An overview of factor models for pricing CDO tranches

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1 An overvew of facor models for prcng CDO ranches Aresk Cousn 1 and Jean-Paul Lauren 14 January 008 Absrac We revew he prcng of synhec CDO ranches from he pon of vew of facor models. Thanks o he facor framework, we can handle a wde range of well-know prcng models. Ths ncludes prcng approaches based on copulas, bu also srucural, mulvarae Posson and affne nensy models. Facor models have become ncreasngly popular snce here are assocaed wh effcen sem-analycal mehods and parsmonous parameerzaon. Moreover, he approach s no resrcve a all n he case of homogeneous cred porfolos. Easy o compue and o handle large porfolo approxmaons can be provded. In facor models, he dsrbuon of condonal defaul probables s he key npu for he prcng of CDO ranches. These condonal defaul probables are also closely relaed o he dsrbuon of large porfolos. Therefore, we can compare dfferen facor models by smply comparng he dsrbuon funcons of he correspondng condonal defaul probables. Keywords: CDOs, boom-up, op-down, large porfolo approxmaons, de Fne s heorem, facor copulas, Archmedean copulas, srucural models, nensy models, mulvarae Posson models. JEL: G13 Inroducon When lookng a he prcng mehodologes for cred dervaves, a srkng feaure s he profuson of compeng approaches, none of hem could be seen as an academc and praconer s sandard. Ths conrass wh equy or neres rae dervaves o se some examples. Despe raher negave apprecaon from he academc world, he ndusry reles on he one facor Gaussan copula for he prcng of CDO ranches, possbly amended wh base correlaon approaches. Among he usual crcs, one can quoe he poor dynamcs of he cred loss process, of he cred spreads and he dsconnecon beween he prcng and he hedgng, whle prcng a he cos of he hedge s a cornersone of modern fnance. Gven he lkelhood of plan sac arbrage opporunes when massagng correlaons whou cauon, he varey and complexy of mappng procedures for he prcng of bespoke porfolos, a purs mgh asser ha base correlaons are smply a way o express CDO ranche quoes. Even from ha mnmal vew, he compuaon of base correlaons from marke quoes s no an easy ask due o he amorzaon scheme of premum legs and he dependence on more or less arbrary assumpons on recovery raes. 1 ISFA Acuaral School, Unversé Lyon 1, Unversé de Lyon, 50 avenue Tony Garner, 69007, Lyon, France, aresk.cousn@unv-lyon1.fr, hp://ares.crone.org ISFA Acuaral School, Unversé Lyon 1, Unversé de Lyon, 50 avenue Tony Garner, 69007, Lyon, France, and BNP-Parbas, lauren.jeanpaul@free.fr, hp://lauren.jeanpaul.free.fr 1

2 Unsurprsngly, here are many ways o assess model qualy, such as he ably o f marke quoes, racably, parsmony, hedgng effcency and of course economc relevance and heorecal conssency. One should keep n mnd ha dfferen models may be suable for dfferen payoffs. As dscussed below, sandard CDO ranche premums only depend upon he margnal dsrbuons of porfolo losses a dfferen daes, and no on he emporal dependence beween losses. Ths may no be he case for more exoc producs such as leverage ranches, forward sarng CDOs. Therefore, copula models mgh be well sued for he former plan vanlla producs whle a drec modellng of he loss process, as n he op-down approach, ackles he laer. Sandard ranches on Traxx or CDX have almos become asse classes on her own. Though he marke drecly provdes her premum a he curren dae, a modellng of he correspondng dynamcs mgh be requred when rsk managng non sandard ranches. Le us remark ha he nformaonal conen of sandard ranches s no fully sasfacory, especally when consderng he prcng of rancheles correspondng o frs losses (e.g. a [ 0,1% ] ranche) or senor ranches assocaed wh he rgh al of he loss dsrbuon. There are also some dffcules when dealng wh shor maury ranches. Whaever he chosen approach, a purely numercal smoohng of base correlaons or a prcng model based nerpolaon, here s usually a lo of model rsk: models ha are properly calbraed o lqud ranches prces may lead o sgnfcanly dfferen prces for non sandard ranches. In he remander of he paper, we wll focus on prcng models for ypcal synhec CDO ranches, eher based on sandard ndexes or relaed o bespoke porfolos and we wll no furher consder producs ha nvolve he jon dsrbuon of losses and cred spreads such as opons on ranches. We wll focus on model based prcng approaches, such ha he premum of he ranche can be obaned by equallng he presen value of he premum and he defaul legs of he ranches, compued under a gven rsk neural measure. A leas, hs rules ou sac arbrage opporunes, such as negave ranchele prces. Thus, we wll leave asde comparsons beween base correlaon and model based approaches ha mgh be mporan n some cases. Though we wll dscuss he ably of dfferen models o be well calbraed o sandard lqud ranches, we wll no furher consder he varous and somemes raher propreary mappng mehodologes ha am a prcng bespoke CDO ranches gven he correlaon smles on sandard ndexes. Such praccal ssues are addressed n Gregory and Lauren (008) and n he references heren. Forunaely, here reman enough models o leave anyone wh an encyclopaedc endency more han happy. When so many academc approaches cones, here s a need o caegorze, whch obvously does no mean o wre down a caalogue. Recenly, here has been a dscusson abou he relave mers of boom-up and opdown approaches. In he acuaral feld, hese are also labelled as he ndvdual and he collecve models. In a boom-up approach, also known as a name per name approach, one sars from a descrpon of he dynamcs (cred spreads, defauls) of he names whn a baske, from whch he dynamcs of he aggregae loss process s derved. Some aggregang procedure nvolvng he modellng of dependence beween he defaul evens s requred o derve he loss dsrbuon. The boom-up approach has some clear advanages over he op-down approach, such as he possbly o

