= Solution
Write $R=1+r$, $m=\mathbb E\xi_1$ and $c=\operatorname{Cov}(S_1,\xi_1)=\mathbb E[(S_1-\mu)(\xi_1-m)]$. The riskless gross return must be nonzero for the requested formulas; in the usual positive-bank-account model $R>0$. If $R=0$, the bank holding cannot affect the terminal payoff, and the later moment equation for pricing $B_0=1$ is impossible. We use the intended nondegenerate case $R\ne0$.
The claim and asset prices are square integrable. Centering separates the bias from the stochastic error:
$$
\mathbb E[(\xi_1-\phi R-\pi\cdot S_1)^2]
=(m-\phi R-\pi\cdot\mu)^2
+\operatorname{Var}(\xi_1)-2\pi^Tc+\pi^TV\pi.
$$
For fixed $\pi$, the unique minimizing bank holding removes the bias. The invertible <covariance matrix> $V$ is positive definite, and completing the square gives
$$
\pi^TV\pi-2\pi^Tc
=(\pi-V^{-1}c)^TV(\pi-V^{-1}c)-c^TV^{-1}c.
$$
Consequently the unique <one-period quadratic hedge> is
$$
\boxed{\pi^*=V^{-1}c,\qquad \phi^*=\frac{m-(\pi^*)^T\mu}{R}.}
$$
Its minimum expected squared error is $\operatorname{Var}(\xi_1)-c^TV^{-1}c$. This is a least-squares hedge, not necessarily exact <claim replication>.
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