Is the Jump-Diffusion Model a Good Solution for Credit Risk Modeling? The Case of Convertible Bonds

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1 MPRA Mnch Peronal RePEc Archve I he Jmp-Dffon Model a Good olon for red Rk Modelng? The ae of onverble ond Tm Xao I. May 03 Onlne a hp://mpra.b.n-menchen.de/47366/ MPRA Paper No , poed 3. Jne 03 4:6 UT

2 I he Jmp-Dffon Model a Good olon for red Rk Modelng? The ae of onverble ond Tm Xao Rk Qan, apal Marke, I, Torono, anada ATRAT Th paper arge ha he redced-form jmp dffon model may no be approprae for cred rk modelng. To correcly vale hybrd defalable fnancal nrmen, e.g., converble bond, we preen a new framework ha rele on he probably drbon of a defal jmp raher han he defal jmp elf, a he defal jmp ally nacceble. The model qe accrae. A prevalng belef n he marke ha converble arbrage manly de o converble nderprcng. Emprcally, however, we do no fnd evdence pporng he nderprcng hypohe. Inead, we fnd ha converble have relavely large poon gamma. A a ypcal converble arbrage raegy employ dela-neral hedgng, a large pove gamma can make he porfolo hgh profable, epecally for a large movemen n he nderlyng ock prce. Key Word: jmp dffon model, hybrd fnancal nrmen, converble bond, converble nderprcng, converble arbrage, defal me approach, defal probably neny approach, ae prcng, cred rk modelng. The vew expreed here are of he ahor alone and no necearly of h ho non. Addre correpondence o Tm Xao, Rk Qan, apal Marke, I, 6 ay ree, 0 h Floor, Torono, ON M5J 8, anada; emal: m_yxao@yahoo.com

3 . Inrodcon A company can rae capal n fnancal marke eher by ng eqe, bond, or hybrd ch a converble bond. From an nveor perpecve, converble bond wh embedded oponaly offer ceran benef of boh eqe and bond lke he former, hey have he poenal for capal apprecaon and lke he laer, hey offer nere ncome and afey of prncpal. The converble bond marke of prmary global mporance. There a rch lerare on he bjec of converble bond. Argably, he fr wdely adoped model among praconer he one preened by Goldman ach 994 and hen formalzed by Tvero and Fernande 998. The Goldman ach olon a mple one facor model wh an eqy bnomal ree o vale converble bond. The model conder he probably of converon a every node. If he converble ceran o reman a bond, hen dconed by a rky dcon rae ha reflec he cred rk of he er. If he converble ceran o be convered, hen dconed by he rk-free nere rae ha eqvalen o defal free. Tvero and Fernande 998 arge ha n pracce one ally nceran a o wheher he bond wll be convered, and h propoe dvdng converble bond no wo componen: a bond par ha bjec o cred rk and an eqy par ha free of cred rk. A mple decrpon of h model and an eay nmercal example n he conex of a bnomal ree can be fond n Hll 003. Grmwood and Hodge 00 ndcae ha he Goldman ach model ncoheren becae ame ha bond are cepble o cred rk b eqe are no. Ayache, Foryh, and Vezal 003 conclde ha he Tvero-Fernande model nherenly nafacory de o nrealc ampon of ock prce beng naffeced by bankrpcy. To correc h weakne, Dav and Lchka 999, Anderen and ffm 004, loomberg 009, and arr and Lneky 006 ec., propoe a jmp-dffon model o explore defalable ock prce dynamc. They all beleve ha nder a rk-

4 neral meare he expeced rae of rern on a defalable ock m be eqal o he rk-free nere rae. The jmp-dffon model characerze he defal me/jmp drecly. The jmp-dffon model wa fr nrodced by Meron 976 n he marke rk conex for modelng ae prce behavor ha ncorporae mall day-o-day dffve movemen ogeher wh larger randomly occrrng jmp. Over he la decade, people aemp o propagae he model from he marke rk doman o he cred rk arena. A he hear of he jmp-dffon model le he ampon ha he oal expeced rae of rern o he ockholder eqal o he rk-free nere rae nder a rkneral meare. Alhogh we agree ha nder a rk-neral meare he marke prce of rk and rk preference are rrelevan o ae prcng ee Hll 003 and hereby he expecaon of a rk-free ae grow a he rk-free nere rae, we are no convnced ha he expeced rae of rern on a defalable ae m be alo eqal o he rk-free rae. We arge ha nlke marke rk, cred rk acally ha a gnfcan mpac on ae prce. Th why reglaor, ch a Inernaonal Acconng andard oard IA, ael ommee on ankng pervon, ec. reqre fnancal non o repor a cred vale adjmen VA n addon o he rk-free mark-o-marke MTM vale o reflec cred rk ee Xao 03b. y defnon, a VA he dfference beween he rk-free vale and he rky vale of an ae/porfolo bjec o cred rk. VA mple ha he rk-free vale hold no be eqal o he rky vale n he preence of defal rk. A a maer of fac, we wll prove ha he expeced rern of a defalable ae nder a rk-neral meare acally grow a a rky rae raher han he rk-free rae. Th conclon very mporan for rky valaon. ecae of her hybrd nare, converble bond arac dfferen ype of nveor. Epecally, converble arbrage hedge fnd play a domnan role n prmary e of converble deb. In fac, beleved ha hedge fnd prchae 70% o 80% of he converble deb offered n prmary marke. A prevalng belef n he marke ha converble arbrage manly de o converble nderprcng Ammann, Knd and Wlde 003, alamo 003, ho, Gemanky and Tooke 009, Loncark, Here, rk-free mean free of cred rk, b no necearly of marke rk

