PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION ALEXANDER HERBERTSSON AND RÜDIGER FREY Absrac. We derive pracical forulas for CDS index spreads in a credi risk odel under incoplee inforaion. The facor process driving he defaul inensiies is no direcly observable, and he filering odel of Frey & Schid 2012) is used as our seup. In his fraework we find a copuaionally racable expressions for he payoff of a CDS index opion which naurally includes he so-called arageddon correcion. The correcion is obained wihou inroducing a change of pricing easure, which is he case in he previous lieraure. A lower bound for he price of he CDS index opion is derived and we provide explici condiions on he srike spread for which his inequaliy becoes an equaliy. The bound is copuaionally feasible and do no depend he noise paraeers in he filering odel. We ouline how o explicily copue he quaniies involved in he lower bound for he price of he credi index opion. Finally, a syseaic sudy is perfored in order o undersand he ipac of various odel paraeers on CDS index opions and on he index iself). 1. Inroducion The developen of liquid arkes for synheic credi index producs such as CDS index swaps has led o he creaion of derivaives on hese producs, os noably credi index opions, soeies also denoed CDS index opions. Essenially he owner of such an opion has he righ o ener a he auriy dae of he opion ino a proecion buyer posiion in a swap on he underlying CDS index a a prespecified spread; oreover, upon exercise he obains he cuulaive loss of he index porfolio up o he auriy of he opion. Credi index opions have gained a lo ineres he las five urbulen years since hey allow invesors o hedge heselves agains broad oveens of CDS index spreads or o rade credi volailiy. To dae he pricing and he hedging of hese opions is largely an unresolved proble. In pracice his conrac is priced by a fairly ad hoc approach: i is assued ha he lossadjused spread of he CDS index a he auriy of he opion is lognorally disribued under a aringale easure corresponding o a suiable nueraire, and he price of he opion is hen copued via he Black forula. Deails are described for insance in Morini & Brigo 2011) or Rukowski & Arsrong 2009). However, beyond convenience here is no jusificaion for he lognoraliy assupion in he lieraure. In paricular, i is unclear Dae: February 11, 2014. The research of Alexander Herbersson was suppored by AMAMEF ravel grans 2206 and 2678, by he Jan Wallanders and To Hedelius Foundaion and by Vinnova. The research of Rüdiger Frey was suppored by Deusche Forschungsgeeinschaf. 1

2 ALEXANDER HERBERTSSON AND RÜDIGER FREY if a dynaic odel for he evoluion of spreads and credi losses can be consruced ha suppors he lognoraliy assupion and he use of he Black forula, and here is no epirical jusificaion for his assupion eiher. In his paper we herefore propose a differen roue for pricing and hedging credi index opions, which is based on a full dynaic credi risk odel. We use a new, inforaionbased approach o credi risk odelling proposed in Frey & Schid 2012) where prices of raded credi derivaives are given by he soluion of a nonlinear filering proble. Frey & Schid 2012) solve his proble using he innovaions approach o nonlinear filering and derive in paricular he Kushner-Sraonovich SDE describing he dynaics of he filering probabiliies. Moreover, hey give ineresing heoreical resuls on he dynaics of he credi spreads and on risk iniizing hedging sraegies. Our paper use he filering odel of Frey & Schid 2012) in order o derive copuaionally pracical forulas for a CDS index under he arke filraion. The arke filraion represens incoplee inforaion since he background facor process driving he defaul inensiies is observed wih noise. Furherore, in his odel we derive copuaionally racable forula for he payoff of a CDS index opion. The forula naurally includes he so-called arageddon correcion and is obained wihou inroducing a change of pricing easure, which is he case in he previous lieraure, see e.g. in Morini & Brigo 2011) or Rukowski & Arsrong 2009). We also derive a lower bound for price of he CDS index opion and provide explici condiions on he srike spread for which his inequaliy becoes an equaliy. The lower bound is copuaionally racable and do no depend on any of he noise paraeers in he filering odel. We hen ouline how o explicily copue he quaniies involved in he lower bound for he price of he credi index opion. Furherore, a syseaic sudy is perfored in order o undersand he ipac of various odel paraeers on hese index opions and on he index iself). Credi index opions have been reaed in several previous papers, see e.g Morini & Brigo 2011), Rukowski & Arsrong 2009), Pedersen 2003), Marin 2012) and Jackson 2005). The idea of using filering echniques in credi risk odelling o price credi derivaives and defaulable bonds is no new. For exaple, Capponi & Cvianic 2009) develops a srucural credi risk fraework which odels he deliberae isreporing by insiders in he fir. In his seing he auhors derive forulas for bond and sock prices which lead o a non-linear filering odel. The odel is calibraed wih Kalan filering and axiu likelihood ehods. The auhors hen apply heir seup o he Parala-case and he paraeers are calibraed agains real daa. The paper Fonana & Runggaldier 2010) considers an inensiy based credi risk odel where defaul inensiies and ineres raes are driven by a parly unobservable facor process. In his seup hey sae forulas for coningen clais given he filraion generaed by he unobservable facor process and he defaul ies. The auhors hen derive a nonlinear filer syse describing he dynaics of he filering disribuion which is needed for pricing he derivaives in heir fraework. The paraeers in he odel are obained via an expeced axiu EM) algorih which includes solving he nonlinear filer syse

