2. Suppose there are only two periods, period 0 and period 1, and three possible states of the world in period 1: a good weather state, a fair weather state, and a bad weather state. Apples are the only product produced in this world, and they cannot be stored from one period to the next. The following abbreviations will be used: PA = apple in the present period (i.e., present apple), GA = good weather apple in the next period, FA = fair weather apple in the next period, BA = bad weather apple in the next period. Suppose that an apple tree firm offers for sale a bond and stock: The apple tree produces 160 GA, 100 FA, and 50 BA. The bond pays 40 GA, 40 FA and 40 BA. The stock pays 120 GA, 60 FA and 10 BA. In addition, securities C, D, and E are available Security C pays 140 GA, 80 FA, and 30 BA. Security D pays 60 GA, 30 FA, and 5 BA. Security E pays 80 GA, 20 FA, and 0 BA. The arbitrage-free price of the bond is 32 PA, and the arbitrage-free price of the stock is 44 PA. Securities C, D, and E are also priced fairly at 60 PA, 22 PA, and 20 PA, respectively. There are no arbitrage opportunities in this market. Note: if you compute the determinant using computer, you may have not an exact result due to numerical accuracy. For instance, if the true answer is 0, you may get a very small number instead but not exactly 0. Round your answers to 4 decimal digits. a) Are the stock, bond, and security C payoffs linearly independent? b) Find the price of the fair weather atomic security. c) Is the market complete? d) Calculate the arbitrage-free price of the apple tree. e) Verify that the principle of value additivity holds. f) Calculate the discount factor. g) Calculate the risk-free interest rate. h) Suppose a new security F is added to the market. Its pays 70 GA, 40 FA and 15 BA. Its price is 30 PA. Are there any arbitrage opportunities? 2. Suppose there are only two periods, period 0 and period 1, and three possible states of the world in period 1: a good weather state, a fair weather state, and a bad weather state. Apples are the only product produced in this world, and they cannot be stored from one period to the next. The following abbreviations will be used: PA = apple in the present period (i.e., present apple), GA = good weather apple in the next period, FA = fair weather apple in the next period, BA = bad weather apple in the next period. Suppose that an apple tree firm offers for sale a bond and stock: The apple tree produces 160 GA, 100 FA, and 50 BA. The bond pays 40 GA, 40 FA and 40 BA. The stock pays 120 GA, 60 FA and 10 BA. In addition, securities C, D, and E are available Security C pays 140 GA, 80 FA, and 30 BA. Security D pays 60 GA, 30 FA, and 5 BA. Security E pays 80 GA, 20 FA, and 0 BA. The arbitrage-free price of the bond is 32 PA, and the arbitrage-free price of the stock is 44 PA. Securities C, D, and E are also priced fairly at 60 PA, 22 PA, and 20 PA, respectively. There are no arbitrage opportunities in this market. Note: if you compute the determinant using computer, you may have not an exact result due to numerical accuracy. For instance, if the true answer is 0, you may get a very small number instead but not exactly 0. Round your answers to 4 decimal digits. a) Are the stock, bond, and security C payoffs linearly independent? b) Find the price of the fair weather atomic security. c) Is the market complete? d) Calculate the arbitrage-free price of the apple tree. e) Verify that the principle of value additivity holds. f) Calculate the discount factor. g) Calculate the risk-free interest rate. h) Suppose a new security F is added to the market. Its pays 70 GA, 40 FA and 15 BA. Its price is 30 PA. Are there any arbitrage opportunities
