Valuation of Credit Default Swap with Counterparty Default Risk by Structural Model *

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1 Appled Mathematcs 6-7 do:436/am Publshed Ole Jauary ( Valuato of Credt Default Swap wth Couterparty Default Rsk by Structural Model * Abstract J Lag * Peg Zhou Yujg Zhou Jume Ma 34 Departmet of Mathematcs ogj Uversty Shagha Cha Delotte ouche omatsu CPA Ltd Shagha Cha 3 Departmet of Appled Mathematcs Shagha Uversty of Face ad Ecoomcs Shagha Cha 4 Departmet of Appled Mathematcs ogj Uversty Shagha Cha E-mal: lag_j@togjeduc Receved Jue 9 ; revsed November 8 ; accepted November 3 hs paper provdes a methodology for valug a credt default swap (CDS) wth cosderg a couterparty default rsk Usg a structural framework we study the correlato of the referece etty ad the couterparty through the jot dstrbuto of them he default evet dscussed our model s assocated to whether the mmum value of the compaes stochastc processes has reached ther thresholds (default barrers) he jot probablty of mmums of correlated Browa motos solves the backward Kolmogorov equato whch s a two dmesoal partal dfferetal equato A closed prcg formula s obtaed Numercal methodology parameter aalyss ad calculato examples are mplemeted Keywords: CDS Spread Couterparty Default Rsk Structural Model PDE Method Mote Carlo Calculato Itroducto A valla credt default swap (CDS) s a kd of surace agast credt rsk he buyer of the CDS s the buyer of protecto who pays a fxed fee or premum to the seller of protecto for a perod of tme If a certa pre-specfed credt evet occurs the seller pays compesato to the buyer he credt evet ca be a bakruptcy of a compay called the referece etty or a default of a bod or other debt ssued by the referece etty I ths paper the credt evet also cludes the default of the protecto seller If there s o credt evet occurs durg the term of the swap the buyer cotues to pay the premum utl the CDS maturty A facal sttuto may use a CDS to trasfer credt rsk of a rsky asset whle cotues to reta the legal owershp of the asset As the rapd growth of the credt default swap market credt default swaps o referece etty are more actvely traded tha bods ssued by the referece ettes here are two prmary types of models of default rsk the lterature: structural models ad reduced form (or *hs work s supported by Natoal Basc Research Program of Cha (973 Program)7CB8493 testy) models A structural model uses the evoluto of a frm s structural varables such as a asset ad debt values to determe the tme of a default Merto s model [] s cosdered as the frst structural model I Merto s model t s assumed that a compay has a very smple captal structure where ts debt has a face value of D ad maturty of tme provdes a zero coupo Merto shows that the compay s equty ca be regarded as a Europea call opto o ts asset wth a strke prce of D ad maturty of A default occurs at f the opto s ot exercsed he secod approach wth the structural framework was troduced by Black-Cox [] ad Logstaff-Schwartz [3] I ths approach a default occurs as soo as the frm s asset value falls below a certa level I cotrast to the Merto approach the default ca occur at ay tme Zhou [45] produces a aalytc result for the default correlato betwee two frms by ths model Usg ths model credt spread wth jump s cosdered by Zhou [6] Reduced form models do ot care the relato betwee default ad frm value a explct maer I cotrast to structural models the tme of default testy models s ot determed va the value of the frm but the Copyrght ScRes

