1. CDO correlation smile and deltas 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?
1 November 2004
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
1 November 2004
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
1 November 2004
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
1 November 2004
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...
1 November 2004
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?
1 November 2004
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
1 November 2004
CDO correlation smile and deltas under different correlations

9. Base correlations Long Short 1 November 2004
CDO correlation smile and deltas under different correlations

10. Base versus compound correlations
1 November 2004
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
1 November 2004
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”?
1 November 2004
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?
1 November 2004
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.
1 November 2004
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
1 November 2004
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
1 November 2004
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
1 November 2004
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)
1 November 2004
CDO correlation smile and deltas under different correlations

19. Andersen & Sidenius 2004, two point model Random 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.
1 November 2004
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
1 November 2004
CDO correlation smile and deltas under different correlations

21. Correlation dynamics when spread changes as well..
1 November 2004
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
1 November 2004
CDO correlation smile and deltas under different correlations

23. 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:
CDS spread
Corr//...
Model 1
Model 2
1 November 2004
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
1 November 2004
CDO correlation smile and deltas under different correlations