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

Transcription

1 Sock Opions and Credi Defaul Swaps: A Join Framework for Valuaion and Esimaion ABSTRACT 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 Poisson process wih a sochasic defaul arrival rae. When defaul occurs, he sock price drops o zero. Prior o defaul, he sock price follows a coninuous process wih sochasic volailiy. The insananeous defaul rae and insananeous diffusion variance rae follow a bivariae coninuous Markov process, wih is dynamics specified o capure he empirical evidence on sock opion prices and credi defaul swap spreads. Under his join specificaion, we derive racable pricing soluions for sock opions and credi defaul swaps. We esimae he join dynamics using sock opion prices and credi defaul swap spreads for four of he mos acively raded reference companies. Our esimaion shows ha for all four reference companies, he defaul rae is much more persisen han he diffusion variance rae under boh he ime series measure and he risk-neural measure. Furhermore, changes in diffusion variance are posiively relaed o conemporaneous and subsequen changes in he defaul rae. Finally, he esimaion reveals ha he marke price of defaul arrival risk is negaive while he marke price of diffusion variance risk is posiive, suggesing ha for individual socks, shoring credi defaul swaps generaes a higher average reurn per uni risk han shoring variance swap conracs. JEL Classificaion: C13; C51; G12; G13. Keywords: Sock opions; credi defaul swaps; defaul arrival rae; reurn variance dynamics; opion pricing; ime-changed Lévy processes. 2

2 Sock Opions and Credi Defaul Swaps: A Join Framework for Valuaion and Esimaion 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 spreads are inherenly linked. Many academic sudies have also empirically documened he posiive link beween credi spreads and sock volailiy a boh he firm level and he aggregae level. 1 Ineresingly, his empirical relaionship has been presaged by classical asse pricing heory. According o he classic srucural model of Meron (1974), corporae bond credi spreads are funcions of financial leverage and firm asse volailiy, boh of which have posiive impacs on he volailiy of he underlying company s sock and hence on sock opion implied volailiies. 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. This negaive skewness is manifesed in he relaive pricing of sock opions across differen srikes. When he Black and Scholes (1973) implied volailiy is ploed agains some measure of moneyness a a fixed mauriy, he average slope of he plo is posiively relaed o he risk-neural skewness of he sock reurn disribuion. Dennis and Mayhew (22) and Bakshi, Kapadia, and Madan (23) examine he negaive skew of he implied volailiy plo for individual sock opions. Recen empirical work, e.g., Cremers, Driessen, Maenhou, and Weinbaum (24), shows ha credi defaul swap raes 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 Poisson process wih a sochasic arrival rae. When defaul occurs, we le he sock price drop o zero. Prior o defaul, we model he sock price by a coninuous process wih sochasic volailiy. The 1 Examples include Bevan and Garzarelli (2), Pedrosa and Roll (1998), Collin-Dufresne, Goldsein, and Marin (21), Bangia, Diebold, Kronimus, Schagen, and Schuermann (22), Capmbell and Taksler (23), Alman, Brady, Resi, and Sironi (24), Bakshi, Madan, and Zhang (24), Ericsson, Jacobs, and Oviedo-Helfenberger (24), Hilscher (24), Consigli (24), and Zhu, Zhang, and Zhou (25).

3 insananeous defaul rae and insananeous diffusion variance rae follow a bivariae coninuous Markov process, wih is join dynamics specified o capure he empirical evidence on sock opion prices and credi defaul swap spreads. Under his join specificaion, we derive racable pricing soluions for sock opions and credi defaul swaps. We hen esimae he join dynamics of defaul rae and diffusion variance rae using sock opion prices and credi defaul swap spreads for four of he mos acively raded reference companies. Our esimaion shows ha for all four reference companies, he defaul rae is more persisen han he diffusion variance rae under boh he ime-series and he risk-neural measures. Furhermore, changes in diffusion variance are posiively relaed o conemporaneous and subsequen changes in he defaul rae. Finally, comparing he ime-series and risk-neural dynamics reveals ha he marke price of he defaul arrival rae risk is negaive, bu he marke price for diffusion variance risk is posiive. Previous works such as Bakshi and Kapadia (23) and Carr and Wu (24b) have sudied he marke price of aggregae variance risk and have found ha a leas for sock indexes, he risk premia are negaive. Our resuls in his paper sugges ha for individual socks, negaive variance risk premia are mainly due o he variance risk generaed from downside jumps. From a pracical perspecive, a negaive aggregae variance risk premium suggess ha selling variance swap conracs generaes posiive average reurns. The negaive defaul risk premium versus he posiive diffusion variance risk premium on he four socks furher sugges ha selling credi insurance using credi defaul swap conracs generaes an even higher rae of average reurn per uni risk. The posiive empirical relaion beween credi defaul swap spreads and sock opion implied volailiies has been recognized only very recenly in he academic communiy. As a resul, effors o heoreically capure his link are only in an embryonic sage. In a recen working paper, Hull, Nelken, and Whie (24) link credi defaul swap spreads and sock opion prices by proposing a new implemenaion and esimaion mehod for he classic srucural model of Meron (1974). As i is well known, his early model is highly sylized as i assumes ha he only source of uncerainy is he firm s asse value. As a resul, sock opion prices and credi defaul swap spreads have changes ha are perfecly correlaed locally. Thus, 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. 2