2 J LIANG E AL 7 frst jump of a exogeously gve jump process he parameters goverg the default hazard rate are ferred from market data hese models ca corporate correlatos betwee defaults by allowg hazard rates to be stochastc ad correlated wth macroecoomc varables Duffe-Sgleto [78] ad Lado [9] provde examples of research followg ths approach here have bee may works o the prcg of credt default swaps Hull - Whte [] frst cosdered the valuato of a valla credt default swap whe there s o couterparty default rsk her methodology s a twostage procedure he frst stage s to calculate the default probabltes at dfferet future tmes from the yelds o bods ssued by the referece etty he secod stage s to calculate the preset value of both the expected future payoff ad expected future paymets o the CDS hey exteded ther study to the stuato where there s possblty of couterparty default rsk ad obtaed a prcg formula wth Mote Carlo smulato [] hey argued that f the default correlato betwee the protecto seller ad the referece etty s postve the default of the couterparty wll result a postve replacemet cost for the protecto buyer Affecto of the correlato o a CDS prcg remas terestg he valuato of the credt default swap s based o computg the jot default probablty of the referece etty ad the couterparty (protecto seller) echcally t s dffcult because correlato betwee the ettes volved the cotract s hard to deal wth Jarrow ad Yldrm [] obtaed a closed form valuato formula for a CDS based o reduced form approach wth correlated credt rsk I ther model the default testy s assumed to be lear the short terest rate Jarrow ad Yu [3] also assumed a ter-depedet default structure that avoded loopg default ad smplfed the payoff structure where the seller s compesato was oly made at the maturty of the swap hey dscovered that a CDS may be sgfcatly overprced f the couterparty default probablty was gored Yu [4] costructed the default processes from depedet ad detcally dstrbuted expoetal radom varables usg the total hazard approach He obtaed a aalytc expresso of the jot dstrbuto of default tmes whe there were two or three frms hs model Leug ad Kwok [5] cosdered the valuato of a CDS wth couterparty rsk usg a cotago model I ther model f oe frm defaults the default testy of aother party wll crease hey cosdered a more realstc scearo whch the compesato paymet upo default of the referece etty was made at the ed of the settlemet perod after default hey also exteded ther model to the three-frm stuato More studes o dfferet kds of CDS such as a basket referece ettes ca be foud from ex [ 5-3] I ths paper we develop a partal dfferetal equato (PDE) procedure for valug a credt default swap wth couterparty default rsk I our model a default evet s supposed to occur at most oe tme whch meas ether referece etty or couterparty may default oce Our work s based o the structural framework where the default evet s assocated to whether the mmum value of stochastc processes (value of the compaes) have reached ther thresholds (default barrers) Usually we choose the compaes lablty as the thresholds [4] We show that the jot probablty of mmums of correlated Browa motos solves the backward Kolmogorov equato whch s a two dmesoal PDE wth cross dervatve term hs equato ca be solved as a summato of Bessel ad Sturm-Louvlle egefuctos he defaultable CDS studed ths paper same as Hull- Whte s s a specal case More complcated features of that kd of CDS are ot cosdered he paper s orgazed as follows I Secto we preset a CDS spread expresso I Secto 3 we establsh a partal dfferetal equato model whch solves the jot probablty dstrbuto of two correlated compaes used Secto uder the some assumptos We obta a explct soluto for ths PDE he ma result of the closed form of the prcg the CDS the follows ad shows ths secto Numercal calculato example tests ad parameter aalyss for our model are collected Secto 4 We coclude the paper Secto 5 CDS Spread wth Couterparty Default Rsk I ths secto frst we aalyze how to value a CDS wth couterparty default rsk Assume that party A holds a corporate bod wth otoal prcpal of $ o seek surace agast the default rsk of the bod ssuer (referece etty B) party A (CDS protecto buyer) eters a CDS cotract ad makes a seres of fxed perodc paymets of the CDS premum to party C (CDS protecto seller) utl the maturty or utl the credt evet occurs I exchage party C promses to compesate party A for ts loss f the credt evet occurs he amout of ths compesato s usually the otoal prcpal of the bod multpled by ( R) where R s the recovery rate as a percetage of the otoal Durg the lfe tme of the CDS a rsk-free terest s appled Assume that the default evet the rsk-free terest rate ad the recovery rate are mutually depedet Defe for the credt default swap Copyrght ScRes