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 3 by using he exended Kalan filer and a linearizaion of he fraework. The odel and esiaion ehod is applied on siulaed daa wih successful resuls. In Frey & Runggaldier 2010) he auhors develops a aheaical fraework for handling filering probles in reduced-for credi risk odels. The res of he paper is organized as follows. Firs, in Secion 2 we give a brief inroducion o how a CDS index works and hen presen a odel independen expression for he so called CDS index spread. Secion 2 also inroduces opions on he CDS index and provides a forula for he payoff such an opion which holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. Then, in Secion 3 we describe he odel used in his paper, originally presened in Frey & Schid 2012). Secion 4 gives a shor recapiulaion of he he Kushner-Sraonovich SDE describing he dynaics of he filering probabiliies in he odels, where we in paricular focus on a hoogeneous porfolio. Nex, Secion 5 describes he ain building blocks ha will be necessary o find forulas for porfolio credi derivaives such as e.g. he CDS index as well as credi index opions. Exaples of such building blocks are he condiional survival disribuion, he condiional nuber of defauls and he condiional loss disribuion. In Secion 6 we use he resuls fro Secion 5 o derive copuaional racable forulas for he CDS index in he odel presened in Secion 3. This will be done in a hoogeneous porfolio. Coninuing, in Secion 7 we give a shor inroducion o opions on a CDS index and derive pracical forulas for he payoff of such opions in he nonlinear filering odell. This convenien forula will hen be used wih Mone Carlo siulaions in order o find good approxiaions o he price of opions on a CDS index in he filering odell. Secion 8 briefly discusses how o esiae or calibrae he paraeers in he filering odel inroduced in Secion 3. Finally, in Secion 9 we calibrae our odel and presen differen nuerical resuls for prices of opions on a CDS index. 2. The CDS index and credi index opions In his secion we will discuss he CDS index and opions on his index. Firs, Subsecion 2.1 gives a brief inroducion o how a CDS index works. Then, in Subsecion 2.2 we ouline odel independen expression for he CDS index spread. Finally, Subsecion 2.3 inroduces opions on he CDS index, soeies denoed by credi index opions, and uses he resul for Subsecion 2.2 o provide a forula for he payoff such an opion which holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. 2.1. Srucure of a CDS index. Consider a porfolio consising of equally weighed obligors. An index Credi Defaul Swap ofen denoed CDS index or index CDS) for a porfolio of obligors, enered a ie wih auriy T, is a financial conrac beween a proecion buyer A and proecion seller B wih he following srucure. The CDS index gives A proecion agains all credi losses aong he obligors in he porfolio up o ie T where < T. Typically, T = + T 0 for T 0 = 3, 5, 7, 10 years. More specific, a each defaul in he porfolio during he period [, T, B pays A he credi suffered loss due o he defaul. Thus, he accuulaed value payed by B o A in he period [, T is he oal credi loss in he porfolio during he period fro o ie T. As a copensaion for his

