Risky Swaps. Munich Personal RePEc Archive. Gikhman, Ilya Independent Research. 08. February 2008

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1 MPR Munch Peronal RePEc rchve Ry Swap Ghman Ilya Independen Reearch 8. February 28 Onlne a hp://mpra.ub.un-muenchen.de/779/ MPR Paper o. 779 poed 9. February 28 / 4:45

2 Ry Swap. Ilya Ghman 677 Ivy Wood Cour Maon OH 454 Emal: Phone: brac. In [ we preened a reduced form of ry bond prcng. he defaul dae a bond eller fal o connue fulfll h oblgaon and he prce of he bond harply drop down. If he face value of he defauled bond for no-defaul cenaro $ hen he bond prce u afer defaul called recovery rae RR. Rang agence and heorecal model are ryng o predc RR for compane or overegn counre. The man heorecal problem wh a ry bond or wh general deb problem preenng he prce gven he RR. The problem of a cred defaul wap CDS prcng omewha an adacen problem. Recall ha he corporae bond prce nverely depend on nere rae. The cred r on a deb nvemen relaed o he lo f defaul occur. There ex a pobly for a ry bond buyer o reduce h cred r. Th can be acheved by buyng a proecon from a proecon eller. The bondholder would pay a fxed premum up o maury or defaul whch one come fr. In exchange f defaul come before maury he proecon buyer wll receve he dfference beween he nally e face value of he bond and RR. Th dfference called lo gven defaul. Th conrac repreen CDS. The counerpary ha pay a fxed premum called CDS buyer or proecon buyer and he oppoe pary he CDS eller. oe ha n conra o corporae bond CDS conrac doe no aume ha buyer of he CDS a holder of he underlyng bond. oe ha underlyng o he wap can be any ae. I called he reference ae or reference eny. Thu CDS a cred nrumen ha eparae cred r from correpondng underlyng eny. Thu he formal ype of he CDS can be decrbed a follow. The buyer of he cred wap pay fxed rae or coupon unl maury or defaul f occur ooner han maury. In cae of defaul proecon buyer delver cah or defaul ae n exchange of he face value of he defauled deb. Thee are nown a cah or phycal elemen correpondngly. Inroducon. The opon valuaon benchmar wa developed by lac Schole and Meron n 7 [2 4. I ue he preen value neuralzed reducon of he underlyng ecury for he nrumen prcng. I wa hghlghed n [5-8 ha he underlyng logc of he benchmar approach n many repec conradc he common ene of he prcng defnon. Indeed eher he lac Schole opon prcng equaon or developed laer he bnomal cheme doe no depend on a

3 real reurn of he underlyng ecury. Therefore hee approache ugge he ame prce for he opon wren on ecure havng equal r characerc volaly and dfferen expeced rae of reurn regardle wheher pove or negave. Thu he benchmar prce he ame for he opon ha promed pove payoff a maury wh probably a cloed o a we wh and wh probably a cloed a we wh [5-7. Recall ha h ncorrec prcng conruced on he bae of he dea nown a elffnancng or no-arbrage. refly he elf-fnancng prcng cheme can be oulned a follow. The prce of a fnancal nrumen can be found ung andard ranacon. Fr borrow fund from he ban a he r free nere rae. Tha gong hor n bond and hen nveng hee fund n he nrumen. The rule of prcng he nrumen ha: he prce of he new nrumen hould have a oal balance equal o a any me n he fuure. Realzaon of uch dea n ochac eng can be acheved by pung he porfolo change over he me equal o. Th raegy repreen no arbrage. Th approach ha explc drawbac. If he nrumen r free hen he elffnancng raegy mae ene. In a ochac eng elf-fnancng cenaro auomacally mple deermnc nerpreaon of he mare prce. Indeed borrowng and nveng fund n a ry mare conan a r o loe nvemen. In anoher word he r of nveng n ochac mare could be decrbed a geng a reurn bellow han wa nally planned. In heory one can aume ha underlyng ecury drbuon and parameer are nown. Fxng he drbuon mgh lead o an opporuny o defne a generalzaon of he clacal arbrage. The ochac arbrage one when an nveor could hope o receve a acal advanage. everhele n fnance dffcul o exerce acal arbrage. Indeed heorecal drbuon ued for model n fnance are mpled and herefore acal arbrage for mpled drbuon alo mpled. From mahemacal pon of vew arbrage a neceary condon of a correc prcng. Tha f prcng correc hen arbrage could no ex bu here no arbrage ay beween ochac oc prce and deermnc bond prce. ow le u formally expre our remar. European call opon prce a oluon of he lac Schole equaon c r x c x 2 σ x c 2 x r c [ T x SE c T x max x K Here r free nere rae r volaly coeffcen σ and re prce K are aumed o be gven conan. The unque oluon of he problem SE adm probablc repreenaon n he form c x exp r T - E max S r T - K where he random proce S r v S r v ; x S r x v T a oluon of he ochac dfferenal equaon d S r r S r d σ S r d w 2

4 on a complee probably pace { Ω F P and w a Wener proce on h pace. ume ha opon underlyng ecury governed by he equaon d S µ µ S µ d σ S µ d w and µ r. The fr remar o he lac Schole prcng ha he oluon of he problem SE could no be called prce. Indeed he noon of he prce commonly aocaed wh he preen value PV of he fuure cah flow. If he cah flow rcly excee oher hen he po prce of he fr cah flow hould be greaer han he po prce of he econd. Oherwe here ex an arbrage. Th prncple can be expreed formally a he Equal Invemen Prncple EIP. Two nvemen opporune are called equal a he momen of me f hey prome equal nananeou rae of reurn. Two nvemen are equal on [ T f hey are equal a any momen of me over h me nerval. Thu equal nvemen are generaed by he equal cah flow and h mple equal po prce. ume ha < r < µ. Then a follow from comparon heorem for one dmenonal ochac dfferenal equaon SDE S r v ; x < S µ v ; x v T wh probably and herefore he cah flow generaed by he real µ-ecury greaer han he cah flow generaed by he r-ecury. Hence he preen value of wo cah flow are alo no equal. Remarably wo nvemen could have equal preen value bu hey could be no equal n accordng o EIP. Th remar how ha he benchmar preen value prncple ncomplee. The EIP wa ued earler n [7 9 for fxed ncome and opon valuaon. oong a he equaon SE one can probably noe ha underlyng ecure havng equal r characerc σ and promng arbrary expeced reurn - are prced by he ame amoun. far a he fuure cah flow of wo procee havng oppoe gn of he expeced reurn afy nequaly S - µ v ; x < S µ v ; x v < hen he preen value of he correponden cah flow over [ T are dfferen. Tha PV T S - µ ; x < PV T S µ ; x Tha mple ha he lac Schole equaon oluon could no be called he prce defned baed on he preen value prncple. Thu hough h funcon doe no preen arbrage n uually acceped ene neverhele h funcon alo can no be aocaed wh he prce noon and herefore canno be ued a he opon prce. There ex oher apec of he dervave prcng we wh o hghlgh here. For he recen generaon of he dervave called cred dervave he r neural eng em from he SE nerpreaon. oe ha ome reearcher are alo ued r neural eng for nere rae dervave prcng. Th problem relae o he probably heory and acually doe no relae o fnance. Th o-called r neural prcng. e u brefly oulne he eence of he problem. Uually reearcher referred o a r neural world when hey wan o emphaze ha n h world real µ-ecury would ranform no neuralzed r-ecury. Mahemacal echnque behnd h ranformaon are nown a Granov heorem. pplcaon h heorem o he random procee S µ S r ae ha meaure m µ m r correpondng hee procee are aboluely connuou o each oher and n parcular 3

