Comparative analysis of CDO pricing models ICBI Risk Management 2005 Geneva 8 December 2005 Jean-Paul Laurent ISFA, University of Lyon, Scientific Consultant BNP Paribas laurent.jeanpaul@free.fr, http://laurent.jeanpaul.free.fr Joint work with X. Burtschell & J. Gregory A comparative analysis of CDO pricing models Beyond the Gaussian copula: stochastic and local correlation Available on www.defaultrisk.com 1
Comparative analysis of CDO pricing models � 1 Factor based copulas � Collective & individual models of credit losses � Semi-explicit pricing � 2 One factor Gaussian copula � Ordering of risks, Base correlation � correlation sensitivities � Stochastic recovery rates � 3 Model dependence / choice of copula � Student t , double t , Clayton, Marshall-Olkin, Stochastic correlation � Distribution of conditional default probabilities � 4 Beyond the Gaussian copula � Stochastic correlation and state dependent correlation � Marginal and local correlation 2
1 Factor based copulas � CDO valuation, credit risk assessment � Only need of loss distributions for different time horizons � Aggregate loss at time t on a given portfolio: ( t ) L � Marginal loss distribution for time horizon t ( ) ( ) → = ≤ ( ) l F l Q L t l ( ) L t � VaR and quantile based risk measures for risk assessment 1 ( ) ( ) ∫ − α α α 1 F v d ( ) L t 0 � Pricing of CDOs only involve options on aggregate loss [ ] ( ) + − K E Q ( ) L t � K attachment – detachment points 3
1 Factor based copulas � Modelling approaches ( t ) � Direct modelling of : collective model L � Dealing with heterogeneous portfolios � non stationary, non Markovian � Aggregation of portfolios, bespoke portfolios? � Risk management of correlation risk? � Modelling of default indicators of names: individual model n ∑ = ( ) 1 L t LGD τ ≤ i t i = 1 i � Numerical approaches � e.g. smoothing of base correlation of liquid tranches 4
1 Factor based copulas � Individual model / factor based copulas � Allows to deal with non homogeneous portfolios � Arbitrage free prices � non standard attachment –detachment points � Non standard maturities � Consistent pricing of bespoke, CDO 2 , zero-coupon CDOs � Computations � Semi-explicit pricing, computation of Greeks, LHP � But… � Poor dynamics of aggregate losses (forward starting CDOs) � Risk management, credit deltas, theta effects � Calibration onto liquid tranches (matching the skew) 5
1 Factor based copulas � Factor approaches to joint default times distributions: � V: low dimensional factor � Conditionally on V, default times are independent. � Conditional default and survival probabilities: � Why factor models ? � Tackle with large dimensions (i-Traxx, CDX) � Need of tractable dependence between defaults: � Parsimonious modelling � Semi-explicit computations for CDO tranches � Large portfolio approximations 6
1 Factor based copulas � Semi-explicit pricing for CDO tranches � Laurent & Gregory [2003] � Default payments are based on the accumulated losses on the pool of credits: n ∑ = = − δ ( ) 1 , (1 ) L t LGD LGD N { } τ ≤ i i i i t i = 1 i � Tranche premiums only involve call options on the accumulated losses ( ) ⎡ ⎤ + − ( ) E L t K ⎣ ⎦ � This is equivalent to knowing the distribution of L(t) 7
1 Factor based copulas � Characteristic function: � By conditioning upon V and using conditional independence: � Distribution of L ( t ) can be obtained by FFT � Similar approaches: recursion, inversion of Laplace transforms i V � Only need of conditional default probabilities p t i V losses on a large homogeneous portfolio p � t � Approximation techniques for pricing CDOs 8
Comparative analysis of CDO pricing models � 2 One factor Gaussian copula � Ordering of risks, Base correlation � correlation sensitivities � Stochastic recovery rates � 3 Model dependence/Choice of copula � Student t , double t , Clayton, Marshall-Olkin, Stochastic correlation � Distribution of conditional default probabilities � 4 Beyond the Gaussian copula � Stochastic correlation and state dependent correlation � Marginal and local correlation 9
