Fixed Income Quaniaive Credi Research November 2003 Valuaion of Porfolio Credi Defaul Swapions Claus M. Pedersen We describe he deails of he CDX and RAC-X porfolio swapion conracs and argue why Black's formulas are inappropriae for heir pricing. We presen a simple, easy o implemen, alernaive model ha prices he swapions using he credi curves of he reference eniies and a single volailiy parameer.
Claus M. Pedersen 1-212-526-7775 cmpeders@lehman.com We discuss he valuaion of opions on porfolio credi defaul swaps wih a focus on sandardized conracs referencing for example he CDX or RAC-X eniies. Firs, we describe he porfolio swap and swapion conracs. Nex, we argue ha Black formulas, he sandard formulas for pricing single-name defaul swapions, are inappropriae for pricing porfolio defaul swapions. Finally, we presen a simple, easy o implemen, alernaive model ha prices porfolio swapions using he credi curves of he reference eniies and a single volailiy parameer. 1 1. INRODUCION Invesors are increasingly finding porfolio credi defaul swaps (also called porfolio CDS, porfolio defaul swaps or, in his repor, simply porfolio swaps) useful for gaining exposure o marke wide credi spreads. Opions on porfolio defaul swaps (porfolio credi defaul swapions, porfolio defaul swapions, or simply porfolio swapions) allow invesors o leverage his exposure and provide a ool for gaining exposure o marke wide credi spread volailiy. Sandardized porfolio defaul swaps referencing he CDX.NA.IG (in his repor simply CDX) and RAC-X NA (in his repor simply RAC-X) eniies are oday rading wih bid-offer spreads as low as 1-2bp. Recenly, CDX and RAC-X swapions have seen increasing rading volume as well. Lehman Brohers is a marke maker in CDX and RAC-X porfolio swaps and swapions. he purpose of his research repor is o inroduce he CDX and RAC-X porfolio swapions and presen a simple model for heir pricing and risk managemen. Such a model already exiss for single-name defaul swapions in he form of a modificaion of Black s formulas for ineres rae swapions. hese Black formulas for defaul swapions provide he values of swapions ha knock ou if he reference eniy defauls before swapion mauriy. o price an opion o buy proecion ha does no knock ou, i is herefore necessary o add o he Black formula price he value of proecion unil swapion mauriy. Porfolio swapions do no knock ou. I has herefore been suggesed o price hese as single-name non-knockou defaul swapions using a credi curve represening he average credi worhiness of he reference eniies. However, such an approach will end o overvalue high srike porfolio swapions compared wih a model ha more explicily models he underlying cashflow. he easies way o see ha Black formulas give mispricing is by examining he pricing of a deep ou-of-he-money payer swapion (his is an opion o buy porfolio proecion a a very high srike spread). If i is priced as suggesed above, is value will be close o he value of proecion unil swapion mauriy. his is incorrec; he price should approach zero as he srike increases. his is because, when he srike is very high, he payer will no be exercised even if a reference eniy has defauled unless he porfolio spread on he non-defauled eniies has widened sufficienly. We propose o direcly model he erminal value of he swapion using a single sae variable which we call he defaul-adjused forward porfolio spread. As long as no defauls have occurred, his spread can reasonably be called he forward porfolio spread since i is he 1 I would like o hank Peer Alpern, Georges Assi, Jock Jones, Roy Mashal, Marco Naldi and Luz Schloegl for discussions and commens. November 2003 Please see imporan analys cerificaion(s) a he end of his repor. 1
