JOINT USC FBE FINANCE SEMINAR and USC MATH DEPT. WORKSHOP presened by Peer Carr TUESDAY, Ocober 17, 26 12: pm - 1:15 pm, Room: DRB-337 Robus Replicaion of Defaul Coningen Claims Peer Carr and Bjorn Flesaker Bloomberg P 731 exingon Avenue, 16h Floor New York, NY 122 Iniial Version: Augus 26, 26 Curren Version: Ocober 13, 26 File Reference: RRDCC3.ex Absrac We show how o replicae he payoffs o a class of defaul-coningen claims by aking saic posiions in a coninuum of credi defaul swaps (CDS) of differen mauriies. Alhough we assume deerminisic ineres raes and a consan recovery rae on he CDS, he replicaion is oherwise robus in ha we make no assumpions on he process riggering defaul. In paricular, we can robusly replicae he payoff o an Arrow Debreu securiy paying one dollar a a fixed dae if a given eniy survives o ha dae. As a consequence, we can deermine risk-neural survival probabiliies from an arbirarily given yield curve and CDS curve. We are graeful o Farshid Asl, Bruno Dupire, David Eliezer, Alireza Javaheri, Rober Jarrow, David ando, Harry ipman, Dilip Madan, eon Taevossian, Arun Verma, and iuren Wu for valuable commens. They are no responsible for any errors.
1 Inroducion The volume in credi defaul swaps coninues o grow. According o he Augus 31, 26 issue of he Wall Sree Journal, he underlying noional now exceeds $17 rillion. Much effor has been spen developing models ha can deermine wha he CDS spread of a given mauriy should be. For example, one migh be ineresed in deermining wha he 5 year CDS spread should be given some model for defaul frequency, some assessmen of marke risk aversion, and he erm srucure of ineres raes. A second line of research consiss of developing he enire CDS spread curve when one or more poins along he curve are given. For example, one migh ry o infer all CDS spreads for o 1 years, given jus he 5 year quoe, he erm srucure of ineres raes, and some assumpions on smoohness. In his paper, we explore a hird line of research. We ake he enire CDS curve as given along wih he erm srucure of ineres raes. We do no use any oher informaion. We hen ry o exrac unique arbirage-free prices of a se of defaul-coningen claims, making as few probabilisic resricions as possible. The se of defaul-coningen claims ha we focus on include any claim which promises o make fixed paymens over ime according o a pre-se schedule. Even hough he arge claims have payoffs closely relaed o hose of a CDS, we find ha some probabilisic resricions are needed. For example, he payoffs o jus he premium leg of a CDS is in our class. Ye in order o exrac he iniial value of he premium leg of a CDS, we will be assuming deerminisic ineres raes and consan recovery raes in his paper. A second objecive of his paper is o indicae he posiions needed o replicae he payoffs of he arge claims. To he exen ha insiuions wish o offer he arge claims ha we focus on, he proper hedging of he ensuing liabiliies requires exac knowledge of posiions required in he hedging insrumens. The hedging insrumens ha we use are a cash accoun and CDS of all mauriies. The posiions in he CDS are saic. To hedge he payoff of a arge claim in our class, he hedger iniially assumes a posiion in zero cos CDS of all mauriies. The hedger also ses up a bank accoun whose iniial balance maches he heoreical value of he arge claim. As calendar ime evolves, CDS values deviae from zero and as a resul, he bank accoun balance deviaes from he heoreical value of he arge claim. Prior o any defaul by he reference eniy, he posiions in he CDS and in he arge claim require cash inflows or ouflows which are financed ou of he bank accoun. The saic naure of our proposed hedge implies ha he iniial CDS posiions are never adjused aferwards. This feaure cus down on ransacion and monioring coss, bu requires iniial liquidiy in more CDS mauriies han a more sandard model-based dynamic hedge. While we assume deerminisic ineres raes and a consan recovery rae on he CDS, our saic hedging sraegy is oherwise robus in ha we make no assumpions on he process riggering defaul. In paricular, our resuls are consisen wih boh reduced form and srucural models of defaul. We solve he replicaion problem by deriving a new erminal value problem which deermines he survival-coningen bank balance when replicaing a claim. This problem consiss of a simple linear second order differenial equaion coupled wih wo erminal condiions. The soluion of his problem also deermines he CDS holdings. A arge payoff of paricular ineres for us is a defaulable annuiy. This claim pays one dollar per year coninuously unil he earlier of a random defaul ime and a fixed mauriy dae. Our 1
