= Solution
The <Brownian martingale representation theorem> argument in part (a) also replicates the bounded put payoff. The discounted <stock> is a true <martingale>, so the elementary terminal payoff identity yields <put-call parity>
$$
\boxed{P(T,K)=C(T,K)-S_0+Ke^{-rT}.}
$$
Set $H(T,K)=-S_0+Ke^{-rT}$. Then $H_T=-rKe^{-rT}$, $H_K=e^{-rT}$, and $H_{KK}=0$. Thus
$$
H_T=\frac12K^2\sigma(T,K)^2H_{KK}-rKH_K.
$$
Linearity of the <Dupire equation> and $P=C+H$ give
$$
\boxed{P_T(T,K)=\frac12K^2\sigma(T,K)^2P_{KK}(T,K)-rKP_K(T,K).}
$$
The initial payoff is $P(0,K)=(K-S_0)^+$, and $P(T,0)=0$. Therefore calls and puts obey the same maturity-strike differential equation, with their respective initial and boundary data.
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