= Put-call symmetry in the Black-Scholes model
{c}
For the <normalized Black-Scholes call function>, completing the square gives $F(v,m)=\Phi(d_1)-m\Phi(d_2)$, with $d_1=(-\log m+v/2)/\sqrt v$ and $d_2=d_1-\sqrt v$. The identities $d_1(v,1/m)=-d_2(v,m)$ and $\Phi(-x)=1-\Phi(x)$ imply
$$
F(v,m)=1-m+mF(v,1/m).
$$
For a <stock> with risk-neutral drift $r$, maturity difference $\tau$, and positive strike $K$, <put-call parity> then gives
$$
P_t(T,K)=Ke^{-r\tau}F\left(\sigma^2\tau,\frac{S_te^{r\tau}}K\right).
$$
The positive sign of $r\tau$ in the reciprocal argument is forced by taking the reciprocal of $Ke^{-r\tau}/S_t$. The identity includes $v=0$ by continuity and requires $m>0$.
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