In the context of the model introduced in Kyle [1147] and discussed in Sect. 10.2, let the couple \((X, P)\) represent the linear equilibrium, as stated in Proposition 10.3. Prove the following claims:

(i) Letting \(\tilde{p}\) denote the equilibrium price, it holds that

\[\operatorname{Var}(\tilde{d} \mid \tilde{p})=\frac{\sigma_{d}^{2}}{2}\]

(ii) The optimal profits of the insider trader (i.e., the profits associated to his optimal demand \(X(d)\) ), conditionally on the observation of the private signal \(\tilde{d}=d\), are given by

\[\frac{(d-\bar{d})^{2}}{4 \lambda}\]

Ex-ante (i.e., before the observation of the realization of \(\tilde{d}\) ), the expected profits of the insider trader are equal to \(\sigma_{d}^{2} /(4 \lambda)\).