= Solution
At zero interest, interchanging spot and strike gives $d_1(K,S_0)=-d_2(S_0,K)$ and $d_2(K,S_0)=-d_1(S_0,K)$. Substitution into the put formula therefore gives
$$
\boxed{C(S_0,K)=P(K,S_0)}.
$$
Scaling both positive arguments by $\alpha$ leaves $\log(S_0/K)$ and both $d$ values unchanged, while multiplying every price prefactor by $\alpha$. Thus
$$
\boxed{C(\alpha S_0,\alpha K)=\alpha C(S_0,K)}.
$$
These are <Put-call symmetry in the Black-Scholes model> and degree-one homogeneity.
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