Valuation and Analysis of Basket Credit Linked Notes with Issuer Default Risk

Transcription

1 Valuaon and Analy of Ba Crd Lnd o wh ur Dfaul R Po-Chng Wu * * Dparmn of Banng and Fnanc Kanan Unvry Addr: o. Kanan Rd. Luchu Shang aoyuan awan R.O.C. E-mal: pcwu@mal.nu.du.w l.: x. 67 Fax:

2 Valuaon and Analy of Ba Crd Lnd o wh ur Dfaul R Abrac h papr xplor a raonabl coupon ra for ba crd lnd no (CL wh ur dfaul r. Bad on h on facor Gauan copula modl h papr propo hr mhod for ncorporang ur dfaul no ba CL prcng. umrcal rul ndca ha ur dfaul r mpac ba CL coupon ra. Furhrmor h coupon ra dffr wh chang n corrlaon rucur among h hr mhod. Fnally on of h hr mhod dnfd a h mo uabl. Kyword: Ba crd lnd no ur dfaul r dfaul corrlaon facor Gauan copula Mon Carlo mulaon.

3 . nroducon A crd lnd no (CL a no for whch h prc or coupon lnd o h crd vn of h rfrnc ny (oblgaon (Anon al. 004; Da 000; Fabozz al A CL lnd o mulpl rfrnc n calld a ba CL. Such a CL can b rucurd by a no and a ba dfaul wap (BDS. h convnonal form of ba CL h h-o-dfaul CL. h CL holdr (h procon llr pay h noonal prncpal o h CL ur (h procon buyr a h ar of h conrac and rcv h coupon paymn unl hr h h dfaul or h conrac maur whchvr occur arlr. f h h dfaul occur bfor conrac maury h CL holdr rcv h rcovrd valu of h rfrnc ny from h CL ur. Ohrw h CL holdr rcv h noonal prncpal bac on conrac maury. n drvav mar h ur dfaul r aracng condrabl anon bcau of h rcn fnancal urmol and collap of larg fnancal nuon. f h CL ur dfaul h CL holdr wll no rcv h rcovrd valu of h rfrnc ny a h crd vn happn nor h noonal amoun a h conrac maury. h coupon paymn alo ca du o h ur dfaul. h ur dfaul rul n a larg lo. hu mporan o ncorpora ur dfaul r n ba CL prcng o oban a raonabl coupon ra. wo man approach x o modlng h dfaul r n h lraur: h rucural and rducd form modl. h rucural modl wa dvlopd by Mron (974 and dfnd crd vn a occurrng whn frm a valu fall blow frm db. h rducd form modl alo nown a h nny modl wa dvlopd by Jarrow and urnbull (995. h modl vw h crd vn a an unxpcd xognou ochac vn and u mar daa o ma h dfaul r. 3

4 Hull and Wh (000 provdd a mhodology for valung crd dfaul wap (CDS whou counrpary dfaul r whn h payoff conngn on h dfaul of a ngl rfrnc ny. Hull and Wh (00 dvlopd a modl of dfaul corrlaon bwn dffrn corpora or ovrgn n. h modl of Hull and Wh an xnon of h rucural modl a crd ndx varabl for ach rfrnc ny and lc corrlad dffuon proc for h crd ndx. hr modl dfn dfaul a h crd ndx fallng blow h prdrmnd dfaul barrr. Mon Carlo mulaon ud o calcula h vanlla CDS and BDS prad gvn h pobly of ur dfaul. Hu and Lo (00 dvlopd a modl o prc h ngl-nam CL wh ur dfaul r ung h framwor of Mron modl. hy dmonrad ha h crd prad of a CL ncra non-lnarly wh dcrang corrlaon bwn h rfrnc ny and h ur. Prcng mul-nam crd drvav rqur a jon drbuon modl of h dfaul m. Howvr whhr ung h rucural or rducd form modl valung h mul-nam crd drvav compuaonally complx. hu h copula funcon (Slar 959 alo nown a h dpndnc funcon whch mplf h maon of h jon drbuon rcnly ha bn wdly ud o prc h mul-nam crd drvav. L ( fr nroducd h copula funcon o dal wh h dpndnc rucur n mul-nam crd drvav prcng. H aumd h dfaul m of rfrnc n o b Poon procd and h dpndnc rucur a a Gauan copula funcon. Fnally L prformd Mon Carlo mulaon o oban h dfaul m. Mahal and ald (003 appld L mhod o analyz how h dfaul probabl of h procon llr and buyr affc BDS prad. Calculang dfaul m ung h copula mhod farly ay. Howvr h 4

