期货期权及其衍生品配套课件全34章Ch04

1、interest rateschapter 4options, futures, and other derivatives 7th international edition, copyright john c. hull 20081types of rates treasury rates libor rates repo rates options, futures, and other derivatives 7th international edition, copyright john c. hull 20082measuring interest rates the compo

2、unding frequency used for an interest rate is the unit of measurement the difference between quarterly and annual compounding is analogous to the difference between miles and kilometersoptions, futures, and other derivatives 7th international edition, copyright john c. hull 20083continuous compoundi

3、ng(page 77) in the limit as we compound more and more frequently we obtain continuously compounded interest rates $100 grows to $100ert when invested at a continuously compounded rate r for time t $100 received at time t discounts to $100e-rt at time zero when the continuously compounded discount ra

4、te is roptions, futures, and other derivatives 7th international edition, copyright john c. hull 20084conversion formulas(page 77)definerc : continuously compounded raterm: same rate with compounding m times per yearoptions, futures, and other derivatives 7th international edition, copyright john c.

5、 hull 20085rmrmrm ecmmrmcln/11zero rates a zero rate (or spot rate), for maturity t is the rate of interest earned on an investment that provides a payoff only at time toptions, futures, and other derivatives 7th international edition, copyright john c. hull 20086example (table 4.2, page 79) options

6、, futures, and other derivatives 7th international edition, copyright john c. hull 20087m aturity(years)zero rate(% cont com p)0.55.01.05.81.56.42.06.8bond pricing to calculate the cash price of a bond we discount each cash flow at the appropriate zero rate in our example, the theoretical price of a

7、 two-year bond providing a 6% coupon semiannually isoptions, futures, and other derivatives 7th international edition, copyright john c. hull 2008833310398390 05 0 50 058 1 00 064 150 068 2 0eeee.bond yield the bond yield is the discount rate that makes the present value of the cash flows on the bon

8、d equal to the market price of the bond suppose that the market price of the bond in our example equals its theoretical price of 98.39 the bond yield (continuously compounded) is given by solving to get y=0.0676 or 6.76%.options, futures, and other derivatives 7th international edition, copyright jo

9、hn c. hull 2008933310398 390 51 01 52 0eeeeyyyy.par yield the par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. in our example we solveoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200810g)compoundin s.

10、a. (with get to87610021002220 . 2068. 05 . 1064. 00 . 1058. 05 . 005. 0.c=ececececpar yield continued in general if m is the number of coupon payments per year, p is the present value of $1 received at maturity and a is the present value of an annuity of $1 on each coupon dateoptions, futures, and o

11、ther derivatives 7th international edition, copyright john c. hull 200811cp ma()100100sample data (table 4.3, page 80) options, futures, and other derivatives 7th international edition, copyright john c. hull 200812bondtime toannualbond cashprincipalmaturitycouponprice(dollars)(years)(dollars)(dolla

12、rs)1000.25097.51000.50094.91001.00090.01001.50896.01002.0012101.6the bootstrap method an amount 2.5 can be earned on 97.5 during 3 months. the 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding this is 10.127% with continuous compounding similarly the 6 month and 1 year rates are

13、 10.469% and 10.536% with continuous compounding options, futures, and other derivatives 7th international edition, copyright john c. hull 200813the bootstrap method continued to calculate the 1.5 year rate we solve to get r = 0.10681 or 10.681% similarly the two-year rate is 10.808%options, futures

14、, and other derivatives 7th international edition, copyright john c. hull 20081496104445 . 10 . 110536. 05 . 010469. 0reeezero curve calculated from the data (figure 4.1, page 82) options, futures, and other derivatives 7th international edition, copyright john c. hull 200815zero rate (%)maturity (y

15、rs)10.12710.46910.53610.68110.808forward rates the forward rate is the future zero rate implied by todays term structure of interest ratesoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200816calculation of forward rates table 4.5, page 83 options, futures,

16、and other derivatives 7th international edition, copyright john c. hull 200817n-yearforward rate zero ratefor n th yearyear (n )(% per annum)(% per annum)13.024.05.034.65.845.06.255.36.5formula for forward rates suppose that the zero rates for time periods t1 and t2 are r1 and r2 with both rates con

17、tinuously compounded. the forward rate for the period between times t1 and t2 isoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200818r tr ttt221121instantaneous forward rate the instantaneous forward rate for a maturity t is the forward rate that applies fo

18、r a very short time period starting at t. it is where r is the t-year rateoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200819rtrtupward vs downward slopingyield curve for an upward sloping yield curve:fwd rate zero rate par yield for a downward sloping yi

19、eld curvepar yield zero rate fwd rateoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200820forward rate agreementa forward rate agreement (fra) is an agreement that a certain rate will apply to a certain principal during a certain future time periodoptions,

20、futures, and other derivatives 7th international edition, copyright john c. hull 200821forward rate agreementcontinued an fra is equivalent to an agreement where interest at a predetermined rate, rk is exchanged for interest at the market rate an fra can be valued by assuming that the forward intere

21、st rate is certain to be realizedoptions, futures, and other derivatives 7th international edition, copyright john c. hull 200822valuation formulas (equations 4.9 and 4.10, pages 86-87) value of fra where a fixed rate rk will be received on a principal l between times t1 and t2 is l(rkrf)(t2t1)exp(-

22、r2t2) value of fra where a fixed rate is paid isl(rfrk)(t2t1)exp(-r2t2) rf is the forward rate for the period and r2 is the zero rate for maturity t2 what compounding frequencies are used in these formulas for rk, rf, and r2?options, futures, and other derivatives 7th international edition, copyrigh

23、t john c. hull 200823duration (page 87-88) duration of a bond that provides cash flow ci at time ti iswhere b is its price and y is its yield (continuously compounded) this leads to options, futures, and other derivatives 7th international edition, copyright john c. hull 200824bectdiytinii1ydbbdurat

24、ion continued when the yield y is expressed with compounding m times per year the expression is referred to as the “modified duration”options, futures, and other derivatives 7th international edition, copyright john c. hull 200825myybdb1dy m1 convexity the convexity of a bond is defined asoptions, f

25、utures, and other derivatives 7th international edition, copyright john c. hull 20082621222)(211ycydbbbetcybbcniytiiithat sotheories of the term structurepage 91-92 expectations theory: forward rates equal expected future zero rates market segmentation: short, medium and long rates determined independently of each other liquid