4 ALEXANDER HERBERTSSON AND RÜDIGER FREY A pays B a fixed fee S, T) uliplied wha is lef in he porfolio a each payen ie which are done quarerly in he period [, T. The fee S, T) is se so expeced discouned cash-flows beween A and B is equal a ie and S, T) is called he CDS index spread wih auriy T. For = 0 i.e. oday ) we denoe S0, T) by ST) and he quaniy ST) can be observed on a daily basis for sandard CDS indexes such as itraxx Europe and he CDX.NA.IG index, for auriies T = 3, 5, 7, 10 years. In order o give a ore explici descripion of he CDS index spread S, T) we need o inroduce soe furher noaions and conceps which is done in he nex subsecion. 2.2. The CDS index spread. In his subsecion we give a quaniaive descripion of he CDS index spread. Firs we need o inroduce soe noaion. Le Ω, G, Q) be he underlying probabiliy space assued in he res of his paper. We se Q o be a risk neural probabiliy easure which exis under raher ild condiion) if arbirage possibiliies are ruled ou. Furherore, le F = F ) 0 be a filraion represening he full arke inforaion a each ie poin. Consider a porfolio consising of equally weighed obligors wih defaul ies τ 1, τ 2...,τ adaped o he filraion F ) 0 and le l 1, l 2,...,l be he corresponding individual credi losses a each defaul ie. Typically l i = 1 φ i )/ where φ i is a consan represening he recovery rae for obligor i. The credi loss L for his porfolio a ie is hen defined as L = i=1 l i1 {τi }. Siilarly, he nuber of defauls in he porfolio up o ie, denoed by N, is N = i=1 1 {τ i }. Noe ha if he individual loss is consan and idenical for all obligors so ha l = l 1 = l 2 =... = l hen he noralized credi loss is given by L = l N. In he res of his paper we will assue ha he individual loss is consan and idenical for all obligors where 1 φ = l = l 1 = l 2 =... = l. Finally, we will assue ha he ineres is consan and given by r and for < s we le B, s) denoe he discoun facor beween s and, ha is B, s) = e rs ). Le T > and consider an CDS index wih auriy T on he porfolio wih loss L a ie. In view of he above noaion we can now define he sochasic) discouned payens V D, T) fro A o B during he period [, T, and V P, T) fro B o A in he iespan [, T, as follows V D, T) = T 4T B, s)dl s and V P, T) = 1 B, n ) 4 n=n 1 N ) n 2.2.1) where n denoes n = 4 + 1 and n = n. We here ephasize ha we have dropped he 4 accrued er in V P, T) and also ignored he accrued preiu up o he firs payen dae in V P, T). The expeced value of he defaul and preiu legs, condiional on he arke inforaion F are given by [ T E B, s)dl s F and 4T 1 B, n ) 1 1 ) 4 E [N n F. 2.2.2) n=n The er V D, T) can be rewrien in a ore pracical for using inegraion by pars see e.g. Theore 3.36, p.107 in Folland 1999)), so ha V D, T) = T B, s)dl s =

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 5 B, T)L T L + T rb, s)l s ds and by applying Fubini-Tonelli on his expressions hen renders [ T T E B, s)dl s F = B, T)E [L T F L + rb, s)e [L s F ds 2.2.3) which will be pracical, as will be seen in Secion 6. In view of srucure of a CDS index described in Subsecion 2.1, he CDS index spread S, T) a ie wih auriy T is defined as S, T) = [ T B, s)dl s F E 1 4T 4 n=n B, n ) 1 1 E [N n F ). 2.2.4) The definiion of S, T) in 2.2.4) is done assuing ha no all obligors have defauled in he porfolio a ie, ha is S, T) is defined on he even {N < }. In he even of a so-called arageddon scenario a ie where N = i.e. all obligors in he porfolio have defauled up o ie ), we see ha he preiu leg V P, T) in 2.2.1) is zero a ie, which obviously akes he definiion of he spread S, T) invalid. Noe ha for = 0 i.e. oday) he quaniy S0, T) can be observed on a daily basis for sandard CDS indexes such as itraxx Europe and he CDX.NA.IG index, for auriies T = 3, 5, 7, 10 years. We reark ha ouline for he CDS index spread presened in his subsecion holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. Consequenly, he filraion F used in his subsecion can be generaed by any credi porfolio odel. 2.3. The CDS index opion. In his subsecion we inroduce opions on he CDS index and discuss how hey work. Then we use he resul for Subsecion 2.2 in order o provide a forula for he payoff of such an opion, which holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. Definiion 2.1. A payer CDS index opion soeies called a pu CDS index opion) wih srike κ and exercise dae wrien on a CDS index wih auriy T is a financial derivaive which gives he proecion buyer A he righ bu no he obligaion o ener he CDS index wih he proecion seller B a ie wih fixed spread κ and auriy T. Moreover, a he exercise dae, he proecion seller B also pays A he accuulaed credi loss occurred during he period fro he incepion ie a ie 0) of he opion o he exercise dae, ha is B pays A he loss L a ie. This is ofen called he fron end proecion. The payoff Π, T; κ) a he exercise ie for a payer CDS index opion is fro he proecion buyer A s poin of given by Π, T; κ) = PV, T) S, T) κ) 1 {N<} + L ) + 2.3.1)

