AFFI conference June, 24, 2003
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1 Basket default swaps, CDO s and Factor Copulas AFFI conference June, 24, 2003 Jean-Paul Laurent ISFA Actuarial School, University of Lyon Paper «basket defaults swaps, CDO s and Factor Copulas» available on AFFI web site «I will survive», technical paper, RISK magazine, june 2003
2 What are we looking for?! A framework where:! One can easily deal with a large number of names,! Tackle with different time horizons,! Compute quickly and accurately:! Basket credit derivatives premiums! CDO margins on different tranches! Deltas with respect to shifts in credit curves! Main technical assumption:! Default times are independent conditionnally on a low dimensional factor
3 Probabilistic Tools: Survival Functions! names! default times! Marginal distribution function! Marginal survival function! Given from CDS quotes! Joint survival function:! (Survival) Copula of default times:! C characterizes the dependence between default times.
4 Probabilistic Tools: Factor Copulas! Factor approaches to joint distributions:! V low dimensional factor, not observed «latent factor»! Conditionally on V default times are independent! Conditional default probabilities! Conditional joint distribution:! Joint survival function (implies integration wrt V):
5 Probabilistic Tools: Gaussian Copulas! One factor Gaussian copula (Basel 2):! independent Gaussian! Default times:! Conditional default probabilities:
6 Probabilistic Tools : Clayton copula! Davis & Lo ; Jarrow & Yu ; Schönbucher & Schubert! Conditional default probabilities! V: Gamma distribution with parameter θ
7 Probabilistic Tools: Simultaneous Defaults! Duffie & Singleton, Wong! Modelling of defaut dates:! simultaneous defaults.! Conditional default probabilities:
8 Probabilistic Tools: Affine Jump Diffusion! Duffie, Pan & Singleton ;Duffie & Garleanu.! independent affine jump diffusion processes:! Conditional default probabilities:
9 Risk Management of Basket Credit Derivatives! Example: six names portfolio! Changes in credit curves of individual names! Amount of individual CDS to hedge the basket! Semi-analytical more accurate than 10 5 Monte Carlo simulations.! Much quicker: about 25 Monte Carlo simulations.
10 Risk Management of Basket Credit Derivatives! Changes in credit curves of individual names! Dependence upon the choice of copula for defaults
11 CDO Tranches «Everything should be made as simple as possible, not simpler»! Explicit premium computations for tranches! Use of loss distributions over different time horizons! Computation of loss distributions from FFT! Involves integration par parts and Stieltjes integrals
12 Credit Loss Distributions! Accumulated loss at t:! Where loss given default! Characteristic function! By conditioning! Distribution of L(t) is obtained by FFT
13 Credit Loss distributions! One hundred names, same nominal.! Recovery rates: 40%! Credit spreads uniformly distributed between 60 and 250 bp.! Gaussian copula, correlation: 50%! 10 5 Monte Carlo simulations
14 Valuation of CDO s! Tranches with thresholds! Mezzanine: pays whenever losses are between A and B! Cumulated payments at time t: M(t)! Upfront premium:! B(t) discount factor, T maturity of CDO! Stieltjes integration by parts! where
15 Valuation of CDO s! One factor Gaussian copula! CDO tranches margins with respect to correlation parameter
16 Risk Management of CDO s! Hedging of CDO tranches with respect to credit curves of individual names! Amount of individual CDS to hedge the CDO tranche! Semi-analytic : some seconds! Monte Carlo more than one hour and still shaky
17 Conclusion! Factor models of default times:! simple computation of basket credit derivatives and CDO s! deal easily with a large range of names and dependence structures
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