4 In his paper, we assume ha prior o defaul, he sock price process is coninuous. Boh he drif and diffusion of his process are sochasic as we assume ha he defaul arrival rae and diffusion variance rae obey a bivariae dynamic process. As a resul, we are able o capure he imperfec posiive correlaion beween sock volailiy and defaul risk. Thus, when compared o effors based on he srucural model of Meron (1974), our conribuion amouns o adding consisen and inerrelaed bu separae dynamics o he relaion beween volailiy and defaul. This separaion of he wo sources of risk allows us o esimae and compare he marke prices of hese risks, a fea ha canno possibly be achieved by daily saic calibraion of simpler models. The dynamic consisency embedded in our specificaion and esimaion is imporan, as i provides a sable framework for dynamic hedging and inegraed invesmen across he wo markes. I also generaes insighs on how differen sources of risks are priced and how hey affec he dynamics of credi spreads and opion prices differenly. The res of he paper is organized as follows. The nex secion proposes a join valuaion framework for sock opions and credi defaul swaps. Secion 2 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 heir implicaions. Secion 5 concludes. 1. Join Valuaion of Sock Opions and Credi Defaul Swaps Consider a reference company ha has a posiive probabiliy of defauling. Le λ() denoe he arrival rae of he defaul even, which we allow sochasic. Le P denoe he ime- sock price for his company, which we assume falls o zero upon defaul. Le (Ω,F,(F ),Q) be a complee sochasic basis and Q be a risk-neural probabiliy measure, under which he sock price dynamics prior o company defaul follows: dp /P = (r() q()+λ())d + v()dw s, (1) where r() and q() denoe he insananeous ineres rae and dividend yield, respecively, which we assume deerminisic, W s denoes a sandard Brownian moion, and v() denoes he insananeous variance rae for he sock diffusion reurn componen, which we also allow sochasic. The incorporaion of λ() in he drif compensaes for he possibiliy of a defaul, so ha he sock price remains a maringale uncondiionally 3

5 under he risk-neural measure. Thus, boh he drif and he diffusion of his pre-defaul sock price process are sochasic Join dynamics of diffusion variance rae and defaul arrival rae We model he join dynamics of he defaul arrival rae and he diffusion reurn variance rae under he risk-neural probabiliy measure Q as follows: dv() = (θ v κ v v())d + σ v v()dw v, (2) dz() = (θ z κ z z() κ zv v())d + σ z z()dw z, (3) λ() = z()+ξv(), (4) ρ sv E[dW s dw v ]/d <, ρ sz E[dW s dw z ] =, ρ zv E[dW z dw v ] =. (5) The specificaions are moivaed by boh empirical evidence and economic raionale: Cremers, Driessen, Maenhou, and Weinbaum (24) find ha opion volailiy predic credi defaul swap spreads. Our own empirical analysis finds similar evidence. Equaion (3) capures his predicabiliy via he cross drif erm, κ zv v(). A negaive esimae for κ zv would indicae ha diffusion reurn variance v posiively predics changes in he risk facor z(), which feeds direcly ino he defaul arrival rae λ() via equaion (4). Anecdoal evidence shows ha when concerns of defaul arise for a company, he sock price for ha company ofen becomes more volaile due o marke jiers. Our empirical analysis also reveals posiive conemporaneous correlaion beween daily changes in opion implied volailiy and credi defaul swap spreads. Equaion (4) capures he posiive conemporaneous correlaion via a posiive loading coefficien esimae ξ beween he defaul arrival rae λ() and he diffusion reurn variance rae v(). When sock price falls, is reurn volailiy ofen increases. A radiional explanaion ha daes back o Black (1976) for his well-documened phenomenon is he leverage effec. Falling sock price 4

6 increases he company s leverage and hence is risk, which shows up in sock reurn volailiy. 2 Equaion (5) capures his phenomenon via a negaive correlaion coefficien ρ sv beween diffusion shocks in reurn and reurn variance. Furhermore, his increased volailiy also raises he defaul arrival rae boh conemporaneously via equaion (4) wih ξ > and subsequenly via equaion (3) wih κ zv <. We assume ρ sz = ρ zv = for parsimony and racabiliy. In marix noaion, we can wrie he bivariae process as dx = (θ κx )d + βx dw, (6) wih x = v z, κ = κ v κ zv κ z, θ = θ v θ z, β = σ2 v σ 2 z, W = W v W z. (7) 1.2. Pricing sock opions Consider a claim ha pays off V (P T ) a expiry T if he company does no defaul before expiry and ϖ a he ime of defaul any ime before expiry. The valuaion of such a coningen claim can be wrien as, ( V (P,K,T) = E [exp +E [ϖ T T ) ] (r(s)+λ(s))ds V (P T ) ( s (r(u)+λ(u))du λ(s)exp ) ] ds, (8) where he expecaion operaor E [ ] is under he risk-neural measure Q and condiional on he filraionf. 2 Various oher explanaions have also been proposed in he lieraure, e.g., Haugen, Talmor, and Torous (1991), Campbell and Henschel (1992), Campbell and Kyle (1993), and Bekaer and Wu (2). 5

7 Now consider he value of a European call opion c(p,k,t) as an example. The erminal payoff is (P T K) + a mauriy T if he company has no defauled ye by ha ime. The payoff is zero oherwise. The valuaion becomes: ( c(p,k,t) = E [exp T = B(,T)E [exp ] (r(s)+λ(s))ds )(P T K) + ( T ] λ(s)ds )(P T K) +, (9) where B(,T) is he ime- discoun facor wih mauriy dae T and he expecaion can be solved by invering he following discouned generalized Fourier ransform, ( φ(u) E [exp T ] λ(s)ds )e iulnp T/P, u D C, (1) where D denoes he subse of he complex plane under which he expecaion is well-defined. Under our dynamic specificaions in equaions (1) o (5), he Fourier ransform is exponenial affine in he bivariae sae vecor x : ) φ(u) = exp (iu(r(,t) q(,t))τ a(τ) b(τ) x, τ = T, (11) where r(,t) and q(,t) denoe he coninuously compounded spo ineres rae and dividend yield rae a ime and mauriy dae T, respecively, and he coefficiens [a(τ),b(τ)] can be solved from he following se of ordinary differenial equaions: a (τ) = b(τ) θ, b (τ) = b (κ M ) b(τ) 1 2 β(b(τ) b(τ)), (12) saring a a() = and b() =, wih denoing he Hadamard produc, and b = (1 iu)ξ+ 1 ( 2 iu+u 2 ) 1 iu, κ M = κ v iuσ v ρ κ zv κ z. (13) Appendix A provides deails of he derivaion. The ordinary differenial equaions can be solved readily using sandard numerical procedures. Given φ(u), opion prices can be obained via fas Fourier inversion (Carr and Wu (24a)). 6