3 easly accoun for name heerogeney: for nsance he rouble wh GMAC and he correspondng wdenng of spreads had a salen mpac on CDX equy ranche quoes. I can be easly seen ha he heerogeney of ndvdual defaul probables breaks down he Markov propery of he loss process. One needs o know he curren srucure of he porfolo, for example he proporon of rsky names, and no only he curren losses o furher smulae appropraely furher losses. Ths ssue s analogous o he well-known burnou effec n morgage prepaymen modellng. The random hnnng approach only provdes a paral answer o he heerogeney ssue: names wh hgher margnal defaul probables acually end o defaul frs, bu he change n he loss nensy does no depend upon he defauled name, as one should expec. The concep of dosyncrac gamma whch s que mporan n he appled rsk managemen of equy ranches s hus dffcul o handle n a op-down approach. Also, a number of models belongng o hs class do no accoun for he convergence o zero of he loss nensy as he porfolo s exhaused. Ths leads o posve, albe small, probables ha he loss exceeds he nomnal of he porfolo. Anoher praccal and paramoun opc s he rsk managemen of CDO ranches a he book level. Snce mos nvesmen banks deal wh numerous cred porfolos, hey need o model a number of aggregae loss processes, whch obvously are no ndependen. Whle such a global rsk managemen approach s amenable o he boom-up approach, remans an open ssue for s conender. On he oher hand, here are some oher major drawbacks when relyng on boom-up approaches. A popular famly whn he boom-up approaches, relyng on Cox processes bores s own burden. On heorecal grounds, fals o accoun for conagon effecs, also known as nformave defauls: defaul of one name may be assocaed wh jumps, usually of posve magnude, of he cred spreads of he survvng names. Though some progress has recenly been compleed, he numercal mplemenaon, especally wh respec o calbraon on lqud ranches s cumbersome. In facor copula approaches, he dynamcs of he aggregae loss s usually que poor, wh hgh dependence beween losses a dfferen me horzons and even comonoonc losses n he large porfolo approxmaon. Thus, facor copula approaches fall no dsrepue when dealng wh some forward sarng ranches where he dependence beween losses a wo dfferen me horzons s a key npu. Neverheless, he prcng of synhec CDO ranches only nvolves margnal dsrbuon of losses and s lkely o be beer handled n he boom-up approach. Snce hs paper s focused on CDO ranches, when dscussng prcng ssues, we wll favour he name per name perspecve. As menoned above, due o he number of prcng models a hand 3, here s he need of a unfyng perspecve, especally wh respec o he dependence beween defaul daes. In he followng, we wll prvlege a facor approach: defaul daes wll be ndependen gven a low dmensonal facor. Ths framework s no ha resrcve snce encompasses facor copulas, bu also mulvarae Posson, srucural models and some nensy models whn he affne class. Moreover, n he homogeneous case, where he names are ndsngushable, on a echncal ground hs corresponds o 3 See Duffe and Sngleon (003), Schönbucher (003), Beleck and Rukowsk (004) or Lando (004) exbooks for a dealed accoun of he dfferen approaches o cred rsk. 3

4 he exchangeably assumpon, he exsence of a sngle facor s a mere consequence of de Fne s heorem as explaned below. From a heorecal pon of vew, he key npus n a sngle facor model are he dsrbuons of he condonal (on he facor) defaul probables. Gven hese, one can unambguously compue CDO ranche premums n a sem-analycal way. I s also farly easy o derve large porfolo approxmaons under whch he prcng of CDO ranche premums reduces o a smple numercal negraon. The facor approach also allows some model axonomy by comparng he condonal defaul probables hrough he so-called convex order. Ths yelds some useful resuls on he orderng of ranche premums. The facor assumpon s also almos necessary o deal wh large porfolos and avod overfng. As an example, le us consder he Gaussan copula; he number of correlaon parameers evolves as n, where n s he number of names, whou any facor assumpon, whle ncrease lnearly n a one facor model. In secon I, we wll presen some general feaures of facor models wh respec o he prcng of CDO ranches. Ths ncludes he dervaon of CDO ranche premums from margnal loss dsrbuons, he compuaon of loss dsrbuons n facor models, he facor represenaon assocaed wh de Fne s heorem for homogeneous porfolos, large porfolo approxmaons and an nroducon o he use of sochasc orders as a way o compare dfferen models. Secon II deals varous facor prcng models, ncludng facor copula models, srucural, mulvarae Posson, and Cox process based models. As for he facor copula models, we deal wh addve facor copula models and some exensons nvolvng sochasc or local correlaon. We also consder Archmedean copulas and evenually perfec copulas ha are mpled from marke quoes. Mulvarae Posson models nclude he so-called common shock models. Examples based on Cox processes are relaed o affne nenses whle srucural models are mulvarae exensons of he Black and Cox frs hng me of a defaul barrer. I Facor models for he prcng of CDO ranches Facor models have been used for a long me wh respec o sock or muual fund reurns. As far as cred rsk managemen s concerned, facor models also appear as an mporan ool. They underle he IRB approach n he Basel II regulaory framework: see Crouhy e al. (000), Fnger (001), Gordy (000, 003), Wlson (1997a, 1997b) or Frey and McNel (003) for some llusraons. The dea of compung loss dsrbuons from he assocaed characersc funcon n facor models can be found n Pykhn and Dev (00). The applcaon of such deas o he prcng of CDOs s dscussed n Gregory and Lauren (003), Andersen, Sdenus and Basu (003), Hull and Whe (004), Andersen and Sdenus (005a) and Lauren and Gregory (005). Varous dscussons and exensons abou he facor approach for he prcng of CDO ranches can be found n a number of papers, ncludng, Fnger (005), Burschell e al. (008). I.1 Compuaon of CDO ranche premums from margnal loss dsrbuons 4

5 A synhec CDO ranche s a srucured produc based on an underlyng porfolo of equally weghed reference enes subjec o cred rsk 4. Le us denoe by n he number of references n he cred porfolo and by ( τ,, 1 τ n ) he random vecor of defaul mes. If name defauls, drves a loss of M E ( 1 δ ) = where E denoes he nomnal amoun (whch s usually name ndependen for a synhec CDO) and δ he recovery rae. M s also referred as he loss gven defaul of name. The key n quany for he prcng of CDO ranches s he cumulave loss L = M D, where D 1 { τ } = 1 = s a Bernoull random varable ndcang wheher name defauls before me. L s a pure jump process and follows a dscree dsrbuon a any me. The cash-flows assocaed wh a synhec CDO ranche only depend upon he realzed pah of he cumulave losses on he reference porfolo. Defaul losses on he cred porfolo are spl along some hresholds (aachmen and deachmen pons) and allocaed o he varous ranches. Le us consder a CDO ranche wh aachmen pon a, deachmen pon b and maury T. I s somemes convenen o see a CDO ranche as a blaeral conrac beween a proecon seller and a proecon buyer. We descrbe below he cash-flows assocaed wh he defaul paymen leg (paymens receved by he proecon buyer) and he premum paymen leg (paymens receved by he proecon seller).. Defaul paymens leg The proecon seller agrees o pay he proecon buyer defaul losses each me hey mpac he ranche [ a, b ] of he reference porfolo. More precsely, he cumulave defaul paymen L on he ranche [ a, b ] s equal o zero f [ a, b ] a L b and o b a f L b. Le us remark ha respec o [ a, b ] L a, o L a f L has a call spread payoff wh + + [, ] L (see Fgure 1) and can be expressed as L a b ( L a) ( L b) [ a, b ] L =. b a a b Fgure 1. Cumulave loss on CDO ranche [ a, b ] wh respec o L L 4 We refer he reader o Das (005) exbook or Kakodkar e al. (006) for a dealed analyss of he CDO marke and cred dervaves cash-flows. 5