srike for which he payer and receiver porfolio swapions are equal in value. If no defauls have occurred a swapion mauriy, he spread is simply he porfolio spread iself. As we explain in deail below, we propose o incorporae defauls direcly ino he spread and hereby avoid modeling he number of defauls which would oherwise be required o explicily model he erminal value of a porfolio swapion. In secion 2, we describe he CDX and RAC-X porfolio swap and swapion conracs. In secion 3, we explain why he Black formulas for defaul swapions should no, as suggesed by ohers, be used o price porfolio swapions. In secion 4, we presen our alernaive valuaion model. Secion 5 concludes. 2. HE PORFOLIO SWAP AND SWAPION CONRACS o value a porfolio swapion i is necessary o undersand he deails of he swapion conrac. he presenaion is focused on CDX bu, as explained below, RAC-X works almos idenically. 2.1. he CDX swap conrac A buyer of CDX proecion is buying credi proecion on 125 fixed reference eniies. If he noional of he CDX swap is 125 million, say, hen buying CDX proecion has he same economic effecs as buying proecion on each reference eniy hrough 125 single-name No-R 2 CDS, each wih a noional of 1 million. A buyer of CDX proecion is obligaed o pay he same premium (called he fixed rae) on all he 125 hypoheical underlying CDS. When a defaul occurs, he CDX proecion buyer mus physically sele o receive he proecion paymen for he defauled eniy. Afer selemen, he relevan hypoheical underlying CDS is eliminaed from he CDX swap, which hen has a noional ha is decreased by 1/125 of he original noional and one less reference eniy. he fixed rae remains unchanged. he paymen daes are he sandard CDS daes (20 h of March, June, Sepember and December). oday here are wo CDX swaps wih 5- and 10-year mauriies. Currenly he fixed rae is 60bp for he 5-year conrac and 70bp for he 10-year conrac. On he 20 h of March and Sepember each year (he roll daes), new CDX swaps will become on-he-run o ensure ha he reference eniies represen he aggregae marke and ha he on-he-run conracs have 5- and 10-year mauriies 3. In addiion o reference eniies and mauriies, he fixed raes will also be changed 4. Changes o new on-he-run CDX swaps have no influence on he exising conracs. he mauriy, he fixed rae, and he reference eniies are all fixed hroughou he life of specific CDX swaps and swapions. Reference eniies may also be eliminaed from he on-he-run CDX swaps beween roll daes, for example, if an eniy defauls. Eliminaed eniies will no be replaced beween roll daes. If an eniy is eliminaed, he relevan hypoheical underlying CDS can be spli off from exising CDX swap conracs o ensure ha hey remain on-he-run. his ensures liquidiy of he CDX swap conrac afer a defaul has been seled. 2 3 No-R means no resrucuring and refers o he fac ha only defaul (bankrupcy and failure o pay) can rigger he CDX conrac. In he US, CDS mainly rade under he Mod-R (modified resrucuring) clause which includes resrucurings as a credi even. For deails see O Kane, Pedersen and urnbull (2003). he reference eniies in he new on-he-run swap are chosen by voing among CDX marke makers. he voing sysem is fundamenally differen from he rules-based mehodology used o deermine he consiuens of he Lehman Brohers Credi Defaul Swap Index. See Berd e al. (2003) for more deails. 4 Afer publicaion of he new reference eniies, CDX marke makers will submi 5- and 10-year quoes, he medians of which will become he new fixed raes. November 2003 2
he value of a CDX swap is driven by he CDS spreads on he 125 reference eniies. When he spreads are high compared wih he fixed rae, he swap has a posiive value o he proecion buyer who herefore pays a CDX price o he proecion seller a conrac iniiaion (when he swap price is negaive is absolue value is paid by he proecion seller o he proecion buyer). he swap price is quoed in he form of a CDX spread which can be readily convered ino he price. he conversion from spread o price can be done via he CDSW calculaor on Bloomberg. he calculaion is o discoun fixed paymens equal o he CDX spread minus he fixed rae, as when calculaing he mark-o-marke on a CDS 5. he conversion can be wrien