VALUATION OF CREDIT DEFAULT SWAPTIONS AND CREDIT DEFAULT INDEX SWAPTIONS

Transcription

1 VALATION OF CREDIT DEFALT SWAPTIONS AND CREDIT DEFALT INDEX SWAPTIONS Marek Rukowski School of Mahemaics and Saisics niversiy of New Souh Wales Sydney, NSW 2052, Ausralia Anhony Armsrong School of Mahemaics and Saisics niversiy of New Souh Wales Sydney, NSW 2052, Ausralia Firs draf: June 20, 2007 This version: July 5, 2008 The research of M. Rukowski was suppored by he ARC Discovery Projec DP

2 2 Valuaion of Credi Defaul Index Swapions 1 Credi Defaul Swaps and Swapions We provide simple and rigorous, albei fairly general, derivaions of valuaion formulae for credi defaul swapions Secion 1 and credi defaul index swapions Secion 2. Resuls of his work cover as special cases he pricing formulae derived previously by Jamshidian [15], Pedersen [19], Brigo and Morini [11], and, more recenly, by Morini and Brigo [18] see also Docor and Goulden [12], Hull and Whie [13], Jackson [14] and Liu and Jaeckel [17]. For a more deailed discussion of exising pricing mehods for credi defaul index swapions, we refer o Armsrong [1] and Armsrong and Rukowski [2]. Mos resuls presened in his work are independen of a paricular convenion regarding he specificaion of he fee and proecion legs and hus hey can be applied in valuaion of oher credi derivaives ha exhibi similar feaures e.g., opions on CDO ranches. In addiion o general represenaions for swapions prices, we derive explici valuaion formulae based on a specificaion of dynamics of a suiably defined spread processes his can be seen as an example of he op-down approach. As an alernaive, in a single name case, one can also produce pricing formulae based on he dynamics of defaul inensiy see, e.g., Brigo and Alfonsi [9], Brigo and Couso [10] or Brigo and El-Bachir [8]. For a credi defaul index swapion, his alernaive boom-up approach would be less pracically appealing, since i would require a specificaion of he join dynamics of a family of individual loss processes for a large porfolio of reference credi names, and he resuling dynamics of he spread process would become raher complicaed. Le us menion ha he issue of hedging is deal wih only marginally in his work. For more deails on hedging of single- and muli-name credi derivaives wih raded credi defaul swaps, he ineresed reader is referred o Bielecki e al. [4] [6]. A credi defaul swap CDS is an over-he-couner conrac beween wo counerparies he proecion buyer and he proecion seller in which proecion agains he risk of defaul by he underlying reference eniy is provided o he buyer. The proecion buyer pays a premium a regular inervals o he proecion seller in order o obain he righ o receive a coningen paymen from he seller following a credi even by he reference eniy. Examples of a credi even include defaul, resrucuring, a raings downgrade, or a failure o pay; for simpliciy, we will refer o a credi even as being a defaul. If no defaul even akes place hen he only cash flows are hose from he proecion buyer, who pays a periodic inervals usually quarerly a predeermined premium κ ermed CDS spread unil he conrac expires. If, however, here is a defaul even, he proecion buyer will cease hose premium paymens a he ime of he even including one final accrual paymen a he defaul ime. If a physical selemen has been agreed upon, he buyer will deliver any deb insrumen known as he reference obligaion of he reference eniy o he seller, in exchange for a cash paymen covering he reference obligaions par value. If, on he oher hand, a cash selemen has been agreed upon, hen he seller will provide a cash paymen equal o he marke value of he reference obligaion o he buyer. Once he loss o he buyer has been covered, he conrac is erminaed. A plain-vanilla credi defaul swapion is a European opion on he value of he underlying forward credi defaul swap. An essenial feaure of his conrac is ha i is cancelled if defaul occurs prior o he swapion s mauriy. I hus can formally be seen as an example of a survival claim cf. Jamshidian [15]. Le us menion ha, due o he presence of he fron-end proecion, his feaure is no longer valid for a credi defaul index swapion examined in Secion 2 of his work and hus his case requires a differen approach. 1.1 Defaul Time and Reference Filraion A sricly posiive random variable τ defined on a probabiliy space Ω, G, Q is called a random ime; i will also be laer referred o as he defaul ime. We inroduce he jump process H = 1 {τ } associaed wih τ and we denoe by H he filraion generaed by his process. We assume ha we are given, in addiion, some reference filraion F and we wrie G = H F, meaning ha G = σh, F for every R +. The filraion G is referred o as o he full filraion, since i is mean o convey he full informaion available o invesors.

3 M. Rukowski and A. Armsrong 3 Remarks 1.1 I is worh sressing ha an addiional filraion is required o address he issue of spread risk. Indeed, credi risk models based he filraion H or on is muli-name exension H = H 1... H n, as defined in Secion 2.1 are only able o handle he jump risk associaed wih defaul evens cf. [4], as opposed o he credi spread risk, which manifess iself by flucuaions of credi spreads for raded credi defaul swaps prior o and beween defaul imes of underlying credi names cf. [5]. Noe ha τ is an H-sopping ime, as well as a G-sopping ime for any choice of a filraion F, bu i is no an F-sopping ime, in general. Le hus he process F be defined by he equaliy, for every R +, F = Qτ F. 1 Le G = 1 F = Qτ > F be he survival process wih respec o he filraion F and le us emporarily assume ha and G > 0 for every R + so ha, in paricular, τ is no an F-sopping ime. The assumpion ha he process G is sricly posiive will be slighly weakened laer on, since i is enough o posulae posiiviy of G for any dae ha precedes he mauriy of a conrac a hand. For any Q-inegrable and F T -measurable random variable Y we have he following well-known resuls see, for insance, Chaper 5 in Bielecki and Rukowski [3] or Jeanblanc and Rukowski [16] The following lemma is also sandard. E Q 1 {T <τ} Y G = 1 {<τ} G 1 E Q G T Y F. 2 Lemma 1.1 Assume ha Y is some G-adaped sochasic process. Then here exiss a unique F- adaped process Ỹ such ha, for every R +, The process Ỹ is ermed he pre-defaul value of he process Y. Y = 1 {<τ} Ỹ. 3 Le us recall ha we may aach o he sricly posiive survival process G he hazard process Γ, defined as Γ = ln G. If he hazard process Γ is absoluely coninuous, so ha Γ = 0 γ u du for some F-predicable process γ, hen we say ha τ admis he F-inensiy process γ or simply, ha γ is he inensiy of defaul. In he special case when he defaul inensiy γ is well defined, some formulae presened below can be represened in erms of his process. Neiher he exisence of he defaul inensiy nor even he coninuiy of he survival process G are required for our furher developmens, however. 1.2 Forward-Sar Credi Defaul Swap We wrie T 0 = T < T 1 < < T J o denoe he enor srucure of a forward-sar credi defaul swap, where: T 0 = T is he CDS incepion dae; T J is he mauriy dae of he CDS; is he jh fee paymen dae for i = 1, 2,..., J. Le 1 A be he indicaor funcion of an even A and le τ be a sricly posiive random variable represening he momen of defaul of he reference credi name. We se βτ = 1 on he even { 1 τ < } and we wrie α j = 1 for every j = 1, 2,..., J. Le B be an F-adaped and sricly posiive process modelling he savings accoun or any oher sricly posiive numeraire. From now on, he underlying probabiliy measure Q is inerpreed as a maringale measure associaed wih he choice of B as he numeraire asse. Le Z be a bounded and F-adaped sochasic process. In pracical implemenaions of Definiion 1.1, i is common o posulae ha Z = 1 δ, where δ is he consan recovery rae.