6 Defaul paymens are smply he ncremen of L [ a, b],.e. here s a paymen of [ a, b] [ a, b] [, ] L L from he proecon seller a every jump me of L a b occurrng before conrac maury T. Fgure shows a realzed pah of he loss process L and consequences on CDO ranche [ a, b ] cumulave losses. L, L [ a, b ] b a b a L [ a, b ] L Fgure. A realzed pah of he reference porfolo losses n blue and he correspondng pah of losses affecng CDO ranche [ a, b ] n red. Jumps occur a defaul mes. For smplcy we furher assume ha he connuously compounded defaul free neres rae r s deermnsc and denoe by B = exp rsds he dscoun facor. 0 Then, he dscouned payoff correspondng o defaul paymens can wren as: T n [ a, b] [ a, b] [ a, b] B dl = ( ) 1 Bτ L τ L τ { τ T }. 0 = 1 Thanks o Seljes negraon by pars formula and Fubn heorem, he prce of he defaul paymen leg can be expressed as: Premum paymens leg T T [ a, b] [ a, b] [ a, b] E B dl = BT E L T r B E L d 0 0. The proecon buyer has o pay he proecon seller a perodc premum paymen (quarerly for sandardzed ndexes) based on a fxed spread or premum S and [, ] proporonal o he curren ousandng nomnal of he ranche b a L a b. Le us denoe by, = 1,, I he premum paymen daes wh I = T and by he lengh of he h perod [, 1 ] (n fracons of a year and wh 0 = 0). The CDO premum [ a, b] paymens are equal o S ( b a L ) a regular paymen daes, = 1,, I. Moreover, when a defaul occurs beween wo premum paymen daes and when affecs he ranche, an addonal paymen (he accrued coupon) mus be made a defaul me o compensae he change n value of he ranche ousandng nomnal. For example, f name j defauls beween 1 and, he assocaed accrued coupon s 6

7 [ a, b] [ a, b] equal o S ( τ j 1)( Lτ L ) j τ j. Evenually, he dscouned payoff correspondng o premum paymens can be expressed as: I [ a, b] [ a, b] B S ( ) ( 1) b a L + B S dl. = 1 1 Usng same compuaonal mehods as for he defaul leg, s possble o derve he prce of he premum paymen leg, ha s I [ a, b] [ a, b] [ a, b] S B ( ) ( 1) ( ( 1) 1) b a E L + B E L B r + E L d. = 1 1 The CDO ranche premum S s chosen such ha he conrac s far a ncepon,.e. such ha he defaul paymen leg s equal o he premum paymen leg. S s quoed n bass pon per annum 5. Fgure 3 shows he dynamcs of cred spreads on he fve year Iraxx ndex (seres 7) beween May and November 007. I s neresng o observe a wde bump correspondng o he summer 007 crss /04/07 07/05/07 1/05/07 04/06/07 18/06/07 0/07/07 16/07/07 30/07/07 13/08/07 7/08/07 10/09/07 4/09/07 08/10/07 /10/07 05/11/07 19/11/07 Fgure 3. Cred spreads on he fve years Traxx ndex (Seres 7) n bps Le us remark ha he compuaon of CDO ranche premums only nvolves he [ a, b] expeced losses on he ranche, E L a dfferen me horzons. These can readly be derved from he margnal dsrbuons of he aggregae loss on he reference 5 Le us remark ha marke convenons are que dfferen for he prcng of equy ranches (CDO ranches [0, b ] wh 0 < b < 1). Due o he hgh level of rsk embedded n hese frs losses ranches, he premum S s fxed beforehand a 500 bps per annum and he proecon seller receve an addonal paymen a ncepon based on an upfron premum and proporonal o he sze b of he ranche. Ths upfron premum s quoed n percenage. 7

8 porfolo. In he nex secon, we descrbe some numercal mehods for he compuaon of he aggregae loss dsrbuon whn facor models. I.3 Compuaon of loss dsrbuons In a facor framework, one can easly derve margnal loss dsrbuons. We wll assume ha defaul mes are condonally ndependen gven a one dmensonal facor V. The key npus for he compuaon of loss dsrbuon are he condonal V p = P τ V assocaed wh names = 1,, n. Exensons defaul probables ( ) o mulple facors are sraghforward bu are compuaonally more nvolved. However he one facor assumpon s no ha resrcve as explaned n Gössl (007) where compuaon of he loss dsrbuon s performed wh an admssble loss of accuracy usng some dmensonal reducon echnques. In some examples dealed below, he facor V may be me dependen. Ths s of grea mporance when prcng correlaon producs ha nvolve he jon dsrbuon of losses a dfferen me horzons such as leverage ranches or forward sarng CDOs. Snce hs paper s focused on he prcng of sandard CDO ranches, whch only nvolve margnal dsrbuons of cumulave losses, omng he me dependence s a maer of noaonal smplcy. Unless oherwse saed, we wll hereafer assume ha recovery raes are deermnsc and concenrae upon he dependence among defaul mes. Two approaches can be used for he compuaon of loss dsrbuons, one based on he nverson of he characersc funcon and anoher one based on recursons. FFT approach The frs approach deals wh he characersc funcon of he aggregae loss whch can be derved hanks o he condonal ndependence assumpon: ul um ( ) 1 ( 1) V ϕ L u = E e = E p e +. 1 n L The prevous expecaon can be compued usng a numercal negraon over he dsrbuon of he facor V. Ths can be acheved for example usng a Gaussan quadraure. The compuaon of he loss dsrbuon can hen be accomplshed hanks o he nverson formula and some Fas Fourer Transform algorhm. Le us remark ha he former approach can be adaped whou exra complcaon when losses gven defaul M, = 1,, n are sochasc bu (jonly) ndependen ogeher wh defaul mes. Ths mehod s descrbed n Gregory and Lauren (003) or Lauren and Gregory (005). Gregory and Lauren (004) nvesgae a rcher correlaon srucure n whch cred references are grouped n several secors. They specfy an ner-secor and an nra-secor dependence srucure based on a facor approach and show ha he compuaon of he loss dsrbuon can be performed easly usng he FFT approach. 8

