Math 373 Test 4 Fall 2015 December 16, 2015

Transcription

1 Math 373 Test 4 Fall 2015 December 16, (3 points) List the three requirements necessary for arbitrage to exist. No Risk No Investment Guaranteed Profit or positive cash flow

2 2. (4 points) Matt enters into a short 9 month forward on the stock of Ozerin Incorporated. Ozerin stock has a current spot price of 82 and does not pay dividends. The forward price is arbitrage free. The annual effective risk free interest rate is 7%. Complete the following payoff and profit table. Show your work to receive full credit. Spot Price at End of 9 Months Payoff Future Value of Cost Profit 75 Forward Price Spot at Expiry = (82)( ) 75 = Cost = 0 FV of Cost = 0 Payoff FV of Cost = =

3 3. (3 points) There are three perspectives on derivatives. Please identify which perspective is represented by each party below. Each perspective will only be used once. Anderson Bank wants to protect itself from a potential change in interest rates. Therefore, Anderson Bank purchases an interest rate swap. Anderson Bank retains Raya Investment Bank to identify a counter party to take the interest rate risk. A professor at Purdue University decides to study this transaction to research the impact of interest rate swaps in reducing the interest rate risk experienced by banks. Anderson Bank represents this perspective: End User Raya Investment Bank represents this perspective: Intermediary The Purdue professor represents this perspective: Economic Observer

4 4. (6 points) You are given the following spot interest rates: Time Spot Rate % % % % % Crum and Associates has a line of credit that will permit it to borrow 400,000 in year one; an additional 200,000 in year 2 for a total of 600,000; and an additional 400,000 in year three for a total of 1 million. Under the loan, the interest rate resets at the beginning of each year and is equal to the one-year spot interest rate at the time of reset. Crum enters into a three year interest rate swap where the notional amount matches the amount available under the line of credit and variable rate matches the rate for the line of credit. Determine the swap rate under Crum s interest rate swap. f r 0.04 [0,1] 1 (1 r ) (1.05) [1,2] 1 r f (1 r ) (1.0575) [2,3] 2 2 (1 r2 ) (1.05) f R Q P f Q P 1 2 (400, 000)(1.04) (0.04) (600, 000)(1.05) ( ) (400, 000)(1.04) (600, 000)(1.05) (1, 000, 000)(1.0575) 3 (1, 000, 000)(1.0575) ( )

5 5. (4 points) Ian wants to buy X shares of Dakota Stock. Dakota stock has a bid price of 100 and a bid-ask spread of 1. Ian can purchase the stock using either Bian Brokers or Simpson Stock Brokerage. Bian charges a flat commission of 150 without regard to the number of shares purchased. Simpson charges a commission that is 0.5% of the purchase price. Determine the values of X for which Ian should use Simpson. Ask Bid BidAskSpread Bian charges commissions of 150. Simpson commissions will be X(101)(0.005) 0.505X Use Simpson if Simpson < Bian 0.505X 150 X X 297

6 6. (4 points) Mark each of the following statements True or False i. A futures contract eliminates credit risk. True or False ii. A forward contract is an exchange traded futures contract. True or False iii. A forward contract is marked to market daily. True or False iv. A forward contract does not have a premium while a futures contract does. True or False

7 7. (6 points) The stock of Liu LTD has a current spot price of 58. The stock pays a quarterly dividend of 2 each quarter with the next dividend due in 2 months. The annual risk free interest rate compounded continuously is 8%. Anne enters into a 10 month forward on Liu LTD. Assuming that there are no transaction costs and no arbitrage, determine the forward price. F e e e e 0, T (0.08)( T ) ( 0.08)( t 1) ( 0.08)( t 2 ) ( 0.08)( t3 [58 2( ) )] F e [58 2( e e e )] (0.08)(10/12) ( 0.08)(2/12) ( 0.08)(5/12) ( 0.08)(8/12) 0,10/12

