Dividend-yield discount shift
= Dividend-yield discount shift
{title2=$q(x,t)=e^{-d(T-t)}p(x,t;r-d)$}
With constant continuous stock-dividend yield $d$, the <risk-neutral measure> gives stock drift $r-d$ but cash claims are discounted at $r$. The same terminal payoff in a no-dividend <Black-Scholes model> with interest rate $r-d$ has the identical stock-transition law but a different discount factor. Their prices therefore obey the displayed identity, under the usual payoff <integrability> conditions.