Valuation of Loan Credit Default Swaps Correlated Prepayment and Default Risks with Stochastic Recovery Rate

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1 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al Valaon of Loan C Dfal Swa Cola Paymn an Dfal Rk wh Sochac Rcovy Ra Yan W Damn of Mahmac, Tongj Unvy 39 Sng Roa, Shangha 9, Chna Tl: E-mal: l.b_fov@yahoo.com.cn Jn Lang (Coonng aho) Damn of Mahmac, Tongj Unvy 39 Sng Roa, Shangha 9, Chna Tl: E-mal: lang_jn5@yahoo.com.cn Rcv: Dcmb 6, Acc: Janay 5, Pblh: Al 5, o:.543/jf.v3n6 URL: h://x.o.og/.543/jf.v3n6 Abac In h a, w ablh an nny ba ml-faco mol o val LCDS. Th cng mol ncooa h molng of fal, aymn an covy k. Ung on faco mol, ngav colaon bwn h fal an aymn nn an ov colaon bwn h fal nny an h lo gvn fal a cb. Th n a an h ho c a chon a h lvan faco. Un h amon, a Cahy oblm of PDE v, whch ha a clo-fom olon. Ba on h olon, nmcal xaml a ov. Kywo: LCDS, Rc Fom Mol, Paymn Rk, Rcovy Rk. Inocon Rcn ya, c mak hav a vgoo vlomn, wh nw oc, ch a CDS, CDO, CPDO a bogh. owv, wh h ong abl, h vav hav alo mo ky onal a wll. Rcn fnancal c mak valng h c oc mo gnfcan. Loan C Dfal Swa (LCDS) a almo ncal o h ana C Dfal Swa (CDS) xc wo fa. F, h fnc oblgaon of a LCDS conac lm on loan; conly, LCDS conac can b cancll. Th man ha cola aymn an fal hol b boh con o c LCDS. Thfo, cng LCDS mo comlca han ong a ana CDS. To al wh h k of LCDS, ffn y of mol hav bn vlo. Zhn W (7) con a ngl nam LCDS, wh ng c famwok, fal an aymn nn an w nvolv. a ngl-faco mol wh common faco an colaon coffcn o c h ngav laonh bwn fal an aymn. owv, n h mol, h aymn nny mgh b ngav, whch manngl fo a ngl nam LCDS. Zho & Lang (, ), Lang & Wang () vlo h mol o val a bak LCDS. Th mol ally q h covy a o b conan. I ha alay bn concl n val ach ha h ochac covy a ha a ngav colaon wh h fal nny. Ecally fo a loan, h covy a ncan. owv, o h aho' knowlg, vy fw a hav nvolv ochac covy a no cng c vav. Dff (5) noc ochac covy a hocally who alcaon. P Dobanzky (8) bl a ml conno-m ochac covy a o cb h laonh bwn CDS an LCDS, wh h cb h ngav colaon bwn fal nny an covy a mly by choong a ngav coffcn. In h a, ba on h mol n n Zhn W (7) an P Dobanzky (8), a cng mol fo LCDS n h c fom famwok ablh. Th ngav colaon bwn fal an aymn k, an ov colaon bwn h fal nny an h lo gvn fal a con n h mol. Snc all 6 ISSN E-ISSN 93-43

