CreditGrades Framework within Stochastic Covariance Models

Transcription

1 Journal of Mahacal Fnanc hp://dxdoorg/036/jf0033 Publshd Onln Novbr 0 (hp://wwwscrporg/journal/jf) CrdGrads Frawork whn Sochasc Covaranc Modls Marcos Escobar Hadrza Aran Lus Sco 3 Dparn of Mahacs Rrson Unvrs orono Canada Roal Bank of Canada orono Canada 3 Dparn of Mahacs Unvrs of orono orono Canada Eal: scobar@rrsonca Had-aran@rbcco sco@ahoronodu Rcvd March 7 0; rvsd Aprl 9 0; accpd Aprl 3 0 ABSRAC In hs papr w sud a ulvara xnson of a srucural crd rsk odl h CrdGrads odl undr h assupon of sochasc volal and corrlaon bwn h asss of h copans h covaranc of h asss follows wo popular odls whch ar non-ovrlappng xnsons of h CIR odl o dnsons grar han on h Wshar procss and h Prncpal coponn procss Undr CrdGrads w fnd quas closd-for soluons for qu opons argnal probabls of dfauls and so ohr ajor fnancal drvavs words: Barrr Opon; Sochasc Covaranc; CrdGrads Mod; Wshar Procss; Prncpal Coponn Procss Inroducon W prsn a srucural crd rsk odl whch consdrs sochasc corrlaon bwn h asss of h copans W allow h covaranc of h asss o follow wo popular sochasc covaranc odls Frs w assu follows a Wshar procss [] hn w assu a Prncpal coponn Procss (s []) boh ar non ovrlappng xnsons of h CIR odl o dnsons grar han on Modlng sochasc corrlaon has dffculs fro h analcal as wll as h saon pon of vw On of h aps o x hs gap bgan wh a papr on Wshar procsss b [] whch followd b a srs of paprs b Gourroux s [3] Svral auhors hav rcnl brough h fnanc coun s anon o h Wshar procss and showd ha h Wshar procss s a good candda for odlng h covaranc of asss A Wshar procss s an affn src posv dfn procss h sra of paprs [3-7] brough h fnanc coun s anon o hs procss as a naural xnson of Hson s sochasc volal odl whch has bn a vr succssful unvara odl for opon prcng and rconsrucon of volal sls and skws h popular of h Hson odl as wll as h prcal vdnc of sochasc corrlaon and volal has conrbud o h rcn popular of h Wshar procss Rsk s usuall asurd b h covaranc arx hrfor Wshar procss can b sn as a ool o odl dnac bhavor of ulvara rsk Va h Laplac ransfor and h dsrbuon of h Wshar procss [3] prcs drvavs wh a gnralzd Wshar sochasc covaranc arx hs approach can b usd o odl rsk n h srucural crd rsk frawork DaFonsca xnds hs approach n [7] o odl h ulvara rsk b a Wshar procss In hs papr svral rsk socks ar consdrd and h prcng probl for on dnsonal vanlla opons and uldnsonal gorc bask opons on h socks s prsnd W adap xsng rsuls abou h Wshar procss o h srucural CrdGrads frawork W gv quas closd-for soluons for qu opons argnal probabls of dfauls and so ohr ajor fnancal drvavs For calculaon of our prcng forulas w ak a brdg bwn wo rcn rnds n prcng hor; fro on sd prcng of barrr opons b [8] and [9] and fro ohr sd h dvlopn of Wshar procss b [] In h scond par of h papr w dvlop a nw odl for crd rsk basd on a odl wh sochasc gnvalus calld prncpal coponn sochasc covaranc o nduc h sochasc no h srucur of volals and corrlaon w assu ha h gnvcors of h covaranc arx ar consan bu h gnvalus ar drvn b ndpndn Cox-Ingrsoll-Ross procsss o prc qu opons on hs frawork w frs ransfor h calculaons fro h prcng doan o h frqunc doan hn w drv a closd forula for h Fourr ransfor of h Grn s funcon of h prcng PDE Fnall w us h hod of ags o Coprgh 0 ScRs

