= One-period quadratic hedge
{title2=$\pi^*=V^{-1}\operatorname{Cov}(S_1,\xi)$}
= One-period least-squares hedging
{synonym}
A one-period quadratic hedge minimizes $\mathbb E[(\xi-\phi R-\pi\cdot S_1)^2]$ over deterministic holdings, where $R\ne0$ is the riskless gross return, $\mu=\mathbb E S_1$, and the <covariance matrix> $V$ is invertible. For a square-integrable claim, the unique solution is $\pi^*=V^{-1}\operatorname{Cov}(S_1,\xi)$ and $\phi^*=(\mathbb E\xi-\pi^*\cdot\mu)/R$. Centering separates the squared bias from the quadratic error. Completing the square gives the excess error as $(\pi-\pi^*)^TV(\pi-\pi^*)$ after the bias is minimized. Unlike <claim replication>, the residual need not vanish.
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