5 where d m d m µ r ω 2 exp { λ [ w T w λ 2 T λ µ r σ Denoe meaure Q on he ame meaurable pace { Ω F wh he help of he equaly d mµ Q ω P dω d m Then he random proce S µ he oluon of he equaon d S r S d σ S d w Q wh he Wener proce w Q w λ on he probably pace { Ω F Q. One could noe ha he meaure Q depend on parameer λ and herefore depend on parameer µ. Moreover calculaon of he expecaon of a funconal gven on r neural world { Ω F Q auomacally conver o he correponden funconal wh repec o he meaure P accordng o he formula E Q f S r * ; x E f S µ * ; x I common o ae ha he oluon of he problem SE can be repreened a expeced value of he funconal over he r neural proce S r on he r neural world { Ω F Q. oe ha funconal a mahemacal erm ha cover varey payoff clae ued n dervave conrac. Tang no accoun he change of varable repreened n above equaly one can ee ha r neural world doe no perform he ranformaon of he real word ecury prce S µ no neuralzed prce S r on r neural world. Tha mean ha commonly aed he r neural eng fal o perform he a of he real world ranformaon of he parabolc equaon wh he real reurn o he parabolc equaon havng r free coeffcen a he fr order dervave wh repec o prce varable. On he oher hand eay o apply he proce S r deermned on he orgnal probably pace n order o preen he oluon of he parabolc equaon SE n he form repreened above. For cred dervave modelng no a mae arng on orgnal probably pace o mae Granov ranformaon and arrve a he r neural world. To follow h way one hould aume a ceran number of ecury nrumen ha compoe a ecury mare. oe ha h number a well a a parcular choce of hee ecure a ubecve heurcally noon. Choong he mare repreenaon one could aume dfferen ochac dynamc on he r neural world. everhele he prcng and accurae calculaon of he r characerc conneced o hee noon would dplay her dependence on ecury mare parameer. Tha n urn rrelevan and ncorrec. well a dffcul o reveal deal of underlyng calculaon loo le ha prmary ofware mply omed Granov r 4

6 deny and ued o-called neuralzed ecury ha of coure mahemacally ncorrec hough can preen cloe quanave or qualave reul f he r free and real reurn are cloed o each oher. CDS cred defaul wap an over-he-couner OTC blaeral fnancal nrumen ued for hedgng defaul r of a ry deb nrumen or a bae of ry deb nrumen. Defaul r a r when one or wo counerpare fal o mae a cheduled paymen. CDS one of he mo popular cla of cred dervave ha allow radng of he counerpary r from one o anoher whou changng ownerhp of an underlyng nrumen. CDS one of he mo lqud nrumen ype n whch one pary called proecon buyer ee for he cred proecon on ry deb uch a corporae bond or loan. Thee are referred o a o reference eny. On he oppoe de of he agreemen anoher pary ha agree o pay n he even of defaul of he reference eny. Th counerpary called a proecon eller. The proecon buyer pay o proecon eller a fee n he form of perodc fxed paymen uually pad emannually or quarerly. he even of defaul he fxed paymen op and he proecon eller pay o he proecon buyer he predeermned amoun. There are wo common way of he CDS elemen. Wh he phycal elemen he proecon buyer delver he defauled reference ecury or equvalen o he proecon eller and n exchange receve he par value of he reference ecury. Oher way he cah elemen. In h cae proecon buyer receve he dfference beween par and recovery value of he ecury. Th amoun called lo gven defaul. cually he phycal elemen more common way of delvery. The mo popular noonal value 5 or mllon dollar wh 5 year unl expraon. Mo of he CDS conrac are ngle name havng a ngle corporae bond underlyng. porfolo name wap wren on a bae of bond. For uch conrac mo popular he fr-o-defaul wap. I ermnae when he fr cred even happen or a he conrac expraon. Oher ype of he wap wren on a bae of he bond n-h o defaul n. Th a CDS conrac ha pay buyer of proecon he dfference beween face value and recovery rae a he n-h defaul even among he reference pool. One probably noed ha he nroduced CDS conrac more lely o he nurance conrac. There a relaonhp beween ry bond evaluaon and CDS prcng. In a ry coupon bond valuaon he face value and coupon paymen can be nerpreed a gven parameer of he expoure for CDS eller. The varable ry bond prce dynamc hen ubec o udy. The CDS valuaon he problem of fndng CDS pread ha he conan rae. Thu he CDS valuaon omewha adacen counerpar o he ry bond prcng. e u now oulne CDS heorecal framewor. We conder wo approache o he CDS valuaon. The fr valuaon approach a andard and baed on comparon preen value of he wo counerpare nvolved n CDS. The man dncon of our approach ha we ue ochac eng n conra o commonly acceped mehod dealng wh expeced value of he cah flow o counerpare. Then we alo conder anoher approach ha ue he opon prcng. oe ha opon-prcng mehod ued n [57 n any repec doe no relae o he lac Schole benchmar dervave prcng [24. ume fr for mplcy ha defaul of he corporae coupon bond occur only a a pecfed ere dae 2. Thee dae can or canno be he coupon paymen dae. e τ ω denoe a random me of defaul and D denoe he defaul even a he dae 5

