Note: A number Z appears in each of the three Problems. It is obtained by dividing the last two digits of your student number by 10 and adding 0.05 to the result. For example, if your student number is 21339, then Z= 39/10 + 0.05 = 3.95. Problem 1. Suppose in the next period there are two possible states of the world, a good weather state and a bad weather state, and that apples are the only product produced in the economy. A present apple is denoted by PA, an apple in the good weather state in the next period by GA, and an apple in the bad weather state in the next period by BA. Two securities are available on the market: a Bond and a Stock. Stock pays 40GA and 20BA and has a price of 26PA. Bond pays 20GA and 20BA. The discount factor is df=1/(1+Z/100). (i) What is the arbitrage-free) price of the Bond? What is the economic interpretation of Z? (ii) Compute the arbitrage-free price of the atomic or time-state securities (i.e., the price of 1GA and 1BA measured in terms of PA). (iii) Suppose an investor wants 10GA and 20BA. Construct a portfolio of the available securities that will provide these time-state claims. Compute its arbitrage-free price. (iv) Construct and graph the opportunity set for future apples per present apple. Mark the locations of the Bond and the Stock in the opportunity set. Where does the opportunity set frontier cut axes? (v) A new security X appears on the market that pays 10GA and 30BA and sells at a price of 15PA. Plot the security X in the opportunity set. Does the appearance of X create arbitrage opportunities? If it does, design a profitable arbitrage strategy. If it does not, explain why it does not. Note: A number Z appears in each of the three Problems. It is obtained by dividing the last two digits of your student number by 10 and adding 0.05 to the result. For example, if your student number is 21339, then Z= 39/10 + 0.05 = 3.95. Problem 1. Suppose in the next period there are two possible states of the world, a good weather state and a bad weather state, and that apples are the only product produced in the economy. A present apple is denoted by PA, an apple in the good weather state in the next period by GA, and an apple in the bad weather state in the next period by BA. Two securities are available on the market: a Bond and a Stock. Stock pays 40GA and 20BA and has a price of 26PA. Bond pays 20GA and 20BA. The discount factor is df=1/(1+Z/100). (i) What is the arbitrage-free) price of the Bond? What is the economic interpretation of Z? (ii) Compute the arbitrage-free price of the atomic or time-state securities (i.e., the price of 1GA and 1BA measured in terms of PA). (iii) Suppose an investor wants 10GA and 20BA. Construct a portfolio of the available securities that will provide these time-state claims. Compute its arbitrage-free price. (iv) Construct and graph the opportunity set for future apples per present apple. Mark the locations of the Bond and the Stock in the opportunity set. Where does the opportunity set frontier cut axes? (v) A new security X appears on the market that pays 10GA and 30BA and sells at a price of 15PA. Plot the security X in the opportunity set. Does the appearance of X create arbitrage opportunities? If it does, design a profitable arbitrage strategy. If it does not, explain why it does not