8 8. (6 points) Madison purchases four S&P 500 futures contracts. The current futures price of the S&P 500 Index is The margin requirement for these contracts is 12.5%. The maintenance margin is 80%. The margin account will earn interest at an annual rate of 7.80% compounded continuously. The futures contracts will be marketed to market weekly. At the end of one week, the S&P 500 futures price is Determine the amount in the margin account after the mark to market. Notional Amount = (4)(250)(1800) 1,800, 000 Margin = (1,800, 000)(0.125) 225, 000 Loss = (4)(250)( ) 16, 000 (0.078)(1/52) Margin After One Week = (225,000) e 225, Margin After One Week = 225, , , Determine the amount of the Margin Call at the end of one week. Maintenace (225, 000)(0.8) 180, 000 Since the Margin Account is not less than 180,000, there is no margin call. Margin Call 0

9 9. (7 points) The FTSE 100 has a current 6 month forward price of The current spot price of the FTSE 100 is The FTSE 100 pays dividends at a continuously compounded rate of 2%. The annual risk free interest rate is 3.5% compounded continuously. Assume there are no transactions costs. Demonstrate that arbitrage exists. Real Forward Price = 5940 Synthetic Forward Price = ( 0.02)(0.5) (0.035)(0.5) 5900e e Since the Real Forward Price does not equal the Synthetic Forward Price, then there is arbitrage. State explicitly what actions you would take to take advantage of this arbitrage. Buy Low (Real Forward) and Sell High (Synthetic Forward) Selling the Synthetic Forward is accomplished by: i. Selling the stock ii. Use the proceeds to buy a zero coupon bond.

10 10. (2 points) Draw the graph of a written straddle.

11 11. The current spot price of the stock of Hannah Company is 42. The stock does not pay a dividend. The premium for a 45 strike one year European-style call on Hannah stock is The annual effective interest rate of 5%. Mark enters into a long synthetic one year forward on Hannah stock using calls and puts with a strike price of 45. (1 point) State explicitly what options Mark buys and/or sells to enter into the synthetic forward. Buy a call with a 45 strike price and sell a put with a 45 strike price. (4 points) Calculate the cost to enter into the synthetic forward. Use the Put Call Parity formula to find the premium for the put. Strike 45 C P S P 42 P i 1.05 Cost = C P (3 points) At the end of one year, the spot price of the stock of Hannah is 50. Determine the profit that Mark will realize. Payoff on call = = 5 Payoff on put = 0 since the spot price is above the strike price FV of Cost = (- 0.86)(1.05) = Profit = Payoff FV of Cost = (- 0.90) = 5.90

12 12. (5 points) Albean Airlines entered into a six-year interest rate swap two years ago. Under the sixyear swap, Albean agreed to a swap rate of 6.75% on a level annual notional amount of 1,000,000 each year for six years. Now there are four years left on the swap agreement. Today, the spot interest rates are: Period of Time Spot Interest Rate 1 Year Years Years Years Years Years Calculate the market value of the swap from Albean s perspective if Albean decided sell it today. f r [0,1] 1 (1 r ) (1.051) [1,2] 1 r f (1 r ) (1.056) [2,3] 2 2 (1 r2 ) (1.051) f (1 r ) (1.06) [3,4] 3 3 (1 r3 ) (1.056) f 1 2 ( )(1.045) ( )(1.051) MarketValue (1, 000, 000) ( )(1.056) ( )(1.06) 28,

13 13. (6 points) The stock of Lyu LTD has a current spot price of 144. A 144 strike European-style 4 month call on Lyu Stock has a premium of The annual effective risk free is 6.25% Nathan purchases a cap on Lyu stock. Determine the payoff and profit if the spot price of Lyu stock is 140 at the end of 4 months. Cap = Short the stock and buy a call Payoff on the stock = - Spot Price = FV of Cost on the stock = (Minus Spot at Zero)(1+i) = (- 144)(1.0625) 4/12 = Profit on Stock = Payoff FV of Cost = ( ) = 6.94 Payoff on Call = 0 since spot at expiry is less than strike price FV of Call = (9.70)(1.0625) 4/12 = 9.90 Profit on Call = Payoff FV of Cost = = Total Payoff = = -140 Total Profit = =

14 14. (3 points) Circle each of the following that are true with regard to a straddle: i. At the time of entering into the straddle, both the put and the call are out of the money. ii. If the put is in the money, then the price of the underlying asset has increased in value since time 0. iii. The maximum gain on a written straddle is larger than the maximum gain on a written strangle provided the underlying asset is the same.

