An Accurate Solution for Credit Value Adjustment (CVA) and Wrong Way Risk

Transcription

1 MPR Munch Personal RePc rchve n ccurae Soluon for Cred alue dusmen C and Wrong Way Rs m ao Rs Quan Caal Mares CIC orono Canada May 203 Onlne a hs://mra.ub.un-muenchen.de/4704/ MPR Paer No osed 20 May 203 2:20 UC

2 n ccurae Soluon for Cred alue dusmen C and Wrong Way Rs m ao Rs Quan Caal Mares CIC orono Canada SRC hs aer resens a new framewor for cred value adusmen C ha s a relavely new area of fnancal dervave modelng and radng. In conras o revous sudes he model reles on he robably dsrbuon of a defaul me/um raher han he defaul me self as he defaul me s usually naccessble. s such he model can acheve a hgh order of accuracy wh a relavely easy mlemenaon. We fnd ha he rces of rsy conracs are normally deermned va bacward nducon when her ayoffs could be osve or negave. Moreover he model can naurally caure wrong or rgh way rs. Key Words: cred value adusmen C wrong way rs rgh way rs cred rs modelng rsy valuaon defaul me aroach D defaul robably aroach DP collaeralzaon margn and neng. he vews exressed here are of he auhor alone and no necessarly of hs hos nsuon. ddress corresondence o m ao Rs Quan Caal Mares CIC 6 ay Sree 0 h loor orono ON M5J 2S8 Canada; emal: m.ao@cic.com or m_yxao@yahoo.com

3 or years a wdesread racce n he ndusry has been o mar dervave orfolos o mare whou ang counerary rs no accoun. ll cash flows are dscouned usng he LIOR curve. u he real ares n many cases haen o be of lower cred ualy han he hyohecal LIOR ary and have a chance of defaul. s a conseuence he Inernaonal ccounng Sandard IS 39 reures bans o rovde a far-value adusmen due o counerary rs. lhough cred value adusmen C became mandaory n 2000 receved a lle aenon unl he recen fnancal crses n whch he rof and loss P&L swngs due o C changes were measured n bllons of dollars. Ineres n C began o grow. Now C has become he frs lne of defense and he cenral ar of counerary rs managemen. C no only allows nsuons o move beyond he radonal conrol mndse of cred rs lms and o uanfy counerary rs as a sngle measurable P&L number bu also offers an ooruny for bans o dynamcally manage rce and hedge counerary rs. he benefs of C are wdely acnowledged. Many bans have se u nernal cred rs radng dess o manage counerary rs on dervaves. he earler wors on C are manly focused on unlaeral C ha assumes ha only one counerary s defaulable and he oher one s defaul-free. he unlaeral reamen neglecs he fac ha boh counerares may defaul.e. counerary rs can be blaeral. rend ha has become ncreasngly relevan and oular has been o consder he blaeral naure of counerary cred rs. lhough mos nsuons vew blaeral consderaons as moran n order o agree on new ransacons Hull and Whe 203 argue ha blaeral C s more conroversal han unlaeral C as he ossbly ha a dealer mgh defaul s n heory a benef o he dealer. C by defnon s he dfference beween he rs-free orfolo value and he rue or rsy or defaulable orfolo value ha aes no accoun he ossbly of a counerary s defaul. he rs-free orfolo value s wha broers uoe or wha radng sysems or models 2

4 normally reor. he rsy orfolo value however s a relavely less exlored and less ransaren area whch s he man challenge and core heme for C. In oher words cenral o C s rsy valuaon. In general rsy valuaon can be classfed no wo caegores: he defaul me aroach D and he defaul robably aroach DP. he D nvolves he defaul me exlcly. Mos C models n he leraure rgo and Caon 2008 Lon and Se 2009 Pyhn and Zhu 2006 and Gregory 2009 ec. are based on hs aroach. lhough he D s very nuve has he dsadvanage ha exlcly nvolves he defaul me. We are very unlely o have comlee nformaon abou a frm s defaul on whch s ofen naccessble see Duffe and Huang 996 Jarrow and Proer 2004 ec.. Usually valuaon under he D s erformed va Mone Carlo smulaon. On he oher hand however he DP reles on he robably dsrbuon of he defaul me raher han he defaul me self. Somemes he DP yelds smle closed form soluons. he curren oular C mehodology Pyhn and Zhu 2006 and Gregory 2009 ec. s frs derved usng D and hen dscrezed over a me grd n order o yeld a feasble soluon. he dscrezaon however s naccurae. In fac hs model has never been rgorously roved. Snce C s used for fnancal accounng and rcng s accuracy s essenal. Moreover hs curren model s based on a well-nown assumon n whch cred exosure and counerary s cred ualy are ndeenden. Obvously can no caure wrong/rgh way rs roerly. In hs aer we resen a framewor for rsy valuaon and C. In conras o revous sudes he model reles on he DP raher han he D. Our sudy shows ha he rcng rocess of a defaulable conrac normally has a bacward recursve naure f s ayoff could be osve or negave. n nuve way of undersandng hese bacward recursve behavours s ha we can hn of ha any conngen clam embeds wo defaul oons. In oher words when enerng an 3

