Higher Order Binaries with Time Dependent Coefficients and Two Factors - Model for Defaultable Bond with Discrete Default Information
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- Linette Clark
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1 po o. IU-MH-03-E--0: on Hgh O n wh Dpnn offcn n wo Fco - Mol fo Dfll on wh Dc Dfl Infoon Hong-hol O Yong-Gon n Dong-Ho Fcl of Mhc Il ng Unv Pongng D. P.. o c: In h cl w con fco-ol fo pcng fll on wh c fl nn n wh h fco ochc f ho poc n f vl poc. h h fl vn occ n n pc nn whn h f vl ch gvn fl pn c nnoncng o n n npc nn h f jp of Poon poc wh gvn fl nn gvn p fncon of vl. hn o pcng ol gvn olvng pol of vl ln PDE wh vl coffcn n nl vl of n p n v nvl wn h wo jcn nnoncng. O n ppoch o hgh o n. f pov h pcng fol of hgh o n wh pnn coffcn n con h ngl on h l p vl. hn ng h pcng fol of hgh n opon n h ngl w gv h pcng fol of fll on n oh c of ogno n nogno fl cov n c p nl. wo : hgh o n opon pnn coffcn fll on fl nn fl ogno nogno c p 00 M : Q9 9G0 9G40 9G50 9G80 F von wn on 9 M 03 h von v on Jn oponng ho: Hong-hol O El ohongchol@hoo.co. Inocon h on fll copo on n c now on of h o pong of cng g n fnncl hc []. wll nown h wo n ppoch o pcng fll copo on on h cl ppoch n h oh on h c fo ppoch. In h cl ho w hn h h fl vn occ whn h f vl no nogh o p h h f vl ch
2 Hong-hol O Yong-Gon n Dong-Ho cn low hhol fl fo h ov. ch fl cn pc n h w cll pc fl. In h c-fo ppoch h fl n npcl vn govn fl nn poc. In h c h fl vn cn occ who n colon wh h f vl n ch fl cll npc fl. In h c-fo ppoch f h fl pol n nvl [ ] hn cll fl nn o hz. fo h ho of h wo ppoch n h vng n hocong cn f o h nocon of [ ]. o h vng n ovco h hocong of cl n c-fo ppoch n ho nf ol of h wo ppoch. [ ]. no n [0 3 4] n ch of nf ol nclng [ ] o p h pc of h on n of h f vl o h l gnl vl o h f vl n h vl of fl nn ogh wh fl n n h whol lf of h on. On h oh hn n [3 0] h ho no h ffcl fo nvo o of h f o now h f fnncl cp fo o c fo pl onc n onh o onc n h onh c. o nnonc ngn n h pcng pol fo fll copo on n h pon h w onl now h f vl n h fl f c nnoncng w on now o n nfoon of h f vl n noh n h fl nn wn h jon wo nnoncng conn n nnonc f vl h fo nnoncng. h coponl o n [3] coc n [0]. h ppoch of [3 0] n of of fll on n nffcn nfoon o h f. I nng o no h gl l [] on pcng pol n pc nfoon wh h chn of fzz hc. h ppoch of [3 0] cn lo n nf ol of cl ol n c fo ol. gl [] cl ol fo fll on wh vl c copon wh h fl cn occ onl whn h f vl no lg nogh o p n copon n ho c copon. In [4] h ho on-fco ol fo fll on wh c fl nn n c fl ng hgh o n opon n h ngl wh h fco h f vl poc. In h c ol h fl vn occ n n pc nn whn h f vl ch cn low hhol - h fl pn c nnoncng o n n npc nn h f jp of Poon poc wh gvn fl nn gvn p fncon of vl pcvl. h con oh nogno n ogno fl cov n h pcng ol olvng pol of nhoogno o hoogno lc-chol PDE wh ffn coffcn n nl vl of n p n v nvl wn h wo jcn nnoncng. In o o l wh h nhoogno l o nogno cov h noc pcl n opon