as: P = PV 01 ( S FR) where FR is he fixed rae, S is he quoed CDX spread, P is he corresponding CDX price, and PV01 is he risky PV01 from oday o swap mauriy, ie, he value of receiving 1bp on he CDX paymen daes unil mauriy of he swap or defaul, whichever occurs firs. he marke sandard for quoing CDX spreads assumes ha he PV01 is calculaed using discoun facors ha have been calibraed o fi a fla CDS curve wih spreads equal o he CDX spread. he calibraion uses a recovery-given-defaul of 40% and defaul-free ineres raes aken from he curren Libor curve. he inrinsic CDX spread he inrinsic CDX spread is he spread quoe ha convers ino he price of buying he CDX equivalen credi proecion hrough 125 single-name CDS conracs, each wih a conracual spread equal o he CDX fixed rae. he inrinsic spread depends on he shape of he individual credi curves, bu will be close o he average CDS spread of he appropriae mauriy across he CDX reference eniies. Deerminaion of he inrinsic spread is made more difficul by he fac ha resrucuring is no included as a credi even in he CDX swap conrac. No-R CDS spreads are rarely available, bu in he US, dealers end o quoe No-R spreads around 5% lower han Mod-R spreads. Inrinsic CDX spreads are usually calculaed from Mod-R spreads ha have been discouned 5% (see O Kane, Pedersen and urnbull (2003)). Aside from possible error inroduced when deermining No-R spreads, a CDX spread may differ from is inrinsic value because of specific shorer erm demand-supply condiions in CDX and/or CDS markes. Because of he No-R feaure, and because of bid-offer spreads in CDS markes, i is generally no feasible o arbirage he differences beween quoed and inrinsic CDX spreads. Currenly, inrinsic CDX spreads end o be higher han quoed CDX spreads implying ha i is cheaper o buy proecion hrough CDX han hrough he underlying single-name CDS. Par of he difference may be a resul of he higher liquidiy of CDX compared o CDS. In oher words, sellers of CDX proecion can be seen as demanding a lower liquidiy premium han sellers of single-name CDS proecion. 5 See O Kane and urnbull (2003) for deails on how o calculae he mark-o-marke of a CDS. November 2003 3
2.2. he CDX swapion conrac A CDX swapion references a specific underlying CDX swap wih a specific mauriy. Usually he CDX swap was on-he-run when he opion was raded. Aside from he underlying CDX swap, he CDX swapion is specified by a swapion mauriy, a srike spread, and a swapion ype. he sandard porfolio swapions are European. he swapion ype is eiher payer or receiver. A payer gives he righ o become a proecion buyer in he underlying CDX swap a he srike spread. A receiver gives he righ o become a proecion seller. If he swapion is exercised, he srike spread is convered ino an exercise price (also called he selemen paymen) using he same calculaion as when convering a CDX spread quoe ino a CDX price. For example, if he srike spread is K, hen he exercise price is: P( K) = γ ( K) ( K FR) where FR is he fixed rae in he underlying CDX swap conrac, and γ(k) is he risky PV01 calculaed a he exercise dae using he same assumpions as if convering a CDX spread quoe of K ino is corresponding CDX price. ha is, γ(k) is he risky PV01 calculaed from a credi curve ha has been calibraed o a fla CDS curve wih spreads equal o he srike spread, K, using a recovery-given-defaul of 40% and he Libor curve a he exercise dae. Consider an example. On November 6, 2003, a CDX payer swapion on he 5-year CDX swap wih a noional of $100 million was raded. he swapion mauriy is March 22, 2004, he srike spread is K = 55bp, and he fixed rae is FR = 60bp. If we assume ha he Libor curve on March 22, 2004 is equal o he forward Libor curve for ha dae observed on November 6, 2003, hen we can find he exercise price. Under his Libor curve, he risky PV01 is γ(k) = 0.0453 ($ per $100 noional). he exercise price is hen 0.0453 (55-60) = -0.227, so if he swapion is exercised, he proecion seller