2 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al a hol b non-ngav, w mol hm a CIR la oc. In h way, w ovcom h hoag of Zhn W' wok, wh h aymn nny mgh b ngav. To al wh ngav colaon, w cb ha h ngav colaon a an nv ooon, whch wa f oo by Ahn an Gao (999) an vlo by W, Jang & Lang, () an Qan, Jang an (). Thn f on af a CIR oc, h oh follow nv CIR oc. Th oblm nc a non-ana aal ffnal qaon whch by & Kznon (8). Common faco n o -faco mol fo aymn, fal an covy a a n a an nflcon nx, whch all follow h CIR oc. Un h amon, cng LCDS moll wh an xlc olon oban. Th of h a oganz a follow. In h nx con, n om bac noaon an amon, w ablh a cng mol fo LCDS. In h h con, w wo-faco mol o cb h laonh of fal, aymn, an covy a. Thn w v PDE oblm an oban a clo-fom of olon fo h a. Fom h olon, h nmcal xaml a gvn n h foh con.. Noaon an Amon Con a ngl-nam LCDS conac. Th conac n bfo x m only f a fal o a aymn vn han. Con a obably ac (, Y, P) an h followng noaon: an : fal an aymn m of h fnc of h conac; R() : covy a f h fnc fal; an : k f n a an nflaon a oc; an : fal an aymn nny a; an C m : coon aymn a an ay con facon fo coon o m. S : LCDS a, T: may m of h conac;. A cbng n [, amng ha an a cononal nnn, hn v P(, v) E [ () P ( ) (, v ) ( ) E [ () v P(, v) E[ (3) Dno h f ong m, ha man, h conac vv nl. On h coon aymn a m, h by hol ay SC m. Song h la coon aymn a bfo, h n val of h coon aymn a ll gvn by ( ) PV a () SC me SC me If h f ong m ng [, +, h conac nwon. In h ca, by (), h n val of h coon accal. Song h la coon aymn a bfo nwon. In h ca, h n val of h coon aymn a ll gvn by PVaccal () SCmE ( ) S C m P ( ) ( ) SC m ( ) E ( ) Th by o ayng coon onc h aymn o fal occ. Thn h n val of h mm lg Pblh by Sc P 6

3 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al gvn by ( ) ( ) PV f() SCm E ( ) E ( ) On h oh han, ng (3), h xc n val of h lo n ca of fal : T vt P(, ) v PVlo ( ) E ( R( )) ( R ) mn(, T ) v v T P(, ) ( ) v T vt ( R) v E ( R) nc, h val of LCDS a a m com: T ( ) E ( R) S (4) ( ) ( ) Cm E ( ) E ( ) 3. LCDS c Molng 3. Dfal nny, aymn nny an covy a A mnon bfo, on cng LCDS, h fal an h aymn k a ngavly cola. If ochac covy a nvolv, alo ngavly cola wh h fal k. In aly, h faco ha affc h fal k, aymn k an covy a comlca. Rach hav hown ha among h faco, h n a h mo moan on. Whn n a ca, boow may fnanc h loan, whch la o hgh aymn a. owv, h aon oo o h fal vn,.. h obably of fal ca. Anoh moan fac nflaon a, whch affc h fal an covy a. Th hgh h nflaon a, h hgh covy an low fal a a. To m h fac, w ak wo-faco mol o cb h laonh of h fal k, aymn k an covy. In a an nflaon a can v a h cola faco of hm a follow: a, a, R, (5) 4 3 wh, a h cfc faco of,, an,, 3, 4. W o ha,,, a nnn. Thy follow CIR oc a follow: k( ) W, (6),,, W (7) k ( ) W, (8) wh,,,,,, ~, ~, ~ a ov conan; W, W, W a nnn ana Bownan moon. Un h amon,,, a man vng oc. I wll know ha, n h conon, k, k,(, ), k,,, a nonngav an h bonay a ogn a naanabl. On h oh han, / follow Inv CIR (ICIR) oc. L y /, by Io fomla: y [ k ( k ) y yw y (9) I ha bn ov ha y abl, an canno qal o f an only f k, ~ >, ~ >. Un h 6 ISSN E-ISSN 93-43