9 Recurson approaches An alernave approach, based on recursons s dscussed n Andersen, Sdenus and Basu (003) and Hull and Whe (004) 6. The frs sep s o spl up he suppor of he loss dsrbuon no consan wdh loss uns. The wdh u of each loss un s chosen such ha each poenal loss gven defaul M can be approxmaed by a mulple of u. The suppor of he loss dsrbuon s hus urned no a sequence l = 0, u,, nmaxu where nmax > n and nmaxu corresponds o he maxmal poenal loss M. Clearly, he smples case s 1 n 1 δ assocaed wh consan losses gven defaul, for nsance M = wh δ = 40% n and n = 15 whch s a reasonable assumpon for sandard ranches. We can hen choose nmax = n. The second sep s performed hanks o he condonal ndependence of defaul evens gven he facor V. The algorhm sars from he condonal loss dsrbuon assocaed wh a porfolo se up wh only one name, hen performs he compuaon of he condonal loss dsrbuon when anoher name s added, and so k q, = 0,, n he condonal probably ha he loss s on. Le us denoe by ( ) h equal o u n he k porfolo where names 1,,, k ( k n ) have been successvely added. Le us sar wh a porfolo se up wh only name number 1 wh condonal defaul probably p, hen 1 V ( ) ( ) ( ) 1 1V q 0 = 1 p, 1 1V q 1 = p, 1 q = 0, > 1. k Assume now ha q (.) has been compued afer successve ncluson of names,,k n he pool. We hen add frm k + 1 n he porfolo wh condonal defaul k 1 V probably p + h. The loss dsrbuon of he ( k + 1) porfolo can be compued wh he followng recursve relaon: k + 1 k+ 1 V k q ( 0) = ( 1 p ) q ( 0 ), k + 1 k+ 1 V k k+ 1 V k q ( ) = ( 1 p ) q ( ) + p q ( 1 ), = 1,, k + 1, k + 1 q ( ) = 0, > k + 1. In he new porfolo, he loss can be u eher by beng u n he orgnal porfolo f frm k + 1 has no defauled or by beng ( 1) u f frm k + 1has defauled. The requred loss dsrbuon s he one obaned afer all names have been added n he 6 Le us remark ha smlar recurson mehods have frs been nvesgaed by acuares o compue he dsrbuon of aggregae clams whn ndvdual lfe models. Several recurson algorhms orgnaed from Whe and Grevlle (1959) have been developed such as he Z-mehod or he Newon mehod based on developmen of he loss generang funcon. 9

10 n pool. I corresponds o ( ) q, = 0,, n. Le us remark ha even hough nermedae loss dsrbuons obvously depend on he orderng of names added n he pool, he loss dsrbuon assocaed o he enre porfolo s unque. The las sep consss of compung he uncondonal loss dsrbuon usng a numercal negraon over he dsrbuon of he facor V. I s sraghforward o exend he laer mehod o he case of sochasc and name dependen recovery raes. However, one of he key ssues s o fnd a loss un u whch boh allows geng enough accuracy on he loss dsrbuon and drvng low compuaonal me. Hull and Whe (004) presen an exenson of he former approach n whch compuaon effors are focused on peces of he loss dsrbuon assocaed wh posve CDO ranche cash flows, allowng he algorhm o cope wh non-consan wdh loss subdvsons. Oher exensons based on approxmaon mehods are dscussed by Pereyakn (006). Oher approxmaon mehods used by acuares n he ndvdual lfe model have also been adaped o he prcng of CDO ranches. For example, De Prsco e al. (005) nvesgae he compound Posson approxmaon, Jackson e al. (007) propose o approxmae he loss dsrbuon by a Normal Power dsrbuon. Glasserman and Suchnabandd (007) propose an approxmaon mehod based on power seres expansons. These expansons express a CDO ranche prce n a mulfacor model as a seres of prces compued whn an ndependen defaul me model, whch are easy o compue. A new mehod based on Sen s approxmaon has been developed recenly by Jao (007) and seems o be more effcen han sandard approxmaon mehods. In praccal mplemenaon, he condonal loss dsrbuon (condonal o he facor) can be approxmaed eher by a Gaussan or a Posson random varable. Then CDO ranche premums can be compued n each case usng an addonal correcor erm known n closed form. When consderng CDO ranches on sandardzed ndces, s somemes convenen o consder a homogeneous cred porfolo. In ha case, he compuaon of he loss dsrbuon reduces o a smple numercal negraon. I.3 Facor models n he case of homogeneous cred rsk porfolos In he case of a homogeneous cred rsk porfolo, all names have he same nomnal E and he same recovery rae δ. Consequenly, he aggregae loss s proporonal o he number of defauls L = E 1 δ N. Le us moreover assume ha defaul N,.e. ( ) mes τ,, 1 τ n are exchangeable,.e. any permuaon of defaul mes leads o he same mulvarae dsrbuon funcon. Parcularly, means ha all names have he same margnal dsrbuon funcon, say F. 10

11 As a consequence of de Fne s heorem 7, defaul ndcaors D,, 1 Dn are Bernoull mxures 8 a any me horzon. There exss a random mxure probably p such ha condonally on p, D,, 1 Dn are ndependen. More formally, f we denoe by ν he dsrbuon funcon of p, hen for all k = 0,, n, 1 n k ( n k ) P( N = k) = p (1 p) ν ( dp) k. 0 As a resul, he aggregae loss dsrbuon has a very smple form n he homogeneous case. Is compuaon only requres a numercal negraon over ν whch can be acheved usng a Gaussan quadraure. Moreover, can be seen ha he facor assumpon s no resrcve a all n he case of homogeneous porfolos. Homogeney of cred rsk porfolos can be vewed as a reasonable assumpon for CDO ranches on large ndces, alhough hs s obvously an ssue wh equy ranches for whch dosyncrac rsk s an mporan feaure. A furher sep s o approxmae he loss on large homogeneous porfolos wh he mxure probably self. I.4 Large porfolo approxmaons As CDO ranches are relaed o large cred porfolos, a sandard assumpon s o approxmae he loss dsrbuon wh he one of an nfnely granular porfolo 9. Ths fcve porfolo can be vewed as he lm of a sequence of aggregae losses on homogeneous porfolos, where he maxmum loss has been normalzed o uny: L n 1 n n = 1 = D, n 1. Le us recall ha when defaul ndcaors D,,,... 1 D n form a sequence of exchangeable Bernoull random varables and hanks o de Fne s heorem, he normalzed loss L converges almos surely o he mxure probably p as he n number of names ends o nfny. p s also called he large (homogeneous) porfolo approxmaon. In a facor framework where defaul mes are condonally ndependen gven a facor V, can be shown ha he mxure probably p concdes wh he condonal defaul probably P ( τ V ) 10. In he cred rsk conex hs dea was frsly pu n pracce by Vascek (00). Ths approxmaon has also been suded by Gordy and Jones (003), Greenberg e al. (004a), Schloegl and O Kane (005) for he prcng of CDO ranches. Burschell e al. (008) compare he prces of CDO ranches based on he large porfolo approxmaon and on exac 7 Aldous (1984) exbook gves a general accoun of de Fne s heorem and some sraghforward consequences. 8 One needs ha he defaul ndcaors are par of an nfne sequence of exchangeable defaul ndcaors. 9 Ths ermnology s aken from he Basel II agreemen as s he sandard approach proposed by he Basel commee o deermnae he regulaory capal relaed o bank cred rsk managemen. 10 The proof reles on a generalzaon of he srong law of large number. See Vascek (00) for more deals. 11