5 compuaonal complxy of h Mon Carlo mulaon wh Gauan copula ncra wh numbr of rfrnc n. h facor copula mhod whch ma h crd vn condonal on ndpndn a varabl wa nroducd o dal wh h problm. Hull and Wh (004 mployd a mul-facor copula modl o prc h h-o-dfaul wap and collaralzd db oblgaon (CDO. Morovr Laurn and Grgory (005 propod on facor Gauan copula o mplfy h dpndnc rucur of rfrnc n and appld h approach o prc BDS and CDO. h papr focu on how o ncorpora ur dfaul r no h ba CL prcng mhod wh h on facor Gauan copula modl. hr dffrn mhod ar propod and numrcal analy prformd o compar hm. h rmandr of h papr organzd a follow. Scon brfly rvw h on facor Gauan copula modl. Scon 3 hn dcrb h propod mhod for ncorporang ur crd vn no ba CL prcng. Subqunly Scon 4 prn h rul of numrcal analy and compar h mhod. Concluon ar fnally drawn n Scon 5.. On Facor Gauan Copula Copula a funcon whch ln h unvara margnal drbuon o hr full mulvara drbuon and can b xprd a follow: C ( u u u = Pr( U u U u U u K K ( whr U ~ U ( 0 = K. Slar (959 provd ha f F ( x x K x a jon mulvara drbuon wh unvara margnal drbuon 5

6 F ( x = K hr x a copula funcon uch ha: ( x x x C( F ( x F ( x F ( x F K = ( K f ach F ( x connuou hn h copula funcon unqu. h dfnon of a Gauan copula a follow: C Ga ( ( u u u = Φ φ ( u φ ( u K φ ( u K ( 3 n R n whr Φ R dno a mulvara cumulav normal (Gauan drbuon R rprn h corrlaon coffcn marx and dmnonal cumulav normal drbuon. φ h nvr funcon of on Ung h rducd form modl ach rfrnc ny dfaul follow a Poon proc. Suppo h crd porfolo conan rfrnc n and h dfaul m ar K rpcvly. a pov random varabl wh drbuon: λ ( > = P( = = K P ( 4 whr λ h hazard ra of h rfrnc ny. h cumulav dfaul probably bfor m : F λ ( = P( = = K ( 5 6

7 Bcau ( ~ U ( 0 F drbuon of dfaul m a follow: applyng h Gauan copula oban h mulvara jon F ( ( = Φ φ ( F ( φ ( F ( K φ ( F ( K ( 6 R n h on facor modl h dfaul m of all rfrnc n dpnd on a common facor Y and frm pcfc r facor ε = K. Y and ε ar ndpndn andard normal varabl. Bad on h abov ng a nw Gauan vcor K can b crad va Choly dcompoon a follow: ( = Y + ε = K ( 7 whr dno h corrlaon coffcn bwn h nw Gauan varabl and h common facor Y. On facor Gauan copula modl wh conan parw corrlaon ha bcom h andard mar modl. L = n Eq. (7 hn h conan parw corrlaon wll b. j F L = φ ( F ( = φ ( F ( K φ ( ( ( = φ( F ( = φ( K F ( φ( = n whch ca F =. By mappng h cumulav normal drbuon bwn dfaul m and Gauan varabl w can mula h dfaul m of h rfrnc ny ung h followng quaon: ( φ( ln = F ( φ( = = K ( 8 λ 7

8 3. Propod Mhod h papr propo hr pobl mhod of ncorporang ur dfaul r no ba CL prcng ung h on facor Gauan copula modl. For comparon h mhod whou ur dfaul r namd mhod A. Manwhl h hr propod mhod wh ur dfaul r ar namd mhod B C and D rpcvly. Aumng ha h normal random varabl corrpondng o h dfaul m of ach undrlyng rfrnc ny (abbrvad hr a h undrlyng varabl h normal random varabl corrpondng o h ur dfaul m (abbrvad a h ur varabl Z and h common facor Y. h dffrn rucur of mhod A o D ar ld blow and hown n Fgur. abl ummarz h paramr of h four mhod. Mhod A: ur dfaul r no condrd. h corrlaon bwn ach and Y a hown n Fgur (A. Mhod B: ur dfaul r condrd. h corrlaon bwn ach and Y bu Z ndpndn of and Y a hown n Fgur (B. Mhod C: ur dfaul r condrd. h corrlaon bwn ach Z and Y a hown n Fgur (C. Mhod D: ur dfaul r condrd bu h common facor Y rplacd by h ur varabl Z. h corrlaon bwn ach and Z a hown n Fgur (D. [nr Fgur abou hr] [nr abl abou hr] 8

