Consider a market with two securities, one of them risk free with rate of return \(R\), and three future states. Suppose that the risky security has a present value \(S_{1}\) and can take three future values \(S_{1}^{1}>S_{1}^{2}>S_{1}^{3}\).

i) Give an "if and only if" condition for no arbitrage.

ii) Show an example of an option \(O\) that can not be replicated (i.e.: there is no hedging strategy). Why is it possible to find \(O\) ?

iii) Show that the present value of \(O\) is not uniquely determined. More precisely, verify that the set of possible present values for \(O\) is an interval.