Exercise $4\mathrm{.}27$ Consider an economy with two traded assets and three possible

states of the world $($i$.$e$.,N=2$ and $S=3),$ with price$-$dividend couple

$D=\left[\begin{array}{cc}1& 5\\ 4& 2\\ 3& 1\end{array}\right],$ and $,p=(\frac{2\mathrm{.}45}{2\mathrm{.}35}).$

$($i$)$ Characterize the set of possible state prices.

$($ii$)$ Determine the space of attainable payoffs.

$($iii$)$ Verify that the arbitrage free price of any attainable payoff does not depend on

the specific state price vector chosen.

$($iv$)$ Determine the interval of arbitrage free prices for the payoff $c=(2,1,3{)}^{TT}.$

$($v$)$ Consider a Call option with strike price $2$ written on the second security. Does

the introduction of such a derivative make the market complete?