Valuation of a Basket Loan Credit Default Swap

Transcription

1 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr Valuaon of a Bak Loan Cr Dfaul Swa Jn Lang (Corronng auhor) Darmn of Mahmac ongj Unvry Shangha 9 PRChna l: x 6 E-mal: lang_jn@ongjucn Yujng Zhou Darmn of Mahmac ongj Unvry Shangha 9 PRChna l: x 6 E-mal: nay_cryal@6com Rcv: Smbr 6 Acc: Smbr 9 o:543/jfrvn h work uor by Naonal Bac Rarch Program of Chna (973 Program) 7CB8493 Abrac h ar rov a mhoology for valung a bak Loan CDS (LCDS) by conrng boh faul an raymn rk Unr o own an nny framwork ung a ngl-facor mol corrla faul an raymn rk ar conr whr h ochac nr ra u o b hr common facor All ochac roc n h mol ar aum o follow CIR roc hrough Fynman-Kac formula w oban a PDE roblm an clo-form oluon Numrcal xaml ar rov Kywor: LCDS Dfaul Praymn PDE mho Clo-form oluon Inroucon Loan-only Cr Dfaul Swa call LCDS n ml ar fnancal nrumn ha rov h buyr an nuranc agan h faul of h unrlyng ynca cur loan I mark wr launch n 6 boh n US an Euro I vlo on h ba of anar Cr Dfaul Swa (CDS) A anar Cr Dfaul Swa a kn of nuranc agan cr rk h buyr of h CDS h buyr of rocon who ay a fx f or rmum o h llr of rocon for a ro of m If h cr vn occur h llr ay comnaon o h buyr If hr no cr vn occur urng h rm of wa h buyr connu o ay h rmum unl h CDS maury Hull-Wh (Hull an Wh ) fr conr h valuaon of a anar cr faul wa whn hr no counrary faul rk An hy xn hr uy o h uaon whr hr obly of counrary faul rk an obanng a rcng formula wh Mon Carlo mulaon (Hull an Wh ) Furhrmor hy vlo h wo fa rocur for valung ranch of collaralz b oblgaon an n o faul CDS whou Mon Carlo mulaon (Hull an Wh 4) Comarng o a anar CDS a LCDS conrac almo h am xc ha I rfrnc oblgaon lm on loan; I can b cancll So ha h rcng of LCDS mu ak no accoun no only faul robabl wh rcovry ra bu alo h raymn robabl h wo robabl ar ngav corrla h rongr h rlaonh bwn faul an raymn h hghr h LCDS ra wll b In h lraur ffrn mol for LCDS hav bn vlo h mol can b claf no wo man cagor known a rucural mol an ruc form mol In h mol faul an raymn ar mol a a funcon of a of a varabl h nny ba mol ruc form on For h rcng LCDS Zhn W (Zhn W 7) conr a ngl nam LCDS whr faul nny an raymn nny wr nvolv H u a ngl-facor mol wh common facor an corrlaon coffcn o c h ngav rlaonh bwn faul an raymn hrough a oubl ochac roc an unr CIR roc h olv h Publh by Scu Pr

