= Claim dividends proportional to the underlying stock
{title2=$V(S,t)=S[ce^{-q(T-t)}+\int_t^Te^{-q(u-t)}f(u)du]$}
In the <Black-Scholes model>, suppose a <stock> pays <continuous dividend yield> $q$, a claim pays cash flow $f(t)S_tdt$, and its terminal <financial payoff> is $cS_T$. If the discounted cash-flow integral is finite, its <risk-neutral pricing> value is the displayed expression. To derive it, substitute $V=S\phi(t)$ into the <Black-Scholes equation with claim dividends>, including stock drift $(r-q)S$. This gives $\phi'-q\phi=-f$, $\phi(T)=c$, which an <integrating factor> solves. The <interest rate> cancels because all payments are proportional to the same <stock>.
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