= Solution
{c}
Differentiate the <dividend> PDE with respect to $S$. The diffusion term contributes $\sigma^2S V_{SS}+\tfrac12\sigma^2S^2V_{SSS}$, and the stock-drift term contributes $(r-\delta)(V_S+SV_{SS})$. With $\Delta=V_S$, the coefficient of $\Delta$ is $(r-\delta)-r=-\delta$. Therefore
$$
\boxed{\Delta_t+(r-\delta+\sigma^2)S\Delta_S
+\tfrac12\sigma^2S^2\Delta_{SS}-\delta\Delta=0.}
$$
The terminal condition is $\Delta(T,S)=\Psi'(S)$ wherever the <contingent claim payoff> is differentiable, with an appropriate weak or limiting interpretation at <contingent claim payoff> kinks. In the next part the squared positive-part <contingent claim payoff> is continuously differentiable, so its terminal <option delta> has no jump.
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