ineres in he pricing and hedging of his claim arises for a leas hree reasons. Firs, he produc of he iniial value of a defaulable annuiy and he known co-erminal CDS rae gives he iniial value of he premium leg of a CDS. As a resul, he iniial value of a defaulable annuiy gives he DV1 of a CDS, i.e. he impac of a one basis poin shif in he CDS spread on he value of he premium leg of a CDS. Thus, he iniial value of a defaulable annuiy can be inerpreed as his risk measure and for breviy, we will frequenly refer o i as such. Second, suppose for each fixed expiry dae, a rader ries o keep he ne noional in he proecion legs of is CDS posiions as close o zero as possible. This sraegy ofen arises for a marke maker in CDS or for an acive rader being on CDS spread levels, raher han on he defaul ime. When he ne noional in he proecion leg is zero, he marke value of he CDS posiion is hen given by he value of he ne noional in he co-erminal premium legs. As he CDS rae flucuaes over ime, he ne fixed paymen is non-zero. Thus, for such a CDS rader, he mark-o-marke of an accumulaed CDS posiion amouns o deermining he value of a posiion in defaulable annuiies from he curren CDS spread curve. Third, we show ha he soluion of he replicaion problem for a defaulable annuiy can be used o deermine he porfolio weighs when replicaing he payoff of an arbirary defaul-coningen claim. In paricular, we show ha he survival-coningen bank balance ha arises when replicaing he payoff of a defaulable annuiy is a Green s funcion, or fundamenal soluion, for he deerminaion of he survival-coningen bank balance when replicaing oher defaul-coningen claims in our class. We also show ha he required CDS posiions become deermined once his bank balance is known. Our idenificaion of he survival-coningen bank balance as a fundamenal soluion furher implies ha he funcion relaing he iniial value of a defaulable annuiy o is mauriy solves an iniial value problem. This iniial value problem also consiss of a simple linear second order differenial equaion, bu now coupled wih wo iniial condiions. Our iniial value problem can be numerically solved o efficienly calculae he erm srucure of DVO1 s from he given CDS spread curve and yield curve. Once one has solved for his erm srucure, oher financially relevan quaniies are quickly deermined. For example, he presen values of eiher leg of a CDS is obained by simply muliplying he DV1 by he known CDS rae. We show ha one can also quickly deermine he risk-neural probabiliy densiy funcion of he defaul ime. Thus a conribuion of his paper is o show ha he replicaion and pricing problems for an arbirary defaul-coningen claim can be expressed in erms of he soluion of he corresponding problems for a defaulable annuiy. Boh problems jus require solving a simple linear second order differenial equaion coupled wih wo boundary condiions. In general, boh evoluion equaions mus be solved numerically, bu we indicae imporan special cases in which hey can be solved in closed form. The remainder of he paper is organized as follows. The nex secion lays ou our assumpions and noaion. The subsequen secion indicaes how he survival-coningen bank balance and he CDS hedge weighs are deermined by a erminal value problem. The secion aferwards gives he financial inerpreaion of he Green s funcion for his problem. The following secion indicaes how his inerpreaion can be used o generae he erm srucure of DV1 s by solving an iniial value problem. The nex secion gives wo expressions for he risk-neural probabiliy densiy funcion of he defaul ime and shows ha he iniial value problem arises by equaing hem. The nex wo 2
secions focus on he hedging and pricing of a uni recovery claim and a survival claim respecively. The penulimae secion presens closed form soluions o our evoluion problems ha arise when eiher he yield curve or he CDS curve are fla. We conclude in he final secion wih a summary and some furher possible exensions. 2 Assumpions and Noaion We consider a finie ime horizon,t] where ime = is he valuaion ime and ime = T is a posiive finie consan. We assume deerminisic ineres raes for simpliciy and we le r(), be he spo ineres rae a ime. One dollar invesed in he money marke accoun a ime grows o e r(u)du dollars wih cerainy a ime. To apply he presen deerminisic ineres rae heory, one can equae he fuure spo rae r() o he forward ineres rae f (). The fac ha he argumen of r() is coninuous is anamoun o assuming ha here exiss a complee erm srucure of defaul-free bonds ou o T. Besides he defaul-free bonds, we assume ha credi defaul swaps for a given reference name also rade. As is well known, a CDS coss zero o ener and