衍生产品市场 课后答案 第六章

Chapter 6 The Wide World of Futures Contracts Question 6 1 The current exchange rate is 0 02 which implies 50 The euro continuously compounded interest rate is 0 04 the yen continuously compounded interest rate is 0 01 Time to expiration is 0 5 years Using the foreign exchange forward formula 6 2 we find 0 04 0 01 0 5 0 01 0 04 0 5 euro yen forward0 020 02 1 0151130 020302 yen euro forward50500 9851149 2556 e e Note that when we use euro yen our home currency is the euro hence when applying the formula we are European It is the price we as Europeans have to pay for the foreign currency the yen Hence we use our home interest rate i e 4 r in the forward exchange formula and we use the yen interest rate as the foreign interest rate i e 1 y r Similarly when we use yen euro our home currency is the yen hence when applying the formula we are Japanese It is the price we as Japanese have to pay for the foreign currency the euro Hence we use our home interest rate i e 1 r in the forward exchange formula and we use the euro interest rate as the foreign interest rate i e 4 y r Question 6 2 The current spot exchange rate is 0 008 the one year continuously compounded dollar interest rate is 5 and the one year continuously compounded yen interest rate is 1 This means that we can calculate the no arbitrage price of a one year yen forward to be 0 05 0 01 dollar yen forward0 0080 008 1 04081080 0083265e We can see that the observed forward exchange rate of 0 0084 is too expensive relative to the fair forward price We therefore go short the forward contract and synthetically create a long forward position buy yen Today At expiration of the contract Description in in yen in in yen Sell forward 0 0 0084 1 Buy yen for 0 0079204 dollar 0 0079204 0 99005 Lend 0 99005 yen 0 99005 1 Borrow 0 0079204 dollar 0 0079204 0 0083265 Total 0 00 0000735 0 72 McDonald Fundamentals of Derivatives Markets Note that when we enter into the short forward position we short 1 yen This is why we buy 0 99005 yen as 01 0 99005 e We could have done an arbitrage by buying 1 yen and shorting 1 01005 01 i e eyen forward With the above arbitrage strategy we ve earned 0 0000735 dollars without any exchange risk or initial investment involved We have exploited the arbitrage opportunity and would like to increase the scale of this strategy as much as possible i e buy as many yen on the spot market and sell them forward With a forward exchange rate of 0 0083 the observed price is too cheap We will buy the forward and synthetically create a short forward position Today At expiration of the contract Description in in yen in in yen Buy forward 0 0 0083 1 Sell yen for 0 0079204 dollar 0 0079204 0 99005 Borrow 0 99005 yen 0 99005 1 Lend 0 0079204 dollar 0 0079204 0 0083265 Total 000 00002650 Therefore we made an arbitrage profit of 0 0000265 dollars Question 6 3 We will do this on a per euro basis Today we borrow a euro at 4 buy 1 25 on the spot market and invest the 1 25 at 5 If T xis next year s exchange rate per euro paying back the euro loan requires 04 T x e dollars Hence our profit in dollars is 05 04 1 251 3141 1 0408 TT ex ex 1 Our zero profit sets the profit equation above equal to zero 1 3141 1 04081 2626 T x 2 Setting1 3 T x in the profit equation yields a profit of 1 3141 1 0408 1 3 0 0390 T x 3 Setting1 22 T x in the profit equation yields a profit of 1 3141 1 0408 1 22 0 0443 T x Question 6 4 1 The Eurodollar futures price is 93 23 i e a 90 day rate of 1 6925 Using Equation 6 3 of text we have a quarterly effective rate LIBOR rate of 91 1191 10093 23 1 7113 100490 r Chapter 6 The Wide World of Futures Contracts 73 2 Since we want to protect ourselves against an interest rate rise we will short 10 million worth of future contracts i e 10 contracts We take a short position as the Eurodollar futures price moves opposite the interest rate 3 We will have to repay principal plus interest on the loan that we are taking from the following June to September Because we shorted a Eurodollar futures we are guaranteed the interest rate we calculated in Part 1 Therefore we have a repayment of 91 10 000 000 1 10 000 000 1 017113 10 171 130r As an example if the 3 month LIBOR rate in June was 1 9 the futures price would be 100 7 6 92 4 and the quarterly effective rate would be 1 9211 Our unhedged borrowing cost interest has gone up by approximately 20 000 however our 10 futures contracts pay 10 100 93 23 92 4 20 750 Question 