# Interest Rate Futures Chapter 6 Options Futures and

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Interest Rate Futures Chapter 6 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 1

Day Count Conventions in the U. S. (Page 129) The interest earned between two dates is: Number of days between dates x Interest earned in reference period Number of dates in reference period Example: Coupon dates are March 1 and September 1. We wish to calculate the interest between March 1 and July 3, with coupon 8% Treasury Bonds: Actual/Actual [(124/184)*4] Corporate Bonds: 30/360 [(122/180)*4] Money Market Instruments: Actual/360 [(124/180)*4] Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 2

Treasury Bond Price Quotes in the U. S Cash price = Quoted price + Accrued Interest since last coupon date Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 3

Example l l l Consider an 11% coupon bond maturing on July 10, 2012 with a price today (March 5, 2007) \$95. 50. The most recent coupon date is January 10 The number of days between January 10 and March 5 is 54, while between January 10 and July 10 is 181 The accrued interest is (54/181)*\$5. 5=\$1. 64 The cash price per \$100 face value is \$97. 14 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 4

Treasury Bond Futures Pages 133 -137 The delivery is any government bond with more than 15 years to maturity, which is not callable within 15 years Cash price received by party with short position = Most Recent Settlement Price × Conversion factor + Accrued interest Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 5

Example Settlement price of bond delivered = 90. 00 l Conversion factor = 1. 3800 l Accrued interest on bond =3. 00 l Price received for bond is (1. 3800× 90. 00)+3. 00 = \$127. 20 per \$100 of principal The party with the short position in one contract would deliver bonds with a face value of \$100. 000 and receive \$127. 20. l Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 6

Conversion Factor The conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semiannual compounding Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 7

Example l l l Consider an 10% coupon bond with 20 years Coupon payments are assumed to be made every 6 months The face value is \$100 When the discount rate is 6% per annum with semiannual compounding: Dividing by the face value gives a conversion factor of 1. 4623 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 8

Cheapest-to-deliver bond Any of a range of eligible bonds can be delivered l The party with the short position receives: Settlement Price × Conversion factor + Accrued interest l The cost of purchasing a bond is Quoted bond price+Accrued Interest l The cheapest-to-deliver bond is the one for which Quoted bond price-(Settlement Price × Conversion factor) is least. l Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 9

Example Quoted bond price 99. 50 143. 50 119. 75 Conversion factor 1. 0382 1. 5188 1. 2615 The most recent settlement price is 93. 25. 1. 99. 50 -(93. 25*1. 0382)=\$2. 69 2. 143. 50 -(93. 25*1. 5188)=\$1. 87 3. 119. 75 -(93. 25*1. 2615)=\$2. 212 Thus, the cheapest-to-deliver is bond 2 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 10

Determining The Futures Price l l We assume that both the cheapest-to-delivery bond and the delivery date are known The Treasury bond futures contract is the futures contract on a security providing the holder with known income: F 0 = (S 0 – I )er. T With I the present value of the coupons during the life of the futures contract Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 11

Example l l l The cheapest-to-deliver is a 12% coupon bond with a conversion factor of 1. 400 The delivery will take place in 270 days. Last coupon date was 60 days ago Next coupon date is in 122 days, and the coupon date thereafter is in 35 days The term structure is flat, the interest rate is 10% The current quoted bond price is \$120 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 12

Example (continued) The cash price is 120+(60/(60+122))*6=121. 978 l A coupon of 6% will be received after 122 days. The present value is 6 e-0. 1*0. 3342=5. 803 l The futures contract lasts for 270 days. The cash futures price is (121. 978 -5. 803) e 0. 1*0. 7397=125. 094 l The quoted futures price is calculated by first subtracting the accrued interest 125. 094 -6*(148/(148+35))=120. 242 The conversion factor is 1. 400. Thus, the quoted futures price should be 120. 242/1. 40=85. 887 l Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 13

