Reference no: EM133558126
Consider a market with three states of nature and three assets. The assets have the following state contingent payoffs: • Asset A: (4, 1, 2) • Asset B: (1, 2, 2) • Asset C: (3, 2, 0) 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) Now assume that state-contingent payoffs for asset C has changed to (2, 4, 4). Answer part (a) again, if possible. If this cannot be done, explain why.
(c) A call option is a financial instrument that gives its holder the right but not the obligation to purchase an asset at a predetermined exercise price. 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. For example, a call option on asset A with exercise price X equal to 1 returns a payoff of (3,0,1). 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, using asset A and call options on asset A with different exercise prices, as well as the risk-free asset.
(d) Show how your answer in part (c) fails to hold if we replace asset A with either asset B or with asset C. Explain why this cannot be done.
(e) A put option is a financial instrument that gives its holder the right but not the obligation to sell an asset at a predetermined exercise price. 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.