Fixed-capital quadratic hedge in a one-period market ($\theta (v)=G^{-1}[\mathrm{Cov}(Y,H)+(\mathbb{E}H-v)\mathbb{E}Y]$)

For a discounted contingent claim $H$ and gain vector $Y$, fix initial capital $v$ and minimize $\mathbb{E}(H-v-{\theta }^{T}Y{)}^2$. Differentiation gives the least-squares normal equations $G\theta =\mathbb{E}[Y(H-v)]$, where $G=\mathbb{E}[Y{Y}^T]$ is assumed invertible. This gives the displayed hedge. Optimizing capital as well instead centers the gains and uses their covariance matrix: ${\theta }^{\ast }=\mathrm{Cov}(Y{)}^{-1}\mathrm{Cov}(Y,H)$ and $v^{\ast }=\mathbb{E}H-({\theta }^{\ast }{)}^T\mathbb{E}Y$. Fixing the initial capital is therefore a different quadratic problem.