2 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr roblm Ba on Zhn W' work Pr Dobranzky (Dobranzky 8) m h raymn nny wh a coffcn varaon o crb h rlaonh bwn LCDS an CDS For rcng a bak LCDS h ruc form mol u mor frqunly u o h cal an comlxy of h ool h ruc form mol can b claf o wo cagor: boom u an o own In a boom u mol h orfolo nny an aggrga of h conun nn In a o own mol h collaral orfolo mol a a whol na of rllng own o nvual conun; h orfolo nny cf whou rfrnc o h conun h conun nn ar rcovr by ranom hnnng h bnf of a o-own aroach mlcy a a rul of no havng o mol h nvual conun of h unrlyng orfolo Kay Gck (Gck 8) conra h wo molng aroach I mhaz h rol of h nformaon flraon a a molng ool SWu LSJang an JLang (Jang an Lang 8) u o-own mol o rcng of MBS wh raymn rk Ung boom u framwork Shk H S Umau an Zhn W (Umau an W 7) u rcng a CDS rfrnc a ool loan crb h faul an raymn by ngl-facor Gauan Coula mol hy oban h ra of h LCD hrough Mon Carlo mulaon Dobranzky an Schoun (Dobranzky an Schoun 7) u ngl-nam Lévy coula o crb h rlaonh bwn faul an raymn In h uy unr o own framwork w conr h valuaon of a LCDS whr h rfrnc ool conr a an ny hu n h mol h faul an h raymn hr bcom lo of h ool by ffrn way whl h nvr raymn bhavor a gan In h followng con w nrouc wo roc of faul an raymn an conruc h mol of LCDS n h connuou m uaon In h hr con w u ngl-facor mol o crb h rlaonh bwn faul an raymn hrough nr ra an ohr CIR roc Morovr w vlo a wo mnonal aral ffrnal quaon Unr h aumon ha h PDE ha raaon rucur oluon h quaon can b ara no on ODE roblm an on Rcca quaon whch can oban a clo-form of h ra In h fourh con numrcal xaml ar hown Molng LCDS A mnon bfor h man on of LCDS a robably ha h loan ray arlr an hnc h nrumn cancllabl Durng h lf of an LCDS conrac wo kn of vn may b rggr hr h unrlyng loan ra or h loan-akr go o faul If a raymn vn wr rggr fr h LCDS woul hav bn cancll If urng h lf of h LCDS conrac a faul vn wr rggr fr h LCDS woul hav faul whn h LCDS ur woul hav o ay h rcovry aju noonal amoun o h LCDS buyr W conr a bak of loan from h a of o-own W no by A h ouanng rncal balanc a m Whou lo of gnraly w aum ha h ouanng balanc n ool a m qual o ha A Any faul vn or raymn vn wll affc n gnral ruc h oal rncal n h ool W no by D a h accumulav amoun of faul an P a h accumulav amoun of raymn whch rul n h cra of oal rncal a m hn for any ADP In our mol a connuou m mol conr W mol faul an raymn by nroucng ochac roc an an ar h fracon of accumulav faul an raymn amoun ovr h ouanng balanc n a un m a m rcvly an ar call nny faul an raymn ra rcvly whl h gn of uncran If a raymn a ohrw a nvr raymn or comlmnary lnng a hrfor for all m h accumulav faul amoun D an A P D ouanng balanc ar non-ngav whl accumulav raymn amoun may ngav om m A ncra cra an P may b ncra or cra A m n a mall m nrval h chang of accumulav amoun of faul an raymn ar D D D an P P P rcvly Accorng o h fnon of an w hav D () A an Plu h quaon () an () P () A D P A( ) ISSN E-ISSN 93-43

3 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr I noc hr ha h manng of an ffrn from h on n rfrnc (Zhn W 7); h aumon of h ngav obly of only avalabl for h bak orfolo whr h o own mol al For h ngl nam on h ngav o no mak n W no A a h ouanng balanc a m whch ha bn known Bcau of ADP hn A af h followng bounary roblm of a ffrnal quaon: A A A A Solvng h quaon (3) gv h followng xron for ( ) (3) u u u A A (4) I no ffcul o ha onc an A ar xlan can b oban Now uo ha h ra ra of LCDS for rocon no wh c If h rocon f of h LCDS ar ayabl connuouly hn rul ha h rn valu of h f lg a m qual h xc valu of all f a B whr A PVF ce A B no h rk-fr coun facor from m o m h rocon lg h xc rn valu of h lo n ca of faul qual PVLo E ( R) D B whr R h rcovry ra whch aum o b a ov conan mallr han Unr h conon of rk nural f h conrac ar a m whr hr no nry f PVF PVLo hu w g E ( R) B D c E A B (5) L r b h rk-fr nr ra h coun facor B ru u hrfor by () u u u D A A (6) akng quaon (4) an quaon (6) no quaon (5) w g h ra u u u r u u u u r u E ( R) A ( ) R E ( ) c u u ru u u u ru u E A E ( ) h h rcng formula Now h roblm urn o fn ho xcaon 3 h oluon of LCDS wh ngl-facor mol 3 h corrlaon bwn faul an raymn h faul an raymn ra ar ngav corrla h mor rlvan h hghr ra ra wll b Now w u ngl-facor mol o crb h rlaonh bwn faul ra an raymn ra ha whr a common facor of corrlaon bwn an ; an ; h corrlaon coffcn whch crb h ngav ar h cfc facor of an rcvly an ar (7) Publh by Scu Pr 3