5 Hor and Veld 009, ec.,.e., he model vale above he oberved marke prce. However, Agarwal, Fng, Loon and Nak 007 and aa, hacko, and Dharan 007 arge ha he exce rern from converble arbrage raege are no manly de o nderprcng, b raher parly de o llqd. alamo 0 beleve ha arbrager n general ake advanage of volaly. A hgher volaly n he nderlyng eqy ranlae no a hgher vale of he eqy opon and a lower converon premm. In fac, converble arbrage far more complcaed, nvolvng akng poon n he converble bond and he nderlyng ae ha hedge ceran rk b leave manager expoed o oher rk for whch hey reap a reward. Th arcle make a heorecal and emprcal conrbon o he dy of converble bond. In conra o he above menoned lerare, we preen a model ha baed on he probably drbon or neny of a defal jmp or a defal me raher han he defal jmp elf, a he defal jmp ally nacceble ee Dffe and Hang 996, Jarrow and Proer 004, ec. We model boh eqe and bond a defalable n a conen way. When a frm goe bankrp, he nveor who ake he lea rk are pad fr. ecred credor have he be chance of eeng he vale of her nal nvemen come back o hem. ondholder have a greaer poenal for recoverng ome her loe han ockholder who are la n lne o be repad and ally receve lle, f anyhng. The defal proceedng provde a jfcaon for or modelng ampon: Dfferen clae of ecre ed by he ame company have he ame defal probably b dfferen recovery rae. Gven h model, we are able o back o he marke prce. Valaon nder or rky model can be olved by common nmercal mehod, ch a, Mone arlo mlaon, ree/lace approache, or paral dfferenal eqaon PDE olon. The PDE algorhm elaboraed n h paper, b of core he mehodology can be ealy exended o ree/lace or Mone arlo. Ung he model propoed, we condc an emprcal dy of converble bond. We oban a daa e from an nvemen bank. The daa e conan 64 converble bond and year of daly marke 3

6 prce a well a aocaed nere rae crve, cred crve, ock prce, mpled lack-chole volale and recovery rae. The mo mporan np parameer o be deermned he volaly for valaon. A common approach n he marke o e he a-he-money ATM mpled lack-chole volaly o prce converble bond. However, mo lqd ock opon have relavely hor marae rarely more han 8 year. A a rel, ome ahor, ch a Ammann, Knd and Wlde 003, Loncark, Hor and Veld 009, Zabolonyk, Jone, Veld 00, have o make do wh horcal volale. Therefore, we egmen he ample no wo e accordng o he me o mary: a hor-mary cla 0 ~ 8 year and a long-mary cla > 8 year. For he hor-mary cla, we e he ATM mpled lack- chole volale for valaon, wherea for he long-mary cla, we calclae he horcal volaly a he annalzed andard devaon of he daly log rern of he la year and hen prce he converble bond baed on h real-world volaly. The emprcal rel how ha he model prce flcae randomly arond he marke prce, ndcang he model qe accrae. Or emprcal evdence doe no ppor a yemac nderprcng hypohe. In fac, people n he marke almo alway calbrae her model o he oberved marke prce ng mpled converble volale. Therefore, nderprcng may no be he man drver of profably n converble arbrage. I efl o examne he bac of he converble arbrage raegy. A ypcal converble bond arbrage employ dela-neral hedgng, n whch an arbrager by a converble bond and ell he nderlyng eqy a he crren dela ee ho, Gemanky and Tooke 009, Loncark, Hor and Veld 009, ec.. Wh dela neral poon, he gn of Gamma mporan. If Gamma negave, he porfolo prof o long a he nderlyng eqy reman able. If Gamma pove, he porfolo wll prof from large movemen n he ock n eher drecon ee omanah 0. We dy he enve of converble bond and fnd ha converble bond have relavely large pove gamma, mplyng ha converble arbrage can make a prof on a large pde and downde movemen of he nderlyng ock prce. nce converble bond are ed manly by ar-p 4

7 or mall compane whle more eablhed frm rely on oher mean of fnancng, he chance of a large movemen n eher drecon very lkely. Even for very mall movemen n he nderlyng ock prce, prof can ll be generaed from he yeld of he converble bond and he nere rebae for he hor poon. Therefore, he large pove gamma of converble conrac cold be one of he man drver n converble arbrage. The re of h paper organzed a follow: The model preened n econ. econ 3 elaborae he PDE approach; econ 4 dce he emprcal rel. The conclon are provded n econ 5. ome nmercal mplemenaon deal and a bnomal ree approach are conaned n he appendce. Model onverble bond can be hogh of a normal corporae bond wh embedded opon, whch enable he holder o exchange he bond ae for he er ock. Depe her poplary and bqy, converble bond ll poe dffcl modelng challenge, gven her hybrd nare of conanng boh deb and eqy feare. Frher complcaon are de o he freqen preence of complex conracal clae, ch a, p, hard call, of call, and oher pah-dependen rgger provon. onrac of ch complexy can only be olved by nmercal mehod, ch a, Mone arlo mlaon, ree/lace approache, or PDE olon. From a praconer perpecve, Mone arlo a la reor and lea preferred mehod, wherea lace or PDE approache ffer from he cre of dmenonaly: The nmber of evalaon and compaonal co ncreae exponenally wh he dmenon of he problem, makng mpraccal o e n more han wo dmenon. Three orce of randomne ex n a converble bond: he ock prce, he nere rae, and he cred pread. A praconer end o echew model wh more han wo facor, a legmae qeon: How can we redce he nmber of facor or whch facor are mo mporan? Grmwood and 5

