Assocaon for Informaon Sysems AIS Elecronc brary (AISe) WICEB 13 Proceedngs Wuhan Inernaonal Conference on e-busness Summer 5-5-13 Prcng Model of Cred Defaul Swap Based on Jump-Dffuson Process and Volaly wh Markov Regme Shf u Xanghua School of Fnance, Zhongnan Unversy of Economcs and aw, xhlu4@16.com Xao Xuepng School of Fnance, Zhongnan Unversy of Economcs and aw, xxao.13@163.com Follow hs and addonal works a: hp://asel.asne.org/whceb13 Recommended Caon Xanghua, u and Xuepng, Xao, "Prcng Model of Cred Defaul Swap Based on Jump-Dffuson Process and Volaly wh Markov Regme Shf" (13). WICEB 13 Proceedngs. 35. hp://asel.asne.org/whceb13/35 Ths maeral s brough o you by he Wuhan Inernaonal Conference on e-busness a AIS Elecronc brary (AISe). I has been acceped for ncluson n WICEB 13 Proceedngs by an auhorzed admnsraor of AIS Elecronc brary (AISe). For more nformaon, please conac elbrary@asne.org.
The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track 611 Prcng Model of Cred Defaul Swap Based on Jump-Dffuson Process and Volaly wh Markov Regme Shf u Xanghua, Xao Xuepng School of Fnance, Zhongnan Unversy of Economcs and aw, Chna Absrac: By nroducng he Jump-Dffuson Process and Markov Regme Shf, he paper explores Mone Carlo smulaon o examne he prcng problem of sngle name Cred Defaul Swaps (CDS), whch he prce of CDS s affeced by boh unpredcable dosyncrac rsk and sysem rsk caused by he macroeconomc change. The sudy shows ha he prce of CDS ncreases as he nensy and he amplude of he Jump-Dffuson Process ncrease. Furhermore, he CDS prce depends on he nal sae and ranson nensy of he volaly of he corporae value, whch he former can reflec he nfluence of macroeconomc suaon. Keywords: Cred Defaul Swap, Jump-Dffuson Process, Markov Regme Shf, Mone Carlo Smulaon 1. INTRODUCTION Cred Defaul Swap (CDS) s one of he smples and wdely used cred dervaves, whch he buyer of CDS pays he fees o he seller regularly o acqure proecon from defaul. When he cred even occurs, he seller has o compensae he buyer for losses. In recen years, CDS develops rapdly and has become one of he mos popular cred dervaves. In November 1, Chnese CDS-- Cred Rsk Mgaon (CRM) was pu forward n ner-bank marke. The CRM wll has far-rangng appled space and appled oulook n rsk managemen feld n he fuure. The prmary problem of cred dervaves s prcng. There are wo knds of models o prce cred dervaves: srucural model and reduced model. Srucural model uses srucural varables such as change of asse value or deb value o model defaul. Wh srong economc background, hs model provdes endogenous defaul probably and recovery rae. Unlke srucural model, reduced model assumes ha defaul even canno be predced. Defaul even s one ump of he exogenous shock. The nensy of ump-dffuson, namely he defaul nensy, can be mpled from marke daa. Respondng o he defaul probably whn he srucural framework, we ry o develop a modfed model o prce CDS. The frs srucural model s pu forward by Meron (1974) whch closed-form soluons for defaul probably and he prce of cred dervaves can be acheved based on he Black-Scholes opon prcng heory. Meron (1976) develops opon prcng when sock prce follows a ump-dffuson process. Black and Cox (1976) modfy assumpons of Meron (1974) s Model o allow defaul ahead of secury maury, whch ndcaes ha shor-erm cred spreads s closed o zero and defaul even can be predced. Zhou(1) comes up wh a new model based on ump-dffuson process whch solves he problem of defaul predcably o some exen and obans posve shor-erm cred spreads. The newes sudy on srucural framework s he nroducon of Markov regme shf no prcng of cred dervaves. Elzalde (7) assumes ha defaul even s relaed o he exernal cred rang and he change of cred rang can be vewed as one regme shf of cred rang sae when prcng CDS. ackbarh, Mao and Morellec (6) nroduce a Markov regme shf for company s cash flow o dscuss he nfluence of busness cycle on company s cash flow and defaul polcy. Chen (1) develops a srucural model and obans closedform soluons for endogenous defaul probably and losses under he opnon ha perodc change of expeced Correspondng auhor. Emal: xhlu4@16.com (u Xanghua), xxao.13@163.com(xao Xuepng)