5 OC dervaves ransacon one ary grans he oher ary an oon o defaul and a he same me also receves an oon o defaul self. In heory defaul may occur a any me. herefore he defaul oons are mercan syle oons ha normally reure a bacward nducon valuaon. Wrong way rs occurs when exosure o a counerary s adversely correlaed wh he cred ualy of ha counerary whle rgh way rs occurs when exosure o a counerary s osvely correlaed wh he cred ualy of ha counerary. or examle n wrong way rs exosure ends o ncrease when counerary cred ualy worsens whle n rgh way rs exosure ends o decrease when counerary cred ualy declnes. Wrong/rgh way rs as an addonal source of rs s rghly of concern o bans and regulaors. Snce hs new model allows us o ncororae correlaed and oenally smulaneous defauls no rsy valuaon can naurally caure wrong/rgh way rs. he res of hs aer s organzed as follows: Secon 2 dscusses unlaeral rsy valuaon and unlaeral C. Secon 2 elaboraes blaeral rsy valuaon and blaeral C. Secon 3 resens numercal resuls. he conclusons are gven n Secon 4.. ll roofs and a raccal framewor ha embraces neng agreemens margnng agreemens and wrong/rgh way rs are conaned n he aendces.. Unlaeral Rsy aluaon and Unlaeral C We consder a flered robably sace 0 P sasfyng he usual condons where denoes a samle sace; denoes a -algebra; P denoes a robably measure; 0 denoes a flraon. he defaul model s based on he reduced-form aroach roosed by Duffe and Sngleon 999 and Jarrow and urnbell 994 whch does no exlan he even of defaul endogenously bu characerzes exogenously by a um rocess. he song or defaul me 4

6 of a frm s modeled as a Cox arrval rocess also nown as a doubly sochasc Posson rocess whose frs um occurs a defaul and s defned as nf : h s s ds 0 where h or h denoes he sochasc hazard rae or arrval nensy deenden on an exogenous common sae and s a un exonenal random varable ndeenden of by I s well-nown ha he survval robably from me o s n hs framewor s defned. s : P s Z ex s h u du 2a he defaul robably for he erod s n hs framewor s defned by s : P s Z s ex s h u du 2b wo counerares are denoed as and. Le valuaon dae be. Consder a fnancal conrac ha romses o ay a 0 from ary o ary a maury dae and nohng before dae. ll calculaons n he aer are from he ersecve of ary. he rs free value of he fnancal conrac s gven by D 3a where D ex r u du 3b where denoes he execaon condonal on he D denoes he rs-free dscoun facor a me for he maury and ru u. denoes he rs-free shor rae a me u Nex we urn o rsy valuaon. In a unlaeral cred rs case we assume ha ary s defaul-free and ary s defaulable. Rsy valuaon can be generally classfed no wo 5