8 1.3. Pricing credi defaul swap spreads For a credi defaul swap conrac iniiaed a ime and wih mauriy dae T, we use S(,T) o denoe he premium (he spread ) paid by he buyer of defaul proecion. Wih coninuous paymen assumpion, he presen value of he premium leg of he conrac is, Premium(,T) = E [S(,T) T ( exp s ) ] (r(u)+λ(u))du ds. (14) Similarly, he presen value of he proecion leg of he conrac is Proecion(,T) = E [w T ( λ(s) exp s ) ] (r(u)+λ(u))du ds, (15) wih (1 w) denoing he recovery rae. Hence, by seing he presen values of he wo legs equal, we can solve for he credi defaul swap spread as S(,T) = [ Ê Ê ] T s E w λ(s)exp( (r(u)+λ(u))du)ds E [ Ê T exp( Ê s (r(u)+λ(u))du)ds ], (16) which can be regarded as a weighed average of he expeced defaul loss. In model esimaion, we discreize he above equaion according o quarerly premium paymen inervals. We se he recovery rae fixed a 4 percen, he average recovery rae for senior unsecured debs and also an indusry sandard number for credi defaul swap valuaion. Under our defaul arrival dynamics specificaion in equaions (2) o (5), we can solve he presen values of he wo legs of he conrac and hence he credi defaul swap spread. The value of he premium leg is Premium(,T) = S(,T) = S(,T) T T s ( ) ] E [exp (r(u)+λ(u))du ds ( s )] B(,s)E [exp b λ x udu ds, (17) 7

9 wih b λ = [ξ,1]. The affine dynamics for he bivariae vecor 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 (2)): Premium(,T) = S(,T) T ) B(,s)exp ( a λ (s ) b λ (s ) x ds, (18) wih a λ (τ) = b λ(τ) θ, b λ (τ) = b λ κ b λ (τ) 1 2 β(b λ(τ) b λ (τ)), (19) saring a a λ () = and b λ () =. Similarly, he presen value of he proecion leg is Proecion(,T) = E [w = w T T ( B(, s)λ(s) exp [ ( ) B(,s)E b λ x s exp s s ( ) λ(u)du ] ds )] b λ x udu ds, (2) which also falls ino he affine srucure and hence also has an exponenial affine soluion: Proecion(,T) = w T ) ) B(, s) (c(s )+d(s ) x exp ( a λ (s ) b λ (s ) x ds, (21) where he coefficiens (a λ (τ),b λ (τ)) are he same as in (19), and he coefficiens (c(τ),d(τ)) can be solved from he following se of ordinary differenial equaions: c (τ) = d(τ) θ, d (τ) = κ d(τ) β(b λ (τ) d(τ)), (22) saring a c() = and d() = b λ. Combining he soluions for he presen values of he wo legs in equaions (17) and (21) solves for he credi defaul swap spread S(,T). 8

10 1.4. Marke prices of risks and ime-series dynamics Our join esimaion idenifies boh he ime-series dynamics and he risk-neural dynamics of he bivariae sae vecor x. To 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 risk level, γ x, wih γ being a diagonal marix. Under his assumpion, he ime-series dynamics are, ) dx = (θ κ P x d + βx dw P, (23) wih κ P = κ βγ, and γ = γ v γ z. (24) 2. Daa and Evidence Boh he sock opion prices and he credi defaul swap spreads are funcions of he bivariae vecor x() = [v(; z()], which joinly deermines he sock diffusion variance and he defaul arrival rae. Therefore, we can use daa on sock opion prices and credi defaul swap spreads o infer he join dynamics Daa descripion We esimae he model using credi defaul swap spreads and sock opions on four reference companies. Bloomberg provides he credi defaul swap spreads quoes from several broker dealers. We use he quoes from differen broker dealers for cross-validaion. Then, we ake he quoes on each series from he mos reliable sources. We choose four companies under which he credi defaul swap quoes have boh a long hisory and frequen updaes. The four companies are: Ford (F), General Moors (GM), Alria Group Inc (MO), and Duke Energy Corp (DUK). For each company, we have credi defaul swap spread series a five fixed mauriies of one, hree, five, seven, and en years. The corresponding sock opions daa are from OpionMerics. Exchange-raded opions on individual socks are American syle and hence he price conains an early exercise premium. OpionMerics uses 9

11 a binomial ree approach o back ou he opion implied volailiy ha explicily accouns for he early exercise premium. We direcly use he implied volailiy from OpionMerics. For model esimaion, we conver he implied volailiy ino European opion prices using he Black-Scholes formula. For each sock, he OpionMerics provides a sandardized implied volailiy surface a fixed Black-Scholes forward delas from 2 o 8 wih a five-dela inerval for boh call and pu opions, and fixed opion mauriies of 3, 6, and 91 days. OpionMerics esimaes he implied volailiy surface via a kernel smoohing approach whenever he underlying quoes are available and lef as missing values when here are no enough quoes o make he smoohing esimaion. Daa a longer mauriies are also available bu only very sparsely. Hence we only use he firs hree mauriies. No-arbirage dicaes ha he implied volailiy compued from pu and call opion prices a he same srike should be he same. Neverheless, he implied volailiy esimaes from OpionMerics are ofen differen from calls and pus a similar srikes, possibly resuling from bid-ask spreads, misalignmens of ineres raes and dividend yields, and/or errors induced by he smoohing mehod and he binomial ree approach in obaining implied volailiies. For esimaion, we ake he average of he wo implied volailiy a each srike and conver hem ino ou-of-money opion prices. To price he credi defaul swap conracs and o conver he implied volailiy ino opion prices, we also need he underlying ineres rae curve. Following sandard pracice in he indusry, 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 12 monhs and swap raes a wo, hree, four, five, seven, and en years. We use a piece-wise consan forward funcion in boosrapping he discoun rae curve Summary saisics Model esimaion uses he common samples of he hree daa ses ha from January 2, 22 o April 3, 24. The daa are available in daily frequency, bu we esimae he model using weekly-sampled daa on every Wednesday o avoid he impacs of weekday effecs. Table 1 repors he summary saisics of he credi defaul swap spreads on he four reference companies. The mean erm srucures of he spreads are relaively fla for all four companies, bu he sandard deviaions of he spreads for all four companies decline wih increasing mauriies. The weekly auocorrelaion esimaes for he spreads range from.9 o.97, showing ha he swap spreads and hence he defaul arrival rae dynamics are highly persisen. 1