61 The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track economc growh, economc nsably and rsk premum affec company s fnancng and defaul polcy. Based on all hese research achevemens, hs paper res o apply he ump-dffuson and Markov regme shf o model he nfluence of macro-economc condons on he volaly of frm value. Under hs condon, we compue company s defaul probably and use drecly o prce sngle name CDS. When modelng he volaly of asse value of he frm, mos exsng papers assume ha s unchangeable whch may no be approprae for he long erm corporae secures. The volaly of asse prce s affeced by company s fnancng and sraegy polces. Durng a long perod, such as 1 years, company managemen wll adus s fnancng and sraegy polces accordng o he change of exernal economy. Thus, s reasonable o assume ha he volaly of asse prce depends on he change of economc suaon. The nfluence of busness suaon can be vewed as sysemac rsk whch all companes face. I causes homogeneous, lasng and neludble nfluences on he whole marke. Excep for sysemac rsk, here s parcular rsk ha one sngle frm has o face, whch we can call unsysemac rsk or specfc rsk. Ths knd of rsk can be descrbed as unpredcable and unconrollable shock o companes, whch s relaed o ndusry, geographc locaon and s own operaons. Therefore, respondng o company s specfc rsk, we nroduce he ump-dffuson process no he model and he change of company s specfc rsk can be vewed as a ump of exogenous Jump-Dffuson. we consder and dsngush company s sysemac and specfc rsk, assumng asse prce follows ump-dffuson process and s volaly has Markov regme shf. Under hs condon, we nspec he change of CDS prce and defaul probably n dfferen rsk sae o acheve more precse CDS prce.. TE MODE.1 Prcng prncple of CDS The cash flow of CDS consss of wo pars. One par s he expeced paymens ha he CDS buyer makes when he defaul even occurs or does no occur. The oher par s he expeced compensaon ha he CDS buyer obans when he defaul even occurs. Suppose ha he fee rae of CDS, s,as a fracon of noonal n Bass Pon per year s pad a daes 1 < < < n = T. ( 1, ) denoes me nerval beween paymen dae 1 and. When he reference eny does no defaul, he expeced paymens ha CDS buyers make can be expressed as equaon (1), where F s noonal prncpal. p% ( ) s he defaul probably before under rsk neural measure. B (, ) s rsk-free dscoun facor. n ( 1, ) 1 %( ) (, ) @ (1) A = F s p B s = 1 When he defaul even occurs, he paymen made by CDS buyer can be defned as: n ( 1, ) B = F s p( ) p( 1) B(, ) s M = 1 % % @ () where s assumed ha he defaul beween he regular paymens always occur exacly n he mddle. The error form hs approxmaon ges smaller as he me sep ges smaller. When he defaul even occurs, he conngen leg payoff, C, can be descrbed as n C = ( 1 R) p % ( ) p % ( 1) B(, ) (3) = 1 A he momen he conrac s sgned, accordng o no-arbrage prcng prncple, he cash flow of he CDS should be zero. Tha s, Therefore, he CDS prce s C = A+ B = s ( + M) (4) C s = + M (5)
The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track 613. Model assumpons and dervaon Assumpon 1: When he volaly of asse prce locaes n regme σ ( =, ), he dynamcs of V s gven by he followng ump-dffuson process dv V = ( µ λν) d + σ dz + ( Π 1) dy =, (6) Where µ denoes he drf rae n dfferen regme, λ s he nensy of ump dffuson process and dz s a sandard Brownan moon. dy s a Posson process wh nensy parameer λ, and Π s he ump amplude wh expeced value equal o ν + 1. We assume ha Π s an..d log-normal random varable, such ha ( µ σ ) ln Π :, (7) N Ths assumpon mples ha σ ν = E [ Π 1] = exp( µ + ) 1 (8) Assumpon : Inal sae of σ can be observed and durng he whole perod, σ a me + 1 only depends on s value a me.tha s Where ( σ + 1 σ, σ 1,, σ ) ( σ σ ) P = = = = = P = = = p (9) 1 + 1 p s he ranson probably form sae o sae durng me nerval. Then he regme shf of σ can be descrbed as a Markov Chan wh wo saes. The ranson probably marx s defned as follows p p P = p p (1) Parcllarly, f he ranson follows Posson process, whn very small me nerval p, whprob.exp( λ ) : 1 λ = 1, whprob.1 exp( λ ) : λ,we have Where λ, =, denoes he rae of leavng sae. e l denoe he me spen n sae,hen he exponenal law holds: λ Pl ( ) > = e, =, (1) 1 In addon, he expeced duraon of regme s ( λ) and he average fracon of me spen n ha regme s λ ( ) 1 λ + λ. Assumpon 3: There exss a posve hreshold value K for he reference eny a whch fnancal dsress occurs. The frm connues o operae and o be able o mee s conracual oblgaons as long as V (11) > K. Oherwse, defauls on all of s oblgaons mmedaely and some form of corporae resrucurng akes place. Defne X = V / K as he rao beween frm asse value and hreshold value. The defaul even occurs when X 1. Assumpon 1 and he defnon ha X = V / K yeld mmedaely: dx / X = ( µ λν ) d + σdz + ( Π 1) dy =, (13) Under rsk neural probably measure, usng he Io emma, we have dln X = ( r σ / λν) d+ σdz+ lnπ dy =, (14) Under no arbrage condon, we know ha he prce of a dervave secury sasfes he paral dfferenal equaon (15) n one sngle sae.