2 30 M ESCOBAR E AL fnd h prc of h qu opons Sa hod s usd o fnd closd forulas for argnal probabls of dfauls and CDS prcs Insprd b h sandard sochasc volal odls sarng fro Hson s papr [0] and h work on [] h applcaons of h prncpal coponn odl n crd rsk s sudd h an da s o dnf h covaranc arx b s gnvcors and gnvalus In h abov paprs auhors hav assud ha h gnvcor of h covaranc arx ar consan bu h gnvalus follow a CIR procss hs pls a sochasc srucur for h corrlaon bwn h asss [] prcs h collaralzd db oblgaons undr h Mron s odl usng a r approach and h prncpal coponn odl Mron s odl [] s h frs srucural crd rsk odl proposd whch consdrs h copan s qu as an opon on h fr s ass hr has bn nurous xnsons for h Mron s odl n h lraur ncludng ncorporang arl dfauls sochasc nrs ras sochasc dfaul barrrs and jups n h ass s prc procss A splr approach was jonl dvlopd b CrdMrcs JP Morgan Goldan Sachs and Dusch Bank calld h CrdGrads odl hs can b sn as a parcular cas of Mron wh zro o aur h splc allows for closd for xprsson on so drvavs as shown n our papr whch can no b found closd for undr Mron or Black Cox srucural fraworks W xnd h CrdGrads odl usng sochasc covaranc Wshar procss focusng on h rol of sochasc corrlaon h prforanc of a copan s usuall onord b obsrvng s qu s volal or h CDS sprad CrdGrads odl can b consdrd as a down-and-ou barrr crd rsk odl hs ans ha dfaul s rggrd f h valu of h ass rachs a cran lvl dnfd b h rcovr par of h db [] has xndd h CrdGrads odl o prc qu opons b nroducng h qu as a shfd log-noral procss [9] has xndd [] da b bddng h Hson s volal no h odl and prcng qu drvav Boh [9] and [] odls ar unvara crd rsk odls W xnd h Crd- Grads odl b us of h Wshar and PC procsss o dnsons grar han on plnng sochasc corrlaon no h dnacs of h asss W gv quas closd forulas for qu drvavs basd on hs odls hs papr s organzd as follows: n Scon w us Wshar procss as a candda o odl h covaranc arx of h asss prcs whn a CrdGrads odl h prcng probl for so drvavs on h qus s drvd n Scon Scon 3 prsns and uss h Prncpal coponn procss for h covaranc arx of h asss prcs whn h sa srucural frawork h prcng probl s drvd n Scon 3 Scon concluds h proofs ar gvn n h appndx h CrdGrads Wshar Procss In hs scon w nroduc our Wshar CrdGrads odl CrdGrads odl s a vrson of h Mron odl jonl dvlopd b CrdMrcs JP Morgan Goldan Sachs and Dusch Bank h orgnal vrson of h CrdGrads odl assus ha volal s drnsc W xnd CrdGrads odl b ans of sochasc covaranc Wshar procss Our odl allows corrlaon and volal b sochasc B consdrng h sochasc covaranc Wshar procss w hav or flxbl and dgr of frdo n h argnal whl analc racabl s prsrvd whn xndng CIR procss o Wshar procss W frs prsn Wshar procss of ngr dgr of frdo and drv hr arx sochasc dffrnal quaon whch lar on wll gv a naural rprsnaon of Wshar procss wh fraconal dgr of frdo A Wshar procss wh ngr dgr of frdo s a su of ndpndn n-dnsonal Ornsn-Uhlnbck procss W frs rnd h foral dfnon blow: Dfnon: Consdr k U k of