12 Table 2 repors he summary saisics of sock opion implied volailiies a he hree fixed mauriies and 13 fixed pu-opion delas for each of he four reference companies. For each company and a each opion mauriy, he implied volailiies a low srikes (low pu delas) are on average higher han he implied volailiies a high srikes, generaing a negaively sloped average implied volailiy smirk across moneyness. The sandard deviaions of he implied volailiy series are also larger for ou-of-money pus han for ou-of-money calls, bu he difference is smaller han he difference in he mean esimaes. The weekly auocorrelaion for he volailiy series range from.69 o.93, indicaing ha he implied volailiies are persisen, bu less so han he credi defaul swap spreads. Figure 1 plos he average implied volailiy smirk a he hree fixed mauriies as a funcion of he pu opion dela. For all he four reference companies and under all hree fixed mauriies, he average implied volailiy smirk is negaively skewed, corresponding o a negaively skewed risk-neural sock reurn disribuion. The hree lines in each panel, which correspond o he hree opion mauriies, say closely o one anoher, suggesing ha he condiional risk-neural disribuion of he sock reurn reains similar shapes a he hree condiioning horizons. Generically, our model specificaion can generae he negaive skewness from wo sources: (1) a posiive probabiliy of defaul (λ() > ) and (2) a negaive correlaion beween he reurn Brownian moion componen and is insananeous variance rae (ρ sv < ). [Figure 1 abou here.] 2.3. Co-movemens beween opion implied volailiies and credi defaul swap spreads Figure 2 overlays he daily ime series of he credi defaul swap spreads (solid lines) wih he daily ime series of a-he-money (5 dela) sock opion implied volailiies a he hree fixed opion mauriies (dashed lines) for he four chosen reference companies. We observe apparen common movemens for wo ypes of ime series for each company. The co-movemens are he mos obvious during financial disresses for he company, during which he wo ses of ime series boh spike up. [Figure 2 abou here.] 11

13 To quanify he co-movemens, we choose he five-year credi defaul swap spread series (S ) and he hree-monh 5-dela implied volailiy series (AT MV ) and run he following regression on he daily series: S = a+bat MV + e. (25) Table 3 repors he parameer esimaes, -saisics, and R 2 of he regression for each of he four reference companies. Figure 3 shows he corresponding scaer plos and fied lines. The significanly posiive slope esimaes confirm he posiive co-movemens beween he wo series. Neverheless, we also observe ha he R 2 esimaes of he regressions vary significanly across differen reference companies. The esimaes are larges a 82 percen for Ford, and lowes a only 36 percen for Alria Group. The variaions across differen companies and he low R 2 esimaes in some insances sugges ha alhough reurn variance and defaul arrival share common movemens, hey also have heir own independen movemens. The relaive srengh of co-movemens may vary across differen reference companies. 3 From he modeling perspecive, i is imporan o capure no only he common movemens beween he wo markes bu also he idiosyncraic movemens in each marke. Our bivariae risk dynamics in equaions (2) o (5) can accommodae differen degrees of common and idiosyncraic movemens. [Figure 3 abou here.] Given he high persisence for boh credi swaps spreads and implied volailiies, we also sudy how he daily changes of one series is correlaed wih he daily changes of anoher series. Figure 4 plos he crosscorrelograms a differen leads and lags beween daily changes in he five-year credi defaul swap spread and he hree-monh a-he-money implied volailiy. The dash-doed lines in each panel denoe he 95 percen confidence band. For all four reference companies, we idenify significanly posiive conemporaneous correlaions beween he daily changes of he wo series. Neverheless, he correlaion esimaes are all below.5, indicaing again ha he wo markes boh share common movemens and possess independen characerisics. 3 Using oher mauriies for he wo series generae similar resuls ha are available upon reques. 12

14 Furhermore, for all four companies, lagged values (up o a week) of he a-he-money implied volailiies significanly and posiively predic movemens in he credi defaul swap spreads. One poenial reason for his predicabiliy is he liquidiy difference beween he wo markes. The sock opions are exchangelised. The quoes are possibly updaed more frequenly han he over-he-couner quoes on he credi defaul swap spreads. Irrespecive of he underlying source, our dynamic specificaion capures boh he posiive conemporaneous correlaion via a posiive loading coefficien ξ and he predicive relaion via a negaive value for κ zv. [Figure 4 abou here.] 3. Join Esimaion of Reurn Variance and Defaul Arrival Dynamics We esimae he bivariae risk dynamics joinly using boh credi defaul swap spreads and sock opions. We cas he model ino a sae-space form and esimae he model using he 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 using an Euler approximaion of he ime-series dynamics in equaion (23): x = θ + ϕv 1 + βv 1 ε, (26) where ϕ = exp( κ P ) denoes he auocorrelaion coefficien wih being he lengh of he discree ime inerval, and ε denoes an iid bivariae sandard normal innovaion. We sample he daa weekly for he esimaion, hence = 7/365. The condiional covariance marix of he sae vecor is a diagonal marix wih sae-dependen diagonal elemens: Q = diag βv 1, where diag denoes a diagonal marix wih he diagonal elemens given by he vecor inside he bracke. 13

