cdo correlation smile and deltas under different correlations

CDO correlation smile anddeltas under different correlations

Jens Lund1 November 2004

1 November 2004 CDO correlation smile and deltas under different correlations

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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?


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Standardized CDO tranches

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

9%

3%6%

12%

100%

3% equity

Mezzanine

22%


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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

iT

1

1

0( ) 1 exp( ( ) )

tiF t u du


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Prices in the market has a correlation smile In practice: Correlation depends on product, 7-oct-2004, 5Y iTraxx Europe Tranche Maturity

Compound correlation

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Tranche


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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...


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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?


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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


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Base correlationsBase correlation

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Detachment pointShort

Long


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Base versus compound correlationsCorrelation

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Base corr


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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

Delta 0%-3% 3%-6% 6%-9% 9%-12% 12%-22%Compound correlation 21.8 8.5 3.0 2.0 0.9Base correlation 21.8 5.8 2.4 1.8 1.0


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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”?


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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?


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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.


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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


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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


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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

21i iX aM a Z

a


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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)

21i iX aM a Z


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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.

( )i iX a M M vZ m

for ( )for Ma MM


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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

Copula with a smile

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RCDM base


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Correlation dynamicswhen spread changes as well..

Correlation dynamics

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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

Delta 0%-3% 3%-6% 6%-9% 9%-12% 12%-22%Compound correlation 21.8 8.5 3.0 2.0 0.9Base correlation 21.8 5.8 2.4 1.8 1.0RCDM 32.1 2.3 0.4 0.4 0.9


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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

( , ) ( , )V VV


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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

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