4 4 Valuaion of Credi Defaul Index Swapions Definiion 1.1 The forward credi defaul swap issued a ime s [0, T ], wih uni noional, proecion paymen Z and F s -measurable spread κ is deermined by is discouned payoff, which equals D = P κa for every [s, T ], where he discouned payoffs of he proecion leg and he fee leg are given by P = B Z τ Bτ 1 1 {T τ TJ } 4 and A = B J j=1 α j B 1 1 {Tj<τ} + B B 1 τ τ T βτ 1 {T <τ TJ } 5 respecively. The fair price a ime [s, T ] of a forward credi defaul swap for he proecion buyer equals S κ = E Q D G = E Q P G κe Q A G. Brigo and Morini [18] examine also a simplified version D of he acual discouned payoff D, which is given by he following expression D = 1 δb J j=1 α j B 1 1 {Tj 1 <τ } κb J j=1 α j B 1 1 {Tj <τ} = P κā. 6 We do no require any specific convenion of his kind for our furher purposes, however. In fac, all resuls of his work can be applied o any paricular convenion regarding he proecion and fee legs. I is only required ha all cash flows occur afer he incepion dae T and he knock ou feaure, ha is, he propery ha he conrac becomes void if defaul occurs prior o or a T. Formally, i is sufficien o assume ha he discouned payoff a ime [0, T ] of he examined conrac has he form D = P κa, 7 where P and A are discouned payoffs of cerain T -survival claims, meaning ha, for every [0, T ], P = 1 {T <τ} P, A = 1 {T <τ} A. 8 To conclude, he specific formulae for processes P and A are no relevan and hus he resuls presened in he sequel can be applied o alernaive varians of credi defaul swaps or oher conracs wih similar feaures. In wha follows, we only require ha he inequaliy G > 0 holds for every [0, T 1 ], so ha, in paricular, we have ha G T1 = Qτ > T 1 F T1 > 0. Lemma 1.2 The price a ime [s, T ] of he forward CDS issued a ime s [0, T ] saisfies S κ = 1 {<τ} G 1 E Q D F = 1 {<τ} S κ, 9 where he pre-defaul price saisfies S κ = P κã, where in urn P = G 1 E Q P F, à = G 1 E Q A F. 10 If he discouned payoffs P and A are given by 4 and 5, respecively, hen P = G 1 J B E Q Z τ B 1 j=1 τ 1 {Tj 1<τ } F 11 and à = G 1 J B E Q α j B 1 1 {Tj<τ} + Bτ 1 τ T βτ 1 1 {T0<τ T J } F. 12 j=1

5 M. Rukowski and A. Armsrong 5 Proof. Recall ha by definiion S κ = E Q D G = E Q P G κe Q A G. By combining 2 wih 7 and 8, we hus obain and E Q P G = 1 {<τ} G 1 E Q P F = 1 {<τ} P E Q A G = 1 {<τ} G 1 E Q A F = 1 {<τ} Ã, so ha he proof of 10 is compleed. To esablish equaliies 11 and 12, i suffices o make use of 11 and explici represenaions 4 and 5 for P and A. The quaniy P is he pre-defaul value a ime [s, T ] of he proecion leg per uni of he nominal, whereas à represens he pre-defaul value a ime [0, T ] of he fee leg per one basis poin of he spread. The laer is frequenly referred o as he pre-defaul presen value of a basis poin of he CDS, bu i is also known as he risky PVBP or he CDS annuiy. I is worh noing ha neiher P nor à depend on he iniiaion dae s of a forward CDS. 1.3 Pre-defaul Fair Forward CDS Spread Since he forward CDS conrac is erminaed a defaul wih no paymens, he fair or par forward CDS spread is only defined prior o defaul. I is hus naural o inroduce he concep of he pre-defaul fair forward CDS spread, raher han he fair forward CDS spread. Definiion 1.2 The pre-defaul fair forward CDS spread a ime [0, T ] is he F -measurable random variable κ such ha S κ = 0. The following resul is a simple consequence of Lemma 1.2. Lemma 1.3 The pre-defaul fair forward CDS spread saisfies, for any [0, T ], J κ = P E Q j=1 Z τ B 1 τ 1 {Tj 1 <τ } F = J à E Q j=1 α jb 1 1 {Tj<τ} + Bτ 1, 13 τ T βτ 1 1 {T0<τ T J } F where he second equaliy holds if he discouned payoffs P and A are given by 4 and 5, respecively. The price of he forward CDS issued a s [0, T ] wih an F s -measurable spread κ admis he following represenaion, for every [s, T ], S κ = 1 {<τ} à κ κ. 14 Proof. Since κ is F -measurable, from Lemma 1.2, we obain ha S κ = P κ Ã. I is hus clear ha he firs equaliy in 13 holds, provided ha à > 0. nder he sanding assumpion ha G T1 > 0, i can be deduced easily from 12 ha à > 0 for every [0, T ] recall ha à does no depend on s. Indeed, using 12, we obain à G 1 B E Q α1 B 1 T 1 1 {T1 <τ} F = G 1 B E Q α1 B 1 T 1 Qτ > T 1 F T1 F = G 1 B E Q α1 B 1 T 1 G T1 F > 0. For he second equaliy in 13, we make use of 11 and 12. To derive 14, i is enough o observe ha S κ = S κ S κ = P κã P κ à = Ãκ κ, where we have used he equaliy S κ = 0 cf. Definiion 1.2.

