= Gaussian quadratic hedge
{c}
{title2=$\pi^*=\mathbb E[\nabla g(S_1)]$}
If $S_1$ has a <multivariate normal distribution> with mean $\mu$ and invertible covariance $V$, and $g$ has bounded gradient, <Gaussian integration by parts> gives $\operatorname{Cov}(S_1,g(S_1))=V\mathbb E[\nabla g(S_1)]$. Consequently the risky holdings of its <one-period quadratic hedge> equal the expected payoff gradient. The initial cost is $R^{-1}\mathbb E[g(S_1)]+(S_0-R^{-1}\mu)\cdot\mathbb E[\nabla g(S_1)]$. Bounded gradient ensures linear growth and square integrability, even if the payoff is not bounded.
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