6 ALEXANDER HERBERTSSON AND RÜDIGER FREY where PV, T) = E [V P, T) F so ha 4T PV, T) = 1 B, n ) 4 n=n 1 1 ) E [N n F 2.3.2) which follows fro he definiion of V P, T) in Equaion 2.2.1). Noe ha he CDS index a ie is enered only if here are any nondefauled obligors lef in he porfolio a ie, which explains he presence of he indicaor funcion of he even {N < } in he expression for he payoff Π, T; κ) in 2.3.1). However, he fron end proecion L will be paid ou by A a ie even if {N = }. In paricular, since we in his paper assue ha he individual loss is consan 1 φ and idenical for all obligors, and since L = 1 φ)n we have ha L 1 {N=} = 1 φ)1 {N=} and in view of 2.2.1) or 2.3.2) i holds ha PV, T)1 {N=} = 0 so Π, T; κ) can hen be rewrien as Π, T; κ) = PV, T) S, T) κ) + L ) + 1 {N<} + 1 φ)1 {N=}. 2.3.3) Fro 2.3.1) or 2.3.3) we conclude ha li Π, T; κ)1 {N <} = 0 2.3.4) κ and Π, T; κ)1 {N=} = L 1 {N=} = 1 φ)1 {N=} for all κ. 2.3.5) So in paricular we have ha li Π, T; κ)1 {N =} = L 1 {N=} = 1 φ)1 κ {N=}. 2.3.6) Thus, in general i will no hold ha li κ Π, T; κ) = 0 alos surely, i.e. he even li κ Π, T; κ) = 0 will no happen wih probabiliy one. The price C 0, T; κ) of payer CDS index opion a incepion ie 0 i.e. oday) wih srike κ and exercise dae wrien on a CDS index wih auriy T, is due o sandard risk neural pricing heory given by C 0, T; κ) = e r E [Π, T; κ). 2.3.7) In view of 2.3.1) he quaniy C 0, T; κ) can be expressed as [ PV ) C 0, T; κ) = e r + E, T) S, T) κ) 1{N<} + L or alernaively using 2.3.3), as 2.3.8) C 0, T; κ) = e r E [ PV, T) S, T) κ) + L ) + 1 {N<} + 1 φ)e r Q [N = 2.3.9) where we reind he reader ha L = 1 φ)n. Fro 2.3.9) we iedialy conclude ha li C 0, T; κ) = 1 φ)e r Q [N = 2.3.10) κ which is in line wih he resuls in 2.3.4) and 2.3.6). Also noe ha he resuls in his secion holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. Recall ha in he sandard Black-Scholes odel he call opion

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 7 price converges o zero as he srike price converges o infiniy bu due o he fron end proecion his will no hold for payer CDS index opion, as is clearly seen in Equaion 2.3.9) and 2.3.10). By using he expression for he CDS index spread S, T) given by 2.2.4) we can rewrie he payoff Π, T; κ) in 2.3.2) as Π, T; κ) = [ T E B, s)dl s F κ 1 4T B, n ) 4 n=n 1 1 ) E [N n F 1 {N<} + L +. 2.3.11) The odel ouline for payer CDS index opion presened in his subsecion holds for any fraework odelling he dynaics of he defaul ies in he underlying credi porfolio. Consequenly, he filraion F used in his subsecion can be generaed by any credi porfolio odel. 3. The odel In his secion we shorly recapiulae he odel of Frey & Schid 2012). Thus, we will consider a reduced-for odel driven by an unobservable background facor process X odelling he rue sae of he econoy. For racabiliy reasons X is odelled as finie-sae Markov chain. The facor process X is no direcly observable. Insead odel quaniies are given as condiional expecaion wih respec o he so called arke filraion F M = F M ) 0. The filraion F M is generaed by he facor process X plus noise, which will be specified in deail below. Inuiively speaking, his eans ha he odel quaniies are observed given an incoplee hisory of he sae of he econoy. Furherore, in he odel of Frey & Schid 2012) he defaul ies of all obligors are condiionally independen given he inforaion of he facor process X. This seup is close o he one found in e.g. Graziano & Rogers 2009). Frey & Schid 2012) rea he case wih sochasic recoveries in a general heoreical seing. In his paper we will ake a siplified approach and only consider deerinisic recoveries, which up o he recen credi crises has been considered as sandard in he credi lieraure. Considering sochasic recoveries in our seup will inroduce challenges in he esiaion rouine. 3.1. The facor process. In his secion we inroduce he odel ha we will consider under he full inforaion. Le X be a finie sae coninuous ie Markov chain on he sae space S X = {1, 2,..., K} wih generaor Q. Le F X = σx s ; s ) be he filraion generaed by he facor process X. Consider obligors wih defaul ies τ 1, τ 2...,τ and le he appings λ 1, λ 2...,λ be he corresponding F X defaul inensiies, where λ i : S X R + for each obligor i. This eans ha each defaul ie τ i is odeled as he firs jup of a Cox-process, wih inensiy λ i X ). I is well known see e.g. Lando 1998)) ha given an i.i.d sequence {E i } where