7 caegores: he defaul me aroach D and he defaul robably nensy aroach DP..e. he D nvolves he defaul me exlcly. If here has been no defaul before me he value of he conrac a s he ayoff. If a defaul haens before.e. a recovery ayoff s made a he defaul me as a fracon of he mare value 2 gven by where s he defaul recovery rae and s he mare value a defaul. Under a rs-neural measure he value of hs defaulable conrac s he dscouned execaon of all he ayoffs and s gven by D D 4 where Y s an ndcaor funcon ha s eual o one f Y s rue and zero oherwse. lhough he D s very nuve has he dsadvanage ha exlcly nvolves he defaul me/um. We are very unlely o have comlee nformaon abou a frm s defaul on whch s ofen naccessble. Usually valuaon under he D s erformed va Mone Carlo smulaon. he DP reles on he robably dsrbuon of he defaul me raher han he defaul me self. We dvde he me erod no n very small me nervals and assume ha a defaul may occur only a he end of each very small erod. In our dervaon we use he aroxmaon y y ex for very small y. he survval and he defaul robables for he erod are gven by h h ˆ : ex 5a h h ˆ : ex 5b he bnomal defaul rule consders only wo ossble saes: defaul or survval. or he one-erod economy a me he asse eher defauls wh he defaul 2 Here we use he recovery of mare value RM assumon. 6

8 robably or survves wh he survval robably. he survval ayoff s eual o he mare value and he defaul ayoff s a fracon of he mare value:. Under a rs-neural measure he value of he asse a s he execaon of all he ayoffs dscouned a he rs-free rae and s gven by ex r ˆ ˆ ex y 6 where y r h r c denoes he rsy rae and c h he shor cred sread. Smlarly we have s called ex y 2 7 Noe ha y ex s -measurable. y defnon an -measurable random varable s a random varable whose value s nown a me. ased on he ang ou wha s nown and ower roeres of condonal execaon we have ex ex y y ex y ex 0 y 2 2 y recursvely dervng from forward over and ang he lm as he rsy value of he asse can be exressed as 8 aroaches zero ex y u du 9 We may hn of y u as he rs-adused shor rae. uaon 9 s he same as uaon 0 n Duffe and Sngleon [999] whch s he mare model for rcng rsy bonds. Usng he DP we oban a closed-form soluon for rcng an asse subec o cred rs. Oher good examles of he DP are he CDS model roosed by J.P. Morgan 999 and a more generc rsy model resened by ao 203a. 7

9 In heory a defaul may haen a any me.e. a rsy conrac s connuously defaulable. hs Connuous me Rsy aluaon Model s accurae bu somemes comlex and exensve. or smlcy eole somemes refer he Dscree me Rsy aluaon Model ha assumes ha a defaul may only haen a some dscree mes. naural selecon s o assume ha a defaul may occur only on he aymen daes. orunaely he level of accuracy for hs dscree aroxmaon s well nsde he ycal bd-as sread for mos alcaons see O Kane and urnbull rom now on we wll focus on he dscree seng only bu many of he ons we mae are eually alcable o he connuous seng. or a dervave conrac usually s ayoff may be eher an asse or a lably o each ary. hus we furher relax he assumon and suose ha may be osve or negave. In he case of 0 he survval value s eual o he ayoff and he defaul ayoff s a fracon of he ayoff. Whereas n he case of 0 he conrac value s he ayoff self because he defaul rs of ary s rrelevan for unlaeral rsy valuaon n hs case. herefore we have Prooson : he unlaeral rsy value of he sngle-aymen conrac n a dscree-me seng s gven by where 0a D 0 0b Proof: See he aendx. Here can be regarded as a rs-adused dscoun facor. Prooson says ha he unlaeral rsy valuaon of he sngle ayoff conrac has a deendence on he sgn of he ayoff. If he ayoff s osve he rsy value s eual o he rs-free value mnus he dscouned oenal loss. Oherwse he rsy value s eual o he rs-free value. 8

10 Prooson can be easly exended from one-erod o mulle-erods. Suose ha a defaulable conrac has m cash flows. Le he m cash flows be reresened as m wh aymen daes m. ach cash flow may be osve or negave. We have he followng rooson. Prooson 2: he unlaeral rsy value of he mulle-aymen conrac s gven by m a 0 where 0 and D 0 b Proof: See he aendx. he rsy valuaon n Prooson 2 has a bacward naure. he nermedae values are val o deermne he fnal rce. or a dscree me nerval he curren rsy value has a deendence on he fuure rsy value. Only on he fnal aymen dae m he value of he conrac and he maxmum amoun of nformaon needed o deermne he rs-adused dscoun facor are revealed. he couled valuaon behavor allows us o caure wrong/rgh way rs roerly where counerary cred ualy and mare rces may be correlaed. hs ye of roblem can be bes solved by worng bacwards n me wh he laer rsy value feedng no he earler ones so ha he rocess bulds on self n a recursve fashon whch s referred o as bacward nducon. he mos oular bacward nducon valuaon algorhms are lace/ree and regresson-based Mone Carlo. or an nuve exlanaon we can os ha a defaulable conrac under he unlaeral cred rs assumon has an embedded defaul oon see Sorensen and oller 994. In oher words one ary enerng a defaulable fnancal ransacon acually grans he oher ary an oon o defaul. If we assume ha a defaul may occur a any me he defaul oon s an 9