15 We consruc he measuremen equaions based on credi defaul swap spreads and sock opions, assuming addiive, normally-disribued measuremen errors: y = h(x ;Θ)+e, (27) where y denoes he observed series and h(x ;Θ) denoes he corresponding model value as a funcion of he sae vecor x and model parameers Θ. Specifically, he measuremen equaion conains five defaul credi swap spread series and 39 opion series, h(x ;Θ) = S(x, + τ s ;Θ) O(x, + τ O,δ;Θ), τ s = 1,3,5,7,1 years τ O = 3,6,91 days;δ = 2,25,,8, (28) where S(x, + τ s ) denoes he model value of he credi defaul swap spreads a ime and mauriy τ s as a funcion of he sae vecor x and model parameers Θ, O(x, + τ O,δ;Θ) denoes he model value for ouof-money opions a ime, ime-o-mauriy τ O, and dela δ, as a funcion of he sae vecor x and model parameers Θ. We scale he ou-of-money opion prices by heir Black-Scholes vega. There are missing values on boh he credi swap daa and he implied volailiy surface. Our esimaion algorihm readily handles missing observaions. The erm e in (27) denoes he measuremen errors, wih is covariance marixr. We assume ha he five credi defaul spread series generae iid normal pricing errors wih he same error variance σ 2 s. We also assume ha he pricing errors on all he opions (scaled by heir vega) are also iid normal wih error variance σ 2 O. For model esimaion, le x,v,y,a denoe he ime-( 1) ex ane forecass of ime- values of he sae vecor, he covariance of he sae vecor, he measuremen series, and he covariance of he measuremen series, respecively, and le x and V denoe he ex pos updaes on he sae vecor and is covariance a he ime based on observaions (y ) a ime. The sae-propagaion equaion is Gaussian-linear, bu he measuremen equaion in (27) is nonlinear. We use he unscened Kalman filer o handle he nonlineariy. The filer uses a se of deerminisically chosen poins o mach no only he mean and variance, bu also he higher momens of he sae disribuion. 14

16 Le k = 2 denoe he number of saes and le ζ > denoe a conrol parameer, a se of 2k + 1 sigma vecors χ i are generaed according o he following equaions, χ, = x, χ,i = x ± (k+ ζ)( V +Q ) j, j = 1,,k; i = 1,,2k, wih he corresponding weighs w i given by, w = δ/(k+ ζ), w i = 1/[2(k+ ζ)], i = 1,,2k. These sigma vecors form a discree disribuion wih w i being he corresponding probabiliies, such ha he mean, covariance, skewness, and kurosis of his disribuion are x, V +Q,, and k+ ζ, respecively. Given he sigma poins, he predicion seps are given by: χ,i = θ + ϕχ,i ; x +1 = V +1 = y +1 = A +1 = and he filering updaes are given by 2k i= 2k i= 2k i= 2k i= w i (χ,i ); w i (χ,i x +1 )(χ,i x +1 ) ; (29) w i h ( χ,i ) ; w i [ h ( χ,i ) y+1 ][ h ( χ,i ) y+1 ] +R, x +1 = x +1 + K +1 (y +1 y +1 ); V +1 = V +1 K +1 A +1 K +1, (3) wih K +1 = S +1 ( A+1 ) 1 ; S+1 = 2k i= w i [ χ,i v +1 ][ h ( χ,i ) y+1 ]. We refer o Julier and Uhlmann (1997) for general reamens of he unscened Kalman filer. 15

17 Based on he prediced mean and covariance on he observaions, we consruc he weekly log-likelihood funcion assuming normally disribued forecasing errors, l +1 (Θ) = 1 2 log A 1 2 ( (y +1 y +1 ) ( ) ) 1(y+1 A +1 y +1 ). (31) Then, he model parameers are chosen o maximize he log likelihood of he daa series, which is a summaion of he weekly log likelihood values, Θ argmax L (Θ,{y } N =1 ), wih L (Θ,{y } N N 1 =1 ) = l +1 (Θ), (32) Θ where N denoes he number of weeks in our sample. The procedure esimaes 13 model parameers: = Θ [κ v,κ z,κ P v,κ P z,θ v,θ z,σ v,σ z,ξ,κ zv,ρ sv,σ 2 s,σ 2 O]. (33) 4. Join Dynamics and Pricing of Reurn Variance and Defaul Arrival Risks Firs, we summarize he performance of our join valuaion model on credi defaul swap spreads and sock opions on he four reference companies. Then, from he srucural parameer esimaes we discuss he join dynamics and pricing of he diffusion variance risk and defaul arrival risk Performance analysis Table 4 repors he explained variaion and prediced variaion for he 39 opion series for each company. The explained variaion is a measure of pricing performance, defined as one minus he sample variance of he pricing error e over he sample variance of he original series y. I is analogous o an R 2 measure for a nonlinear regression. The model s pricing performance on sock opions is relaively uniform across all four companies. The explained variaions are over 9 percen for near he money opions across all socks. The performance remains reasonably well for ou-of-money pu opions, bu become significanly worse for ou-of-money call opions (high pu dela). In he sock opions marke, ou-of-money pu opions are more acively raded han ou-of-money call opions. The quoes on he ou-of-money pu opions are also 16