Ornsn-Uhlnbck procsss k k k du AU d dw as an ndpndn s whr A and ar nn arcs wh nvrbl hn a Wshar procss of dgr s dfnd as k k k U U k whr U s h ranspos of h vcor U Io s la can b usd o fnd a dffuson SDE for h procss k k k k k k k U U U U U U k d d d d d A A d k k k k k W d U U d W As can b sn h drf r of h SDE abov conans bu h dffuson par conans h rs k k U and U sparal [3] show ha also sasfs h followng arx SDE d A A ddw d W W s an n n sandard Brownan oon a- whr rx h Dnacs of h Asss h asss ar dfnd on a probabl spac F Coprgh 0 ScRs

3 M ESCOBAR E AL 305 whr F s h nforaon up o and s 0 h rsk-nural asur quvaln o h ral-world h asur P L s assu ha h fr s ass prc pr shar s gvn b A Hr w rvw h rsuls rgardng h dnacs of h asss wh sochasc covaranc Wshar procss As bfor assu h asss prcs follow h ulvara ral-world odl A dlna r D d dw d M M dw Hr h vcor n s consan and D s a src posv dfn arx h log-prc procss has h drf r d and h quadrac vara- E dlna rd on V dlna d Mo rovr w assu ha h Brownan oons drvng h asss and h Brownan oons drvng h Wshar procss ar uncorrlad r D 0 accouns for rsk pru (undr rsk nural asur) For h ranson dsrbuon of A h gvn A and w hav ln A A h h u N ln A r D d u du h u and h uncond onal probabl funcon can b found b h Ingraon ovr h dsrbuon funcon of u du Up o now w hav dscrbd h dnacs of h as- ss prcs wh sochasc covaranc srucur cong fro h Wshar procss h ass's prc for a fr s no drcl obsrvd fro h ark hs lads us o srucural crd rsk odls whch nroduc h qu as a for of a drvav on h ass of h copan hn h rol of h odl s n conncng h qu ark o h dfaul vn For xapl [3] proposs a dnacs for h asss and labls n h Mron s odl whr h qu s dfnd as a call opon on h ass wh labl as h srk prc hs s a drc xnson of h Mron odl wh ulvara sochasc volal In ha cas h prc of a bond has quas closd-for forulas basd on h closd forulas for h condonal Laplac ransfor of h jon prc-covaranc procss W prfr no o us hs odl for svral rasons Frs w found h CrdGrads odl or popular n fnancal arks bcaus of s abl o lnk h srucural frawork o qu drvavs On h ohr hand h odl proposd b [3] has on coon Wshar procss and on dsnc Wshar procsss for ach ass hrfor h nubr of parars for h odl s hgh and h calbraon s xrl llposd bcaus of h hgh dgr of frdo posd b h nubr of parars nsd h odl W found ha n h cas of wo copans assung onl on Wshar procss drvng h covaranc arx gvs a farl flxbl odl o capur ark s bhavor whl a h sa provds a fwr nubr of parars Nx w nroduc h qu procss basd on h CrdGrads prspcv rahr han Mron s prspcv akng advanag of h flxbl of h Crd- Grads odl hs wll nrch h crd rsk odlng wh possbl of arl dfaul and also a sraghforward lnk bwn crd rsk and h qu ark W wll drv a forula for h nfnsal gnraor of h jon qu-covaranc procss blow hs opraor wll pla an poran rol n h paral dffrnal quaon of h qu opon s prc Now ha w hav dnfd h dnacs of h asss w xplan h chans of h CrdGrads odl As bfor w assu ha h h fr s valu A s drvn b h dnacs da dag A r d Id dw d M M d dz () whr for so n and M s a ngav dfn arx W assu ha as ss ar drvn b h Brownan oon W h covaranc arx of h asss whch follows a Wshar procss s drvn b h Brownan oon Z and