provides proecion agains he loss arising from a defaul over a specified ime period. In reurn for his proecion, he long side of he CDS makes fixed paymens periodically. For simpliciy, we assume ha he paymens on he premium leg of he CDS occur coninuously over ime, raher han discreely. In pracice, he fixed paymens are made quarerly, bu an accrued ineres paymen is made whenever defaul occurs beween one of he quarerly paymen daes. As a resul, he coninuous paymen assumpion has lile impac on he analysis, while permiing he use of differenial calculus raher han he difference calculus. Throughou he paper, we assume ha he recovery rae on he insured bond is a known consan. In realiy, recovery raes are random and he impac of covariance beween he recovery rae and defaul incidence is he major concern of an emerging line of research. Our primary jusificaion for he consan recovery rae assumpion is simpliciy. The difference beween uniy and he recovery rae is called he loss given defaul, denoed by (, 1]. If defaul occurs a some random ime τ, hen all CDS mauring afer τ have value a ime τ. Once he liquidaing paymen is made a τ, he ex-dividend CDS have zero value aferwards. We assume ha a ime, an invesor can also ake posiions of any size in CDS of any mauriy up o a fixed horizon T. In pracice, only several discree mauriies are available, bu marke makers are willing o quoe on any mauriy ou o some fixed horizon. Our assumpion ha a coninuum of mauriies are available for rading again faciliaes he use of differenial calculus raher han he difference calculus, while having lile qualiiaive impac on he conclusions. We le S (u) denoe he iniially observed CDS spread for all mauriies u,t]. We consider he problem of replicaing a claim which promises o pay an infiniesimal cash flow c()d coninuously for all,t]. For example, he promised coupon rae migh be c() =1,u], for u,t], i.e. one dollar per year unil he mauriy dae u T. For ease of exposiion, we do no allow discree cash flows o be promised a any ime, bu we will show how o exend our analysis so as o deal wih such claims. 3
We allow for he possibiliy ha he issuer of he arge claim may defaul a any ime. If defaul occurs a some random ime τ,t], hen he promised cash flows sop and he claim is worh is recovery value R(τ), where he recovery funcion R(),,T] is known a ime. The specificaion of he promised coupon rae c() and he recovery funcion R() on he domain,t] compleely deermines he payoffs o he arge claim. When we refer o arbirary defaulconingen claims in he sequel, we have in mind he ensemble of all claims generaed by any choice of coninuous funcions c() and R(). For example, if c() =1 (,u),u,t] and R() =on (,T), hen he arge claim is a defaulable annuiy mauring a u, i.e. a claim ha pays one dollar per year unil he earlier of he defaul ime τ and is fixed mauriy dae u,t]. If defaul occurs a some random ime τ before u, hen he defaulable annuiy becomes worhless a τ. Thus, he acual cash flow from he defaulable annuiy a each insan (,T)is1 (,τ u) d. One reason for our focus on he replicaion of defaulable annuiies is ha he payoffs from he premium leg of a CDS are proporional o he payoff from a defaulable annuiy. In fac, a saic posiion in S (T ) unis of he defaulable annuiy has he same payoffs as he premium leg of a CDS of mauriy T, when paymens are made coninuously. I follows ha he iniial value of his premium leg is jus he produc of he iniial value of he defaulable annuiy and he iniial CDS spread. The paymens arising from he proecion leg of a T mauriy CDS also arise in our framework by seing c() =andr() =, 1) for,t]. By he definiion of he CDS rae, he iniial value of his proecion leg maches he iniial value of he premium leg. Thus, he payoffs from a defaulable annuiy, he premium leg, and he proecion leg all arise in our framework. While i migh a firs appear ha a leas one of hese example payoffs arises from a posiion in jus a CDS of mauriy T, his appearance is misleading. In fac, he resuls of he nex secion imply ha he replicaion of each of he hree payoffs requires posiions in cash and CDS of all mauriies up o T. 3 Replicaion Via Backward Equaion Recall ha in our seing, he payoffs from an arbirary defaul-coningen claim wih mauriy T are deermined by he promised coupon rae c(),,t] and a recovery funcion R(),,T]. The acual cash flow received a each insan of ime,t] is random and given by c()1( > τ)+r()δ( τ)]d, where τ is he random defaul ime. In his secion, we focus on deermining he replicaing sraegy in cash and CDS for his arge claim. e M() be he amoun of money kep in he money marke accoun a ime,t], given no defaul prior o. For breviy, we refer o M() as he survival-coningen bank balance as of he fuure ime. e Q(u)du be he noional amoun in CDS of mauriy u,t] ha he invesor wries a ime. If Q(u) is posiive for some mauriy u, hen he invesor is selling proecion a ha mauriy. Our sign convenion resuls in he wo conrols M and Q boh being nonnegaive when he arge funcions c and R are boh nonnegaive. Accordingly, he language used o convey inuiion for our mahemaical resuls will accord wih all quaniies being nonnegaive, bu he mahemaical resuls exend o all quaniies being real. We inerpre he conrols M and Q as generalized funcions. This inerpreaion allows impulse 4