8 Hodge 00 condc a envy dy and fnd ha accraely modelng he eqy proce appear crcal. Th why all converble bond model n he marke capre, a a mnmm, he dynamc of he nderlyng eqy. nce converble bond are ed manly by ar-p or mall compane whle more eablhed frm rely on oher mean of fnancng, cred rk play an mporan role n he valaon. Grmwood and Hodge 00 frher noe ha he nere rae proce of econd order mporance. mlarly, rennan and chwarz 980 conclde ha he effec of a ochac nere rae on converble bond prce o mall ha can be negleced. Frhermore, Ammann, Knd, and Wlde 008 noce ha he overall prcng benef of ncorporang ochac nere rae wold be very lmed and wold no jfy he addonal compaonal co. For hee reaon, mo praccal converble model n he marke do no ake ochac nere rae no accon. We conder a flered probably pace, F, F 0, P afyng he al condon, where denoe a ample pace, F denoe a -algebra, P denoe a probably meare, and 0 F denoe a flraon. The rk-free ock prce proce can be decrbed a d r d dw where denoe he ock prce, r denoe he rk-free nere rae, denoe he volaly, W denoe a Wener proce. The expecaon of eqaon d r d where E F he expecaon condonal on he F. E F Eqaon ell ha n a rk-neral world, he expeced rern on a rk-free ock he rk-free nere rae r proce.,.e., he dconed ock prce nder he rk neral meare a marngale 6

9 Nex, we rn o a defalable ock. The defalable ock proce propoed by Dav and Lchka 999, Anderen and ffm 004, and loomberg 009, ec., gven by r h d ˆ dw du d 3 where U an ndependen Poon proce wh du wh probably h d and 0 oherwe, h he hazard rae or he defal neny, he ock prce mmedaely before any jmp a me. The expecaon of du E du F h d. The expecaon of eqaon 3 gven by d r h d h d r d E F 4 I hown n eqaon 4 ha he expeced rern of a defalable ock grow a he rk-free rae. Eqaon 3 a mpler veron of he Meron Jmp-dffon model where he nmber of jmp. The jmp-dffon model wa fr propoed n he conex of marke rk, whch narally exhb hgh kewne and lepokro level and capre he o-called mpled volaly mle or kew effec. Ederngon and Lee 993 fnd ha he marke end o have overreacon and nderreacon o he ode new. The jmp par of he model can be nerpreed a he marke repone o ode new. If here no any ode new, he ae prce change accordng o a geomerc rownan moon. nce he marke prce of rk and rk preference are rrelevan o ae prcng whn he marke rk conex, he expeced rae of rern o he ockholder eqal o he rk-free rae nder a rk-neral meare. However, we wonder wheher approprae o propagae he jmp-dffon model drecly from he marke rk doman o he cred rk doman, a cred rk acally mpac he valaon of ae. Th why fnancal non are reqred by reglaor o repor VA. In fac, we wll how n he followng dervaon ha he expeced rern of a defalable ae nder he rk-neral meare acally eqal o a rky rae nead of he rk-free nere rae. The conclon very mporan for rky valaon. 7

10 The world of cred modelng dvded no wo man approache: rcral model and redced-form or neny model. The rcral model regard defal a an endogeno even, focng on he capal rcre of a frm. The redced-form model do no explan he even of defal endogenoly, b nead characerze exogenoly a a jmp proce. In general, rcral model are baed on he nformaon e avalable o he frm' managemen, ch a he frm ae and lable; whle redced-form model are baed on he nformaon e avalable o he marke, ch a he frm bond prce or cred defal wap D prema. Many praconer n he cred radng arena have ended o gravae oward he redced-from model gven her mahemacal racably. The redcedform model can be made conen wh he rk-neral probable of defal backed o from corporae bond prce or D pread/prema. In he redced-form model, he oppng or defal me of a frm modeled a a ox arrval proce alo known a a dobly ochac Poon proce whoe fr jmp occr a defal and defned a, nf : h, d 5 0 where h or h, denoe he ochac hazard rae or arrval neny dependen on an exogeno common ae, and a n exponenal random varable ndependen of I well-known ha he rvval probably from me o n h framework defned by. p, : P, Z exp h d 6 The defal probably for he perod, n h framework defned by q, : P, Z p, exp h d 7 We conder a defalable ae ha pay nohng beween dae and T. Le V and V T denoe vale a and T, repecvely. Rky valaon can be generally clafed no wo caegore: he defal me approach DTA and he defal probably neny approach DPA. 8

11 The DTA nvolve he defal me explcly. If here ha been no defal before me T.e., T, he vale of he ae a T V T. If a defal happen before T.e., T, a recovery payoff made a he defal me a a fracon of he marke vale 3 gven by V where he defal recovery rae and V he marke vale a defal. Under a rk-neral meare, he vale of h defalable ae he dconed expecaon of all he payoff and gven by V E D, T V T T D, V T F 8 where Y an ndcaor fncon ha eqal o one f Y re and zero oherwe, and D, denoe he ochac rk-free dcon facor a for he mary gven by D, exp r d 9 Alhogh he DTA very nve, ha he dadvanage ha explcly nvolve he defal me/jmp. We are very nlkely o have complee nformaon abo a frm defal pon, whch ofen nacceble. Ually, valaon nder he DTA performed va Mone arlo mlaon. The DPA rele on he probably drbon of he defal me raher han he defal me elf. We dvde he me perod, T no n very mall me nerval and ame ha a defal may occr only a he end of each very mall perod. In or dervaon, we e he approxmaon y exp y for very mall y. The rvval and he defal probable for he perod, are gven by h h pˆ : p, exp 0 h h qˆ : q, exp The bnomal defal rle conder only wo poble ae: defal or rvval. For he oneperod, economy, a me he ae eher defal wh he defal probably q, or rvve wh he rvval probably p,. The rvval payoff eqal o he 3 Here we e he recovery of marke vale RMV ampon. 9

