= Solution
{c}
For an ex-dividend stock, holding $\Delta$ shares earns both $\Delta\,dS$ and the dividend gain $\Delta\delta S\,dt$. Freeze $\Delta=V_S$ locally in a self-financing hedge that is long the claim and short these shares. <Itô's formula> cancels its Brownian exposure, leaving drift $V_t+\tfrac12\sigma^2S^2V_{SS}-\delta SV_S$. Its value is $V-SV_S$, so absence of arbitrage requires this drift to equal $r(V-SV_S)$. Rearrangement gives the <Black-Scholes equation with continuous stock dividends>:
$$
V_t+(r-\delta)SV_S+\tfrac12\sigma^2S^2V_{SS}-rV=0.
$$
With the given parameters,
$$
\boxed{V_t+0.04SV_S+0.02S^2V_{SS}-0.05V=0,\qquad V(T,S)=\Psi(S).}
$$
The physical drift $0.1$ cancels because the hedge removes the stock-price risk; the dividend payment remains in the stock's total gain.
Back to article page