3 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on cll ngl of -h n o nohng n ng on h pcng fol of fll copo on. h ppoch of [4] o ol c l wh h on of [0] h nl ffnc h n [0] h h h now h f vl onl n h c nnoncng n h fl nn wn wo jcn nnoncng n h f vl n h fo nnoncng. noh ffn pon h [4] con n of nnoncng [0] con onl nnoncng. conn of [4] w h con wo fco - ol fo pcng fll on wh c fl nn n wh h fco ochc f ho poc n f vl poc. O pcng ol gvn olvng pol of vl PDE wh vl coffcn n nl vl of n p n v nvl wn h wo jcn nnoncng. hogh h chng of n h nfo no vl hoogno o nhoogno lc-chol PDE wh ffn pnn coffcn n nl vl of n p. h coffcn pnnc h ffn pon fo [4]. H w ncon h pol of hgh o n wh pnn coffcn vn f h f n voll of ho n f vl poc ll conn. hfo w f pov h pcng fol of hgh o n wh pnn coffcn n con h ngl on h l p vl. hn ng h pcng fol of hgh n opon n h ngl w gv h pcng fol of fll on n oh c of ogno n nogno fl cov n c p nl. Fnll w no h nng o h h G copon opon ppoch n [] fo pcng of fll on wh c copon pn n cl ppoch h chn hgh n h. h n of h cl ognz follow. In con w con hgh o n wh pnn coffcn n h pop. In con 3 w h pol fo fll on n pov h pcng fol n c p nl. In con 4 w pov h ch of h poof of pcng fol fo fll on.. Hgh O n wh Dpnn offcn F w pln hgh o on n n wh f vn n voll h olon o h pol. n. cll h -o-nohng n o 3
4 Hong-hol O Yong-Gon n Dong-Ho n n no. h olon o h pol. n.3 cll h ch-o-nohng n o on n n no. n n on n cll h f o n opon. If nc w wll no o wh h coffcn n of lc-chol on. plcl ncl n h noon. L h 0 0 n n h n h o o on n n opon n n n n l fn. L n n.4 n 0 n 0 n n 0 n 0 n h olon o h pol. n.4 cll h n-h o n n no n. h olon o h pol. n.5 cll h 0 n 0 n n. 0 n n-h o on n n no 0 n w pov h pcng fol of n on n wh pnn coffcn. o h n w n h followng popoon. f.6 L. h h nonngv conn M n ch h ln M 0. hn h olon of. n.6 pov follow : f 0 z ln z f z z H ln z 0 z f z z Poof. I wll nown h h olon o lc-chol on wh pnn coffcn n cn on plcng n n h olon pnon of lc-chol on wh conn coffcn n 4
5 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on 5 no n. Ung h fc n h popoon pg 49 n [] w oon hv.7. w of c poof follow. n [8] n. w h chng of vl n nnown fncon U. hn. chng o f U U U If w chng vl no 0 0 ˆ hn w hv 0. ˆ 0 ˆ 0 0 f U U U h h lc-chol on wh conn coffcn 0 0 n n h w ppl h popoon pg 49 n [] o g h pnon of U. nng o ognl vl n nnown fncon w g.7. QED ho. h Pcng Fol of Hgh O n Opon wh Dpnn offcn h pc of hgh o on n n wh f ho vn n voll follow. o '.9 ] / p[ ' / / ln o ' '.0 z z z ' ' 3 o.
6 Hong-hol O Yong-Gon n Dong-Ho 6 p ' / ln/ j j j. H n h j j gvn follow:. fo noh 0 j j Poof. j h olon o h pol. n.6 whn f {. If w f { no h con fol of.7 w oon g h fol fo of.9. ll f w f { no h f fol of.7 w oon g h fol fo of.9. j h olon o h pol. n.6 whn n f {. If w f { n h fol fo of.9 no h con fol of.7 n pn h ngl wh h clv on fncon of v nol on w g h fol fo of.0. ll f w f { n h fol fo of.9 no h f fol of.7 n pn h ngl wh h clv on fncon of v nol on w g h fol fo of.0. In h c of w ncon o pov.. h. hol fo n. hn fo n n n f. whn n n n n.
