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A constant-proportion portfolio holds a fixed fraction $\gamma$ of its portfolio wealth in the stock. In the Black-Scholes model with bank rate $\rho$, initial wealth $V_0$, and no consumption, its value isThe stock holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma )V_t$ is invested in the bank account. The Itô formula verifies $d{V}_t/V_t=[\rho +\gamma (\mu -\rho )]dt+\gamma \sigma d{W}_t$ when the physical stock drift is $\mu$.

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.