12 marke vale V and he defal payoff a fracon of he marke vale: V. Under a rk-neral meare, he vale of he ae a he expecaon of all he payoff dconed a he rk-free rae and gven by exp r pˆ qˆ V Eexp y V E F V F where y r h r c denoe he rky rae and c h hor cred pread. mlarly, we have called he V E exp y V F 3 Noe ha y exp F -mearable. y defnon, an F -mearable random varable a random varable whoe vale known a me. aed on he akng o wha known and ower propere of condonal expecaon, we have V E exp E exp y V F y E exp y E exp 0 y V F V F y recrvely dervng from forward over T and akng he lm a rky vale of he ae can be expreed a F 4 approache zero, he T V E exp y d V T F 5 Ung he DPA, we oban a cloed-form olon for prcng an ae bjec o cred rk. Oher good example of he DPA are he D model propoed by J.P. Morgan 999 and a more generc rky model preened by Xao 03a. The dervaon of eqaon 5 ake no accon all cred characerc: pobly of a jmp o defal and recovery rae. I ell ha a defalable ae nder he rk-neral meare grow a a rky rae. The rky rae eqal o a rk-free nere rae pl a cred pread. If he ae a bond, he eqaon he ame a Eqaon 0 n Dffe and ngleon 999, whch he marke model for 0

13 prcng rky bond. The marke bond model ay ha he vale of a rky bond obaned by dconng he promed payoff ng he rk-free nere rae pl he cred pread 4. Under a rk-neral meare he marke prce of rk and rk preference are rrelevan o ae prcng ee Hll 003 and hereby he expecaon of a rk-free ae grow a he rk-free nere rae. However, cred rk acally ha a gnfcan mpac on ae prce. Th he reaon ha reglaor, ch a IA and, reqre fnancal non o repor a VA n addon o he rkfree MTM vale o reflec cred rk. In ae prcng heory, he fndamenal no-arbrage heorem do no reqre expeced rern o be eqal o he rk free rae, b only ha prce are marngale afer dconng nder he nmerare. For rk-free valaon, people commonly e a rk-free bond a he nmerare, wherea for rky valaon, hey hold chooe an aocaed rky nmerare o reflec cred rk. The expeced rern ha of he nmerare. If a company fle bankrpcy, boh bond and ock go no a defal a. In oher word, he defal probable for boh of hem are he ame.e., eqal o he frm probably of defal. he recovery rae are dfferen becae he ockholder are he lowe prory n he l of he akeholder n he company, wherea he bondholder have a hgher prory o receve a hgher percenage of nveed fnd. The defal proceedng provde a jfcaon for or modelng ampon: Dfferen clae of ecre ed by he ame company have he ame defal probably b dfferen recovery rae. Accordng o eqaon 5, we propoe a rky model ha embed he probably of he defal jmp or defal neny raher han he defal jmp elf no he prce dynamc of an ae. The ochac dfferenal eqaon DE of a defalable ock defned a where r h d dw y d dw d 6 he recovery rae of he ock and y r h he rky rae. 4 There a lqdy componen n he bond pread. Th paper, however, foce on cred rk only.

14 For mo praccal problem, zero recovery a defal or jmp o zero nrealc. For example, he ock of Lehman roher fell 94.3% on epember 5, 008 afer he company fled for haper bankrpcy. mlarly, he hare of General Moor GM plnged 3% on Jne, 009 afer he frm naed haper bankrpcy. A good framework hold flexbly allow people o ncorporae dfferen recovery ampon no rky valaon. Eqaon 6 he drec dervaon of eqaon 5. The formla allow dfferen ampon concernng recovery on defal. In parclar, 0 repreen he aon where he ock prce jmp o 0, and correpond o he rk-free cae. The expecaon of eqaon 6 E d r h d F 7 Eqaon 7 ay ha he expeced rern of a ock bjec o cred rk eqal o a rky rae raher han he rk-free rae. The rky rae reflec he compenaon nveor receve for bearng cred rk. 3. PDE Algorhm The nmercal olon of or rky model can be obaned by eher PDE mehod, ree approache, or Mone arlo mlaon. In h paper, we nrodce he PDE procedre, b of core he mehodology can be ealy exended o he ree/lace or Mone arlo algorhm. The defalable ock prce proce gven by r q h d dw d dw d 8 where q he dvdend and r q h. The valaon of a converble bond normally ha a backward nare nce here no way of knowng wheher he converble hold be convered who knowledge of he fre vale. Only on he mary dae, he vale of an nrmen and he decon raegy are clear. If he converble ceran o be convered, behave lke a ock. If he converble no convered a an nermedae node, we are

15 ally nceran wheher he connaon vale hold be reaed a a bond or a ock, becae n backward ndcon he crren vale ake no accon he rel of all fre decon and ome fre vale may be domnaed by he ock or by he bond or by boh. Therefore, we arrange he valaon o ha he vale of he converble a each node dvded no wo componen: a componen of bond and a componen of ock,.e. L, G,, where G, denoe he eqy par of he converble bond and, denoe he bond par of he converble. ppoe ha G, ome fncon of and. Applyng Io Lemma, we have G G G G dg d dw 9 nce he Wener proce nderlyng and G are he ame, we can conrc he followng porfolo o ha he Wener proce can be elmnaed. X G G 0 Therefore, we have dx G G G dg d d In conra o all prevo de, we beleve ha he defalable eqy hold grow a he rky rae ncldng dvdend, wherea he eqy par of he converble bond hold earn he rky rae of rern excldng dvdend,.e., r G G G h Gd d dx d o ha he PDE of he eqy componen gven by G G G r q h r h G 0 3 mlarly applyng Io Lemma o he bond par of he converble,, we oban d d dw 4 3