614 The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track ( ) ( ).5 σ XGXX + r λν XGX rg+ λ E GX ( Π, T) G XT, = GT =, (15) where GX, GXX mean he frs and he second paral dervaves respecvely. e τ represen he me when a defaul occurs. Mahemacally, τ = nf { X 1, > } (16) Whou regard o he regme shf, Abrahams(1986) and Zhou 1 pon ou, he closed-form soluon for equaon (15) does no exs, neher nor he defaul probably. Zhou 1 and Wang and Chen(3) provde he closed-form soluon for a smplfed model where s assumed ha he defaul only occurs a maury dae T. Namely,. Under hs condon, he defaul probably can be wren as τ = T ( λt)( λt) ln (.5 ) exp X + r σ λν T + µ Q FT ( 1 X) = Q( XT 1) = N (17) =! σ T + σ Where Q denoes rsk neural probably measure. Because here s no general closed-form soluon for he equaon (15) and he correspondng defaul probably, we urn o use Mone Carlo smulaon o compue he frm defaul probably and he correspondng CDS prce under ump-dffuson process ha s prce volaly has he propery of Markov regme shf. e = T n, where T denoes he maury dae, n denoes he oal number of me nervals. Accordng o equaon (14), we have ln X ln X, = x 1 + y = 1,,, n (18) where x : N r σ λν T n, σ T / n (19) (( ) ) σ shf s sae wh he probably presened before. : N ( µ, σ ) (), wh prob.1 λ T / n y = (1) 1, wh prob. λ T / n Equaon (1) s smlar o equaon (11). Boh of hem are he smplfed form of Posson dsrbuon. They hold because n a very small me perod, here s no more han one ump can occur and he dffuson process canno move a large dsance almos surely..3 Mone Carlo smulaon We now descrbe a Mone Carlo approach o value CDS based on he former prncple. Procedures of valung he CDS as follows: Sep 1: Dvde he me nerval [,T] no n equal subperods for suffcenly large n, n can be one day. I can be deermned accordng o he maury dae T. W = 1,,..., W Sep : Take Mone Carlo smulaons by repeang he followng sub-procedures for ( ) mes o compue he defaul probably. Typcally, one can choose W beween 1, and,. 1)For each me pon generae random varable σ as equaon (11),hen generae ndependen random vecors ( x,, y ), where x,, y respecvely follows he dsrbuon expressed by equaon (19) 1. ) A nal me, le X = X, X s exogenously gven. The nal sae of σ s observable. ln X can be compued as equaon (18). 3) Fnd he mnmum ln X, f ln X mn, defaul even occurs. 4) e W denoe he sum he roues n whch defaul even occur, hen he defaul probably can be
The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track 615 expressed asw W. Sep3: Compue he CDS prce accordng o he equaon (1) 3 5. 1) Follow sep, calculae he defaul probably before, presen value of expeced paymens when defaul occurs or does no occur and expeced reparaon, where B(, ) = exp( r). ) Sum he presen value a each pon, we oban A, BC, and he CDS prce can be compued by equaon (5). 3. NUMERICA ANAYSIS 3.1 Calbraon of parameers In order o make analyss easy, he relaed parameers are denoed as follows:.5 1) Mean value of lognormal dsrbuon µ =, he varance ln Π : N,.5 and ν =.1331. we have ( ) ) The rsk-free rae.5 defaul hreshold σ =. Accordng o equaon (7) (8), r =. Recovery rae R =.4.The nal rao beween frm asse value and X =. 3) The volaly of asse prce n dfferen sae: σ =.45, σ =.5. 4) The ranson nensy n dfferen sae, λ =.8, λ =.4. When he economy s n he downurn (hgh volaly sae), he governmen has he movaon o ake powerful macro-conrol measures o smulae economc growh. Thus, s reasonable o assume ha he ranson nensy s larger when he economy s n hgh volaly sae. Accordng o former conen, he expeced duraon of regme s 1 λ =.5 and he expeced duraon of regme s 1 λ = 1.5. Durng 1 years, he average me spen n regme s 3.3 years and he average me spen n regme s 6.7 years. 