8 ALEXANDER HERBERTSSON AND RÜDIGER FREY E i is exponenially disribued wih paraeer one, such ha all {E i } are independen of F X, hen { } τi = inf > 0 : λ i X s )ds E i. 3.1.1) Hence, for any T we have Q [ τ i > F X T 0 = exp 0 ) λ i X s )ds 3.1.2) and hus [ ) Q [τ i > = E exp λ i X s )ds. 3.1.3) 0 Noe ha he defaul ies are condiionally independen, given F X. The saes in S X = {1, 2,..., K} are ordered so ha sae 1 represens he bes sae and K represens he wors sae of he econoy. Consequenly, he appings λ i ) are chosen o be sricly increasing in k {1, 2,..., K}, ha is λ i k) < λ i k + 1) for all k {1, 2,..., K 1} and for every obligor in he porfolio. 3.2. The arke filraion and full inforaion. In his subsecion we forally inroduce he arke filraion, ha is he inforaion observed by he arke paricipans. Recall ha he prices of all securiies are given as condiional expecaions wih respec o his filraion. We also shorly discuss he full inforaion F = F ) 0, which is he bigges filraion conaining all oher filraions, where Ω, G, P) wih G = F will be he underlying probabiliy space assued in he res of his paper. Here P is he so called physical probabiliy easure soeies also denoed he saisical or real probabiliy easure). Le Y,i denoe he rando variable Y,i = 1 {τi } and Y be he vecor Y = Y,1,...,Y, ). The filraion F Y = σy s ; s ) represens he defaul porfolio inforaion a ie, generaed by he process Y s ) s 0. Furherore, le B be a one-diensional Brownian oion independen of X ) 0 and Y ) 0 and le a ) be a funcion fro {1, 2,..., K} o R. Nex, define he process Z as Z = 0 ax s )ds + B. 3.2.1) We here reark ha Frey & Schid 2012) allows for ulivariae Brownian oion B in 3.2.1) as well as a vecor valued apping a ) wih sae diension as B and in he nuerical sudies of Frey & Schid 2012) hey use a one-diensional Brownian oion B. In his paper we resric ourselves o only one source of randoness in he noise represenaion 3.2.1). Exending o several sources of randoness in 3.2.1) will in principle no change he ain ideas in he esiaion ehod. Inuiively Z represens he noisy hisory of X and he funcional for of Z given by 3.2.1) is a represenaion ha is sandard in he nonlinear filering heory, see e.g. Davis & Marcus 1981). Following Frey & Schid 2012), we define he arke filraion F M = F M ) 0 as F M = F Y F Z. 3.2.2)

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 9 We se he full inforaion F = F ) 0 o be he bigges filraion conaining all oher filraions wih G = F. We can for exaple le F be given by F = F X F Y F B 3.2.3) where F B) 0 is he filraion generaed by he Brownian oion B. Noe ha F X a subfilraion of F Z, and siilarly, FB is no conained in F Z. 4. Applying he Kushner-Sraonovic SDE in he credi risk odel is no In his secion we sudy he Kushner-Sraonovic SDE in our filering odel. We use he sae noaion as in Frey & Schid 2012). Firs, define π k as he condiional probabiliy of he even {X = k} given he arke inforaion F M a ie, ha is π k = Q [ X = k F M 4.1) and le π R K be a row-vecor such ha π = π 1,.. ).,πk. In he sequel, for any F -adaped process U we le Û denoe he opional projecion of Û ono he filraion F M, ha is Û = E [ U F M. To his end, we have for exaple Nex, define M,i and µ as M,i λ i X ) = E [ λ i X ) F M âx ) = E [ ax ) F M = Y,i µ = Z τi 0 0 K = λ i k)π k K = ak)π k. λ i X s )ds for i = 1,..., 4.2) âx s ) ds In Frey & Schid 2012) i is shown ha M,i is an F M -aringale, for i = 1, 2,..., and ha µ is a Brownian oion wih respec o he filraion F M. Thus, he vecor M = M,1,...,M, ) is an F M -aringale. These resuls have been proven previously when considered separaely, i.e. for pure diffusion filering probles, see e.g. Davis & Marcus 1981), and pure jup process filering process, see e.g Bréaud 1981). Furherore, Frey & Schid 2012) also proves he following proposiion, which is a version of he Kushner- Sraonovic equaions, adoped o he filering odels presened in his paper originally developed in Frey & Schid 2012)). Proposiion 4.1. Wih noaion as above, he processes π k saisfies he following K- diensional syse of SDE-s, K dπ k = Q l,k π l d + γk π )) dm + α k π ) dµ, 4.3) l=1