18 more reliable and consisen in general. On he oher hand, he ou-of-money call opion prices ofen show significan idiosyncraic variaions. The model capures he behavior of ou-of-money pu opions well. The prediced variaion is a measure of predicive performance, defined as one minus he sample variance of he predicive error over he sample variance of he original series. As expeced, he prediced variaion is significanly smaller han he explained variaion, suggesing ha he opions series conain a significan porion of unpredicable movemens. The decline is he mos noable for ou-of-money call opions. The prediced variaion for some ou-of-money call opion series even become negaive, suggesing no predicabiliy a all for hese series. To obain a beer picure on he pricing performance, we conver he model-implied opion prices ino he Black-Scholes implied volailiies and redefine he pricing errors as he difference in percenage poins beween he observed series and model values of he implied volailiies. Table 5 repors he sample esimaes of he mean, sandard deviaion, and weekly auocorrelaion of he pricing errors on implied volailiies. The mean pricing errors are fairly small and show no obvious paerns across moneyness and mauriies. The sandard deviaion ranges from one o four percenages poins. Comparing hese esimaes o he mean implied volailiy esimaes in Table 2 poin o an average pricing error of less han en percen. The weekly auocorrelaions of he pricing errors in implied volailiies are also much lower han he weekly auocorrelaions of he original series. Table 6 repors he explained variaion and prediced variaion for he five credi defaul swap spread series for each of four reference companies. The model s pricing performance is good for Ford and Duke Energy, as he explained variaions on all series are over 8 percen. However, he model performance is relaively poor for General Moors, and even worse for Alria Group. The model explains less han 5 percen of variaion for mos of he series on hese wo companies. Inspecing he ime series plos in Figure 2, we observe ha for General Moors, he five credi defaul swap spread series diverge dramaically afer January 23 o generae a very seep erm srucure from a virually fla erm srucure before 23. This dramaic erm srucure change eiher comes from economic forces or from a mere fac of more frequen quoe updaing in he second half of he daa. Irrespecive of he underlying reasons, our wo-facor model seems o have difficulies fiing he whole erm srucure of credi defaul swap spreads and he opions daa. The model performs well on all he opions series, and 17

19 also reasonably well on shor-erm credi defaul swap spreads, bu i performs poorly on he long-erm swap spreads. For Alria Group, he credi defaul swap daa are no updaed as frequenly before 23 whereas he opions daa are acively quoed and raded. I is poenially due o his difference ha he model parameers are geared o price he opions marke beer han he credi defaul swap spreads. Table 7 repors he sample esimaes of he mean, sandard deviaion, and weekly auocorrelaion of he pricing errors on he credi defaul swap spreads in percenage poins. Corresponding o he lower explained variaion, he sandard deviaions and weekly auocorrelaions are relaively large for he pricing errors on he credi spreads, especially for General Moors and Alria Group The join dynamics of reurn variance and defaul arrival raes Table 8 repors in panel A he maximum likelihood esimaes and -saisics of he srucural parameers ha conrol he join dynamics of he diffusion variance rae and he defaul arrival rae. The join dynamics differ across differen companies. Neverheless, several common feaures emerge from he esimaes. Firs, he esimaes for he risk-neural mean-revering coefficiens (κ v,κ z ) and heir ime-series counerpars (κ P v,κ P z ) show ha he defaul arrival rae is much more persisen han he diffusion variance rae under boh he risk-neural measure Q and he ime-series measure P. Their persisence difference is larger under he risk-neural measure. The difference in saisical persisence suggess ha he diffusion variance raes are srongly meanrevering and hence predicable, bu ha he defaul arrival rae is slow in revering back o is long-run mean level. The small esimaes for κ P z sugges ha i is difficul o predic he defaul arrival changes based on is pas values as he movemens of he defaul risk facor z are close o ha of a random walk. The difference in risk-neural persisence indicaes ha he wo facors (v,z ) have differen loading paerns across he erm srucure of opions and credi defaul swap spreads. Shocks on he diffusion variance rae affec he shor-erm opions and credi defaul swap spreads, bu dissipae quickly as he opion and swap mauriy increases. In conras, shocks on he more persisen defaul arrival rae las longer across he erm srucure of opions and credi spreads. 18

20 If we define he credi spread a a mauriy τ as he difference beween he coninuously compounded spo rae on a reference company and he corresponding spo rae in he benchmark eurodollar marke, under our dynamic model specificaion his spread is affine in he sae vecor, [ ] [ ] aλ (τ) bλ (τ) Spread(, τ) = + x, (34) τ τ wih he coefficiens [a λ (τ),b λ (τ)] given by he ordinary differenial equaions in (19). Hence, b λ (τ)/τ measures he conemporaneous response of he credi spread erm srucure o uni shocks on he wo risk facors. Figure 5 plos his response as a funcion of he credi spread mauriy. The solid lines denoe he response o he defaul risk facor z whereas he dashed lines denoe he response o he diffusion variance facor v. The higher persisence in z dicaes ha is impac declines more slowly as mauriy increases han he impac of he more ransien facor v. [Figure 5 abou here.] The second common feaure for he four reference companies is on how he defaul arrival rae ineracs dynamically wih he reurn variance rae. The esimaes for he loading coefficien ξ are posiive for all four companies, suggesing ha posiive shocks on he reurn variance rae increase he defaul arrival rae conemporaneously. The esimaes for he cross erm κ zv are all negaive, suggesing ha he reurn variance rae also posiively predic defaul arrival rae movemens. These esimaes sugges posiive co-movemens beween sock opion implied volailiies and credi defaul swap spreads. Finally, for all four companies, he esimaes for he insananeous correlaion beween sock reurn and reurn variance ρ sv are negaive, consisen wih classic sories on he leverage effec. Figure 6 plos he exraced ime series on he reurn variance rae (solid line) and he defaul arrival rae (dashed line), wih scales on he lef and righ hand sides of he y-axis, respecively. The exraced ime series show common rends ha mach he ime series plos of he credi defaul swap spreads and implied volailiies in Figure 2. The plos for all four companies show a spike for boh he variance rae and he defaul arrival rae in lae 22, revealing he financial sress during ha period. [Figure 6 abou here.] 19

