Examples o 10%with semiannual compounding is equivalent to 2In(1.05=9.758% with continuous compounding e% with continuous compounding is equivalent to 4(e0.08/4-1)=8.08% With quarterly compounding Rates used in option pricing are nearly always expressed with continuous compounding Options, Futures, and other Derivatives 8th Edition Copyright O John C Hull 2012
Examples 10% with semiannual compounding is equivalent to 2ln(1.05)=9.758% with continuous compounding 8% with continuous compounding is equivalent to 4(e 0.08/4 -1)=8.08% with quarterly compounding Rates used in option pricing are nearly always expressed with continuous compounding Options, Futures, and Other Derivatives 8th Edition, Copyright © John C. Hull 2012 11
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' Options, Futures, and other Derivatives 8th Edition Copyright o John C. Hull 2012
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 Options, Futures, and Other Derivatives 8th Edition, Copyright © John C. Hull 2012 12
Example Table 4.2, page 87) Maturity(years) Zero rate(cont. comp. 0.5 50 1.0 58 1.5 6.4 2.0 68 Options, Futures, and other Derivatives 8th Edition Copyright C John C Hull 2012
Example (Table 4.2, page 81) Options, Futures, and Other Derivatives 8th Edition, Copyright © John C. Hull 2012 13 Maturity (years) Zero rate (cont. comp. 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8
Bond pricing To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate o In our example, the theoretical price of a two year bond providing a 6%coupon semiannually is 0.05×0.5 +3e 0.058×1.0 0.064×1.5 +3e +103e 0.068×2.0 98.39 Options, Futures, and other Derivatives 8th Edition Copyright o John C. Hull 2012 14
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 twoyear bond providing a 6% coupon semiannually is Options, Futures, and Other Derivatives 8th Edition, Copyright © John C. Hull 2012 14 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 − − − − + + + = . . . . . . . .