12 compuaons. The large porfolo approxmaon can also be used o compare CDO ranche premums on fne porfolos. I.5 Comparng dfferen facor models Exchangeably of defaul mes s a nce framework o sudy he mpac of dependence on CDO ranche premums. We have seen ha he facor approach s legmae n hs conex and we have exhbed a mxure probably p such ha, gven p, defaul ndcaors D,, 1 Dn are condonally ndependen. Thanks o he heory of sochasc orders, s possble o compare CDO ranche premums assocaed wh porfolos wh dfferen mxure probables. Le us compare wo porfolos wh defaul ndcaors,, * * D1 Dn and D,, 1 Dn and wh (respecvely) * * mxure probables p and p. If p s smaller han p n he convex order 11, hen n he aggregae loss assocaed wh p, L = M D s smaller han he aggregae loss assocaed wh * p, n * * M D = 1 L = 1 = n he convex order 1. See Cousn and Lauren (007) for more deals abou hs comparson mehod. Then, can be proved (see Burschell e al. (008)) ha when he mxure probables ncrease n he convex order, [0, b ] equy ranche premums decrease and [ a,100%] senor ranche premums ncrease 13. II) A revew of facor approaches o he prcng of CDOs In he prevous secon, we sressed he key role played by he dsrbuon of condonal probables of defaul when prcng CDO ranches. Loosely speakng, specfyng a mulvarae defaul me dsrbuon amouns o specfyng a mxure dsrbuon on defaul probables. We hereafer revew a wde range of popular defaul rsk models facor copulas models, srucural, mulvarae Posson, and Cox process based models hrough a meculous analyss of her mxure dsrbuons. II.1 Facor copula models In copula models, he jon dsrbuon of defaul mes s coupled o s onedmensonal margnal dsrbuons hrough a copula funcon C 14 : P τ,, τ = C F,, F. ( 1 1 n n ) ( 1 ( 1 ) n ( n )) 11 Le X and Y be wo scalar negrable posve random varables. We say ha X precedes Y n + + convex order f E[ X ] E[ Y ] for all 0 K. 1 Losses gven defaul M,, 1 M n mus be jonly ndependen from,, * * D1 Dn and D,, 1 Dn. a b wh 0 < a < b < 1, s no possble o nfer such a comparson 13 As for he mezzanne ranche [, ] = and E ( X K ) E ( Y K ) resul. For example, s well known ha he presen value of a mezzanne ranche may no be monoonc wh respec o he compound correlaon. 14 For an nroducon o copula funcons wh applcaons o fnance, we refer o Cherubn e al. (004) exbook. 1

13 In such a framework, he dependence srucure and he margnal dsrbuon funcons F are can be handled separaely. Usually, he margnal defaul probables ( ) nferred from he cred defaul swap premums on he dfferen names. Thus, hey appear as marke npus. The dependence srucure does no nerfere wh he prcng of sngle name cred defaul swaps and s only nvolved n he prcng of correlaon producs such as CDO ranches. In he cred rsk feld, hs approach has been nroduced by L (000) and furher developed by Schönbucher and Schuber (001). Facor copula models are parcular copula models for whch he dependence srucure of defaul mes follows a facor framework. More specfcally, he dependence srucure s drven by some laen varables V,, 1 Vn. Each varable V s expressed as a bvarae funcon of a common sysemc rsk facor V and an dosyncrac rsk facor V : ( ) V = f V, V, = 1,, n, where V and V, = 1,, n are assumed o be ndependen. In mos applcaons, he specfed funcon f, he facors V and V, = 1,, n are seleced such ha laen varables Consequenly, V, = 1,, n form an exchangeable sequence of random varables. V, = 1,, n mus follow he same dsrbuon funcon, say H. ( ) 15 where F are he dsrbuon funcons of defaul mes and H he margnal dsrbuon of laen 1 Evenually, defaul mes are defned by τ = F H ( V ) varables V, = 1,, n. For smplcy, we wll hereafer resrc o he case where he margnal dsrbuons of defaul mes do no depend upon he name n he reference porfolo and denoe he common dsrbuon funcon by F. In a general copula framework, compuaon of loss dsrbuons requres n successve numercal negraons. The man neres of facor copula approach les n s racably as compuaonal complexy s relaed o he facor dmenson. Hence, facor copula models have been nensely used by marke parcpans. In he followng, we wll revew some popular facor copula approaches. Addve facor copulas The famly of addve facor copulas s he mos wdely used as far as he prcng of CDO ranches s concerned. In hs class of models, he funcon f s addve and laen varables V,, 1 Vn are relaed hrough a dependence parameer ρ akng values n [0,1] : 1 ρ, 1,, V = ρv + V = n. 15 Le us remark ha defaul mes n a facor copula model can be vewed as frs passage mes n a mulvarae sac srucural model where V, = 1,, n correspond o some correlaed asse values and where F ( ) drves he dynamcs of he defaul hreshold. In fac, defaul mes can be expressed as 1 { V H ( F ( ) )} τ = nf 0, = 1,, n. 13