mus pay $227 housand o he proecion buyer. his cash ransfer of $227 housand is independen of he number of defauls ha may have occurred before swapion mauriy and he CDX spread a swapion mauriy. he only uncerainy abou he amoun arises from uncerainy abou he Libor curve a he exercise dae. When he swapion is exercised, he proecion buyer has in effec bough proecion on all reference eniies, including hose ha may have defauled before swapion mauriy. he proecion buyer can hen immediaely sele for proecion paymen on any defauled eniies. he erminal value of he swapion herefore depends on he recoveries on defauled eniies and he mark-o-marke on he CDX swap wih he defauled eniies eliminaed. Depending on he mauriy of he swapion and he ime from he opion rade dae o he following roll dae, he CDX swap wih he defauled eniies eliminaed may or may no be on-he-run a opion mauriy. However, given he relaively shor swapion mauriies usually seen (mos swapions rade wih mauriies of six monhs or less), his CDX swap should be liquid hroughou he life of he swapion. Le us exend he example above o deermine he erminal value of he swapion. Assume ha one reference eniy defauled before March 22, 2004, and he cheapes deliverable obligaion issued by he defauled eniy rades a $45 (per $100 noional). Also assume ha he quoed marke CDX spread on March 22, 2004, on he CDX swap wih he defauled eniy eliminaed is 75bp. Finally, assume as above ha he Libor curve on March 22, 2004, is he forward Libor curve for ha dae observed on November 6, 2003. In his case, he payer swapion will be exercised and he swapion seller mus make an iniial $227 housand cash paymen o he swapion buyer a exercise (see deails above). Afer exercise, he swapion November 2003 4
buyer can deliver $800 housand of noional of he deliverable obligaion menioned above and receive a $800 housand cash paymen. he cos of he deliverable is $360 housand. he las sep in deermining he erminal value of he swapion is o find he mark-o-marke of he CDX swap wih he defauled eniy eliminaed. he noional of his CDX swap is $99.2 million and he marke value o he swapion buyer of his posiion urns ou o be $675 housand (found as 0.0450 (75-60) $99.2 million/100, given ha he risky PV01 calculaed from a fla curve of 75bp is 0.0450). Alogeher he payer swapion has a oal value of $1.342 million (227+800-360+675). Noice ha for his paricular example, he effec of one defaul is abou he same as a 10bp widening in he CDX spread on he non-defauled eniies. 2.3. he RAC-X swap and swapion conracs RAC-X swap and swapion conracs work in he same way as he CDX swap and swapion conracs described above. he differences are he number of reference eniies, he ideniy of he reference eniies, he fixed rae, and he procedure for deermining new reference eniies. We are no aware of any oher differences wih pricing implicaions. here are currenly 100 RAC-X reference eniies he composiion of which is scheduled for change every hree monhs (around he sandard CDS daes, ie, he 20 h of March, June, Sepember, and December). A change o new on-he-run swap conracs occurred in Sepember 2003 when 34 eniies were replaced. he fixed rae is currenly 100bp in boh he old and he new RAC-X swap conracs. On November 6, 2003, RAC-X was rading a 53 and a RAC-X proecion seller would have o make a significan upfron paymen a conrac iniiaion (more han $2 million on a conrac wih a $100 million noional). Currenly, here are on-he-run RAC-X swaps wih 5- and 10-year mauriies. he remainder of his repor uses generic language, and he issues apply o boh CDX and RAC-X swapions. We use erminology such as porfolio swap, porfolio swapion and porfolio spread. 