Suppose there are two states of nature in the future. In the asset market, there are the two contingent claims, one for each state. Theres a third security with payoffs x = (3, 6). Suppose the prices of the contingent claims are p(c(s1)) = 1/3 and p(c(s2)) = 2/3 , while the price of the third security is 6.

(a) What is the payoff matrix X of the asset market? Is the market complete or incomplete? Why? Is there any redundant security?

(b) What is the price vector p of the securities market?

(c) The law of one price says that the securities with the same payoffs must have the same price. Does the LOOP hold here? Why or why not?

(d) Find an arbitrage portfolio. Show that one can possibly receive some positive payoffs without any cost or risk using the arbitrage portfolio.