6 6 Valuaion of Credi Defaul Index Swapions I is worh noing ha he pre-defaul fair forward CDS spread depends on he curren dae, he enor srucure of he forward CDS, he erm srucure of ineres raes, he survival process G and he recovery rae δ. I is well defined for any marke convenions regarding he paymens srucures for he proecion and fee legs of a forward CDS. I is only required ha he quaniy à is non-zero; in fac, i is ypically a sricly posiive process. In paricular, he firs equaliy in 13 is universal and hus all foregoing general resuls are valid for any convenion regarding he iming and amouns of cash flows of a forward CDS. In fac, afer a minor, bu essenial, modificaion i will be also valid for he forward credi defaul index swap. We will show ha i suffices o replace he defaul ime τ wih he momen of he las defaul in a reference credi porfolio for deails, see Secion 2.3. In pracical implemenaions of he pricing formula 14, one needs o compue he quaniy Ã, since he marke quoe for his erm is no readily available. The compuaion of à hinges on he concep of he implied risk-neural defaul probabiliies, which for a single-name case are obained from marke quoes for CDS spreads for differen mauriies, i.e., from he curren CDS spread curve. 1.4 Credi Defaul Swapions Le us now focus on plain-vanilla opions relaed o a forward credi defaul swap. A credi defaul swapion gives is holder, who pays an upfron fee, he righ bu no he obligaion o buy or sell he proecion on a prearranged single-name CDS. I hus can be seen as a call or pu opion wih srike zero wrien on he marke value of he underlying CDS a he opion s expiry dae. An imporan feaure of a credi defaul swapion is he knock ou feaure. For a single-name swapion, if he sole reference eniy of he underlying CDS defauls a ime τ before he opion is exercised hen he opion is knocked ou. This means ha he credi defaul swapion is nullified and hus erminaes wih zero value. We will laer see ha his is differen o he case of a credi defaul index swapion where, if several bu no all of he many reference eniies defaul, he swapion will coninue unil is mauriy. I is posulaed hroughou ha he underlying conrac is he forward CDS issued a ime s [0, T ] wih an F s -measurable spread κ, as specified by Definiion 1.1. We assume ha he exercise dae of he swapion is T, ha is, he swapion expires eiher before or a he sar dae T 0 = T of he underlying forward CDS. Of course, we make he obvious assumpion ha s [0, ]. Definiion 1.3 The credi defaul swapion o ener a forward CDS wih an F s -measurable spread κ a a fuure dae T has he payoff a mauriy equal o C = S κ +. I is worh noing ha he credi defaul swapion is knocked ou if defaul occurs prior o or a mauriy. This feaure is already encoded in he payoff C, since we have ha C = S κ + = 1 {<τ} S κ + = 1{<τ} S κ The price a ime [s, ] of he European claim C is given by he risk-neural valuaion formula C = B E Q B 1 S κ + G = B E Q 1 {<τ} B 1 à κ κ + G, 16 where we used 14 in he second equaliy. The nex lemma furnishes a represenaion for he price of a credi defaul swapion in erms of he reference filraion F. Lemma 1.4 The price a ime [s, ] of he credi defaul swapion equals C = 1 {<τ} B G 1 E Q G B 1 à κ κ + F. 17

7 M. Rukowski and A. Armsrong 7 Proof. The random variable Y = B 1 Ã κ κ + is manifesly F -measurable. Hence he equaliy is an immediae consequence of formulae 2 and 16. The pricing formula 17 can be furher simplified by a suiable change of a probabiliy measure. Generally speaking, we follow here he ideas of Jamshidian [15], Brigo [7] and Brigo and Morini [18]. I is worh sressing, however, ha neiher he so-called hypohesis H nor any addiional assumpions on he hazard process G of τ are required in he derivaion of pricing formulae for he credi defaul swaps and swapions. The only assumpion we make is ha he survival process G is sricly posiive, bu possibly disconinuous. Le us define an equivalen probabiliy measure Q on Ω, F by posulaing ha he Radon- Nikodým densiy of Q wih respec o Q equals d Q dq = cg B 1 Ã =: η, Q-a.s. 18 Observe ha he process η = cg B 1 Ã, [s, ], is a sricly posiive F-maringale under Q, since η = cg B 1 Ã = c E Q B 1 A F = c E Q X F, where he random variable X = B 1 A is independen of [s, ] cf. 5. Therefore, for every [s, ], d Q F = E Q η F = η, Q-a.s. dq The quaniy c = E Q G B 1 Ã 1 is simply he normalizing consan, which ensures ha E Q η = 1, so ha Q given by 18 is indeed a probabiliy measure on Ω, F. Lemma 1.5 The price a ime [s, ] of a credi defaul swapion saisfies C = 1 {<τ} C, where he pre-defaul price of he swapion equals C = Ã E eq κ κ + F. 19 Proof. sing 17, we obain C = 1 {<τ} B G 1 E Q G B 1 Ã κ κ + F = 1 {<τ} Ã η 1 E Q η κ κ + F = 1 {<τ} Ã E eq κ κ + F, where he las equaliy is an immediae consequence of he absrac Bayes formula. The nex lemma shows he change of he probabiliy measure from Q o Q is crucial. I shows ha he drif erm in he dynamics of he fair forward CDS spread κ under Q will no appear in he pricing formula for he credi defaul swapion. Lemma 1.6 The pre-defaul fair forward CDS spread κ, [0, ] is a sricly posiive F-maringale under Q. Proof. The produc κη is manifesly an F-maringale under Q, since i saisfies, for every [0, ], κ η = cκ G B 1 J Ã = cg B 1 P = c E Q Z τ B 1 j=1 τ 1 {Tj 1 <τ } F. By he well-known resul, his implies ha κ is an F-maringale under Q.

8 8 Valuaion of Credi Defaul Index Swapions 1.5 Black Formula for Credi Defaul Swapions Le us assume ha F is he Brownian filraion, specifically, ha i is generaed by some Brownian moion W defined on he underlying probabiliy space Ω, G, Q. I is no essenial for our purposes o assume ha W is a Brownian moion wih respec o he filraion G. In oher words, we need no posulae ha he hypohesis H is saisfied by filraions F and G. Recall ha he process κ, [0, ] is a sricly posiive, F-maringale under Q. Since W is a Brownian moion under Q and Q is equivalen o Q on Ω, F, he sandard argumens can be used o show ha κ admis he following inegral represenaion κ = κ σ u κ u d W u, [0, ], 20 where W is a Brownian moion under Q and σ is some F-predicable process. Proposiion 1.1 Assume ha he volailiy σ of he pre-defaul fair forward CDS spread is a posiive funcion. Then he pre-defaul price of he credi defaul swapion wih an F s -measurable srike κ equals, for every [s, ], C = à κ N d + κ,, κn d κ,, 21 or, equivalenly, where We also have ha C = P N d + κ,, κã N d κ,,, 22 d ± κ,, = lnκ /κ ± 1 2 σ 2 u du 1/2 σ 2 u du. d C /à = N d + κ,, dκ. 23 Proof. In view of 19, we obain C = à E eq κ κ + F = à E eq κ κ + κ = à κ N d + κ,, κn d κ,, where he las equaliy follows by sandard compuaions. Equaliy 23 follows by an applicaion of Iô s formula o Hedging of Credi Defaul Swapions To ge he simples hedging sraegy for a credi defaul swapion, we assume ha a forward CDS issued a ime v [0, s] wih an F v -measurable rae κ is raded. In oher words, we assume ha he raded forward CDS and he forward CDS underlying he swapion have he same covenans, bu hey are issued a possibly differen daes and hus hey have differen spreads, in general. Of course, he mos naural candidae for he forward CDS used for hedging he swapion is he underlying forward CDS, bu his paricular choice is no necessary. I is apparen from 10 ha à can be seen as he pre-defaul value a ime [0, T ] of a paricular porfolio of defaulable bonds wih zero recovery, referred o hereafer as he swap porfolio. We assume ha he swap porfolio is raded. Noe ha if defaul occurs a some dae [0, T ], he wealh of his porfolio falls o zero. Of course, he same propery holds also for he CDS and he credi defaul swapion. Hence in wha follows i suffices o focus on he dynamics of he pre-defaul value of he swapion and he pre-defaul wealh of a hedging porfolio. Le A be he price process of he swap porfolio a ime [0, T ]. Formally, we se A = 1 {<τ} Ã. Recall also ha S κ = 1 {<τ} S κ.

