Create Presentation
Download Presentation

Download Presentation
## CDO correlation smile and deltas under different correlations

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**CDO correlation smile anddeltas 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 • Are deltas consistent? CDO correlation smile and deltas under different correlations**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**• 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**• 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?**• 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**• 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**• 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 correlations**Short Long CDO correlation smile and deltas under different correlations**Base versus compound correlations**CDO correlation smile and deltas under different correlations**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?**• 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?**• 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…**• 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…**• 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...**• “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**• 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**• 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 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?**• 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 dynamicswhen spread changes as well..**CDO correlation smile and deltas under different correlations**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**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**• 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