11 mercan syle oon. mercan oons normally have bacward recursve naures and reure bacward nducon valuaons. he smlary beween mercan syle fnancal oons and mercan syle defaul oons s ha boh reure a bacward recursve valuaon rocedure. he dfference beween hem s n he omal sraegy. he mercan fnancal oon sees an omal value by comarng he exercse value wh he connuaon value whereas he mercan defaul oon sees an omal dscoun facor based on he oon value n me. he unlaeral C by defnon can be exressed as m 0 C D 2 Prooson 2 rovdes a general form for rcng a unlaeral defaulable conrac. lyng o a arcular suaon n whch we assume ha all he ayoffs are nonnegave we derve he followng corollary: Corollary : If all he ayoffs are nonnegave he rsy value of he mulle-aymens conrac s gven by m 0 3a where 0 and D 3b he roof of hs corollary s easly obaned accordng o Prooson 2 by seng 0 snce he value of he conrac a any me s also nonnegave. he C n hs case s gven by m C D 4 0 he curren oular C model e.g. euaon 7 n Pyhn and Zhu 2007 and euaon 3 n Gregory 2009 s ue dfferen from above eher euaon 2 or euaon 4. s a maer of fac he curren C model has never been rgorously roved. In order o reflec 0

12 he economc value of counerary cred rs o measure he rof and loss of a ban and o rovde roer ncenves o raders a good C model mus be no only rgorous and accurae bu also feasble o mlemen. 2. laeral Rsy aluaon and laeral C here s amle evdence ha cororae defauls are correlaed. he defaul of a frm s counerary mgh affec s own defaul robably. hus defaul correlaon and deendence arse due o he counerary relaons. Defaul correlaon can be osve or negave. he effec of osve correlaon s usually called conagon whereas he laer s referred o as comeon effec. wo counerares are denoed as and. he bnomal defaul rule consders only wo ossble saes: defaul or survval. herefore he defaul ndcaor Y for ary = follows a ernoull dsrbuon whch aes value wh defaul robably and value 0 wh survval robably.e. P { Y 0} and P Y } {. he margnal defaul dsrbuons can be deermned by he reduced-form models. he on dsrbuons of a bvarae ernoull varable can be easly obaned va he margnal dsrbuons by nroducng exra correlaons. Consder a ar of random varables Y Y ha has a bvarae ernoull dsrbuon. he on robably reresenaons are gven by 00 : P Y 0 Y 0 5a 0 : P Y 0 Y 5b 0 : P Y Y 0 5c : P Y Y 5d where 2 Y : Y Y where denoes he defaul correlaon coeffcen and denoes he defaul covarance.

13 able. Payoffs of a blaerally defaulable conrac hs able dslays all ossble ayoffs a me. In he case of 0 here are a oal of four ossble saes a me : oh and survve wh robably 00. he conrac value s eual o he ayoff. defauls bu survves wh robably 0. he conrac value s where reresens he non-defaul recovery rae 3. =0 reresens he one-way selemen rule whle = reresens he wo-way selemen rule. survves bu defauls wh robably 0. he conrac value s where reresens he defaul recovery rae. v oh and defaul wh robably. he conrac value s where denoes he on recovery rae when boh ares and defaul smulaneously. smlar logc ales o he case of 0. Sae Y 0 Y 0 Y Y 0 Y 0 Y Y Y Commens & survve defauls survves survves defauls & defaul Probably Payoff here are wo defaul selemen rules n he mare. he one-way aymen rule was secfed by he early ISD maser agreemen. he non-defaulng ary s no oblgaed o comensae he defaulng ary f he remanng mare value of he nsrumen s osve for he defaulng ary. he wo-way aymen rule s based on curren ISD documenaon. he non-defaulng ary wll ay he full mare value of he nsrumen o he defaulng ary f he conrac has osve value o he defaulng ary. 2