9 M. Rukowski and A. Armsrong 9 Le φ = φ 1, φ 2 be rading sraegy on [s, ], where φ 1 and φ 2 are G-predicable processes. The wealh of φ equals, for any [s, ], V φ = φ 1 S κ + φ 2 A and hus he pre-defaul wealh saisfies, for any [s, ], Ṽ φ = φ 1 S κ + φ 2 Ã. Of course, he equaliy V φ = 1 {<τ} Ṽ φ holds for any [s, ]. Therefore, i suffices o examine a hedging sraegy on he inerval [s, s τ ]. Pu anoher way, we posulae ha rading is sopped eiher a defaul or mauriy, whichever comes firs. In view of Lemma 1.1, i hus suffices o consider F-predicable processes φ 1 and φ 2 represening he pre-defaul values of he corresponding G-predicable processes, in he sense of Lemma 1.1. A sraegy φ is required o be self-financing, in he sense ha dṽφ = φ 1 d S κ + φ 2 dã. I is easy o show by Iô s formula ha he relaive pre-defaul wealh saisfies dṽφ/ã = φ 1 d S κ/ã. 24 Proposiion 1.2 The replicaing sraegy φ for he credi defaul swapion wih he erminal payoff C = S κ + is given by, for any [s, s τ ], Proof. On he one hand, we have cf. 23 φ 1 = N d + κ,,, C = φ 1 S κ + φ 2 Ã. 25 d C /Ã = N d + κ,, dκ. On he oher hand, since S κ = P κã and κ is a fixed random variable, we obain and hus, in view of 24, d S κ/ã = dκ κ = dκ dṽφ/ã = φ 1 d S κ/ã = φ 1 dκ. I hus is apparen ha he sraegy φ given by 25 is self-financing and is pre-defaul wealh saisfies Ṽφ = C for any [s, ]. As already menioned above, i defaul occurs prior o or a mauriy hen he wealh of he hedging porfolio falls o zero and he same feaure is enjoyed by he value of he credi defaul swapion. In view of 24, we also have ha, for any [s, ], C /Ã = C s /Ãs + s φ 1 u d S u κ/ãu = C s /Ãs + Hence hedging can also be inerpreed in erms of he forward CDS. s φ 1 u dκ u. 26 In pracice, hedging can also be achieved by aking posiions a any dae in he marke CDS, ha is, he jus-issued CDS wih he spread κ. However, an explici represenaion for his hedging sraegy is raher cumbersome in he coninuous-ime se-up, since one needs o deal wih a coninuum of raded asses.

10 10 Valuaion of Credi Defaul Index Swapions 2 Credi Defaul Index Swaps and Swapions A credi defaul index swap CDIS is a sandardized conrac ha is based upon a fixed porfolio of reference eniies. The ever increasing rade of index credi derivaives has been esimaed a being upwards of S$90 billion annually. The wo main indices o which CDSs are referenced are CDX, referring o companies wihin Norh America and itraxx, which refers o companies wihin Europe and Asia. We look a he case of he CDX, alhough itraxx and oher indices have very similar characerisics. A is concepion, he CDX is referenced o n = 125 fixed companies ha are chosen by marke makers. These 125 reference eniies are specified o have equal weighs wihin he CDX. If we assume each has a nominal value of one hen, because of he equal weighing, he oal noional would be 125. In effec, one CDX provides on average he same proecion o ha of 125 single-name CDSs upon he same reference eniies. By conras o a sandard single-name CDS, he buyer of he CDX provides proecion o he marke makers. In oher words, by purchasing a CDX from marke makers he invesor is no receiving proecion, raher hey are providing i o he marke makers. In exchange for he proecion he invesor is providing, he marke makers pay he invesor a periodic fixed premium, oherwise known as he credi defaul index spread. Such sandardized conracs promoe liquidiy wihin he derivaives marke. Typically, he recovery rae δ [0, 1] is predeermined and consan for all reference eniies in he index. By purchasing he index he invesor is agreeing o pay he marke makers 1 δ for any defaul ha occurs before mauriy. Tha is, following a defaul he invesor has o cover he loss incurred, which is achieved by paying o he marke maker he amoun of 1 δ. Following his, he nominal value of he CDX is reduced by one. Once a removal has aken place here is no replacemen of he defauled firm. This process repeas afer every defaul and he CDX coninues on unil mauriy. The sandard mauriies of a CDX are five and en years wih paymens occurring quarerly. However, in more recen imes, hree and seven-year producs have been inroduced. New CDXs are defined semi-annually and he fixed rae, reference eniies and mauriies are reconfigured by he marke makers according o curren marke condiions. Such changes do no aler pre-exising conracs. itraxx and oher credi defaul index swaps operae analogously wih he CDX, wih he only disincions being in he conrac deails he premium, he number and choice of reference eniies and reconfiguraion procedures. 2.1 Defaul Times and Reference Filraion As we are now working wihin he muli-name case, he appropriae noaion needs o be inroduced. Le G be he filraion generaed by he reference filraion F and filraions H 1,..., H n, where H i is he filraion generaed by he defaul indicaor H i = 1 {τi } of he ih credi name. We may now define H = H 1... H n and we may represen G as follows G = H F = H 1... H n F. This decomposiion of he filraion G is no suiable for efficien compuaions based on a sandard reduced-form approach summarized in Secion 1.1, however. As a more viable alernaive, we firs inroduce he sequence τ 1... τ n of ordered defaul imes associaed wih he original sequence of defaul imes τ 1,..., τ n. In fac, since we will only deal wih underlying models in which simulaneous defauls are excluded, we may assume, wihou loss of generaliy, ha he ordering above is sric. We hus have G = H n F, where H n is he filraion generaed by he indicaor process H n = 1 {τn } of he las defaul and F = F H 1... H n 1. For breviy, in wha follows we will wrie τ = τ n o denoe he random ime when all firms in a given porfolio are in defaul. The usefulness of his paricular decomposiion of he full filraion in he conex of valuaion of credi defaul index swapions was noed independenly by Armsrong and Rukowski [2] and Morini and Brigo [18].

