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

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

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

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

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

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

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

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