14 3 Suose ha a fnancal conrac ha romses o ay a from ary o ary a maury dae and nohng before dae where. he ayoff may be osve or negave.e. he conrac may be eher an asse or a lably o each ary. ll calculaons are from he ersecve of ary. me here are a oal of four ossble saes shown n able. he rsy value of he conrac s he dscouned execaon of he ayoffs and s gven by he followng rooson. Prooson 3: he blaeral rsy value of he sngle-aymen conrac s gven by D K 0 0 6a where 6b 6c Proof: See he aendx. We may hn of K as he rs-adused dscoun facor. Prooson 3 ells us ha he blaeral rsy rce of a sngle-aymen conrac can be exressed as he resen value of he ayoff dscouned by a rs-adused dscoun facor ha has a swchng-ye deendence on he sgn of he ayoff. Usng a smlar dervaon as n Prooson 2 we can easly exend Prooson 3 from one-erod o mulle-erods. Suose ha a defaulable conrac has m cash flows. Le he m cash flows be reresened as wh aymen daes where = m. ach cash flow may be osve or negave. he blaeral rsy value of he mulle-aymen conrac s gven by Prooson 4: he blaeral rsy value of he mulle-aymen conrac s gven by m K 0 7a

15 where 0 and K D 0 0 7b where and are defned n Prooson 3. Proof: he roof s smlar o Prooson 2 by relacng wh K. Prooson 4 says ha he rcng rocess of a mulle-aymen conrac has a bacward naure snce here s no way of nowng whch rs-adused dscounng rae should be used whou nowledge of he fuure value. Only on he maury dae he value of he conrac and he decson sraegy are clear. herefore he evaluaon mus be done n a bacward fashon worng from he fnal aymen dae owards he resen. hs ye of valuaon rocess s referred o as bacward nducon. here s a common msconceon n he mare. Many eole beleve ha he cash flows of a defaulable fnancal conrac can be rced ndeendenly and hen be summed u o gve he fnal rsy rce of he conrac. We emhasze here ha hs concluson s only rue of he fnancal conracs whose ayoffs are always osve. In he cases where he romsed ayoffs could be osve or negave he valuaon reures no only a bacward recursve nducon rocedure bu also a sraegc selecon of dfferen dscoun facors accordng o he mare value n me. hs couled valuaon rocess allows us o caure correlaon beween counerares and mare facors. he blaeral C of he mulle-aymen conrac can be exressed as m D K C Numercal Resuls In hs secon we resen some numercal resuls for C calculaon based on he heory descrbed above. rs we sudy he mac of margn agreemens on C. he esng 4

16 orfolo consss of a number of neres rae euy and foregn exchange dervaves. he number of smulaon scenaros or ahs s he me buces are se weely. If he comuaonal reuremens exceed he sysem lm one can reduce boh he number of scenaros and he number of me buces. he me buces can be desgned fne-granulary a he shor end e.g. daly and hen weely and coarse-granulary a he far end e.g. monhly and hen yearly. he raonale s ha he calculaon becomes less accurae due o he accumulaed error from smulaon dscrezaon and nhered errors from calbraon of he underlyng models such as hose due o he change of macro-economc clmae. he collaeral margn erod of rs s assumed o be 4 days 2 wees. We use a CIR Cox-Ingersoll-Ross model for neres rae and hazard rae scenaro generaons; a modfed GM Geomerc rownan Moon model for euy and collaeral evoluons; and a K lac Karasns model for foregn exchange dynamcs. he resuls are resened n he followng ables. able 2 llusraes ha f ary has an nfne collaeral hreshold hreshold H.e. no collaeral reuremen on he C value ncreases whle he H ncreases. able 3 shows ha f ary has an nfne collaeral hreshold H he C value acually decreases whle he hreshold H ncreases. hs reflecs he blaeral mac of he collaerals on he C. he mac s mxed n able 4 when boh ares have fne collaeral hresholds. able 2. he mac of collaeral hreshold hs able shows ha gven an nfne denoes he collaeral hreshold of ary and Collaeral hreshold H on he C H he C ncreases whle H ncreases where H H denoes he collaeral hreshold of ary. H 0. Ml 5. Ml 20. Ml Infne C

