Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation

Transcription

1 Journal of Financial Economerics Advance Access published July, 9 Journal of Financial Economerics, 9, 4 Sock Opions and Credi Defaul Swaps: A Join Framework for Valuaion and Esimaion Peer Carr Bloomberg LP and New York Universiy Liuren Wu Baruch College, CUNY absrac We propose a dynamically consisen framework ha allows join valuaion and esimaion of sock opions and credi defaul swaps wrien on he same reference company. We model defaul as conrolled by a Cox process wih a sochasic arrival rae. When defaul occurs, he sock price drops o zero. Prior o defaul, he sock price follows a jump-diffusion process wih sochasic volailiy. he insananeous defaul rae and variance rae follow a bivariae coninuous process, wih is join dynamics specified o capure he observed behavior of sock opion prices and credi defaul swap spreads. Under his join specificaion, we propose a racable valuaion mehodology for sock opions and credi defaul swaps. We esimae he join risk dynamics using daa from boh markes for eigh companies ha span five secors and six major credi raing classes from B o AAA. he esimaion highlighs he ineracion beween marke risk (reurn variance) and credi risk (defaul arrival) in pricing sock opions and credi defaul swaps. (JEL: C3, C5, G, G3) We hank George auchen (he edior), he associae edior, wo anonymous referees, Gurdip Bakshi, Philip Brian, Bjorn Flesaker, Dajiang Guo, Pa Hagan, Harry Lipman, Bo Liu, Sheikh Pancham, Louis Sco, and paricipans a Bloomberg, Baruch College, MI, he 5 Credi Risk Conference a Wharon, he 3h annual conference on Pacific Basin Finance, Economics, and Accouning a Rugers Universiy, he 6 Norh American Winer Meeing of he Economeric Sociey a Boson, and he Credi Derivaive Symposium a Fordham Universiy, for commens. Liuren Wu acknowledges parial financial suppor from Baruch College, he Ciy Universiy of New York. Address correspondence o Liuren Wu, Zicklin School of Business, Baruch College, One Bernard Baruch Way, B-5, New York, NY, or liuren.wu@baruch.cuny.edu doi:.93/jjfinec/nbp C he Auhor 9. Published by Oxford Universiy Press. All righs reserved. For permissions, please journals.permissions@oxfordjournals.org.

2 Journal of Financial Economerics keywords: credi defaul swaps, defaul arrival rae, opion pricing, reurn variance dynamics, sock opions, ime-changed Lévy processes Markes for boh sock opions and credi derivaives have experienced dramaic growh in he pas few years. Along wih he rapid growh, i has become increasingly clear o marke paricipans ha sock opion implied volailiies and credi defaul swap (CDS) spreads are posiively linked. Furhermore, when a company defauls, he company s sock price ineviably drops by a sizeable amoun. As a resul, he possibiliy of defaul on a corporae bond generaes negaive skewness in he probabiliy disribuion of sock reurns. his negaive skewness is manifesed in he relaive pricing of sock opions across differen srikes. When he Black and Scholes (973) implied volailiy is ploed agains some measure of moneyness a a fixed mauriy, he slope of he plo is posiively relaed o he risk-neural skewness of he sock reurn disribuion. Recen empirical works, for example, Cremers e al. (8), show ha CDS spreads are posiively correlaed wih boh sock opion implied volailiy levels and he seepness of he negaive slope of he implied volailiy plo agains moneyness. In his paper, we propose a dynamically consisen framework ha allows join valuaion and esimaion of sock opions and credi defaul swaps wrien on he same reference company. We model company defaul as conrolled by a Cox process wih a sochasic arrival rae. When defaul occurs, he sock price drops o zero. Prior o defaul, we model he sock price by a jump-diffusion process wih sochasic variance. he insananeous defaul rae and he insananeous variance rae follow a bivariae coninuous Markov process, wih is join dynamics specified o capure he empirical evidence on sock opion prices and CDS spreads. Under his join specificaion, we propose a racable valuaion mehodology for sock opions and CDS conracs. We esimae he join dynamics of he defaul rae and he variance rae using four years of sock opion prices and CDS spreads for eigh reference companies ha span five secors and six major credi raing classes from B o AAA. Our esimaion shows ha for all eigh companies, he defaul rae is more persisen han he variance rae under boh saisical and riskneural measures. he saisical persisence difference manifess differen degrees of predicabiliy. he risk-neural difference suggess ha he defaul rae has a more long-lasing impac on he erm srucure of opion implied volailiies and CDS spreads han does he variance rae. he esimaion also highlighs he ineracion beween marke risk (sock reurn variance) and credi risk (defaul arrival) in pricing sock opions and CDS, especially for companies wih significan defaul probabiliies. Shocks o he variance rae have a relaively uniform impac on he implied volailiy skew along he moneyness dimension, whereas he impacs of shocks o he defaul arrival rae are larger on opions a low srikes han on opions a high srikes. Along he opion mauriy dimension, he impac of variance rae shocks declines wih increasing opion mauriy, whereas he impac of he defaul risk increases wih

