(i) In the context of a single-period financial market with $d$ traded assets, what is an arbitrage? What is an equivalent martingale measure?

A simple single-period financial market contains two assets, $S^{0}$ (a bond), and $S^{1}$ (a share). The period can be good, bad, or indifferent, with probabilities $1 / 3$ each. At the beginning of the period, time 0 , both assets are worth 1 , i.e.

$$S_{0}^{0}=1=S_{0}^{1}$$

and at the end of the period, time 1 , the share is worth

$$S_{1}^{1}= \begin{cases}a & \text { if the period was bad, } \\ b & \text { if the period was indifferent } \\ c & \text { if the period was good }\end{cases}$$

where $a<b<c$. The bond is always worth 1 at the end of the period. Show that there is no arbitrage in this market if and only if $a<1<c$.

(ii) An agent with $C^{2}$ strictly increasing strictly concave utility $U$ has wealth $w_{0}$ at time 0 , and wishes to invest his wealth in shares and bonds so as to maximise his expected utility of wealth at time 1 . Explain how the solution to his optimisation problem generates an equivalent martingale measure.

Assume now that $a=3 / 4, b=1$, and $c=3 / 2$. Characterise all equivalent martingale measures for this problem. Characterise all equivalent martingale measures which arise as solutions of an agent's optimisation problem.

Calculate the largest and smallest possible prices for a European call option with strike 1 and expiry 1, as the pricing measure ranges over all equivalent martingale measures. Calculate the corresponding bounds when the pricing measure is restricted to the set arising from expected-utility-maximising agents' optimisation problems.