17 able 3. he mac of collaeral hreshold hs able shows ha gven an nfne denoes he collaeral hreshold of ary and Collaeral hreshold H on he C H he C decreases whle H ncreases where H H denoes he collaeral hreshold of ary. H 0. Ml 5. Ml 20. Ml Infne C able 4. he mac of he boh collaeral hresholds on he C he C may ncrease or decrease whle boh collaeral hresholds change where he collaeral hreshold of ary and reflecs he fac ha he collaerals have blaeral macs on he C. Collaeral hreshold Collaeral hreshold H denoes H denoes he collaeral hreshold of ary. hs H 0. Ml 5. Ml 20. Ml Infne H 0. Ml 5. Ml 20. Ml Infne C Nex we examne he mac of wrong way rs. Wrong way rs occurs when exosure o a counerary s adversely correlaed wh he cred ualy of ha counerary whle rgh way rs occurs when exosure o a counerary s osvely correlaed wh he cred ualy of ha counerary. Wrong/rgh way rs as an addonal source of rs s rghly of concern o bans and regulaors. Some fnancal mares are closely nerlned whle ohers are no. or examle CDS rce movemens have a feedbac effec on he euy mare as a radng sraegy commonly emloyed by bans and oher mare arcans consss of sellng a CDS on a reference eny and hedgng he resulng cred exosure by shorng he soc. On he oher hand Moody s Invesor s Servce 2000 resens sascs ha sugges ha he correlaons beween neres raes and CDS sreads are very small. 6

18 o caure wrong/rgh way rs we need o deermne he deendency beween counerares and o correlae he cred sreads or hazard raes wh he oher mare rs facors e.g. eues commodes ec. n he scenaro generaon. We use an euy swa as an examle. ssume he correlaon beween he underlyng euy rce and he cred ualy hazard rae of ary s. he mac of he correlaon on he C s show n able 5. he resuls say ha he C ncreases when he absolue value of he negave correlaon ncreases. able 5. he mac of wrong way rs on he C hs able shows ha he C ncreases whle he negave correlaon ncreases n he absolue value. We use an euy swa as an examle and assume ha here s a negave correlaon beween he euy rce and he cred ualy of ary. Correlaon 0-50% -00% C Concluson hs arcle resens a framewor for rcng rsy conracs and her Cs. he model reles on he robably dsrbuon of he defaul um raher han he defaul um self because he defaul um s normally naccessble. We fnd ha he valuaon of rsy asses and her Cs n mos suaons has a bacward recursve naure and reures a bacward nducon valuaon. n nuve exlanaon s ha wo counerares mlcly sell each oher an oon o defaul when enerng no an OC dervave ransacon. If we assume ha a defaul may occur a any me he defaul oons are mercan syle oons. If we assume ha a defaul may only haen on he aymen daes he defaul oons are ermudan syle oons. oh ermudan and mercan oons reure bacward nducon valuaons. 7

19 8 ased on our heory we roose a novel cash-flow-based framewor see aendx for calculang blaeral C a he counerary orfolo level. hs framewor can easly ncororae varous cred mgaon echnues such as neng agreemens and margn agreemens and can caure wrong/rgh way rs. Numercal resuls show ha hese cred mgaon echnues and wrong/rgh way rs have sgnfcan macs on C. endx. Proofs Proof of Prooson : Under he unlaeral cred rs assumon we only consder he defaul rs when he asse s n he money. ssume ha a defaul may only occur on he aymen dae. herefore he rsy value of he asse a s he dscouned execaon of all ossble ayoffs and s gven by D D a where 0 D b Proof of Prooson 2: Le 0. On he frs aymen day le denoe he rsy value of he asse excludng he curren cash flow. ccordng o Prooson he rsy value of he asse a s gven by 0 2a where D 2b Smlarly we have

20 9 Noe ha 0 s -measurable. ccordng o he ang ou wha s nown and ower roeres of condonal execaon we have y recursvely dervng from 2 forward over m where m m we have m 0 5 Proof of Prooson 3: We assume ha a defaul may only occur on he aymen dae. me here are four ossble saes: boh and survve 2 defauls bu survves 3 survves bu defauls and 4 boh and defaul. he on dsrbuons of and are gven by 5. Deendng on wheher he ayoff s n he money or ou of he money a we have D K D a where 6b 6c. raccal framewor for calculang blaeral C We develo a raccal framewor for calculang blaeral C a counerary orfolo level based on he heory descrbed above. he framewor ncororaes neng and margn agreemens and caures rgh/wrong way rs.