10 ALEXANDER HERBERTSSON AND RÜDIGER FREY where γ k π)) = γ1 kπ),...,γk π)) wih π = π 1, π 2,...,π ) and he coefficiens γi kπ) are appings given by γi k π) = λ i k) πk K ), n=1 λ in)π 1 1 i 4.4) n and α k π ) = π k ak) K π ), n an) 1 k K. 4.5) n=1 The K-diensional SDE-syse parly uses he vecor noaion for he M vecor. However, as will be seen below, i will be beneficial o rewrie his SDE on coponen for, especially when we consider hoogeneous credi porfolios. Thus, le us rewrie 4.3) on coponen for, so ha K dπ k = Q l,k π l d + γi k π )dm,i + α k π )dµ 4.6) l=1 i=1 Nex, le us consider a hoogeneous credi porfolio, ha is, all obligors are exchangeable so ha λ i X ) = λx ) and γi kπ ) = γ k π ) for each obligor i and define N as N = Y,i = 1 {τi }. 4.7) i=1 Furherore, define λ as λ = λ1),...,λk)) and le e k R be a row vecor where he enry a posiion k is 1 and he oher enries are zero. For a hoogeneous porfolio he resuls of Proposiion 4.1 can be siplified o he following corollary. Corollary 4.2. Consider a hoogeneous credi porfolio wih obligors. Then, wih noaion as above, he processes π k saisfy he following K-diensional syse of SDE-s, i=1 dπ k = γk π )dn + π Qe k γ k π )λ N ) ) d + α k π )dµ 4.8) where γ k π ) and α k π ) are given by ) λk) γ k π ) = π k π λ 1 and α k π ) = π k ak) K n=1 ) π n an). 4.9) Proof. Firs, fro 4.2) we have dm,i = dy,i 1 λ {τi >} i X )d = dy,i 1 K {τi >} λ ik)π k d which in 4.6) iplies ha K dπ k = π Qe k d + γi k π )dy,i γi k π )1 {τi >} λ i k)π k d + α j π )dµ. 4.10) i=1 i=1 Since λ i X ) = λx ) and γi kπ ) = γ k π ) for all obligors i, and recalling ha N denoes N = i=1 Y,i = i=1 1 {τ i } so ha i=1 1 {τ i >} = N, we can afer soe copuaions rewrie 4.10) as dπ k = γ k π )dn + π Qe k γ k π )λ N ) ) d + α j π )dµ

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 11 λk) π λ 1 ) ak) where γ k π ) and α k π ) are given by γ k π ) = π k and α k π ) = π k K n=1 πn an)). Fro he SDE 4.8) in Corollary 4.2 we clearly see ha he dynaics of he condiional probabiliies π k conains a drif par, a diffusion par and a jup par. The diffusion par is due o he dµ coponens and he jup par is due o he defauls in he porfolio, given by he differenial dn. he process X he defaul process N he probabiliy π 1 2 1.5 A realizaion of he process X 1 0 0.5 1 1.5 2 2.5 3 ie A realizaion of he poin process N 20 10 0 0 0.5 1 1.5 2 2.5 3 ie A rajecory of π 1 siulaed wih he Kushner Sraaonovich SDE 1 0.5 0 0 0.5 1 1.5 2 2.5 3 ie in years) π 1 X N Figure 1. A siulaed rajecory of X, N and π 1 where K = 2 and = 125. Figure 1 visualizes a siulaed pah of π 1 given by 4.8) in Corollary 4.2 in an exaple where K = 2 and = 125, using ficive paraeers for Q and λ assuing ak) = c lnλk) for a consan c. Fro he hird Figure 1 we clearly see ha π 1 has jup, drif and diffusion pars. The firs and second subfigures in Figure 1 shows he corresponding rajecories for X and N. Noe how he defauls presened by N cluser as X swiches o sae 2, represening he worse econoic sae aong {1, 2}. 5. The ain building blocks In his secion we describe he ain building blocks ha will be necessary o find forulas for porfolio credi derivaives such as e.g. he CDS index. Exaples of such building

12 ALEXANDER HERBERTSSON AND RÜDIGER FREY blocks are he condiional survival disribuion, he condiional nuber of defauls and he condiional loss disribuion. The condiional expecaions are wih respec o he noisy arke inforaion F M defined in Equaion 3.2.2) in Subsecion 3.2. Recall ha Y,i denoes he rando variable Y,i = 1 {τi }, Y = Y,1,...,Y, ) and N and L are given by N = i=1 1 {τ i } and L = 1 i=1 1 φ i)1 {τi } where φ i is he recovery rae for obligor i. Our ain ask in his secion is o find he following quaniies Q [ [ τ i > T F M, E NT F M and E [ L T F M where T >. These expressions will be useful when deriving forulas for he forward saring CDS index spread S, T) as well as oher forward saring credi derivaives in Secion 6 and Secion??. 5.1. The condiional survival disribuion. In his subsecion we sudy he condiional survival disribuion Q [ τ i > T F M for T > in he filering odel. To do his we need o inroduce soe noaion. If X is a finie sae Markov jup process on S X = {1, 2,..., K} wih generaor Q, hen, for a funcion λx) : S X R we denoe he arix Q λ = Q I λ where I λ is a diagonal-arix such ha I λ ) k,k = λk). Furherore, we le 1 be a colun vecor in R K where all enries are 1. The following heore is a perquisie for all oher resuls in his paper and is herefore a core resul. Theore 5.1. Consider a credi porfolio specified as in Secion 3 and le λ i X ) be he F X -inensiy for obligor i. If T hen, wih noaion as above Q [ τ i > T F M = 1{τi >}π e Q λ i T ) 1 5.1.1) where he arix Q λi = Q I λi is defined as above. Proof. Since T >, hen E [ [ 1 {τi >T } F = E 1{τi >T } F X F Y i = 1{τi >}E[e T λ i X s)ds F X 5.1.2) where he firs equaliy is due o he fac ha condiionally on X, hen τ i is independen of τ j for j i. The second equaliy follows fro a sandard resul for he firs jup ie of a Cox-process, see e.g. p.102 in Lando 1998), Corollary 9.1 in McNeil, Frey & Ebrechs 2005) or Corollary 6.4.2 in Bielecki & Rukowski 2001). [ Since T > and due o he Markov propery of X we can rewrie he quaniy E e T λ i X s)ds F X as E [e T λ i X s)ds F X = E [e T λ i X s)ds X = which iplies ha recall ha F X is no a subfilraion of F M ) [ E E [e T λ i X s)ds F X F M = K K E [e T λ i X s)ds X = k 1 {X=k} E [e T λ i X s)ds X = k π k 5.1.3)

