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Take $r>0$, $\sigma >0$ and proportional continuous dividend yield $\delta \ge 0$, the usual perpetual-option setting. Write $L$ for the prescribed downward exercise trigger, with $0<L<X$. Above the trigger there is no dependence on calendar time, so the Black-Scholes equation with continuous stock dividends reduces toSubstitute $V=S^{\beta }$. The characteristic equation and its roots areSince the root product is $-2r/{\sigma }^2<0$, ${\beta }_{-}<0<{\beta }_{+}$. The boundary condition $V(S)\to 0$ as $S\to \infty$ excludes the positive-power solution. Value matching at the trigger requires $V(L)=X-L$, giving the fixed-rule priceThe lower branch means exercise immediately if the starting price is already below the trigger. Above it, the formula also equals ${\mathbb{E}}_Q[e^{-r{\tau }_L}(X-L)1_{\{{\tau }_L<\infty \}}]$, where ${\tau }_L$ is the first passage to $L$. To verify this expectation, apply Itô formula to the bounded stopped discounted solution on $S\ge L$; the generator equation makes it a martingale before the hit, and the residual term on surviving paths vanishes as the time horizon tends to infinity because $r>0$. A fixed trigger is a specified exercise rule, not yet the optimized American price.

Assume $\sigma >0$, $x>0$ and sufficient $C^{2,1}$ regularity to use the Itô formula. Under the physical measure write $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ and let the bank account be $R_t=e^{\rho t}$. If $V_t=f(S_t,t)$ is a self-financing portfolio with ${\Delta }_t$ stock units and ${\beta }_t$ bank-account units, thenThe Itô formula gives $df=(f_t+\mu S_t{f}_x+\frac{1}{2}{\sigma }^2{S}_t^2{f}_{xx})dt+\sigma S_t{f}_{x}d{W}_t$. Equating diffusion terms forces ${\Delta }_t=f_x(S_t,t)$; equating drifts and eliminating ${\beta }_t{R}_t=f-S_t{f}_x$ gives the Black-Scholes equationConversely, if this equation holds, take $\Delta =f_x$ and $\beta =(f-x{f}_x)/R$. The same comparison shows $df=\Delta dS+\beta dR$, so the constructed portfolio is self-financing. This is a local replication equivalence for smooth functions on the state space; global admissibility and a terminal condition specify a unique economically admissible claim price.

With continuous stock-dividend yield $\theta$, holding ${\Delta }_t$ stock units earns $\theta {\Delta }_t{S}_{t}dt$ in addition to their price changes. A self-financing portfolio reinvests that income, soThe same diffusion comparison gives $\Delta =f_x$, and the drift comparison yields the Black-Scholes equation with continuous stock dividendsUnder the risk-neutral measure, the ex-dividend stock consequently has drift $\rho -\theta$, while cash is discounted at $\rho$.

For the last claim, distinguish the terminal payoff function $F$ from a time-dependent price. Put $\tau =t_0-t$. The price with yield $\theta$ is $q(x,t)=e^{-\rho \tau }\mathbb{E}[F(x{e}^{(\rho -\theta -{\sigma }^2/2)\tau +\sigma \sqrt{\tau }Z})]$, with $Z$ a standard normal random variable. In a no-dividend model with interest rate $\rho -\theta$, the stock-transition law is identical but its discount factor is $e^{-(\rho -\theta )\tau }$. Thus, whenever the payoff expectation is finite, the dividend-yield discount shift givesEquivalently substitute this expression in the dividend equation: $q_t=e^{-\theta \tau }(p_t+\theta p)$, and the equation reduces exactly to the no-dividend Black-Scholes equation with rate $\rho -\theta$, with the same terminal data.

A self-financing portfolio changes its holdings using only its existing wealth. If it holds $g_t$ shares and $h_t$ units of a traded bond or bank account $B_t$, its value $V_t=g_t{S}_t+h_t{B}_t$ obeys the gain identityThe changes in holdings do not add or withdraw capital. Set $T=t_0$, $B_t=e^{-\rho (T-t)}$, so $d{B}_t=\rho B_{t}dt$, and let $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$. Write $V_t=p(S_t,t)$ and suppress the arguments of the smooth holdings $g,h$.