Eurodollar Futures (Page 137 -142) A Eurodollar is a dollar deposited in a bank outside the United States l Eurodollar futures are futures on the 3 -month Eurodollar deposit rate (same as 3 -month LIBOR rate) l One contract is on the rate earned on \$1 million l A change of 1 basis point or 0. 01% in a Eurodollar futures quote corresponds to a contract price change of \$25 : 1. 000*0. 0001*0. 25=25 l When there is an increase in 1 bp, a trader who is long 1 contract gains \$25, while the short loses \$25 Options, Futures, and Other Derivatives 6 Edition, Copyright © John C. Hull 2005 l th 6. 14

Eurodollar Futures continued l l A Eurodollar futures contract is settled in cash When it expires (on the third Wednesday of the delivery month) the final settlement price is 100 minus the actual three month deposit rate: 100 - R Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 15

Example l l Suppose you buy (take a long position in) a contract on November 1 The contract expires on December 21 The prices are as shown How much do you gain or lose a) on the first day, b) on the second day, c) over the whole time until expiration? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 16

Example Date Nov 1 Quote 97. 12 Nov 2 97. 23 Nov 3 96. 98 ……. …… Dec 21 97. 42 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 17

Example continued l l l If on Nov. 1 you know that you will have \$1 million to invest on for three months on Dec 21 the contract locks in a rate of 100 - 97. 12 = 2. 88% In the example you earn 100 – 97. 42 =2. 58% on \$1 million for three months (=\$6, 450) and make a gain day by day on the futures contract of 30×\$25 =\$750 Total gain: \$6. 450+\$750=\$7. 200 which is equal to 1. 000*0. 25*0. 0288 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 18

Formula for Contract Value (page 138) If Q is the quoted price of a Eurodollar futures contract, the value of one contract is 10, 000[100 -0. 25(100 -Q)] l In the above example, the settlement price of 97. 12 corresponds to a contract price of 10. 000[100 -0. 25*(100 -97. 12)]=\$992. 800 l The final contract price is 10. 000[100 -0. 25*(100 -97. 42)]=\$993. 550 The difference is \$750. This is the gain of an investor with long position l Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 19

Forward Rate Agreements vs Futures Interest rates l l Futures is settled daily where forward is settled once Futures is settled at the beginning of the underlying three-month period; forward is settled at the end of the underlying threemonth period Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 20

Forward Rates and Eurodollar Futures continued Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 21

Convexity Adjustment when s=0. 012 (Table 6. 3, page 141) Maturity of Futures 2 Convexity Adjustment (bps) 3. 2 4 12. 2 6 27. 0 8 47. 5 10 73. 8 Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 22

Extending the LIBOR Zero Curve l l LIBOR deposit rates define the LIBOR zero curve out to one year Eurodollar futures can be used to determine forward rates (using the convexity adjustment) and the forward rates can then be used to bootstrap the zero curve Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 23

Example so that If the 400 day LIBOR rate has been calculated as 4. 80% and the forward rate for the period between 400 and 491 days is 5. 30 the 491 days rate is 4. 893% Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 24

Duration Matching l l This involves hedging against interest rate risk by matching the durations of assets and liabilities It provides protection against small parallel shifts in the zero curve Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 25

Use of Eurodollar Futures l l One contract locks in an interest rate on \$1 million for a future 3 -month period How many contracts are necessary to lock in an interest rate for a future six month period? Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 26

Duration-Based Hedge Ratio FC Contract price for interest rate futures DF Duration of asset underlying futures at maturity P Value of portfolio being hedged DP Duration of portfolio at hedge maturity Since ΔP=-P DP Δy, and it is approximately true that ΔFC=-FC DF Δy, the number of contracts required is: Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 27

Example l l l It is August. A fund manager has \$10 million invested in a portfolio of government bonds with a duration of 6. 80 years and wants to hedge against interest rate moves between August and December The manager decides to use December T-bond futures. The futures price is 93 -02 or 93. 0625 and the duration of the cheapest to deliver bond is 9. 2 years The number of contracts that should be shorted is If Interest rates go up, a gain will be made on the short futures, but a loss will be made on the bond portfolio. If interest rates decrease, the opposite is true Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 28

Limitations of Duration-Based Hedging l l Assumes that only parallel shift in yield curve take place Assumes that yield curve changes are small Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull 2005 6. 29