9 3. Prcng ba CL whou ur Dfaul R Aum a h-o-dfaul CL nvolvng rfrnc n whch h noonal prncpal of ach rfrnc ny on dollar. h coupon ra c. h coupon (h noonal prncpal mulpld by h coupon ra pad annually and h paymn da ar = K. h maury da of h ba CL. Furhrmor h h dfaul m and < < L <. Morovr δ h rcovry ra of h h dfaul rfrnc ny. hu δ dno h rdmpon procd (h noonal prncpal mulpld by h rcovry ra whch h ur pay o h ba CL holdr on h h dfaul. h dcoun ra r %. Fnally Q dno h r-nural probably maur and ( rprn an ndcaor funcon. h valu of a h-o-dfaul CL can b rprnd a follow: CL = E Q r r r c ( < + δ ( + ( > = ( 9 L h abov quaon qual on h quaon can b rwrn a: c E = E Q Q = ( < r r [ δ ( ( > ] r ( 0 Rarrangng Eq. (0 can yld h far coupon ra a h ar of h CL a follow: 9

10 0 [ ] < > = = r Q r r Q E E c ( ( ( δ ( By ung W run of Mon Carlo mulaon o prc h ba CL h numraor of Eq. ( : [ ] = > W r r W ( ( δ ( whr δ dno h rcovry ra of h h dfaul rfrnc ny a h h mulaon and rprn h h dfaul m a h h mulaon. h dnomnaor of Eq. ( : = = < W r W ( ( 3 hrfor h far valu of h coupon ra c : [ ] = = = < > = W r W r r c ( ( ( δ ( 4 3. Prcng ba CL wh ur Dfaul R n uaon nvolvng ur dfaul r ncary o condr whhr h ur dfaul bfor or afr h h dfaul. h papr dfn ˆ a h ur

11 dfaul m a h h mulaon and δˆ a h ur rcovry ra. h CL holdr g bac h rcovrd valu of h rfrnc oblgaon f h h dfaul occur bfor boh h ur dfaul m ˆ and h maury da. f h ur dfaul bfor h h dfaul and h maury da h ur wll no provd h CL holdr wh h rdmpon procd and op h coupon paymn. n h uaon h noonal prncpal mulpld by h ur rcovry ra rurnd o h CL holdr. o oban all of h noonal prncpal bac boh h h dfaul m and h ur dfaul m mu b lar han h conrac maury da. hu h valu of a h-o-dfaul CL wh ur dfaul r mu b modfd a follow: ( ( ( ( < + < + < + < = = ˆ mn( mn( ˆ ˆ mn( ˆ ˆ mn( ˆ r r r r Q c E CL δ δ ( 5 h dfnon of δ and ar a n Eq. (. h far valu of h coupon ra c wh ur dfaul r : ( ( ( ( = = = < < < < = W r W r r r c ˆ ˆ mn( ˆ mn( mn( ˆ ˆ mn( ˆ δ δ ( 6 4. umrcal Analy h papr adop a fv-yar ba CL wh hr rfrnc n a an xampl of numrcal analy. Aum all hr rfrnc n hav noonal

12 prncpal on dollar hazard ra 5% and rcovry ra 30%. Furhrmor aum h coupon pad annually h hazard ra and rcovry ra of h ur % and 30% rpcvly. Dcoun ra ar oband by boorappng from h govrnmn bond daa. Sxy-houand run of Mon Carlo mulaon ar xcud o calcula h coupon ra for mhod A o D and h rul ar hown n abl o abl 5 and Fgur. [nr abl abou hr] [nr abl 3 abou hr] [nr abl 4 abou hr] [nr abl 5 abou hr] [nr Fgur abou hr] h corrlaon coffcn rprn h corrlaon bwn ach rfrnc ny and h common facor (or h ur dfaul r. h corrlaon bwn h rfrnc n and j whch qual povly corrlad o. A hown n Fgur (A h coupon ra of h fr-o-dfaul ( = CL ngavly corrlad wh bcau h probably of h fr-o-dfaul occurrng ncra a dcra. Convrly a hown n Fgur (B and Fgur (C h coupon ra of h cond ( = and hrd-o-dfaul ( =3 CL povly corrlad wh bcau h probably of h jon dfaul ncra a ncra. Fgur alo how ha h coupon ra wh ur dfaul r n mhod B o D ar hghr han ho whou ur dfaul r n mhod A. Furhrmor whn

