Comparative analysis of CDO pricing models

Transcription

1 Comparatve analyss of CDO prcng models ICBI Rsk Management 2005 Geneva 8 December 2005 Jean-Paul Laurent ISFA, Unversty of Lyon, Scentfc Consultant BNP Parbas laurent.jeanpaul@free.fr, Jont work wth X. Burtschell & J. Gregory A comparatve analyss of CDO prcng models Beyond the Gaussan copula: stochastc and local correlaton Avalable on

2 Comparatve analyss of CDO prcng models Factor based copulas Collectve & ndvdual models of credt losses Sem-explct prcng 2 One factor Gaussan copula Orderng of rsks, Base correlaton correlaton senstvtes Stochastc recovery rates 3 Model dependence / choce of copula Student t, double t, Clayton, Marshall-Olkn, Stochastc correlaton Dstrbuton of condtonal default probabltes 4 Beyond the Gaussan copula Stochastc correlaton and state dependent correlaton Margnal and local correlaton 2

3 Factor based copulas CDO valuaton, credt rsk assessment Only need of loss dstrbutons for dfferent tme horzons Aggregate loss at tme t on a gven portfolo: L(t) Margnal loss dstrbuton for tme horzon t l ( l) = Q( L( t l) F ) L( t ) VaR and quantle based rsk measures for rsk assessment F L ( t )( α ) v( α ) dα 0 Prcng of CDOs only nvolve optons on aggregate loss K attachment detachment ponts [( L( t K) + ] E Q ) 3

4 Factor based copulas Modellng approaches Drect modellng of L(t) : collectve model Dealng wth heterogeneous portfolos non statonary, non Markovan Aggregaton of portfolos, bespoke portfolos? Rsk management of correlaton rsk? Modellng of default ndcators of names: ndvdual model L Numercal approaches n ( t) = LGD τ t = e.g. smoothng of base correlaton of lqud tranches 4

5 Factor based copulas Indvdual model / factor based copulas Allows to deal wth non homogeneous portfolos Arbtrage free prces non standard attachment detachment ponts Non standard maturtes Consstent prcng of bespoke, CDO 2, zero-coupon CDOs Computatons Sem-explct prcng, computaton of Greeks, LHP But Poor dynamcs of aggregate losses (forward startng CDOs) Rsk management, credt deltas, theta effects Calbraton onto lqud tranches (matchng the skew) 5

6 Factor based copulas 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 6

7 Factor based copulas Sem-explct prcng for CDO tranches Laurent & Gregory [2003] Default payments are based on the accumulated losses on the pool of credts: n Lt () = LGD, LGD = N( δ) = { 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) 7

8 Factor based copulas Characterstc functon: By condtonng upon V and usng condtonal ndependence: Dstrbuton of L(t) can be obtaned by FFT Smlar approaches: recurson, nverson of Laplace transforms Only need of condtonal default probabltes V p t p t V losses on a large homogeneous portfolo Approxmaton technques for prcng CDOs 8

9 Comparatve analyss of CDO prcng models 2 One factor Gaussan copula Orderng of rsks, Base correlaton correlaton senstvtes Stochastc recovery rates 3 Model dependence/choce of copula Student t, double t, Clayton, Marshall-Olkn, Stochastc correlaton Dstrbuton of condtonal default probabltes 4 Beyond the Gaussan copula Stochastc correlaton and state dependent correlaton Margnal and local correlaton 9

10 2 One factor Gaussan copula One factor Gaussan copula: ndependent Gaussan, Default tmes: F margnal dstrbuton functon of default tmes Condtonal default probabltes: 0

11 2 One factor Gaussan copula Equty tranche premums are decreasng wrt ρ General result (use of stochastc orders theory) Equty tranche premum s always decreasng wth correlaton parameter Guarantees unqueness of «base correlaton» Monotoncty propertes extend to Student t, Clayton and Marshall-Olkn copulas

12 2 One factor Gaussan copula ρ = 00% Equty tranche premums decrease wth correlaton Does ρ = 00% correspond to some lower bound? ρ = 00% corresponds to «comonotonc» default dates: ρ = 00% 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 2

13 2 One factor Gaussan copula Par-wse correlatons Par-wse correlaton senstvtes for CDO tranches Can be computed analytcally See Gregory & Laurent, «In the Core of Correlaton», Rsk ρ 2 ρ 2. ρ + δ j. ρj + δ Parw se Correlaton Senstvty (Senor Tranche) Hgher correlaton senstvtes for rsker names (senor tranche) PV Change Credt spread 2 (bps) Credt spread (bps) 3

14 4 2 One factor Gaussan copula Intra Inter sector correlatons, name, s() sector W s() factor for sector s() W global factor Allows for ratngs agences correlaton matrces Analytcal computatons stll avalable for CDOs Increasng ntra or ntersector correlatons decrease equty tranche premums Does not explan the skew.. m m m m m m β β β β β β γ γ β β β β β β 2 () () () () s s s s W W W λ λ = + s s s V W V 2 ) ( ) ( ) ( ρ ρ + =

15 2 One factor Gaussan copula 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% 5% 0% 5% 50% 70% 0% 0-3% 3-6% 6-9% 9-2% 2-22% Tranche 5

