【正文】 more elaborate model involves simulating the credit rating changes in each counterparty. ? This enables the credit losses arising from both credit rating changes and defaults to be quantified Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Structural Model Approach ? Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995), Zhou (1997) etc ? Company defaults when the value of its assets falls below some level. ? The default correlation between two panies arises from a correlation between their asset values Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Measure 2 ? Based on a Gaussian copula model for time to default. ? Define tA and tB as the times to default of A and B ? The correlation measure, rAB , is the correlation between uA(tA)=N1[QA(tA)] and uB(tB)=N1[QB(tB)] where N is the cumulative normal distribution function Options, Futures, and Other Derivatives, 5th edition 169。P, January 20xx, p626) Year End R ating Init Rat AAA AA A BBB BB B CCC D ef AAA 93 .66 3 0. 00 AA 6. 9 4 0. 01 A 0. 04 BBB 0. 24 BB 1. 08 B 5. 94 CCC 25. 26 D ef 100 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Volatilities s ?? s sE V VE EV V N d V0 0 1 0? ? ( )This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Possible Reasons for These Results ? The liquidity of corporate bonds is less than that of Treasury bonds ? Bonds traders may be factoring into their pricing depression scenarios much worse than anything seen in the last 20 years Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Cumulative Average Default Rates (%) (Table , page 619。 Expected Recovery Rate=30%) Bond Life Coupon (%) Yield (%) 1 2 3 4 5 10 Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Probability of Default 0 . 0 2 5 9 2 4 a n d 0 . 0 2 5 2 9 4 , 0 . 0 2 1 6 6 2 , 0 . 0 1 4 9 0 6 ,0 . 0 0 4 9 9 4 , a r e 5 a n d 4, , 3 2, 1, ye a r sin d e f a u l t of i e sp r o b a b i l i t e x a m p l e , o u r in 0 . 5R a t eR e c IfR a t e R e L o ss% E x p .D e f of P r o bL o ss% E x p . R a t e ) R e (1 D e f . of P r o b .????Options, Futures, and Other Derivatives, 5th edition 169。 continuously pounded) M a tu r ity( y e a r s)Ri sk f r e ey ieldCorpo r a teb o n d y ield1 5% 5 .2 5 %2 5% 5 .5 0 %3 5% 5 .7 0 %4 5% 5 .8 5 %5 5% 5 .9 5 %Options, Futures, and Other Derivatives, 5th edition 169。Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example (Zero coupon rates。 20xx by John C. Hull Recovery Rates (Table , page 614. Source: Moody’s Investor’s Service, 20xx) Cla s s Me an (%) SD (%) Seni or Secured Seni or Unsecured Seni or Subo rdinated Subo rdinated 8 Junio r Subo rdinated Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Sample Data (Riskfree Rate=5%。 20xx by John C. Hull Historical Data Historical data provided by rating agencies are also used to estimate the probability of default Options, Futures, and Other Derivatives, 5th edition 169。 t h e of 2 . 4 7 % or l o se to e x p e ct w et h a t m e a n s T h i s????eOptions, Futures, and Other Derivatives, 5th edition 169。w h e r esssOptions, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull OneYear Transition Matrix (Samp。 20xx by John C. Hull Measure 1 continued Denote QA(T) as the probability that pany A will default between time zero and time T, QB(T) as the probability that pany B will default between time zero and time T, and PAB(T) as the probability that both A and B will default. The default correlation measure is ])()(][)()([)()()()(22 TQTQTQTQTQTQTPTBBAABAABAB?????Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Modeling Default Correlations Two alternatives models of default correlation are: ? Structural model approach ? Reduced form approach Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Correlation between Credit Rating Changes ? The correlation between credit rating changes is assumed to be the same as that between equity prices ? We sample from a multivariate normal distribution and use the result to determine the rating change (if any) for each counterparty