2) , std = T σ^2 ).
Let’s standardize it to get
Z := (log S_T – mean)/std ~ N(0,1)
Pr = Pr (Z > (log K – mean)/std ) = Pr (Z < (mean – log k)/std) = N( (mean – log k)/std) = N(d2)