15 15. (7 points) The stock of Wang Corporation has a current spot price of 200. You can purchase the following 6 month European-style call and put options. Strike Price Call Option Premium Put Option Premium You purchases a collar on Wang Corp. Determine the maximum profit that you could earn. (Hint: You can calculate the risk free interest rate from the information given.) Collar is Buy a Put and Sell a call with a higher strike price. Therefore, we will buy the put with a strike of 195 and sell the call with a strike of 205. We need the risk free interest rate. K C P S (1 i) i (1 i) If you know the graph, the maximum profit is if the spot at expiry is zero. Payoff on Put = 195; FV of Cost of Put = (9.15)( )=9.42; Profit = = Payoff on Call = 0; FV of Cost of Call = (-14.58)( ) = ; Profit = 0 - (-15.01) =15.01 Total Profit = =

16 16. (2 points) Mark each of the following statements True or False i. Hedging adds value to a company if it has a concave profit curve True or False ii. Cost is a reason that companies do not hedge. True or False

17 17. The stock of Weakland and Associates has a current spot price of 250. Jimin can purchase the following 6 month European-style call options. Strike Price Call Option Premium The annual risk free interest rate is 5% compounded continuously. Jimin purchases a bear spread using the above calls. (4 points) Determine the maximum profit that Jimin could earn. Bear Spread is Buy a Call with a K=260 and sell a call with a K=240. If you know the graph, the maximum profit is if the spot at expiry is less than 240. We will use zero. (0.05)(0.5) Payoff on 260 K Call= 0; FV of Cost of Call = (16.00) e 16.41; Profit = (0.05)(0.5) Payoff on 240 K Call = 0; FV of Cost of Call = ( 26.17) e 26.83; Profit = 0 ( 26.83) Total Profit (3 points) Determine the range of spot prices at expiry for which Jimin will have a positive profit. For every movement of 1 in the price above 240, the profit will decrease by 1. Therefore, X = 240 X =

18 18. (5 points) The Shanghai Composite Index has a current spot price of The Index pays dividends continuously at an annual rate of 1.5%. The annual risk free interest rate compounded continuously is 8%. Jie purchases a prepaid forward for delivery in 9 months. Assuming that there are no transaction costs and no arbitrage, determine the price of the prepaid forward. F S e F e P T P (0.015)(9/12) 0, T 0 0,0.75 (3500)

19 19. (6 points) You are given the following spot interest rates and the forward prices of silver: Time Spot Rate Forward Silver Price % % % % % Today, Chen Silver Miners enters into a deferred swap on silver. Under the swap, Chen agrees to sell 20,000 ounces of silver at the end of 4 years as well as sell 20,000 ounces at the end of 5 years. Determine the swap rate on silver. 4 5 Q P f (20, 000)(1.0625) (13.50) (20, 000)(1.065) (13.00) R QP (20, 000)(1.0625) (20, 000)(1.065)

20 20. (3 points) There are three styles of options. For each of the situations below, list the style of option being described. The same style of option may be used multiple times. a. Edward enters into an agreement with Simon which gives him the right but not the obligation to purchase Intel stock from Simon at any time during the next 6 months. Style of Option: American Style b. Henry and Renyuan enter into an agreement under which Henry must buy Renyuan s textbooks at the end of semester for 200 if Renyuan wants to sell them to Henry. Style of Option: European Style c. Nanqi and Yue enter into an agreement. Nanqi agrees to provide Yue with a new iphone if she requests a new phone and pays 500 for the phone. Yue has the right to request a new phone at the end of each month over the next 4 months. However, Yue can only make the request once. Style of Option: Bermuda Style

21 21. A stock has a current spot price of 100 and does not pay dividends. Based on the expected return on the stock, the expected stock price in one year is 112. The one year forward price on the stock is (1 point) Calculate the Cost of Carry for the stock: Cost to Carry = F0,1 S0 = = 4.25 (1 point) Calculate the Forward Premium on the stock: F 0,1 Forward Premium S (1 point) Define credit risk. Credit risk is the risk that one party in a deal will not keep their end of the deal.