16 Le conrc a porfolo o ha we can elmnae he Wener proce a follow Th, we have Y 5 dy d d d 6 The defalable eqy hold grow a he rky rae ncldng dvdend, whle he bond par of he converble bond grow a he rky rae of he bond. oneqenly, we have b 7 r h d r q h d dy d The PDE of he bond componen r q h r h 0 b 8 Eqaon 3 and 8 are copled hrogh approprae fnal and bondary condon reflecng he erm of each ndvdal converble and need o be olved mlaneoly. onverble bond ofen ncorporae varo addonal feare, ch a call and p provon. The fnal condon a mary T can be generalzed a T G T mn 0, 0, T, f T mn P, max P c p, N oherwe P, maxp, N, f mn P, maxp, N c p T c p oherwe where N denoe he bond prncpal, denoe he copon, P c denoe he call prce, P p denoe he p prce and denoe he converon rao. The fnal condon ell ha he converble bond a he 9 30 mary eher a deb or an eqy. The pde conran a me [ 0, T] G, 0 f G 0, Pp G 0, Pc ~ ~ G G, mn Pc, max P ~ ele f L ~ ele f L p ~, L P P ele p c 3 4

17 where ~ L ~ ~ G he connaon vale of he converble bond, ~ he connaon vale of he bond componen and G ~ he connaon vale of he eqy componen. Eqaon 3 ay ha he converble eher n he connaon regon or one of he hree conran called, p or convered. 4. Emprcal rel Th econ preen he emprcal rel. We e wo year of daly daa from epember 0, 00 o epember 0, 0,.e., a oal of 5 obervaon day. Th propreary daa are obaned from an nvemen bank. They con of converble bond conrac, marke oberved converble prce, nere rae crve, cred crve, ock prce, mpled lack-chole volale, and recovery rae. We only conder he converble oandng drng he perod and wh ffcen prcng nformaon. A a rel, we oban a fnal ample of 64 converble bond and a oal of 64 5 = 85,608 obervaon. None of he converble n h ample acally defaled drng he me wndow. A of epember 0, 0, he ample repreen a famly of converble bond wh a me o mary rangng from monh o 36.6 year, and ha an average remanng mary of 4.35 year. The hogram of conrac on epember 0, 0 for varo mary clae gven n Fgre. onverble bond prce oberved n he marke wll be compared wh heorecal prce nder dfferen volaly ampon. The ample egmened no wo e accordng o he me o mary: a hor-mary cla 0 ~ 8 year and a long-mary cla > 8 year. We fr elec a converble bond from each grop: a 7-year or 5-year oandng conrac and a 0-year or 7-year oandng conrac hown n Table. Fgre. Hogram of converble bond by me o mary Th hogram dvde he converble bond n or ample, a of epember 0, 0, no dfferen bn accordng o he me o mary. The x-ax repreen he mary n year and he y-ax repreen he 5

18 nmber of converble n each bn. A mary bn of n cover conrac wh a me o mary rangng from n- year o n year. Hogram of conerble by me o mary freqency >8 mary Table. onverble ond We hde he er name accordng o he ecry polcy of he nvemen bank, b everyhng ele ahenc. In he marke, eher a converon prce or a converon rao gven for a converble bond, where converon rao = face vale of he converble bond / converon prce. onverble bond ae a 7-year converble ae a 0-year converble Ier X company Y company Noonal of bond Annal copon rae Paymen freqency emannal emannal Ing dae Jne 9, 00 Jne 5, 009 Mary dae Jne 5, 07 Jne 5, 09 onveron prce rrency UD UD Day con 30/360 30/360 6

19 ne day convenon Followng Followng P prce - 00 a Jne 0, 04 Le valaon dae be epember 0, 0. An nere rae crve he erm rcre of nere rae, derved from oberved marke nrmen ha repreen he mo lqd and domnan nere rae prodc for ceran me horzon. Normally he crve dvded no hree par. The hor end of he erm rcre deermned ng he London Inerbank Offered Rae LIOR. The mddle par of he crve conrced ng Erodollar fre ha reqre convexy adjmen. The far end derved ng md wap rae. The LIOR-fre-wap crve preened n Table. We boorap he crve and ge he connoly componded zero rae. Table : UD LIOR-Fre-wap rve Th able dplay he clong prce a of epember 0, 0. Inrmen Name Prce epember 9, 0 LIOR % epember 0 Erodollar 3 monh December 0 Erodollar 3 monh March 03 Erodollar 3 monh Jne 03 Erodollar 3 monh epember 03 Erodollar 3 monh December 03 Erodollar 3 monh March 04 Erodollar 3 monh year wap rae % 3 year wap rae % 4 year wap rae 0.60% 5 year wap rae 0.894% 6 year wap rae.0537% 7 year wap rae.738% 8 year wap rae.4678% 9 year wap rae.6360% 7

