cdo correlation smile and deltas under different correlations
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CDO correlation smile and deltas under different correlations. Jens Lund 1 November 2004. 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

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Presentation Transcript
outline
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

standardized cdo tranches
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

reference gaussian copula model
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

prices in the market has a correlation smile
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

why do we see the smile
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

compound correlations
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

base correlations
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

base correlations9
Base correlations

Short

Long

CDO correlation smile and deltas under different correlations

base versus compound correlations
Base versus compound correlations

CDO correlation smile and deltas under different correlations

delta hedges
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

what does the smile mean
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

is base correlations a real solution
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

in theory
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

in practice however
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

so the search for better copulas has started
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

implementation of gaussian copula
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

copulas with a smile some posibilities
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

andersen sidenius 2004 two point model random correlation dependent on market
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

can these models explain the smile
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

correlation dynamics when spread changes as well
Correlation dynamicswhen spread changes as well..

CDO correlation smile and deltas under different correlations

delta with model generating smile
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

different models has different deltas
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

conclusion
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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