Consider a market with three states of nature and three assets. The assets have the following state contingent vectors of payoffs:
asset A: (2, 5, 7) ,asset B: (2, 4, 4) ,asset C: (1, 0, 2).
Assume that all assets may be sold short.
(a) Show how to synthetically construct the Arrow–Debreu securities, as well as the risk-free asset using assets A, B and C.
(b) A call option with exercise price X on an asset pays max(as − X, 0) in state s, where as is the asset payoff in state s. Suppose that only asset
A exists in this market (not B or C), but that call options on asset A may also be bought or sold with any desired nonnegative exercise price X. Show how to synthetically construct the Arrow–Debreu securities,as well as the risk-free asset.
(c) Show how your answer in part (b) fails to hold if we replace asset A with either asset B or with asset C. Explain why this cannot be done.
(d) A put option with exercise price X on an asset pays max(X − as , 0) in state s, where as is the asset payoff in state s. Show how asset C together with the purchase or sale of put options can be used to synthetically construct the Arrow–Debreu securities. Explain why the same cannot be done if we replace asset C with asset B.