= Black-Scholes equation with claim dividends
{c}
{title2=$p_t+\rho xp_x+\tfrac12\sigma^2x^2p_{xx}-\rho p+k=0$}
If the claim holder receives a cash-flow rate $k(S_t,t)$ before maturity, the <self-financing> hedge matches the claim's cum-dividend gain $dp+kdt$. Its stock holding is the <option delta> $p_x$ and its bank-account value is $p-xp_x$. Comparing <Itô formula> with portfolio gains gives the displayed equation, with terminal value equal to the terminal payoff. The positive source term $k$ represents dividends paid by the claim; dividends on the underlying stock would instead alter the stock's pricing drift. <Risk-neutral pricing> includes both the discounted terminal payoff and the integral of discounted interim payments.
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