Credit Derivatives: From the simple to the more advanced

1. Credit Derivatives:From the simple to the more advanced Jens Lund 2 March 2005

2. Outline • CDS • Hazard Curves • CDS pricing • Credit Triangle • Index CDS • Basket credit derivatives, n-to-default, CDO • Standardized iTraxx tranches, implied correlation • Gaussian copula model • Correlation smile • Pricing of basket credit derivatives • Implementation strategies • Subjects not mentioned • Conclusion Credit Derivatives: From the simple to the more advanced

3. CDS Cash flow Premium leg: Protection buyer Protection seller Continues until maturity or default Spread Only in the event of default Protection leg: a) cash settlement Protection buyer Protection seller 100 - Recovery b) physical settlement Bond Protection buyer Protection seller 100 Credit Derivatives: From the simple to the more advanced

4. Hazard curves • A distribution of default times can be described by • The density f(t) • The cumulative distribution function • The survival function S(t) = 1-F(t) • The hazard (t) = f(t)/S(t) • Interpretation • P(T in [t,t+dt[)  f(t)dt • P(T in [t,t+dt[|T>t)  (t)dt • Conections (t) f(t) Credit Derivatives: From the simple to the more advanced

5. CDS Pricing Model • CDS pricing models takes a lot of input • Length of contract • Risk free interest rate structure • Default probabilities of the reference entity for any given horizon • Expected recovery rate • Conventions: day count, frequency of payments, date roll etc. • PV of the CDS payments: Payment in the event of default Discount factor Probability of default at time t Premium payments Accrual factor Discount factor Survival probability Credit Derivatives: From the simple to the more advanced

6. Credit Triangle - What Determines the Spread? Assume hazard rate is constant Assume premium is paid continuously Credit Derivatives: From the simple to the more advanced

7. Index CDS • Simply a collection of, say, 100, single name CDS. • Each name has notional 1/100 of the index CDS notional. • Spread is lower than average of CDS spreads: • Intuition: the low spreads are paid for a longer time period than the high spreads. • PV01n = value of premium leg for name n • Not correlation dependent Credit Derivatives: From the simple to the more advanced

8. First-to-Default Basket • Alternative to buying protection on each name • Usually cheaper than buying protection on the individual names • Pays on the first (and only the first) default • Spread depends on individual spreads and default correlation Premium leg: Basket buyer Basket seller First-to-default Spread Continues until the first default or until maturity Protection leg: Basket buyer Basket seller 100 – Recovery on defaulted asset Only in the event of default, and only the first default Credit Derivatives: From the simple to the more advanced

9. Standardized CDO tranches 100% • iTraxx Europe • 125 liquid names • Underlying index CDSes for sectors • 5 standard tranches, 5Y & 10Y • First to default baskets, options • US index CDX • Has done a lot to provide liquidity • in structured credit • Reliable pricing information available  • Implied correlation information 88% Super senior 22% Mezzanine 12% 9% 6% 3% 3% equity Credit Derivatives: From the simple to the more advanced

10. Reference Gaussian copula model • N credit names, i = 1,…,N • Default times: ~ •  curves bootstrapped from CDS quotes • Ti correlated through the copula: • Fi(Ti) = (Xi) with X = (X1,…,XN)t ~ N(0,) • Note: Fi(Ti) = (Xi)  U[0,1] •  correlation matrix, variance 1, constant correlation  • In model: correlation independent of product to be priced Credit Derivatives: From the simple to the more advanced

11. Prices in the market has a correlation smile • In practice: • Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe • Tranche • Maturity Credit Derivatives: From the simple to the more advanced

12. Why do we see the smile? • Spreads not consistent with basic Gaussian copula • Different investors in • different tranches have • different preferences • If we believe in the Gaussian model: • Market imperfections are present and we can arbitrage! • However, we are more inclined to another conclusion: • Underlying/implied distribution is not a Gaussian copula Credit Derivatives: From the simple to the more advanced

13. Compound correlations • The correlation on the individual tranches • Mezzanine tranches have low correlation sensitivity and • even non-unique correlation for given spreads • No way to extend to, say, 2%-5% tranche • or bespoke tranches • What alternatives exists? Credit Derivatives: From the simple to the more advanced