20 0 year wap rae.785% year wap rae.0334% 5 year wap rae.783% 0 year wap rae.478% 5 year wap rae.5790% 30 year wap rae.64% The eqy nformaon and recovery rae are provded n Table 3. To deermne hazard rae, we need o know he oberved marke prce of corporae bond or D prema, a he marke andard pracce o f he mpled rk-neral defal nene o hee cred enve nrmen. The corporae bond prce are nfornaely no avalable for compane X and Y, b her D prema are obervable a hown n Table 4. Ually he D marke lead he bond marke, n parclar drng cr aon. Lqdy n he bond marke ypcally dryng p drng a fnancal cr. Demand for nrance agan defal rk, on he oher hand, ncreae f he er experencng fnancal re. oneqenly, prce and pread derved from he D marke end o be more relable. ad dfferenly, D on reference ene are ofen more acvely raded han bond ed by he reference ene. Unlke oher de ha e bond pread for prcng ee Tvero and Fernande 998, Ammann, Knd and Wlde 003, Zabolonyk, Jone, and Veld 00, ec., we perform rky valaon baed on cred nformaon exraced from D pread. Gven he recovery rae and he D prema, we can compe he hazard rae va a andard calbraon proce ee J.P. Morgan 00. Table 3. Eqy and recovery nformaon Th able dplay he clong ock prce and dvdend yeld on epember 0, 0, a well a he recovery rae ompany X ompany Y ock prce Dvdend yeld.55% 3.95% ond recovery rae 40% 36.4% 8

21 Eqy recovery rae % % Table 4. D prema Th able dplay he clong D prema a of epember 0, 0. Name ompany X ompany Y 6 monh D pread year D pread year D pread year D pread year D pread year D pread year D pread year D pread year D pread year D pread The mo mporan np parameer o be deermned he volaly for valaon. A common approach n he marke o e he a-he-money ATM mpled lack-chole volaly o prce converble bond. For he 5-year oandng converble bond cae n Table, we fnd he ATM mpled lack-chole volaly 3.87%, and hen prce he converble bond accordngly. The rel are hown n Table 5. Or analy acally ndcae an overprcng of 0.4%. For he 7-year oandng converble bond cae n Table, however, mo lqd ock opon have relavely hor marae rarely more han 8 year. Therefore, ome ahor, ch a Ammann, Knd and Wlde 003, Loncark, Hor and Veld 009, Zabolonyk, Jone, Veld 00, have o make do wh horcal volale. mlarly, we calclae he horcal volaly a he annalzed andard devaon of he daly log rern of he la year from epember 0, 00 o epember 0, 0, and hen vale he converble bond baed on h real-world volaly. The rel hown n Table 5 repor an nderprcng of.07%. The e rel demonrae ha he model prce are very cloe o he marke prce, ndcang ha he model qe accrae. 9

22 Table 5. Model prce v. marke prce Th able how he dfference beween he model prce and he marke prce of he converble bond nder dfferen volaly ampon, where Dfference = Model prce / Marke oberved prce. The converble bond are defned n Table. ae a 7-year converble ae a 0-year converble Type of volaly ATM mpled lack-chole volaly Annalzed horcal volaly Vale of volaly 3.87% 8.07% Model prce Marke oberved prce Dfference -0.4%.07% Any model can be ed o calclae aocaed mpled volale baed on oberved marke prce. Ung he mpled volale a np o he model, he model prce can exacly mach he marke prce. The calbraon can be done by choong he mpled converble volaly ha mnmze he m of he qared dfference beween he marke prce and he model prce. The calbraed converble volale and he aocaed model prce are dplayed n Table 6. Table 6. Impled converble volale Th able dplay he mpled converble volale and he aocaed model prce, where Dfference = Model prce / Marke oberved prce. The converble bond are defned n Table. ae a 7-year converble ae a 0-year converble Impled converble volaly 3.55% 5.9% Model prce Marke oberved prce Dfference 0 0 0

23 We repea h exerce for all conrac on all obervaon day. For any hor-mary converble bond, we e he ATM mpled lack-chole volaly for prcng, wherea for any longmary converble bond, we perform valaon va he horcal volaly. The rel are preened n Table 7. Table 7. ac of nderprcng for dfferen mary clae An obervaon correpond o a prce napho of a converble bond a a ceran valaon dae. Underprcng referred o a he model prce mn he marke prce. Mary Obervaon Underprcng Mean % d % Max % Mn % 8 year > 8 year Nex, or ample paroned no bample accordng o he moneyne of converble. The moneyne meared by he rao of he converon vale o he eqvalen ragh bond vale or he nvemen vale. The nderprcng of each daly obervaon wh repec o he degree of moneyne hown n Table 8, where moneyne beween 0 and 0.9 correpond o o-of-he-money; moneyne beween 0.9 and. repreen arond-he-money; and moneyne hgher han. relaed o n-hemoney. Table 8. ac of nderprcng for dfferen moneyne clae The moneyne meared by dvdng he converon vale hrogh he aocaed ragh bond vale. An obervaon correpond o a napho of he marke ed o prce a converble bond a a ceran valaon dae. Underprcng Moneyne Obervaon Mean % d % <

24 > From Table 8, can be een ha he model prce flcae randomly arond he marke prce omeme overprced and omeme nderprced, ndcang he model qe accrae. Emprcally, we do no fnd ppor for preence of a yemac nderprcng a ndcaed n prevo de ee arayannopolo and Kalmpall 003, Ammann, Knd and Wlde 003, ec.. If here no nderprcng, how ha he arbrage raegy been ccefl n he pa? Maybe converble arbrage no olely baed on nderprcng In a ypcal converble bond arbrage raegy, he arbrager enal prchang a converble bond and ellng he nderlyng ock o creae a dela neral poon. The nmber of hare old hor ally reflec a dela-neral or marke neral rao. I well known ha dela neral hedgng no only remove mall dreconal rk b alo capable of makng a prof on an explove pde or downde breako f he poon gamma kep pove. A ch, dela neral hedgng grea for nceran ock ha are expeced o make hge breako n eher drecon. nce converble bond are ed manly by ar-p or mall compane, he chance of a large movemen n eher drecon very lkely. Even for very mall movemen n he nderlyng ock prce, prof can ll be generaed from he yeld of he converble bond and he nere rebae for he hor poon. We calclae he dela and gamma vale for he wo deal decrbed n able. The Greek v. po eqy prce are ploed n Fgre ~ 5. I can be een ha he dela ncreae wh he nderlyng prce n Fgre and 4. A low marke level, he converble behave lke her ragh bond wh very mall dela. A he ock prce ncreae, converon become more lkely. A ceran marke level he

