 Solution

LetPart (c) has $d(y)=\overline{y}-y$. Comparing the optimized values with and without the contract, $V_0^{\mathrm{swap}}=-e^{-\gamma x-T\kappa -\gamma T(\overline{y}-y)}$ and $V_0=-e^{-\gamma x-T\kappa }$, shows that $y>\overline{y}$ makes the swap value strictly more negative. Thus The investor prefers not to enter when $y>\overline{y}$. He is indifferent at equality and prefers entry for a smaller payment.

The periodic utility indifference payment for Gaussian income equals the mean income less the cost of hedging its correlated component and the exponential-utility penalty for the residual unhedgeable variance. Comparing unoptimized strategies would not justify this price threshold.