7 defned on a probably pace { Ω F P. Thu D { ω : τ ω.... e T be a maury of he bond. oe ha alo poble o aume ha < T hough for mplcy we pu T. If a cenaro ω D hen a proecon buyer pay a fxed perodc premum $q a he dae -. We aume ha he premum q doe no be pad a he dae of defaul. On he oppoe de of he wap conrac he proecon eller would compenae he loe of he proecon buyer a he dae of defaul. Th compenaon value an amoun lo gven defaul. There are everal reaonable poble o defne he value of lo gven defaul GD. For nance can be defned a [ F { τ ω [ F R q T ; ω { τ ω [ F T R q T ; ω { τ ω Here R q T ; ω he value of he ry coupon bearng bond a he dae and maury T. The fr equaly adm ha he value of he ry bond a he defaul even be equal o recovery rae. Therefore GD defned a he promed face value mnu recovery. In he econd equaon he GD he dfferenal beween bond face value and value u before defaul. The la value correpond o he cae when defaul deb n eher form delvered o he proecon eller n exchange o he r free bond wh he equal face value and maure. oe ha h cae alo cover he cheape-o-delvery delvery opon a far a T could be ued a arbrary momen n he pa. Indeed le T ; denoe he r free bond prce a he dae ued a wh maury T. Pung T mn{ : T ; we noe ha he hrd equaly can be nerpreed a he cheape-o-delvery elemen. The preen value of he fuure paymen condered by defnon a he po value of he fuure paymen. Th reducon uually preened n he form n whch he fuure paymen mulpled by he zero coupon r-free bond havng maury a he pecfed dae n he fuure. Thu he dcouned value he eller po prce. The ere of he fuure paymen receved over a parcular me perod a coupon paymen wll accumulae by he bond buyer a he maury or a he defaul dae whch one come up earler. Thu h oher preen value can be deermned a a preen value of he oal balance a he dae ha he mnmum beween maury or defaul dae. Th preen value relae o he bond buyer po prce and by conrucon a random varable. On he oher hand he bond buyer can buy he bond and go hor on he ame amoun equal o dcouned coupon fuure paymen. In h cae ha loo le omewha unrealc one need o ue he benchmar preen value for he bond buyer valuaon. We conder uch cae a unrealc becaue n a wo pary deal he ecury eller a pary loong for fnancng whle ecury buyer repreen an nveor. Therefore n cae when he nveor buyng a ecury goe hor geng cah a mxed raegy combnng nveor and fnancer raegy. I mgh be reaonable way bu hould be more accuraely pecfed. In h paper we conder mare deal pare behavor n her common ene. Secury buyer are nveor nereed n geng maxmum reurn on her nvemen whle ecury eller are loong for fnancng. The CDS value by defnon he value of a perodc paymen q for whch he balance of he cah flow o counerpare equal o. Th value q can be alo nerpreed a a 6

8 coupon or premum bu commonly called alo CDS pread. Hence he CDS pread a fxed rae q durng he lfeme of he CDS conrac for whch he cah flow o eher counerpary. The preen value of he cah flow over he lfeme CDS o he proecon eller [ q - - F R T; ω τ ω T q [ - T τ ω T The oluon of he equaon a random varable q q b ω equal o q b [ τ ω [ F R - [ F R T; ω - T; ω τ ω τ ω τ ω T T - T 2 The value q b he exac oluon of he CDS prcng problem. oe ha from 2 follow n parcular ha for a cenaro ω aocaed wh -defaul even he value of he premum a hould be expeced. There no need o buy proecon for no defaul over he lfeme of he ry bond cenaro. Then for ω D { τ ω here only one erm n denomnaor and numeraor ha no equal o and n h cae q b ω τ ω [ F R T;ω and q b ω for ω { τ ω T. Summng up hee equale we arrve a he equaly 2. Remar. The benchmar CDS valuaon model [4 reduce he CDS pread noon o he brea-even value ha mae he expeced value of he cah flow o counerpare be equal. I eay o chec ha he expeced value of he exac oluon of a lnear algebrac equaon wh random coeffcen doe no concde wh he oluon of he problem n whch real random cah flow are replaced by he expecaon of hee cah flow. ede ha dealng wh expecaon of he cah flow one would lo he mare r expoure mpled by he random coeffcen. One can alo noe ha proecon buyer and proecon eller have aymmerc 7

9 expoure o he cred r. Indeed for no defaul cenaro proecon buyer pad perodc premum over he lfeme of he CDS wherea proecon eller doe no pay anyhng o he proecon buyer. The preen value of hee cah flow preened below by he fr erm of he equaly 3. If defaul occurred a he dae hen he proecon buyer would receve from proecon eller lo gven defaul amoun equal o F R T ; ω n exchange for he fxed perodc paymen q proecon buyer ha pad on dae <. For nance he preen value o he proecon buyer equal o f [ F R T; ω q - τ ω 3 q τ ω T The perodc coupon mpled by equaon 3 q ω [{ - [ F [ F - R R τ ω T ; ω T ; ω τ ω τ ω τ ω T 4 The r analy of he CDS conrac can be eablhed a follow. Recall ha he mare r of he proecon eller and proecon buyer are dfferen and can be deermned ung formula 2 4. Counerpary r depend on defaul me drbuon. There are varey aumpon regardng defaul me drbuon mgh be reaonable o apply ee for example [. In addon gven D he proecon buyer r a depend on fuure rae over perod [ < n formula 4. In hee formula we gnored proecon eller r of defaul on clam amoun a he dae of defaul. far a he eller and he buyer valuaon formula are dfferen hen mae ene o preen hee r formula eparaely. When wo counerpare come o he agreemen regardng premum value hen h value hould be appled o emae he mare r for eher pary of he conrac. Thu a elemen mare prce µ q on CDS conrac mple he r for boh counerpare. e µ q denoe a mare pread. Then mare r can be decrbed a follow. The proecon buyer r a probably ha he mare pread value µ q exceed q b ω.e. P { q b ω < µ q. Th probably repreen he meaure of he chance ha he mare prce exceed he exac conrac prce. Recall ha for each cenaro ω D he value R T ; ω defned by 8

10 recovery rae [. Th recovery rae can be found baed on formula 4. If reaonable o ue bnomal drbuon a an approxmaon of he me of defaul we can ee ha P{ τ p p P{ τ p Here he value - p denoe he probably of defaul. Ung hee formula one can eay calculae any acal characerc of he wap. Some CDS varaon. Though CDS conrac he mo popular cred dervave nrumen here are everal mporan varaon of he andard CDS on he cred mare. e u fr conder a conan maury defaul wap CMDS conrac. Goldman Sach fr nroduced h conrac n997. CMDS conrac almo dencal o he andard CDS. The prmary dfference relae o he premum leg of he conrac. Recall ha for he andard CDS premum leg a pecfed conan coupon pad perodcally o he proecon eller unl earler dae beween defaul or maury. For a CMDS conrac everal parameer hould be pecfed n advance. Thee are he lengh of he conan maury a equence of ree dae when he prevou mare pread replaced by he new on-he-run CDS mare pread wh he pecfed maury and a percenage facor ha would be appled for. Th percenage facor alo nown a a parcpaon rae. Thu gven avalable nformaon regardng CDS conrac he valuaon of CMDS conrac he problem of calculaon unnown percenage facor. Fr noe ha he proecon leg paymen he ame for he eher ype of he conrac CDS or CMDS and equal o [ F R T; ω τ ω modfcaon preened for CMDS conrac can be uded a follow. e be he bor Eurodollar rae a dae wh maury. I ha been defned a he mple nere rae for Eurodollar depo a wh maury. The cah flow from proecon buyer o he proecon eller for CDS and CMDS conrac are dfferen and can be wren n he form [ τ ω - - τ ω T q [ τ ω - q H - q H τ ω T p correpondngly. Here p unnown conan repreenng percenage facor and q H he po CDS pread a he dae wh maury H. Proecon eller receve paymen on cheduled dae pror o defaul or maury whch one come fr. Recall ha bor rae ued here a a dcoun facor. Tang no accoun ha he proecon leg he ame for eher CDS or CMDS conrac we remar ha he cah flow from proecon buyer hould be equal 9