3 CARR &WU Sock Opions and Credi Defaul Swaps 3 i. For companies wih significan defaul probabiliies, he conribuions of he defaul rae and he variance rae are comparable in magniude in cerain segmens of he implied volailiy surface, in paricular a long mauriies and low srikes. he posiive empirical relaion beween sock opion implied volailiies and CDS spreads has been recognized only recenly in he academic communiy. As a resul, effors o capure his linkage heoreically are only in an embryonic sage. Hull, Nelken, and Whie (4) link CDS spreads and sock opion prices by proposing a new implemenaion and esimaion mehod for he classic srucural model of Meron (974). As is well known, his early model is highly sylized as i assumes ha he only source of uncerainy is a diffusion risk in he firm s asse value. As a resul, sock opion prices and CDS spreads have changes ha are perfecly correlaed locally. hus, he empirical observaion ha implied volailiies and swap spreads someimes move in opposie direcions can only be accommodaed by adding addiional sources of uncerainy o he model. When compared o effors based on he srucural model of Meron (974), our conribuion amouns o adding consisen, inerrelaed, bu separae dynamics o he relaion beween volailiy and defaul. he CDS conracs and he sock opions conain overlapping informaion on he marke risk and he credi risk of he company. Our join valuaion and esimaion framework explois his overlapping informaional srucure o provide beer idenificaion of he dynamics of he sock reurn variance and defaul arrival rae. he esimaion resuls highligh he inerrelaed and ye disinc impacs of he wo risk facors on he wo ypes of derivaive securiies. Also relaed o our work is a much longer lis of sudies on he linkages beween he primary equiy and deb markes. hese sudies can be classified ino wo broad approaches. he firs is he srucural modeling approach proposed by Meron (974), who sars wih a dynamic process (geomeric Brownian moion) for he firm s asse value and reas he deb and equiy of he firm as coningen claims on he firm s asse value. he oher approach is ofen ermed as reduced-form, exemplified by anoher classic paper of Meron (976), who recognizes he direc impac of corporae defaul on he sock price process and assumes ha he sock price jumps o zero and says here upon he random arrival of a defaul even. Meron uses he firs approach o analyze he company s capial srucure and is impac on credi spreads, bu he chooses he laer o analyze he impac of corporae Various modificaions and exensions on he deb srucure, defaul riggering mechanisms, firm value dynamics, and implemenaion procedures have been proposed in he lieraure. Prominen examples include Black and Cox (976), Geske (977), Ho and Singer (98), Ronn and Verma (986), iman and orous (989) Kim, Ramaswamy, and Sundaresan (993), Longsaff and Schwarz (995), Leland (994, 998), Anderson and Sundaresan (996), Anderson, Sundaresan, and ychon (996), Leland and of (996), Briys and de Varenne (997), Mella-Barral and Perraudin (997) Garbade (999), Fan and Sundaresan (), Duffie and Lando (), Goldsein, Ju, and Leland (), Zhou (), Acharya and Carpener (), Huang and Huang (3), Hull, Nelken, and Whie (4), Bhamra, Kuehn, and Srebulaev (7), Buraschi, rojani, and Vedolin (7), Chen, Collin-Dufresne, and Goldsein (8), and Cremers, Driessen, and Maenhou (8). Exensions and esimaions of he jump-o-defaul-ype models include Das and Sundaram (4), Carr and Linesky (6), Le (7), and Carr and Wu (8b).

4 4 Journal of Financial Economerics defaul on sock opions pricing. Our work belongs o he laer approach as we focus on he dynamic linkages beween he wo (equiy and credi) derivaives markes. he remainder of his paper is organized as follows. he nex secion proposes a join valuaion framework for sock opions and CDS. Secion describes he daa se and summarizes he sylized evidence ha moivaes our specificaion. Secion 3 describes he join esimaion procedure. Secion 4 presens he resuls and discusses he implicaions. Secion 5 concludes. JOIN VALUAION OF SOCK OPIONS AND CREDI DEFAUL SWAPS We consider a reference company ha has posiive probabiliy of defauling. Le P denoe he ime- sock price for his company. We assume ha he sock price P is sricly posiive prior o defaul and falls o zero upon defaul. Le (Q, F,(F ), Q) be a complee sochasic basis defined on he risk-neural probabiliy measure Q. We assume ha, prior o defaul, he company s sock price is governed by he following sochasic differenial equaion under he risk-neural measure Q, dp /P = (r q + λ ) d + v dw P + (e x )(μ(dx, d) π(x)dxv d), () R where P denoes he ime- pre-jump level of he sock price; r and q denoe he insananeous ineres rae and dividend yield, respecively, which we assume evolve deerminisically over ime; λ denoes he risk-neural arrival rae of he defaul even; and v denoes he insananeous variance rae ha conrols he inensiy of boh he Brownian movemen W P and he jump movemen in sock price prior o defaul. he incorporaion of λ in he drif of he sock price process compensaes for he possibiliy of a defaul, so ha he forward price of he sock remains a maringale uncondiionally under he risk-neural measure. he las erm in Equaion () under he inegral denoes a jump maringale, wih μ(dx, d) couning he number of jumps of size x and π(x)v dxd being is compensaor. he inegral is over all possible jump sizes, defined on he whole real line excluding zero, R. Condiional on he insananeous variance rae level v,he arrival rae of jumps of size x is conrolled by π(x), which we specify as ζ e x/v+ x, x > π(x) = ζ e x /v x, x <, () where ζ conrols he average arrival rae scale and (v +, v ) conrol he average sizes of upside and downside jumps, respecively. Wih v fixed, he π(x) specificaion describes he variance-gamma Lévy jump process sudied in Madan, Carr, and Chang (998). In his model, he jump arrival rae declines monoonically as he absolue jump size declines. he singulariy of he arrival rae a he origin leads o an infinie number of jumps wihin any finie ime inerval. We use his highfrequency jump componen o describe he disconinuous sock price movemens