conrols o be applied a a fixed ime such as T when a claim maures. We now show ha he wo conrols M() and Q() are uniquely deermined by wo equaions. We refer o he firs of hese equaions as he recovery maching condiion: M() Q(u)du = R(),,T]. (1) If defaul occurs a a candidae fuure ime, hen (1) says ha he he wo conrols M( ) and Q( ) are iniially chosen so ha he value a in he money marke accoun and from all of he ousanding CDS sum o he arge recovery value R(). Differeniaing (1) w.r.. implies: M () =R () Q(),,T]. (2) Thus, he change in he survival-coningen bank balance increases due o any increase in he required recovery amoun and decreases as he liabiliy from he expiring CDS rolls off. Solving (2) for Q() implies: Q() = 1 R () M ()],,T]. (3) Since he loss given defaul and he recovery funcion R(),,T] of he arge claim are boh given, (3) indicaes ha he rae Q(u) a which CDS are wrien a each mauriy u is deermined once one firs deermines he survival-coningen bank balance M(). The second equaion governing he wo conrols is he self-financing condiion: M () =r()m()+ S (u)q(u)du c(),,t]. (4) In words, he change in value of he survival-coningen bank balance arises from ineres earned on he previous balance, premium inflows due o all of he ousanding shor posiions in CDS, less he wihdrawal required o finance he promised coupon rae of he arge claim. Differeniaing (4) w.r.. implies: M () =r()m ()+r ()M() S ()Q() c (),,T]. (5) Subsiuing (3) in (5) and re-arranging implies ha M() solves he following linear second order ordinary differenial equaion (ODE): M M () r()+ S ] () M () r ()M() =f(),,t], (6) where he forcing funcion f() is given by: f() c () S () R (),,T]. (7) For defaul-coningen claims mauring a T, a unique soluion for M arises on he domain (,T) by imposing wo erminal condiions: M(T )=, (8) 5
and lim M () = c(t ). (9) T The homogeneous erminal condiion (8) arises due o our ineres only in claims ha have coninuous cash flows over ime. The lef slope condiion (9) arises from leing T in (4) and subsiuing in (8). For arbirarily given yield curves and CDS curves, one mus numerically solve he erminal value problem (6) o (9) for M(),,T) using finie differences. However, we laer show ha he problem can be solved in closed form if eiher he forward rae curve or he CDS curve is iniially fla. Wheher i is obained by finie differences or a formula, he scalar M() has he imporan financial inerpreaion as he iniial value of he arge claim wih mauriy T, because all of he CDS used in he replicaing porfolio are iniially cosless. Furhermore, (3) gives he rae a which CDS are iniially wrien for each mauriy (,T). Thus, we have a complee soluion o he problem of replicaing a arge claim of mauriy T using a money marke accoun and CDS of all mauriies up o T. To illusrae our soluion procedure, consider he erminal value problem ha arises when replicaing a defaulable annuiy of mauriy T. e M a (; T ) denoe he survival-coningen bank balance a,t] when replicaing his claim. Seing c() =1 (,T ) and R() in (6) o (9) implies ha M a (; T ) solves he homogeneous ODE: M a (; T )=, (,T), (1) subjec o he erminal condiions: M a (T ; T )=, (11) and lim T M a(; T )= 1. (12) Since T is fixed, M a (; T ) is he scalar indicaing he iniial value of he defaulable annuiy mauring a T. From (3), he rae a which CDS are iniially wrien is: Q a (; T ) M a (; T ) (13) for each mauriy,t]. So long as ineres raes are nonnegaive, i can be shown ha M a(; T ) for each,t]. As a resul, we have Q a (; T ) for each,t], which explains our sign convenion. 4 Green s Funcion We reurn o he general problem of replicaing a claim wih promised coupon rae c() and recovery funcion R() enering hrough he forcing erm f() defined in (7). I is well known ha he general soluion can be expressed in erms of he Green s funcion g(; u), defined as a funcion solving: g(; u) =δ( u),,t], (14) 6
