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.
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