13 h corrlaon coffcn pov h coupon ra n mhod D ar lowr han ho n mhod B and C. Manwhl whn h corrlaon coffcn ngav h coupon ra n mhod D ar hghr han ho n mhod B and C. hu h curv of coupon ra n mhod D ar aymmrc. h followng xplor h abov phnomnon. Whn h rfrnc ny dfaul h CL holdr rcv h rcovrd valu of h rfrnc ny and lo h coupon ncom afr h rfrnc ny crd vn. h ur dfaul xhb a mlar ffc. Whn h ur dfaul h CL holdr rcv h rcovrd valu of h CL and lo h coupon ncom afr h ur dfaul vn. h ur dfaul may occur bfor or afr h rfrnc ny dfaul. Whn h ur dfaul occur afr h rfrnc ny dfaul h conrac rmnad a h dfaul m of h rfrnc ny and h CL holdr rcv h rcovrd valu and lo h followng coupon ncom. hu h lo of a CL wh ur dfaul r wll b dncal o ha whou ur dfaul r. On h ohr hand whn h ur dfaul arlr han h rfrnc ny dfaul h conrac rmnad mulanouly wh h ur dfaul. Suppo h rcovry ra of h ur and h rfrnc ny ar dncal. h CL holdr hn rcv h am rcovrd valu a whn h rfrnc ny dfaul fr. Howvr h CL holdr no only lo h coupon ncom afr h rfrnc ny dfaul m bu alo ho bwn h ur and rfrnc ny dfaul m. hu h lo of a CL wh ur dfaul r xcd ha whou ur dfaul r. h uaon cau h oal r of h CL wh ur dfaul r o b hghr han ha whou ur dfaul r. hrfor h coupon ra of h CL wh ur dfaul r hghr han ha whou ur dfaul r. Whn h ur and rfrnc ny dfaul ar hghly povly corrlad 3

14 hr movmn ar almo dncal mang h dfaul m of h ur and rfrnc n clo oghr. A h corrlaon approach on h jon dfaul of h ur and h rfrnc ny bcom mor lly and h dfaul m bcom clor oghr. n xrm ca for xampl whn h corrlaon coffcn on h ur wll dfaul mulanouly wh h rfrnc ny dfaul. hu h ur dfaul do no nflunc h oal r of h CL. On h ohr hand a h corrlaon approach ngav on h dfaul m bcom mor dprv. Ohr hng bng qual h mpac of ur dfaul r dcra wh ncrang clon of dfaul m. hrfor h dffrnc of h coupon ra bwn h wh and whou ur dfaul r uaon rduc a h corrlaon approach on. h dfaul corrlaon bwn h rfrnc n and h ur zro n mhod B n mhod C and n mhod D rpcvly. Whn h corrlaon pov.. whn 0 hn 0. hu h coupon ra follow h ranng mhod B > mhod C > mhod D a hown n Fgur. Whn h corrlaon ngav.. whn 0 hn 0. hu h coupon ra follow h ranng mhod D > mhod B > mhod C. Morovr n mhod B and C h coupon ra curv ar ymmrc. Howvr n mhod D h coupon ra curv ar aymmrc dpndng on whhr h dfaul corrlaon pov or ngav. hrfor ung h ur varabl a h common facor n h facor copula framwor wll nclud mor nformaon abou h dfaul corrlaon bwn h rfrnc n and h ur. 5. Concluon o oban h mo raonabl coupon ra ur dfaul r n ba CL prcng mu b condrd. h papr propo hr mhod for ncorporang h 4

15 ur dfaul r no ba CL prcng ung on facor Gauan copula modl. h analycal rul how ha h coupon ra oband by all hr propod mhod wh ur dfaul r ar hghr han h mhod whou ur dfaul r. hu ur dfaul r ncra h ba CL coupon ra. Furhrmor among h propod mhod bcau mhod D drcly a accoun of h dfaul corrlaon bwn h rfrnc n and h ur h pov or ngav ffc of h dfaul corrlaon fully rflcd n h coupon ra. hrfor mhod D h mo prfrabl modl for prcng ba CL wh ur dfaul r. Rfrnc Anon M. J. Fabozz F. J. Choudhry M. and Chn R. (004 Crd drvav: nrumn applcaon and prcng John Wly and Son. Da S. (000 Crd Drvav and Crd Lnd o John Wly and Son. Fabozz F. J. Dav H. A. and Choudhry M. (007 Crd-Lnd o A Produc Prmr h Journal of Srucurd Fnanc Hu C. H. and Lo C. F. (00 Effc of A Valu Corrlaon on Crd-Lnd o Valu nrnaonal Journal of horcal and Appld Fnanc Hull J. and Wh A. (000 Valung Crd Dfaul Swap : o Counrpary Dfaul R Journal of Drvav Hull J. and Wh A. (00 Valung Crd Dfaul Swap : Modlng Dfaul Corrlaon Journal of Drvav 8 -. Hull J. and Wh A. (004 Valuaon of a CDO and an nh o Dfaul CDS whou Mon Carlo Smulaon Journal of Drvav 8-3. Jarrow R. and urnbull S. (995 Prcng Drvav on Fnancal Scur 5

