26.16 Relaxing assumptions This analysis assumes constant interest rates, and known recovery rates and claim amounts If default events, risk-free rates, and recovery rates are independent, results hold for stochastic interest rates and uncertain recovery rates providing the recovery rate is set equal to its expected value in a risk- neutral world Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.16 Relaxing Assumptions • This analysis assumes constant interest rates, and known recovery rates and claim amounts • If default events, risk-free rates, and recovery rates are independent, results hold for stochastic interest rates, and uncertain recovery rates providing the recovery rate is set equal to its expected value in a riskneutral world, R ˆ
Extending the analysis to allow Defaults at Any Time The analysis can be extended to allow defaults at any time It is important to distinguish between the default probability density and the hazard rate The default probability density, q( is defined so that qost as the probability of default between times t and t+ot as seen at time zero The hazard rate is the probability of default between times t and t+ot conditional on no earlier default Ih(r)dr (t)=h()e Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.17 • The analysis can be extended to allow defaults at any time • It is important to distinguish between the default probability density and the hazard rate • The default probability density, q(t) is defined so that q(t)dt as the probability of default between times t and t+dt as seen at time zero • The hazard rate is the probability of default between times t and t+dt conditional on no earlier default = − t h d q t h t e 0 ( ) ( ) ( ) Extending the Analysis to Allow Defaults at Any Time
What Should We Use as the a.18 Clain△ mount The best assumption seems to be that the claim amount for a bond equals the face value plus accrued interest---not he no-default value Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.18 What Should We Use as the Claim Amount The best assumption seems to be that the claim amount for a bond equals the face value plus accrued interest --- not the no-default value
26.19 Sample data( risk-free rate=5% Expected Recovery rate=30%) Bond Life Coupon (%)Yield(%) 7.0 6.6 7.0 6.7 2345 7.0 68 7.0 69 7.0 7.0 10 7.0 7.2 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.19 Sample Data (Risk-free Rate=5%; Expected Recovery Rate=30%) Bond Life Coupon (%) Yield (%) 1 7.0 6.6 2 7.0 6.7 3 7.0 6.8 4 7.0 6.9 5 7.0 7.0 10 7.0 7.2
26.20 Implied Default Probabilities assuming That Default Can Happen on Bond Maturity Dates Table 26.5, page 617) Time Claim =No-Claim=Face (yrs) Def Value Val+Accr Int 0.0224 0.0224 12345 0.0249 0.0247 0.0273 0.0269 0.0297 0.0291 0.0320 0.0312 10 0.1717 0.1657 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 26.20 Implied Default Probabilities Assuming That Default Can Happen on Bond Maturity Dates (Table 26.5, page 617) Time (yrs) Claim = NoDef Value Claim=Face Val+Accr Int 1 0.0224 0.0224 2 0.0249 0.0247 3 0.0273 0.0269 4 0.0297 0.0291 5 0.0320 0.0312 10 0.1717 0.1657