11 M. Rukowski and A. Armsrong 11 We will be ineresed in evens of he form { τ } and { τ > } for a fixed. Morini and Brigo [18] refer o hese evens as he armageddon and he no-armageddon evens, respecively. We decided o use insead he erms collapse even and he pre-collapse even, respecively. The even { τ } corresponds o he oal collapse of he reference porfolio, in he sense ha all underlying credi names defaul eiher prior o or a ime. Similarly as in Secion 1.1 cf. formula 1, we sar by defining he auxiliary process F, which is now given by he following expression, for every R +, F = Q τ F. Le us denoe by Ĝ = 1 F = Q τ > F he corresponding survival process wih respec o he filraion F and le us emporarily assume ha he inequaliy Ĝ > 0 holds for every R +. Then for any Q-inegrable and F T -measurable random variable Y we have ha cf. 2 E Q 1 {T <bτ} Y G = 1 {<bτ} Ĝ 1 E Q ĜT Y F. 27 For he reader s convenience, le us sae he following immediae consequence of Lemma 1.1. Lemma 2.1 Assume ha Y is some G-adaped sochasic process. Then here exiss a unique Fadaped process Ŷ such ha, for every [0, T ], The process Ŷ is ermed he pre-collapse value of he process Y. Y = 1 {<bτ} Ŷ Forward-Sar Credi Defaul Index Swap We wrie T 0 = T < T 1 < < T J o denoe he enor srucure of he forward-sar CDIS, where: T 0 = T is he incepion dae; T J is he mauriy dae; is he jh fee paymen dae for j = 1, 2,..., J. Le α j = 1 for every j = 1, 2,..., J. As before, B is an F-adaped or, a leas, Fadaped and sricly posiive process represening he price of he savings accoun or any oher sricly posiive numeraire and he underlying probabiliy measure Q is inerpreed as a maringale measure associaed wih he choice of B as he numeraire asse. Definiion 2.1 The discouned cash flows for he seller ha is, for he proecion buyer of he forward CDIS issued a ime s [0, T ] wih an F s -measurable spread κ are, for every [s, T ], where and D n = P n κa n, 29 P n = 1 δb n A n = B J j=1 i=1 α j B 1 B 1 τ i 1 {T <τi T J } 30 n 1 1{Tj τ i } are discouned payoffs of he proecion leg and he fee leg per one basis poin, respecively. The fair price a ime [s, T ] of a forward credi defaul index swap for he proecion buyer equals i=1 S n κ = E Q D n G = E Q P n G κe Q A n G. 31

12 12 Valuaion of Credi Defaul Index Swapions Of course, for s = T, i.e. when he issuance dae coincides wih he incepion dae, he forward CDIS becomes he spo CDIS. From 30 and 31, we obain he following simple bu crucial properies of P n and A n : P n = 1 {T <bτ} P n, A n = 1 {T <bτ} A n. 32 Le us noe ha he quaniies P n and A n are well defined for any [0, T ] and hey do no depend on he issuance dae s of he forward CDIS under consideraion. Remarks 2.1 Le us make few commens on he scope of he mehod presened in wha follows. As in he single name case, only he equaliies 32 are essenial for our mehod o work, as opposed o he exac specificaion of he payoffs P n and A n, which is of a minor imporance. Therefore, he same approach can be applied o oher convenions regarding muli-name credi defaul swaps and no only o he sandard forward CDIS, as specified in Definiion 2.1. The cash flows of he proecion leg of he forward credi defaul index swap are somewha similar o summing he cash flows of n individual single-name forward CDS cash flows. Indeed, he righ-hand side in 30 is simply he discouned sum of consan proecion payous 1 δ for all he reference eniies which have defauled during he lifeime of he forward CDIS, ha is, beween he incepion dae T and he mauriy dae T J. The fee leg is somewha differen, however. The premium paymen of he forward CDIS decreases following every defaul. This is because he consan premium is only paid on he nominal value of he remaining eniies for which he invesor is being provided proecion. As he nominal value reduces afer every defaul, so does he invesor s premium paymen which decreases in proporion o he change in nominal value. In he case of a porfolio of single-name CDSs, he oal nominal also decreases afer each defaul, bu he oal premium paid afer defauls depends also on he ideniies of defauled names, since spreads of individual CDSs are ypically differen. For breviy, we will wrie J o denoe he reduced nominal a ime [s, T ], as given by he formula n J = 1 1{ τi }. 33 i=1 In wha follows, we only require ha he inequaliy Ĝ > 0 holds for every [s, T 1 ], so ha, in paricular, ĜT 1 = Q τ > T 1 F T1 > 0. The proof of he following pricing resul is exacly he same as he proof of Lemma 1.2. I is based on formulae 27 and 32. Lemma 2.2 The price a ime [s, T ] of he forward CDIS saisfies S n κ = 1 {<bτ} Ĝ 1 E Q D n F = 1 {<bτ} Ŝ n κ, 34 where he pre-collapse price of he forward CDIS saisfies Ŝn κ = P n κân, where in urn or, more explicily, and P n P n = Ĝ 1 E Q P n F, Â n = Ĝ 1 E Q A n F 35 = 1 δĝ 1 Â n = Ĝ 1 B E Q n B E Q J Bτ 1 i 1 {T <τi T J } i=1 j=1 F 36 α j B 1 J Tj F. 37 The process Ân may be hough of as he pre-collapse presen value of receiving risky one basis poin on he forward CDIS paymen daes on he residual nominal value J Tj. Similarly, he process P n represens he pre-collapse presen value of he proecion leg of he conrac.

