Chapter4InterestRatesandDuration期权期货及其衍生市场 厦门大学

1、 Chapter 4 Interest Rates and Duration Types of Rates ? Treasury rates ? LIBOR rates ? Repo rates Zero 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 T Example ? Maturity(years)0.51.01.52.0Zero Rate(% cont comp)

2、5.05.86.46.8Bond 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 two-year bond providing a 6% coupon semiannually is ?0.05?0.5?0 .058?1 .0?0.064?1 .5?3 e?3 e 3 e?0.068?2.0 ?103 e?98 .39 Bond Yield(到期

3、收益率) ? The bond yield is the discount rate that makes the present value of the cash flows on the bond 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 is given by solving ?y?0.5?y?1 .0?y?1 .5?y?2.0

4、 3 e?3 e?3 e?103 e?98 .39 to get y=0.0676 or 6.76%. 问题:请问投资者按市价购买该债券并持有到期,其实际收益率等于多少? Par Yield(平价收益率) ? 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 solve ?0.05?0.5c?0.058?1.0c?0.064?1.5ce?e?e222 c?0.068?2.0? ?100?e?

5、100?2?to get c=6.87Par 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 date,then c 100?A?100 P,或m (100?100 P)mc?A问题 ? 假设你是财政部国债司司长,你的目标是使国债发行实际收入尽量等于计划收入,你应如何确

6、定国债的票面利率? Determining Zero Rate ? Sample Data Bond Principal (dollars) 100 100 100 100 100 Time to Maturity (years) 0.25 0.50 1.00 1.50 2.00 Annual Coupon (dollars) 0 0 0 8 12 Bond Price (dollars) 97.5 94.9 90.0 96.0 101.6 100 2.75 10 99.8 The Bootstrap Method ? An amount 2.5 can be earned on 97.5 d

7、uring 3 months. ? The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding ? This is 10.13% with continuous compounding 0 .102564ln( 1?)?0 .10127 4? Similarly the 6 month and 1 year rates are 10.47% and 10.54% with continuous compounding ? The Bootstrap Method continued ? To calcul

8、ate the 1.5 year rate we solve ?0.1047?0.5?0.1054?1 .0?R?1 .54e?4e?104 e?96? ? to get R = 0.1068 or 10.68% ? Similarly the two-year rate is 10.81% The Bootstrap Method continued ? The cash flows of the sixth bond are: 3 months later $5 9 months later $5 1.25 years later $5 1.75 years later $5 2.25 y

9、ears later $5 2.75 years later $105 The Bootstrap Method continued ? Using interpolation Method(线性插值法)to find the other zero rates: 9 months 10.505%(=(10.47%+10.54%)/2) 1.25years 10.61% 1.75years 10.745% ? So the present value of the first four cash flows is: ?.1013?.25?.10505?.75?.1061?1 .25?.10745

10、?1 .755 e?5 e?5 e?5 e?18 .018 The Bootstrap Method cont. ? The present value of the last two cash flows is: 99.8-18.018=81.782 ? Let the 2.75-year zero rate is R,using interpolation method we can find the 2.25-year zero rate: .10812/3+R/3=.0721+R/3 ? So we have: 5 e?(.0721?R/3 )?2.25?e?R?2?81 .782?

11、Solved by trial and error: R=0.1087 Zero Curve Calculated from the Data 12? 1110.469 1010.681 10.536 10.808 10.87 10.127 900.511.522.5Forward Rates ? The forward rate is the future zero rate implied by todays term structure of interest rates Calculation of Forward Rates ? n ) Year ( 1 2 3 Zero Rate

12、for (% per annum) 10.0 10.5 10.8 11.0 11.4 Forward Rate (% per annum) n th Year an n -year Investment for 4 5 11.0 11.1 11.6 11.5 Formula for Forward Rates ? Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded. ? The forward rate for the perio