14 From wha was saed n prevous secons, he condonal defaul probably or mxure probably p can be expressed as: ( ( )) 1 ρv + H F p = H. 1 ρ In mos applcaons, V and V, = 1,, n belong o he same class of probably dsrbuons whch s chosen o be closed under convoluon. The mos popular form of he model s he so-called facor Gaussan copula whch reles on some ndependen sandard Gaussan random varables V and V, = 1,, n and leads o Gaussan laen varables V,, 1 Vn. I has been nroduced by Vascek (00) n he cred rsk feld and s known as he mulvarae prob model n sascs 16. Thanks o s racably, he one facor Gaussan copula has become he fnancal ndusry benchmark despe of some well known drawbacks. For example, s no possble o f all marke quoes of sandard CDO ranches of he same maury. Ths defcency s relaed o he so-called correlaon skew. An alernave approach s he Suden- copula whch embeds he Gaussan copula as a lm case. I has been consdered for cred rsk ssues by a number of auhors, ncludng Andersen e al. (003), Embrechs e al. (003), Frey and McNel (003), Mashal e al. (003), Greenberg e al. (004b), Demara and McNel (005), Schloegl and O Kane (005). Neverheless, he Suden- copula feaures he same defcency as he Gaussan copula. For hs reason, a number of addve facor copulas such as he double- copula (Hull and Whe (004)), he NIG copula (Guegan and Houdan (005)), he double-nig copula (Kalemanova e al. (007)), he double Varance Gamma copula (Moosbrucker (006)) and he α -sable copula (Prange and Scherer (006)) have been nvesgaed. Oher heavy-aled facor copula models are dscussed n Wang e al. (007). For a comparson of facor copula approaches n erms of prcng of CDO ranches, we refer o Burschell e al. (008). We plo n Fgure 4 he mxure dsrbuons assocaed wh some of he prevous addve facor copula approaches. Le us recall ha mxure dsrbuons correspond o he loss dsrbuon of large homogeneous porfolos (see Secon I.4). 16 The mulvarae prob model s a popular exenson of he lnear regresson model n sascs. For a descrpon of he model wh applcaons o economercs, we refer he reader o Goureroux (000). 14

15 ndependence comonoonc Gaussan double 4/4 double NIG 1/1 Fgure 4 The graph shows he cumulave densy funcons of he mxure probably p for he Gaussan, he double- (4/4) and he double NIG (1/1) facor copula approaches. The margnal defaul probably s F( ) =.96% and we choose ρ = 30% as he correlaon beween defauls. Evenually, we also plo he mxure dsrbuons assocaed wh he ndependence case ( ρ = 0 ) and he comonoonc case ( ρ = 1 ). Sochasc correlaon Sochasc correlaon models are oher exensons of he facor Gaussan copula model. In hs approach, he dependence parameer s sochasc. The laen varables are hen expressed as: V V V n, = ρ + 1 ρ, = 1,, where V and V, = 1,, n are ndependen sandard Gaussan random varables and ρ, = 1,, n are dencally dsrbued random varables akng values n [0,1] and ndependen from V, V, = 1,, n. A suable feaure of hs approach s ha he laen varables V, = 1,, n follow a mulvarae Gaussan dsrbuon 17. Ths eases calbraon and mplemenaon of he model. Le us remark ha n hs framework, defaul mes are exchangeable. Then, he condonal defaul probably p can be expressed as: ( F( ) ) 1 1 ρv + Φ p = Φ G ( d ρ ), 0 1 ρ where G denoes he dsrbuon funcon of ρ, = 1,, n and Φ s he Gaussan cumulave densy funcon. 17 Thanks o he ndependence beween ρ, V, V, = 1,, n, gven ρ, V follows a sandard Gaussan dsrbuon. Thus, afer an negraon over he dsrbuon of ρ, he margnal dsrbuon of V s also sandard Gaussan. 15

16 Burschell e al. (008) nvesgaed a wo saes sochasc correlaon parameer. Tavares e al. (004) also nvesgae a model wh dfferen saes ncludng a possbly caasrophc one. I has been shown by Burschell e al. (007) ha a hree saes sochasc correlaon model s enough o f marke quoes of CDO ranches for a gven maury. In her framework, he sochasc correlaon parameers ρ, = 1,, n have also a facor represenaon: ρ = 1 B 1 B ρ + B where B s, B,, 1 Bn ( )( ) s s are ndependen Bernoull random varables ndependen from V, V, 1,, n s s p = P B =, = 1,, n, defaul mes are comonoonc ( V = V ) wh probably p s, ndependen V = V ) wh probably (1 p ) p and have a sandard Gaussan facor ( =. Consequenly, f we denoe by p = P( B = 1) and ( 1) represenaon wh probably (1 p )(1 p). Mean-varance Gaussan mxures s s In hs class of facor models, laen varables are smply expressed as mean-varance Gaussan mxures: V = m V + σ V V = n, ( ) ( ), 1,, where V and V, = 1,, n are ndependen sandard Gaussan random varables. Two popular CDO prcng models have been derved from hs class, namely he random facor loadng and he local correlaon model. The random facor loadng model has been nroduced by Andersen and Sdenus (005b). In hs approach, laen varables are modelled by: ( { V < e} { V e} ) V = m + l 1 + h 1 V + νv, = 1,, n, where l, h, e are some npu parameers such ha l, h > 0. m and ν are chosen such ha E[ V ] = 0 and = E V 1. Ths can be seen as a random facor loadng model, snce he rsk exposure l1{ } + h1 V < e { V e} s sae dependen. I s conssen wh emprcal researches showng ha defaul correlaon changes wh respec o some macroeconomc random varables (see Das e al. (006) and references heren). The condonal defaul probably can be wren as: 1 ( ( ( )) ( 1{ } 1 < { } ) ) 1 p = Φ H F m l + h V V e V e, ν where H s he margnal dsrbuon funcon of laen varables V, = 1,, n. Le us remark ha conrary o he prevous approaches, laen varables here are no Gaussan and he dsrbuon funcon H depends upon he model parameers. We compare n Fgure 5 he mxure dsrbuon funcons obaned under a random facor loadng model and a hree saes sochasc correlaon model. 16

17 ndependence comonoonc hree saes correlaon RFL Fgure 5 The graph shows he mxure dsrbuon funcons assocaed wh he hree saes sochasc correlaon model of Burschell e al. (007) and he random facor loadng model of Andersen and Sdenus (005b). The margnal defaul probably, F( ) =.96% holds o be he same for boh approaches. As for he sochasc correlaon model, he parameers are respecvely p s = 0.14, p = 0.81, ρ = 58%. As for he random facor loadng model, we ook l = 85%, h = 5% and e =. The graph also shows he mxure dsrbuon funcons assocaed wh he ndependence and he comonoonc case. Lke he hree saes verson of he sochasc correlaon model, hs approach has he ably o f perfecly marke quoes of sandardzed CDO ranche spreads for a gven maury. The local correlaon model proposed by Turc e al. (005) s assocaed wh he followng paramerc modellng of laen varables: ( ) ρ V = ρ V V + 1 ( V ) V, = 1,, n, where V and V, = 1,, n are ndependen sandard Gaussan random varables and (.) 0,1. ρ (.) s known as he local ρ s some funcon of V akng values n [ ] correlaon funcon. The condonal defaul probables can be wren as: 1 ρ ( V ) V + H ( F( ) ) p = Φ, 1 ρ ( V ) where H s he margnal dsrbuon funcon of laen varables V, = 1,, n. The local correlaon can be used n a way whch parallels he local volaly modellng n he equy dervaves marke. Ths consss n a non paramerc calbraon of ρ (.) on marke CDO ranche premums. The local correlaon funcon has he advanage o be a model based mpled correlaon when compared o some sandard marke pracce such as he compound and he base correlaon. Moreover, here s a smple relaonshp beween ρ (.) and marke compound correlaons 17