where u,t]. Since is a second order differenial operaor, we also need wo boundary condiions o uniquely deermine a Green s funcion. For his purpose, we choose: g(t ; u) =, and lim T g(; u) =. (15) These homogeneous erminal condiions cause g(; u) o vanish for (u, T ). We noe ha he ODE (14) arises from (6) by seing c() =1 (,u) and R() for,t]. Thus, he Green s funcion g(; u) is conneced wih he replicaion of a defaulable annuiy mauring a u. In fac, a soluion o he erminal value problem consising of he ODE (14) and he erminal condiions (15) is: { M a (; u) if,u] g(; u) = (16) if (u, T ). Thus, for fixed u, g(; u) is he survival-coningen bank balance a,u] when replicaing he u mauriy defaulable annuiy. ike M, he funcion g is in general only obained numerically, bu i can be obained in closed form in special cases. In conras o M, we seek he dependence of g(; u) on u raher han. Once g(; u) is known as a funcion of u for some, hen he survival-coningen bank balance which arises when replicaing an arbirary defaul-coningen claim can be expressed in erms of i: M() = f(u)g(; u)du,,t]. (17) I follows from (3) and (11) ha he rae a which CDS are wrien a each mauriy can also be expressed in erms of he Green s funcion: Q() = 1 R () f(u) ] g(; u)du,,t]. (18) Recall ha all of he CDS used in he replicaing porfolio are iniially cosless. Hence, evaluaing (17) a = relaes he iniial value of he arge claim wih mauriy T o he Green s funcion g(; u). M() = f(u)g(; u)du,,t]. (19) Thus, he valuaion of a arge claim reduces o deermining g(; u) for all u,t]. Given he above definiion of g, i migh seem ha he ODE (14) has o be numerically solved once for each level of u. Forunaely, he nex secion shows ha anoher ODE governs g when i is considered as a funcion of u. As a resul, only a single ODE needs o be numerically solved o obain he arbirage-free price of an arbirary defaul-coningen claim. 5 Pricing via he Forward Equaion The las secion showed ha g(; u) is he survival-coningen bank balance a,u] when replicaing he u mauriy defaulable annuiy. Thus, for each fixed u, he number g(; u) is he 7
iniial value of he defaulable annuiy of mauriy u,t]. This secion shows ha his laer observaion can be used o efficienly deermine he whole erm srucure of iniial values of a defaulable annuiy, g(; u),u,t], from he given iniial CDS curve S (u),u,t] and he given iniial forward rae curve r(u),u,t]. The appendix proves he well known resul (see eg. Sakgold2] page 2), ha g saisfies he adjoin ODE when considered as a funcion of is second variable u, i.e.: g(; u) =δ( u), (2) where is he following linear differenial operaor: Du 2 + r(u)+ S ] (u) Du 1 + S (u) D u, (21) and D u denoes differeniaion w.r.. u. The domain for he inhomogeneous ODE (2) is he square (, u),t],t]. Equaions (16) and (2) imply ha he survival-coningen bank balance solves he homogeneous ODE: DuM 2 a (; u)+ r(u)+ S ] (u) D u M a(; u)+ S (u) M a(; u) =, (22) on he domain u (, T ) and imposing he iniial condiions: and: lim u M a (; ) =, (23) u M a(; u) =1. (24) In general, he iniial value problem (22) o (24) mus be solved numerically using finie differences. However, we laer show ha he problem can be solved in closed form if eiher he forward rae curve or he CDS curve is iniially fla. Recall ha for a finie-lived claim, he ODE (6) was solved by propagaing he erminal condiions (8) and (9) backwards in calendar ime from he erminal ime T. In conras, (22) is solved by propagaing he iniial condiions (23) and (24) forward in he mauriy dae u from ime. As a resul, we refer o (6) as he backward ODE and we refer o (22) as he forward ODE. The backward ODE governs he survival-coningen bank balance as a funcion of and i applies o a wide class of payoffs. Hence is soluion a ime = yields he iniial price of a wide class of claims. In conras, he forward ODE governs he survival-coningen bank balance as a funcion of u and i applies only when replicaing a defaulable annuiy. The reason for he greaer scope of he backward ODE is ha he applicaion of any linear operaor in he parameer u leaves he backward equaion unchanged, while i changes he forward equaion. I is he applicaion of he inegral operaor wih f as kernel ha causes he backward ODE o hold for any claim. Recall ha for any arge claim, he iniial bank balance when replicaing he payoff wih cash and CDS is jus he iniial value of he arge claim. Applying his resul o a defaulable annuiy, le 8
