Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 6 / a

Write $R=1+r$, $m=\mathbb{E}{\xi }_1$ and $c=\mathrm{Cov}(S_1,{\xi }_1)=\mathbb{E}[(S_1-\mu )({\xi }_1-m)]$. 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 $R\ne 0$.

The claim and asset prices are square integrable. Centering separates the bias from the stochastic error:For fixed $\pi$, the unique minimizing bank holding removes the bias. The invertible covariance matrix $V$ is positive definite, and completing the square givesConsequently the unique one-period quadratic hedge isIts minimum expected squared error is $\mathrm{Var}({\xi }_1)-c^T{V}^{-1}c$. This is a least-squares hedge, not necessarily exact claim replication.