18 mpled from CDO ranchles 18 (margnal compound correlaon) as explaned n Turc e al. (005) or Burschell e al. (007). Bu he rouble wh hs approach s ha he exsence and unqueness of a local correlaon funcon s no guaraned gven an admssble loss dsrbuon funcon possbly nferred from marke quoes. Archmedean copulas Archmedean copulas have been wdely used n cred rsk modellng as hey represen a drec alernave o he Gaussan copula approach. In mos cases, here exss an effecve and racable way of generang random vecors wh hs dependence srucure. Moreover, Archmedean copulas are nherenly exchangeable and hus adm a facor represenaon. Marshall and Olkn (1988) frs exhb hs facor represenaon n her famous smulaon algorhm. More precsely, each Archmedean copula can be assocaed wh a posve random facor V wh nverse Laplace ransform ϕ (.) (and Laplace ransform ϕ 1 (.) ). In hs framework, he laen varables can be expressed as: 1 lnv V = ϕ, = 1,, n, V where V, = 1,, n are ndependen unform random varables. Then, he jon,, 1 n s he ϕ -Archmedean copula 19. In parcular, each laen varable s a unform random varable. Then he condonal defaul probably can be wren as: p = exp( ϕ ( F ( ) ) V ). dsrbuon of he random vecor ( V V ) Le us remark ha he prevous framework corresponds o fraly models n he relably heory or survval daa analyss 0. In hese models, V s called a fraly snce low levels of V are assocaed wh shorer survval defaul mes. The mos popular Archmedean copula s probably he Clayon copula. In a cred rsk conex, has been consdered by, among ohers, Schönbucher and Schuber (001), Gregory and Lauren (003), Lauren and Gregory (003), Madan e al. (004), Frend and Rogge (005). In addon, Rogge and Schönbucher (003), Schloegl and O Kane (005) have nvesgaed oher Archmedean copulas such as he Gumbel or he Frank copula. 18 CDO ranches [ a, a + 1%] wh 0 a < 1 19 A random vecor ( V,, 1 Vn ) 1 P( V1 v1,, Vn vn ) = ϕ ( ϕ ( v1 ) + + ϕ ( vn )) follows a ϕ -Archmedean copula f for all v, 1. n [ ], vn 0,1 n : 0 We refer he reader o Hougaard (000) exbook for an nroducon o mulvarae survval daa analyss and a dealed descrpon of fraly models. 18

19 Copula Generaor ϕ Parameer Clayon θ 1 θ 0 θ ( ) θ Frank ln ( 1 θ e ) ( 1 e ) Gumbel ln ( ) θ 1 * R Table 1 Some examples of Archmedean copulas wh her generaors. In Fgure 6, we compare he mxure dsrbuon funcons assocaed wh a Clayon copula and a Gaussan facor copula. The dependence parameer θ of he Clayon copula has been chosen o ge he same equy ranche premums as wh he one facor Gaussan copula model ndependence comonoonc Fgure 6 The graph shows he mxure dsrbuon funcons assocaed wh a Clayon copula and a facor Gaussan copula. F( ) =.96%, ρ = 30%, θ = Gaussan Clayon I can be seen ha he dsrbuon funcons are very smlar. Unsurprsngly, he resulng premums for he mezzanne and senor ranches are also very smlar n boh approaches 1. Perfec copula approach As we saw n prevous secons, much of he effor has focused on he research of a facor copula whch bes fs CDO ranche premums. Le us recall ha specfyng a facor copula dependence srucure s equvalen o specfyng a mxure probably p. Hull and Whe (006) explo hs remark and propose a drec esmaon of he mxure probably dsrbuon from marke quoes. In her approach, for he sake of nuon on spread dynamcs, he mxure probably s expressed hrough a hazard rae random varable λ wh a dscree dsrbuon: P τ λ = λ = 1 exp λ, k = 1,, L. ( k ) ( k ) 1 See Burschell e al. (008), Table 8, for more deals abou correspondence beween parameers and assumpons on he underlyng cred rsk porfolo. 19

20 Then, defauls occur accordng o a mxure Posson process (or a Cox Process) wh π = P λ = λ can hazard rae λ. Once a grd has been chosen for λ, he probably k ( k ) be calbraed n order o mach marke quoes of CDO ranches. Hull and Whe (006) have shown ha hs las sep s no possble n general. Consequenly, hey allow recovery rae o be a decreasng funcon of defaul raes, as suggesed n some emprcal researches such as Alman e al. (005). II. Mulvarae srucural models Mulvarae srucural or frm value models are mul-name exensons of he socalled Black and Cox model where he frm defaul me corresponds o he frs passage me of s asse dynamcs below a ceran hreshold. Ths approach has frs been proposed by Arvans and Gregory (001) (chaper 5) n a general mulvarae Gaussan seng for he prcng of baske cred dervaves. More recenly, Hull e al. (005) nvesgae he prcng of CDO ranche whn a facor verson of he Gaussan mulvarae srucural model. In he followng, we follow he laer framework. We 0,T. Ther asse are concerned wh n frms whch may defaul n a me nerval [ ] dynamcs V, 1 V are smply expressed as n correlaed Brownan moons:, n V = ρv + 1 ρ V, = 1,, n,,, where V, V, = 1,, n are ndependen sandard Wener processes. Defaul of frm s rggered whenever he process V falls below a consan hreshold a whch s here assumed o be he same for all names. The correspondng defaul daes are hen expressed as: τ = nf 0 V a, = 1,, n. {, } Clearly, defaul daes are ndependen condonally on he process V. Le us remark ha as he defaul ndcaors are exchangeable, he exsence of a mxure probably s guaraned hanks o he de Fne s heorem. We are hus n a one facor framework, hough he facor depends on he me horzon conrary o he facor copula case. No mxure dsrbuon can be expressed n closed form n he mulvarae srucural model. Bu, s sll possble o smulae losses on a large homogeneous porfolo (and hen approxmae he mxure probably p ) n order o esmae he mxure dsrbuon. Fgure 7 shows ha he laer happens o be very smlar o he one generaed whn a facor Gaussan copula model. Ths s no surprsng gven he resul of Hull e al. (005) where CDO ranche premums are very close n boh frameworks. Moreover, he facor Gaussan copula can be seen as he sac counerpar of he srucural model developed above. 0

