n1 := N(d1)

n2 := N(d2)

Note n1 and n2 are both about probability distributions, so they always assume some probability measure. By default, we operate under (not the physical measure but) the risk-neutral measure with the money-market account as the “if-needed, standby” numeraire.

- n2 is the implied probability of stock finishing above strike, implied from various live prices. RN measure.

- n1 is the same probability but implied under the share-measure. Therefore,

ST *N(d1) would be the weighted average payoff (i.e. expected payoff) of the asset-or-nothing call, under share-measure.

St * N(d1) would be the PV of the payoff, i.e.

*current price of asset-or-nothing call*. Note as soon as we talk about price, it is automatically measure-independent.
Remember n2 is between $0 and $1 so it reminds us of ... the binary call. I think this is the weighted average payoff of the binary call. RN measure. Therefore,

N(d2) exp(-Rdisc T) -- If we discount that weighted average payoff to Present Value, we get the

**current price of the binary call**. Note all prices are measure-independent.
N(d1) is also delta of the Vanilla call, measure-independent. Given call's delta, using PCP, we work out put's delta (always negative) = 1-N(d1) = -N(-d1)

The pdf N’(d1) appears in the gamma and vega formulas, measure-independent, i.e.

gamma = N' (d1) * some function of (S, t)

vega = N' (d1) * some other function of (S, t)

Notice we only put K (never S) in front of N(d2)

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