 One-period quadratic hedge

A one-period quadratic hedge minimizes $\mathbb{E}[(\xi -\varphi R-\pi \cdot S_1{)}^2]$ over deterministic holdings, where $R\ne 0$ 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 ${\pi }^{\ast }=V^{-1}\mathrm{Cov}(S_1,\xi )$ and ${\varphi }^{\ast }=(\mathbb{E}\xi -{\pi }^{\ast }\cdot \mu )/R$. Centering separates the squared bias from the quadratic error. Completing the square gives the excess error as $(\pi -{\pi }^{\ast }{)}^{T}V(\pi -{\pi }^{\ast })$ after the bias is minimized. Unlike claim replication, the residual need not vanish.