《金融期货与期权》（英文版） Chapter 5 Interest Rate Markets

5.1 Interest rate Markets Chapter 5 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.1 Interest Rate Markets Chapter 5

5.2 Types of Rates Treasury rates ·L| BOR rates Repo rates Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.2 Types of Rates • Treasury rates • LIBOR rates • Repo rates

53 Zero rates a zero rate(or spot rate), for maturity T is the rate of interest earned on an nvestment that provides a payoff only at time t Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.3 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

5.4 Example(Table 5.1, page 95) Maturity Zero Rate (years)(% cont comp) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.4 Example (Table 5.1, page 95) Maturity (years) Zero Rate (% cont comp) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8

5.5 Bond Pricing o 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 e +3e +3e +103e-000×20=9839 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.5 Bond 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 3 3 3 103 98 39 0 05 0 5 0 058 1 0 0 064 1 5 0 068 2 0 e e e e −  −  −  −  + + + = . . . . . . . .

5.6 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 our example equals its theoretical price of 98.39 The bond yield is given by solving Fe xo 5 +3e-y×10 +3e-y×15+103e-y×20=9839 y×1.5 to get y=0.0676 or 6.76% Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.6 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 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 + + + = . . . .

5.7 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 solve C e005×0.5C-005810,C064×15 +-e +100+-e -0.068×2.0 100 to get c=6.87(with s a compoundin g) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.7 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 solve to get 6 87 (with s.a. compoundin g) 100 2 100 2 2 2 0.068 2.0 0.0 5 0.5 0.058 1.0 0.064 1.5 c= . e c e c e c e c  =      + + + + −  −  −  − 

5.8 Par yield continued In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date (100-1004)r Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.8 Par Yield continued In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date A d m c (100 −100 ) =

Sample data for determining the 5.9 Zero Curve(Table 5.2, page 97) Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) years)(dollars)(dollars) 100 0.25 97.5 100 0.50 0008 94.9 100 1.00 90.0 100 1.50 96.0 100 2.00 12 101.6 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Sample Data for Determining the 5.9 Zero Curve (Table 5.2, page 97) 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

5.10 The bootstrapping the zero Curve 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 10.469% and 10.536%with continuous compounding Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.10 The Bootstrapping the Zero Curve • 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 10.469% and 10.536% with continuous compounding