21 wo ares are denoed as and. ll calculaons are from he ersecve of ary. Le he valuaon dae be. he C comuaon rocedure consss of he followng ses... Rs-neural Mone Carlo scenaro generaon One core elemen of he radng cred rs modelng s he Mone Carlo scenaro generaon mare evoluon. hs mus be able o run a large number of scenaros for each rs facor wh flexbly over arameerzaon of rocesses and reamen of correlaon beween underlyng facors. Cred exosure may be calculaed under real robably measure whle C or rcng counerary cred rs should be conduced under rs-neural robably measure. Due o he exensve comuaonal nensy of rcng counerary rs here wll nevably be some comromse of lmng he number of mare scenaros ahs and he number of smulaon daes also called me buces or me nodes. he me buces are normally desgned fne-granulary a he shor end and coarse-granulary a he far end. he deals of scenaro generaon are beyond he scoe of hs aer..2. Cash flow generaon or ease of llusraon we choose a vanlla neres rae swa as neres rae swas collecvely accoun for around wo-hrds of boh he noonal and mare value of all ousandng dervaves. ssume ha ary ays a fxed rae whle ary ays a floang-rae. ssume ha here are M me buces 0... M n each scenaro and N cash flows n he samle swa. Le consder scenaro frs. or swale here are four moran daes: he fxng dae f he sarng dae s he endng dae e and he aymen dae. In general hese daes are no concdenly a he smulaon me buces. he me relaonsh beween swale and he smulaon me buces s llusraed n gure. 20

22 Ineres rae curve smulaed a f Raes f s erms e gure : n neres rae swale hs fgure llusraes he me relaonsh beween an neres rae swale and he smulaon me buces. he floang leg of he swale s rese a he fxng dae f wh he sarng dae s he endng dae e and he aymen dae. he smulaon me buces are. he smulaed neres rae curve s sarng a f.... oh fxed rae aymens and floang-rae aymens occur on he same aymen daes. he cash flow of swale s deermned a he fxng dae f ha s assumed o be beween he smulaon me buces and. rs we need o creae an neres rae curve observed a f by nerolang he neres rae curves smulaed a and va eher rownan rdge or lnear nerolaon. he lnear nerolaon s he execaon of he rownan rdge. hen we can calculae he ayoff of swale a scenaro as ; R N f s e s e 2

23 where N denoes he noonal; f ; s e denoes he smly comounded forward rae rese a f for he forward erod s e ; denoes he accrual facor or day coun s e fracon for he erod s e and R denoes he fxed rae. he cash flow amoun calculaed by s ad on he aymen dae. hs value should be allocaed no he neares revous me buce as: ~ D 2 where D denoes he rs-free dscoun facor based on he neres rae curve smulaed a. Cash flow generaon for roducs whou early-exercse rovson s ue sraghforward. or early-exercse roducs one can use he aroach roosed by Longsaff and Schwarz 200 o oban he omal exercse boundares and hen he ayoffs. 3. ggregaon and neng agreemens fer generang cash flows for each deal we need o aggregae hem a counerary orfolo level a each scenaro and each me buce. he cash flows are aggregaed by eher neng or nonneng based on he neng agreemens. neng agreemen s a rovson ha allows he offse of selemen aymens and reces on all conracs beween wo counerares. noher moran use of neng s he close-ou neng ha allows he offse of close-ou values. or neng we add all cash flows ogeher a he same scenaro and he same me buce o recognze offseng. he aggregaed cash flow under neng a scenaro and me buce s gven by ~ ~ 3 22

24 or nonneng we dvded cash flows no osve and negave grous and add hem searaely. In oher words he offseng s no recognzed. he aggregaed cash flows under nonneng a scenaro and me buce are gven by ~ m l ~ ~ l m f f l m Margn or collaeral agreemens Under a margn agreemen he collaeral s called as soon as he counerary exosure rses above he gven collaeral hreshold H or more recsely above he hreshold H lus mnmum ransferable amoun M. ha would resul n reducon of exosure by he collaeral amoun held. Conseuenly here would be no exosure above he hreshold H f here were no me lags beween collaeral callng osng ludang and closng ou. However hese lags whch are acually he margn erod of rs do exs n racce. he collaeral can derecae or arecae n value durng hs erod. he lags exose he ban wh addonal exosure above he hreshold whch s normally referred o as collaeralzed exosure. Clearly he longer he margn erod of rs s he larger he collaeralzed exosure s. or a more dealed dscusson on collaeralzaon see ao 203b. ssume ha he collaeral margn erod of rs s. he collaeral mehodology consss of he followng rocedures: node. rs for any me buce we nroduce an addonal collaeral me node. Second we comue he orfolo value a scenaro and a collaeral me hen we calculae he collaeral reured o reduce he exosure a as 23

