c) (3 p.) Note that

\phi(s)=(s^{2}-4Ks+4K^{2})\mathbf{1}_{\{s\leq K\}}+s^{2}\mathbf{1}_{\{s\geq K\}}=s^{2}+4K(K-s)\mathbf{1}_{\{s\leq K\}}.

Hence,

\Pi(\phi(S_{T})) =e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_{T}^{2}]+4Kp(S_{0};T,K) =e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_{T}^{2}]+4K(e^{-rT}K-S_{0}+c(S_{0};T,K)),

where $p(S_{0};T,K)$ is the price of a standard put option and we applied the put-call parity (alternatively the same expression can be obtained by further re-writing the expression for $\phi$ ). The price of $c(S_{0};T,K)$ is obtained from a). Moreover,

\mathbb{E}^{\mathbb{Q}}\big[S_{T}^{2}\big]=\mathbb{E}^{\mathbb{Q}}\Bigg[\Big(S_{0}e^{(r-\sigma^{2}/2)T+\sigma\widetilde{W}_{T}}\Big)^{2}\Bigg]=S_{0}^{2}e^{(2r-\sigma^{2})T}\mathbb{E}^{\mathbb{Q}}\Big[e^{2\sigma\widetilde{W}_{T}}\Big]=S_{0}^{2}e^{(2r+\sigma^{2})T},

where $\widetilde{W}\sim^{\mathbb{Q}}\mathcal{N}(0,T)$ and the last equality was obtained by computing the expected value as in Question 4 on the exam from August 2019 (or simply applying the formula for the Moment Generating Function from the formula sheet).