A (u) M a (; u) denoe he iniial value (DV1) of he defaulable annuiy mauring a u,t]. Evaluaing (22) o (24) a = implies ha A (u) solves he homogeneous ODE: A (u)+ r(u)+ S ] (u) A (u) (u)+s A (u) =, (25) on he domain u (,T) and imposing he iniial condiions: A () =, (26) and: lim A u (u) =1. (27) 6 Financial Derivaion of Forward Equaion In he las secion, he Green s funcion g(; u) was used o derive a forward equaion (22) governing he survival coningen bank balance when replicaing a defaulable annuiy. By considering his equaion a =, we developed a forward equaion for he iniial value A of a defaulable annuiy considered as a funcion of is mauriy dae u. This secion shows ha his forward ODE arises direcly as a consequence of no arbirage. We assume ha he defaul ime τ has a probabiliy densiy funcion (PDF) and our approach is o find wo equivalen expressions for i. We sar from he observaion ha S (u)a (u) is he spo value of he premium leg of a CDS of mauriy u. By he definiion of S (u), S (u)a (u) is also he spo value of he proecion leg of a CDS of mauriy u. Dividing his value by produces he erm srucure of iniial values for a claim paying one dollar a he defaul ime τ if τ < u and zero oherwise. We refer o his defaul-coningen claim as a uni recovery claim. The nex secion shows how o replicae he payoff of his uni recovery claim. In his secion, we jus observe ha S (u)a (u) is he spo value of he uni recovery claim. Differeniaing w.r.. u implies ha S (u)a (u) is he spo value of a claim wih he payoff δ(τ u) au. Fuure valuing his spo value u gives he defaul ime PDF: Q{τ du} = e u r(v)dv u S (u)a (u). (28) A second observaion is ha ha A (u) is he spo value of a survival claim, i.e. a claim ha pays one dollar a u if he reference eniy survives unil hen. As a resul, he complemenary disribuion funcion of he defaul ime is is forward price: Differeniaing boh sides w.r.. u and negaing implies ha: Q{τ u} = e u r(v)dv A (u). (29) Q{τ du} = r(u)e u r(v)dv A (u) e u r(v)dv A (u). (3) 9
Equaions (28) and (3) boh express he risk-neural PDF of he defaul ime in erms of he ex ane unknown annuiy value. Equaing hem yields he forward equaion (25) governing he iniial value A (u) of a defaulable annuiy, considered as a funcion of is mauriy dae u: A (u)+ r(u)+ S (u) ] A (u)+s (u) A (u) =, u,t]. (31) Subjecing he funcion A (u) o he wo iniial condiions (26) and (27) uniquely deermines i. Eiher of (28) or (3) hen implies ha he risk-neural PDF of τ is also deermined. I follows ha any claim ha pays f(τ) a ime τ can be uniquely priced relaive o he iniial yield curve and CDS curve. The claim can be finie-lived or even perpeual, so long as he payoff does no lead o infinie value. While we have assumed deerminisic ineres raes and a consan recovery rae on he CDS, he deerminaion of he defaul ime PDF is oherwise robus in ha we have made no assumpions on he process riggering defaul. In paricular, our resuls are consisen wih boh reduced form and srucural models. Equaions (28) and (3) can be regarded as analogs of he Breeden and izenberger1] resul linking he risk-neural PDF of he underlying sock price o he second srike derivaive of a call. Boh equaions link he risk-neural PDF of he defaul ime o parial derivaives of observables, wih he main difference being ha A (u) only becomes observable once an iniial value problem is solved. While our resul is no presenly as explici as he B resul, we will laer show ha our resul can be made compleely explici if eiher he CDS curve or yield curve is iniially fla. 7 Replicaing he Payoff of a Uni Recovery Claim The las secion showed ha S (u)a (u) is he spo value of he uni recovery claim. In his secion, we develop he replicaing sraegy. Recall ha a defaulable annuiy mauring a T produces a coninuous cash flow a each,t] of 1(τ > )d, where τ is he random defaul ime. Also recall ha his payoff is replicaed by keeping he survival-coningen bank balance a M a (; T ) for each,t] and also shoring Q a ()d CDS of each mauriy,t]. For each fixed T, he funcion M a of is deermined by he erminal value problem (1) o (12), while he funcion Q a of is deermined by (13). Consider scaling hese holdings by S (T ). Since he original holdings produce a coninuous cash flow a each,t] of 1(τ > )d, he scaled holdings produce a coninuous cash flow a each,t]of S (T ) 1(τ > )d. In conras, a long posiion in one CDS of mauriy T produces a coninuous cash flow a each,t]ofδ(τ ) S (T )1(τ >)]d. Hence, if we now add a a long posiion in 1 CDS of mauriy T o he scaled holdings, hen he combined posiion produces a coninuous cash flow a each,t]ofδ(τ )d. This is he payoff of a uni recovery claim, i.e., a claim paying one dollar a he defaul ime τ if τ<tand zero oherwise. eing M u (; T ) denoe he survival-coningen bank balance a ime,t] when replicaing a uni recovery claim, we conclude ha: M u (; T )= S (T ) M a(; T ),,T]. (32) 1