14. Base correlations • Started in spring 2004 • Quote correlation on all 0%-x% tranches • Prices are monotone in correlation, i.e. uniqueness • 2%-5% tranche calculated as: • Long 0%-5% • Short 0%-2% • Can go back and forth between base and compound correlation • Still no extension to bespoke tranches Credit Derivatives: From the simple to the more advanced

15. Base correlations Short Long Credit Derivatives: From the simple to the more advanced

16. Base versus compound correlations Credit Derivatives: From the simple to the more advanced

17. Is base correlations a real solution? • No, it is merely a convenient way of describing prices on CDO tranches • An intermediate step towards better models that exhibit a smile • No general extension to other products • No smile dynamics • Correlation smile modelling, versus • Models with a smile and correlation dynamics Credit Derivatives: From the simple to the more advanced

18. Implementation of Gaussian copula • Factor decomposition: • M, Zi independent standard Gaussian, • Xi low  early default • FFT/Recursive: • Given T: use independence conditional on M and calculate loss distribution analyticly, next integrate over M • Simulation: • Simulate Ti, straight forward • Slower, in particular for risk, but more flexible • All credit risk can be calculated in same simulation run as the basic pricing Credit Derivatives: From the simple to the more advanced

19. 100% 12% 9% 6% 3% Pricing of CDOs by simulation • 100 names • Make, say, 100000 simulations: • Simulate default times of all 100 names • Price value of cash-flow in that scenario • Do it all again, 100000 times • Price = average of simulated values 88% Super senior Mezzanine 3% equity Credit Derivatives: From the simple to the more advanced

20. Default time simulationHazard and survival curve S = exp(-H*time) Credit Derivatives: From the simple to the more advanced

21. From Gaussian distribution to default time Credit Derivatives: From the simple to the more advanced

22. 1000 simulations 2 names 2 dimensions In general 100 names Gaussian/Normal distribution Transformed to survival time Correlation between 2 names Credit Derivatives: From the simple to the more advanced

23. Credit Derivatives: From the simple to the more advanced

24. Credit Derivatives: From the simple to the more advanced

25. Credit Derivatives: From the simple to the more advanced

26. Credit Derivatives: From the simple to the more advanced

27. Credit Derivatives: From the simple to the more advanced

28. Default Times, Correlation = 1Companies Have Different Spreads • Spread Company A = 300 • Spread Company B = 600 • Correlation = 1 • Note that when A defaults we always know when B defaults... • …but note that they never default at the same time • B always defaults earlier Credit Derivatives: From the simple to the more advanced

29. Correlation • High correlation: • Defaults happen at the same quantile • Not the same as the same point in time! • Corr = 100% • First default time: look at the name with the highest hazard (CDS spread) • Low correlation: • Defaults are independent • Corr = 0% • First default time: Multiplicate all survival times: 0.95^100 = 0.59% • Default times: • Always happens as the marginal hazard describes! Credit Derivatives: From the simple to the more advanced

30. Copula function • Marginal survival times are described by the hazard! • AND ONLY THE HAZARD • It doesn’t depend on the Gaussian distribution • We only look at the quantiles in the Gaussian distribution • Copula = “correlation” describtion • Describes the co-variation among default times • Here: Gaussian multivariate distribution • Other possibilities: T-copula, Gumbel copula, general Archimedian copulas, double T, random factor, etc. • Heavier tails more “extreme observations” • Copula correlation different from default time correlation etc. Credit Derivatives: From the simple to the more advanced

31. Subjects not mentioned • Other copula/correlation models that explains the correlation smile • CDO hedge amounts, deltas in different models • CDO behavior when credit spreads change • Details of efficient implementation strategies • Flat correlation matrix or detailed correlation matrix? • Etc. etc. Credit Derivatives: From the simple to the more advanced

32. Conclusion • Still a lot of modelling to be done • In particular for correlation smiles • The key is to get an efficient implementation that gives accurate risk numbers • Market is evolving fast • New products • Standardized products • Documentation • Conventions • Liquidity Credit Derivatives: From the simple to the more advanced