【正文】 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 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 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 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。(),([)(22???r??r?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])()(][)()([)()(])。 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 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。 20xx by John C. Hull Measure 1 ? One monly used default correlation measure is the correlation between 1. A variable that equals 1 if pany A defaults between time 0 and time T and zero otherwise 2. A variable that equals 1 if pany B defaults between time 0 and time T and zero otherwise ? The value of this measure depends on T. Usually it increases at T increases. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Transition Matrix Consistent With Default Probabilities A B C Default A % % % % B % % % % C % % % % Default % % % 100% This is chosen to minimize difference between all elements of Mi1 d1 and the corresponding cumulative default probabilities implied by bond prices. Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Example Cumulative probability of default 1 2 3 4 5 A % % % % % B % % % % % C % % % % % Suppose there are three rating categories and riskneutral default probabilities extracted from bond prices are: 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 Reducing Credit Exposure (page 625) ? Collateralization ? Downgrade triggers ? Diversification ? Contract design ? Credit derivatives Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull The Loss Given Default (LGD) ? For derivatives we need to distinguish between a) those that are always assets, b) those that are always liabilities, and c) those that can be as