Differentiate $p=xg+B_{t}h$:The Itô formula minus the proposed portfolio gains givesUniqueness of the local-martingale and finite-variation decomposition shows that self-financing requires $p_x=g$. Since $S_t$ has full support on $(0,\infty )$ for each $t>0$, smoothness gives the corresponding functional identity for every positive $x$, and continuity extends it to the interval endpoints. ThusDifferentiating this identity once more in $x$ gives $g_x+x{g}_{xx}+B_t{h}_{xx}=0$, so $p_{xx}=g_x$. Substitute both identities into the drift condition to obtainConversely these equations give $p_x=g$, $p_{xx}=g_x$ and zero drift and diffusion in the displayed gain difference. They therefore prove the claimed self-financing conditions for smooth stock and bond holdings in both directions.

Using $B_{t}h=p-xg$ in the gain identity and setting $g=p_x$, self-financing impliesThis is the Black-Scholes equation, independent of the physical drift $\mu$. Conversely, if a smooth $p$ satisfies it, define its delta hedge byThe preceding Itô formula verifies that these holdings are self-financing.

For prescribed arbitrary holdings, the value equation alone is insufficient. The corrected equivalence is the Black-Scholes value equation needs delta-compatible holdings: a specified portfolio is self-financing if and only if its value solves the equation and $g=p_x$. Indeed $p=0$, $g=1$, $h=-x/B_t$ satisfies the value equation but its proposed gains are $d{S}_t-\rho S_{t}dt$, which have nonzero Brownian part while $dp=0$. Thus the printed final equivalence is valid as an existence assertion for the delta hedge, not for every decomposition of the same value. No change to the valid two holdings equations is needed.

With continuous underlying stock dividends, the stock's gain is $d{S}_t+\theta S_{t}dt$, so self-financing becomes $d{V}_t=g_t(d{S}_t+\theta S_{t}dt)+h_{t}d{B}_t$. Under the risk-neutral measure the ex-dividend stock drift is $(\rho -\theta )S_t$. Matching its diffusion still gives $g=p_x$, and matching its gain drift gives the Black-Scholes equation with continuous stock dividendsOnly the coefficient of $x{p}_x$ changes. The $-\rho p$ discount term remains, because the claim is financed at the bank rate; there is no separate source term for a dividend paid by the claim itself. The dividends here are paid by the underlying stock.

Substitute the nonzero product $V=f(S)g(t)$ into the Black-Scholes equation with continuous stock dividends and divide by $fg$ on intervals where it is nonzero. The result isThe left side depends only on time and the right side only on spot, so both equal a constant $\lambda$. ThusThese are the separated solutions of the Black-Scholes equation. The zero solution is included separately, so division loses no relevant case.

For the unheaded parity request, the terminal difference of the power-call and power-put payoffs is $S_T^{\beta }-K^{\beta }$. Under the risk-neutral measure, $S_T=S\exp [(r-q-{\sigma }^2/2)\tau +\sigma (W_T-W_t)]$, where $\tau =T-t$. The Gaussian exponential moment gives ${\mathbb{E}}_Q[S_T^{\beta }\mid S_t=S]=S^{\beta }{e}^{{\lambda }_{\beta }\tau }$ with ${\lambda }_{\beta }=\beta (r-q)+{\sigma }^2\beta (\beta -1)/2$. Discounting the difference proves power put-call parity:For $\beta =1$ this is the familiar dividend-adjusted put-call parity, $S{e}^{-q\tau }-K{e}^{-r\tau }$.

In the Black-Scholes equation with continuous stock dividends, separation of variables gives $g^{\prime }=(r-\lambda )g$ and ${\sigma }^2{S}^2{f}^{\prime \prime }/2+(r-q)S{f}^{\prime }-\lambda f=0$. Substituting $f=S^{\alpha }$ gives the displayed eigenvalue. A stationary solution uses the two roots of ${\sigma }^2\alpha (\alpha -1)/2+(r-q)\alpha -r=0$. For positive volatility and $r>0$, their product is negative, so they are distinct real roots.