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Unformatted text preview: FBE459: Financial Derivatives Solutions to Spring 2011 Midterm Exam 1. (20 points) A stock is expected to pay a dividend of $1 per share in 2 months and $1 in 5 months. The stock price is $100, and the risk-free rate of interest is 6% per annum with continuous compounding . 1.a. (5 points) What should be the forward price for a contract deliverable in 6 months? Do you think this market is an example of contango or backwardation? What is the initial value of the forward contract? SOLUTION: The present value of the income of the security (I) is given by: I = 1.e-0.06x2/12 + 1.e-0.06x5/12 = $ 1.965 The arbitrage-free forward price is: F 0,6M = (S-I).e r.T = (100-1.965).e 0.06x6/12 = $ 101.020 At current market conditions, you would conclude that Forward Price (F 0,T ) > Spot (S ). Thus this market is in CONTANGO. The initial value of the forward contract is (by design) ZERO. 1.b. (5 points) Another market participant quotes you the 6-month forward price at $105. What arbitrage opportunity exists? If one exists, explicit all cash flows for the arbitrage. SOLUTION: The actual futures price is above the arbitrage-free value of 101.02, thus the forward is too high relative to the underlying stock. The correct arbitrage strategy is: - “Buy low”: Buy the underlying stock by borrowing money - “Sell high”: Sell the forward contract Here are the arbitrage profits you can make: Today In 2 months In 5 months In 6 months, Forward is settled Buy S Borrow PV(F) Borrow PV(D 1 ) Borrow PV(D 2 ) -100 +105e-0.06.6/12 = +101.9 +1e-0.06.2/12 = +0.99 +1e-0.06.5/12 = +0.975 +1 -1 +1 -1 S T-105 Short F 0 105 – S T Profit +3.862 0 0 0 FBE 459 Midterm Exam: Spring 2011 [SOLUTIONS] Page 2 1.c. (5 points) Suppose a customer wishes to enter a long forward contract. If you sell him the contract, outline how you would hedge your resulting short forward position by using stocks and any borrowing or lending? SOLUTION: To hedge a short forwards position you would take the same position when shorting the forwards in an arbitrage strategy [as in question 1.b)]. The hedge strategy when you sell the forward contract is: - Buy the shares underlying the index + Borrowing money 1.d. (5 points) Suppose you enter a short position in the forward contract at the “fair” price you determined in part a. above. Three months later, the stock price is at 105 and the risk-free interest rate is still 6%. What should be the forward price prevailing now for 3 months remaining maturity? What would be the value of your original short forward position? SOLUTION: There is now only $1 dividend remaining. The present value of the income (I) is given by: I = 1.e-0.06x2/12 = $ 0.990 The arbitrage-free forward price is now: F 3M,6M = (S-I).e r.T = (105-0.990).e 0.06x3/12 = $ 105.58 Since the forward price has gone down (from $ 101.020 to $ 105.58), we are loosing money on a short forward position. The rationale is that we agreed to sell the asset in 6M at F 0,6M = 101.02 when now the “fair” would have been F= 101....
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