25 converble are ceran o be convered. In h cae, he converble are mlar o he nderlyng eqe and he dela are eqal o he nmber of hare.e., converon rao. The gamma dagram n Fgre 3 and 5 have a frown hape. The gamma are he hghe when he converble are a-he-money. I nve ha when he ock prce re or fall, prof ncreae becae of favorably changng dela. For h reaon, converble bond are very good canddae for dela neral hedgng. Relavely large pove gamma of converble cold be one of he man drver of profably n converble arbrage. Fgre. Dela v. nderlyng prce for a 7-year converble bond Th graph how how he dela of he 7-year converble bond decrbed n Table change a he nderlyng ock prce change. Dela of he 7-year converble bond v. nderlyng prce dela eqy prce Fgre 3. Gamma v. nderlyng prce for a 7-year converble bond Th graph how how he gamma of he 7-year converble bond decrbed n Table change a he nderlyng ock prce change. 3

26 Gamma of he 7-year converble v. nderlyng prce gamma eqy prce Fgre 4. Dela v. nderlyng prce for a 0-year converble bond Th graph how how he dela of he 0-year converble bond decrbed n Table change a he nderlyng ock prce change. Dela of he 0-year converble v. nderlyng prce 8 6 dela Eqy prce Fgre 5. Gamma v. nderlyng prce for a 0-year converble bond 4

27 Th graph how how he dela of he 0-year converble bond decrbed n Table change a he nderlyng ock prce change. Gamma of he 0-year converble v. nderlyng prce 0.6 gamma eqy prce 5. onclon Th paper am o vale hybrd fnancal nrmen e.g., converble bond whoe vale may mlaneoly depend on dfferen ae bjec o cred rk n a proper and conen way. The movaon for or model ha f a company goe bankrp, all he ecre ncldng he eqy of he company defal. The recovery realzed n accordance wh he prory eablhed by he ankrpcy ode. In oher word, dfferen ecre have he ame probably of defal, b dfferen recovery rae. Or dy how ha rky ae prcng qe dfferen from rk-free ae prcng. In fac, he expecaon of a defalable ae acally grow a a rky rae raher han he rk-free rae. Therefore, he redced form jmp dffon model ha ame ha he expeced rae of rern on a rky ae m be eqal o he rk-free nere rae may no be approprae. 5

28 We propoe a hybrd framework o vale rky eqe and deb n a nfed way. The model rele on he probably drbon of he defal jmp raher han he defal jmp elf, becae he defal jmp normally nacceble. The model qe accrae for prcng converble bond. Emprcally, we do no fnd evdence pporng a yemac nderprcng hypohe. We alo fnd ha converble bond have relavely large pove gamma, mplyng ha converble arbrage can make a prof on a large pde and downde movemen of he nderlyng ock prce. A. Nmerc mplemenaon for PDE Appendx In h econ, we decrbe he nmercal mehod ed o olve dcree form of 3 and 8. Le x ln and defne backward me a T. The eqaon 3 and 8 can be rewren a 0 G r q h x G r q h x x r h 0 b G x r h G 0 A A The eqaon A and A can be approxmaed ng rank-ncolon rle. We dcreze he x o be eqally paced a a grd of node 0 ~ M. A he mary, 9 and 30. A any me +, he bondary condon are G T and T are deermned accordng o G r h 0.5 r q h G b r h 0.5 r q h b when x 0 A3 G M M 0 M when x A4 Then, we condc he backward ndcon. The procedre a follow. For = penlmaetme o crrentme 6

29 // deermne accral nere and call/p prce // deermne bondary node // e he POR Projeced cceve Over Relaxaon mehod o oban he connaon vale of he bond componen componen G ~, applyng he conran 3. ~ and he connaon vale of he eqy EndFor The vale a node[0][y] he converble bond prce where he eqy prce a node[0][y] eqal o he marke prce. A. nomal ree algorhm A bnomal ree mehod eqvalen o an explc dfference cheme. ppoe ha he ock prce wll eher move p o he vale wh probably p or down o he vale d wh probably p d p. A he bnomal ree a dcree approxmaon o he conno drbon of eqaon 6, he expecaon and varance of he dcree drbon hold be eqal o hoe of he conno drbon. Th mehod commonly referred o a he momen machng echnqe. To mach he expecaon, we have or where E / p p d exp y A5 p p d exp y A6 y r q c r q h A7 where q he dvdend. To mach he varance, we ge or Var expy exp / p p d exp y A8 7

30 p p d exp y exp A9 olvng eqaon A6 and A9 accordng o he al ree-ymmery condon: = /d, we oban expy exp p exp y d d exp y exp y exp 4exp y A0 A exp y d exp expy exp y exp 4exp y A There are many way o approxmae eqaon A and A. The mo well-known one he ox, Ro, and Rbnen 979 ype approxmaon ha p o order accracy and gven by exp A3 d exp A4 Eqaon A0, A3 and A4 pecfy he bnomal rky ree parameer ha are ed o map he conno ock prce dynamc no he lace repreenaon. ppoe ha here a converble bond. Le conrc a radng raegy H, o hold n of he rky ock and n of he rky bond. A me he converble bond vale where he bond componen and he ock componen; he ock vale ; and he bond vale. A me, he bond vale become exp yb where y r he rky rae of he bond; he ock vale become eher or d ; and he b h b converble vale ha wo poble ocome: or E 8