5 CARR &WU Sock Opions and Credi Defaul Swaps 5 during normal marke condiions, in conras o he rare, bu caasrophic, defaul even ha is conrolled by a Cox process wih arrival rae λ. he pre-defaul sock price dynamics in Equaion () is carefully specified o mach he observed sock price behaviors. Several sudies have found ha high-frequency, infinie-aciviy jumps perform beer han low-frequency, finieaciviy jumps in capuring boh he ime-series behavior of sock and sock index reurns (Carr e al., ; Li, Wells, and Yu, 8) and he cross-secional behavior of sock index opions (Carr and Wu, 3a; Huang and Wu, 4). In line wih such evidence, we include an infinie-aciviy jump componen in he sock price dynamics. Quesions arise on wheher a diffusion componen is sill needed once an infinie-aciviy jump is incorporaed ino he dynamics. In a pure Lévy seing, Carr e al. () and Carr and Wu (3a) find ha i is difficul o idenify a diffusion componen in addiion o an infinie-aciviy jump. On he oher hand, Carr and Wu (3b) consruc a simple and robus es on he presence of jumps and diffusion componens based on he asympoic opion price behavior as he opion mauriy approaches zero. hey find ha a diffusion componen is always presen and priced in he S&P 5 index opions, whereas he addiional conribuion of a jump componen varies over ime. Recenly, odorov and auchen (8) consruc an aciviy signaure funcion from discree observaions of a coninuous process, and show ha he asympoic properies of he funcion as he sampling frequency increases can be used o make inferences on he aciviy behavior of he underlying process. Esimaing he signaure funcion on dollar/mark exchange raes, hey also find supporing evidence for a diffusion componen in addiion o jumps. Hence, we specify a jump-diffusion insead of a pure jump specificaion. We allow boh he insananeous variance rae v and he defaul arrival rae λ o be sochasic, and we model heir join dynamics under he risk-neural probabiliy measure Q as dv = (θ v κ v v ) d + σ v v dw v, (3) λ = βv + z, (4) dz = (θ z κ z z ) d + σ z z dw z, E dw z dw P = E [dw z dw v ] = (5) ρ = E dw P dw v /d. (6) he specificaions are moivaed by boh empirical evidence and economic jusificaion. I is well documened ha sock reurn volailiy is sochasic. We use a square-roo process in Equaion (3) o model he dynamics of he insananeous variance rae of he sock reurn prior o defaul. here is evidence ha credi spreads of a company are posiively relaed o he equiy reurn volailiies of he same company. 3 Equaion (4) capures his posiive relaion hrough a posiive loading coefficien β beween he defaul arrival rae λ and he variance rae v.i is also imporan o accommodae he realiy ha credi spreads someimes move 3 See, for example, Collin-Dufresne, Goldsein, and Marin (), Campbell and aksler (3), Bakshi, Madan, and Zhang (6), Consigli (4), and Zhu, Zhang, and Zhou (5).

6 6 Journal of Financial Economerics independen of he sock and sock opions marke. We use z o capure his independen credi risk componen, wih is dynamics conrolled by an independen square-roo process specified in (5). Finally, when he sock price falls, is reurn volailiy ofen increases. A radiional explanaion ha daes back o Black (976) is he leverage effec. So long as he face value of deb is no adjused, a falling sock price increases he company s leverage and hence is risk, which shows up in sock reurn volailiy. 4 Equaion (6) capures his phenomenon via a negaive correlaion coefficien ρ beween diffusion shocks in reurn and diffusion shocks in reurn variance.. Pricing Sock Opions Consider he ime- value of a European call opion c (P, K, ) wih srike price K and expiry dae. he erminal payoff of he opion is (P K ) + if he company has no defauled by ha ime, and is zero oherwise. he value of he call opion canbewrienas c (P, K, ) = E exp (r s + λ s ) ds (P K ) +, (7) where E [ ] denoes he expecaion operaor under he risk-neural measure Q and condiional on he filraion F. Given he deerminisic ineres rae assumpion, we have c (P, K, ) = B (, ) E exp λ s ds (P K ) + (8) wih B (, ) denoing he ime- value of a defaul-free zero-coupon bond paying one dollar a is mauriy dae. he expecaion can be solved by invering he following discouned generalized Fourier ransform of he pre-defaul sock reurn, ln(p /P ), iuln(p /P ) φ (u) E exp λ s ds e, u D C, (9) where D denoes he subse of he complex plane under which he expecaion is well defined. Under he dynamics specified in Equaions () (6), he Fourier ransform is exponenial affine in he bivariae risk facor x [v, z ] : φ(u) = exp(iu(r(, ) q(, ))τ a(τ) b(τ) x ), τ =, () where r(, ) and q(, ) denoe he coninuously compounded spo ineres rae and dividend yield a ime and mauriy dae, respecively, and he 4 Various oher explanaions have also been proposed in he lieraure; for example, Haugen, almor, and orous (99), Campbell and Henschel (99), Campbell and Kyle (993), Bekaer and Wu (), and Carr and Wu (8a).

7 CARR &WU Sock Opions and Credi Defaul Swaps 7 ime-homogeneous coefficiens [a(τ), b(τ)] are given by θ κ M v η v v η a(τ) = ln ( e v τ ) + η v κv M τ σv η v θ z η z κ z η + ln ( e zτ ) + (η z κ z )τ, () σ η z z [ ] b v ( e ηvτ ) b z ( e ηzτ ) b(τ) =, () η v η κ M ( e ηvτ ) η z (η z κ z )( e ηzτ ) v wih κv M = κ v iuσ v ρ, η v = (κv M ) + σv b v, η z = (κ z ) + σz b z, b z = iu, and b v = ( iu)β + (iu + u )+ζ (ln( iuv + )( + iuv ) iuln( v + )( + v )). Appendix A provides deails of he derivaion. Given φ(u), opion prices can be obained via fas Fourier inversion (Carr and Wu, 4).. Pricing Credi Defaul Swap Spreads he mos acively raded credi derivaive in he over-he-couner marke is a CDS wrien on a corporae bond. he proecion buyer pays a fixed premium, called he CDS spread, o he seller periodically over ime. If a cerain pre-specified credi even occurs, he proecion buyer sops he premium paymens and he proecion seller pays he par value in reurn for he corporae bond. he CDS spread is se a incepion so ha he conrac is cosless o ener. As a resul, he expeced value of he premium paymen leg is se equal o he expeced value of he proecion leg. 5 Consider a CDS conrac iniiaed a ime and wih mauriy dae. Le S(, ) denoe he fixed premium rae paid on his conrac by he buyer of defaul proecion. Assuming one dollar noional and coninuous paymens for simpliciy, we can wrie he presen value of he premium leg of he conrac as Premium(, ) = E S(, ) exp (r u + λ u )du ds (3) wih r and λ denoing he insananeous benchmark ineres rae and defaul arrival rae. Furher, assuming ha upon defaul, he underlying corporae bond recovers a fixed fracion w of is par value, we can wrie he presen value of he proecion leg as Proecion(, ) = E ( w) λ s exp (r u + λ u )du ds. (4) s s 5 For companies wih high defaul probabiliies, he indusry ofen swiches o anoher convenion, under which he proecion buyer pays an upfron fee o he proecion seller wih he periodic premium paymen fixed a 5 basis poins per annum of he noional amoun. A he ime of his wriing, he Norh America CDS marke is going hrough furher reforms o increase he fungibiliy and o faciliae cenral clearing of he conracs. he convenion on virually all conracs is swiching o fixed premium paymens of eiher or 5 basis poins, wih upfron fees o sele he value differences beween he wo legs.