3 8 J LIANG E AL : Maturty of the credt default swap; R: Recovery rate o referece oblgato; r : Rsk eutral terest rate; w : otal paymets per year made by the CDS buyer (party A) per $ of otoal prcpal; : t Rsk eutral probablty desty of default by the referece etty ad o default by the couterparty; : t Rsk eutral probablty desty of default by the couterparty ad o default by the referece etty A valla CDS cotract usually specfes two potetal cash flow streams - a fxed premum leg ad a cotget leg O the premum leg sde the buyer of protecto makes a seres of fxed perodc paymets of the CDS premum utl maturty or utl a credt evet occurs O the cotget leg sde the protecto seller makes a sgle paymet the case of the credt evet he value of the CDS cotract to the protecto buyer at ay gve pot of tme s the dfferece betwee the expected preset value of the cotget leg whch s the protecto buyer expects to receve ad that of the fxed leg whch he expects to pay or Value of CDS = E PV cotget leg () E PV fxed premum leg Smlar to the valla CDS we assume that the paymets are made at dates t < t < < t = Let t be the tme terval betwee paymets dates the the paymet made every tme s w t I practce the paymets are usually made quarterly therefore t =5 he CDS paymets cease whe ether the referece etty or the couterparty defaults If a credt evet occurs at tme < deote the paymets dates precsely before ad after the default tme by t ad t Whe ths credt evet occurs exactly at oe of the paymets dates let t he we have t < t Frst we aalyze the fxed premum leg sde As we assumed the credt evet occurs at most oce hat s there are three cases as follows Case the credt evet s that the referece etty defaults at tme he the preset value of all paymets s = rt rt w te w t e := wa we Case the credt evet s that the couterparty defaults at tme he there s o fal accrued paymet ad the preset value of all paymets s w te = wa( ) = rt Case 3 ether the referece etty or the couterparty defaults pror to maturty tme hs tme the preset value of the paymets s wa Usg the default probablty destes of t ad t the total expected preset value of the premum leg s w a e a d wa O the cotget leg sde f the referece etty defaults at tme the preset value of the payoff form the CDS s gve by r R e where s the lqudato perod he expected payoff s t r R e d Accordg to () the value of the CDS at tme t s t R e r d w a e a d wa he value of the swap at orgato must be equal to zero he CDS spread s s the value of w whch makes the value of the CDS equal to zero hus r Re d a e a d a s = () he varable s s referred as the credt default swap spread or the CDS spread It s the total of the paymets per year as a percetage of the otoal prcpal I expresso () the jot probablty destes are stll ukow We wll focus o how to obta these probablty destes followg sectos 3 Modellg ad Soluto I ths secto we preset several mathematcal theorems whch are ecessary for the valuato of the CDS wth the couterparty default rsk I order to descrbe the correlato betwee the referece etty ad the couterparty we study the jot probablty dstrbuto fuctos of the mmum values of two correlated Browa motos he followg are Basc Assumptos for our model ) Iterest rate r s costat; ) Frm s asset value V t follows a geometrc Browa process wth costat drft r ad volatlty uder the rsk eutral measuremet Copyrght ScRes

4 J LIANG E AL 9 dv t V t where cov dw t dw t = dt wth beg a costat; = rdt dw t = 3) Frm defaults as soo as ts asset value V () t reaches the default barrer D I ths paper we use the Black-Cox type default barrer whch s D t = t Fe where F ad are gve costats respectvely (see []); 4) he credt evet occurs at most oce ad a credt evet oc- D F m =l =l V V curs whe X reaches m Defe the rug mmum of X X = X s m ts by I order to get the probablty desty eeded () defe := Prob < X m Xt x Xt x hus u x x u x x t X m > = = t s the probablty of the evet that 3 Default Probablty X defaults (e X reaches m ) ad X does ot default tll tme Our ma theorem dsplays the probablty dstrbuto fuctos of the extreme values of two V t ake X =l t t e V the X = ad correlated Browa motos