What exactly is the relationship between BSM and risk-neutral probability?

All three are correct, just at different levels of abstraction. Let me untangle them.

Level 1 — $N(d_2)$ as a number:

For the BSM call formula $c = S_0 \cdot e^{-qT} \cdot N(d_1) - K \cdot e^{-rT} \cdot N(d_2)$, the number $N(d_2)$ is the risk-neutral probability that $S_T > K$. That is, it is the probability that the call expires in-the-money under a specific probability measure called $Q$.

So if $d_2 = 0.15$, then $N(0.15) \approx 0.5596$ means "the call has a 55.96% chance of finishing ITM under $Q$."

Level 2 — The risk-neutral measure $Q$ as a probability measure:

In probability theory, a measure is a way of assigning probabilities to events. The real world has a measure $P$ that reflects investors' actual beliefs about returns. The risk-neutral measure $Q$ is a different probability assignment in which:

• The stock's expected return equals the risk-free rate ($E^Q[S_T] = S_0 \cdot e^{(r-q) \cdot T}$)
• The variance of log returns equals $\sigma^2 T$ (same as in $P$)

$Q$ is constructed by tilting $P$ toward the risk-free rate. Under $Q$, every traded asset has the same expected return — $r$ — so it is called "risk-neutral" because no risk premium is built in. This is a mathematical construct, not a description of investor preferences.

Level 3 — Why pricing under $Q$ gives the right answer:

A famous theorem (Girsanov, fundamental theorem of asset pricing) says: in a frictionless market with no arbitrage, every derivative's price equals its expected payoff under $Q$, discounted at $r$.

Connecting back to BSM:

For a European call with payoff $\max(S_T - K, 0)$:

If you compute that expectation under the lognormal distribution implied by GBM with drift $r$ and volatility $\sigma$, you get exactly:

The $N(d_2)$ inside is the $Q$-probability that $S_T > K$. The $N(d_1)$ inside is the same probability under a different but equivalent measure called the stock-numeraire measure $Q_S$.

Where students get confused:

• "Risk-neutral" does NOT mean investors are risk-neutral. Investors in the real world demand risk premia. $Q$ is a pricing tool, not a description of preferences.
• The probability the option expires ITM under the real-world measure $P$ is generally HIGHER than $N(d_2)$, because real-world expected returns are higher than $r$. $Q$ understates ITM probability, but it correctly prices the option.
• For a put, $N(-d_2)$ is the $Q$-probability of finishing ITM.

For the exam:

If the exam asks "the risk-neutral probability that the call expires in the money," the answer is $N(d_2)$, not $N(d_1)$. If it asks "the probability that the call expires ITM under the real-world measure," that is a curriculum nuance — most CFA questions give you risk-neutral as the assumed measure.