8 8 Journal of Financial Economerics By equaing he presen values of he wo legs, we can solve for he CDS spread S(, ) ha ses he conrac value o zero a iniiaion: s E s ( w) λ s exp (r u + λ u )du ds S(, ) =, (5) E exp (r u + λ u )du ds which can be regarded as a weighed average of he expeced defaul loss. Under he dynamics specified in Equaions (3) (6), we can solve for he presen values of he wo legs of he CDS. he value of he premium leg can be wrien as Premium(, ) = S(, ) B(, s)e exp b λ x u du ds (6) wih b λ = [β,]. he affine dynamics for he bivariae risk facors x and he linear loading funcion b λ dicae ha he presen value of he premium leg is an exponenial affine funcion of he sae vecor (Duffie, Pan, and Singleon, ): Premium(, ) = S(, ) B(, s)exp a λ (s ) b λ (s ) x ds, (7) where he affine coefficiens can be solved analyically: s θ v η v κ v η v τ a λ (τ) = ln ( e ) σv η v + (η v κ v )τ θ z η z κ z η + ln ( e zτ ) σ η z + (η z κ z )τ, (8) z β( e ηvτ ) ( e ηzτ ) b λ (τ) =, (9) η v (η κ v ) ( e ηvτ ) η z (η z κ z )( e ηzτ ) wih η v = (κ v ) + σv β and η z = (κ z ) + σz. he presen value of he proecion leg can be wrien as Proecion(, ) = ( w) B(, s)e b λ x s exp b λ x u du ds, () which also allows for an affine soluion Proecion(, ) = ( w) B(, s)(c λ (s ) + d λ (s ) x ) exp( a λ (s ) b λ (s ) x ) ds, () where he coefficiens (a λ (τ), b λ (τ)) are he same as in (7), and he coefficiens (c λ (τ), d λ (τ)) can also be solved analyically by aking parial derivaives agains (a λ (τ), b λ (τ)) wih respec o mauriy τ: c λ (τ) = a λ (τ)/ τ, d λ (τ) = b λ (τ)/ τ. () s

9 CARR &WU Sock Opions and Credi Defaul Swaps 9 able Lis of companies. Equiy icker Company name Secor Credi raing a C Ciigroup Inc. Financial AA DUK Duke Energy Corporaion Uiliies BBB F Ford Moor Company Consumer Cyclical BB FNM Fannie Mae Financial AAA GM General Moors Corporaion Consumer Cyclical B IBM Inernaional Business Services A Machines Corp MO Alria Group, Inc. Consumer Non-Cyclical BBB A& Inc. Services A/BBB a During our sample period, from May o May 6. Combining he soluions for he presen values of he wo legs in Equaions (6) and () leads o he CDS spread S(, ). When we esimae he model, we discreize he above equaion o accommodae quarerly premium paymens..3 Marke Prices of Risks and ime-series Dynamics Our esimaion procedure idenifies boh he ime-series dynamics and he riskneural dynamics of he bivariae sae vecor x = [v, z ]. o derive he ime-series dynamics for he bivariae vecor x under he saisical measure P, we assume ha he marke prices of risks are proporional o he corresponding risk level. Under his assumpion, he ime-series dynamics are dv = θ v κ P v d + σ v v dw vp, dz = θ z κ P z d + σ z z dw zp, (3) v z where κv P = κ v σ v γ v and κz P = κ z σ z γ v, wih (γ v, γ z ) denoing he wo propor ional marke price of risk coefficiens on he wo risk sources (W v, W z ). DAA AND EVIDENCE We collec daa on CDS spreads and sock opion prices for eigh reference companies from May 8, o May, 6. he choices of he sample period and he company lis are largely deermined by daa availabiliy and coverage. he eigh companies are Ciigroup Inc. (C), Duke Energy Corporaion (DUK), Ford Moor Company (F), Fannie Mae (FNM), General Moors Corporaion (GM), Inernaional Business Machines Corp (IBM), Alria Group, Inc. (MO), and A& Inc. (). able liss he eigh companies, including heir equiy ickers, company names, he secors ha hey belong o, and heir credi raings during our four-year sample period. he eigh companies span six major raing classes from B o AAA, and cover five differen secors including Financials, Uiliies, Consumer Cyclical,