3. BLACK FORMULAS FOR DEFAUL SWAPIONS Black formulas for defaul swapions are simple heoreically consisen formulas for valuing single-name defaul swapions. he formulas give values of swapions ha knock ou if a defaul occurs before swapion mauriy. he knockou feaure is no relevan for receiver swapions (opions o sell proecion), as hey will never be exercised afer a defaul. For payer swapions (opions o buy proecion), i is necessary o add o he Black formula price he value of proecion from he swapion rade dae o he swapion mauriy. o inroduce noaion, le >0 be he swapion mauriy and le M > be he mauriy of he underlying CDS. Le PV01 (, M ) be he value a ime of a securiy ha pays a 1bp annual flow saring a he firs CDS dae afer ime and ending a M or defaul, whichever occurs firs. Similarly, le PVP (, M ) be he value a ime of a securiy ha pays par minus he recovery-given-defaul a he ime of defaul if defaul occurs beween ime and ime M. Given a Libor curve and an issuer CDS curve a ime, PV01 (, M ) and PVP (, M ) can be found using he Jarrow-urnbull credi pricing framework wih a piecewise linear hazard rae. Furhermore, for le: PVP (, M ) F (, M ) = PV 01 (, ) M November 2003 5
F (, M ) is he -forward-saring spread on a CDS ha maures a ime M. he Black formulas are based on he assumpion ha: F (, M 1 2 ) = F0 (, M )exp σ + σ ε 2 where ε is a sandard normal random variable under he risk neural measure, corresponding o F (, M ) being lognormally disribued wih volailiy σ. he formulas are 6 : PS RS ( F (, ) N( d ) K N( )) KO 0 d = PV 010 (, M ) 0 M 1 2 ( K N( d ) F (, ) N( )) = PV 010 (, M ) 2 0 M 1 0 d 2 log( F0 (, M ) / K) + σ / 2 2 d1 =, d2 = d1 σ σ where K is he srike, PS 0 KO is he knockou payer value and RS 0 is he receiver value. he value of a payer ha does no knock ou is: KO + PS0 = PS0 FEP0 (0, ) where FEP (0,) (fron-end proecion) is he value a ime of a conrac ha pays par minus he recovery-given-defaul a swapion mauriy,, if defaul occurs beween ime 0 and ime. Deermining he forward spread, he forward PV01, and he fron-end proecion, requires a CDS curve for he issuer. he shape of he curve is significan in he valuaion. I has been suggesed ha he above mehodology can be used o value a porfolio swapion, using a CDS curve ha represens he average crediworhiness of he reference eniies. As an approximaion we could, for example, for each mauriy calculae he average spread across reference eniies and shif he averages o ensure ha he 5- and 10-year poins mach he quoed porfolio spreads. We could also adjus he averages in such a way ha if all he reference eniies had he adjused average curve, hen he inrinsic porfolio spreads would equal he quoed porfolio spreads. Alhough his approach is appealing because of is simpliciy, i is no heoreically sound. I is easy o see his by examining he pricing of payer swapions as he srike increases. For a very high srike, he above approach will give a price close o FEP, he value of he fron-end proecion. When he srike is infinie, however, he price should be 0, since such a swapion should no be exercised even if mos eniies defauled 7. If all reference eniies had he same CDS curve, hen he approach would correcly price he difference beween he payer and he receiver (his will become clearer in he nex secion). From ha perspecive, he fron-end proecion has o be included in he payer price. However, his causes he decomposiion ino he payer and receiver prices o become skewed owards high prices for high srike swapions. When he reference eniies do no have he same CDS curves he pricing of he difference beween he payer and he receiver also breaks down. Using he average curve will end o overvalue a long-payer shor-receiver posiion compared wih he approach we sugges in he 6 7 For more deails on how o derive he formulas, see he secion on modeling credi opions in he Lehman Brohers Guide o Exoic Credi Derivaives and he references given here. heoreically he swapion could be exercised if all eniies defauled and had a combined recovery of less han 40%. November 2003 6