4 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al conon, h ogn CIR oc ll man man vng an nonngav. Th aon h am o. 3. Th olon n CIR oc Takng qaon (5) no qaon (4), by nocng ha h nnnc among,, an, h cng fomla (4) n o: S wh 4 [( ) ( ) E a a a V T 4 3 M 4 [( ) a a VM C E 4 [( ) a a 4 ( ) E ( a a ) Obvng h qaon, w can fn h oblm can b olv a long a h followng h xcaon a v:. 4 E E ( a ) [( ) 4 a a 4 [( ) a a E E E E 4 [( ) 4 a a E E E E 4 [( ) a a E a E E E. 4 [( ) a a E E [( ) 4 a a E E E E 3. 4 E3 E ( a a ) [( ) a a 4 4 [( ) ( ) E a a E E E () Pblh by Sc P 63

5 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al 4 [( ) 4 a a E E E E 4 [( ) a a E a E E E 4 [( ) a a E a E E E Th, w j n o olv h xcaon n h followng y: ( ) a a E, E, E wh ( ) follow CIR oc. Thn E, E, E3 can b x by h abov h xcaon, hn h qaon (9) olv. L olv h h xcaon of (). Fom h mho noc fom of W (7), f, fn a z I, ( a, E, hn E I a z E I a z a, (, ), a z z, (, ) z a z L x (, ) I, ( az, ) E oblm: ( x) x xxx ax, zx x (, ). So ha h oblm ha affn c olon oblm o oban: ( ) ( ) ( z ) ( A () ln, ( z ) ( ( ) ( ) ( z ) ( B (). ( ) ( ) ( z ) ( wh () () (3), by Fynman-Kac fomla, x (, ) h olon of h followng PDE x (, ) A () xb () (4), hn A() an B() can b olv fom ODE a a, (5) Th h olon of () : ( ) ( ) ( z ) ( I, ( a, ( z ) ( Fom (3), com ( ) ( ) x ( z ) ( ( ) ( ) ( z ) (. 64 ISSN E-ISSN 93-43

6 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al E x ( ) ( ) ( ) ( ) a ( ) ( ) ( ) ( ) ( ) An ( ) ( ) ( ) ( ) x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x a ( ) ( ). E, wh an a fn n (5). Now, l olv h h xcaon of (). an Kznov (6) hav ov ha f follow CIR oc, h xcaon can b v n clo fom n m of a confln hygomc fncon. ( x ) Dfn x vx (, ) E f( x ), ng Fynman-Kac fomla, v(x, ) af h followng PDE oblm: v ( x) v x xv xx ( x ) v, x vx (, ) f( x). L (, ) ax b vx xwx (, ), wh a, b a coffcn o b mn, an hn w(x, ) af h followng qaon: w b( a ) x wx xwxx bb ( ) ( a a ) x( b ) ( aba b) w x ( abab ) L a /a, an bb ( )/b o olv a an b. Now l wx (, ) x (, ), hn (x, ) af h followng PDE oblm: ' ' ( x) x xxx, (7) ax b x (, ) f( x ) x. wh ' ' a, ( b)/( a ). Th olon : (,, ) ( ) ay b x f y y Gyxy (,,, ) (8) wh q c( ) Gyx (,,, ) c ( ) I q (( ) ) An ' ' ' q k ( ) ( ) c ', q, cx, cy, I (( ) ) ( ). ( ) q k ( ) k! ( qk) f(y) can b x by ' ay ' b y, o ay b y x (,, ) f( y ) y Gyxy (,,, ) y Gyxy (,,, ) (6) Pblh by Sc P 65