10 Journal of Financial Economerics Services, and Consumer Non-Cyclicals. hus, he eigh companies ha we choose cover a wide specrum of credi raings and indusry secors.. Daa Descripion We obain he CDS spread quoes from several broker dealers. We cross-validae he numbers and ake he quoes from he mos reliable source. he consruced daase includes six ime series for each company a six fixed erms: one, wo, hree, five, seven, and years. he sock opions daa are from OpionMerics. Exchange-raded opions on individual socks are American-syle and hence he price reflecs an early exercise premium. OpionMerics uses a binomial ree o back ou he opion implied volailiy ha explicily accouns for his early exercise premium. We esimae our model specificaion based on heir implied volailiy esimaes. A each ime and mauriy, we ake he implied volailiy quoes of ou-of-he-money opions (call opions when he srike is higher han he spo, and pu opions when srike is lower han he spo) and conver hem ino European opion values based on he Black and Scholes (973) pricing formula. Processing he opions daa involves careful consideraions and delicae choices. Normally, wo opions are available a each mauriy and srike: one call and he oher pu. For European opions, pu-call pariy dicaes ha he pu and he call a he same mauriy and srike have he same ime value and hus he same implied volailiy. For model esimaion, i suffices o pick one of hem as he wo opions conain idenical informaion abou he underlying sock price dynamics. When he wo opions quoes deviae from pu-call pariy due o measuremen errors or marke fricions (such as shor-sale consrains), aking a weighed average of he ime values or implied volailiies of he wo opions can be a useful way o reduce measuremen noise. Since ou-of-he-money opions are more acively raded han in-he-money opions, he quoes on ou-of-he-money opions are usually more reliable. hus, he weigh should be higher on he ou-of-he-money opion han on is in-he-money counerpar. he exac weighing scheme becomes an empirical issue and can vary across markes. he American feaure of he single name opions adds anoher layer of complexiy. Direcly using he model o generae American values is numerically difficul and compuaionally inensive. A commonly used shorcu is o exrac he Black Scholes implied volailiy from he price of an American opion and use he implied volailiy o compue a European opion value for he same mauriy dae and srike. Pu-call pariy does no hold for American opions, nor does i need o hold for he European opion values ha we compued from he American opion prices. Apar from measuremen errors and marke fricions, in-he-money opions ofen have a higher chance of being exercised early and hence have a shorer effecive mauriy A en-year opion o be exercised omorrow only has an effecive mauriy of one day lef. he implied volailiy esimae on each opion reflecs he volailiy over he effecive mauriy horizon. hus, when he implied volailiy has a nonfla erm srucure, he wo implied volailiy esimaes from

11 CARR &WU Sock Opions and Credi Defaul Swaps he American pu and call will no be he same. In his case, we choose o use he ou-of-he-money opion implied volailiy excep for near-he-money conracs. 6 As discussed earlier, ou-of-he-money opions are more acively raded and he quoes are usually more reliable. Furhermore, for American opions, he effecive mauriy of he ou-of-he-money opion is closer o he mauriy of he conrac. hus, he de-americanizaion procedure inroduces smaller approximaion errors for ou-of-he-money opions. o price he CDS conracs and o conver he implied volailiy ino opion values, we also need he underlying ineres rae curve. Following sandard indusry pracice, we use he ineres rae curve defined by he Eurodollar LIBOR and swap raes. We download LIBOR raes a mauriies of one, wo, hree, six, nine, and monhs and swap raes a wo, hree, four, five, seven, and years. We use a piecewise consan forward funcion in boosrapping he discoun rae curve.. Summary Saisics of CDS Spreads able repors he summary saisics of he CDS spreads on he eigh reference companies. Panel A repors he sample averages of he CDS spreads a each of he six mauriies and for each of he eigh companies. A each mauriy, he average spread varies grealy across he eigh reference names. he average spread is he lowes for he AAA-raed Fannie Mae, followed by AA-raed Ciigroup and hen by A-raed IBM. he average spreads on hese A-level companies are less han 5 basis poins across all six mauriies. he nex group are he BBB-raed companies, including Duke Energy, Alria, and A&, wih he credi spreads averaging beween 5 and basis poins. Finally, BB-raed Ford and B-raed General Moors have much higher average spreads over our sample period, ranging from hree o five percenage poins. For all eigh companies, he average spreads a long mauriies are much higher han he average spreads a shor mauriies. he differences generae seeply upward sloping mean erm srucures on he CDS spreads. Panel B of able repors he sandard deviaion esimaes on he CDS spread series. he sandard deviaion esimaes are similar in magniude o he average spreads, suggesing large hisorical variaions on each series. he only excepion is he spread on Fannie Mae, he sandard deviaion esimaes of which are much smaller han he already low mean spread esimaes. We conjecure ha he implici governmen guaranee on he agency deb no only lowers he credi spread level, bu also makes he spread sable over ime. he erm srucures of he sandard deviaions show differen shapes for differen companies, upward sloping for he hree high-raing companies C, FNM, and IBM, downward sloping for DUK, GM, MO, and, and hump-shaped for F. If he credi spread were driven by one srongly mean-revering risk facor, we would expec he sandard deviaions o be lower a longer mauriies and hence 6 We apply equal weigh o he wo a-he-money implied volailiies, bu le he weigh decline rapidly as he opion becomes in-he-money.

12 Journal of Financial Economerics able Summary saisics of credi defaul swap spreads. CDS spreads, in basis poins Years C DUK F FNM GM IBM MO Panel A: Mean Panel B: Sandard deviaion Panel C: Auocorrelaion he saisics are based on weekly sampled daa (every Wednesday) from May 8, o May, 6; observaions for each series. he sandard deviaion erm srucure o be downward sloping. he differen erm srucure shapes observed in CDS spreads sugges ha credi risk facors can be highly persisen under he risk-neural measure. Panel C of able shows ha he weekly auocorrelaion esimaes on he CDS spreads are also very high, ranging from.86 o.98. he high esimaes sugges ha he CDS spreads (and hence credi risk facors) are also highly persisen under he saisical measure. Figure plos he ime series of CDS spreads a seleced mauriies of one year (solid lines), five years (dashed lines), and years (dash-doed lines). Each panel is for one company. Seven of he eigh chosen companies have experienced dramaic credi spread variaions during our sample period. he CDS spreads have spiked for hese companies a leas once during our sample period. he one excepion is Fannie Mae, which shows he sabilizing effec of he implici governmen guaranee. he hree lines in each panel also reveal he CDS erm srucure and is variaions. he long-erm CDS spreads are on average wider han he shor-erm CDS spreads, especially during calm periods; bu he erm srucure can become downward sloping when he CDS spread level spikes.

