Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 27I / b

With continuous dividend yield $D$, it is the discounted total gain from the stock, including dividends, that must be a martingale. The ex-dividend stock drift under the risk-neutral measure is therefore $r-D$. Replace $r-{\sigma }^2/2$ in the exponential representation by $r-D-{\sigma }^2/2$ and repeat the chain rule. The discount rate remains $r$, givingThe terminal payoff condition is unchanged.