Strict positive covariance with an increasing payoff ($\mathrm{Cov}(g(Z),Z)>0$)

If $Z$ is nondegenerate, $g$ is strictly increasing and the relevant products are integrable, an independent copy $Z^{\prime }$ gives $2\mathrm{Cov}(g(Z),Z)=\mathbb{E}[(g(Z)-g(Z^{\prime }))(Z-Z^{\prime })]$. The integrand is positive exactly when $Z\ne Z^{\prime }$, so the covariance is strictly positive. The same argument applies within each conditional law and yields a positive terminal stock hedge for an attainable increasing payoff.