13 M. Rukowski and A. Armsrong Pre-Collapse Fair CDIS Spread Since he forward CDIS is erminaed a he momen of he nh defaul wih no furher paymens, i makes sense o define he forward CDS spread only prior o τ. I is hus naural o inroduce he concep of he pre-collapse fair forward CDIS spread, raher han he fair forward CDS spread. Definiion 2.2 The pre-collapse fair forward CDIS spread a ime [0, T ] is he F -measurable random variable κ n such ha Ŝn κ n = 0. The following resul, which is a counerpar of Lemma 1.3, is a sraighforward consequence of Lemma 2.2. I is worh noing ha he quaniy κ n is well defined for every [0, T ] and, manifesly, i does no depend on he issuance dae s. Lemma 2.3 Assume ha ĜT 1 = Q τ > T 1 F T1 > 0. Then he pre-collapse fair forward CDIS spread saisfies, for every [0, T ], n κ n = P n 1 δ E Q i=1 B 1 τ i 1 {T <τi T J } F =  n J E Q j=1 α jb J Tj F The price of he forward CDIS admis he following represenaion, for every [0, T ], S n κ = 1 {<bτ}  n κ n κ. 39 Proof. The proof is essenially he same as he proof of Lemma 1.3. We firs noe ha, since κ n is F -measurable, we have ha Ŝn κ = P n κ n Ân. Moreover, under he sanding assumpion ha Ĝ T1 > 0, i can be deduced easily from 37 ha  > 0 for every [0, T ]. Indeed, 37 yields, for every [0, T ],  n Ĝ 1 B E Q α1 B 1 T 1 J T1 F Ĝ 1 B E Q α1 B 1 = Ĝ 1 B E Q α1 B 1 T 1 Q τ > T 1 F T1 F = Ĝ 1 T 1 1 {T1 <bτ} F B E Q α1 B 1 T 1 Ĝ T1 F > 0. I is hus clear ha he firs equaliy in 38 is valid. For he second equaliy, we make use of 36 and 37. To derive 39, i is enough o observe ha Ŝ n κ = Ŝn κ Ŝn κ = P n κân P n κ n Ân = Ân κ n κ, where we have used he equaliy Ŝn κ n = 0 cf. Definiion Marke Convenion for Valuing a CDIS nforunaely, a marke quoe for he quaniy Ân, which is essenial in marking-o-marke of a CDIS, is no direcly available. The marke convenion for approximaion of he value of A n hinges on he following bold posulaes: all firms are idenical from ime onwards homogeneous porfolio; herefore, we jus deal wih a single-name case, so ha eiher all firms defaul or none; he implied risk-neural defaul probabiliies are compued using a fla single-name CDS curve wih a consan spread equal o κ n. nder his se of convenional posulaes, he righ-hand side in 37 is approximaed using he marke convenion ha  n J P V κ n,

14 14 Valuaion of Credi Defaul Index Swapions where P V κ is he risky presen value of receiving one basis poin a all CDIS paymen daes calibraed o a fla CDS curve wih spread equal o κ n, where κ n is he quoed CDIS spread a ime. Consequenly, he convenional marke formula for he value of he CDIS wih fixed spread κ reads, on he pre-collapse even { < τ}, Ŝ κ = J P V κ n κ n κ. 40 In paricular, if he credi defaul index swap was issued a ime 0 wih he spread κ n 0 marked-o-marke value a ime equals hen is Ŝ κ n 0 = 1 {<bτ} P V κ n J κ n κ n Le us sress ha he quaniy P V κ n is compued as if i was a single-name case, no a muliname. For his very reason, we underline he imporance of his sep in he marke convenion for he CDIS value as well as for he credi defaul index swapion examined in he foregoing secion. As we shall see in ha follows, from he heoreical viewpoin, i is much easier o work wih formula 39, raher han wih he convenional expression Marke Payoff of a Credi Defaul Index Swapion In Secion 1.4, we have examined he valuaion of opions on a single-name credi defaul swap. This is now exended o opions on a credi defaul index swap, referred o as credi defaul index swapions. Credi defaul index swapions are European opions and hus can only be exercised a expiry a he prese exercise spread κ. Sandard conracs have mauriies of eiher hree or sixmonhs. For example, in a sandard CDX swapion conrac he specifics would be: he underlying CDX, he expiry dae, he srike level κ and he ype payer or receiver. Le us firs describe he marke convenion regarding he payoff of he payer credi defaul index swapion. We refer o Pedersen [19] for more deails and commens. I is assumed here ha he credi defaul index swap was issued a ime 0, wih he consan spread κ n 0 and κ n represens he corresponding marke quoe a ime. Definiion 2.3 The convenional marke formula for he payoff a mauriy T of he payer credi defaul index swapion wih srike level κ reads C = 1 {<bτ} P V κ n J κ n κ n 0 1 {<bτ} P V κnκ κ n 0 + L +, 42 where he reduced nominal J is given by 33 and L sands for he loss process for our porfolio so ha, for every R +, n L = 1 δ 1 {τi }. Le us make some commens regarding he swapion s payoff. Noe firs ha in Definiion 2.3 we se our underlying forward CDIS o have incepion dae T 0 = T and mauriy T J. However, he losses from he porfolio are compued from ime 0 onwards. Hence he holder of he swapion has he righ o ener he underlying forward CDIS a ime and, if his opion is exercised, he also gains proecion agains losses from he porfolio beween ime 0 and ime. The key difference beween his cash flow and he credi defaul swapion cash flow is ha here we no longer deal wih he knock ou feaure, even afer he nh defaul. This lack of knock ou proves o be difficul in he valuaion and hedging of index swapions. No longer may he sandard Black formula be used o price he credi defaul index swapions as his formula only works for opions ha do knock ou a defaul such as single-name credi defaul swapions. The marke convenion 42 is due o he fac ha he swapion has physical selemen and he CDIS wih spread κ is no raded. If he swapion is exercised, is holder akes a long posiion in he i=1

15 M. Rukowski and A. Armsrong 15 on-he-run index and is compensaed for he difference beween he value of he on-he-run index and he value of he non-raded index wih spread κ, as well as for defauls ha occurred in he inerval [0, ]. Recall ha P V κ is he risky presen value a ime of receiving one basis poin a all CDIS paymen daes calibraed o a fla single-name CDS curve wih spread equal o κ. I is worh observing ha P V κ is random only in he ineres raes ypically, forward LIBORs, whereas P V κ n is random in boh ineres raes and he index spread κn. In order o make P V κ compleely deerminisic, one may use a common assumpion ha he ineres rae for some fuure dae lies on he curren forward curve for ha same dae. nder his addiional assumpion, he quaniy P V κ n will only be random via is dependence on he index spread. 2.6 Pu-Call Pariy for Credi Defaul Index Swapions For he sake of breviy, le us denoe, for any fixed κ > 0, fκ, L = L 1 {<bτ} P V κnκ κ n 0. Then he payoff of he payer credi defaul index swapion enered a ime 0 and mauring a equals C = 1 {<bτ} P V κ n +, J κ n κ n 0 + fκ, L whereas he payoff of he corresponding receiver credi defaul index swapion saisfies P = 1 {<bτ} P V κ n +. J κ n 0 κ n fκ, L This leads o he following equaliy, which holds a mauriy dae C P = 1 {<bτ} P V κ n J κ n κ n 0 + fκ, L. 2.7 Model Payoff of a Credi Defaul Index Swapion The acual payoff 42 of a credi defaul index swapion is raher difficul o handle analyically, in general. The advanage of his formula is ha i is based on marke daa and i is easy o implemen. The major drawback is ha i is inernally inconsisen since he quaniies P V κ n and P V κ are compued on he basis of a single-name case and hus are no consisen wih any model for defaul imes τ 1,..., τ n. For his reason, we will consider in wha follows he simplified version of he swapion s payoff. Definiion 2.4 The model payoff of he payer credi defaul index swapion enered a ime 0 wih mauriy dae and srike level κ equals or, more explicily cf. 39 C = To formally derive 44 from 42, i suffices o posulae ha C = S n κ + L {<bτ} Â n κ κ + L P V κn P V κ J Ân. We will firs use represenaion 43 o esablish, in Secion 2.10, he pricing formula for a credi defaul index swapion, which was proposed recenly by Morini and Brigo [18]. Subsequenly, in Secion 2.12, we will use 44 o jusify he marke pricing formula for a credi defaul index swapion. The crucial difference beween he wo approaches is ha he marke pricing formula 58 refers o he fair CDIS spread κ n, whereas he model formula 55 of Morini and Brigo [18], as well as he relaed Pedersen s [19] formula, are based on he loss-adjused fair CDIS spread κ a, which will be inroduced in Secion 2.9 below.

