= Derivative-weighted call delta martingale
{title2=$M_t=Z_t\pi_t$}
Suppose $a'$ is bounded and let $dZ_t=Z_ta'(S_t)dW_t$, $Z_0=1$. The <Novikov condition> makes $Z$ a strictly positive true <martingale>. For $\pi_t=U(t,S_t)$ solving the <call delta equation for a driftless local volatility diffusion>, the product formula gives
$$
d(Z_t\pi_t)=Z_t\bigl(a(S_t)U_S(t,S_t)+a'(S_t)U(t,S_t)\bigr)dW_t.
$$
The two drift terms cancel through <quadratic covariation>. Thus $Z\pi$ is a <local martingale>. If it is a true <martingale> on the closed horizon, its terminal value is $Z_T\mathbf1_{\{S_T\ge K\}}$, so
$$
\pi_t=\frac{\mathbb E[Z_T\mathbf1_{\{S_T\ge K\}}\mid\mathcal F_t]}{Z_t},\qquad0\le\pi_t\le1.
$$
The upper bound uses $\mathbb E[Z_T\mid\mathcal F_t]=Z_t$. Equivalently, the <option delta> is the terminal-exceedance probability under the measure with density $Z_T$.
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