16 Subjc o Crd R Journal of Fnanc Laurn J. P. and Grgory J. (005 Ba Dfaul Swap CDO and Facor Copula Journal of R L D.. (999 h Valuaon of Ba Crd Drvav CrdMrc Monor Aprl L D.. (000 On Dfaul Corrlaon: A Copula Funcon Approach Journal of Fxd ncom Mahal R. and ald M. (003 Prcng Mul-nam Dfaul Swap wh Counrpary R Quanav Crd Rarch ovmbr -7. Mron R. C. (974 On h Prcng of Corpora Db: h R Srucur of nr Ra Journal of Fnanc Slar A. (959 Foncon d rparon a n dmnon lur marg Publ. n. Sa. Unv. Par

17 abl Summary of h paramr of h four mhod. Paramr Sng (Corrlaon wh undrlyng varabl Undrlyng ur Mhod dfaul r Common ur condrd? Facor Varabl Varabl Y Z j A o condrd B Y ndpndn C Y D Y o condrd 7

18 abl Coupon ra undr varou corrlaon coffcn n mhod A ( =: fr-o-dfaul : cond-o-dfaul 3: hrd-o-dfaul. Mhod A = = = % 5.69% % % 4.974% 3.05% % 4.703%.649% % %.3993% % 4.476%.39% % %.07% % %.080% % %.9809% % 3.697%.9499% % %.9359% % %.9470% % %.979% % 3.887%.07% % 4.04%.0883% % 4.55%.05% % 4.409%.3793% % %.679% % 4.938%.9783% % 5.95% % 8

19 abl 3 Coupon ra undr varou corrlaon coffcn n mhod B ( =: fr-o-dfaul : cond-o-dfaul 3: hrd-o-dfaul. Mhod B = = = % 5.939% 4.58% % % 3.688% % 5.407% 3.3% % 5.587% % % 4.937%.898% % %.779% % %.6958% % %.6480% % %.664% % 4.34%.600% % 4.364%.633% % 4.463%.639% % 4.568%.6843% % 4.707%.7558% % %.8783% % 5.096% % % 5.337% 3.985% % % 3.655% % % 4.37% 9

20 abl 4 Coupon ra undr varou corrlaon coffcn n mhod C ( =: fr-o-dfaul : cond-o-dfaul 3: hrd-o-dfaul. Mhod C = = = % 5.585% 3.767% % 5.490% 3.363% % % 3.44% % %.9853% % %.8735% % %.7835% % %.777% % %.6597% % 4.377%.65% % 4.34%.600% % %.60% % 4.405%.630% % %.6668% % %.74% 0.5.5% 4.749%.83% % %.947% % % 3.099% % % 3.344% % 5.336% % 0

21 abl 5 Coupon ra undr varou corrlaon coffcn n mhod D ( =: fr-o-dfaul : cond-o-dfaul 3: hrd-o-dfaul. Mhod D = = = % 6.338% 4.389% % % 3.778% % % % % % 3.86% % %.955% % %.83% % %.7359% % %.6849% % 4.43%.6497% % %.677% % 4.344%.698% % 4.37%.64% % %.6595% % %.6870% % %.744% % %.89% % 4.738%.944% % % 3.49% % 5.959% 3.590%

22 Undrlyng = Common Facor Y (A ndpndn ur Z ndpndn Undrlyng = Common Facor Y (B ur Z Undrlyng = Common Facor Y (C Undrlyng = ur Z (D Fgur Rlaonhp bwn h varabl n mhod A o D: (A mhod A; (B mhod B; (C mhod C; (D mhod D.

23 5.00% 4.00% Coupon Ra 3.00%.00%.00% 0.00% 9.00% 8.00% 7.00% Corrlaon A B C D (A 6.50% Coupon Ra 6.00% 5.50% 5.00% 4.50% A B C D 4.00% 3.50% Corrlaon (B 5.00% 4.50% Coupon Ra 4.00% 3.50% 3.00% A B C D.50%.00%.50% Corrlaon (C Fgur Coupon ra of h h-o-dfaul CL n mhod A o D: (A =; (B =; (C =3. 3