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In the Black-Scholes model, suppose a stock pays continuous dividend yield $q$, a claim pays cash flow $f(t)S_{t}dt$, and its terminal financial payoff is $c{S}_T$. If the discounted cash-flow integral is finite, its risk-neutral pricing value is the displayed expression. To derive it, substitute $V=S\varphi (t)$ into the Black-Scholes equation with claim dividends, including stock drift $(r-q)S$. This gives ${\varphi }^{\prime }-q\varphi =-f$, $\varphi (T)=c$, which an integrating factor solves. The interest rate cancels because all payments are proportional to the same stock.

A self-financing replicating strategy must fund both the claim's price change and its outgoing cash flow:Apply Itô's formula to $V(t,S_t)$. Matching the diffusion gives $\Delta =V_s$. Matching drift and eliminating $\beta B=V-s{V}_s$ gives the Black-Scholes equation with claim dividendsThe physical drift cancels because stock risk is hedged. The cash-flow term has a plus sign because it is a payment to the holder.

Use a self-financing portfolio which holds $\Delta =V_S$ shares and $V-S{V}_S$ in the bank account, withdrawing the claim's cash flow $Cdt$. A share's total gain includes the continuous dividend yield, so its gain is $dS+DSdt$. Matching the claim's total gain givesBy the Itô formula,The delta hedge cancels the Brownian motion term. Comparing the remaining drifts yields the Black-Scholes equation with claim dividends, with the stock's own dividend yield included:This gain calculation accounts for rebalancing through the self-financing portfolio condition. The physical expected-return parameter $\mu$ cancels, as required by risk-neutral pricing.

Take a non-dividend-paying stock with physical dynamics $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ and continuous-time bank account $B_t=e^{\rho t}$. Write $T=t_0$ and assume the candidate claim price $p(x,t)$ is $C^{2,1}$ before maturity, with enough growth control for the hedge to be admissible. The terminal condition is $p(x,T)=f(x)$.

The Itô formula gives the claim's price changeA self-financing portfolio worth $p$ with $\Delta$ shares has bank-account value $p-\Delta S_t$. Its gain isTo replicate the claim, the Brownian coefficients must agree, forcing the delta hedge $\Delta =p_x$. Equating the remaining drift coefficients cancels the physical drift $\mu$ and yieldsThe cancellation explains why the Black-Scholes equation contains the interest rate, not the physical stock drift. Conversely, if an admissible solution of this equation is given, holdings $p_x$ in the stock and $(p-x{p}_x)/B_t$ in the bank account have value $p$ and gains exactly $dp$. They are therefore a self-financing portfolio replicating the terminal payoff. Absence of arbitrage forces the claim to have this price. Merely decomposing the current value into arbitrary holdings would not justify self-financing; the gain identity and delta hedge are essential.

Now suppose the claim holder receives the cash rate $k(S_t,t)$ before $T$. The price $p$ is the ex-dividend value, so its total gain is $dp+kdt$. The replicating portfolio pays the same amounts out of its wealth, with no external injections. Thus its gain identity isAgain the Brownian coefficients force $\Delta =p_x$. Comparing drifts now gives the Black-Scholes equation with claim dividends:In particular the cash-flow source has a positive sign in the left-hand side. It is not an underlying-stock dividend yield and does not replace the drift $\rho x$ by a dividend-adjusted drift.

For another justification, under the risk-neutral measure $Q$, discount the claim's cum-dividend gain. The Itô formula shows thatexactly when the displayed equation holds. Subject to the usual integrability making this a true martingale, conditional expectation at maturity givesPositive interim payments therefore add positive value, which also checks the source sign. For instance, a constant payment rate $k_0$ and zero terminal payoff have price $k_0(1-e^{-\rho (T-t)})/\rho$, interpreted as $k_0(T-t)$ at $\rho =0$; substitution satisfies $p_t-\rho p+k_0=0$.