Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 6 / c / Solution

For an ex-dividend stock, holding $\Delta$ shares earns both $\Delta dS$ and the dividend gain $\Delta \delta Sdt$. 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 $V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}-\delta S{V}_S$. Its value is $V-S{V}_S$, so absence of arbitrage requires this drift to equal $r(V-S{V}_S)$. Rearrangement gives the Black-Scholes equation with continuous stock dividends:With the given parameters,The physical drift $0.1$ cancels because the hedge removes the stock-price risk; the dividend payment remains in the stock's total gain.