Assume there are two possible states of the world: 1 and 2. There are two assets, a

Assume there are two possible states of the world: ω1 and ω2. There are two assets, a risk-free asset returning Rf in each state, and a risky asset with initial price equal to 1 and date–1 payoff x˜. Normalize the unit of measurement of the risk-free asset so that its price is 1 at date 0 and its payoff per share is Rf . Let Rd = ˜x(ω1) and Ru = ˜x(ω2). Assume without loss of generality that Ru > Rd.

(a) What inequalities between Rf , Rd, and Ru are equivalent to the absence of arbitrage opportunities?

(b) Assuming there are no arbitrage opportunities, compute the unique vector of state prices, and compute the unique risk-neutral probabilities of states ω1 and ω2.

(c) Suppose another asset is introduced into the market that pays max(x˜ −K,0) for some constant K. Compute the price at which this asset should trade, assuming there are no arbitrage opportunities.