Consider a one-period market model with $d$ risky assets and one risk-free asset. Let $S_{t}$ denote the vector of prices of the risky assets at time $t \in\{0,1\}$ and let $r$ be the interest rate.

(a) What does it mean to say a portfolio $\varphi \in \mathbb{R}^{d}$ is an arbitrage for this market?

(b) An investor wishes to maximise their expected utility of time-1 wealth $X_{1}$ attainable by investing in the market with their time- 0 wealth $X_{0}=x$. The investor's utility function $U$ is increasing and concave. Show that, if there exists an optimal solution $X_{1}^{*}$ to the investor's expected utility maximisation problem, then the market has no arbitrage. [Assume that $U\left(X_{1}\right)$ is integrable for any attainable time-1 wealth $X_{1}$.]

(c) Now introduce a contingent claim with time-1 bounded payout $Y$. How does the investor in part (b) calculate an indifference bid price $\pi(Y)$ for the claim? Assuming each such claim has a unique indifference price, show that the map $Y \mapsto \pi(Y)$ is concave. [Assume that any relevant utility maximisation problem that you consider has an optimal solution.]

(d) Consider a contingent claim with time-1 bounded payout $Y$. Let $I \subseteq \mathbb{R}$ be the set of initial no-arbitrage prices for the claim; that is, the set $I$ consists of all $p$ such that the market augmented with the contingent claim with time- 0 price $p$ has no arbitrage. Show that $\pi(Y) \leqslant \sup \{p \in I\}$. [Assume that any relevant utility maximisation problem that you consider has an optimal solution. You may use results from lectures without proof, such as the fundamental theorem of asset pricing or the existence of marginal utility prices, as long as they are clearly stated.]