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 13 where we used he noaion π k = Q[ X = k F M. By using Theore A.1 in Appendix A we ge E [e T λ i X s)ds X = k = e k e Q λ i T ) 1 5.1.4) where he arix Q λi is defined as previously for ore on hese deails, see in Appendix A). So 5.1.4) in 5.1.3) yields [ E E [e T λ i X s)ds F X F M = K e k e Q λ i T ) 1π k = π e Q λ i T ) 1 5.1.5) where π is a row-vecor such ha π = ) π 1,..., π K. Nex, noe ha E [ [ [ 1 {τi >T } F M = E E 1{τi >T } F F M [ = 1 {τi >}E E [e T λ i X s)ds F X F M = 1 {τi >}π e Q λ i T ) 1 where he second equaliy is due o 5.1.2) and he hird equaliy follows fro 5.1.5). Thus, for T we conclude ha Q [ τ i > T F M = 1{τi >}π e Q λ i T ) 1 which proves he heore. 5.2. The condiional nuber of defauls. In his subsecion we derive pracical expressions for E [ N F M. We consider an hoogeneous credi porfolios where λi X ) = λx ) so ha Q λi = Q λ for each obligor i. Recall ha N = i=1 1 {τ i }. The ain essage of his subsecion is he following proposiion. Proposiion 5.2. Consider an exchangeable credi porfolio wih obligors in a odel specified as in Secion 3. Then, for T and wih noaion as above E [ N T F M = N ) π e Q λ T ) 1. 5.2.1) Proof. Le T > and firs noe ha E [N T F = E [ 1 {τi >T } F = i=1 i=1 1 {τi >}E [e T λ i X s)ds F X 5.2.2) where he las equaliy is due o Equaion 5.1.2) in Theore 5.1. Furherore, in a hoogeneous porfolio we have λ i X s )[ = λx s ) for all obligors i and his in 5.2.2) iplies ha E [N T F = N ) E e T λx s)ds F X. Thus, by using E [ N T F M = E [ E [N T F F M and following siilar arguens as in Theore 5.1 we conclude afer soe copuaions ha E [ N T F M = N ) π e Q λ T ) 1 which proves he proposiion. A siilar proof can be found for inhoogeneous porfolios.

14 ALEXANDER HERBERTSSON AND RÜDIGER FREY 5.3. The condiional porfolio loss: The case wih consan recovery. This is rivial for hoogeneous porfolios, given he resuls fro Subsecion 5.2. To see his, recall ha N = i=1 1 {τ i } and L = 1 i=1 1 φ i)1 {τi } where φ i are consans and in a hoogeneous porfolio we have φ 1 = φ 2 =... = φ = φ so ha L = 1 φ) N. Thus, E [ L T F M 1 φ) = E[ N T F M 5.3.1) where E [ N T F M is explicily given in Subsecion 5.2 for hoogeneous porfolios. To be ore specific, 5.3.1) wih Proposiion 5.2 yields E [ L T F M = 1 φ) 1 1 N ) ) π e Q λ T ) 1. 5.3.2) Siilar resuls can also be obained in an inhoogeneous porfolio boh wih idenical or differen recoveries. 6. The CDS index in he filering odel In his secion we apply he resuls fro Secion 5 ogeher wih Subsecion 2.2 o find forulas for he CDS index spreads in he odels inroduced in Secion 3. This will be done in a hoogeneous porfolio. We will assue ha he ineres is consan and given by r and for < s we le B, s) denoe B, s) = e rs ). Furherore, recall ha he oal credi loss in a hoogeneous porfolio wih consan recovery φ is given by L = 1 φ) N where N = i=1 1 {τ i }. We consider he odel specified in Secion 3 and sae he following proposiion. Proposiion 6.1. Consider an CDS index porfolio in he filering odel. Then, wih noaion as above [ T E B, s)dl s F M = 1 φ) 1 N ) 1 π e Q λ T ) I + r Q λ ri) 1) e rt ) r Q λ ri) 1) 1 ) 6.1) and so if N < we have E [ V P, T) F M 1 = 1 N ) 4T π e Qλn ) 1e rn ) 4 n=n S, T) = 1 φ) 1 π e Q λ T ) I + r Q λ ri) 1) e rt ) r Q λ ri) 1) 1 ) 1 4 4T n=n π e Q λ n ) 1e rn ). 6.2) 6.3)

