Petit déjeuner de la finance

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1 Beyond the Gaussan copula: stochastc and local correlaton for CDOs Pett déjeuner de la fnance 12 Octobre 2005 Jean-Paul Laurent ISFA, Unversté Claude Bernard à Lyon Consultant scentfque, BNP-Parbas A comparatve analyss of CDO prcng models Beyond the Gaussan copula: stochastc and local correlaton dsponbles sur 1

2 Beyond the Gaussan copula One factor Gaussan copula Factors models, sem-analytcal computatons Orderng of rsks, Base correlaton Gaussan extensons, correlaton senstvtes Stochastc recovery rates Model dependence/choce of copula Student t, double t, Clayton, Marshall-Olkn, Stochastc correlaton Calbraton methodology, emprcal results Dstrbuton of condtonal default probabltes Beyond the Gaussan copula Margnal compound correlaton Stochastc correlaton and state dependent correlaton Local correlaton 2

3 Sem explct prcng, condtonal default probabltes Factor approaches to jont default tmes dstrbutons: V: low dmensonal factor Condtonally on V, default tmes are ndependent. Condtonal default and survval probabltes: Why factor models? Tackle wth large dmensons (-Traxx, CDX) Need of tractable dependence between defaults: Parsmonous modellng Sem-explct computatons for CDO tranches Large portfolo approxmatons 3

4 Sem explct prcng, condtonal default probabltes Sem-explct prcng for CDO tranches Laurent & Gregory [2003] Default payments are based on the accumulated losses on the pool of credts: n Lt () = LGD1, LGD = N(1 δ) = 1 { τ t} Tranche premums only nvolve call optons on the accumulated losses E ( L() t K ) + Ths s equvalent to knowng the dstrbuton of L(t) 4

5 Sem explct prcng, condtonal default probabltes Characterstc functon: By condtonng upon V and usng condtonal ndependence: Dstrbuton of L(t) can be obtaned by FFT Or other recurson technque Only need of condtonal default probabltes V p t p t V losses on a large homogeneous portfolo Approxmaton technques for prcng CDOs 5

6 Sem explct prcng, condtonal default probabltes One factor Gaussan copula: ndependent Gaussan, Default tmes: F margnal dstrbuton functon of default tmes Condtonal default probabltes: 6

7 One factor Gaussan copula CDO margns (bps pa) equty mezzanne senor Wth respect to correlaton Gaussan copula 0% Attachment ponts: 3%, 10% 10% names Unt nomnal Credt spreads 100 bp 30% 50% years maturty 70% %

8 One factor Gaussan copula Equty tranche premums are decreasng wrt ρ General result Equty tranche premum s always decreasng wth correlaton parameter See Burtschell et al [2005] for more detals about stochastc orders Guarantees unqueness of «base correlaton» Monotoncty propertes extend to Student t, Clayton and Marshall-Olkn copulas 8

9 One factor Gaussan copula: extreme cases ρ = 100% Equty tranche premums decrease wth correlaton Does ρ = 100% correspond to some lower bound? ρ = 100% corresponds to «comonotonc» default dates: ρ = 100% s a model free lower bound for the equty tranche premum ρ = 0% Does ρ = 0% correspond to the hgher bound on the equty tranche premum? ρ = 0% corresponds to the ndependence case between default dates The answer s no, negatve dependence can occur Base correlaton does not always exsts 9

10 One factor Gaussan copula and extensons Gaussan extensons Parwse correlaton senstvtes for CDO tranches Can be computed analytcally See Gregory & Laurent, «In the Core of Correlaton», Rsk 1 ρ 21 ρ ρ + δ j. ρj + δ 1 Parw se Correlaton Senstvty (Senor Tranche) Postve senstvtes (senor tranches) PV Change Credt spread 2 (bps) Credt spread 1 (bps) 10

11 One factor Gaussan copula and extensons Gaussan extensons 2 V = ρ W + 1 ρ V Intra & ntersector correlatons, name, s() sector s() k() s() 2 W k() factor for sector k() Ws () = λs() W + 1 λs() Ws() W global factor Allows for ratngs agences correlaton matrces 1 β1 β1 Analytcal computatons stll β1 1 β1 γ β 1 avalable for CDOs 1 β1 1 Increasng ntra or ntersector.. correlatons decrease equty 1 1 β tranche premums m β m Does not explan the skew γ β m 1 β m β m β m 1 11