7 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on... n hn fo h con fol of.7... pov follow:... n... n n n n ln z 0 z... n... n z z z. n... n H z... n h pc of h nlng n -h o n opon n n h ncon-pon n. w hv... n n z n z n nn.... n n If w h l no h ov ngl pnon n pn h ngl wh h clv on fncon of n-v nol on w cn g h f fol of. fo = n. h l fo on n h con fol of. ll pov. QED.. h ho gnlzon of h coponng l of [5 ]. In ho h clv on fncon of v nol on wh n vco [0 0] n covnc [ ] ol n h ofw Ml n h clv on fncon of -v nol on wh zo n vco n covnc j j wh j j j j j. ch pcl fncon cn l clcl n fncon ppl n n ofw fo hcl clclon fo pl Ml. w con lon wn pc of hgh o n wh conn ffnc wn f n vn. Fo h fol.9.0 n. whn conn w hv:. n [4] w cn pov h followng l. h poof n o.. L. Ingl of n o nohng h g conno fncon of [ ]. L 7
8 Hong-hol O Yong-Gon n Dong-Ho g F 0 hn h olon of. n.3 gvn follow: 0 g F H F o F. 3. h Pol of Dfll on wh Dc Dfl Infoon h Pcng Fol n p nl 3. h Pol pon: ho follow h lw 3. n h nl ngl n n n poc. 0 0 nnoncng n h of o copo on wh fc vl n of cnc. Fo v 0 h npc fl pol n [ ] [ ]. H h fl nn conn. 3 h f vl follow goc own oon n h nl ngl n n n poc n E. h f connol p o vn n conn fo n of f vl. 4 h pc fl onl gvn n h pc fl vn occ whn H. conn h flc h n of n fl f zo copon on pc. 5 h pc fl cov gvn h npc fl cov n h cov 0 conn. Eogno cov. 5 h pc fl cov gvn h npc fl cov n{ n h cov 0 conn n 0 conn h flc h n of Enogno cov. H h on wh h pc fl cov n npc cov gvn n ffn fo o vo h pol of png o hn h cn pc of f on 8
9 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on 9 fl cov whn h npc fl vn occ. 6 In h nvl h pc of o copo on gvn ffcnl ooh fncon 0. Pol: Fn h pnon of h pc fncon 0 n h ov pon. 3. h Pcng Mol Un h pon h pc of fl f on h olon o h followng pol h olon gvn. 3.3 H n ffnl gvn pnn on h pcfc ol of ho [6]. Fo pl f h ho follow h c ol h f h coffcn n ll conn h hn n pcvl gvn follow:. 3.4 [6] fo n n Ho-L ol Hll-h ol n I ol. ccong o [6] n h ov pon h pc of fll on wh conn fl nn n npc fl cov f h followng PDE: 0. hfo f w l n hn h pc ol of o on gvn follow:. 0 { { 0 3.5
10 Hong-hol O Yong-Gon n Dong-Ho 0 H h fl cov ffnl gvn whh w choo o h pon 5 o 5. Un h pon 5 h ol follow:. 0 { { Un h pon 5 h ol follow:. 0 { { 0 n{ h Pcng Fol ho. Eogno cov Un h pon n 6 h pc of o on pn follow:. 0 / / 3.8 H ] [. 3.0 n gvn n h pc of -h o on n wh 0- f -vn n -voll. h ho.. h ho gnlzon of h ho of [4] no h c of
11 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on ochc f. h f w l conn n = w hv h ho of [4]. h fnncl nng of h pcng fol 3.8 v cl whn n j h h on of h ho n [4]. / h vvl pol f h [ h h pol wh whch no fl vn occ n h nvl ] [ n / h n pol f h [ h h pol wh whch fl vn occ n h nvl ] [ whn. h fol 3.8 cn wn follow:. 0 / 3. h fnncl nng of 3. h h f of 3. h cn pc of h p o gvn o on hol gl of fl occ o no n h con h llownc pnn on h vvl pol f. ho 3. Enogno cov Un h pon n 6 h pc of o on pov follow: 0 /. 3. H n gvn n 3.0 n n 0 h pc of -h o on n n wh 0- f -vn n -voll. h ho. 3. h ho 3 gnlzon of h ho of [4] no h c of ochc