he probablty destes of t ad t ca be obtaed drectly from dx t = dt dw t u x x Lemma he jot probablty (3) satsfes backward where = r he default barrers chage to Kolmogorov equato u u u u u u u = = t x x x x xx x x tm m (4) um x t= x x tm m ux m t= x tm ux x = x x m m where m m Proof Usg Itô's formula (see ex [5]) deote = = = u X X t u x x t t X t X t t u u u u u u s x x xx t u t u dw dw x x s s x x Assume that u s the soluto of backward Kolmogorov equato (4) so u u u u u = s x x x u u x xx = he t ux t Xt t= ux x u dws x ad ds x (3) t u dw s (5) Eu X X t u x x Defe the frst passage tme t t = (6) =f s X s m X s > m s Let t = we fd (6) s = u x x Eu X t Xt t Copyrght ScRes

5 J LIANG E AL s s ; s s = u m X s p m X s; x x ds u X m s p X m s x x ds m m u p ; x x d d where ; p X t Xt t x x s the trasto probablty of beg at state Xt X t at tme t gve that t starts at x x at tme Notce here the above equato s also held for ux x t < t < t f oly chage the low lmt of the tegrato Because of the boudary ad fal-tme codtos (4) we get ux x = EuX t Xt t = pm X ; s s x x ds Accordg to the defto of pm Xs s; x x defed at the ed of (7) ux x s the probablty defed (3) at tme Now let us solve PDE (4) Frst we make the followg trasformato to elmate the drft terms Let = t ad where ax axb u x x = e p x x a = a = (7) b= aa p x x satsfes p p p p = x x x x am axb pm x = e (8) px m = px x = he Next we elmate the cross-partal dervatve ad ormalze the Browa motos by a sutable trasformato of coordates ths dea was troduced by He etc ([6] Defe ew coordates z ad z as the followg satsfes z z x m x m = x = m he qzx x zx x = px x (9) () q q q = z z ql = ql e qz z = ama z m b = where L = z z z = L = z z z = z () Because the boudary codtos are more coveetly expressed polar coordates we troduce r correspodg to z z as z r = z z ta = () z thus ad obta q q q q = r r r r qr = qr e qr = ama rs m b q r satsfes = Defe a ew fucto f r ama rs m b = e the gr = qr f r solves g g g g = bf r r r r r gr = gr = gr = f r (3) (4) (5) I order to solve PDE (5) we cosder the Gree s fucto Gr ; r of ths problem whch satsfes G G G G = (6) r r r r Gr = Gr = Gr = rr Lemma he soluto of PDE (6) s r Gr ; r = Copyrght ScRes

6 J LIANG E AL r r rr e I = s s (7) Proof We try to fd separable solutos to ths equato the form of G r = M r (8) Pluggg (8) to (6) we fd that M M M M r = r r r r Dvde the prevous equato by M r we fd M M M = = M r r r r r (9) Sce the left sde of (9) s a fucto of ad the rght sde s a fucto of r ad so t must be a costat Deote ths costat by ad we have = Ke O the other had M r satsfes equato M M M M = r r r r () M r = M r = hs s a Sturm-Louvlle problem We try to fd separable solutos the form of M r = RrΘ Pluggg ths to (6) we get R R r r r = Θ () R R Θ wth boudary codtos Let RrΘ = RrΘ = = the Θ Θ k Θ solves Θ k Θ = () Θ = Θ = It s easy to see that Θ = Ask Bcos k Cosderg the boudary codtos we have B = ad As k = Because Θ s o-zero soluto we kow that A ad k = = hus the egefuctos cosstet wth the boudary codtos are Θ = C s = Fally cosder the radal part of the soluto whch satsfes rr rr r k R= Rr Deotg = r we get the stadard form of Bessel s equato d R dr d d k R = he well kow fudametal solutos of ths Bessel's equato s ad J x = k = x Γ k Γ k J p xcosp J px k x= lm pk s p Y k dverges ad we requre Y Y Sce R to be bouded the soluto k y s ot permtted Hece the geeral radal part of the soluto s R r Jk r Sum up R rθ he = over we have = M r R r Θ = = C J rs = = G r M r = = KC e J rs Because K s a costat we ca defe A = KC Itegral the prevous equato over we obta the geeral soluto to PDE (6) for Gr as = s G r A e J r d = (3) Now we try to fd the coeffcet A whch ft the tal codto Gr = rr Mul- tply the prevous equato at = by s m ad tegrate over we fd Copyrght ScRes

