 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 paritySet $H(T,K)=-S_0+K{e}^{-rT}$. Then $H_T=-rK{e}^{-rT}$, $H_K=e^{-rT}$, and $H_{KK}=0$. ThusLinearity of the Dupire equation and $P=C+H$ giveThe 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.