12 Hong-hol O Yong-Gon n Dong-Ho f. h f w l conn / n n = hn w hv h ho of [4]. h fnncl nng of h h lv pc of o on n nvl wh pc o h f zo copon on. 3.4 p nl In h con w wll ll h ffc of vl p nclng cov voll of f vl h lv pc of h f vl n c. on c p. h c p fn h ffnc wn h l of fll on n fl-f on n gvn h followng pon: ln ln. In h c of ogno cov con n ho h c p f l wh h of [4]. H w con h c of nogno cov con n ho 3. In h c h c p ffnl gvn n v nvl. ln / ln / L 3 6nn c fo clclon of follow: ho ol p: 0.379* c ol f vl poc p: vn = 0.05 voll =.0 = / = 00 colon of ho n f vl: = pcvl fl nn n h nvl 0 ] ] 0 0 [ [ 00 fl cov : = = 0.5 / 50. wll nlz : plo chngng on of = / n n png h n of on h ov. In wh follow h fg how h nc of cov l n c of c p. h fg how h nc of voll of f vl l n nc of c p. h on h whn nc h f vl flc o ol n h o of fl whch l n nc of c p. h fg 3 how h nc of colon wn f vl n ho l n nc of c p. h fg 4 how h nc of f vl l n c of c p. h fg 5 6 n 7 how h n h nvl clo o h nc of h fl nn l n nc of c p n oh gon h ccnc no o pl. h fg 8 9 n 0 how h h ffc of fl on c p ffn n h nvl [0 ] n [ ]. h on of ch copl of h ffc of fl nn n fl n h fol 3.3.
13 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on Fg. Plo : whn = = Fg. Plo : whn = Fg 3. Plo : whn =
14 Hong-hol O Yong-Gon n Dong-Ho Fg 4. Plo : whn = / = Fg 5. Plo : whn 0 = Fg 6. Plo : whn 0 = 0.0 =
15 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on Fg 7. Plo : whn 0 = Fg 8. Plo : whn = Fg 9. Plo : whn =
16 Hong-hol O Yong-Gon n Dong-Ho 6 Fg 0. Plo : whn =00 = h Poof of h Pcng Fol h Poof of ho. Un h pon n 6 h pc ol of o on gvn 3.6. In 3.6 w chng of n 0 / /. 4. H h pc of fl f zo copon on gvn 3.3. If w 4. no h f on of 3.6 no h / / n con h on 3. on hn w hv 0 ] [. Dv h wo hn n l hn h pol 3.6 chng o h followng on nonl pol: 0 0 { { H. h chng of nnown fncon = + = 0 4.3
17 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on 7 o hv. 0 0 { { H. hn ng h o on pc pov 3.8. h on 4.4 cll h on fo h vvl pol f h [. 4.4 of lc-chol on j l 4. n [4] h h coffcn no conn. o w o ho n of h ho of [5 n ]. ow w olv h pol 4.4. hn { { h pcng pol of n opon wh coffcn who p poff conon of on n n. h fnon of on n w hv H gvn h fol.9 of h ho. Fo o fh ppo ng h lon. w w 4.6 h pc of on n n wh h coffcn 0 : In o o olv 4.4 whn w n o w 4.6 h pc of on n n wh h coffcn j no n h 3 n [4]. 0. ] [ 4.8 hn ng wn follow: 0. { ] { { [
18 Hong-hol O Yong-Gon n Dong-Ho 8 h pcng pol of n opon wh coffcn who p poff conon of on n n. h fnon of con o n w hv 0. ] [ H gvn h fol.0 of h ho. Fo o fh ppo ng h lon. w w h pc of on n n wh h coffcn 0 : 0. ] 0 0 [ ncon w hv 3.9. nng o ognl vl hogh 4. n 4.3 hn w hv h fol 3.8. QED h Poof of ho 3. Un h pon n 6 h pc ol of o on gvn 3.7. In 3.7 w chng of n 4. hn w hv 0 0 { { 0 0 n{ 4. h 4. l pol wh h pol 4.5 n [4]. h onl ffnc h h 4. of nl vl pol fo nhoogno lc-chol on wh pnn coffcn h 4.5 n [4] of nl vl pol fo nhoogno lc-chol on wh conn coffcn. If w follow h w of olvng 4.5 n [4] ng o ho l n h lon. hn w cn g h fol 3.3. hn nng o h ognl vl n h nnown fncon ng 4. w cn oon on h fol 3.. h l o. QED 5. onclon pov h pcng fol of hgh o n opon wh pnn coffcn ho. h gnlzon of h coponng l of [5 ]. n w gnlz h ngl fol of hgh o n opon on h l p vl no h c wh pnn coffcn l.