11 for hee conrac oo. Thu wrng he equaly n whch he value of nflow equal o he value of ouflow from proecon buyer o he proecon eller a he dae of defaul we ee ha p ω q ω [ τ ω - q - H - - τ ω T q H 5 Remar. I mae ene o hghlgh a echncal problem ha relae o he dervave valuaon. Th a common for fnance valuaon pracce of he replacemen ochac cah flow by expecaon. Th reducon canno be alway acceped whou crcal remar. Someme for a parcular problem mgh mae ene. On he oher hand can have no ene. e u conder llurave example. e y be an unnown parameer and he preen value of cah flow o and from an nveor can be modeled [ y and 5 3. If we conder he equaon generaed by expeced cah flow we arrve a he oluon << y 5 3 On he oher hand he exac oluon of he problem y 5w w and expecaon doe no concde wh <<y. Th hghlgh he pon ha replacemen ochac flow by expecaon can lead o he crude problem oluon. Thu n he cae when a prcng problem adm an exac oluon advanced reducon of he problem o he expeced flow mgh be even mahemacally ncorrec. We alo llurae h pon of vew bellow. The equy defaul wap EDS wa launched by JP Morgan Chae n 23 hough he fr equy wap wa when moco Penon exchanged fxed rae on Japanee oc ndex exced n 99. a CDS benchmar EDS conrac exchange varable rae on a conan rae unl maury or a cred even whch one come fr. oc or a bae of oc could be a provder of he varable rae. For example a oc bae can be referred o a raded ndex or a compoed vrual ndex. Thu EDS reference on equy mare raher han o cred mare. Remar. Fr le u mae a commen relaed o he equy wap ES a conrac prcng wh zero chance of defaul. Such conrac fr ared o rade a he lae 8. The prcng model of he ES are well nown [3. The problem : gven ochac an equy prce S ω o derve a fxed rae R of he wap conrac. The benchmar formula ha wa developed ung elf-fnancng and no arbrage general prncple que mple. Followng [3 he ES pread value equal o

12 R n n Here T a Treaury bond prce a he me and maury T and T T $. Though n [3 wa remared ha urprngly he level of oc rrelevan n deermnng he value of wap he correc concluon wa no provded probably baed on wdely prevalen over fnancal communy fah n perfecon of he valuaon mehod. Thee mehod mgh be reaonable when he mare conued by ecure whch prce are ubec o elffnancng no arbrage valuaon. In ochac eng ecury prce S a gven random proce whch prce doe no governed by h rule mpoble o expec ha dervave would be governed by elf-fnancng and no arbrage prncple ha relae o he defaul free governmen reaure. In [3 wa alo noed ha he oc prce S doe effec on wap prcng relae o he mare praccal acvy bu a we ee ha doe no appeared n he above heorecal formula. Th an example ha how when common ene follow behnd he fah n he mehod. From our pon of vew he formula provded above ncorrec. Indeed he nroduced above formula ugge he ame fxed rae pread value for equy wap on dfferen oc for whch expeced reurn over a pecfed perod equal for example o 3% 5% % or -%. One alo can remar ha h formula doe no depend on volaly of he oc. The volaly of he oc can be equal or no. I obvou ha prcng mehod ha lead o he above formula for he fxed rae R ncorrec regardle of populary or mplcy. e u brefly oulne oher framewor of he ES prcng. In conra o elf-fnancng mehod we aume ha an nveor ha a needed volume of fund for nvemen. There are wo nvemen opporune nveng n fxed or varable leg. ume ha mare provde complee nformaon o mare parcpan. Tha an nveor free o chooe long or hor bae on reurn analy. mple approxmae formula for he fxed rae R can be receved f one equae he rae of reurn for boh de of he r free conrac over he perod [ T. I lead o he equaon S R $ $ S From whch follow ha R [ S S [ S S In h dervaon we dd no ae no accoun he equy wap he rule ha ae he only need amoun of he ranacon change hand. In h cae he opon prcng mehod a prece approach ha provde a correc reducon of he cah flow o a po prce. ellow we oulne he applcaon of he opon prcng approach o he ry wap prcng.

13 efore wrng general formula le u conder a mple numerc example ha llurae ypcal equy wap ranacon. Pu he noonal prncpal be equal o. Then e of ranacon can be pecfed by he able Dae 2 3 T Floang rae: S Fxed rae: R Tranacon value [6/2--2 [3/ [2/3--2 The calculaon how he cah flow o he holder of he varable equy rae. The varable rae of reurn exchanged for a fxed rae mulpled by he noonal prncpal. he dae here no cah change hand. Then a 2 amoun 25 gone from floang leg holder o he fxed rae holder and a he maury T amoun of goe from fxed rae holder o he counerpary. ow n general cae le u aume ha S a ochac proce. Denoe a counerpary ha receve a fxed rae and pay floang. Counerpary an oppoe leg: receve floang rae and pay fxed. Then he ochac cah flow from counerpare can be repreened n he form I ω S [ ω - S ω S R { ω - S ω S ω S ω R where { and he ymbol expree a funconal dependence on me. The cah flow o he I S ω [ R ω - S ω S { ω - S ω S ω S ω R The value of he wap defned by he need cah flow o and from counerpary a he conrac naon. Thu he problem o preen dae- reducon of he cah flow I and I. We ue opon prcng mehod nroduced n [5. Th approach conen wh he nvemen equaly defnon gven above. Followng [5 he call and pu European opon prcng equaon are ST S {ST K C T ST C S ST S {ST < K P T S T P S Here K a nown re prce and European call and pu payoff a expraon dae T are defned a 2

14 C T S T max{ S T - K P T S T max{ K - S T correpondngly. Th approach repreen oher opon prce defnon and doe no concde wh lac Schole prcng n wo maor ue. Fr h approach doe no advce he ame dervave prce for wo nrumen havng he ame volaly and dfferen expeced rae of reurn whn [ - µ µ where µ an arbrary conan. The econd ha doe no relevan o elf-fnancng prcng. Inveor have fund and hey mae a decon where o nve fund baed on expecaon of he fuure reurn regardle where he fund have been receved. In ochac mare he decon could lead an nveor eher o prof or lo. Selffnancng prcng approach neglec prof - lo and bnd prcng wh r free rae only. The oluon of he equaon are C S P S S ST S ST C T ST {ST K P T ST {ST < K EO e u conder paymen I ω I ω generaed by he exchange floang rae S / S for fxed rae R. Thu ung opon prcng oluon we enable o produce he value of he cah flow a a follow I ω S S ω S ω S ω ω [ S S ω ω R S { S ω ω R I ω S S ω S ω S ω ω [ R S S ω ω S { S ω ω < R oe ha cenaro n whch ndcaor conan equaly gn can be omed a far a he correpondng erm n he um equal o. The value R for whch he rgh hand de of he above formula are equal repreen a oluon of he equy wap prcng problem. Th oluon of he problem can be wren n a mple compac form. Indeed he equaly of he wo cah flow a reul I ω I ω Ung equaly { Q x - { Q x we ee ha he fxed rae a random varable equal o 3

