Consider an economy with three possible states of the world and three traded securities (i.e., \(S=N=3\) ), with dividend matrix given by

\[D=\left[\begin{array}{lll}3 & 4 & 1 \\2 & 2 & 4 \\4 & 5 & 2\end{array}\right]\]

(i) Verify that the market is complete.

(ii) Given the price vector \(p=(2.85,3.45,2.35)^{\top}\), determine the state price vector.

(iii) Determine the risk free rate \(r_{f}\) implicit in the price-dividend couple \((p, D)\).

(iv) Determine the portfolio \(z^{c}\) which replicates the payoff \(c=(1,4,3)^{\top}\) and verify that the value at time \(t=0\) of such a portfolio coincides with the arbitrage free price of the payoff \(c\) computed in terms of the state price vector \(m\).

(v) Determine the arbitrage free price of a Call option written on the second security with strike price 2 . Verify that the value at time \(t=0\) of the replicating portfolio \(z^{\text {call }}\) coincides with the arbitrage free price of the Call option.