eing Q u (; T ) denoe he rae a which CDS are iniially wrien for mauriy,t], when replicaing a uni recovery claim, we conclude ha: Q u (; T )= δ( T ) + S (T ) Q a(; T )1,T ],. (33) I can be shown ha his replicaing sraegy arises from seing c() = and R() =1,T ] in he backward ODE (6). 8 Replicaing and Pricing Survival Claims Given no defaul up o ime, he value a ime of a defaulable annuiy mauring a T can be represened as: V da (T )=M a (; T )+ Q a (u; T )V cds (u; S (u))du, (34) where V cds (u; k) is he unknown value a ime of a CDS, mauring a u (, T ) and wih fixed paymen rae k. Recall ha he survival claim pays off one dollar a is mauriy if he firm survives unil hen, and pays zero oherwise. Differeniaing he payoff of a defaulable annuiy w.r.. is mauriy T produces he payoff of a survival claim wih mauriy T. Differeniaing (34) w.r.. T implies: T V da (T )= T M a(; T )+Q a (T ; T )V cds (T ; S (T )) + T Q a(u; T )V cds (u; S (u))du. (35) The HS of (35) is he value a ime of a survival claim, given no defaul up o ime. Since (35) holds for all imes (,T) he RHS represens he survival-coningen value a of he replicaing porfolio. In paricular, he survival-coningen bank balance a ime when replicaing a survival claim is given by M T a(; T ). Evaluaing (13) as T implies ha: M lim Q a (; T )= lim a(; T ) = 1 T T, from (12). Hence, replicaing he payoff of a survival claim requires going shor 1 CDS of mauriy T and also wriing Q T a(u)du CDS of mauriy u for each mauriy u (,T). The erm srucure of survival claim values is given by A (T ) for T. 9 Closed Form Soluions When replicaing he defaulable annuiy, he survival-coningen bank balance saisfies he backward ODE (6), while he iniial value of he defaulable annuiy saisfies he forward ODE (25). This secion shows ha boh ODE s have closed form soluions when eiher he iniial CDS curve S (T ),T is fla in T or he iniial forward rae curve r(t ),T is fla in T. As a resul, we also ge closed form soluions for he Green s funcion, and hence he price and hedge of any defaul-coningen claim. 11
9.1 Fla CDS Curve In his subsecion, we assume ha he iniial CDS spread curve is fla a S. In his case, he backward ODE (1) governing he survival-coningen bank balance M a (; T ) when replicaing a defaulable annuiy of mauriy T simplifies o: 2 2M a(; T ) r()+ S The ODE is inegrable since he HS inegraes: { M a(; T ) r()+ S ] } M a (; T ) ] M a(; T ) r ()M a (; T )=, (,T). (36) =, (,T). (37) Subjecing (36) o he erminal condiions (11) and (12) and solving implies ha he closed form soluion for he survival-coningen bank balance M a (; T ) is given by: M a (; T )= e y(; )+ S ]( ) d,,t], (38) where: y(; r(v)dv ) (39) is he yield o mauriy a ime, T] of a defaul-free bond wih mauriy u, T ]. Subsiuing (38) in (13) implies ha he closed form soluion for he rae Q a (; T ) a which CDS are iniially wrien when replicaing a defaulable annuiy is given by: Q a (; T )= 1 r()+ S ]e y(; )+ S ]( ),,T]. (4) When he CDS curve is fla, (16) and (38) imply ha he Green s funcion g(; u) is given by: g(; u) = { u e r(v)+ S ]dv d, if,u] if (u, T ). (41) Evaluaing (38) a = implies ha if he iniial CDS spread curve is fla a S, hen he iniial value of a defaulable annuiy of mauriy T is given by: A (T ) M a (; T )= e y(; )+ S ] d. (42) Thus, he defaulable annuiy is simply valued by discouning each dollar received a ime u back o ime. The rae used for discouning a dollar received a some ime u (,T) is he sum of he yield o mauriy y(; u) and he recovery-adjused CDS spread S. Equaion (42) explains why each fixed paymen rae S is ofen referred o as a spread. 12
When he CDS spread is consan, he forward ODE (25) for A (u) simplifies o: A (u)+ r(u)+ S ] A (u) =, u,t]. (43) The soluion of (43) subjec o he iniial condiions (26) and (27) is given by (42) evaluaed a T = u. Hence, (29) implies ha he risk-neural survival probabiliy is exponenial: Q{τ u} = e S u. (44) Thus, he risk-neural probabiliy of defaul is he same as would arise in a reduced form model of defaul wih he hazard rae consan a S. 9.2 Fla Forward Rae Curve In his subsecion, we rever o a deerminisic CDS spread curve S (u),u, bu now we assume ha he iniial yield curve is fla a r. I follows ha he iniial forward rae curve is fla a r. We assume no arbirage and as we have already assumed deerminisic ineres raes, i follows ha he spo rae is consan over ime a r. Under a consan spo rae, he backward ODE (1) governing he survival-coningen bank balance M a () when replicaing a defaulable annuiy simplifies o: 2 2M a(; T ) r + S () The soluion of (45) subjec o he erminal condiions (11) and (12) is: M a (; T )= Subsiuing (46) in (13) implies: e T u ] M a(; T )=, (,T). (45) r+ S (v) r+ S (v) ] dv du,,t]. (46) Q a (; T )= e,,t]. (47) When he spo rae is consan a r, (16) implies ha he Green s funcion g(; u) is given by: { u e ] u g(; u) = r+ S (v) dv d, if,u] (48) if (u, T ). Evaluaing (46) a = implies ha under a consan ineres rae r, he iniial value of a defaulable annuiy of mauriy T is given by: A (T ) M a (; T )= 13 ] dv e ŷ(u;t )+r](t u) du, (49)