15 R ω - S [ S - S S ω ω ω ω [ - S S ω ω 6 Th he defnon of he fxed rae of he equy wap. On he oher hand when counerpare agree abou a parcular prce hey are ubec o r. For example le counerpare agreed o apply a value <R a a conracual fxed rae of he wap. For example h value can be aocaed wh he expecaon of he R ω. The value of he r pulaed by a chance ha he real world rae of reurn o counerpare for he choen fxed rae value bellow ha mpled by he exac value R ω. e <R be he fxed rae for he equy wap. Then he wap value hen by defnon equal o I S ω I [ S ω S R S ω ω 7 Formula 6 and 7 repreen a oluon of he equy wap prcng aumng chance of defaul. ow le u conder he equy defaul wap EDS. Fr we clarfy he ue of he erm. Recall ep of he bond valuaon. We fr perform r free valuaon formula. Then we defned ry bond prce [ and hen CDS conrac repreen a co of he defaul proecon of he bond. Here we u defned he prce of he -defaul wap. Followng bond program he econd ep hould be a prcng of a ry wap.e. he wap ha adm defaul. The la ep would be a prcng of he proecon fxed rae. The laer ep could be named a cred defaul wap over he underlyng ry wap. We begn wh he prcng of he ry wap. The cah flow o counerpare are cheduled a he dae 2 can be wren n he form I I { τ ω S ω - S τ ω S { S ω { τ ω ω - S ω R S ω d { τ ω ω 2 I 2 < R S [ [ d R d I ω - S S ω ω { τ ω T S S ω ω - ω R ω S d ω Here he ymbol { denoe ndcaor of he cenaro ha ranacon ae place a he dae. Thee formula how ha a he cheduled equence of he ree dae counerpare exchange her rae. receve fxed and pay equy floang rae wherea 4

16 receve floang and pay fxed rae. he defaul here are no ranacon. oe ha poble o add o hee formula he erm ha cover recovery paymen from defauled de n exchange for full or a poron of paymen from proecon eller. In h cae would be a rlaeral cred conrac. pplyng he opon prcng mehod ued above for ranacon for every cenaro ω { ω : τ ω 2 and hen ummng hem up we arrve a he formula for wap fxed rae and wap value R I d ω - 2 [ - {τ ω τ ω T R I S S { [ τ ω T ω ω - 2 [ I S S ω Here expreon R ω and I - I are defned above n formula 6 7 for he r free equy wap. Thu he ochac pread R Rd pulaed by he pobly of defaul. In he rucural approach defaul defned a he fr momen of me when company oc value n bellow of a ceran fracon q of he nal prce. In h cae defaul me would be defned a τ ω mn { : S / S < q {τ ω T Hence f defaul occurred durng he lfeme of he equy wap hen he rae q a hrehold whch eparae defaul from no defaul perod. Gven a drbuon of he random proce S * one can fnd an approprae approxmaon of he defaul me drbuon. ow le u conder valuaon of he premum whch proecon buyer hould pay o proecon eller n order o receve a complee compenaon n he cae of defaul. Fr we need o pecfy a reaonable value of clam a he defaul even. If q he defaul barrer hen a clam amoun n he dcree me eng could be defned a F [ Q - S τ ω / S 9 where F a noonal prncpal and Q Q q nally pecfed rae ha can be eher a conan uch a poon of R or a pecfed funcon dependng on. For example Q can be repreened by a parcular ndex value. Cred defaul wap on a ry equy wap a value of a fxed rae premum ha hould be pad perodcally by proecon buyer o proecon eller. Proecon buyer can be a recever of he floang rae n EDS or no. The prcng problem o derve h premum n exchange for proecon 9 delvered by proecon eller a defaul. Th problem cloe o CDS valuaon problem. e R R denoe premum value from proecon buyer and S ω S I ω ω {τ ω T R ω {τ ω 8 5

17 proecon eller perpecve correpondngly. When he cred even comng he proecon eller pay amoun 9 o he proecon buyer. The me of defaul can be wren n he dcree me eng n he form τ ω { τ ω S { S q S { S q If ω { τ ω hen proecon buyer mae regular paymen FR on he cheduled ree dae. On he equy leg proecon eller pay he amoun 9 a he dae τ ω for any ω for whch τ ω < T. If here no cred even over he lfeme of he EDS hen proecon eller accumulae he cheduled paymen unl he wap expraon and doe no pay defaul compenaon. Th reaonng lead o he equaon {τ ω FR {τ ω T FR {τ ω F [ Q S S { τ ω T From whch follow ha R ω {τ ω - {τ ω - [ Q S S [ Q {τ ω T {τ ω S S On he oher hand compare he paymen from proecon buyer perpecve one can eay fgure ou ha R ω {τ ω {τ ω - - [ Q S S {τ ω T - 6

18 S [ Q - S { τ ω Remar. In formula f we deal wh expeced value hen he la erm on he lef-hand de would be non-zero erm. Denoe <<R a eller premum when ochac cah flow replaced by expeced value. Then premum a deermnc conan equal o << R P{τ S E {τ [ Q S P{τ T Tha doe no concde wh expeced value of he expecaon of he rae R ω. n ae-wap conrac a conrac ha ranform he dfference beween po prce and he face value of he ry bond n a ere of fuure paymen n whch each paymen repreen perodcally adued ondon Inerban Offered Rae IOR plu a conan pread. The ae-wap mare an mporan egmen of he Cred Dervave Mare aached o IOR rae whch uually nerpreed a ry rae. e company ell o a counerpary Y company Z ry zero-coupon bond for par a dae and hen ener o he nere rae wap IRS payng fxed rae $c o he counerpary Y. Denoe < R T he mare prce of he ry -coupon bond a. If here no defaul durng he lfeme of he bond he company Y would receve he face value of he bond $ a he bond maury T. On he oher hand on he predeermned dae he IRS fxed paymen of $c would be pad by o Y. In reurn Y pay o varable IOR rae plu a pread. Conder he cae when cred even mgh occur only when underlyng bond defaul. Th approxmaon mgh be realc f defaul r of he company Z more gnfcan han he -Y counerpary r. everhele n general boh de of he conrac can defaul on a parcular ranacon and poble o udy h general cae oo. earng n mnd decrbed above rucure he correpondng ae wap ranacon can be formally deermned a follow. e R T ; ω denoe he ry coupon bond prce a wh maury T. Company ell ry bond for < R T o he company Y a he dae. The r of he deal geng lower reurn han mpled by he prce < R T. Th r dcued n deal n [. Thu he cah flow o he company Y c c { for a cenaro ω for whch τ ω 2. If τ ω T hen 7