12 One factor Gaussan copula and extensons Gaussan extensons Intra & ntersector correlatons, name, s() sector ρ systemc correlaton Accountng for sector dversfcaton n rsk assessment Rsk measures based on unexpected losses, α = 99.9% V = ρ W + 1 ρ V 2 s() k() s() W = ρw + 1 ρ W 2 s() s() ζ (VaR) κ (Expected Shortfall) ρ = 100% (Basel II) 6,1% 6,9% ρ = 50% (multfactor model) 4,6% 5,0% Relatve varaton -25% -27% 12

13 One factor Gaussan copula and extensons VaR, Expected Shortfall and systemc correlaton fg. 5 : VaR and ES as a functon of systemc correlaton 8% 7% 6% 5% 4% 3% VaR ES 2% 1% 0% 0% 8% 15% 23% 30% 38% 45% 53% 60% 68% 75% 83% 90% 98% systemc correlaton Rsk measures change almost lnearly wrt to systemc correlaton Basel II: no sector dversfcaton Sector dversfcaton lessens captal requrements See Aggregaton and credt rsk measurement n retal bankng, Chabaane et al [2003] 13

14 One factor Gaussan copula and extensons VaR and ntrasector correlaton fg. 6 : VaR senstvty to a one 1% error on correlaton 4,5% 4,0% 3,5% 3,0% 2,5% 2,0% 1,5% 1,0% 0,5% 0,0% mult Basel Elastcty of VaR wrt ntrasector correlaton parameters ρj ζ ζ ρ Lnes 1 and 2 correspond to subportfolos wth hghest credt qualty J 14

15 One factor Gaussan copula and extensons Correlaton between default dates and recovery rates One factor Gaussan copula for default dates Losses Gven Default also have a one factor structure: Merton type LGD: max 0,1 ( ) e µ + σξ A two factor Gaussan model wth factors Ψ,ξ Correlaton between defaults & recoveres and amongst recoveres Ψ = ρψ+ 1 ρψ See Credt Rsk Assessment and Stochastc LGD's: an Investgaton of Correlaton Effects n Recovery Rsk: The Next Challenge n Credt Rsk Management, Rsk Books ξ = βξ + 1 βξ 15

16 One factor Gaussan copula and extensons Correlaton between default dates and recovery rates VaR and ES as a functon of correlaton parameters β η 0% 20% 40% 60% 80% 100% 0% 158,9% 161,0% 164,2% 162,5% 159,3% 145,9% 154,8% 160,2% 165,4% 164,7% 162,4% 152,1% 20% 157,5% 175,4% 182,6% 186,8% 186,0% 172,8% 153,9% 175,6% 183,7% 188,6% 192,5% 179,8% 40% 160,2% 194,1% 207,9% 211,8% 212,6% 205,7% 156,0% 196,6% 211,6% 218,7% 219,5% 217,2% 60% 158,2% 207,4% 227,0% 238,9% 240,8% 234,1% 155,2% 210,3% 231,1% 243,0% 249,2% 243,4% 80% 159,6% 223,1% 244,1% 257,4% 264,5% 260,5% 156,0% 229,4% 249,4% 265,1% 271,2% 273,4% 100% 158,1% 238,9% 262,7% 276,5% 283,3% 286,8% 153,9% 246,4% 268,0% 287,3% 296,3% 296,6% Takng nto account correlaton between default events and LGD leads to a substantal ncrease n VaR and Expected Shortfall 16

17 One factor Gaussan copula and extensons Correlaton between default dates and recovery rates Correlaton smle mpled from the correlated recovery rates Not as mportant as what s found n the market 35% 30% Impled Correlaton 25% 20% 15% 10% 5% 50% 70% 0% 0-3% 3-6% 6-9% 9-12% 12-22% Tranche 17

18 Model dependence / choce of copula Stochastc corrrelaton copula ndependent Gaussan varables B = 1 correlaton ρ, B = 0 correlaton β ( 1 ) ( )( ) ρ ρ β β V = B V + V + B V + V τ = F 1 ( Φ( V )) ( ()) β ( ()) 1 1 V ρv +Φ F t V +Φ F t pt = pφ + (1 p) Φ ρ 1 β 18