5) The ump nensy λ =.5. S = 1means he nal value of volaly σ =.5, respondng o economc boom perod. S = means he nal value of volaly σ =.45,respondng o economc downurn perod. S = means he suaon where he nfluence of Markov regme shf s gnored. For comparson, we assume ha when S =, he volaly s me weghed mean of ha wh regme shf. Namely λ λ σs = σ + σ () λ + λ λ + λ Accordng o equaon (), we have.3 σ =. S In he laer analycal process, all parameers are deermned as presened above whou specfcaon. 3. Jump dffuson and CDS prce 3..1 Jump amplude and CDS prce As shown n fgure 1,when low volaly regme s he nal sae, he defaul probably ncreases as σ ncreases. Accordng o equaon (18) and (),when σ =,he asse prce follows general dffuson process. Gven X =, s known ha a dffuson process has a connuous sample pah and canno cross a boundary from somewhere else nsananeously. Therefore, he probably of defaulng on very shor erm s zero, so s he CDS prce. As he ump volaly ncreases, he ump sze becomes bgger and s possble he asse prce drops dramacally. Then defaul even occurs, whch makes he CDS prce posve n a shor-erm perod. Fgure shows ha, due o he exsence of ump dffuson, he CDS prce s posve n shor erm perod. In hs way, he flaw of Meron(1976) and Black and Cox(1976) model ha defaul even can be predced and
616 The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track cred spreads s zero n shor erm s modfed. Defaul Propably..15.1.5 σ = σ =.5 σ =.5 CDS Prce (bass) 16 14 1 8 6 4 σ = σ =.5 σ =.5 4 6 8 1 4 6 8 1 Fgure 1 Jump amplude and defaul probably C D S P re c (b as s ) 15 5 λ= λ=.5 λ=.1 Fgure Jump amplude and CDS prce 4 6 8 1 3.. Jump nensy and CDS prce Assumng hgh volaly regme s he nal sae, Fgure 3 llusraes ha he larger he ump nensy, he hgher he CDS prce. The nfluence of ump nensy on CDS prce s smlar o ha of ump amplude. When λ =,asse prce follows dffuson process. Under hs condon, he shor erm cred spreads s zero. As ump nensy becomes larger, mes of ump ncreases and shor erm cred spreads becomes larger. C DS Prec (bass) 18 16 14 1 8 6 4 λ= λ=.5 λ=.1 4 6 8 1 Fgure 3 Jump nensy and CDS prce 3.3 Regme shf and CDS prce 3.3.1 Sae ranson nensy and CDS prce As shown n Fgure 4, gven he nal sae S = 1, he CDS prce ncreases accordng o he ncrease of
The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track 617 sae ranson nensy. Whle n Fgure 5,gven he nal sae S =, he CDS prce decreases accordng o he ncrease of sae ranson nensy. In Fgure 4, gven S = 1, as he sae ranson nensy s small, he nal value of low volaly wll connue affec asse prce for a long me. Thus, he probably of defaul s small and so s he CDS prce. As he ranson nensy ncreases, he nfluence of nal volaly sae on asse prce becomes weak and he CDS prce ncreases. Smlar wh he analyss of Fgure 4, wh he nal sae of hgh volaly, he ncrease of ranson nensy weakens he nfluence of nal sae of hgh volaly and he CDS prce becomes lower. 1 C DS Prce (bass) 8 6 4 λ =.4, λ =. λ =.8, λ =.4 λ =4, λ = CDS Prece (Bass) 8 6 4 λ =.4, λ =. λ =.8, λ =.4 λ =4, λ = 4 6 8 1 4 6 8 1 Maury(year) Fgure 4 CDS prce when S = 1 Fgure 5 CDS prce when S = 3.3. Comprehensve relaon beween nal sae, ranson nensy and CDS prce Fgure 6,7 and 8 show he nfluence of nal sae on CDS prce under he suaon of dfferen ranson nensy. In Fgure 6, he CDS prce s he larges wh nal sae of hgh volaly, he second s he prce whou regme shf wh s volaly value equal o he average of regme and weghed by me spen n each regme. The prce s smalles wh nal sae of low volaly. Compared wh Fgure 6, Fgure 7 shows ha as he