19 8 oe ha he face value $ receved by he company Y f here no defaul of he bond unl T can alo be pad by he company Z hough we udy he cae when pad by. The cah flow from he company Y o could be preened a follow e u conder a more general when nveor pay a nown recovery proecon of - o Y a he defaul dae. In order o avod arbrage opporuny he value of he cah flow o and from he counerpary Y hould be equal and herefore Here he lef hand de of h equaon repreen cah flow pad o and rgh hand de pad by counerpary Y. There are wo way o dcoun fuure paymen. One way aached o US Treaury rae and oher aached o he IOR rae. There alo a pobly o ue pecfed or a conan maury ample. e D T denoe a dcoun facor over [ T. Then he pread value from he counerpary Y perpecve equal o T. τ f... dae a c c 2... τ f... dae a c c he dae a T R < < < { c c T { { γ γ T {τ { γ γ {τ { T R { { γ γ [ {τ { γ γ T { T [ {τ - - < < <

20 9 τ { D T R D c D ω T τ { D T R D c T D - { < < 2 3. oe ha mare daa of he pread mple r for eher counerpary. Th r value expreed by probably of recevng reurn le han mpled by he mare daa. In conra o he pread formula preened above one can apply reducon of he cah flow o he fuure dae τ T mn { τ T and hen calculae preen value. Then Hence - 2 c D D τ T c D T D T τ c D D τ c T c D T D T τ c c D τ D c T c D [ T D T τ c c D τ D

21 τ T D T D - T c and herefore ω 2 τ D D D - D c - τ T D D D - T D c - T oe ha IOR rae ued by floang leg of he nere rae wap could be expreed n dfferen way. For example a he dae he bor rae l ; H one can apply for he nex perod [ where H T a fxed maury. bor rae could be aumed n h formula eher ochac or deermnc. Wh a oal reurn wap TRS wo counerpare and Y exchange her cah flow. The lfeme of he TRS conrac pulaed by a ry bond ued by a hrd pary company Z. e u aume ha bond mgh defaul only a he momen 2. If here no defaul durng he lfeme of he TRS conrac hen he face value of he bond of $F pad by he company o Y a he bond maury T. If defaul occurred a hen pay o Y a he dae 2 * <R T bond prce a he dae ; * a pecfed coupon paymen $c ; * [ R T ; ω R - T ; ω { R T ; ω R T ; ω ; * he recovery rae - a defaul dae. In reurn f defaul occurred a company Y pay o a he dae 2 * he bor rae - pecfed by he prevou me perod plu pread ; * [ R T ; ω R T ; ω { R T ; ω < R T ; ω ; * he par value of he bond $F a T f here no defaul. Fr noe ha a reference oblgaon ha can be any ae mple a r. Th a r ha mare prce of he reference oblgaon doe no concde wh he fuure cah flow mple by h prce. oe ha mlar o ae wap underlyng r can be condered eparaely [. We begn wh he andard PV prcng emae ued uually for conrucon an approxmae oluon of he valuaon problem. Seng o be equal he preen value of he cah flow o and from Y we arrve a he equaly 2

22 < R R T F D T T; ω R τ { D T; ω τ T D - c D [ c oe ha f defaul occur a hen by defnon we aumed ha Y pad he noonal and receved recovery rae and here no oher ranacon for h cenaro. From equaon follow ha TRS pread equal o ω - τ D τ - D < R - D FD T τ T [ c T R D T; ω R T; ω 2 τ T D D c We arrve a he more accurae formula of he TRS f we replace he erm D on D T D - T on he rgh hand de 2 The nex a wap hybrd conrac nown a a cred-lned noe C. Th a funded ype of cred dervave whch ynhecally combne wo nrumen a corporae bond and a andard CDS. feme of he C ued by a company Y defned by he lfeme of he corporae bond. Th bond aumed o be ued by a company Z. If here no defaul of he company Z bond hen he company Y mae a fxed perodc coupon paymen o a C buyer a pecfed paymen dae and a prncpal C a he bond maury T. If a cred even uch a defaul or banrupcy doe occur before C maury T hen he C conrac ermnaed. The nex coupon paymen are no pad and C holder wll receve value on defauled bond. long wh C he company Y ener no CDS deal. Though can be done before or laer he C ue dae we uppoe for mplcy ha he naon dae of C and CDS are he ame. Wh CDS conrac he company Y a proecon eller who wll pay a lo gven defaul a he dae of defaul n exchange for perodc premum unl defaul and recovery rae a he defaul dae. Th recovery rae goe o C nveor and CDS premum paed on o he C nveor o ncreae yeld on Z company noe. The C holder delver eher he bond of he company Z or he amoun repreenng he mare value of he defauled bond o he C uer Y and n reurn Y pay lo gven defaul o. Recall ha hee ype of elemen are nown a phycal and cah correpondngly. 2

23 e Q T ; ω denoe he C prce a he dae. Then ung ynhec prcng one can pl Q T ; ω prce no wo componen. The PV of he one componen he cah flow from C uer Y o he C buyer τ [ c τ T [ T c Oher cah flow em from CDS ranacon. The CDS cah componen for counerpary Y are recovery rae pad by Y o he CDS proecon buyer. In exchange Y receve defauled bond or he equvalen cah a he dae of defaul and fxed rae coupon from CDS buyer. Thu he PV of he CDS ranacon τ { [ - CDS τ T CDS If here no defaul he C uer would pay C coupon and receve CDS perodc paymen CDS. Thu approxmaon of he C prce repreened by he PV reducon τ { - [ CDS c 3 τ T { [ CDS c T From whch follow ha c CDS τ - τ T T Th repreenaon of he C pread baed on PV reducon. dmng a parcular defaul me drbuon eay o preen calculaon of he expeced value of he C premum a well a r characerc. Tang no accoun 2 4 we can perform more accuraely repreenaon of he C prce 22