16 3 Model dependence / choce of copula Stochastc correlaton copula ndependent Gaussan varables B = correlaton ρ, B = 0 correlaton β ( ) ( )( ) 2 2 ρ ρ β β V = B V + V + B V + V τ = F ( Φ( V )) ( ()) β ( ()) V ρv +Φ F t V +Φ F t pt = pφ + ( p) Φ 2 2 ρ β 6

17 3 Model dependence / choce of copula Student t copula = + V = W X τ = 2 X ρv ρ V 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 + ( ()) /2 ρv W tν F t =Φ 2 ρ 7

18 3 Model dependence / choce of copula Clayton copula V = ψ lnu V τ = F V ( ) ψ () s = + ( ) s / θ V: Gamma dstrbuton wth parameter θ U,, U n ndependent unform varables Condtonal default probabltes (one factor model) ( ( θ )) V p = exp V F( t) t 8

19 3 Model dependence / choce of copula Double t model (Hull & Whte) /2 /2 ν 2 2 ν 2 = ρ + ρ V V V ν ν V,V are ndependent Student t varables wth ν and ν degrees of freedom τ = ( ) ( ) F H V where H s the dstrbuton functon of V /2 2 H F t V ν = ν 2 2 ρ ν /2 ( () ) ρ V ν p t ν t 9

20 3 Model dependence / choce of copula Shock models (multvarate exponental copulas) Marshall-Olkn copulas Modellng of default dates: VV exponental wth parameters α, 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 )), q = S ( t) V t V> ln S ( t) α 20

21 3 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 Reprce mezzanne and senor CDO tranches Gven the ftted parameter Look for departures from the Gaussan copula Look for ablty to explan the correlaton skew 2

22 3 Model dependence / choce of copula CDO margns (bps pa) equty mezzanne senor Wth respect to correlaton Gaussan copula 0% Attachment ponts: 3%, 0% 0% names Unt nomnal Credt spreads 00 bps 30% 50% years maturty 70% %

23 3 Model dependence / choce of copula ρ 0% 0% 30% 50% 70% 00% Gaussan Clayton Student (6) Student (2) 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) 23

24 3 Model dependence / choce of copula ρ 0% 0% 30% 50% 70% 00% Gaussan Clayton Student (6) Student (2) 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 24

25 3 Model dependence / choce of copula Why do Clayton and Gaussan copulas provde same premums? Loss dstrbutons depend on the dstrbuton of condtonal default probabltes V ( θ ( )) V ρv +Φ ( F () t ) pt = exp V F( t) pt =Φ 2 ρ Dstrbuton of condtonal default probabltes are close for Gaussan and Clayton 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 0,5 0, 0,05 0 0,05 0, 0,5 0,2 0,25 0,3 0,35 0,4 0,45 0, ,00 0,05 0,0 0,5 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, Clayton Gaussan MO ndependence comonotonc stoch. 25

26 3 Model dependence / choce of copula mpled compound correlaton 40% 35% 30% 25% 20% 5% 0% Market Gaussan doubl e t 4/ 4 clayton exponent al t-student 2 St och. 5% 0%

27 4 Beyond the Gaussan copula Stochastc correlaton Latent varables 2 V = ρv + ρ V, =,, n ρ, stochastc correlaton, ρ = ( B )( B) ρ + B s s QB ( = ) = q), systemc state, s QB ( = ) = q, dosyncratc state Condtonal default probabltes s ( Ft ()). V, B s = 0 Φ ρv pt = ( q) Φ + qf( t), F( t) default probablty 2 ρ p. V, B s = t = V Φ F t, comonotonc ( ()) 27

28 4 Beyond the Gaussan copula 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 = ( B )( B) ρ + B s s V= mvv ( ) + σ ( VV ), =,, n 2 V = ρ( V) V + ρ ( V) V ( ) V = m+ l + h V + νv V< e V e 28

29 4 Beyond the Gaussan copula 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 29

30 4 Beyond the Gaussan copula Margnal compound correlaton Compound correlaton of a [ α, α ] tranche Dgtal call on aggregate loss obtaned from condtonal default probablty dstrbuton Need to solve a second order equaton zero, one or two margnal compound correlatons 30

31 4 Beyond the Gaussan copula Margnal compound correlatons: Wth respect to attachment detachment pont Stochastc correlaton vs RFL zero margnal compound correlaton at the expected loss 3

32 4 Beyond the Gaussan copula Local correlaton obtaned from condtonal default probablty dstrbuton Fxed pont algorthm Local correlaton at step one: rescaled margnal compound correlaton Same ssues of unqueness and exstence as margnal compound correlaton 32

33 4 Beyond the Gaussan copula 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 33

34 4 Beyond the Gaussan copula Local correlaton assocated wth stochastc correlaton model Wth respect to factor V Correlatons of for hgh-low values of V (comonotonc state) Possbly two local correlatons leadng to the same prces As for RFL, rather rregular pattern 34

35 Concluson Analyss of dependence through factor models Usefulness of stochastc orders Correlaton senstvtes, base correlatons Matchng the correlaton skew Condtonal default probablty dstrbutons are the drvers Beyond the Gaussan copula Stochastc, local & margnal compound correlaton Further work Matchng term structure of correlaton skews Integratng factor copulas and ntensty approaches 35