7 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al q c ( q ) c M( q, q, ), q ( c) ( q) c wh an n on f(y), ( ) k k M (, vz, ) z confln hygomc fncon an k k!( v) k ( ) k ( ) ( k), k,( ). 4. Nmcal Examl So fa, w hav v h olon of h cng qaon (4). Now ng h olon, w gv om nmcal xaml n h mol, wh w ak =. An h aam a hown a follow:.3,.3, h.3,.3,.5,.5,.5, h.5,.5,.5,.5, h.5, x.,., x., h.,,.,.3,., a, a. 3 4 Fg how h laonh of LCDS a wh,, an cvly. I obvo ha h LCDS a nca a, o nca. Tha, h LCDS a nca a nca, bca nca an ca whn nca. I alo how ha h ffc of an lay mo moan ol han on of an. Fg how h laonh of LCDS a wh h, h, h an h cvly. Th fg nca ha h LCDS a ca a h, h o h nca. Tha, h LCDS a ca a $h$ nca, bca ca an nca whn h nca. An can b obv fom Fg an Fg ha h ffc of h nflcon lay a mo moan ol han ha of h n. Fg 3 nca ha h n of LCDS a wh ochac covy a q ffn fom h on wh h conan covy a. I gg ha h ochac oy of covy a gv h LCDS Sa volaly an h LCDS a bcom mo vayng. W alo fn ha h LCDS a ha a hm y ha whch mla o h on of many ochac n oc. 5. Conclon In h a, cng LCDS con. W vlo an nny ba ml-faco mol, whch ncooa h jon molng of fal, aymn an covy k. W nf wo colaon a k faco ha may xlan h colaon among fal nny, aymn nn an covy a. Th wo common faco a am o b n an nflcon a, whch follow CIR oc. Th c n ha h a a non-ngav an afy ov o ngav colaon. A clo-fom olon oban. Som nmcal calclaon xaml a gvn, fo whch w g mo c vw of h laonh among h aam. Th gah of h mol how ha h nflcon mac h LCDS a mo han h n o. Rfnc Ahn, D.. & Gao, B. (999). A aamc nonlna mol of m c ynamc, Rvw of Fnancal S,, h://x.o.og/.93/f/.4.7 Dobanzky P. & Schon W., (8). Gnc Lvy On-Faco Mol fo h Jon Molng of Paymn an Dfal: Molng LCD, Tchncal Ro 8-3, Scon of Sac, K.U. Lvn. h://x.o.og/ Dobanzky, P. (8). Jon Molng of CDS an LCDS Sa wh Cola Dfal an Paymn Inn an wh Sochac Rcovy Ra, Tchncal Ro 8-4, Scon of Sac, K.U. Lvn. h://x.o.og/ Dff, D. (5). C k molng wh affn oc. Jonal of Bankng an F-nanc. h://x.o.og /.6_j.jbankfn.5..6, T. R. & Kznon, A. (8). Exlc fomla fo Lalac anfom of ochac ngal, Makov Poc. Rla Fl, 4, Lang, J. & Wang, T. (). Valaon of a loan-only c fal wa wh ngavly cola fal an aymn nn, o aa n Innaonal Jonal of Com Mahmac. Lang, J. & Zho, Y. (). Valaon of a anch Loan C Dfal Swa Inx, Tchnology an Invmn, 66 ISSN E-ISSN 93-43

8 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al PP h://x.o.og/.436/..45 Qan,., Jang, L. &, C. (8). Exlc fomla fo cng of Mogag-Back Sc n a ca of aymn a ngavly cola o n, Mahmac Fnanc. Shk,., Uma, S. & W, Z. (7). Valaon of Loan CDS an CD, wokng a, Sanfo Unvy W, Z. (7). Valaon of Loan CDS n Inny Ba Mol, wokng a, Sanfo Unvy. h://x.o.og/ W, S., Jang, L. & Lang, J. Pcng of Mogag-Back Sc wh Raymn Rk, o aa n Innaonal Jonal of Thocal an Al Fnanc Zho, Y. & Lang, J. (). Valaon of a Bak Loan C Dfal Swa, Innaonal Jonal of Fnancal Rach.-9. h://x.o.og/.543/jf.vn Fg. Tm c of h LCDS a fo ffn. Pblh by Sc P 67

9 Innaonal Jonal of Fnancal Rach Vol. 3, No. ; Al Fg. Tm c of h LCDS a fo ffn h Fg 3. LCDS a wh ffn covy a. 68 ISSN E-ISSN 93-43