CDO correlation smile and deltas under different correlations

1. CDO correlation smile anddeltas under different correlations Jens Lund 1 November 2004

2. Outline • Standard CDO’s • Gaussian copula model • Implied correlation • Correlation smile in the market • Compound correlation  base correlation • Delta hedge amounts • New copulas with a smile • Are deltas consistent? CDO correlation smile and deltas under different correlations

3. Standardized CDO tranches 100% • iTraxx Europe • 125 liquid names • Underlying index CDSes for sectors • 5 standard tranches, 5Y & 10Y • First to default baskets • 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 CDO correlation smile and deltas under different correlations

4. 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,) •  correlation matrix, variance 1, constant correlation  • In model: correlation independent of product to be priced CDO correlation smile and deltas under different correlations

5. Prices in the market has a correlation smile • In practice: • Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe • Tranche • Maturity CDO correlation smile and deltas under different correlations

6. 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 • We will not go further into why we see a smile, but rather look at how to model it... CDO correlation smile and deltas under different correlations

7. 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? CDO correlation smile and deltas under different correlations

8. 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 CDO correlation smile and deltas under different correlations

9. Base correlations Short Long CDO correlation smile and deltas under different correlations

10. Base versus compound correlations CDO correlation smile and deltas under different correlations

11. Delta hedges • CDO tranches typical traded with initial credit hedge • Conveniently quoted as amount of underlying • index CDS to buy in order to hedge credit risk • Base correlation: find by long/short strategy • Base and compound correlation deltas are different CDO correlation smile and deltas under different correlations

12. What does the smile mean? • The presence of the smile means that the Gaussian copula does not describe market prices • Otherwise the correlation would have been constant over tranches and maturities • How to fix this “problem”? CDO correlation smile and deltas under different correlations

13. Is base correlations a real solution? • No, it is merely a convenient way of describing prices • An intermediate step towards better models that exhibit a smile • No smile dynamics • Correlation smile modelling, versus • Models with a smile and correlation dynamics • So how to find a solution? CDO correlation smile and deltas under different correlations

14. In theory… • Sklar’s theorem: • Every simultaneous distribution of the survival times with marginals consistent with CDS prices can be described by the copula approach • So in theory we should just choose the correct copula, i.e. • Choose the correct simultaneous distribution of Xi. CDO correlation smile and deltas under different correlations

15. In practice however… • So far we have chosen from a rather limited set of copulas: • Gaussian, T-distribution, Archimedian copulas • A lot of parameters (correlation matrix) which we do not know how to choose • None of these have produced a smile that match market prices • Or the copulas have not provided the “correct” distributions CDO correlation smile and deltas under different correlations

16. So the search for better copulas has started... • “Better” means • describing the observed prices in the market for iTraxx • produces a correlation smile • has a reasonable low number of parameters • one can have a view on and interpret • has a plausible dynamics for the correlation smile • constant parameters can be used on a range of • tranches • maturities • (portfolios) • Start from Gaussian model described as a 1 factor model CDO correlation smile and deltas under different correlations

17. 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 CDO correlation smile and deltas under different correlations

18. Copulas with a smile, some posibilities • Start from factor model: • Let M and Zi have different distributions • Hull & White, 2004: Uses T-4 T-4 distributions, seems to work well • Let a be random • Correlate M, Zi, a and RR in various ways • Andersen & Sidenius, 2004 • Changes weight between systematic M and idiosyncratic Zi • Limits on variations as we still want nice mathematical properties • Distribution function H for Xi needed in all cases: Fi(Ti) = H(Xi) CDO correlation smile and deltas under different correlations

19. Andersen & Sidenius 2004, two point modelRandom Correlation Dependent on Market • Let a be a function of the market factor M • , m ensures var=1, mean=0 • Interpretation for > and  small: • Most often correlation is small, • but in bad times we see a large correlation. • Senior investors benefit from this. CDO correlation smile and deltas under different correlations

20. Can these models explain the smile? • Yes, they are definitely better at describing market prices than many alternative models • E.g. • 2 = 0.5 • 2 = 0.002 • () = 0.02 CDO correlation smile and deltas under different correlations

21. Correlation dynamicswhen spread changes as well.. CDO correlation smile and deltas under different correlations

22. Delta with model generating smile • Deltas differ between models: • Agreement on delta amounts requires model agreement • New “market standard copula” will emerge? • Will have to be more complex than the Gaussian CDO correlation smile and deltas under different correlations

23. CDS spread Model 1 Model 2 Corr//... Different models has different deltas... • This does not necessarily imply any inconsistencies • On the other hand it might give problems! • Different parameter spaces in • different models give • different deltas • We want models with stable parameters •  Makes it easier to hedge risk • Beware of parameters, say , moving when other parameters move: CDO correlation smile and deltas under different correlations

24. Conclusion • The market is still evolving fast • More and more information available • Models will have to be developed further • Smile description • Smile dynamics • Delta amounts • Bespoke tranches (no implied market) • Will probably go through a couple of iterations CDO correlation smile and deltas under different correlations