7 J LIANG E AL m r rs = A d m e Jm r (4) Notcg the completeess relato of xj d = s ax Js bx x a b a rj r ad tegrate over r multply equato (4) by m r m Am= e s Jm r Pluggg ths expresso to Gr (3) we get r Gr = m e s s J rj rd = (5) Usg the fact [7] that a b c x ab 4c s s s xe J ax J bx dx = e I c c (5) ca be smplfed to (7) Wth Gree s fucto Gr ; r ad the boudary ad tal codtos the soluto of PDE (5) ca be expressed as = ; F Gr ; r f r drd q r d G r r bf r dr d where F (6) F = r r < (7) f r e he soluto of PDE (3) s q r = g r f r ( ama ( rs m ) b) = (8) Returg to the orgal coordates ad varables x x t we get ax axb t axa x b t = axa x b t u x x t = e p x x t e q z z t = e q r t (9) where z z r are defed (9) () ad () hat s we have heorem he soluto of the tal boudary problem (4) has a closed form soluto (9) assocated by (6) (4) ad (7) By ow we have already obtaed the probablty of that compay defaults ad compay does ot defaults betwee tme t ad Chage t to to t here comes the probablty betwee tme ad t I order to obta the probablty vx x t of that compay defaults ad compay does ot default we oly eed to chage the postos of the parameters of the two compaes such as F V u x x t Now apply our result to the spread Formula () whe = I the valuato formula of () what we eed are the default probablty desty fuctos of ad whle we oly have the probablty fuctos herefore we eed to modfy () I fact otce that a ad e are pecewse cotuous fuctos ad o every pece r r a = e = t e = e re Itegrate the umerator ad deomator of () by parts ad we have where r r r umerator = R e d = R e r e d t = t t = deomator = a e d = a d a t a ed a d a a e t = t = t r t e re d t a a = t u ad v = = (3) for u ad v are solved ths subsecto 3 Survval Probablty π( xxt ) o calculate the credt default swap spread s we stll eed to study the jot survval probablty of x x t for x > m x > m Copyrght ScRes

8 J LIANG E AL 3 x x t Same as heorem the probablty of x x t =Prob X > m X > m X t = x X t = x the soluto of PDE where m m ad = = t x x x x xx x x tm m m x t= x tm x m t= x tm x x = x xm m I the prevous secto we get the solutos to these two PDEs the doma m m where m m Compare the boudary ad fal codtos of PDE (3) (3) (33) the soluto to (3) ca be wrtte as the lear combato of the other two u u u u u u u = = t x x x x xx um x t= ux m t= ux x = v v v v v v v = = t x x x x xx vm x t= vx m t= vx x = x x t ux x t vx x t = (34) = r r s (3) (3) (33) Set t = we get the probablty of whch was defed Secto x x = = (35) hus the CDS spread () ca be rewrtte as R e r e d s = t t r t a e t e red a t t a Remark It s a specal case of our model that the CDS wth the couterparty default whe the correlato of the couterparty ad referece etty are depedet Remark he same method ca be used to the prcg the CDS for a basket referece ettes I ths case the PDE model s smpler as the boudares codto are all equal to So that t has o problem caused by the sgularty ear () However f the basket has a bg umber of referece ettes the closed form soluto of the PDE s dffcult to be obtaed 33 Ma Result (36) Combe the prevous two subsecto we obta the all probabltes requred Formula (36) herefore we obta the ma theorem of ths paper preseted as follows: heorem (ma theorem) Uder the Basc Assumpto ()-(4) the credt default swap spread wth couterparty default rsk s gve by (36) where the formula the probablty are gve by (3) solvg the problem (4); are solved as the same way; s gve by (35) 4 Numercal Aalyss So far we have derved the three probabltes Secto Wth these we ca calculate the CDS spread by (9) Eve though we have a so called closed or sem-closed form soluto but the calculato of the form s stll ot trval he expresso of the form cludes tegrato Copyrght ScRes

9 4 J LIANG E AL ad fte seral as well as a specal fucto he drect calculato s ot easy to udertake ad the result s usually ot satsfyg hs because that the value of the tegrad cocetrates a very small area ad ths area s movg as the chage of the tme t So that the dfferece approxmato geeral wll make the result value very small Here we troduce a algorthm of Mote Carlo method to evaluate the form (9) It souds that there s o dfferece from the oe to calculate CDS spread by drect Mote Carlo method however t s really dfferet wth ad wthout the closed form soluto We wll see t the later Usg Mote Carlo to calculate the closed form fact we oly eed to kow how to calculate the frst tegrato of the Formula (6) he steps to do t are as follows: ) Represetg the tegrato wth the excepto of the