华东师范大学：《金融工程》英文版 Chapter 4 Interest rates and duration(久期)

4.1 Interest Rates and duration(久期) Chapter 4 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.1 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Interest Rates and Duration(久期) Chapter 4

4.2 Types of rates Treasury rates(国债利率)- regarded as risk-free rates LIBOR rates(London Interbank Offer ate)(伦敦银行同业放款利率) generally higher than Treasury zero rates Repo rates(回购利率) -slightly higher than the Treasury rates Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.2 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Types of Rates • Treasury rates（国债利率）—regarded as risk-free rates • LIBOR rates (London Interbank Offer rate) (伦敦银行同业放款利率)–generally higher than Treasury zero rates • Repo rates (回购利率)—slightly higher than the Treasury rates

4.3 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. In practice. it is usually called zero-coupon interest rate(零息票利率) Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.3 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University 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. In practice, it is usually called zero-coupon interest rate (零息票利率)

4.4 Example (Table 4.1, page 89) Maturity Zero Rate (years)(% cont comp) 0.5 5.0 1.0 58 1.5 64 2.0 68 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.4 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Example (Table 4.1, page 89) Maturity (years) Zero Rate (% cont comp) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8

4.5 Bond pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate In our example (page 89), the theoretical price of a two-year bond with a principal of $100 providing a 6% coupon semiannually is Be 0.05×0.5 +3e005800+3e-04.5 +103e 0.068×2.0 $98.39 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.5 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Bond Pricing • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate • In our example (page 89), the theoretical price of a two-year bond with a principal of $100 providing a 6% coupon semiannually is 103 $98.39 3 3 3 0.068 2.0 0.05 0.5 0.058 1.0 0.064 1.5 + = + + −  −  −  −  e e e e

4.6 Bond yield 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 ex0.5 +3ey10+3e-y 103ey×20 9839 to get y=0.0676 or 6.76% Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.6 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Bond Yield • 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 to get y=0.0676 or 6.76%. 3 3 3 103 98 39 0 5 1 0 1 5 2 0 e e e e − y − y − y − y + + + = . . . .

4.7 Par yield The par yield(面值收益率) for a certain maturity is the coupon rate that causes the bond price to equal its face value(ie. The principal). The bond is usually assumed to provide semIannual coupons In our example we solve 0.05×0.5 0.058×1.0 0.064×1.5 +一e +100+ 0.068×2.0 100 to get c=6.87 Tang Yincai, C 2003, Shanghai Normal University

4.7 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Par Yield • The par yield (面值收益率) for a certain maturity is the coupon rate that causes the bond price to equal its face value (ie. The principal). The bond is usually assumed to provide semiannual coupons. • In our example we solve c e c e c e c e c= . 2 2 2 100 2 100 687 0 05 0 5 0 058 1 0 0 064 1 5 0 068 2 0 −  −  −  −  + + + +       = . . . . . . . . to get

4.8 Par 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(年金)of$10 n each coupon date 100=A+100P→/c(100-100P)m Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.8 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Par 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(年金) of $1 on each coupon date ➔ c P m A = (100 − 100 ) 100 100P m c = A +

4.9 Sample data (Table 4.2, page 91) Bond Time to Annual Principal Maturity Coupon Price (dollars) years)(dollars)(dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 0 90.0 100 1.50 96.0 100 2.00 101.6 Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.9 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University Sample Data (Table 4.2, page 91)) Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6

4.10 The Bootstrap Method (息票剥离法) used to determine zero rates 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 10.13% with continuous compounding Similarly the 6 month and 1 year rates are 10. 47% and 10.,54% with continuous compounding Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University

4.10 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Bootstrap Method (息票剥离法) --used to determine zero rates • 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 10.13% with continuous compounding • Similarly the 6 month and 1 year rates are 10.47% and 10.54% with continuous compounding