19 Model dependence / choce of copula Student t copula Embrechts, Lndskog & McNel, Greenberg et al, Mashal et al, O Kane & Schloegl, Glkes & Jobst = + V = W X τ = 2 X ρv 1 ρ V 1 F t V ( ν ( )) VV ndependent Gaussan varables, ν 2 follows a dstrbuton W χ ν Condtonal default probabltes (two factor model) p V, W t + ( ()) 1/2 1 ρv W tν F t =Φ 2 1 ρ 19

20 Model dependence / choce of copula Clayton copula Schönbucher & Schubert, Rogge & Schönbucher, Frend & Rogge, Madan et al V lnu = ψ V τ = F ( ) 1 ψ () s = 1+ Marshall-Olkn constructon of archmedean copulas ( ) s 1/ V: Gamma dstrbuton wth parameter θ U 1,, U n ndependent unform varables Condtonal default probabltes (one factor model) V θ ( ( θ )) V p = exp V 1 F( t) t 20

21 Model dependence / choce of copula Double t model (Hull & Whte) 1/2 1/2 ν 2 2 ν 2 = ρ + 1 ρ V V V ν ν V,V are ndependent Student t varables wth ν and ν degrees of freedom τ = ( ) ( ) F H V 1 where H s the dstrbuton functon of V 1/2 2 H F t V ν = ν ρ 1 ν 1/2 ( () ) ρ V ν p t ν t 21

22 Model dependence / choce of copula Shock models (multvarate exponental copulas) Duffe & Sngleton, Gesecke, Elouerkhaou, Lndskog & McNel, Wong Modellng of default dates: VV exponental wth parameters α,1 α, Default dates S τ = S margnal survval functon Condtonally on V τ are ndependent. Condtonal default probabltes V = ( V V ) mn, ( exp mn ( V, V )) 1, q = 1 S ( t) V t V> ln S ( t) 1 α 22

23 Model dependence / choce of copula Calbraton procedure One parameter copulas Ft Clayton, Student t, double t, Marshall Olkn parameters onto CDO equty tranches Computed under one factor Gaussan model Or gven market quotes on equty trances Reprce mezzanne and senor CDO tranches Gven the prevous parameter 23

24 Model dependence / choce of copula CDO margns (bps pa) equty mezzanne senor Wth respect to correlaton Gaussan copula 0% Attachment ponts: 3%, 10% 10% names Unt nomnal Credt spreads 100 bp 30% 50% years maturty 70% %

25 Model dependence / choce of copula ρ 0% 10% 30% 50% 70% 100% θ ρ 14% 39% 63% 100% 6 2 ρ 12 22% 45% 67% 100% ρ t(4)-t(4) 0% 12% 34% 55% 73% 100% ρ t(5)-t(4) 0% 13% 36% 56% 74% 100% ρ t(4)-t(5) 0% 12% 34% 54% 73% 100% ρ t(3)-t(4) 0% 10% 32% 53% 75% 100% ρ t(4)-t(3) 0% 11% 33% 54% 73% 100% α 0 28% 53% 69% 80% 100% Table 5: correspondence between parameters 25

26 Model dependence / choce of copula ρ 0% 10% 30% 50% 70% 100% Gaussan Clayton Student (6) Student (12) t(4)-t(4) t(5)-t(4) t(4)-t(5) t(3)-t(4) t(4)-t(3) MO Table 6: mezzanne tranche (bps pa) 26

27 Model dependence / choce of copula ρ 0% 10% 30% 50% 70% 100% Gaussan Clayton Student (6) Student (12) t(4)-t(4) t(5)-t(4) t(4)-t(5) t(3)-t(4) t(4)-t(3) MO Table 7: senor tranche (bps pa) Gaussan, Clayton and Student t CDO premums are close 27

28 Model dependence / choce of copula Why Clayton and Gaussan copulas provde same SL premums? Loss dstrbutons depend on the dstrbuton of condtonal default probabltes 1 V ( θ ( )) V ρv +Φ ( F () t ) pt = exp V 1 F( t) pt =Φ 2 1 ρ Dstrbuton of condtonal default probablltes are close for Gaussan and Clayton 1 0,95 0,9 0,85 0,8 0,75 0,7 0,65 0,6 0,55 0,5 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0, ,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 Clayton Gaussan MO ndependence comonotonc stoch. 28