tegrad ake A r f r s cf r e as a de- sty fucto where A = c s a costat such that A r c d f r e dr d = s F where f r whch s o-egatve as s defed (8) By smple calculato we obta ba A s c = am a m b e e Now rewrte the tegrato E s measured respect to f : ; d G r r bf r dr d F b A r s = E Gr ; r e c where G s defed (5) ) Radom umbers fetchg I our case the three-dmesoal radom X Y Z has a jot desty f r We sample ths radom varable X Y Z from U 3 for = the followg way: a) Frst the margal desty fucto respect to r s f r = f r d d a A e a A r s = s ad the margal dstrbuto fucto s r a A us ( )= s F r a A e du aars = e he geerate uform radom umber U ad set X = lu a A s b) Secodly for gve X = r the codtoal desty fucto b f r b e f r= = b f r e he margal desty fucto respect to s f r= f rd = ad ts dstrbuto fucto s u F r= du = he geerate uform radom umber U ad set Y = U c) hrdly the jot margal dstrbuto fucto wth respect to r ad s f r = f r d aa rs = a A s e the for gve X = r Y = the codtoal desb f r be ty fucto f 3 r = = ad b f r e b e ts dstrbuto fucto s F 3 r = b e he geerate uform radom umber U 3 ad set b Z = l e U3 b herefore we geerate the th radom sample X Y Z wth desty fucto f r 3) By the method above obta three-dmesoal radom sample X Y Z the replace r ad put t to the tegrad b G r ; r s e A r c For = repeat the process tmes (eg = as requred) the fd the mea value to fd approxmated the expectato It may argue that f use Mote Carlo method why just smulate drectly o the orgal Formula ()? he Fgure ca aswer ths questo Cosder practcal examples Assumg there are two compaes B ad C wth tal values of V B =$7 mllo ad V C = $ mllo; volatltes of them are B = = C = = 3 respectvely; recover rate R =3; correlato =7; = ad the default barrers are $4 mllo ad $6 mllo respectvely Copyrght ScRes

10 J LIANG E AL 5 I Fgure method meas the CDS spread s obtaed by smulatg drectly o the orgal Formula () method meas the CDS spread s calculated by our closed form soluto wth the tegral evaluated by Mote Carlo method he method s repeated tmes usg computer tme secod whle the method s repeated tmes usg computer tme 787 secod We ca see that the calculato by our soluto coverges much faster tha drectly smulate the orgal formula As less as / tmes the result of the method s much better tha the method Now use the closed form soluto by Mote Carlo smulate tmes to calculate the tegral we ca aalyss the parameters of R ad respectvely he other parameters are chose as above he left fgure ad the rght oe show the mpact of correlato coeffcet maturty tme ad recovery rate R o CDS spread wo fgures Fgure show ther relatoshp I the upper fgure of Fgure CDS spreads are greater for swaps wth loger maturtes he lower oe llustrates the extet to whch CDS spreads deped o the recovery rate Whe the recovery rate becomes larger the payoff wll get smaller Hece the CDS spread s gettg smaller whe recovery rate gettg larger Both of them show that the spread goes dow as the correlato goes up Fgure 3 cofrms that CDS spread creases wth expred tme ad decreases wth recover rate R whe the correlato s fxed Fgure 4 show what kd of the rules for the volatltes of the two compaes he behavors of them affect to the CDS spread dfferet way Suppose that the other parameters are fxed If the volatlty of the Compay B s larger whch meas the probablty of the default goes larger as well t results that the CDS spread s more expasve O the other had f the volatlty of the Compay C s larger whch meas the probablty of the falure of the CDS payoff s larger t results CDS spread s cheaper Fgure CDS spread wth couterparty rsk vs correlato ρ varyg (upper) ad R (lower) Fgure 3 CDS spread wth couterparty rsk vs tme varyg R Fgure 5 s a three-dmesoal surface of the value for the probablty of uv B V C =5 respect to V V ad B 5 Coclusos C Fgure CDS spread wth couterparty rsk by two methods I ths paper we have troduced a PDE methodology for modelg default correlatos We assume that the value of compaes follow correlated geometrc browa motos Whe the asset value of a compay reaches a predefed barrer a credt evet called default occurs Copyrght ScRes