31 9 d E d d correpondng o eher an p movemen or a down movemen n he ock prce. The dconed porfolo hold replcae he dconed converble bond 5, whch yeld exp exp exp y y y b A5 exp exp exp y y y d d b d A6 olvng for, yeld exp y y d d b d d A7 exp exp y d d y d d b d d A8 For a elf-fnancng porfolo, he nal wealh needed o fnance h raegy omeme called he manfacrng co of he conngen clam exp exp y p p y p p b d d A9 where p defned n A0. We pl eqaon A9 no an eqy eqaon and a bond eqaon, and ge exp y p p d A0 exp y p p b d A Eqaon A0 and A ell ha he far prce of an eqy componen or a bond componen eqal o he expeced vale of fre payoff dconed by he aocaed rky rae. The expeced vale calclaed ng he correpondng vale from he laer wo node p or down weghed by he ranon probable. 5 Unlke he rk-free ree, he rky ree re o mach he dconed vale of he replcang porfolo o he dconed vale of he converble bond n order o cach cred rk properly.

32 For eay replcaon, we e a very mple converble bond decrbed n Table A. The nderlyng bond a zero copon bond. The dvdend 0. The converon rao 5,.e., he bond can be exchanged for 5 hare of he company ock a any me drng he nne monh. The hazard rae fla a h 0.03/ Table A. A mple 9-monh converble bond The nderlyng bond a zero-copon bond. onveron cold happen a any me. Mary T 9 monh Noonal N 00 onveron rae 5 hare rao all prce P c P prce P p 8 a 3, 6, 9 monh 8 a 3, 6, 9 monh po ock prce 0 Impled volaly of he converble 30% Inere rae r 0.0 ond pread c 0.03 ond recovery rae b 0.6 ock recovery rae 0.05 Fr, we conrc he bnomal rky ree, whch ame ha he ock prce evolon compoed of a nmber of mall bnomal movemen. We dvde me from he valaon me o mary no 3 lce,.e., he me ep 3 monh or 0.5 year, hown n Fgre A. The nmber of me ep o a node n he ree we defne a j. The nmber of me he ae prce ha gone p o reach a node we defne a. The fr node n he ree agned j=0, =0. The ock follow he movemen defned n eqaon A3 and A4: exp. 68, exp , wh ranon probable defned n eqaon A0: p [exp r q h d ]/ d , 30

33 p p The rky rae of he bond and eqy are y r h and d y r q h Valaon performed eravely, arng a each of he fnal node, and hen workng backward hrogh he ree oward he fr node valaon dae. The vale comped a each age he vale of he opon a he pon n me. The payoff a mary gven by V j, T max j, T, mn Pc, max Pp, N A Eqaon A how ha we fr e wheher he bond hold be p. Then we e wheher he bond hold be called. Fnally we e wheher converon opmal. The opmal raegy a any b b nermedae node gven by V j, ~ max, mn P, max P, V A3 j, c p j, where ~ V j, he connaon vale ha can be comped accordng o eqaon A0 and A. In Fgre A, he op nmber a each node repreen he ock prce; he econd nmber repreen he vale of he eqy componen; he hrd nmber repreen he vale of he bond componen; and he forh nmber repreen he oal vale of he converble. A he fnal node 9 monh, he payoff defned n eqaon A. For example, a node 3, 3, he ock prce $3.37, beer o conver he nrmen no eqy and receve he converon prce $ han o ge he noonal of $00. Th, we e he bond componen o 0 and he eqy componen o $ A mlar calclaon appled o he oher fnal node. Fgre A. nomal ree for prcng a converble bond The converble bond defned n Table A. The me ep 3 monh. We defne he nmber of me ep a j and he nmber of relave ree poon pck a. The fr node n he ree agned j=0, =0. The op nmber a each node he ock prce; he econd nmber he vale of he eqy 3

34 componen; he hrd nmber he vale of he bond componen; and he forh nmber he oal vale of he converble , ,, ,,, ,3 3, 3, 3,0 Nex, we condc he backward ndcon proce. Le go o he penlmae node 6 monh. A node,, accordng o eqaon A0 he eqy par of he connaon vale comped a D exp , The bond par of he connaon vale 0 accordng o A. Therefore, he oal connaon vale ~ V, $35. The converble hold be called fr a he call prce $8, and hen hold be convered a he converon prce The eqy componen a node, worh $35 and he vale of he bond componen 0. mlarly we can compe he oher penlmae node. worh Then, we go o he node a 3 monh. A node,, he eqy par of he connaon vale D exp , 3

35 The bond componen of he connaon vale worh exp , The oal connaon vale Obvoly, he bond hold be called a he call prce $8. Therefore, we agn $8 o he bond componen and 0 o he eqy componen. Fnally, we reach he valaon dae. The vale of he eqy componen D The vale of he bond componen exp , exp , The fnal prce of he converble Reference Agarwal, V., W. Fng, Y. Loon and N. Nak, 007, Lqdy provon n he converble bond marke: analy of converble arbrage hedge fnd, FR-workng paper. Ammann, M., Knd, A., and Wlde,., 003, Are converble bond nderprced? An analy of he French marke, Jornal of ankng & Fnance 7, Ammann, M, Knd, A., and Wlde,., 008, mlaon-baed prcng of converble bond, Jornal of emprcal fnance, 5, Anderen, L. and ffm, D., 004, albraon and mplemenaon of converble bond model, Jornal of ompaonal Fnance, 7, -34. Ayache, E., Foryh, P. A., and Vezal, K. R., 003, The valaon of converble bond wh cred rk, Jornal of Dervave,,

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