Consider a market with two assets, a riskless bond and a risky stock, both of whose initial (time-0) prices are $B_{0}=1=S_{0}$. At time 1 , the price of the bond is a constant $B_{1}=R>0$ and the price of the stock $S_{1}$ is uniformly distributed on the interval $[0, C]$ where $C>R$ is a constant.

Describe the set of state price densities.

Consider a contingent claim whose payout at time 1 is given by $S_{1}^{2}$. Use the fundamental theorem of asset pricing to show that, if there is no arbitrage, the initial price of the claim is larger than $R$ and smaller than $C$.

Now consider an investor with initial wealth $X_{0}=1$, and assume $C=3 R$. The investor's goal is to maximize his expected utility of time-1 wealth $\mathbb{E} U\left[R+\pi\left(S_{1}-R\right)\right]$, where $U(x)=\sqrt{x}$. Show that the optimal number of shares of stock to hold is $\pi^{*}=1$.

What would be the investor's marginal utility price of the contingent claim described above?