24 τ { - [ τ T [ - [ T CDS - T CDS c - c 3' oe ha n h adumen he paymen made by he counerpary Y are dcouned by he andard PV reducon whle receved paymen fr are ummng up a he earle of maury or defaul me ung fuure rae. Then h cumulave um dcouned o he dae. The correponden formula for C coupon can be wren n he form c τ [ - - CDS - τ T T - T CDS e u conder a C prcng bearng n mnd on effec of he counerpary r. aed on a erm of CDS conrac and a value of he lo gven defaul clear ha he proecon eller credworhne hould be ang no accoun for more accurae prcng. Thu aume ha proecon eller mgh alo defaul on proecon ranacon. Denoe D p τ ω a defaul even when proecon eller fal on delvery recovery $ F R τ ω T ; ω o he proecon buyer a he dae of defaul. Therefore f underlyng ecury defaul a.e. ω { τ ω here ex a chance D p ha proecon eller fal o fulfll CDS oblgaon. Subcrp p here and for he proecon eller. Thu F R T ; ω { τ ω p F R T ; ω { τ ω D p F R T ; ω { τ ω [ - D p Here p denoe a recovery rae on defaul of he ranacon lo gven defaul from he proecon eller o he proecon buyer. Subuon of he rgh hand de of h equaly n 2 and 4 lead o he formula refnemen ha ae no accoun proecon eller cred r. In h cae for a cenaro ω { τ ω D p 2.. proecon buyer lo 23

25 p F R T;ω {τ ω D p The correponded value of he pread wll be reduced a follow q c b [ F - R - T ; ω [ p D p {τ ω 2' Thu he pread deermned by he proecon buyer fxed rae regular paymen wll be reduced a far a here ex a chance ha proecon eller wll no pay he proecon paymen n full. The correponden eller expoure wll alo reduced and equal o c [ F R T ; ω q [ p D p { τ ω - 4' oe ha for calculaon of he mean or varance of he random varable 2 and 4 we need o now he on condonal drbuon of he random vecor R T ; ω D p condonng on { τ ω. The even D p can be aumed o be ndependen on he even { τ ω a lea for he fr order approxmaon neverhele he drbuon R T ; ω n general hould correlae wh he even { τ ω. Therefore for an accurae modelng one need a realc aumpon regardng h condonal drbuon. ow le u conder he connuou me CDS conrac. ume ha coupon pad on he fxed dae 2. Then he balance equaon 3 hould be adued ang no accoun connuou drbuon of he defaul even. earng n mnd accrual nere and a pobly o defaul of he proecon eller on delvery lo gven defaul occurred a he dae equaly 3 can be rewren n he form lm { τ ω [ { [ F R T; ω ε n [ p D q [ - [ R ; ω R ; ω q τ ω T 24

26 Here < < 2 < < n a ub-paron of he nerval [ and ε. Denoe d ; ω he dcoun nere rae of he ry bond over he nerval [. Then he erm q [ R ; ω - R ; ω q [ d ; ω - d ; ω - / 36 he accrued nere unnown a he dae. oe ha f defaul occurred exacly a hen from h formula follow ha accrued nere wll no be pad. If he chance ha defaul occur a naon dae doe no equal o hen he erm equal o F - R T { τ ω mu be added o he lef hand de of he above formula. In h cae loo reaonable o pay he fr coupon paymen $q a. In he lm wa aumed ha proecon elemen occurred mmedaely afer defaul alhough for oher varaon h formula can be ealy adued. The pread formula can be receved from he above equaon. Indeed q lm ε n n [ F R {τ ω [ T; ω [ { - p [ R D p ; ω - R { τ ω ; ω {τ ω T τ ω [ F {τ ω [ R τ ω T; ω [ { [ - p D R τ ω p { τ ω T ; ω - R ; ω Here n denomnaor on he rgh hand de of he above equaly he erm ha correpond o no defaul cenaro can be omed a far a he value of he numeraor for uch cenaro equal o. Indeed for no defaul cenaro no reaonable for he buyer o pay a proecon. Floang rae ry bond. e u brefly oulne r free conrac called floang rae bond. ede he populary of h conrac alo mporan wh applcaon o he valuaon of he floang leg of he nere rae wap. The valuaon mehod nroduced below follow [9. 25

27 Conder an nerval [ T and le < < < T be he nere rae ree dae and aume ha he ep ε - doe no depend on. e be he floang rae a whch appled over he perod [. For wrng mplcy aume ha noonal prncpal $. Oherwe he value of ranacon hould be proporonally changed. The floang nere rae ha would be appled for regular paymen from he conrac buyer o eller are repreened n he able bellow Dae 2 T Floang flow - ε ε - - ε oong a h able one can ee ha one-dollar a dae equal o Hence n parcular $ $[ - T T $ - 2 $ T [ - T $ [ - 2 Therefore he cumulave cah flow o he bond buyer over he me perod [ T can be calculaed n he bacward of me arng from he dae T o. I yeld $ $ 2 2 $ T [ - T $ $ 2 2 $ - [ - 2 $ [ $ Thee calculaon prove ha $ nveed n he r free ecury a he dae generae a floang rae cah flow. Thu from bond eller perpecve he bond buyer payng $ a wll receve equvalen cah paymen over he perod T. The varably of he nere rae here doe no affec h valuaon. Tha Dae 2 T Floang flow - ε - ε - [ ε Th floang rae bond valuaon ha ued for he preen value reducon n order o ufy he prcng model. Thu floang bond eller receve $ a he dae o. Inveng and payng coupon - a 2 - and a he bond maury T he amoun of would exhaued he upfron fundng of $. oe ha h conrucon acually doe no depend on T and ε and herefore can be appled for arbrary T and ε. On he oher hand bond buyer emae he fuure value of he conrac ung formula FlT T [ 26

28 Th formula preen he dae-t value of he floang bond paymen and he ε-compoundng nere rae formula hould be appled for he emaon of he bond value. To avod arbrage over [ T one hould expec ha he floang bond and he -coupon bond ued by he ame Governmen hould provde he ame rae of reurn. Tha mple n parcular ha The oluon of h equaon Fl T / Fl / T Fl T [ T Rae are unnown a he dae herefore mgh mae ene o nerpre hee unnown a a equence of he random varable. ume ha underlyng o he floang bond conrac a ry bond. Denoe F λ T λ { ; 2 he cah flow generaed by he equence of paymen pad a 2 and - pad a T. Then by defnon $ F λ T for any λ and T regardle wheher he rae are random or deermnc. eller of he ry floang bond would pay λ-ree floang nere rae paymen unl defaul or maury whch one come fr. ume ha a he defaul dae he bond eller would pay a pecfed rao <. In reurn a floang bond nveor pay $ a naon of he conrac. The floang bond valuaon problem o derve he value of he upfron premum $ gven he nonrandom recovery rae and a drbuon of he defaul me. ond buyer pay upfron $ and receve from he bond eller he cah flow equal o τ [ τ T τ [ F λ τ T [ F λ T - [ F λ τ τ < T Thu upfron premum value ω [ F λ τ τ < T In a CDS conrac wren on a floang rae bond a proecon buyer buy a proecon ha would cover poble defaul loe. e recovery rae of he ry floang bond and defaul occur a he dae hen a proecon eller hould rembure a he dae or a he nex dae he lo -. On he oher hand he proecon buyer would pay a fxed premum unl earle beween he dae of defaul or maury T. ume ha dae of defaul and he recovery rae. Then he lo λ occurred a he dae equal o 27