11 6 J LIANG E AL ca be exteded to the valuato of ay credt dervatve whe the payoff s based o defaults by two compaes he shortage of the model s lmted by the dmeso t s dffcult to exted the method to a basket CDS wth a large portfolo 6 Ackowledgemets he authors would lke to express the thaks to Prof Lshag Jag for the helpful dscussos ad suggestos 7 Refereces Fgure 4 CDS spread wth couterparty rsk vs tme varyg σ (upper) ad σ (lower) Fgure 5 s a three-dmesoal surface of the value for the probablty of uv B V C =5 respect to V B ad V B he essetal part s to derve the jot default probablty as the soluto to a partal dfferetal equato hs soluto s more computatoally effcet tha tradtoal smulato for orgal formula or lattce techques to the equato We appled the default probabltes solved from the PDE to the valuato of credt default swaps wth couterparty default rsk he model [] R Merto O the Valug of Corporate Debt: he Rsk Structure of Iterest Rates Joural of Face Vol 9 No 974 pp do:37/97884 [] F Black ad J Cox Valug Corporate Securtes: Some Effects of Bod Ideture Provsos Joural of Face Vol 3 No 976 pp do:37/ 3667 [3] F Logstaff ad E Schwartz A Smple Approach to Valug Rsky Fxed ad Floatg Rate Debt Joural of Face Vol 5 No pp do:37/ 3988 [4] C Zhou A Jump-Dffuso Approach to Modelg Credt Rsk ad Valug Defaultable Securtes Face ad Ecoomcs Dscusso Seres Workg Paper Board of Goverors of the Federal Reserve System Washgto DC 997 [5] C Zhou A Aalyss of Default Correlato ad Multple Defaults Revew of Face Studes Vol 4 No pp do:93/rfs/4555 [6] C Zhou he erm Structure of Credt Spreads wth Jump Rsk Joural of Bakg & Face Vol 5 No pp 5-4 do:6/s () 68- [7] D Duffe ad K J Sgleto Modelg erm Structures of Defaultable Bods Revew of Facal Studes Vol No pp do:93/rfs/4 687 [8] D Duffe ad K J Sgleto Credt Rsk Prceto Uversty Press Prceto 3 [9] D Lado O Cox Processes ad Credt Rsky Securtes Revew of Dervatves Research Vol No pp 99- do:7/bf5333 [] J Hull ad A Whte Valug Credt Default Swaps I: No Couterparty Default Rsk Joural of Dervatves Vol 8 No pp 9-4 do:395/jod 395 [] J Hull ad A Whte Valug Credt Default Swaps II: Modelg Default Correlatos Joural of Dervatves Vol 8 No 3 pp - do:395/jod 3953 [] R Jarrow ad Y Yldrm A Smple Model for Valug Default Swaps whe Both Market ad Credt Rsk are Copyrght ScRes

12 J LIANG E AL 7 Correlated Joural of Fxed Icome Vol No 4 pp 7-9 do:395/jf3938 [3] R Jarrow ad F Yu Couterparty Rsk ad the Prcg of Defaultable Securtes Joural of Face Vol 56 No 5 pp do:/ [4] F Yu Correlated Defaults ad the Valuato of Defaultable Securtes Proceedgs of d Iteratoal Coferece o Credt Rsk Motréal 5-6 Aprl 4 pp -3 [5] S Y Leug ad Y K Kwok Credt Default Swap Valuato wth Couterparty Rsk he Kyoto Ecoomc Revew Vol 74 No 5 pp 5-45 [6] R Jarrow ad S M urbull Prcg Dervatves o Facal Securtes Subject to Credt Rsk Joural of Face Vol 5 No 995 pp do:37/ 3939 [7] J Hull ad A Whte Valuato of a CDO ad th to Default CDS wthout Mote Carlo Smulato Joural of Dervatves Vol No 4 pp 8-3 do: 395/jod [8] J Hull M Predescu ad A Whte he Relatoshp betwee Credt Default Swap Spreads Bod Yelds ad Credt Ratg Aoucemets Joural of Bakg & Face Vol 8 No 4 pp do: 6/jjbakf46 [9] J Hull M Predescu ad A Whte he Valuato of Correlato-Depedet Credt Dervatves Usg a Structural Model Joural of Credt Rsk Vol 6 No 3 pp 99-3 [] M Kjma ad Y Muromach Credt Evets ad the Valuato of Credt Dervatves of Basket ype Revew of Dervatves Research Vol 4 No pp do:3/a: [] M Kjma ad Y Muromach Valuato of a Credt Swap of the Basket ype Revew of Dervatves Research Vol 4 No pp 8-97 do:3/a: [] M Wse ad V Bhasal Correlated Radom Walks ad the Jot Survval Probablty Workg Paper Caltech ad PIMCO pp -3 [3] P Zhou ad J Lag Aalyss of Credt Default Swap Appled Mathematcs-JCU Vol 7 pp 3-34 [4] K Gesecke ad L R Goldberg he Market Prce of Credt Rsk Workg Paper Corell Uversty New York 3 pp -9 [5] J Hull Optos Futures ad Other Dervatves 7th Edto Pretce Educato Upper Saddle Rver 9 [6] H He P W Kerstead ad J Rebholz Double Lookbacks Mathematcal Face Vol 8 No pp -8 do:/ [7] I S Gradshtey ad I M Ryzhk able of Itegrals Seres ad Products Academc Press New York 98 Copyrght ScRes