4 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr hr nnn roc h movaon an facor ha affc h raymn ar comlx Rarchr hav hown ha h raymn of loan affc by nr ra macroconomc facor an aonal facor an o on Among hm h nr ra h mo moran on Whn nr ra cra borrowr rfnanc hr loan whch l o hghr raymn ra h man ha whn h currn morgag nr ra blow h conrac ra o a cran xn h borrowr can ak loan from ohr bank o ray h xng loan A long a h ra rach a cran lvl an covr h ranacon co uffcnly h borrowr wll hav a rong ncnv o rfnanc Bu hr an oo uaon for faul vn h co of borrowng ncra wh h nr ra ncrang whch ncra h lklhoo of faul Alo faul ha clo rlaon wh h borrowr' fnancal conon macroconomc an o on Maurng h facor uabl o u h nr ra a h common facor of faul ra an raymn ra So w ak r for om hu w hav If r an mnon bfor ar ov roc h 3 h oluon unr CIR roc Now w gv a mol o crb h bhavor of r roc of CIR r (8) r (9) wll b a nonngav roc bu may b ngav om m a w r ( r ) r W r W aum ha follow h ochac ( ) W whr ar all ov conan; W W follow anar Brownan moon Unr h aumon r ar man rvrng roc h aramr ar h long-run-man ; ar h man-rvron ra; h volaly I wll know ha unr h conon r an ar nonngav an hr bounary a orgn ar unaanabl(x Lnky4) akng quaon (8) an quaon (9) no quaon (7) an conrng h nnnc among r an h rcng formula (6) chang o 33 Solvng rcng quaon ( u u ru) u ( R) E c ( u u ru) u E ru u uu u u ( R) E r E E ru u uu u u E E E ru u uu u u E E E ru u uu u u E E E () r an ar all CRI roc n orr o calcula h ra w hav o olv wo xcaon On 4 ISSN E-ISSN 93-43

5 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr a E uu a uu E h ohr whr a conan an follow h CIR roc af ( ) W h xcaon n quaon () can b xr a h wo xcaon for xaml ru u a uu E E ru u a uu E r E r a r If w can valu h wo xcaon h rcng quaon () wll b olv a r Now l u olv h wo xcaon E a uu an E a uu Dfn I a z E a uuz () hn an I Now h roblm urn o olv E I a z a uu z ( ) z a uu E I a z z az () (3) ux I az E L () a uuz ung h Fynman-Kac formula (x Jang 8) can b oban ha n h ara { } ux () af h PDE roblm u xux xuxx axu (4) zx ux ( ) Suo ha h roblm ha affn rucur oluon ux ( ) A() an B () afy h followng ODE ym (5) (6): A () xb () akng no h quaon (4) obanng ha A ' () B() A( ) (5) ' B () B() B () a B( ) z W can olv () B nc quaon (6) a Rcca quaon an () A can b aly go onc () B oban hrough calculaon w hav (6) Publh by Scu Pr 5

6 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr whr a an a hu h oluon of quaon () I ( a ( ) ( ) ( z ) ( A () In ( z ) ( ( z ) ( z ) B () A() xb() ( ) ( ) ( ) ( ) ( z ) ( z ) ( ) ( ) z z ( z ) ( ( ) ( ) ( ) ( ) x ( z ) ( ( ) ( ) ( z ) ( whr a an hrfor a (7) Alo E I a z a uu z ( ) z x x E I a z a uu z ( ) ( ) ( ) ( ) x ( ) ( ) whr a an a 34 Numrcal Examl So far w hav rv h oluon of h rcng quaon () Hr w gv om xaml unr h ngl-facor mol Now w ak R 3 Fgur -3 how h ha of h LCDS Sra unr ffrn aramr h aramr an hr uro ar ummarz n h abl h n abl rrn h aramr varabl <abl abou hr> Fgur how h ha of LCDS rm rucur whn w fx h common facor r whch h cfc facor of h corrlaon aramr 5 an vary h man an nal valu for whch h cfc facor of W can ha h LCDS Sra of boh fgur ncra a or ncra Suo h ohr aramr ar fx If h nal valu or man of cfc facor of go u whch man h robably of faul go largr a wll rul ha h LCDS Sra mor xnv 6 ISSN E-ISSN 93-43

