4. (25 points) Consider a financial market in which investors can trade a risky stock with price So at time and borrow or lend at the risk-free rate r > 0 (one-period simple interest rate). Consider an one-period market model with three states: 2 = {u,m,d) with S > S > SY > 0 and probabilities Pu = P[S1 = $1] > 0, P=P[S1 = Sr] >0, and pa=P[S1 = $i] = 1 - Pu Pr >0. (a) (5 points) Is the market complete? (b) (8 points) Show that the risk-neutral probability measure is not uniquely determined. What is the maximum possible range of the risk-neutral prob- ability = f[s = S. (u)]? 2 (c) (6 points) Consider a European call option with maturity at time 1 and strike price K = Sr. Use your answers in part (b) to find the range of possible prices for this European call option at time zero. (d) (6 points) Suppose that the European call option considered in (c) is traded liquidly in the market with the price c, which is in the range of the risk- neutral prices you found in (c). Is the risk-neutral measure uniquely deter- mined? 4. (25 points) Consider a financial market in which investors can trade a risky stock with price So at time and borrow or lend at the risk-free rate r > 0 (one-period simple interest rate). Consider an one-period market model with three states: 2 = {u,m,d) with S > S > SY > 0 and probabilities Pu = P[S1 = $1] > 0, P=P[S1 = Sr] >0, and pa=P[S1 = $i] = 1 - Pu Pr >0. (a) (5 points) Is the market complete? (b) (8 points) Show that the risk-neutral probability measure is not uniquely determined. What is the maximum possible range of the risk-neutral prob- ability = f[s = S. (u)]? 2 (c) (6 points) Consider a European call option with maturity at time 1 and strike price K = Sr. Use your answers in part (b) to find the range of possible prices for this European call option at time zero. (d) (6 points) Suppose that the European call option considered in (c) is traded liquidly in the market with the price c, which is in the range of the risk- neutral prices you found in (c). Is the risk-neutral measure uniquely deter- mined