21 4.3. Differen marke prices for diffusion variance risk and defaul arrival risk The difference beween he mean-revering coefficiens under he wo measures defines he marke prices of he wo sources of risks. Specifically, he marke prices of diffusion reurn variance risk is given by γ v = (κ v κ P v)/σ v and he marke price of he defaul risk facor z is given by γ z = (κ z κ P z )/σ z. Based on he parameer esimaes and covariance marix, we compue he wo marke prices for each company and repor he esimaes and -saisics in panel B of Table 8. The esimaes for all four companies show negaive marke price for he defaul arrival rae risk, bu posiive marke price for he diffusion variance risk. The exan empirical sudies, e.g., Bakshi and Kapadia (23) and Carr and Wu (24b), use sock and sock index opions and he underlying ime series reurns o sudy he oal reurn variance risk premia. They find ha he risk premia are negaive for some socks, and highly negaive for sock indexes. Consisen wih he negaive variance risk premia, selling opions and dela hedge, or direcly selling variance swap conracs, generaes posiive average excess reurns. Under our model specificaion, we decompose he oal variance risk ino wo componens: risk in he defaul arrival rae and risk in he diffusion variance rae. By using boh he credi defaul swap daa and sock opions daa, we are able o separae he wo sources of risks and idenify heir respecive marke prices. Our esimaion suggess ha for he four socks, negaive variance risk premia only come from he variance generaed from negaive jumps, no from he variance generaed from he diffusion componen of he sock reurn. From an invesmen perspecive, our resuls sugges ha selling credi insurance hrough he credi defaul swap marke generaes an even higher average reurn per uni risk han selling variance swap conracs. 5. Summary and Conclusions Based on documened evidence on he join movemens beween credi defaul swap spreads and sock opion implied volailiies, we propose a dynamically consisen framework for he join valuaion and esimaion of sock opions and credi defaul swaps wrien on he same reference company. We model he company defaul by a Poisson process wih sochasic arrival rae, and assume ha he sock price falls o zero upon defaul. We model he pre-defaul sock price as following a coninuous process wih sochasic volailiy. We assume ha he defaul arrival rae and diffusion variance rae follow a bivariae process wih 2

22 dynamic ineracions ha mach he empirical evidence linking sock opion implied volailiies and credi defaul swap raes. Imporanly, our dynamic specificaion allows boh common movemens and independen characerisics beween he wo markes. Under his join specificaion, we derive racable pricing soluions for sock opions and credi defaul swaps. We hen esimae he join dynamics of he diffusion variance rae and he defaul arrival rae using daa on sock opion implied volailiies and credi defaul swap raes for four of he mos acively raded reference companies. The esimaion shows ha he defaul arrival rae is much more persisen han he diffusion variance rae under boh he saisical measure and he risk-neural measure. The saisical persisence difference suggess differen degrees of predicabiliy. The risk-neural difference in persisence suggess ha he defaul arrival rae has a more long-lasing impac on he erm srucure of opion volailiies and credi defaul swap spreads han does he diffusion variance. The esimaion also suggess ha he diffusion variance feeds posiively ino he hazard rae, boh conemporaneously and dynamically. I is hese posiive dynamic ineracions ha generae he observed comovemens beween credi defaul swap spreads and sock opion implied volailiies. Finally, comparing he parameer esimaes ha conrol he saisical and risk-neural dynamics shows ha he marke price for he defaul arrival risk is negaive, bu he marke price for he diffusion variance risk is posiive, suggesing differen prices for differen sources of reurn variance risk. 21

23 Appendix. Generalized Fourier ransform of sock reurns To derive he generalized Fourier ransform: ( φ(u) E [exp we use he language of sochasic ime change of Carr and Wu (24a) and define T ] λ(s)ds )e iulnp T /P, u D C, (A1) T T v(s)ds, T z T z(s)ds, T λ T λ(s)ds =T z + ξt. Then, condiional on no defaul during he ime horizon [,T], wih τ = T, we can wrie he log sock reurn as ln(p T /P ) = (r(,t) q(,t))τ+t λ +W s T 1 2 T, (A2) where r(,t) and q(,t) denoe he coninuously compounded spo ineres raes and dividend yields of he relevan mauriy. The log reurn goes o negaive infinie as he sock price falls o zero when defaul occurs. The discouned generalized Fourier ransform becomes ( φ(u) = E [exp T λ + iu(r(,t) q(,t))τ+iut λ + iuwt s 1 )] 2 iut = E [exp (iuw st + 12 ) ( u2 T exp T λ + iu(r(,t) q(,t))τ+iut λ 1 2 iut 1 )] 2 u2 T [ ( = exp(iu(r(,t) q(,t))τ)e M exp (1 iu)t λ 1 ( iu+u 2 ) )] T 2 [ ( ( = exp(iu(r(,t) q(,t))τ)e M exp (1 iu)t z (1 iu)ξ+ 1 ( iu+u 2 )) )] T, 2 where he new measure M is defined by ( dm dq = exp iuwt s u2 T ), under which he drif of he wo dynamic processes change o: µ M v = θ v κ v v()+iuσ v ρ sv v(), µ M z = θ z κ z z() κ zv v()+iuρ sz σ z v()z(). 22

24 Hence, by assuming ρ sz =, we mainain he affine srucure for z() and v(). Then, we have φ(u) = exp(iu(r(,t) q(,t))τ)e M [ ( exp T )] b x s ds, wih b = (1 iu)ξ+ 1 ( 2 iu+u 2 ) 1 iu, dx = ( θ+κ M ) x d + βxdw M, κ M = κ v iuσ v ρ. κ zv κ z We hus can derive he exponenial affine soluion according o Carr and Wu (24a). 23

