In a single$-$period market, there are two securities initially priced at $S_0^1=$$$1,S_0^2=$$$4.$ For period one, there are three possible world states $=\{{}_1,{}_2,{}_3\},$ where the securities are expected to have the corresponding values:

$S_1^1({}_1)=$$$0,S_1^2({}_1)=$$$1$

$S_1^1({}_2)=$$$1,S_1^2({}_2)=$$$3$

$S_1^1({}_2)=$$$1,S_1^2({}_3)=$$$5$

a$.$ Is this market arbitrage free? Explain your reasoning.

b$.$ A bank is planning to introduce a risk$-$free bond in this market. The bond has an initial value of $$3,$ and an interest rate $r,$ i$.$e$.,$ by buying one bond at time $0,$ you will make $$3(1+r)$ at time $1.$ For what values of $r,$ can this bond be introduced into to the market without creating an arbitrage opportunity. Explain your reasoning.