29 Matchng the correlaton skew Tranches Market Gaussan Clayton Student (12) t(4)-t(4) Stoch. MO [0-3%] [3-6%] [6-9%] [9-12%] [12-22%] Table 17: CDO tranche premums Traxx (bps pa) Tranches Market Gaussan Clayton Student (12) t(4)-t(4) Stoch. MO [0-3%] [0-6%] [0-9%] [0-12%] [0-22%] Table 18: equty tranche CDO tranche premums Traxx (bps pa) 29

30 Matchng the correlaton skew mpled compound correlaton 40% 35% 30% 25% 20% 15% 10% Market Gaussan doubl e t 4/ 4 clayton exponent al t-student 12 St och. 5% 0%

31 Beyond the Gaussan copula: stochastc and local correlaton Stochastc correlaton Latent varables 2 V = ρv + 1 ρ V, = 1,, n ρ, stochastc correlaton, ρ = (1 B )(1 B) ρ + B s s QB ( = 1) = q), systemc state, s QB ( = 1) = q, dosyncratc state Condtonal default probabltes s ( Ft ()) 1. V, B s = 0 Φ ρv pt = (1 q) Φ + qf( t), F( t) default probablty 2 1 ρ p. V, B s = 1 t = V Φ F t 1, comonotonc 1 ( ()) 31

32 Beyond the Gaussan copula: stochastc and local correlaton Stochastc correlaton Sem-analytcal technques for prcng CDOs avalable Large portfolo approxmaton can be derved Allows for Monte Carlo ρ, qs, qleads to ncrease senor tranche premums State dependent correlaton Local correlaton Turc et al Random factor loadngs Andersen & Sdenus ρ = (1 B )(1 B) ρ + B s s V= mvv ( ) + σ ( VV ), = 1,, n 2 V = ρ( V) V + 1 ρ ( V) V ( 1 1 ) V = m+ l + h V + νv V< e V e 32

33 Beyond the Gaussan copula: stochastc and local correlaton Dstrbuton functons of condtonal default probabltes stochastc correlaton vs RFL Wth respect to level of aggregate losses Also correspond to loss dstrbutons on large portfolos 33

34 Beyond the Gaussan copula: stochastc and local correlaton Margnal compound correlatons: Wth respect to attachment detachment pont Compound correlaton of a αα, tranche [ ] Stochastc correlaton vs RFL 34

35 Beyond the Gaussan copula: stochastc and local correlaton Margnal compound correlaton Can be obtaned from the dstrbuton functon of condtonal default probabltes Need to solve a second order equaton There mght be zero, one or two margnal compound correlatons Assocated wth the same condtonal default probabltes Always a zero margnal compound correlaton at the expected loss 35

36 Beyond the Gaussan copula: stochastc and local correlaton Local correlaton Can be obtaned from the condtonal default probablty dstrbuton Need to solve for a functonal equaton Fxed pont algorthm Step one: solvng for a second order equaton smlar to the one gvng margnal compound correlaton Local correlaton at step one: rescaled margnal compound correlaton Same ssues of unqueness and exstence 36

37 Beyond the Gaussan copula: stochastc and local correlaton Local correlaton assocated wth RFL (as a functon of the factor) Jump at threshold 2, low correlaton level 5%, hgh correlaton level 85% Possbly two local correlatons 37

38 Beyond the Gaussan copula: stochastc and local correlaton Local correlaton assocated wth stochastc correlaton model Wth respect to factor V Correlatons of 1 for hgh-low values of V (comonotonc state) Possbly two local correlatons leadng to the same prces As for RFL, rather rregular pattern 38

39 Beyond the Gaussan copula: stochastc and local correlaton Checkng for the convergence of the fxed pont algorthm Good news: convergence at step one 39

40 Beyond the Gaussan copula: stochastc and local correlaton Market fts: stochastc correlaton model 40

41 Beyond the Gaussan copula: stochastc and local correlaton Calbraton hstory (from 15 Aprl 2005) Impled correlaton, mpled dosyncratc and systemc probabltes Trouble n fttng durng the crss Snce then, decrease n systemc probablty 41

42 Concluson Analyss of dependence through Gaussan models CDO premums, Rsk measures Stochastc orders, base correlatons Analytcal technques, large portfolo approxmatons Matchng the skew wth second generaton models RFL, double t Condtonal default probablty dstrbutons are the drvers Technque can be extended to structural or ntensty models Beyond the Gaussan copula Stochastc, local & margnal compound correlaton Prcng bespoke portfolos, CDO squared wth a consstent model 42