13、d between times T1 and T2 is R2T2?R1T1 T2?T1Instantaneous Forward Rate ? The instantaneous forward rate(RF) for a maturity T is the forward rate that applies for a very short time period starting at T. It is ?RR?T ?T where R is the T-year rate ? Because RF=R2+(R2-R1)T1/(T2-T1) Upward vs Downward Slo

14、ping Yield Curve ? For an upward sloping yield curve: Fwd Rate Zero Rate Par Yield ? For a downward sloping yield curve Par Yield Zero Rate Fwd Rate Forward Rate, Zero Rate and Par Yield ? Interest Rate Forward Rate Par Yield Zero Rate T Theories of the Term Structure ? Expectations Theory: forward

15、rates equal expected future zero rates ? Market Segmentation: short, medium and long rates determined independently of each other ? Liquidity Preference Theory: forward rates higher than expected future zero rates 远期利率协议远期利率协议 ? 远期利率协议是空方承诺在未来的某个时刻（T时刻）将一定数额的名义本金（A）按约定的合同利率（rK）在一定的期限（T*-T）贷给多方的远期协议

16、。 ?AerK(T*?T)远期利率协议的定价远期利率协议的定价 ?多方（即借入名义本金的一方）的现金流为： T时刻：A T*时刻： rk(T*?T)Ae这些现金流的现值即为远期利率协议多头的价值。 远期利率协议的定价（远期利率协议的定价（2） ? 为此，我们要先将T*时刻的现金流用T*-T期限的远期利率贴现到T时刻，再贴现到现在时刻t。 ? f?Ae?r(T?t)?AerK(T*?T)?e?r(T*?T)?e?r(T?t)?(rK?r)( T*?T)?r(T?t)?Ae?1?e?Day Count Conventions in the U.S. ? Treasury Bonds: Actual

17、/Actual (in period) ? Corporate Bonds:30/360 ? Money Market Instruments: Actual/360 Treasury Bond Price Quotes in the U.S ? Cash price = Quoted price + Accrued Interest Treasury Bill Quote in the U.S. ? If Y is the cash price of a Treasury bill that has n days to maturity, the quoted price is: 360 (

18、100?Y) n? Its called discount rate(折扣率). Its the annualized dollar return expressed as a percentage of the par value. Its not the same as the rate of return investors really earned. 中长期国债期货的定价 ?中长期国债属附息票债券，属支付已知现金收益的证券，因此公式（3.4）和（3.5）适用于中长期国债期货的定价。只是由于其报价和交割制度的特殊性，使这些公式的运用较为复杂而已。 ? 以下我们以美国芝加哥交易所的长期国

19、债期货为例来说明其定价问题，其结论也适用于中期国债期货。 长期国债现货和期货的报价与现长期国债现货和期货的报价与现金价格的关系金价格的关系 ?长期国债期货的报价与现货一样，以美元和32分之一美元报出，所报价格是100美元面值债券的价格，由于合约规模为面值10万美元，因此90 25的报价意味着面值10万美元的报价是90，781.25美元。 ? 应该注意的是，报价（净价）与购买者所支付的现金价格（全价）是不同的。全价（即期货价格）与净价的关系为： 全价=净价+上一个付息日以来的累计利息 交割券与标准券的转换因子交割券与标准券的转换因子 ?芝加哥交易所规定，空头方可以选择期限长于15年且在15年内不

20、可赎回的任何国债用于交割。由于各种债券息票率不同，期限也不同，因此芝加哥交易所规定交割的标准券为期限15年、息票率为8%的国债，其它券种均得按一定的比例折算成标准券。这个比例称为转换因子（Conversion Factor ）。 ?转换因子等于面值为100美元的各债券的现金流按8%的年利率（每半年计复利一次）贴现到交割月第一天的价值，再扣掉该债券累计利息后的余额。在计算转换因子时，债券的剩余期限只取3个月的整数倍，多余的月份舍掉。如果取整数后，债券的剩余期限为半年的倍数，就假定下一次付息是在6个月之后，否则就假定在3个月后付息。转换因子由交易所计算并公布。 空方收到的现金 ?算出转换因子后，我

