Problem 0.1. Assume the three security market – assets S1 and S2, and the
riskless bond B with continuously compounded rate r.
A Swap contract calls for the following:
1. The buyer X pays the seller Y an amount q to enter into the contract at
time t = 0
2. The seller agrees to exchange one unit of asset S1 for one unit of asset S2
at time t = T .
The share prices of assets S1 and S2 at times t = 0 are S1(0) and S2(0). The
share prices of the assets at time t = T are S1(T ) and S2(T ) respectively.
Determine the fair market value q of the contract in two ways:
• by an arbitrage argument
• using the Fundamental Theorem
Problem 0.2. Consider a single period discrete market with two freely traded
assets, a Bond B and a Stock S, and three terminal states ω1, ω2, ω3. Assume
that the t = T share price of Stock in scenario ωi is di, and that d1 < d2 < d3.
Let r be the riskless rate of return, and S0 the share price of Stock at t = 0
• Show that this market is incomplete.
• Create a derivative security with unattainable payoff (for which there is
no replicating portfolio in the assets)
• Show that the state-price vector for the market is not unique.
• Show that the set of possible market prices of this derivative security
(defined by the state price vector) is an interval of real numbers.
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