16 16 Valuaion of Credi Defaul Index Swapions 2.8 Loss-Adjused CDIS Since L 0 and, obviously, L = 1 {<bτ} L + 1 { bτ} L, he payoff 44 can also be represened as follows C = S n κ + 1 {<bτ} L { bτ} L = S a κ + + C L, 45 where we denoe S a κ = S n κ + 1 {<bτ} L, C L = 1 { bτ} L. The quaniy S a κ represens he payoff a ime of he loss-adjused forward CDIS, which is formally defined as follows. Definiion 2.5 The discouned cash flows for he seller of he loss-adjused forward CDIS ha is, for he buyer of he proecion are, for every [0, ], D a = P a κa n, where P a = P n + B B 1 1 {<bτ}l. I is essenial o observe ha he payoff D a is he -survival claim, in he sense ha D a = 1 {<bτ} D a. Le us noe ha any oher adjusmens o he payoff P n or A n of he CDIS is also admissible, provided ha he propery P a = 1 {<bτ}p a or Aa = 1 {<bτ}a a holds. Therefore, if we wish o define a paricular adjusmen of he fair CDIS spread for any dae [0, ], we only need o ensure ha he modified proecion and fee payoffs are -survival claims, Lemma 2.4 The price of he loss-adjused forward CDIS equals, for every [0, ], S a κ = 1 {<bτ} Ĝ 1 E Q D a F = 1 {<bτ} Ŝ a κ, where he pre-collapse price saisfies Ŝa κ = P a κân, where in urn P a = Ĝ 1 E Q P a F, Â n = Ĝ 1 E Q A n F or, more explicily, and P a = Ĝ 1 B E Q 1 δ Â n = Ĝ 1 n i=1 B 1 τ i B E Q J 1 {T <τi T J } + 1 {<bτ} B 1 j=1 α j B 1 J Tj F. L F 2.9 Pre-Collapse Loss-Adjused Fair CDIS Spread We are in a posiion o define he fair loss-adjused forward CDIS spread. Definiion 2.6 The pre-collapse loss-adjused fair forward CDIS spread a ime [0, ] is he F -measurable random variable κ a such ha Ŝa κ a = 0. Then we have he following resul, which corresponds o Lemma 2.3.

17 M. Rukowski and A. Armsrong 17 Lemma 2.5 Assume ha ĜT 1 = Q τ > T 1 F T1 > 0. Then he pre-collapse loss-adjused fair forward CDIS spread saisfies, for every [0, ], E Q 1 δ n F κ a = P a  n = i=1 B 1 τ i 1 {T <τi T J } + 1 {<bτ} B 1 L E Q J j=1 α jb 1 J Tj F. 46 The price of he forward CDIS admis he following represenaion, for every [0, T ], S a κ = 1 {<bτ}  n κ a κ. 47 Proof. The proof of his resul is essenially he same as he proof of Lemma 2.3 and hus i is omied Model Pricing Formula for Credi Defaul Index Swapions I is easy o check ha he model payoff 44 of he credi defaul index swapion can be represened as follows C = 1 {<bτ}  n κ a κ { bτ} L. 48 This equaliy should be seen as he loss-adjused simplificaion of formula 45. The price a ime [0, ] of he claim C is hus given by he risk-neural valuaion formula C = B E Q B 1 C G = B E Q 1{<bτ} B 1 Ân κ a κ + G + B E Q 1{ bτ} B 1 L G. sing he filraion F, we can obain a more explici represenaion for he firs erm in he formula above, as he following resul shows. Lemma 2.6 The price a ime [0, ] of he payer credi defaul index swapion equals C = 1 {<bτ} B Ĝ 1 E Q Ĝ B 1 Ân κ a κ + F + B E Q 1 { bτ} B 1 L G. 49 Proof. The random variable Y = B 1 Ân κa κ+ is manifesly F -measurable and Y = 1 {<bτ} Y. Hence he equaliy is an immediae consequence of formula 27. Le us firs noe ha, on he collapse even { τ} we have ha 1 { bτ} B 1 L = B 1 n1 δ and hus he pricing formula 49 reduces o C = B E Q 1{ bτ} B 1 L G = n1 δeq B 1 G = n1 δb, T, 50 where B, T is he price a of -mauriy risk-free zero-coupon bond. This case is hus easy o handle, so ha i will no be considered in wha follows. and Le us hus concenrae on he pre-collapse even { < τ}. We now have C = C a + C L, where C a = B Ĝ 1 E Q Ĝ B 1 Ân κ a κ + F 51 C L = B E Q 1{ bτ>} B 1 L F, where he las equaliy follows from he well known fac ha on { < τ} any G -measurable even can be represened by an F -measurable even, in he sense ha for any even A G here exiss an even  F such ha 1 {<bτ} A = 1 {<bτ}  cf. Lemma 2.1. The compuaion of C L relies only on he knowledge of he risk-neural condiional disribuion of τ given F and he erm srucure of ineres raes, since on he even { τ > } we have ha B 1 L = B 1 n1 δ.

