Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 28J / b / 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 givesScaling both positive arguments by $\alpha$ leaves $\log (S_0/K)$ and both $d$ values unchanged, while multiplying every price prefactor by $\alpha$. ThusThese are Put-call symmetry in the Black-Scholes model and degree-one homogeneity.