Problem set 1
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Fin Eco II (2010-’11) Problem set 1 Due: 25 Jan 2011, 08:00 hrs January 18, 2011 Instructions: Collaboration on these exercises is permitted; however, I expect everyone who has submitted an answer to a question to be able to defend that answer. I encourage collaboration, not copying. Please sketch your reasoning clearly, especially since I will award partial marks for reasoning, even if the final answer is incorrect. Please ensure that your handwriting is legible. All problem sets are due before the beginning of the first practical session dealing with that problem set. This deadline will be strictly enforced - late submissions may not be accepted, and will be penalised. Question 1.1. Assume that in the following market there are three possible states of nature. The following securities are traded • Security 1: (1.1, 1.1, 1.2), Price of security 1, p 1 = 1.00 • Security 2: (1.1, 1, 0.9), Price of security 2, p 2 = 0.95 • Security 3: (1.1, 1.05, 1.3), Price of security 3, p 3 = 1.00 Is the market complete? If yes, compute the prices of the Arrow-Debreu securities. Show that it is possible to construct Arrow-Debreu prices by replicating the payoffs of contingent claims using the complex securities. (1 point) Question 1.2. (a) You are given that the current exchange rate JPY/MXN is 7.24, i.e., you get 7.24 JPY for every MXN. The six-month forward JPY/MXN exchange rate is 7.42. What is the difference in the six-month interest rates in Mexico and Japan? (0.25 points) (b) The 3 × 6 FRA rate in Mexico is 12.5% and the price of a zero-coupon bond maturing in three months in Mexico with face value 100 Pesos is 99 Pesos today. What is the six-month interest rate in Mexico? What is the interest rate in Japan, considering your result in part (a)? (0.75 points) Question 1.3. Assume that at time t + 1 there are 3 possible states of nature. The following securities are traded today: • Security 1’s payoffs: (2; 1; 3); Price of security 1, p 1 = 2.1 • Security 2’s payoffs: (1; 1; 1); Price of secuirty 2, p 2 = 0.9 (a) Is the market complete? Are there arbitrage opportunities? Would it be useful to write an option on the first security? Compute the risk-free rate. (0.5 points) (b) Assume there exists a call option on Security 1 with strike price K = 2. Its price at time t is 0.4. What are the Arrow-Debreu securities prices? What is the price in period t of a security which pays 4 in the first state, 2 in the second and 6 in the third state next period? (0.5 points) 1
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Transcript of Problem set 1
Fin Eco II (2010-’11) Problem set 1
Due: 25 Jan 2011, 08:00 hrs
January 18, 2011
Instructions:
Collaboration on these exercises is permitted; however, I expect everyone who has submitted an answer to a questionto be able to defend that answer. I encourage collaboration, not copying. Please sketch your reasoning clearly,especially since I will award partial marks for reasoning, even if the final answer is incorrect. Please ensure thatyour handwriting is legible. All problem sets are due before the beginning of the first practical session dealingwith that problem set. This deadline will be strictly enforced - late submissions may not be accepted, and will bepenalised.
Question 1.1.
Assume that in the following market there are three possible states of nature. The following securities are traded
• Security 1: (1.1, 1.1, 1.2), Price of security 1, p1 = 1.00
• Security 2: (1.1, 1, 0.9), Price of security 2, p2 = 0.95
• Security 3: (1.1, 1.05, 1.3), Price of security 3, p3 = 1.00
Is the market complete? If yes, compute the prices of the Arrow-Debreu securities. Show that it is possible toconstruct Arrow-Debreu prices by replicating the payoffs of contingent claims using the complex securities. (1point)
Question 1.2.
(a) You are given that the current exchange rate JPY/MXN is 7.24, i.e., you get 7.24 JPY for every MXN. Thesix-month forward JPY/MXN exchange rate is 7.42. What is the difference in the six-month interest rates in Mexicoand Japan? (0.25 points)
(b) The 3 × 6 FRA rate in Mexico is 12.5% and the price of a zero-coupon bond maturing in three months inMexico with face value 100 Pesos is 99 Pesos today. What is the six-month interest rate in Mexico? What is theinterest rate in Japan, considering your result in part (a)? (0.75 points)
Question 1.3.
Assume that at time t + 1 there are 3 possible states of nature. The following securities are traded today:
• Security 1’s payoffs: (2; 1; 3); Price of security 1, p1 = 2.1
• Security 2’s payoffs: (1; 1; 1); Price of secuirty 2, p2 = 0.9
(a) Is the market complete? Are there arbitrage opportunities? Would it be useful to write an option on thefirst security? Compute the risk-free rate. (0.5 points)
(b) Assume there exists a call option on Security 1 with strike price K = 2. Its price at time t is 0.4. What arethe Arrow-Debreu securities prices? What is the price in period t of a security which pays 4 in the first state, 2 inthe second and 6 in the third state next period? (0.5 points)
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Question 1.4.
In this exercise, you will incorporates market imperfections in the pricing of a futures contract. Let’s begin withthe case of no market imperfections: assume that the price at t0 of corn futures for delivery at t1 is Ft0 (one cornfuture is for one bushel of corn). The spot price (the price at t0) of a bushel of corn is S0. The cost of carry (thecost, expressed as a percentage of the price of corn, of storing the corn from t0 to t1) is c. Show that in the absenceof arbitrage, Ft0 = S0(1 + c). (0.25 points)
(a) Assume that there are no transactions costs in the futures markets, but that the spot market has a cost Tfor every Euro of transaction. What are the price bounds for Ft0 under no arbitrage? (0.25 points)
(b) Assume that the cost of carrying the physical asset is funded by either borrowing or lending, and that thelending rate is cL while the borrowing rate is cB . Find the arbitrage bounds for Ft0 under no arbitrage. (0.50points)
Question 1.5.
These are simple practice questions to test your familiarity with well-known concepts and terminology that we willbe using in the course. If you encounter difficulties in understanding the terminology, check any standard textbook(or Google it if you like!); if you still have problems, send me an E-mail.
(a) Here are some alternative investments you are considering for one year. (i) Bank A promises to pay 8% onyour deposit compounded annually. (ii) Bank B promises to pay 8% on your deposit compounded daily. (iii) BankC promises to pay 8% on your deposit compounded continuously. Compare the effective annual rate (EAR) onthese investments. (0.25 points)
(b) Suppose a hedge fund manager earns 1% per trading day. There are 250 trading days per year. (i) Whatwill be your annual yield on $100 invested in her fund if she allows you to reinvest in her fund the 1% you earn eachday? (ii) What will be your annual yield assuming she puts all of your daily earnings into a zero interest-bearingaccount and pays you everything earned at the end of the year? (0.25 points)
(c) Suppose you bought a five-year zero-coupon Treasury bond with face value $1000 for $800. (i) What isthe yield to maturity (annual compounding) on the bond? (ii) Assume the yield to maturity on comparable zerosincreases to 7% immediately after purchasing the bond and remains there. Calculate your annual return (holdingperiod yield) if you sell the bond after one year. (0.50 points)
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