Derivation: Analytic Greeks from the Black-Scholes Formula

The exam hands you the Black–Scholes call and put formulae and asks you to derive the Greeks by differentiation. The differentiation looks messy — $d_1$ and $d_2$ both depend on $S$, $\sigma$, $r$ — but every derivation collapses once you prove one identity that makes all the $\phi$-terms cancel. We prove that identity first, then derive delta, gamma, vega and rho in sequence. Work with the dividend-yield-$\delta$ forms throughout; set $\delta = 0$ to recover the mock-exam case.