13 CARR &WU Sock Opions and Credi Defaul Swaps 3 C 5 DUK.8 4 CDS (%).6.4 CDS (%) F FNM.8 CDS (%) CDS (%) CDS (%) CDS (%) CDS (%) CDS (%) GM MO IBM Figure he ime series of CDS spreads a seleced mauriies of one year (solid lines), five years (dashed lines), and years (dash-doed lines). Each panel is for one company.

14 4 Journal of Financial Economerics.3 Summary Saisics of Sock Opion Implied Volailiies he exchange-lised sock opions are quoed a fixed srike prices and expiraion daes. As calendar ime passes and he underlying sock price changes, he moneyness and ime-o-mauriy of each conrac also change. o analyze he crosssecional behavior of he opions across differen levels of moneyness and mauriy, we perform nonparameric regressions on he opion implied volailiies agains he ime o mauriy (τ) and a sandardized moneyness measure d ln(k /P )/(IV τ), where K denoes he srike price and IV denoes he Black Scholes implied volailiy of he opion. We perform nonparameric regression using an independen bivariae Gaussian kernel and a defaul choice of bandwidh ha is proporional o σ x N /6, wih N being he number of observaions and σ x being he sandard deviaion of he regressor x. Figure plos he nonparamerically esimaed mean implied volailiy surface for he eigh companies, one company in each panel. Compared o he large cross-secional variaion of he average CDS spreads across he eigh companies, he average implied volailiy levels vary wihin a narrower range of % 7%. he eigh mean implied volailiy surfaces also share similar shapes. he mean implied volailiy exhibis he well-documened smile paern along he moneyness dimension a shor mauriies, bu his smile gradually becomes a negaively sloped skew a longer mauriies. 7 I has been well appreciaed ha he implied volailiy smiles and skews along he moneyness direcion are direc resuls of condiional non-normaliy in he underlying sock reurns under he risk-neural measure. he posiive curvaure of he smile reflecs fa ails (posiive excess kurosis) in he risk-neural reurn disribuion, whereas he negaive slope of he implied volailiy skew indicaes negaive skewness in he risk-neural reurn disribuion. Under our model specificaion, negaive risk-neural reurn skewness can come from hree sources: (i) posiive probabiliy of defaul (λ >), (ii) asymmery in he high-frequency jump componen (v >v + ), and (iii) negaive correlaion beween he reurn Brownian moion componen and is insananeous variance rae (ρ <). o analyze he ime-series behavior of implied volailiy, we inerpolae o creae implied volailiy esimaes a fixed levels of moneyness and mauriy a each dae. We firs perform a local quadraic regression of he implied variance on he sandard moneyness measure d a each observed mauriy and dae. he local quadraic regression generaes no only he inerpolaed implied variance, bu also he slope and curvaure esimaes of he locally quadraic fi a each moneyness poin. hen, a each fixed moneyness level, we perform linear inerpolaion along he mauriy dimension on he oal variance and he implied variance slope o generae he implied variance and he implied variance skew a fixed mauriies. able 3 repors he summary saisics of he inerpolaed implied volailiy ime series a seleced moneyness levels and mauriies. We choose wo mauriies, 7 Dennis and Mayhew () and Bakshi, Kapadia, and Madan (3) have examined he negaive skew of he implied volailiy plo for individual sock opions.

15 CARR &WU Sock Opions and Credi Defaul Swaps 5 C DUK Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) Implied volailiy (%) F GM 4 3 FNM IBM MO Figure Mean implied volailiy surface across moneyness and ime o mauriy. he mean implied volailiy surface as a funcion of ime o mauriy τ and a sandardized moneyness measure d is esimaed nonparamerically wih an independen bivariae Gaussian kernel. Each panel represens one company.

16 6 Journal of Financial Economerics able 3 Summary saisics of inerpolaed implied volailiy series. Implied volailiies, in percenages Days d C DUK F FNM GM IBM MO Panel A: Mean Panel B: Sandard deviaion Panel C: Auocorrelaion Enries repor he sample esimaes of he mean, sandard deviaion, and weekly auocorrelaion on inerpolaed implied volailiy series a seleced fixed moneyness levels and mauriies. We firs perform a local quadraic nonparameric regression of he implied variance on moneyness o obain implied variance a fixed moneyness levels for observed mauriies. he moneyness of each opion is defined as d ln(k /P )/(IV τ), where K denoes he srike, P he sock price level, IV he implied volailiy esimae, and τ he ime o mauriy. hen, a each fixed moneyness level, we inerpolae across he mauriy dimension using piecewise linear inerpolaion on he oal variance. he saisics on he inerpolaed series are based on weekly sampled daa (every Wednesday) from May 8, o May, 6; observaions for each series. one a he shor end a one monh (3 days), and he oher a he long end a one year (36 days). A each of he wo mauriies, we choose hree moneyness levels a d =,,. Approximaely speaking, d = corresponds o a srike ha is one sandard deviaion below he curren spo price level and d = corresponds o a srike ha is one sandard deviaion above he curren spo price level. Panel A repors he sample average of he implied volailiy levels. he average implied volailiy levels are mosly in he range of % 5%, wih he averages for BB-raed Ford and B-raed General Moors higher han he averages for he oher companies wih higher credi raings. A each mauriy and for each company, he average implied volailiy a d = is higher han he average implied