where: ŷ(u, T ) u S (v) dv T u. (5) Curiously, (49) indicaes ha under a consan ineres rae, he defaulable annuiy is now valued by discouning each dollar received forward o T raher han backward o. The rae used for discouning a dollar received a some ime u (,T) is he sum of he consan ineres rae r and ŷ(u; T ) defined in (5). To undersand why his curious resul holds, we noe ha when he spo riskfree rae is consan a r, he forward ODE (25) for A (u) simplifies o: A (u)+ r + S (u) ] A (u)+ S (u) A (u) =, u,t], (51) or equivalenly: { u u A (u)+ r + S ] } (u) A (u) =, u,t]. (52) Suppose we consider a dual economy in which ime ˆ runs backward from T, i.e. ˆ T and in which he CDS spread is consan a r, while he spo ineres rae a ime ˆ is deerminisically given by S (T ). In such an economy, he yield o mauriy a he iniial ime ˆ = wih erm T u is given by ŷ(u; T ). By comparing (37) and (52), we see ha he laer equaion is recognized as he backward ODE in he dual economy. Hence, he soluion (49) o (52) is jus he soluion (42) o (37) wih ime running backwards from T and he wo curves swiched. The soluion of (51) subjec o he iniial condiions (26) and (27) is given by (49) evaluaed a T = u. Differeniaing (49) w.r.. T implies ha he iniial value of a survival claim of mauriy T is: ( A (T ) = 1 r + S ) (T ) A (T ), = 1 ra (T ) U (T ), (53) where U (T ) S (T ) A (T ) is he spo value of a uni recovery claim. Fuure valuing (53) and using (29) implies ha he risk-neural survival probabiliy is given by: 1 Fla Forward and CDS Curve Q{τ T } = e rt 1 ra (T )+U (T )]. (54) The Green s funcion in (48) simplifies if boh he forward rae curve and he CDS spread are fla: 1 e r+ S ](u ), if,u] g(; u) = r+ S (55) if (u, T ). 14
11 Affine Forward and CDS Curve In his secion, we suppose ha he inial forward rae curve and he iniial CDS curve are boh affine in erm. We show ha he Green s funcion can be expressed in erms of confluen hypergeomeric funcions. 12 Summary and Fuure Research We showed how o replicae he payoffs o a class of defaul-coningen claims by aking saic posiions in a coninuum of CDS of differen mauriies. Alhough we assumed deerminisic ineres raes and a consan recovery rae on he CDS, he replicaion was oherwise robus in ha we made no assumpions on he process riggering defaul. We also derived a new erminal value problem which deermines CDS holdings and we illusraed is specificaion for defaulable annuiies, uni recovery claims, and survival claims. We furhermore derived a new iniial value problem which deermined he DVO1 of a CDS. In general, he backward and forward differenial equaions in hese problems mus be solved numerically, bu we indicaed imporan special cases in which hey can be solved in closed form. Fuure research can proceed in a leas four direcions. Firs, one can look a oher specificaions of he forward rae curve and CDS spread curve ha lead o closed form soluions of eiher he backward or forward equaion. For example, if he CDS curve and he forward rae curve are boh affine in he erm, hen one can ransform eiher evoluion equaion ino Kummer s ODE. Second, one can explore wheher he resuls of he curren analysis exend o he case where he loss given defaul,, is deerminisic raher han consan. Third, one can ry o address sochasic ineres raes. Finally, one can consider calibraing o oher insrumens such as off marke CDS or defaulable zero coupon corporae bonds. When a bond is insured by a CDS, he recovery rae can be allowed o be sochasic and unknown since i drops ou of he combined payoff. In he ineress of breviy, hese exensions are bes lef for fuure research. Appendix Recall from (6) and (14) ha he Green s funcion g(u; v) solves: g(u; v) =δ(u v), (56) where: Du 2 r()+ S ] () Du 1 r ()Du (57) is a second order linear differenial operaor. Muliplying (56) by g(; u) and inegraing u from o T implies: g(; u)g(u; v)du = g(; v), (58) 15
by he sifing propery of dela funcions. By he definiion of he adjoin operaor: g(; u)g(u; v)du = δ(u )g(u; v)du, (59) where is given in (21) and he sifing propery of dela funcions has been used on he righ hand side. I follows ha: g(; u) δ(u )]g(u; v)du =, (6) for all v. Bu his can only be rue if: for all, u,t]. g(; u) =δ(u ) (61) 16
References 1] Breeden, D. and R. izenberger, 1978, Prices of Sae Coningen Claims Implici in Opion Prices, Journal of Business, 51, 621 651. 2] Sakgold I., 1979, Green s Funcions and Boundary Value Problems, John Wiley & Sons, New York. 17