21、们就可算出空方交割100美元面值的债券应收到的现金： ? 空方收到的现金=期货净价?交割债券的转换因子+交割债券的累计利息 确定交割最合算的债券确定交割最合算的债券 ?由于转换因子制度固有的缺陷和市场定价的差异决定了用何种国债交割对于双方而言是有差别的，而空方可选择用于交割的国债多达30种左右，因此空方应选择最合算的国债用于交割。 ?交割最合算债券就是购买交割券的成本与空方收到的现金之差最小的那个债券。 ?交割差距=债券净价+累计利息（期货净价?转换因子）+累计利息 =债券净价（期货净价?转换因子） 国债期货价格的确定国债期货价格的确定 ?由于国债期货的空方拥有交割时间选择权和交割券种选择权，

22、因此要精确地计算国债期货的理论价格也是较困难的。但是，如果我们假定交割最合算的国债和交割日期是已知的，那么我们可以通过以下四个步骤来确定国债期货价格： ?根据交割最合算的国债的净价，算出该交割券的全价。 ?根据交割券的全价算出交割券期货的理论全价。 ?根据交割券期货的全价算出交割券期货的净价。 ?将交割券期货的净价除以转换因子即为标准券期货净价，也是标准券期货的全价。 Eurodollar Futures ? The variable underlying the 3-month Eurodollar futures is the 3-month Eurodollar interest rat

23、e(LIBOR) applicable to a 90-day period rdbeginning on the 3 Wed. of the delivery month. ? If Z is the quoted price of a 3-month Eurodollar futures contract, the value of one contract is 10,000100-0.25(100-Z )(注意期货与现货报价的不同). ? A change of one basis point or 0.01 in a Eurodollar futures quote correspo

24、nds to a contract price change of $25 Eurodollar Futures continued ? A Eurodollar futures contract is settled in cash ? When it expires (on the third Wednesday of the delivery month) Z is set equal to 100R(the 90 day Eurodollar interest rate) and all contracts are closed out Forward Rates and Eurodo

25、llar Futures ? Eurodollar futures contracts last out to 10 years ? For Eurodollar futures lasting beyond two years we cannot assume that the forward rate equals the futures rate Forward Rates and Eurodollar Futures continued A convexity adjustment often made is12Forward rate=Futures rate?t1t22where

26、t1 is the time to maturity of the futures contract, t2 is the maturity of the rate underlying the futures contract(90 days later than t1) and ? is thestandard deviation of the short rate changesper year (typically ? is about 0 .012 )Duration ? Duration of a bond that provides cash flow c i at time t

27、 i is ? yti n ?c e? where B is its price & y is its yield (continuously compounded) ? This leads to B ?i? 1?t i?iB?B? ?D?y Duration Cont. ? Its because: n ?B?i?1cie?ytin?B?yti? ?ticie? ?BDi?1?y?B? ?D?yBDuration Continued ? When the yield y is expressed with annual compounding, we have Macaulay D

28、uration(DM): ct t (1?y)DM?tt?1BnDuration cont. c1c2cn? B?2n1?y(1?y)(1?y) ?B(?1 )c1(?2 )c2(?n)cn?23n?1?y(1?y)(1?y)(1?y)1ntct? ?t1?yt?1(1?y)?B11so,? ?DM?yB1?yDuration Cont. ? Let 1?D?DM 1?y ? D is called modified duration. ? We have: ?B ? ?D?yBDuration Cont. ? Generally,when y is expressed with a compounding frequency of m times per year, we have: ?B B?D?yDMwhere ,D?1?y/mDuration Matching ? This involves hedging against interest rate risk by matching the durations of assets and liabilities ? It provides protec