wo Brownan oons W and Z ar uncorrlad W assu ha h S s h f r s qu prc pr shar B s h h fr s db pr shar and R s h h fr s rcovr ra B has a drnsc growh ra r d whr r s h rsk fr nrs ra and d s h dvdnd ld for h h fr h rcovr par of h db D RB s h dfaul barrr for h ass hrfor h dfaul basd on CrdGrads odl s gvn b nf 0 A D In h frawork of CrdGrads odl h qu s valu s gvn b S A D 0 In rs of h qu h dfaul can b wrn as nf 0S 0 Zro s an absorbng sa for h qu procss whch aks h prcng of h qu opon slar o prcng of down-and-ou opons sudd b [] B usng S A D h dnacs of D and quaon () h qu follows a shfd log-noral SDE W wll us h noaon dag x vc x and I dnong dagonal arx and vcor wh lns x and h arx of ons rspc- Coprgh 0 ScRs

4 306 M ESCOBAR E AL vl ds dag S r d Id dag S D d W d M M d dz No ha h soluon of h dnacs abov can rach ngav valus bu no bfor h soppng W forc suffcn condons on h Wshar procss o ak an rvrng For our purposs w assu M s ngav dfn and for so n Morovr whou loss of gnral w as- W frs drv h nfnsal gn- su raor of h jon procss S hs opraor wll appar n h prcng PDE for qu opons and h probabls of dfaul n h nx scon Proposon : h nfnsal gnraor of h jon procss S s gvn b A r d S S S S D S S D S r M M D DD whr D r s h rac of a arx and j j w v usd h noaon rds VcrdS and S D Vc S S Drvav Prcng; Analcal Rsuls D S In hs scon w ackl h prcng probl of our crd rsk odl W wll us h fourr ransfor and hod of ags o solv h prcng probl for Europan calls and pus on h qu Equ Call Opons h prc of a Europan Call opon on h qu s cal- xpcaon of h culad b dscounng h rsk-nural paoff a aur Snc opon could b rwrn as () S S h prc of h call No ha S wh h abov dnacs s allowd o gan ngav valus bu no pror o h soppng Evn hough gh s unrasonabl o allow S hav ngav valus hs dosn affc an of h prcng forulas snc whnvr h procss S s nvolvd s followd b h runcang facor ( as n Equaons (3) and (0) for h paoffs of call and pu opons) Vcall S E xp rs d ss S E xp r s ds S S h prc of a sngl na drvav on on of h qus sasfs h paral dffrnal quaon V A rv 0 S Spcall h prc of an qu call opon s gvn b h PDE V S D VS S r d SV S A V rv 0 (3) V00 V SS whr A s h nfnsal gnraor of h jon S procss S gvn b h proposon W frs S D chang h varabls b x ln D D a ln and D xp sds V S G x r o ransfor h PDE D o G Gx A G 0 x G x () x a G00 V x o us h hod of ags w nd o lna h drf r frs hnc w chang h varabls b x a and a U G x hn PDE (3) ransfors o U U 0 A U U 8 U 00 U0 h PDE (5) s our rfrnc PDE o solv h prcng probl for qu opons on S W hav h followng proposon for h Fourr ransfor of h Grn s funcon of PDE Proposon : h Fourr ransfor of h Grn s funcon of PDE s gvn b ky A k rb k q Y d k (6) whr j j B k k k A kr Bu kdu 0 (5) Coprgh 0 ScRs