29 λ $[ $ $ $ Thu proecon eller paymen o proecon buyer can be repreened by he lo funcon n he form of lo gven defaul λ λ τ τ τ T On he oher de of he conrac cah flow o proecon eller from proecon buyer q [ - τ τ T oe ha by defnon we pu q ω for ω { ω : τ ω. I follow from he fac ha f defaul occur mmedaely afer conrac naon and herefore coupon wll no be pad here no reaon o defne coupon value. From la wo equale follow ha premum a random varable equal o q ω 2 τ τ T τ T - τ 4 The erm τ T n denomnaor can be omed a far a for uch cenaro numeraor equal o. oe n parcular ha momen of he pread can be wren n compac form - n n n E ω P τ 5 where n 2. Remar. In conemporary cred dervave reearch commonly acceped n floang leg calculaon replacng random cah flow by expecaon. I eem mporan o noe ha he value of he pread receved wh uch popular approach doe no concde wh he fr momen of he exac oluon. Indeed he pread value ha follow from equal expeced cah flow o and from a counerpary of he CDS wren on he ry floang bond conrac << q - 2 P τ - P τ T P τ T 28

30 The addonal erm P τ T can be eher mall or large mang h approxmaon laer cae crude and baed. Counerpary R of he Inere Rae Swap. Follow [9 le u fr recall a valuaon model of nere rae wap IRS wh chance of defaul. andard IR wap a wo pary conrac. The counerpary mae fxed emannual or quarerly paymen o counerpary. The magnude of each fxed paymen uually a pre-pecfed percen of he noonal prncpal. In reurn counerpary pay floang rae paymen o. ll paymen are made n he ame currency and only need amoun pad. e < < < T be ree dae q and l * * denoe a fxed and a floang IOR nere rae correpondngly and $ he noonal prncpal. The fxed flow lne n he able below repreen he cheduled paymen hould be made by counerpary o and he floang lne he cheduled paymen of he counerpary o. Dae 2 T Fxed flow q q q Floang flow l ε l ε l - - ε where ε -. Recall ha only need paymen are pad. If he noonal prncpal $F hen all enre n lne hould be mulpled by he F o preen he real cah ream. e u recall ome mporan pon of he wap valuaon. The domec r free rae uually ue for calculaon. If for a parcular cenaro q l hen he paymen of q - l would made by o a. Th real world amoun would be held unl maury unnown a. Thu cumulave fuure value a he IRS maury pad o counerpary and are [ q [ l l q l q q < l T T 6 correpondngly. I alo mgh have ene o replace he fuure value by he PV reducon. The PR wap rae by defnon he rae q for whch he preen value of he all fxed de paymen equal o he preen value of floang paymen. In anoher word a value q for whch he value of he wap a. Mulplyng boh expreon n 6 by he ame facor T we arrve a 29

31 q l T T 7 Defnon. The value of he wap u T he dfference beween fxed and floang de value a he dae u [ T. Though he preen value fuure ll a benchmar concep n ae prcng hould be clear ha preen an approxmaon of he general Equal Invemen Prncpal prcng concep formulaed above. Moreover along wh r neuralzaon concep defne opon prce ncorrecly. I wa demonraed n deal n [5-7. e u very brefly llurae he prmary defcency of he Preen Value prcng. e < < 2 T be he dae of rade. ume for nance ha a dae he r free nere rae erm rucure 4.% T 4.2% and T T. In h cae wo counerpare are ubec o he mare r. Th r ha he bond buyer loe expeced reurn f occurred ha bond exerced a. Th mple example how necey of he ochac modelng of he fuure nere rae. ow we apply opon-prcng approach for he r free IRS valuaon. We wll employ he opon valuaon mehod [5-7 for he wap valuaon. Recall ha he cah flow o he counerpare and can be repreened n he form C P λ λ [ l [q q {l l {l q < q 8 where l l - λ { 2. We nerpre he value l a a varable poron of he dollar value a. Funcon C T P T n 8 repreen a dcree me cah flow n whch each erm on he rgh hand de are payoff of he call and pu opon correpondngly on varable nere l wh he re q. We apply he opon prcng formula follow [5-7. e u recall he opon prce defnon. The European call and pu opon prce C x P x a dae on underlyng ecury S u u and S x wh re prce K and maury T defned above by EO. Th defnon lead o C P λ λ l l l l [ l [ q q { l l { l q < q 9 3

32 Thee formula preen he value of dervave nrumen nown a floorle and caple conrac. Hence he wap value by defnon he dfference λ λ C P l l [ l q 2 ow he formula 7 can be nerpreed a an approxmaon of he wap pread value preened by he equaon 9. Indeed we ee ha lef-hand de n 6 7 conan all paymen o and from pary. ddonal dffcule n preenng acal calculaon are he floang forward fuure rae l l. Hypohecal log-normal mpled drbuon commonly ued for opon valuaon. far a horcal daa avalable he acal e can provde quanave lelhood of he mpled drbuon. If he wap value gven hen fxed leg rae q equal o q l l l Pung n h formula we oban he value q ha repreen he fxed rae CDS pread ha equae floang paymen. ow we conder he cae when one or wo counerpare of he IRS are ubec o cred r. e u aume fr ha floang rae payer ubec o defaul. e he only dae 2 can be he dae of defaul. The pary mgh defaul before or afer he dae he underlyng bond defaul. The defaul of he counerpary mean ha fal o delver he proecon amoun [ l q o. ume ha recovery rae of he pary mpled by rang a conan <. If pary defaul a pay o counerpary he fracon of he amoun due [ l - q. Thu he cah flow from o gven ha defaul pror o C λ { [ τ ω τ ω τ ω τ ω [ [ l q {τ ω {τ ω [ l q Here τ ω denoe he defaul me of he underlyng ecury. oe h me of defaul τ ω and τ ω can be correlaed. Th cah flow above ha wo componen. One when he defaul of underlyng ecury comng before he defaul of. The econd componen of he cah flow correpond o cenaro when defaul earler han underlyng ecury. Our a o preen fxed rae of he wap. Th value hould be found a naon dae. correpondng reducon ha could be ued he opon valuaon mehod. Gven ω { ω : τ ω here 3

33 are wo muually excluve cenaro { ω : τ ω and { ω : τ ω for whch we can apply formula 9. Then C { [ τ τ τ τ τ - l l τ [ l q l l l [ l q q l q 2 τ T τ T C λ where he value of C λ gven by 9. oe ha he cah flow from o a he dae gven ha mgh defaul P { [ τ τ τ τ - l l [ q l q l τ τ l l [ q l q l The wap value a a random varable dependng on a cenaro and me. Th value equal o S C - P The random value of q q ω for whch S he defaul wap rae. The formula for he counerpary ry IRS wap rae q can be preened n analyc form. Indeed ang no accoun equaly l - q l q - q - l l < q l - q one can eay fgure ou ha he oluon of he problem C P can be wren n he form 32