21 Srucural correl = 30% Gaussan correl = 30% Srucural correl = 60% Gaussan correl = 60% Fgure 7 The graph shows emprcal esmaon of one year mxure dsrbuons correspondng o srucural models wh correlaon parameers ρ = 30% and ρ = 60%. The barrer level s se a a = such ha he margnal defaul probably (before = 1 year) s he same n boh approaches and s equal o F( ) = 3.94%. We hen make a comparson wh he mxure dsrbuon assocaed wh facor Gaussan copula models wh he same correlaon parameers and he same defaul probably. The rouble wh he frs passage me models s ha compuaon of CDO ranche premums exclusvely reles on Mone Carlo smulaons and can be very me consumng. Kesel and Scherer (007) propose an effcen Mone Carlo esmaon of CDO ranche spreads n a mulvarae jump-dffuson seng. Oher conrbuons such as Lucano and Schouens (006), Baxer (007) and Wlleman (007) nvesgae he classcal Meron model where defaul a a parcular me occurs f he value of asses s below he barrer a ha parcular pon n me. In hs framework, defaul ndcaors a me are ndependen gven he sysemc asse value V and sem-analycal echnques as explaned n par 1 can be used o compue CDO ranche premums. Moreover, several emprcal researches clam ha he Meron srucural model s a reasonable approxmaon of he more general Black-Cox srucural model when consderng he prcng of CDO ranches. Lucano and Schouens (006) consder a mulvarae Varance Gamma model and show ha can be easly calbraed from marke quoes. Baxer (007) proposes o model he dynamcs of asses wh mulvarae Lévy processes based on he Gamma process and Wlleman (007) nvesgae a mulvarae srucural model as n Hull e al. (005) and adds a common jump componen n he dynamc of asses. II.3 Mulvarae Posson models These models orgnae from he heory of relably where hey are also called shock models. In mulvarae Posson models, defaul mes correspond o he frs jump 1 n N,, N. For example, when he nsans of a mulvarae Posson process ( ) Posson process N jumps for he frs me, rggers he defaul of name. The dependence beween defaul evens derves from he arrval of some ndependen 1

22 sysemc evens or common shocks leadng o he defaul of a group of names wh a gven probably. For he sake of smplcy, we lm ourselves o he case where only wo ndependen shocks can affec he economy. In hs framework, each defaul can be rggered eher by an dosyncrac faal shock or by a sysemc bu no necessarly faal shock. The Posson process whch drves defaul of name can be expressed as: where N and N = + j j= 1 N N B N are ndependen Posson processes wh respecvely parameer λ and λ. We furher assume ha B, = 1,, n, j 1 are ndependen Bernoull j random varables wh mean p ndependen of N and N, = 1,, n. Evenually, defaul mes are descrbed by: τ = nf 0 N > 0, = 1,, n. { } The background even (new jump of N ) affecs each name (ndependenly) wh probably p. A specfcy of he mulvarae Posson framework s o allow for more han one defaul occurrng n small me nervals. I also ncludes he possbly of some Armageddon phenomena where all names may defaul a he same me, hen leadng o faen he al of he aggregae loss dsrbuon as requred by marke quoes. Le us sress ha defaul daes are ndependen condonally on he process N, whle defaul ndcaors D,, 1 Dn are ndependen gven N. By he ndependence of all sources of randomness, N, = 1,, n are Posson processes wh parameer λ + pλ. As a resul, defaul mes are exponenally dsrbued wh he same parameer. I can be shown ha he dependence srucure of defaul mes s he one of he Marshall-Olkn copula (see Lndskog and McNel (003) or Elouerkhaou (006) for more deals abou hs copula funcon). The Marshall-Olkn mulvarae exponenal dsrbuon (Marshall and Olkn (1967)) has been nroduced o he cred doman by Duffe and Sngleon (1998) and also dscussed by L (000) and Wong (000). More recenly, analycal resuls on he aggregae loss dsrbuon have been derved by Lndskog and McNel (003) whn a mulvarae Posson model. Some exensons are presened by Gesecke (003), Elouerkhaou (006), Brgo e al. (007a, 007b). In hs mulvarae Posson model, defaul mes and hus defaul ndcaors are exchangeable. The correspondng mxure probably can be expressed as: N ( ) ( ) p = 1 1 p exp λ. As n he case of mulvarae srucural models, we are sll n a one facor framework, where he facor depends on he me horzon. We plo n Fgure 4, he dsrbuon funcon assocaed o a Mulvarae Posson model. As he mxure probably s a dscree random varable, s dsrbuon funcon s sepwse consan. N B j s assumed o be equal o zero when 0 j= 1 N =.

23 ndependence comonoonc Gaussan Mulvarae Posson Fgure 8 The graph shows he mxure dsrbuon funcons assocaed wh a Mulvarae Posson model wh λ = 0.5%, λ = % and p = 5%. These parameers have been chosen such he margnal defaul probably before = 5 years s F( ) =.96%. For he sake of comparson, we also plo he mxure dsrbuon funcon of he facor Gaussan copula wh ρ = 30%. II.4 Affne nensy models In affne nensy models, he defaul dae of a gven name, say, corresponds o he frs jump me of a doubly sochasc Posson process 3 wh nensy λ. The laer follows an affne jump dffuson sochasc process whch s assumed o be ndependen of he hsory of defaul mes: here are no conagon effecs of defaul evens on he survval name nenses. Le us remark ha, gven he hsory of he process λ, survval dsrbuon funcons of defaul daes can be expressed as: P ( τ λs, 0 s ) = exp λsds, = 1,, n 4. 0 In affne models, dependence among defaul daes s concenraed upon dependence among defaul nenses. In he followng, we follow he approach of Duffe and Gârleanu (001) where he dependence among defaul nenses s drven by a facor represenaon: λ = ax + x, = 1,, n. a s a non negave parameer accounng for he mporance of he common facor and governng he dependence. The processes x, x, = 1,, n are assumed o be ndependen copes of an affne jump dffuson (AJD) process. The choce of AJD 3 Also know as a Cox process. 4 Condonally on he hsory of defaul nensy λ, he defaul daeτ s he frs jump me of a non homogeneous Posson process wh nensy λ. Moreover, as far as smulaons are concerned, defaul mes are ofen expressed usng some ndependen unformly dsrbued random varables U,, 1 U n ndependen of defaul nenses: τ = nf 0 exp λsds U, = 1,, n. 0 3