5 M ESCOBAR E AL 307 wh M xp k I M Now ha w hav found h Fourr ransfor of h Grn s funcon of h prcng PDE w solv h prcng probl for an qu call opon b h hod of ags Proposon 3: h prc of a call opon on Sj wh aur da and srk prc s gvn b j V S D xp r s d s Z ln S D ln D r s d s ds b ln D ln D r s d s ds and h funcon b Z Z s dfnd b A k rb k 0 cos k cos b k d s k (7) wh A k and B k gvn as n proposon For larg valus of k h ngrand n (7) s xponnall dcrasng whch aks as o valua h ngral nurcall R ark : As w hav nond bfor our rsul covrs [9] as a spcal cas If n h dnacs of h ass () w assu n and for h parars w l M and proposons and 3 ld B k Now o fnd and b proposon w hav k E E s a arx wh whr k k k hrfor k B k k k h funcon A k can b found b ngraon fro hs gvs h prc of qu call opon n h prsnc of Hson sochasc volal (as n [9] Equaons (35)-(37)) h prc of a Europan pu opon on h qu s calculad b dscounng h rsk-nural xpcaon of h paoff a aur Slarl o paoff of h call opon on can chck ha S S hrfor h prc of h pu opon could b rwrn as pu V S E xp r s ds S S E xp r s d s S S (0) Equaons (3) and (0) gv h pu-call par for h qu opons call pu V S V S call E xp r s ds S S E xp r s ds S S V S0 xp r s d s Survval Probabls and Crd Dfaul Swaps Suppos P S s h survval probabl for h h copan S P S S 0 hn usng Fnan-ac forula P S sasfs h paral dffrnal quaon P AS P 0 h Proposon : h survval probabl for h fr s gvn b A krb k ksnk P S d k 0 k wh funcons A k and B k as n Equaon (7) Crd dfaul swaps ar on of h os popular crd drvavs radd n h ark A CDS provds procon agans h dfaul of a fr known as rfrn c n h bur of h conrac pas prodc pan s calld CDS sprads unl h dfaul or aur Coprgh 0 ScRs

6 308 M ESCOBAR E AL da In rurn h sllr of h CDS provds h bur wh h unrcovrd par of h noonal f dfaul occurs h valuaon probl of a CDS s hn o gv h CDS sprad a valu such ha h conrac bgns wh a zro valu hs ans ha h valu of h floang lg and h fxd lg should concd whn h conrac s wrn Assu ha h CDS sprad s dnod b Sp h prodc pans occur a 0 0 N h noonal s N h of dfaul s dnod b τ and h rcovr ra s h consan R h fxd lg of h CDS s h valu a 0 of h cash flow corrspondng o h pans h bur aks Wh h abov noaon w hav N rsds Fxd E SpN SpN 0 N 0 rsds () h CDS sprad Sp s chosn such ha h conrac has a far valu a 0 B sng h fxd lg qual o h floang lg h quaons and pl () On h ohr hand h floang lg whch s h valu of h procon cash flow a 0 s N rsds Floang E R 0 Sp N 0 rsds R N rsds 0 N rsds 0 R 3 h CrdGrads Prncpal Coponn Modl In hs scon w frs prsn an sochasc gnvalu procss whch s usd for h covaranc of h asss procss h scon hn covrs prcng of drvavs usng h CrdGrads odl W frs rnd h foral dfnon blow: Dfnon : h nsananous sochasc covaranc follows a Prncpal Coponn Modl f: A A (3) whr s a d d dagonal arx whos lns V ar ral valud CIR procss dfnd for p b: d d dz () h Z s ar ndpndn on-dnsonal Brownan oon and E s an orhogonal consan arx W also assu 0 for p and whou los of gnral 0 h an ngrdn of hs ulvara procss s a fal of on-dnsonal sochasc procsss for h gnvalus W assu for splc Hson-p procsss bu hs approach works for ohr knd of procsss h condons 0 nsur saonar rgodc and xng condons for h on-dnsonal procsss (s []) h consrans 0 nsur ha h gnvalus proc ss wll kp on avrag h sa ordr bu hr pahs could vnuall cross ovr hs ordrng on avrag allows us o kp h gnvalus wh gras an rvrng lvls whl droppng h lss sgnfcan ons 3 h Dnacs of h Asss h W assu ha h fr s valu A s drvn b h dnacs da dag A r d Id dw whr W M A M n EDE wh Each E D dag ; j n follows a CIR procss of h p d d dz In h wo asss cas h abov dnacs follows dw da A r d d dw d d dw da A r d W whr h gnvalus of h covaranc procss follow d d dz d d dz (5) And assung as h angl ha h frs gnvcor aks wh h ral axs h gnvcor arx E s gvn b sn sn cos E sn W assu ha asss ar drvn b h Brownan oon W h covaranc arx of h asss s drvn b h Brownan oon Z and wo Brownan oons W and Z ar uncorrlad h rason w ak h ndpndnc assupon bwn sock and s volal s ha closd for forulas for h valu of doubl-ba r- Coprgh 0 ScRs

