【導(dǎo)讀】Chapter26. CreditRisk. CreditRatings. IntheS&Pratingsystem,AAAisthe. BBB,BB,B,andCCC. Aaa,Aa,A,Baa,Ba,B,andCaa. grade”。ratings. world. TypicalPattern. Spread. over. Treasuries. Maturity. Baa/BBB. A/A. Aa/AA. Aaa/AAA. TheRisk-FreeRate. therisk-freerate. defaults. Example(Zerocouponrates;Maturity. Risk-free. yield. Corporate. bondyield. 15%5.25%. 25%5.50%. 35%5.70%. 45%5.85%. 55%5.95%. Examplecontinued. One-yearrisk-freebond(principal=$1)sellsfor. One-yearcorporatebond(principal=$1)sellsfor. orata%discount. e???00510951229..e???0052510948854..Examplecontinued. bondexpectstolose. or%inthefirsttwoyears. expectedlossis%. ee. e. 0052005502. 00520009950. Examplecontinued. successiveyearsare%,%,%,%,and. %. SummaryofResults. Maturity. Cumul.Loss.%. Loss. DuringYr(%). 10.24970.2497. 20.99500.7453. 32.07811.0831. 43.34281.2647. 54.63901.2962. RecoveryRates. ClassMean(%)SD(%). SeniorSecured. SeniorUnsecured. SeniorSubordinated. Subordinated8. JuniorSubordinated. 0.004994,are5and4,,32,1,yearsindefaultof. RateRe. Loss%Exp.DefofProb. Loss%Exp.Rate)Re(1Def.ofProb.

　　

【正文】 is often used in practice because it focuses on the things we are most interested in (Whether a default happens and when it happens) ? Suppose that we wish to simulate the defaults for n panies . For each pany the cumulative probabilities of default during the next 1, 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Use of Gaussian Copula continued ? We sample from a multivariate normal distribution for each pany incorporating appropriate correlations ? N 1() = , N 1() = , N 1() = , N 1() = , N 1() = Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Use of Gaussian Copula continued ? When sample for a pany is less than , the pany defaults in the first year ? When sample is between and , the pany defaults in the second year ? When sample is between and , the pany defaults in the third year ? When sample is between 1,55 and , the pany defaults in the fourth year ? When sample is between and , the pany defaults during the fifth year ? When sample is greater than , there is no default during the first five years Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Measure 1 vs Measure 2 n o r m a l tem u l t i v a r i a be to a ss u m e d be ca n t i m e s su r v i v a l dt r a n sf o r m e b e ca u se co n si d e r e d a r e co m p a n i e sm a n y w h e nu se to e a si e r m u ch is It 1. M e a su r e t h a n h i g h e rt l y si g n i f i ca nu su a l l y is 2 M e a su r ef u n ct i o n . ond i st r i b u t iy p r o b a b i l i t n o r m a l b i v a r i a t e cu m u l a t i v e t h e is w h e r ea n d:v e r sa v i ce a n d 2 M e a su r e f r o m ca l cu l a t e d be ca n 1 M e a su r eMTQTQTQTQTQTQTuTuMTTuTuMTPBBAABAABBAABABBAAB])()(][)()([)()(])。(),([)(])。(),([)(22???r??r?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 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 Reduced Form Approach ? Lando(1998), Duffie and Singleton (1999), Jarrow and Turnbull (20xx), etc ? Model the hazard rate as a stochastic variable ? Default correlation between two panies arises from a correlation between their hazard rates Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Pros and Cons ? Reduced form approach can be calibrated to known default probabilities. It leads to low default correlations. ? Structural model approach allows correlations to be as high as desired, but cannot be calibrated to known default probabilities. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Credit VaR (page 630) Credit VaR asks a question such as: What credit loss are we 99% certain will not be exceeded in 1 year? Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Basing Credit VaR on Defaults Only (CSFP Approach) ? When the expected number of defaults is m, the probability of n defaults is ? This can be bined with a probability distribution for the size of the losses on a single default to obtain a probability distribution for default losses enn?mm!Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Enhancements ? We can assume a probability distribution for m. ? We can categorize counterparties by industry or geographically and assign a different probability distribution for expected defaults to each category Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Model Based on Credit Rating Changes (Creditmetrics) ? A 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 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