nex secion. his is mainly caused by he concaviy in he value of he proecion leg of a CDS when seen as a funcion of he CDS spread level. 4. A MODEL FOR PORFOLIO DEFAUL SWAPIONS o explicily model he erminal value of a porfolio swapion i is necessary o model boh he recovery on defauled eniies and he spread on he porfolio swap wih he defauled eniies eliminaed. As a simplificaion, we sugges incorporaing defauls direcly ino he porfolio spread o arrive a a defaul-adjused forward porfolio spread ha can be used as a single sae variable o generae he erminal value of he swapion. Alernaively, he erminal value could be modeled direcly, bu i is no clear which disribuion would be reasonable. On he oher hand, we find i reasonable o use as a firs approximaion a lognormal disribuion for he defaul-adjused forward porfolio spread a swapion mauriy. 4.1. he pu-call pariy he firs sep in valuing a porfolio swapion is o idenify and value he underlying. We refer o he underlying as he defaul-adjused forward porfolio swap. I is differen from a regular knockou forward porfolio swap because proecion for defauls ha occur before swapion mauriy are paid a swapion mauriy. he defaul-adjused forward porfolio swap can be viewed as a porfolio of non-knockou forward saring CDS on each reference eniy. A non-knockou forward saring CDS is a combinaion of a regular knockou forward saring CDS and fron-end proecion from oday o swapion mauriy. he conracual spread in he CDS is he fixed rae in he porfolio swap, denoed FR. he value of such a forward saring CDS a ime is: V i i i M M + = PVP (, ) PV 01 (, ) FR FEP (0, ) where an I superscrip indicaes ha he values are for he i h reference eniy. is he swapion mauriy and M is he mauriy of he CDS. he PVP and PV01 noaion was explained in he previous secion. FEP i (0,) is he value a ime of a conrac ha pays par minus he recovery-given-defaul if he i h eniy defauls beween ime 0 and ime. I is imporan o noe ha if he i h eniy defauled beween 0 and, hen FEP i (0,) is par minus recovery discouned on he ime Libor curve from o. he value a ime of he defaul-adjused forward porfolio swap is V 1 N i = V i= 1 N where N is he number of reference eniies. he facor 1/N is used because everywhere he noional of a conrac is assumed o be par unless oherwise specified. If he swapion is exercised, he proecion buyer mus pay he exercise price o he proecion seller. he exercise price is specified by he srike spread, K, bu also depends on he Libor curve a opion mauriy. For valuaion purposes, we assume ha Libor raes are deerminisic. We can herefore use oday s forward Libor curve a swapion mauriy o calculae he exercise price. I is given by: P( K ) = γ ( K)( K FR) (1) i November 2003 7
where γ(k) is he risky PV01 from o M calculaed on a credi curve ha has been fied o a fla CDS erm srucure wih spreads equal o K, using a recovery-given-defaul of 40% (see secion 2.2 for deails). Because he fron-end proecion values a, FEP i (0,), incorporae defauls beween 0 and, we can wrie he erminal swapion values as: PS ( K) = max{ V P( K),0} (2) RS ( K) max{ P( K) V =,0} (3) where PS is he payer and RS is he receiver. We see ha PS (K) RS (K) = V P(K) and we ge he pu-call pariy: = (4) PS0( K) RS0( K) V0 P( K) D(0, ) where D(0,) is he Libor discoun facor from 0 o. When pricing he defaul-adjused forward porfolio swap, i is imporan ha he porfolio swap iself is priced correcly off he individual credi curves, ie, he inrinsic porfolio spread should equal he curren marke porfolio spread. In pracice, his requires ha he individual curves are adjused as explained in secion 2.1.1. 