 Minimum-norm one-period pricing weight

The initial cost of the one-period quadratic hedge is $\mathbb{E}[{\rho }^{\ast }\xi ]$. This pricing weight satisfies $\mathbb{E}[{\rho }^{\ast }R]=1$ and $\mathbb{E}[{\rho }^{\ast }{S}_1]=S_0$. Among square-integrable weights satisfying these equations it has the smallest squared norm:For another admissible weight $\rho$, its difference from ${\rho }^{\ast }$ is orthogonal to $1$ and all coordinates of $S_1$, hence to ${\rho }^{\ast }$. The Pythagorean theorem in an inner-product space gives $\mathbb{E}[{\rho }^2]=\mathbb{E}[({\rho }^{\ast }{)}^2]+\mathbb{E}[(\rho -{\rho }^{\ast }{)}^2]$. This weight can be signed; positivity, and hence interpretation as a state-price density, requires additional assumptions.