7 M ESCOBAR E AL 309 rr opons and qu opons ar no avalabl whn h ass and s volal ar corrlad as pond ou b [8 9] h nfnsal gnraor of h jon procss S AS appars n h prcng PDE Hr w fnd a foula for hs opraor o us for our prcng purposs n h nx scon Snc S A D h qu sasfs h sochasc dffrnal quaon ds r d S d S D dw A S can b dvdd no hr rs rlad o h sock s opraor h covaranc opraor and hr jon opraor A A A A S S S Snc dz and dw ar ndpndn h las r s zro Fro h dnacs of h qu w know ha A S W r d SW S S D jj WS S j r d SWS S D jj W SS j (6) And fro h classcal rsuls rgardng h nfnsal gnraor of h CIR procss A W W W hrfor f W s a drvav on h frs undrl- ng ass onl w hav A S W r d SW S D W W W S jj SS j In h nx scon w drv closd forulas for h prc of qu opons and argnal probabls of dfaul 3 Drvav Prcng; Analcal Rsuls In a odl wh wo undrlngs h frs ass follows h followng procss: dw Z Z da A r d d dw d d d d d d W wll show nx h prcs of svral drvavs as sn fro a crd prspcv 3 Equ Call Opons Calculang qu opon prcs s ssnal o calbra h sochasc corrlaon CrdGrads odl snc hs odl uss h nforaon avalabl fro h qu opons o sa h parars of h odl Lar w wll us h voluonar algorh hod o ach h horcal rsuls of our xndd CrdGrads odl wh h ark daa On of h advanags of h CrdGrads odl copard o Mron s odl s h sragh forward lnk aks wh h qu opon arks h prc of h qu opon can b calculad b dscounng h paoff funcon a h aur h onl subl pon hr n prcng hs opons ls n h spcfc dnacs of h qu slf and h possbl of dfaul for h copan In Black-Schols odl h sock follows gorc Brownan oon whch s a srcl posv procss wh a log-noral dsrbuon and nvr hs zro In h CrdGrads odl qu s odld as a procss sasfng a shfd log-noral dsrbuon whch hs h sa zro whn h copan dfauls Bcaus of h absorbng propr of h sa zro for h qu procss hr s a rsblanc n prcng h qu opons and h prcng of h downand-ou opons B consdrng h barrr condon for qu h paoff of an qu call opon s gvn b S hrfor h prc of an qu call opon can b wrn as: xp d S E S rs ss V S E r s s S call rs xp d s S Slarl h paoff of an qu pu opon s S hrfor h prc of an qu pu opon s gvn b: Vpu S E xp d S E xp r s ds S S Equaon (8) gv h pu-call par for h qu opons: Vcall S Vpu S (8) Vcall S0xp rsds h followng proposon gvs a closd for soluon for h prc of an qu call opon on h frs ass Proposon C5 and Equaon (8) gv h prc of an Coprgh 0 ScRs