25 0 H H f H f f H H H 5 where H H M s he collaeral hreshold of ary and H H M s he negave collaeral hreshold of ary. ach ban has s own collaeral smulaon mehodology ha smulaes he collaeral value evolvng from o over he margn erod of rs. he deals of collaeral smulaon are beyond he scoe of hs aer. ssume ha he collaeral value has already been calculaed. cash flow Nex we comue he change amoun of he collaerals beween and as : 6 Snce our C mehodology s based on cash flows we model collaeral as a reversng a. nally he oal cash flow a s gven by ~ 7 5. C Calculaon fer aggregang all cash flows va neng and margn agreemens one can rce a orfolo n he same manner as rcng a sngle deal. We assume ha he reader s famlar wh he regresson-based Mone Carlo valuaon model roosed by Longsaff and Schwarz 200 and hus do no reea some well-nown rocedures for brevy. he rs-free value a scenaro s gven by m D 0 8 he fnal rs-free orfolo value s he average execaon of all scenaros gven by m D

26 buce he rsy valuaon rocedure s erformed eravely sarng a he las effecve me m and hen worng bacwards owards he resen. We now he value of he orfolo a he fnal effecve me buce whch s eual o he las cash flow.e. D m m. D ased on he sgn of and Prooson 4 we can choose a roer rs-adused m dscoun facor and hen use he well-nown regresson aroach roosed by Longsaff and D Schwarz 200 o esmae m from cross-seconal nformaon n he smulaon by usng leas suares. We conduc he bacward nducon rocess erformed by eravely rollng bac a seres of long ums from he fnal effecve me buce reach he valuaon dae. hen he resen value a scenaro s m 0 K 0 m across me nodes unl we 9 he fnal rue/rsy orfolo value s he average execaon of all scenaros gven by D m 0 K 0 0 C s by defnon he dfference beween he rs-free orfolo value and he rue or rsy or defaulable orfolo value gven by D K m 0 0 D C Reference rgo D. and Caon laeral counerary rs valuaon wh sochasc dynamcal models and alcaon o Cred Defaul Swas Worng aer. Duffe Darrell and Mng Huang 996 Swa raes and cred ualy Journal of nance

27 Duffe Darrell and Kenneh J. Sngleon 999 Modelng erm srucure of defaulable bonds Revew of nancal Sudes Gregory Jon 2009 eng wo-faced over counerary cred rs RISK Hull J. and Whe. 203 C and wrong way rs forhcomng nancal nalyss Journal. Jarrow R.. and Proer P Srucural versus reduced form models: a new nformaon based ersecve Journal of Invesmen Managemen Jarrow Rober. and Suar M. urnbull 995 Prcng dervaves on fnancal secures subec o cred rs Journal of nance Lon. and Se Cred value adusmen for cred defaul swas va he srucural defaul model Journal of Cred Rs Longsaff rancs. and duardo S. Schwarz 200 alung mercan oons by smulaon: a smle leas-suares aroach he Revew of nancal Sudes Moody s Invesor s Servce 2000 Hsorcal defaul raes of cororae bond ssuers J. P. Morgan 999 he J. P. Morgan gude o cred dervaves Rs Publcaons. O Kane D. and S. urnbull 2003 aluaon of cred defaul swas xed Income Quanave Cred Research Lehman rohers QCR Quarerly 2003 Q/Q

28 Pyhn Mchael and Seven Zhu 2007 gude o modelng counerary cred rs GRP Rs Revew July/ugus Sorensen. and. oller 994 Prcng swa defaul rs nancal nalyss Journal ao. 203a he mac of defaul deendency and collaeralzaon on asse rcng and cred rs modelng Worng aer. ao. 203b n economc examnaon of collaeralzaon n dfferen fnancal mare Worng aer. 27