An agent has expected-utility preferences over his possible wealth at time 1 ; that is, the wealth $Z$ is preferred to wealth $Z^{\prime}$ if and only if $E U(Z) \geqslant E U\left(Z^{\prime}\right)$, where the function $U: \mathbb{R} \rightarrow \mathbb{R}$ is strictly concave and twice continuously differentiable. The agent can trade in a market, with the time-1 value of his portfolio lying in an affine space $\mathcal{A}$ of random variables. Assuming cash can be held between time 0 and time 1 , define the agent's time-0 utility indifference price $\pi(Y)$ for a contingent claim with time-1 value $Y$. Assuming any regularity conditions you may require, prove that the map $Y \mapsto \pi(Y)$ is concave.

Consider a market with two claims with time-1 values $X$ and $Y$. Their joint distribution is

$$\left(\begin{array}{l} X \\ Y \end{array}\right) \sim N\left(\left(\begin{array}{l} \mu_{X} \\ \mu_{Y} \end{array}\right),\left(\begin{array}{ll} V_{X X} & V_{X Y} \\ V_{Y X} & V_{Y Y} \end{array}\right)\right) .$$

At time 0 , arbitrary quantities of the claim $X$ can be bought at price $p$, but $Y$ is not marketed. Derive an explicit expression for $\pi(Y)$ in the case where

$$U(x)=-\exp (-\gamma x)$$

where $\gamma>0$ is a given constant.