【正文】 it by D ? Use Merton’s model to calculate a theoretical probability of default ? Use historical data or bond data to develop a onetoone mapping of theoretical probability into either realworld or riskneutral probability of default. 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 assets or liabilities ? What is the loss in each case? Options, Futures, and Other Derivatives, 5th edition 169。 20xx by John C. Hull Netting (page 624) Netting clauses state that is a pany defaults on one contract it has with a financial institution it must default on all such contracts ????????????NiiNiiVRVR110,m a x)1()0,m a x ()1( to f r o m l o ss t h e ch a n g e s T h i sOptions, 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 OneYear Transition Matrix (Samp。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 RiskNeutral Transition Matrix ? A riskneutral transition matrix is necessary to value derivatives that have payoffs dependent on credit rating changes ? A riskneutral transition matrix can (in theory) be determined from 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。 20xx by John C. Hull Matrix Implied Default Probability ? Let M be the annual rating transition matrix and di be the vector containing probability of default within i years ? d1 is the rightmost column of M ? di = M di1 = Mi1 d1 ? Number of free parameters in M is number of ratings squared 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 Credit Default Correlation ? The credit default correlation between two panies is a measure of their tendency to default at about the same time ? Default correlation is important in risk management when analyzing the benefits of credit risk diversification ? It is also important in the valuation of some credit derivatives 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 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 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 Use of Gaussian Copula ? The Gaussian copula measure