25 References Alman, E. I., B. Brady, A. Resi, and A. Sironi, 24, The Link Beween Defaul and Recovery Raes: Theory, Empirical Evidence and Implicaions, Journal of Business, forhcoming. Bakshi, G., and N. Kapadia, 23, Dela-Hedged Gains and he Negaive Marke Volailiy Risk Premium, Review of Financial Sudies, 16(2), Bakshi, G., N. Kapadia, and D. Madan, 23, Sock Reurn Characerisics, Skew Laws, and he Differenial Pricing of Individual Equiy Opions, Review of Financial Sudies, 16(1), Bakshi, G., D. Madan, and F. Zhang, 24, Invesigaing he Role of Sysemaic and Firm-Specific Facors in Defaul Risk: Lessons From Empirically Evaluaing Credi Risk Models, Journal of Business, forhcoming. Bangia, A., F. X. Diebold, A. Kronimus, C. Schagen, and T. Schuermann, 22, Raings Migraion and he Business Cycle, Wih Applicaion o Credi Porfolio Sress Tesing, Journal of Banking and Finance, 26(2-3), Bekaer, G., and G. Wu, 2, Asymmeric Volailiies and Risk in Equiy Markes, Review of Financial Sudies, 13(1), Bevan, A., and F. Garzarelli, 2, Corporae Bond Spreads and he Business Cycle: Inroducing GS-Spread, Journal of Fixed Income, 9(4), Black, F., 1976, The Pricing of Commodiy Conracs, Journal of Financial Economics, 3, Black, F., and M. Scholes, 1973, The Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, 81, Campbell, J. Y., and L. Henschel, 1992, No News is Good News: An Asymmeric Model of Changing Volailiy in Sock Reurns, Review of Economic Sudies, 31, Campbell, J. Y., and A. S. Kyle, 1993, Smar Money, Noise Trading and Sock Price Behavior, Review of Economic Sudies, 6(1), Capmbell, J. Y., and G. B. Taksler, 23, Equiy Volailiy and Corporae Bond Yields, Journal of Finance, 63(6), Carr, P., and L. Wu, 24a, Time-Changed Lévy Processes and Opion Pricing, Journal of Financial Economics, 71(1), Carr, P., and L. Wu, 24b, Variance Risk Premia, working paper, Bloomberg and Baruch College. Collin-Dufresne, P., R. S. Goldsein, and J. S. Marin, 21, The Deerminans of Credi Spread Changes, Journal of Finance, 56(6),

26 Consigli, G., 24, Credi Defaul Swaps and Equiy Volailiy: Theoreical Modelling and Marke Evidence, working paper, Universiy Ca Foscari. Cremers, M., J. Driessen, P. J. Maenhou, and D. Weinbaum, 24, Individual Sock Opions and Credi Spreads, Yale ICF Working Paper 4-14, Yale School of Managemen. Dennis, P., and S. Mayhew, 22, Risk-neural Skewness: Evidence from Sock Opions, Journal of Financial and Quaniaive Analysis, 37(3), Duffie, D., J. Pan, and K. Singleon, 2, Transform Analysis and Asse Pricing for Affine Jump Diffusions, Economerica, 68(6), Ericsson, J., K. Jacobs, and R. Oviedo-Helfenberger, 24, The Deerminans of Credi Defaul Swap Premia, working paper, McGill Universiy. Haugen, R. A., E. Talmor, and W. N. Torous, 1991, The Effec of Volailiy Changes on he Level of Sock Prices and Subsequen Expeced Reurns, Journal of Finance, 46(3), Hilscher, J., 24, Is he Corporae Bond Marke Forward Looking?, working paper, Harvard Universiy. Hull, J., I. Nelken, and A. Whie, 24, Merons Model, Credi Risk and Volailiy Skews, working paper, Universiy of Torono. Julier, S. J., and J. K. Uhlmann, 1997, A New Exension of he Kalman filer o Nonlinear Sysems, working paper, Universiy of Oxford, Sydney, Ausralia. Meron, R. C., 1974, On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes, Journal of Finance, 29(1), Pedrosa, M., and R. Roll, 1998, Sysemaic Risk in Corporae Bond Yields, Journal of Fixed Income, 8(1), 7 2. Zhu, H., Y. Zhang, and H. Zhou, 25, Equiy Volailiy of Individual Firms and Credi Spreads, working paper, Bank for Inernaional Selemens. 25

27 Table 1 Summary Saisics on credi defaul swap spreads Enries repor he sample esimaes of he mean, sandard deviaion, and weekly auocorrelaion on he credi defaul swap spreads (in percenages) a five fixed mauriies for each of he four reference companies. The saisics are based on weekly sampled daa from January 2, 22 o April 28, 24. Mauriy Mean: F GM MO DUK Sandard Deviaion: F GM MO DUK Auocorrelaion: F GM MO DUK

28 Table 2 Summary saisics on sock opion impled volailiies Enries repor he sample esimaes of he mean, sandard deviaion, and weekly auocorrelaion on he implied volailiies (in percenages) a 13 fixed delas and hree fixed mauriies for four reference companies. The saisics are based on weekly sampled daa from January 2, 22 o April 28, 24. Dela Mean: F 1m F 2m F 3m GM 1m GM 2m GM 3m MO 1m MO 2m MO 3m DUK 1m DUK 2m DUK 3m Sandard Deviaion: F 1m F 2m F 3m GM 1m GM 2m GM 3m MO 1m MO 2m MO 3m DUK 1m DUK 2m DUK 3m Auocorrelaion: F 1m F 2m F 3m GM 1m GM 2m GM 3m MO 1m MO 2m MO 3m DUK 1m DUK 2m DUK 3m

29 Table 3 Regressing credi defaul swap spreads on sock opion implied volailiies Enries repor he parameer esimaes, absolue magniudes of -saisics (in parenheses), and R 2 for he following regression on four reference companies: S = a+bat MV + e, where S denoes he five-year credi defaul swap spreads in percenage poins and AT MV denoes he hree-monh a-he-money implied volailiy in percenage poins for he same reference company s sock opions. Daa are daily from January 2, 22 o April 3, 24. Companies a b R 2 F ( ).95 ( ) GM.225 ( 15.7 ).57 ( ) 59.2 MO.16 (.79 ).62 ( 9.2 ) 36.4 DUK -1.2 ( 6.47 ).87 ( )