8 30 M ESCOBAR E AL qu pu opon hs rsul s an ssnal ool o calbra h odl n h nx scon Proposon 5: h prc of a call opon on S wh aur da and srk prc s gvn b: W Sj D xp rsds Z wh A k B k b Z cos cos 0 k k b k B k k ds A k ln D S ln rsd d D s s D bln rsd d D s s k 3 Survval Probabls and Crd Dfaul Swaps Slar chnqus can b usd o fnd h argnal probabls of dfaul Suppos P S s h survval probabl for h copan h P S S 0 S Usng h Fnan-ac forula P S sasfs h paral dffrnal quaon P AS P 0 wh boundar condons P 0 0 and P S W hav h fo llowng proposon for h survval probabls h Proposon 6: h survval probabl for h fr s gvn b A k B k j j j ksn k P S dk 0 k nowng h probabl of h dfaul o n can fnd h CDS sprad for h undrlng copan Assu ha h CDS sprad s dnod b S h prodc pans occur a h noonal s 0 0 N N h of dfaul s dnod b and h rcovr ra s h consan R h fxd lg of h CDS s h valu a 0 of h cash f low corrspondng o h pans h bur aks Wh h abov noaon w hav N rsds Fxd E SN 0 N ds r s SN 0 0 h CDS sprad S s chosn such ha h conrac has a far valu a 0 B sng h fxd lg qual o h floang lg h Equaon (8) pl On h ohr hand h floang lg whch s h valu of h procon cash flow a 0 s N rsds Floang E R 0 N rsds R S Concluson N ds R r s 0 N rsds 0 W prsnd a srucural crd rsk odl whch consdrs sochasc corrlaon bwn h asss of h copans h sochasc of h volal and corrlaon cos fro frs a Wshar procss and hn a prncpal coponn sochasc covaranc procss whch drvs h covaranc arx of h asss o odl crd rsk w us h so calld CrdGrads odl Usng h affn proprs of h jon log-prc and volal procss w solvd h prcng probl of h qu opons W usd our analcal chnqus o drv quas closd-for soluon for qu opons probabl- s of dfauls an d prcs of CDSs ssud b h copans REFERENCES [] M-F Bru Wshar Procsss Journal of horcal Probabl Vol No 99 pp [] M Escobar B Goz L Sco and R Zags Prcng of a CDO on Sochascall Corrlad Undrlngs uanav Fnanc Vol 0 No pp [3] C Gourroux J Jasak and R Sufana Drvav Prcng wh Mulvara Sochasc Volal: Applcaon o Crd Rsk Workng Papr 00 [] C Gourroux and R Sufana h Wshar Auorgrssv Procss of Mulvara Sochasc Volal Econorcs Vol 50 No 009 pp 67-8 Coprgh 0 ScRs

9 M ESCOBAR E AL 3 do:006/jjcono00806 [0] S L Hson A Closd-For Soluon for Opons wh Sochasc Volal wh Applcaons o Bond and Cur- rnc Opons Rvw of Fnancal Suds Vol 6 No 993 pp do:0093/rfs/637 [] R C Mron On h Prcng of Corpora Db: h Rsk Srucur of Inrs Ras Journal of Fnanc Vol 9 No 97 pp 9-70 [] R Sacar and C Fngr Incorporang Equ Drva- h- vs no h Crdgrads Modl RskMrcs Group apa 005 [3] H Abou-andl G Frlng V Ionscu and G Jank Marx Rcca Equaons n Conrol and Sss or Sprngr Brln 003 do:0007/ [] M Grassll and C bald Solvabl Affn r Srucur Modls Mahacal Fnanc Vol 8 No 00 pp [5] J DaFonsca M Grassll and F Ilpo Esang h Wshar Affn Sochasc Corrlaon Modl Usng h E- prcal C haracrsc Funconk Workng Papr ESILV RR [6] J DaFonsca M Grassll and C bald A Mulfacor Volal Hson Modl uanav Fnanc Vol 8 No pp [7] J DaFonsca M Grassll and C bald Opon Prcng Whn Corrlaons Ar Sochasc: An Analcal Frawork Rvw of Drvavs Rsarch Vol 0 No 007 pp 5-80 [8] A Lpon Mahacal Mhods fo r Forgn Exchang: A Fnancal Engnrs Approach World Scnfc Sngapor 00 [9] A Spp Exndd Crdgrads Modl wh Sochasc Volal and Jups Wlo Magazn 006 pp 50-6 Coprgh 0 ScRs

