Analytical Greeks for European Options

1. Why is $\Delta_{\text{call}} = N(d_1)$ rather than $N(d_2)$, given that $N(d_2) = \mathbb{Q}(S_T > K)$ is the risk-neutral probability of the call expiring in the money?

2. By put-call parity, which of the following pairs of Greeks are identical for European calls and puts with the same strike, maturity, and underlying?

3. The Black-Scholes PDE written in Greek notation is $\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta - rC = 0$. For a delta-hedged call with zero net delta, the daily P&L from a spot move $dS$ and a theta accrual $dt$ is:

4. Vanna is defined as $\partial^2 C / \partial S\,\partial\sigma$. Which of the following is the correct expression?

5. For a deep in-the-money European call near expiry ($\tau \to 0$, $S \gg K$), Gamma approaches zero while Delta approaches 1.

6. A long OTM call position has positive vanna ($-n(d_1)d_2/\sigma > 0$ since $d_2 < 0$). In equity markets, spot and implied vol are negatively correlated (the leverage effect). What is the sign of the vanna contribution to the daily P&L of this position on a day when the market falls?