17 CARR &WU Sock Opions and Credi Defaul Swaps 7 able 4 Co-movemens of credi spreads wih sock opion implied volailiies and implied variance skews. Days d C DUK F FNM GM IBM MO Panel A: Implied volailiy Panel B: Negaive implied variance skew Enries repor he cross-correlaion esimaes beween weekly changes in he average CDS spreads for each company and weekly changes in he sock opion implied volailiy (panel A) and he negaive of he implied variance skew (panel B) across differen mauriies and moneyness. We use a simple average of he six CDS series as an average credi spread series for each company. o obain he implied volailiy and skew series a fixed ime o mauriies and moneyness, we firs perform a local quadraic regression of he implied variance on he moneyness d o obain implied variance and is slope a fixed moneyness levels for observed mauriies. hen, a each fixed moneyness level, we linearly inerpolae on oal variance and he skew o obain he implied variance and skew a fixed ime o mauriies. he saisics are based on weekly sampled daa (every Wednesday) from May 8, o May, 6; observaions for each series. volailiy a d =, consisen wih he negaively sloped skew observed in Figure. A each fixed moneyness, he average implied volailiy does no show much variaion across he wo mauriies. By conras, he sandard deviaion esimaes repored in panel B show significanly smaller magniudes a longer mauriies. he downward sloping sandard deviaion erm srucure is consisen wih mean-revering variance risk dynamics under he risk-neural measure. he las panel (panel C) in able 3 repors he weekly auocorrelaion esimaes of he implied volailiy series. he esimaes are high, suggesing ha he implied volailiy series are highly persisen in heir ime-series dynamics..4 Co-Movemens beween Opion Implied Volailiies and CDS Spreads o analyze how a company s CDS spreads co-move wih he company s sock opions, able 4 measures he cross-correlaion of he weekly changes in he average CDS spread for a company wih weekly changes in he sock opion implied volailiy levels (panel A) and he implied variance skews (panel B) of he same

18 8 Journal of Financial Economerics company a differen levels of moneyness and mauriy. For each company and a each dae, we use a simple average of he six CDS quoes a he six mauriies o represen he average CDS spread for he company. he esimaes in panel A show ha he correlaions beween he credi spreads and he sock opion implied volailiies are universally posiive across all companies, all mauriies, and all moneyness levels. For each company, he correlaion esimaes are in general higher a low srikes (d = ) han a high srikes (d = ). he esimaes are also higher a one year han a one monh mauriies. Under our model specificaion, his posiive correlaion can come from wo major sources. Firs, he posiive loading coefficien β in Equaion (4) generaes a direc posiive linkage beween sock reurn variance and he defaul arrival rae. Second, he defaul arrival rae iself conribues posiively o he opion implied volailiy. he implied variance skew esimaes are predominanly negaive, especially a long mauriies and low srikes. Panel B of able 4 repors he correlaion esimaes beween weekly changes in he credi spread and weekly changes in he negaive of he implied variance skew a differen mauriies and moneyness. he esimaes are again universally posiive, suggesing ha when a company s credi spread widens, is implied variance skew becomes more negaively skewed. Overall, he implied variance skew a longer mauriies show higher correlaion wih he credi spread. In Figure 3, we overlay he ime series of he average CDS spread (solid line) wih he one-year sock opion implied volailiy a d = (dashed line) and he negaive of he one-year implied variance skew a d = (dash-doed line), one panel for each company. o accommodae he scale differences in he same plo, we normalize each ime series o have uni sample sandard deviaion. he comparaive ime-series plos show ha for each company, he CDS series show posiive co-movemens wih boh he implied volailiy and he implied variance skew ime series. hey also show variaions independen of one anoher. Boh he correlaion esimaes in able 4 and he ime-series plos in Figure 3 show ha he credi spread is inricaely relaed o he equiy opions marke. he linkages ask for a dynamically and inernally consisen heoreical framework o joinly model he dynamics and pricing of defaul arrival raes and sock reurn variance. Our model does jus ha, and i accommodaes boh he posiive comovemens and heir separae variaions hrough he bivariae specificaions in Equaions (3) (5). 3 JOIN ESIMAION OF MARKE AND CREDI RISK DYNAMICS We esimae he bivariae risk dynamics x = [v, z ] joinly using boh CDS spreads and sock opions. We cas he model ino a sae-space form and esimae he model using a quasi-maximum likelihood mehod. In he sae-space form, we regard he bivariae risk vecor as he unobservable saes and specify he sae propagaion equaion as an Euler approximaion of he

19 CARR &WU Sock Opions and Credi Defaul Swaps 9 6 C 7 DUK Normalized ime series Normalized ime series Normalized ime series Normalized ime series F FNM GM IBM Normalized ime series Normalized ime series Normalized ime series Normalized ime series MO Figure 3 Co-movemens of CDS spreads wih sock opion implied volailiies and implied variance skews. Each panel represens one company. he hree lines in each panel denoe he normalized ime series of he average CDS spreads (solid lines), he one-year implied volailiies (dashed lines), and he negaive of he implied variance skew (dash-doed lines). Each ime series is normalized o have uni sample sandard deviaion.