ranson nensy ncreases, he dfference beween dfferen nal sae decreases. Comparng Fgure 6 and 7 wh Fgure 8, a neresng fndng s ha wh furher ncrease of ranson nensy, he dfference beween hem becomes smaller bu he prce whou regme shf becomes smalles of all. In Fgure 6,he average ranson mes s beween 4 and 8, he nal sae domnaes he CDS prce. When ranson nensy ncreases, as shown n Fgure 7, he nfluence of nal sae becomes weak and he effec of regme shf ncreases. Ulerorly, n Fgure 8, he average ranson mes s beween and 4. The mpac of regme shf prevals over ha of nal sae and he CDS prce wh regme shf s larger han ha whou regme shf. Ths phenomenon can be explaned as follows: when he asse prce volaly has he propery of regme shf, whaever he nal sae, he frm defaul probably s always larger ha he suaon whou regme shf as long as he ranson nensy s large enough. As we know, regme shf of sae can be vewed as he volaly of asse prce. If he ranson nensy s large enough, he probably ha asse prce changes dramacally s larger han he suaon of consan volaly. Under hs condon, s easer o produce lower exremes. Thus, boh he defaul probably and he CDS prce become larger.
618 The Twelfh Wuhan Inernaonal Conference on E-Busness Innovaon Managemen and IT Busness Value Track 1 C D S Prce (bass) 8 6 4 S=1 S= S= 4 6 8 1 C D S P rce (b ass) 8 6 4 S=1 S= S= 4 6 8 1 C D S P rce (b ass) 8 6 4 S=1 S= S= 4 6 8 1 Fgure 6 CDS prce whenλ =.4, λ =. Fgure 7 CDS prce whenλ =.8, λ =.4 Fgure 8 CDS prce whenλ = 4, λ= 4. CONCUDING REMARKS Ths paper develops a new CDS prcng model under srucural framework. In our model, he asse prce of he reference eny follows a ump-dffuson process and s volaly has he Markov regme shf. We provde complee Mone Carlo procedure o smulae CDS prce. The smulaon resuls show ha heump-dffuson process makes defaul unpredcable and shor erm cred spreads become posve whch s dfferen from he dffuson process. Compared wh he CDS prce under he suaon where he volaly s consan, he CDS prce under Markov regme shf s much dfferen. The dsance beween hem depends on nal sae and ranson nensy. The former can reflecs he nfluence of busness suaon changes on CDS prce. Ths s realsc when valung a medum-and long-erm CDS. I can be a good referrence for CDS prcng. ACKNOWEDGEMENT Ths research was suppored by he Naonal Naural Scence Foundaon of Chna under Gran 7111154. REFERENCES [1] Abrahams J..(1986). A survey of recen progress on level crossng problems. Communcaons and Neworks: A Survey of Recen Advances, Sprnger-Verlag. [] Black F., Cox J..(1976). Valung corporae secures: some effecs of bond ndenure provsons. Journal of Fnance, 31(): 351-367. [3] Chen.. (1). Macroeconomc condons and he puzzles of cred spreads and capal srucure. Journal of Fnance, 65(6): 171-1. [4] Da Q., Sngleon K.J. and We Y.. (7). Regme shfs n a dynamc erm srucure model of U.S. Treasury Bond yelds. The Revew of Fnancal Sudes, (5):1669-176. [5] Elzalde A.. (6). From Basel I o Basel II: An analyss of he Three Pllars. Workngpaper 7_wp74, CEMFI. [6] ackbarh D., Mao J. and Morellec E..(6). Capal srucure, cred rsk, and macroeconomc condons. Journal of Fnancal Economcs, 8(3): 519-55. [7] Meron R..(1974). On he prcng of corporae deb: he rsk srucure of neres raes. Journal of Fnance, 9(): 449-469. [8] Meron R.(1976). Opon prcng when underlyng sock reurns are dcounnuous. Journal of Fnancal Economcs, 3(1-): 15-144. [9] Wang Q., Chen J.. (3). A sudy on prcng model of cred defaul swap based on ump-dffuson process. Sysems Engneerng, 1: 79-83. (n Chnese) [1] Zhou C..(1).The erm srucure of cred spreads wh ump rsk. Journal of Bankng and Fnance,5(11):15-4.