4.2. Pricing he payers and receivers o price he swapions, we propose a sochasic model for he value of he defaul-adjused forward porfolio swap a swapion mauriy, V, and hus for he erminal swapion values. We specify he disribuion of V hrough wha we call he defaul-adjused forward porfolio spread, denoed X. For any ime beween 0 and, X is given by: V = γ ( X )( X FR) D(, ) (5) he funcion γ and he consan FR were defined in he previous secion. D(,) is he Libor discoun facor from o. When K = X 0 hen V 0 = γ(k)(k FR)D(0,) = P(K)D(0,), and from he pu-call pariy PS 0 (K) = RS 0 (K). In words: he defaul-adjused forward porfolio spread is he srike for which he value of he payer is equal o he value of he receiver. Because of his relaionship we can simply call he defaul-adjused forward porfolio spread he forward porfolio spread when no defauls have ye occurred. So X 0 is he forward porfolio spread a ime 0 for ime. If no defauls have occurred a, hen he defaul-adjused forward porfolio swap is he porfolio swap iself, and V is he value o he proecion buyer of all he underlying CDS. By he definiion of X (V = γ(x )(X FR)), X is he porfolio spread a. his is consisen wih he above erminology of calling X he forward porfolio spread a for when no defauls have occurred before. When specifying he disribuion of X, we mus ensure ha he defaul-adjused forward porfolio swap is priced correcly in he model. X will be specified under he risk neural measure (where D(,) is he numeraire price a ime ). Specifically, we mus ensure ha: [ ] V = 0 D(0, ) E V or equivalenly, using (1) and (5): November 2003 8
[ P X )] P( X ) E = ( 0 We assume ha X is lognormally disribued wih: 1 2 X = X 0 exp ( µ σ ) + σ ε 2 where ε is a sandard normal random variable. µ is he drif and σ he volailiy of he defaul adjused forward porfolio spread. We choose his specificaion because of is simpliciy and because a lognormal disribuion is fairly consisen wih empirical evidence. I is no difficul o imagine ha he X process may conain jumps. X may jump, for example, if here is a surprise defaul of one of he reference eniies. On he oher hand, an eniy usually defauls only when is spreads are already high, in which case he jump in he porfolio spread a he acually ime of defaul may be small. In general, he lognormal disribuion in our porfolio conex seems more appropriae han assuming a lognormal spread for a single eniy. he drif µ is he free parameer used o ensure ha E[P(X )] = P(X 0 ). he funcion P is well behaved, and can be very well approximaed using a polynomial spline. Even a simple second-order aylor approximaion of P around X 0 works well as long as he variance of X is no oo high. In pracice, he drif will be close o 0. Once he drif has been fixed, and we have he approximaion of he funcion P, i is sraighforward o price he swapions by discouning he expeced erminal value using he disribuion of X. Subsiuing (5) and (1) ino (2) and (3), and using he fac ha P is an increasing funcion, we ge: [( P( X ) P( K)) ( X K )] [( P( K) P( )) ] PS ( K) = D(0, ) E 1 0 RS ( K) = D(0, ) E 1 0 X ( K ) he equaions can be solved in closed form when P is approximaed by a polynomial spline. 4.3. Pricing examples We illusrae by pricing CDX swapions ha maure on March 22, 2004. he underlying CDX swap maures on March 20, 2009. We did he valuaion on November 6, 2003, a a ime when he CDX spread was 56. X Figure 1. Adjused CDS spreads (in bp) on he CDX reference eniies on November 6, 2003 Mauriy 6M 1Y 2Y 3Y 4Y 5Y 7Y Average 35.6 39.0 44.1 48.8 52.3 55.6 61.3 Maximum 206 228 241 264 277 281 295 Median 22 24 29 33 38 40 46 Minimum 4 4 4 9 9 11 15 he firs sep in pricing he swapions is o choose he bes possible CDS spread curves for he 125 reference eniies. Usually i suffices o base he curves on yeserday s closing levels. he curves can hen be adjused o curren marke levels by using liquid CDS spreads (eg, parallel shifing based on he 5-year CDS spreads). Afer he individual curves have been adjused o he curren single-name marke, all he curves mus be adjused so ha he November 2003 9