20 Journal of Financial Economerics ime-series dynamics in Equaion (3): [ ] θ e κp v v σv v x = + θ e κ P x + z z σ z z ε, (4) where ε denoes an iid bivariae sandard normal innovaion and = 7/365 denoes he sampling frequency. We consruc he measuremen equaion based on CDS spreads and sock opions, assuming addiive, normally disribued measuremen errors y = h(x ; 8) + e, (5) where y denoes he observed series and h(x ; 8) denoes he corresponding model value as a funcion of he sae vecor x and model parameers 8. Each day, he measuremen equaion conains six CDS spread quoes a six differen mauriies. We scale each CDS series by is sample average and hen assume ha he pricing errors on he six scaled CDS series are iid normal wih variance σ C. he number of opion observaions varies across differen daes and differen companies. he esimaion includes opions wih a minimum of days o expiraion and srike prices wihin wo sandard deviaions of he spo ( d ). he average number of opions per day included in he esimaion ranges from 5 for Ford o 59 for General Moors. he opion mauriy for each company ranges from he minimum requiremen of days o abou.5 years, wih he median mauriy varying from 9 o 9 days for differen companies. We scale he ou-of-he-money European opion values a each srike price and mauriy by is Black Scholes vega, and assume ha he scaled pricing errors on all opions are iid normal wih variance σo. When boh he sae propagaion equaion and he measuremen equaion are Gaussian and linear, he Kalman (96) filer generaes efficien forecass and updaes on he condiional mean and covariance of he sae vecor and he measuremen series. In our applicaion, he sae propagaion equaion in (4) is Gaussian and linear, bu he measuremen equaion in (5) is nonlinear. We use he unscened Kalman filer (Wan and van der Merwe, ) o handle he nonlineariy. he unscened Kalman filer approximaes he poserior sae densiy using a se of deerminisically chosen sample poins (sigma poins). hese sample poins compleely capure he mean and covariance of he Gaussian sae variables, and when propagaed hrough he nonlinear funcions in he measuremen equaion, capure he poserior mean and covariance of he CDS spreads and opion prices accuraely o he second order for any nonlineariy. Appendix B provides he echnical deails for he filering mehodology. We consruc he log-likelihood value assuming normally disribued forecasing errors. Furhermore, since we use differen numbers of opions and CDS spreads in he esimaion, he esimaed dynamics end o bias oward he marke wih more daa observaions. o correc for his bias and o assign approximaely equal weighs o boh markes, we separaely calculae he weekly likelihood

21 CARR &WU Sock Opions and Credi Defaul Swaps values on he opions (l O ) and he CDS spreads (l C ), and we divide he wo likeli O C hood values by he number of opion (n )andcds(n ) observaions in ha week, respecively. hen, we maximize he sum of he rescaled log-likelihood values over he whole daa series o esimae he model parameers: N C O 8 arg max l C (8)/n + l C (8)/n, (6) 8 = where 8 denoes 6 model parameers o be esimaed: 8 [κ v, κ z, κv P, κz P, θ v, θ z, σ v, σ z, β, ρ, ζ, v +, v, w, σ C, σ O ] and N = denoes he number of weeks in our sample. 4 RESULS AND DISCUSSION Firs, we summarize he performance of our join valuaion model on CDS spreads and sock opions on he eigh reference companies. hen, from he esimaes of he model parameers, we discuss he join dynamics and pricing of he reurn variance risk and defaul arrival risk, and heir impacs on he CDS spread erm srucure and he opion implied volailiy surface. 4. Performance Analysis able 5 repors he summary saisics of he pricing errors on he credi defaul swap spreads and opion implied volailiies. We repor he pricing errors on he CDS spreads a each of he six fixed erms and he pricing errors on he implied volailiies as one pooled series for each company. he pricing errors are defined as he differences beween he daa observaions (CDS spreads and opion implied volailiies, boh in percenage poins) and he corresponding model values. Panel A of able 5 repors he sample averages of he pricing errors. he mean pricing errors for he CDS spreads do no show any obvious srucure, excep for Alria and A&, where he mean errors show a posiive mean bias across all mauriies. he mean bias is abou 5 basis poins for Alria and 6 basis poins for A&. he mean biases on he opion implied volailiies are all posiive bu small, less han half a volailiy percenage poin for all companies, and less han one-enh of a volailiy poin for Alria and A&. o learn how he mean opion pricing errors vary across he opion moneyness and mauriy specrum, we perform nonparameric regressions on he pricing errors as a funcion of moneyness d and mauriy, using he same mehodology as we have done o obain he mean implied volailiy surface in Figure. Figure 4 plos he mean pricing error surfaces, one panel for each company. he shapes of he mean pricing error surface vary across differen reference companies, wih no sysemaic paern. he larges mean pricing errors come from Alria and A&, wih he mean errors negaive a low srikes (negaive d), bu posiive a high srikes (posiive d). he biases are sronger a longer mauriies. he mean bias paern across he moneyness dimension suggess ha he observed implied volailiies are no as negaively skewed as he model-implied values. Under our model

22 Journal of Financial Economerics able 5 Summary saisics of pricing errors on CDS spreads and opion implied volailiies. C DUK F FNM GM IBM MO Panel A: Mean pricing errors IV Panel B: Mean absolue pricing errors IV Panel C: Explained percenage variaion IV Enries repor he mean pricing error (panel A), mean absolue pricing error (panel B), and explained variaion (panel C) on CDS spreads and opion implied volailiies. he pricing errors are defined as he difference beween he observed CDS spreads and implied volailiies, boh in percenage poins, and heir model-implied values. he explained variaion is defined as one minus he raio of he variance of he pricing error o he variance of he original series. he saisics on he credi defaul swap spreads are a each of he six fixed erms, and he saisics on he implied volailiies is on one pooled series across all mauriies and srikes for each company. specificaion, a highly negaive skew a long opion mauriies can be generaed from a high defaul arrival rae; ye, he posiive mean pricing errors on he CDS spreads on hese wo companies sugges ha he esimaed defaul arrival raes are no high enough o mach he observed CDS spreads. aken ogeher, he CDS and he opions markes for he wo companies show a degree of pricing ension wihin our modeling framework: he CDS spreads imply higher defaul arrival raes han hose revealed from he opion implied volailiy skews. Panel B of able 5 repors he mean absolue pricing errors. he mean absolue pricing errors on he CDS spreads are below five basis poins for he high-raing

23 CARR &WU Sock Opions and Credi Defaul Swaps 3 C DUK Mean pricing error (%) Mean pricing error (%) Mean pricing error (%) Mean pricing error (%) F.5.5 Mean pricing error (%) Mean pricing error (%) Mean pricing error (%) Mean pricing error (%) FNM GM MO IBM Figure 4 Mean pricing error in implied volailiy across moneyness and ime o mauriy. he pricing error is defined as he difference beween he observed implied volailiy and he corresponding model values in volailiy percenage poins. he mean pricing error as a funcion of ime o mauriy τ and a sandardized moneyness measure d is esimaed nonparamerically wih an independen bivariae Gaussian kernel. Each panel represens one company.