19 Hgh O n wh Dpnn offcn n wo Fco-Mol fo Dfll on on h pcng fol of wo fco - ol fo fll on wh c fl nn n c fl n oh c of ogno n nogno cov ho n ho 3 ng h pcng fol of hgh o n opon wh pnn coffcn. 3 In fh h ho cn ngl ppl o gnlzon of h of [] no h pcng fll copon on n conng ol wh h cl ppoch n h c fo ppoch. cnowlgn: ho hn nono v oo fo c no whch hlp o h von n o copl. fnc [] gl. ophnv cl Mol fo Dfll F-nco on Qnv Fnnc : DOI:0.080/ [] gl E. n gl. on Pcng n Ipc Infoon Oponl ch Innonl Jonl DOI: 0.007/ [3] ll L. G Pcll Pcng fll on: nw ol conng cl nfoon wh h c-fo ppoch Mch 008 ong P DOI: 0.39/n [4] Yhng n n ojn Pcng opo on wh oh Epc n Unpc Dfl Jonl of ongj Unvl cnc ol.35 o. 7 Jl 007 pg [5] chn P. h Pcng of l-p oc Qnv Fnnc DOI: 0.088/ /4//009 [6] hc L. n L El-Jhl -nlcl pcng of fll on n gnlng jpfl ol Jonl of oponl Fnnc pg [7] hc L. n Ln El-Jhl Pcng fll on: l-w ppoch wn cl n c-fo ol Qnv Fnnc ol 6 I pg 43 ~ 53 DOI: 0.080/ [8] Jng L-hng Mhcl Mol n Mho of Opon Pcng ol cnfc [9] O Hong-chol n ng nlcl Pcng of Dfll on wh ochc Dfl Inn pg ~35 Dvv Jonl 05/005 DOI:0.39/n.7360 v ppn v:303.98[-fn.p] [0] O Hong-chol Pcng opo Dfll on ng Dcl F l c n n l n Innonl po n ooon of h 65 h nnv of h Fonon of Il ng Unv Mhc p 0- Jch 000 9
20 Hong-hol O Yong-Gon n Dong-Ho Pongng DP Plhng Ho of Il ng Unv pp Jch 00 n l wh poof n v ppn pp.- v: [-fn.p] [] O Hong-chol n Mn-chol Hgh O n Opon n Mlpl Ep Eoc Elconc Jonl of Mhcl nl n pplcon ol. Jl 03 pp hp://j.6.n/ol_pp/ol_pp_c_w.h [] O Hong-chol n Mn-chol h Pcng of Mlpl Ep Eoc v ppn pp -6 v:30.339[-fn.p] [3] O Hong-hol n -Mn n n ng Pcng opo Dfll on wh F Dc Dclon of F l pg ~4 Dvv Jonl 05/005 DOI: 0.39/n.7356 [4] O Hong-chol Dong-Ho ong-hn n Jo Jong-Jn: Ingl of Hgh n Opon n Dfll on wh Dc Dfl Infoon v ppn pp -7 v: v4[-fn.P] [5] lon M. pcng wh oh pc n npc fl ppl Fnncl Econoc L ol. 3 o pg 5-30 DOI: 0.080/ [6] lo P. Dvv: h ho n Pcc of Fnncl Engnng John l & on. Inc
Pricing Arithmetic Average Reset Options With Control Variates
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