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Write $T=t_0$ and $b(t)=e^{-\rho (T-t)}$ for the price of the unit zero-coupon bond maturing at $T$, so $db=\rho bdt$. Let the stock satisfy $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ with $\sigma >0$. Assume the value function $f(x,t)=xg(x,t)+b(t)h(x,t)$ is $C^{2,1}$ and the holdings have enough smoothness for the calculations below. The self-financing portfolio condition is the gain identityThe Itô formula expresses the left side asMatching the diffusion terms gives $f_x=g$, or $x{g}_x+b{h}_x=0$. Differentiating $f_x=g$ gives $f_{xx}=g_x$. Also $f_t=x{g}_t+b{h}_t+\rho bh$. Matching the remaining drift terms consequently gives the two necessary conditionsThey are sufficient because substituting them into the Itô formula restores exactly the gain identity. Necessity first holds along the stock process; its positive lognormal density for every $t>0$, together with continuity of the coefficient functions, gives the displayed identities for all $x>0$ and $0<t<T$, and continuity extends them to the endpoints.

Equivalently, the smooth value of any self-financing portfolio satisfies the Black-Scholes equationIndeed, the gain identity after imposing $g=f_x$ has drift $\mu x{f}_x+\rho (f-x{f}_x)$, which gives this equation. Conversely, if $f$ satisfies this equation, takeThe Itô formula then verifies that these holdings form a self-financing portfolio of value $f$. This converse asserts existence of the correct holdings. It does not certify an arbitrary decomposition of $f$ into stock and bond holdings: for example, $f=0$, $g=1$, $h=-x/b$ solves the value equation but fails $g=f_x$. The Black-Scholes value equation needs delta-compatible holdings precisely for this reason.

For a constant-proportion portfolio, the stock value must satisfy $xg=\gamma f$. Since $g=f_x$, integration of $x{f}_x=\gamma f$ gives $f(x,t)=C(t)x^{\gamma }$. Substituting into the Black-Scholes equation givesThus, for initial portfolio wealth $V_0$ and stock price $S_0$, the complete explicit solution isHere $g_t$ counts shares and $h_t$ counts maturity-$T$ bonds; the initial cost is $V_0$. Directly, the self-financing gain equation is $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$, which also verifies the displayed value. Constant proportions concern the values of the holdings, not constant numbers of shares or bonds. The nondegenerate volatility assumption is necessary for the preceding diffusion matching; with $\sigma =0$, one only obtains a gain condition along the deterministic stock path.

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.

Write $T=t_0$ and $B_t=e^{-\rho (T-t)}$ for the risk-free asset paying one at $T$. Under the physical measure, $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$, with $\sigma >0$, while $d{B}_t=\rho B_{t}dt$. A self-financing portfolio with predictable stock and bond holdings $g_t,h_t$ has value $V_t=g_t{S}_t+h_t{B}_t$ and gainsRebalancing therefore adds no outside money and withdraws none. This gains equation, rather than merely the value decomposition, defines self-financing.

Let $p(x,t)=xg(x,t)+B_{t}h(x,t)$. The Itô formula givesThe prescribed portfolio gains instead have diffusion coefficient $\sigma S_{t}g$ and drift $\mu S_{t}g+\rho B_{t}h$. Equality of diffusion coefficients requires $p_x=g$, equivalentlyUnder this condition, $p_{xx}=g_x$. Equality of drifts is then $p_t+\frac{1}{2}{\sigma }^2{x}^2{g}_x=\rho B_{t}h$. Since $p_t=x{g}_t+B_t{h}_t+\rho B_{t}h$, this becomesThese conditions are also sufficient by the same Itô formula. Necessity initially holds along the stock process. Its positive transition density on $(0,\infty )$ for every positive time, together with continuity of the coefficient functions, extends the identities to all $x>0$ and then to the time endpoints by continuity.

For a self-financing portfolio, substitute $g=p_x$ and $B_{t}h=p-x{p}_x$ into the drift identity to obtain the Black-Scholes equationConversely, a smooth solution of this equation gives a self-financing portfolio by choosing the delta hedgeIts value is $p$, and both coefficient identities follow directly.

The printed deduction needs this holdings qualification: the Black-Scholes value equation needs delta-compatible holdings. A value function alone cannot make an arbitrary prescribed decomposition self-financing. As a counterexample, take $p=0$, $g=1$, $h=-x/B_t$. The value function solves the Black-Scholes equation, but this zero-value portfolio has gains $d{S}_t-\rho S_{t}dt$, which contain the nonzero term $\sigma S_{t}d{W}_t$. Thus for the originally specified $g,h$, the exact equivalence is the pricing equation together with $g=p_x$, not the pricing equation alone.

For a claim paying the holder at rate $k(S_t)$, let $p$ denote its remaining, ex-dividend value. A replicating portfolio pays out $k(S_t)dt$, so its net wealth equation is $dp=gdS+hdB-k(S_t)dt$. The delta hedge is still $g=p_x$, but the Black-Scholes equation becomesIf there is a terminal payoff $f$, set $p(x,T)=f(x)$; with no terminal payment set it to zero. The corresponding risk-neutral valuation isEach intervening payment is discounted from its own payment time.