= Minimum-norm one-period pricing weight
{title2=$\rho^*=R^{-1}+(S_0-R^{-1}\mu)^TV^{-1}(S_1-\mu)$}
The initial cost of the <one-period quadratic hedge> is $\mathbb E[\rho^*\xi]$. This pricing weight satisfies $\mathbb E[\rho^*R]=1$ and $\mathbb E[\rho^*S_1]=S_0$. Among square-integrable weights satisfying these equations it has the smallest squared norm:
$$
\mathbb E[(\rho^*)^2]=R^{-2}+(S_0-R^{-1}\mu)^TV^{-1}(S_0-R^{-1}\mu).
$$
For another admissible weight $\rho$, its difference from $\rho^*$ is orthogonal to $1$ and all coordinates of $S_1$, hence to $\rho^*$. The <Pythagorean theorem in an inner-product space> gives $\mathbb E[\rho^2]=\mathbb E[(\rho^*)^2]+\mathbb E[(\rho-\rho^*)^2]$. This weight can be signed; positivity, and hence interpretation as a <state-price density>, requires additional assumptions.
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