6 5 1 The Eurodollar futures price is 94 65 i e a 90 day rate of 1 3375 Using Equation 6 3 of text we have a quarterly effective rate LIBOR rate of 91 1191 10093 23 1 3524 100490 r 2 Since we want to protect ourselves against an interest rate fall we will long 10 million worth of futures contracts i e 10 contracts We take a long position as the Eurodollar futures price moves opposite the interest rate 3 We will receive principal plus interest on our loan that is made from the following June to September Because we are long in Eurodollar futures we are guaranteed the interest rate we calculated in Part 1 Therefore we receive a repayment of 91 10 000 000 1 10 000 000 1 013524 10 135 240r Question 6 6 1 As the hint suggests we have to increase our short position in the September Eurodollar futures contract by the 91 day rate The Eurodollar futures is a way of locking in 3 month rates hence locking in a 6 month rate requires borrowing for 3 months at the 91 day June rate call it J r and then borrowing 1 J r at the 91 day September rate Hence we have to go short 1 J r more proportionally Using a similar calculation as 6 4 a and 6 5 a the two 91 day rates are 1 3875 J r and 1 3 S r Note that increasing our position of 10 contracts by 1 3875 is not technically feasible as the contracts have 1 million principle 2 We have locked in a 182 day rate of 1 013875 1 0130012 705 74 McDonald Fundamentals of Derivatives Markets Question 6 7 A storer of corn must be compensated for the storage costs as well as interest forgone For example the holder could sell corn for 3 per bushel on the spot market invest the 3 at 1 This would give 3 cents interest and would save 4 cents in storage Hence the January forward price must be 3 07 for there to be storage If it was less than 3 07 no one would store it if it was greater than 3 07 cash and carry arbitrage would be possible Similarly the February price must cover two months interest and storage We must also cover the interest on the first month s storage cost Mathematically 2 0 2 12 3 1 01 04 1 01 04 3 1407 F Note we could use the January price of 3 07 and argue that one could buy in January at 3 07and store it in February They must be rewarded with the interest and storage cost i e 0 2 12 3 07 1 01 04 3 1407 F March has three months interest and storage costs to cover Mathematically 32 0 3 12 3 1 01 04 1 01 04 1 01 04 3 2121F We could have also used February s price 0 3 12 3 1407 1 01 04 3 2121F Question 6 8 We will do this slightly different than the text s example to demonstrate that the scale of the trade can be different Instead of buying 0 9727 ounces on the spot market our cash and carry will buy one ounce on the spot market Specifically we borrow 400 at 5 for two years buy one ounce of gold lend it at a lease of 1 384 and short 0 01384 2 1 0281e ounces forward at 430 per ounce This has zero cost and in two years we receive 05 2 1 0281 4304000e Note that holding gold i e not lending it in our cash and carry strategy will lose money for we can only short one unit forward which implies a cash flow of 05 2 43040012 070e Question 6 9 1 The spot price of gold is 300 00 per ounce With a continuously compounded annual risk free rate of 5 and at a one year forward price of 310 686 we can calculate the lease rate according to formula 6 8 0 0 1310 686 ln0 05ln0 015 300 T F r TS Chapter 6 The Wide World of Futures Contracts 75 2 Suppose gold cannot be loaned Then our cash and carry arbitrage is Transaction Time 0 Time T 1 Short forward 0 310 686 T S Buy gold 300 T S Borrow 0 05 300 315 38 Total 0 4 6953 The forward price has an implicit lease rate If we try to engage in a cash and carry arbitrage but do not have access to the gold loan market we will incur a loss 3 We will use our cash and carry as in 6 8 however this time we will tail our position and purchase 015 0 9851e ounces on the spot market with borrowed money and lend the 0 9851 ounces In this case we will receive 1 ounce of gold in one year Hence we short one ounce forward We have the following cash and carry payoff table Transaction Time 0 Time T 1 Short forward 0310 686 T S Buy tailed gold 295 5336 T S position lend 0 015 Borrow 0 05 295 5336 310 686 Total 00 Therefore we now break even since the forward was fairly priced taking the implicit lease rate into account this result should not surprise us Question 6 10 The spot price of a widget is 70 00 With a continuously compounded annual risk free