= Black-Scholes value equation needs delta-compatible holdings
{c}
{title2=$g=p_x,\quad hB=p-xp_x$}
A smooth value function satisfying the <Black-Scholes equation> admits a <self-financing portfolio> with the displayed <delta hedge>. An arbitrary decomposition $p=xg+Bh$ need not be self-financing. For example $p=0$, $g=1$, $h=-x/B$ solves the value equation but has portfolio gains $dS-\rho S\,dt$, which are not zero when volatility is positive. For specified holdings, self-financing is equivalent to the value equation together with $g=p_x$.
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