Now we have the hedging instruments: $S_{1}$ , the asset price at time $1$ , and $N$ , the reinsurance company’s payment for all persons alive at time $1$ . Let

$Z:=\binom{S_{1}}{N}$

be the vector of hedging instruments. The insurance company’s liability at time $1$ is still $S_{1}N$ , so the optimal quadratic hedge is $h_{0}+\boldsymbol{h}^{\top}Z$ where $\boldsymbol{h}=\Sigma_{Z}^{-1}\Sigma_{S_{1}N,Z}$ and $h_{0}=\mathbb{E}[S_{1}N]-\boldsymbol{h}^{\top}\mathbb{E}[Z]$ . We compute

\Sigma_{Z}=\begin{pmatrix}\mathrm{Var}(S_{1})&\mathrm{Cov}(S_{1},N)\\ \mathrm{Cov}(S_{1},N)&\mathrm{Var}(N)\end{pmatrix}=\begin{pmatrix}\mathrm{Var}(S_{1})&0\\ 0&\mathrm{Var}(N)\end{pmatrix}\,,

by independence of $S_{1}$ and $N$ , and

\Sigma_{S_{1}N,Z}=\begin{pmatrix}\mathrm{Cov}(S_{1}N,S_{1})\\ \mathrm{Cov}(S_{1}N,N)\end{pmatrix}=\begin{pmatrix}\mathbb{E}[N]\mathrm{Var}(S_{1})\\ \mathbb{E}[S_{1}]\mathrm{Var}(N)\end{pmatrix}\,,

where the last equality comes from computations similar to those of the first question. Therefore

$\boldsymbol{h}=\Sigma_{Z}^{-1}\Sigma_{NS_{1},Z}$

\begin{array}{l} = \left( \begin{array}{cc} \operatorname{Var}(S_1)^{-1} & 0 \\ 0 & \operatorname{Var}(N)^{-1} \end{array} \right) \left( \begin{array}{c} \mathbb{E}[N] \operatorname{Var}(S_1) \\ \mathbb{E}[S_1] \operatorname{Var}(N) \end{array} \right) \\ = \left( \begin{array}{c} \mathbb{E}[N] \\ \mathbb{E}[S_1] \end{array} \right) \end{array}

and

\begin{array}{l} h_0 = \mathbb{E}[S_1 N] - (\mathbb{E}[N], \mathbb{E}[S_1]) \left( \begin{array}{c} \mathbb{E}[S_1] \\ \mathbb{E}[N] \end{array} \right) \\ = \mathbb{E}[N] \mathbb{E}[S_1] - \mathbb{E}[N] \mathbb{E}[S_1] - \mathbb{E}[S_1] \mathbb{E}[N] \\ = -\mathbb{E}[S_1] \mathbb{E}[N]. \end{array}

We then compute

\mathbb{E}[S_1] = \mathbb{E}[S_0 e^{\ln(S_1 / S_0)}] = S_0 e^{\mu + \frac{\sigma^2}{2}},

so we get $\boldsymbol{h} = (np, S_0 e^{\mu + \frac{\sigma^2}{2}})^\top$ and $h_0 = -np S_0 e^{\mu + \frac{\sigma^2}{2}}$ . We conclude that the optimal quadratic hedge is a long position in $np$ units of the traded asset, a long position in $S_0 e^{\mu + \frac{\sigma^2}{2}}$ units of the reinsurance company's asset, and a short position in $np S_0 e^{\mu + \frac{\sigma^2}{2}}$ units of the zero-coupon bond.