porfolio of he 125 single-name CDS is priced according o he marke CDX spread of 56, ie, so ha he inrinsic CDX spread is 56. his adjusmen also incorporaes he fac ha CDX is No-R. o price he swapions, we mus choose a spread volailiy. he spread volailiy in our model is no he same as he spread volailiy o be used wih he Black formula. We model he defaul-adjused forward porfolio spread which incorporaes he defauls ha occur before swapion mauriy. he volailiy in our model should herefore be higher han he volailiy used in he Black formulas. Based on marke prices, we choose a defaul-adjused forward porfolio spread of 55%. Figure 2 shows he compued payer and receiver values. In parenheses nex o a price is he implied Black volailiy when he average adjused CDS spread curve of he reference eniies is used in he Black formula. Figure 2. CDX swapion prices (in $ per $100 par) and implied Black volailiies off he average adjused CDS spread curve of he reference eniies Srike (in bp) 45 50 55 60 65 70 75 Discouned exercise price -0.68-0.45-0.23 0 0.23 0.45 0.67 Payer price (implied Black vol.) 0.79 (35%) 0.63 (40%) 0.49 (41%) 0.38 (41%) 0.29 (39%) 0.22 (37%) 0.16 (31%) Receiver price (implied Black vol.) 0.08 (52%) 0.15 (52%) 0.24 (51%) 0.35 (50%) 0.48 (48%) 0.64 (47%) 0.80 (45%) here are wo main lessons o draw from Figure 2. For high srikes, implied Black volailiies for payer swapions decrease as he srike increases. he implied Black volailiies are no he same for payers and receivers wih he same srike. Figure 2 shows ha, according o our model, he implied Black volailiy for he payer swapions should decrease as he srike increases for high srikes. he effec is very noiceable, especially when comparing payers wih srikes of 70 and 75. When he srike is 75, he value of he payer according o our model is very close he value of fron-end proecion priced off he adjused average CDS curve (0.16 versus 0.14). When he srike is 80, he payer value is oo low for he implied Black volailiy o be defined. We also see ha he Black volailiies implied from our model are no he same for payers and receivers wih he same srike. his shows ha he defaul-adjused forward porfolio swap is no priced correcly off he adjused average CDS spread curve. According o he pu-call pariy he payer price minus he receiver price plus he discouned exercise price is equal o he value of he defaul-adjused forward porfolio swap. In he example above, he defauladjused forward porfolio swap has a value of 0.03. In he Black formulas, he value of he payer minus he value of he receiver does no depend on he volailiy. If he discouned exercise price is subraced from ha difference, we arrive a he defaul-adjused forward porfolio swap value priced off he adjused average CDS curve. he value is 0.09, which is 0.06 oo high according o he individual curves. I is his mispricing ha gives he differen implied Black volailiies for he payers and receivers. November 2003 10
5. SUMMARY We presened he deails of he CDX and RAC-X swap and swapion conracs, and inroduced marke-consisen erminology o discuss heir pricing. We argued ha Black s formulas for single-name defaul swapions are inappropriae for pricing porfolio swapions, wih he mispricing especially noiceable for deep ou-of-he-money payer swapions. We presened an alernaive model ha direcly models he erminal swapion values hrough a defaul-adjused forward porfolio spread. Our mehodology ensures ha a combined longpayer shor-receiver posiion, which is insensiive o spread volailiy, is priced in a manner consisen wih he credi curves of he reference eniies. Finally, we illusraed wih numerical examples ha he price differences beween our model and he Black formulas are significan. REFERENCES Berd, A., A. Desclée, A. Golbin, D. Munves and D. O Kane (2003). he Lehman Brohers Credi Defaul Swap Index, Lehman Brohers, Fixed Income Index Group, Ocober 2003. O Kane, D, M. Naldi, S. Ganapai, A. Berd, C. Pedersen, L. Schloegl and R. Mashal (2003). he Lehman Brohers Guide o Exoic Credi Derivaives, Lehman Brohers, Ocober 2003. O Kane, D, C. Pedersen, and S. urnbull (2003). Pricing he Resrucuring Clause in Credi Defaul Swaps, Lehman Brohers, June 2003. O Kane, D. and S. urnbull (2003). Valuaion of Credi Defaul Swaps. Quaniaive Credi Research Quarerly, Lehman Brohers, April 2003. November 2003 11
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