7 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr <Fgur abou hr> r Howvr Fgur how ha h ha of LCDS rm rucur whn w fx h common facor whch h cfc facor of h corrlaon aramr 5 an vary h man an nal valu for whch h cfc facor of W can ha h LCDS Sra of boh fgur cra a or ncra Suo h ohr aramr ar fx If h nal valu or man of cfc facor of go u whch man h robably of raymn go largr an avanag for h nvor of LCDS an rul ha h LCDS Sra char <Fgur abou hr> Fgur 3 how h mac of ffrn coffcn h LCDS Sra ar grar wh largr ncrang h ngav corrlaon bwn faul an loan canclaon ncra h LCDS Sra <Fgur 3 abou hr> Fgur 4 how h ffrn ah of an Snc ( ) r an ar non-ngav roc An bcau w ak 5 wll b a nonngav roc W can from h lf cur ha h ah all abov zro Bu ngav o r non-ov roc manwhl non-ngav roc hu h gn of uncran If a raymn a ohrw a nvr raymn W can from h rgh cur ha on of h hr ah go blow zro a abou yar four an blow zro nc hn h ohr wo ah ar abov zro <Fgur 4 abou hr> Fgur 5 how h mac of ffrn whn w fx h ohr aramr rrn h mac rngh of common facor If hy ar wo nnn roc h ra low W can ha h ra ncra a ncra On h on han h ncrang of ncra h robably of faul whch ncra h ra On h ohr han nc ngav h ncrang of wll ncra h robably ha raymn bcom ngav whch alo ncra h ra <Fgur 5 abou hr> 4 Concluon In h uy by ung o own framwork w vlo an nny mol for rcng a LCDS conrac rfrnc a bak loan whr nvr raymn allow A clong-form oluon unr ngl-facor mol an corrla CIR roc oban h oluon ay o calculaon o ha ay o aly o h racc Som numrcal calculaon xaml ar rn for whch w g mor rc vw of h ror of h ra funcon an h rlaon among h aramr Rfrnc Dobranzky P (8) Jon Molng of CDS an LCDS Sra wh Corrla Dfaul an Praymn Inn an wh Sochac Rcovry Ra chncal Ror 8-4 Scon of Sac KU Luvn Dobranzky P an Schoun W (8) Gnrc Lvy On-Facor Mol for h Jon Molng of Praymn an Dfaul: Molng LCD chncal Ror 8-3 Scon of Sac KU Luvn Gck K (8) Porfolo Cr Rk: o Down v Boom U Aroach Fronr n Quanav Fnanc: Cr Rk an Volaly Molng Hull J an Wh A () Valung Cr Dfaul Swa I: No Counrary Dfaul Rk Journal of Drvav 8() 9-4 Hull J an Wh A () Valung Cr Dfaul Swa II: Molng Dfaul Corrlaon Journal of Drvav 8(3) - h Hull J an Wh A (4) Valuaon of a CDO an n o Dfaul CDS whou Mon Carlo mulaon Journal of Drvav () 8-3 Publh by Scu Pr 7

8 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr Jang LS (8) Mahmacal Molng an Ca Analy of Fnancal Drvav Prcng Hghr Eucaon Pr Lnky V (4) Comung hng m n for CIR an OU ffuon: alcaon o man-rvrng mol J Comuaonal Fnanc 7 No4 - Shk H Umau S an Zhn W (7) Valuaon of Loan CDS an CD workng ar Sanfor Unvry Zhn W (7) Valuaon of Loan CDS unr Inny Ba Mol workng ar Sanfor Unvry Wu S Jang LS an Lang J (8) Prcng of Morgag-Back Scur wh Raymn Rk workng ar abl Dffrn aramr Fgur r; ; ; ; Puro (lf) ( 3 ; ; ; -5;) Dffrn (rgh) ( 3 ; 5 ; ; -5;) Dffrn (lf) ( 3 ; 5 ; ; -5;) Dffrn (rgh) ( 3 ; 5 ; ; -5;) Dffrn 3 ( 3 ; 5 ; ; ;) Dffrn 4(lf) ( 3 ; 5 ; ; -5;5) Pah of 4(rgh) ( 3 ; 5 ; ; -5;5) Pah of 5 ( 3 ; 5 ; ; -5; ) Dffrn Fgur LCDS Sra v xr m varyng (Lf) an (Rgh) Fgur LCDS Sra v xr m varyng (Lf) an (Rgh) 8 ISSN E-ISSN 93-43

9 wwwcuca/jfr Inrnaonal Journal of Fnancal Rarch Vol No ; Dcmbr Fgur 3 LCDS Sra v xr m varyng Fgur 4 h ffrn ah of (Lf) an (Rgh) Fgur 5 LCDS Sra v xr m varyng Publh by Scu Pr 9