2 One factor Gaussian copula � One factor Gaussian copula : independent Gaussian, � � Default times: � F i marginal distribution function of default times � Conditional default probabilities: 10
2 One factor Gaussian copula ρ � Equity tranche premiums are decreasing wrt � General result (use of stochastic orders theory) � Equity tranche premium is always decreasing with correlation parameter � Guarantees uniqueness of « base correlation » � Monotonicity properties extend to Student t, Clayton and Marshall-Olkin copulas 11
2 One factor Gaussian copula ρ = 100% � � Equity tranche premiums decrease with correlation ρ = � Does correspond to some lower bound? 100% ρ = 100% corresponds to « comonotonic » default dates: � ρ = 100% is a model free lower bound for the equity tranche � premium ρ = 0% � ρ = 0% � Does correspond to the higher bound on the equity tranche premium? ρ = 0% corresponds to the independence case between � default dates � The answer is no, negative dependence can occur � Base correlation does not always exists 12
2 One factor Gaussian copula � Pair-wise correlations ρ ⎛ ⎞ 1 12 ⎜ ⎟ ρ 1 ⎜ ⎟ 21 ⎜ ⎟ 1 ρ + δ ⎜ ⎟ ij � Pair-wise correlation ⎜ . ⎟ ⎜ ⎟ . sensitivities for CDO tranches ⎜ ⎟ ρ + δ ij ⎜ ⎟ 1 ⎜ ⎟ 1 . ⎜ ⎟ ⎜ ⎟ ⎝ . 1 ⎠ � Can be computed analytically � See Gregory & Laurent, « In the Core Pairwise Correlation Sensitivity (Senior Tranche) of Correlation », Risk 0.003 0.002 PV Change 0.002 � Higher correlation sensitivities 0.001 for riskier names (senior 0.001 205 tranche) 0.000 115 Credit spread 25 2 (bps) 65 105 25 145 185 225 265 Credit spread 1 (bps) 13
2 One factor Gaussian copula � Intra Inter sector correlations = ρ + − ρ 2 1 V W V � i, name, s(i) sector ( ) ( ) ( ) i s i s i s i i � W s(i) factor for sector s(i) = λ + − λ 2 1 W W W ( ) ( ) ( ) ( ) s i s i s i s i � W global factor � Allows for ratings agencies correlation matrices ⎛ β β ⎞ 1 ⎜ 1 1 ⎟ � Analytical computations still β β γ ⎜ ⎟ 1 1 1 ⎜ ⎟ available for CDOs β β 1 ⎜ ⎟ 1 1 1 ⎜ ⎟ ⎜ ⎟ � Increasing intra or intersector . ⎜ ⎟ . ⎜ ⎟ correlations decrease equity ⎜ ⎟ 1 ⎜ ⎟ β β 1 tranche premiums ⎜ ⎟ m m γ β β ⎜ ⎟ 1 m m ⎜ ⎟ β β � Does not explain the skew ⎝ 1 ⎠ m m 14
2 One factor Gaussian copula � Correlation between default dates and recovery rates � Correlation smile implied from the correlated recovery rates � Not as important as what is found in the market 35% 30% Implied Correlation 25% 20% 50% 70% 15% 10% 5% 0% 0-3% 3-6% 6-9% 9-12% 12-22% Tranche 15
3 Model dependence / choice of copula � Stochastic correlation copula independent Gaussian variables � ρ B = B = β 1 0 correlation , correlation � i i ( ) ( ) ( ) = ρ + − ρ + − β + − β 2 2 1 1 1 V B V V B V V i i i i i ( ( ) ) − 1 τ = Φ F V i i i ⎛ ( ) ⎞ ⎛ ( ) ⎞ − − − ρ + Φ − β + Φ 1 1 ( ) ( ) V F t V F t ⎜ ⎟ ⎜ ⎟ = Φ + − Φ | i i i V (1 ) p p p ⎜ ⎟ ⎜ ⎟ t − ρ − β 2 2 1 1 ⎝ ⎠ ⎝ ⎠ 16
3 Model dependence / choice of copula � Student t copula ⎧ = ρ + − ρ 2 1 X V V ⎪ i i ⎪ = × ⎨ V W X i i ⎪ ( ) ( ) − τ = 1 F t V ⎪ ν ⎩ i i i , independent Gaussian variables V V � i ν χ 2 follows a distribution � ν W � Conditional default probabilities (two factor model) ( ) ⎛ ⎞ − − − ρ + 1/ 2 1 ( ) V W t F t ⎜ ν ⎟ = Φ⎜ | , i i V W p ⎟ t − ρ 2 ⎝ 1 ⎠ 17
3 Model dependence / choice of copula � Clayton copula ( ) ( ) − θ ψ ⎛ ⎞ ln − 1/ U τ = ψ = + 1 = − ( ) 1 s s F V i ⎜ ⎟ V i i i i ⎝ ⎠ V θ � V: Gamma distribution with parameter � U 1 ,…, U n independent uniform variables � Conditional default probabilities (one factor model) ( ) ( ) − θ = − iV exp 1 ( ) p V F t t i 18
3 Model dependence / choice of copula � Double t model (Hull & White) 1/ 2 1/ 2 ν − ν − ⎛ ⎞ ⎛ ⎞ 2 2 = ρ + − ρ 2 1 ⎜ ⎟ ⎜ ⎟ V V V ν ν i i i i ⎝ ⎠ ⎝ ⎠ V , V are independent Student t variables � i ν ν � with and degrees of freedom ( ) ( ) − τ = 1 F H V i i i i � where H i is the distribution function of V i ⎛ ⎞ ν − 1/ 2 ⎛ ⎞ 2 ( ) − − ρ ⎜ ⎟ 1 ( ) ⎜ ⎟ H F t V 1/ 2 ν ν ⎛ ⎞ i i i ⎝ ⎠ ⎜ ⎟ = | i V ⎜ ⎟ p t ν ⎜ ⎟ ν − t ⎝ ⎠ 2 − ρ 2 1 ⎜ ⎟ ⎜ i ⎟ ⎝ ⎠ 19
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