18 18 Valuaion of Credi Defaul Index Swapions By conras, he compuaion of C a hinges on exacly he same argumens as in he single-name case. Firs, we define an equivalen probabiliy measure Q on Ω, F by posulaing ha he Radon-Nikodým densiy of Q wih respec o Q equals d Q dq = cĝ B 1 Ân, Q-a.s. 52 Le us noe ha he process η = cĝb 1 Â n, [0, ], is a sricly posiive F-maringale under Q, since J Â n = c E Q η = cĝb 1 j=1 α j B 1 J Tj F and we have ha Qτ > F Tj = Ĝ > 0 for every j. Therefore, for every [0, ], d Q F = E Q η dq F = η, Q-a.s. The quaniy c = E Q Ĝ B 1 Ân 1 is he normalizing consan, which ensures ha Q given by 52 is indeed a probabiliy measure on Ω, F. Lemma 2.7 The price a ime [0, ] of he payer credi defaul index swapion on he collapse even { τ} is given by 50. On he pre-collapse even { < τ} i equals C = Ân E bq κ a κ + F + B E Q 1{ bτ>} B 1 L F. 53 Proof. I suffices o examine C a. sing 51 and 52, we obain C a = B G 1 E Q Ĝ B 1 Ân κ a κ + F = Ân η 1 E Q η κ a κ + F = Ân E bq κ a κ + F, where he las equaliy follows from he absrac Bayes formula. The nex lemma esablishes he maringale propery of he process κ a under Q. Lemma 2.8 The pre-collapse loss-adjused fair forward CDIS spread κ a, [0, ], is a sricly posiive F-maringale under Q. Proof. Similarly as in he proof of Lemma 1.6, i suffices o observe ha he produc κ a η saisfies κ a η = cκ a ĜB 1 Â n = cĝb 1 P a = c E Q 1 δ n i=1 B 1 τ i 1 {T <τi T J } + 1 {<bτ} B 1 L F so ha κ a η is an F-maringale under Q. By he well known argumen, we conclude ha κ a is an F-maringale under Q Black Formula for Credi Defaul Index Swapions Our nex goal is o esablish a suiable version of he Black formula for he credi defaul index swapion. To his end, we posulae ha he pre-collapse loss-adjused fair forward CDIS spread saisfies κ a = κ a σ u κ a u dŵu, [0, ], 54

19 M. Rukowski and A. Armsrong 19 where Ŵ is he one-dimensional sandard Brownian moion under Q wih respec o F and σ is an F-predicable process. Le us emphasize ha he assumpion ha he filraion F is he Brownian filraion cf. Secion 1.5 would be oo resricive, since F = F H 1... H n 1 and hus F will ypically need o suppor also disconinuous maringales. Proposiion 2.1 Assume ha he volailiy σ of he pre-collapse loss-adjused fair forward CDIS spread is a posiive funcion. Then he pre-defaul price of he payer credi defaul index swapion equals, for every [0, ] on he pre-collapse even { < τ}, C = Ân κ a N d + κ a,, κn d κ a,, + C L 55 or, equivalenly, where C = P a N d + κ a,, κân N d κ a,, + C L, 56 d ± κ a,, = lnκa /κ ± 1 2 σ 2 u du 1/2 σ 2 u du. Proof. I suffices o focus on C a. In view of 53, we obain C a = Ân E bq κ a κ + F = Â n E bq κ a κ + κ a = Â n κ a N d + κ a,, κn d κ a,, where he las equaliy follows by sandard compuaions. A slighly differen approach o valuaion of a credi defaul index swapion was proposed by Pedersen [19]. The derivaion of Pedersen s formula hinges on he simplificaion of he acual payoff of he swapion combined wih he assumpion ha no every reference names will defaul prior o he swapion s mauriy dae T. Formally, i is enough o posulae ha L = 1 {<bτ} L 57 so ha C L = 0. nder his assumpion, he second erm in pricing formulae 49, 53 and 55 will vanish and hus, for insance, expression 55 will reduce o he sandard Black swapions formula. nder usual circumsances, he probabiliy of all defauls occurring prior o is expeced o be very low, and hus assumpion 57 seems o be reasonable. Morini and Brigo [18] argue, however, ha his assumpion is no always jusified, in paricular, i is no suiable for periods when he marke condiions deeriorae. I is also worh menioning ha since we deal here wih he riskneural probabiliy measure, he probabiliies of defaul evens are known o drasically exceed saisically observed defaul probabiliies, ha is, probabiliies of defaul evens under he physical probabiliy measure. nder assumpion 57, he second erm in 48, and hus also C L, vanish and hus he pricing formula 55 reduces o a single erm. Le us finally menion ha Jackson [14] proposed an alernaive approach o valuaion of a credi defaul index swapion by condiioning on he number of defauls prior o he swapion s mauriy. He obains, under raher sringen model assumpions, he pricing formula in he form of a weighed average of suiable Black s formulae. We do no discuss his mehod here; for a deailed analysis of assumpions underpinning Jackson s approach, which is based on condiioning on he number of defauls ha occur prior o he swapion s expiry, he ineresed reader is referred o Secion 3.5 in [1] or Secion 4.4 in [2] Marke Pricing Formula for Credi Defaul Index Swapions Before concluding his paper, le us briefly examine one of he convenional marke formulae for valuing credi defaul index swapion. Le us emphasize ha pricing formula 58 refers o he quoed fair forward CDIS spread κ n, raher han o is loss-adjused and hus no direcly observed version κ a. To accoun for a poenial loss prior o he swapion s mauriy, a suiable alhough somewha ad hoc adjusmen o he srike level κ is inroduced.

20 20 Valuaion of Credi Defaul Index Swapions Proposiion 2.2 The price of a payer credi defaul index swapion can be approximaed as follows C 1 {<bτ}  n κ n N d + κ n,, κ L N d κ n,,, 58 where d ± κ n,, = lnκn /κ L ± 1 2 in which L = E bq A n 1 L F for [0, ]. σ 2 u du 1/2 σ 2 u du Proof. We merely skech he proof. We sar by approximaing he model payoff, which is given by he expression cf. 44 +, C = 1 {<bτ}  n κ n κ + L in he following way where C 1 {<bτ}  n κ n κ + L +, L = E bq  n 1 L F. This las equaliy may be wrien as Ĉ 1 {<bτ}  n κ n κ L +, where he random variable L is manifesly F -measurable. By applying he risk-neural valuaion and proceeding as in he proof of Proposiion 2.1, we obain he saed formula Concluding Remarks As already observed in Remark 2.1, he approach developed in Secion 2 can be applied o oher varians of muli-name credi defaul swaps and no only o sandard credi defaul index swaps and swapions. In paricular, he valuaion of a single ranche of a synheic collaeralized deb obligaion CDO and he relaed opion can be done along he same lines. For a CDO ranche, he random ime τ = τ n should be replaced by he random ime τ k, where k sands for he minimal number of defaul evens for which he percenage loss process, wih consan jumps, crosses he deachmen poin of he ranche. There is one imporan cavea here. The assumpion ha he ranche spread is a lognormally disribued maringale under he corresponding maringale measure, as defined by a suiable modificaion of formula 52, can only be jusified for pricing of a ranche opion on a sand alone basis. For he consisen simulaneous valuaion of opions on various CDO ranches, we would need o produce firs a mulidimensional arbirage-free model for he ranche spreads wih desired disribuional properies. An imporan issue arising in his conex is he specificaion of drif erms in he join dynamics of ranche spreads under a common maringale probabiliy, which would allow us o value also more complex credi correlaion producs. References [1] A. Armsrong: Valuaion of credi defaul index swapions. Honours projec, School of Mahemaics and Saisics, NSW, [2] A. Armsrong and M. Rukowski: Valuaion of credi defaul index swaps and swapions. Working paper, School of Mahemaics and Saisics, NSW, 2007.