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Unformatted text preview: FBE459: Financial Derivatives Solutions to Spring 2007 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 riskfree 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.e0.06x2/12 + 1.e0.06x5/12 = $ 1.965 The arbitragefree forward price is: F 0,6M = (SI).e r.T = (1001.965).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 6month S&R forward at $105. What arbitrage opportunities exist? SOLUTION: The actual futures price is above the arbitragefree 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 Outlining the arbitrage strategy would be enough for your answer, but 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 +105e0.06.6/12 = +101.9 +1e0.06.2/12 = +0.99 +1e0.06.5/12 = +0.975 +1 1 +1 1 S T105 Short F 0 105 S T Profit +3.862 0 0 0 FBE 459 Midterm Exam: Spring 2007 [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 riskfree 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.e0.06x2/12 = $ 0.990 The arbitragefree forward price is now: F 3M,6M = (SI).e r.T = (1050.990).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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 Spring '08
 Matos

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