10 3 M ESCOBAR E AL Appndx Proof proposon : S A D Snc S ds r d S d S D dw A can b dvdd no Snc dz and zro B []: S S S A A A A dw ar ndpndn h las r s A r M M D DD o fnd A A S n b h dnacs of rd S S S S n S D S D j j j j SSj r d S S S DSS D S Proof proposon : Dfn and subsung no (5) lds X ky A k rb k A rb jj k r M M B BB jj 0 8 (9) No ha h funcons sasfng h ODE abov ( A k and B k) do no dpnd on h varabl So w s 0 o g 0 A r B (0) and hn b subsung (0) no (9) lads o r B jj k r M M B BB jj 0 8 o solv h abov ODE w rarrang h quaon as r H H kb BM MB B B jj k hrfor H sasfs: n Hl l jj k l j Snc h funcon H j s ndpndn of assung o b a zro arx xcp for h l h nr hrfor B BM MB BB k I 0 () hs arx Rca quaon has bn sudd n h lraur (s [3]) and n Affn r srucur odls (s []) ladng o: B k k k A k can b found b ngraon Proof proposon 3: h prvous proposon gvs h Fourr ransfor of h Grn s funcon of h prcng PDE Now no q Y s nvaran wh rspc o h chang of varabls and k k hrfor q Y s an vn funcon wh rspc o Y hs pls ha h Fourr ransfor of h Grn s funcon absorbd a x b s b q q q b B Duhal s forula U b d 0 q A krb k k k b b dk k Wh h consqun changs of varabls d r s sw S a G x U G x D on can conclud ha Z U Proof Proposon : h PDE for survval probabl s: S P S P S D PSS r d S P A P 0 () P 0 0 S D Usng h chang of varabls ln D and P S U h PDE ransfors o U U 0 A U U 8 P 0 P0 0 Coprgh 0 ScRs

11 M ESCOBAR E AL 33 hs PDE s h sa as (5) In Proposon w hav provd ha h aggrgad Grn s funcon for hs PDE s of h for (8) o fnd a boundd soluon rflcd a x 0 w us h hod of ags o wr h absorbd aggrgad Grn s funcon as 0 q q q Now b Duhal s forula 0 U 0 q d A krb k ksnk d k 0 k Usng h chang of varabl P S U on can fnd h survval probabl fro h abov for- ula for U as: A krb k ksn k P S d k 0 k Proof proposon 5: B rsk nural valuaon W sasfs W A rw S 0 whr A S s h nfnsal gnraor of h SDE drvng h qu B subsuon W r d SW S D W W rw 0 W S jj SS j W chang h var- Fro now w drop h ndx abls as whch gvs D S D D x ln ; ln D W S G x xp rsds D G jj Gxx Gx j G 0 G W prfor h scond chang of varabls as xa; a F G x And fnall w prfor h hrd chang of varabls ladng o U jj U j U F U 0 U jj U 8 j W cla ha h Fourr ransfor of h Grn s funcon for h abov PDE s of h for ky A k jb j k j q Y dk (3) B k k ) A k ln( ) k W know ha q Y sasfs h corrspondng q Y no h PDE on gs R- A k and Bj k s whch fnall B k as PDE Pluggng ca ODE s for gvs h funcon u u B k k h rprsnaon for h funcon fro quaon u A k ln C A k cos ln No ha q Y has a srucur nvaran wh rspc o h chang of varabls k k hrfor h Fourr ransfor absorbd a x b s b q b q q h abov xprsson and Duhal s forula lads o: b U q d 0 b A k jbj kj k kb k dk Coprgh 0 ScRs

12 3 M ESCOBAR E AL Snc Z U h rsul follows ky A k j B j k j q Y dk Proof proposon 6: Subsung for h nfnsal gnraor fro Equaon (7) P S solvs and h funcons A k and Bj k ar gvn In P ndd so r d SP ordr o fnd a bou luon rflcd a x 0 w S us h hod of ags o wr h absorb d aggr- as S D jj j P SS gad Grn s funcon (6) 0 0 P P q q q Now b h Duhal s forula P 00 P S 0 U S D q d 0 Usng h chang of varabls ln D A k jbj kj ksnk and P S U h PDE (6) dk 0 k ransfors o j U U j j U U 0 8 jj U j U U In h proof of h proposon 5 w showd ha h Fourr ransfor of h Grn s funcon for h abov PDE s of h for hrfor h survval probabl s gvn b P S U A k j Bj k j ksn 0 k k dk Coprgh 0 ScRs