rate of 5 we can calculate the annualized lease rates for each delivery date T according to the formula 6 8 0 0 1 ln T F r TS T Forward price Annualized lease rate 1 4 70 70 0 0101987 1 2 71 410 0101147 3 4 72 130 0100336 1 72 860 0099555 The lease rate is less than the risk free interest rate this is equivalent to the forward curve being upward sloping which is an example of contango 76 McDonald Fundamentals of Derivatives Markets Question 6 11 1 Buying gas in October at 6 85 with borrowed money at 1 monthly interest storing it for a month and selling it at the November price of 7 50 yields a profit of 7 506 85 1 01Oct Storage Cost0 5815Oct Storage Cost Notice that a similar analysis could be done for any of the back to back months Let i SCbe the storage costs of month i The strategy of using the futures contract to buy in month i and sell in month 1 i using borrowed money would yield to a profit with Month i 1 1 01 ii FF Oct 0 5815 Oct SC Nov 0 5750 Nov SC Dec 0 0315 Dec SC Jan 0 0320 Jan SC Feb 0 1325 Feb SC Mar 0 8320 Mar SC Since storage costs are greater than zero storage is not profitable for December through March 2 From the table above we see that marginal storage costs in October should be approximately 0 5815 and marginal storage costs in November should be approximately 0 5750 If they were greater no one would be willing to store it during the month If they were lower then the storage arbitrage strategy would earn a profit as shown in the table above Question 6 12 If you borrow natural gas you would sell it immediately on the October spot market for 6 50 Investing this at 1 you would have 6 50 1 01 6 565 in November Since you have borrowed natural gas you have to replace it In order to have no risk you must also go long the November forward note you do this today in October In November you have to pay 7 25 to replace the gas you borrowed Hence in the absence of a cash flow from the person you borrowed the gas from you would be losing 7 250 6 565 0 685 Therefore you require a cash payment or 0 685 from the lender of the gas The lender is would be willing to pay this because by lending the gas which will be replaced in November she avoids storage costs In this case there is a negative lease rate convenience yield due to storage costs Question 6 13 If you borrow natural gas you would sell it immediately on the March spot market for 8 00 Investing this at 1 you would have 8 00 1 01 8 08 in April Since you have borrowed natural gas you have to replace it In order to have no risk you must also go long the April forward note you do this today in March In April you have to pay 7 25 to replace the gas you borrowed Hence in the absence of a cash flow between you and the lender you would be making 8 087 250 83 Therefore you must make a cash payment of 0 83 to the lender of the gas Chapter 6 The Wide World of Futures Contracts 77 The lender would demand a payment since instead of lending it to you and getting the gas back in April she could sell it in March for a relatively high price 8 and earn interest Note storage costs are not relevant as storage is not expected to occur In this case there is a positive lease rate convenience yield owners of the asset are reluctant to lend their gas and must be paid a premium if they are to lend it Question 6 14 The spot price of oil is 32 00 per barrel With a continuously compounded annual risk free rate of 2 we can again calculate the lease rate according to the formula 0 0 1 ln T F r TS Time to expiration Forward price Annualized lease rate 3 months 31 37 0 0995355 6 months 30 750 0996918 9 months 30 140 0998436 12 months 29 540 0999906 In this case the lease rate is higher than the risk free rate and the market is in backwardation Question 6 15 The average temperature is simply the sum of the temperatures divided by the number of days 7 60556872605045 7410 758 57 Heating degree days HDD sum absolute deviations from 65 degrees on days that are colder than 65 Hence we sum the deviation from days 1 2 5 6 and 7 HDD5105152055 Cooling degree days CDD sums absolute deviations from 65 degrees on days that are warmer than 65 Hence we sum the deviation from days 3 and 4 CDD3710 Question 6 16 1 By selling gas to customers at a fixed price in advance of the heating season the natural gas company is worried about the case when natural gas prices are abnormally high In this case they are not able to raise their price to their customers and might have to