(a) In the context of a single-period financial market with $n$ traded assets and a single riskless asset earning interest at rate $r$, what is an arbitrage? What is an equivalent martingale measure? Explain marginal utility pricing, and how it leads to an equivalent martingale measure.

(b) Consider the following single-period market with two assets. The first is a riskless bond, worth 1 at time 0 , and 1 at time 1 . The second is a share, worth 1 at time 0 and worth $S_{1}$ at time 1 , where $S_{1}$ is uniformly distributed on the interval $[0, a]$, where $a>0$. Under what condition on $a$ is this model arbitrage free? When it is, characterise the set $\mathcal{E}$ of equivalent martingale measures.

An agent with $C^{2}$ utility $U$ and with wealth $w$ at time 0 aims to pick the number $\theta$ of shares to hold so as to maximise his expected utility of wealth at time 1 . Show that he will choose $\theta$ to be positive if and only if $a>2$.

An option pays $\left(S_{1}-1\right)^{+}$at time 1 . Assuming that $a=2$, deduce that the agent's price for this option will be $1 / 4$, and show that the range of possible prices for this option as the pricing measure varies in $\mathcal{E}$ is the interval $\left(0, \frac{1}{2}\right)$.