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.
To synthetically construct Arrow-Debreu securities and the risk-free asset using assets A, B, and C, we need to create portfolios that replicate the payoffs of these securities. Arrow-Debreu securities pay $1 in one specific state of nature and $0 in all other states. The risk-free asset pays the same in all states.
Arrow-Debreu Securities
Arrow-Debreu Security for State 1: Create a portfolio that consists of 4 units of Asset A, 1 unit of Asset B, and 3 units of Asset C. This portfolio will have a payoff of (4, 1, 3) in the three states. Thus, it replicates the Arrow-Debreu security for state 1.
Arrow-Debreu Security for State 2: Create a portfolio that consists of 1 unit of Asset A, 2 units of Asset B, and 2 units of Asset C. This portfolio will have a payoff of (1, 2, 2) in the three states, replicating the Arrow-Debreu security for state 2.
Arrow-Debreu Security for State 3: Create a portfolio that consists of 2 units of Asset A, 2 units of Asset B, and 0 units of Asset C. This portfolio will have a payoff of (2, 2, 0) in the three states, replicating the Arrow-Debreu security for state 3.
Risk-Free Asset
The risk-free asset should have the same payoff in all states. You can create a portfolio of assets A, B, and C such that the total payoff is the same in all states. For example, a portfolio of (2, 1, 1) will have a constant payoff of (2+1+1=4) in all three states, making it the risk-free asset.
(b) If the state-contingent payoffs for Asset C change to (2, 4, 4), it becomes impossible to construct Arrow-Debreu securities using only assets A, B, and C. This is because the new payoffs for Asset C are the same in all states, and there is no way to create securities that pay $1 in one specific state and $0 in all other states using only these assets. The Arrow-Debreu securities require distinct payoffs for each state, which is not possible with the new payoffs for Asset C.
(c) To synthetically construct Arrow-Debreu securities using Asset A and call options on Asset A with different exercise prices, as well as the risk-free asset:
(d) If we replace Asset A with either Asset B or Asset C, it becomes impossible to construct Arrow-Debreu securities using the same method as in part (c). This is because call options on Asset B or Asset C with any exercise price cannot replicate the Arrow-Debreu securities due to their different state-contingent payoffs. The uniqueness of Asset A’s payoffs allows us to construct Arrow-Debreu securities using call options, but this cannot be done with Assets B or C.
(e) To synthetically construct Arrow-Debreu securities using Asset C and put options:
Arrow-Debreu Security for State 1: Create a portfolio of Asset C and a put option with an exercise price of 2. In state 1, the put option will be exercised, resulting in a payoff of (2 – 2 = 0), and Asset C itself has a payoff of 2. The total payoff will be (2, 0), replicating the Arrow-Debreu security for state 1.
Arrow-Debreu Security for State 2: Create a portfolio of Asset C and a put option with an exercise price of 4. In state 2, the put option will be exercised, resulting in a payoff of (4 – 4 = 0), and Asset C itself has a payoff of 4. The total payoff will be (4, 0), replicating the Arrow-Debreu security for state 2.
Arrow-Debreu Security for State 3: Create a portfolio of Asset C and a put option with an exercise price of 4. In state 3, the put option will be exercised, resulting in a payoff of (4 – 4 = 0), and Asset C itself has a payoff of 4. The total payoff will be (4, 0), replicating the Arrow-Debreu security for state 3.
This construction is possible because the put options can adjust the payoffs of Asset C to match the desired Arrow-Debreu security payoffs in each state.
However, if we replace Asset C with Asset B, this method cannot be used because Asset B has different state-contingent payoffs, and there is no combination of put options that can replicate the Arrow-Debreu securities due to the lack of flexibility in altering Asset B’s payoffs.
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