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 15 Proof. Le us sar wih he defaul leg. If s > hen 5.3.2) renders E [ L s F M = 1 φ) 1 1 N ) ) π e Q λ s ) 1 and using his in 2.2.3) and recalling ha B, s) = e rs ) for s >, we ge [ T E B, s)dl s F M = B, T)E [ T L T F M L + rb, s)e [ L s F M = e rt ) 1 φ) 1 1 N ) ) π e Q λ T ) 1 φ) 1 N T + re rs ) 1 φ) 1 1 N ) ) π e Q λ s ) 1 ds. The inegral in he RHS of 6.4) can be siplified according o T re rs ) 1 φ) 1 1 N ) ) π e Q λ s ) 1 ds = 1 φ) 1 e r1 φ) rt )) where he las equaliy in 6.5) is due o he fac ha T e rs ) e Q λ s ) ds = 1 N ) π e Q λ T ) e rt ) I ) Q λ ri) 1 1 T ds e Q λ ri)s ) ds = e Q λ T ) e rt ) I ) Q λ ri) 1. 6.4) 6.5) Noe ha Q λ ri) 1 exiss since Q λ ri by consrucion is a diagonal doinan arix, iplying ha de Q λ ri) 0 by he Levy-Desplanques Theore. By plugging 6.5) ino 6.4) and perforing soe rivial bu edious copuaions we ge [ T E B, s)dl s F M = 1 φ) 1 N ) 1 π e Q λ T ) I + r Q λ ri) 1) e rt ) r Q λ ri) 1) 1 ) which proves 6.1). To derive he expression for he preiu leg we use 5.2.1) in Proposiion 5.2 wih s > and obain 1 1 E[ N s F M = 1 N ) π e Q λ s ) 1 so ha 4T 1 B, n ) 1 1 4 E[ ) N n F M = 1 1 N ) 4T π e Qλn ) 1e rn ) 4 n=n n=n which proves 6.2). Finally, 6.3) follows fro he definiion in 2.2.4) ogeher wih he expressions for he defaul leg and preiu leg in 6.1) and 6.2).

16 ALEXANDER HERBERTSSON AND RÜDIGER FREY Noe ha he er 1 N / in he righ hand side of boh 6.1) and 6.2) iplies ha he condiional expecaions of he defaul and preiu legs will be zero for he arageddon even N =, which akes he spread S, T) undefined when N =. Fro Proposiion 6.1 we conclude ha given he vecor π, hen he forulas for he defaul and preiu leg in he filering odel as well as he CDS index spread S, T) are copac and copuaionally racable closed-for expressions in ers of π and Q λ. The copuaionally convenien forulas for he forward saring CDS index in Proposiion 6.1 are a prerequisie for convenien esiaion of he paraeers describing he underlying odel. This opic is carefully discussed in he paper XXX lägg in referens ill papper P3 här sedan ). Furherore, Proposiion 6.1 will also help us o find racable forulas for he payoff of ore exoic derivaives wih he CDS index as a underlyer. Exaple of such derivaives are call opions on he CDS index, which we will rea in he nex secion. 7. CDS index opions in he filering odel In his secion we apply he resuls fro Secion 6 and Subsecion 2.3 o derive copuaionally racable forulas for he payoff of a so called CDS index opion in he odel presened in Secion 3. Furherore, we presen a lower bound for price of he CDS index opion and also provide explici condiions on he srike spread for which his inequaliy becoes an equaliy. The lower bound is copuaionally racable and do no depend on any of he noise paraeers in he filering odel inroduced in Secion 3. Finally, we ouline how o explici copue he quaniies involved in he lower bound for he price of he CDS index opion. Saring off by using he explici expressions for he defaul leg and preiu legs given in Proposiion 6.1 we ake he payoff Π, T; κ) in Equaion 2.3.11) ore explici in ers of he quaniies consiuing he nonlinear filering odell presened in Secion 3. This is done in he following lea. Lea 7.1. Consider an CDS index porfolio in he filering odel. Then, he payoff Π, T; κ) for an CDS index opion wih srike κ, exercise dae and auriy T for he underlying CDS index, is given by Π, T; κ) = π [A, T) κb, T) 1 1 N ) ) + 1 φ)n + 7.1) where A, T) and B, T) are defined as A, T) = 1 φ) [I e Q λ T ) I + r Q λ ri) 1) e rt ) + r Q λ ri) 1 and B, T) = 1 4 4T n=n e Q λ n ) e rn ). Proof. By insering he explici expressions for he defaul and preiu legs for he index- CDS spread given by 6.1) and 6.2) ino 2.3.11) we ge afer soe eleenary copuaions ha

Π, T; κ) = = 1 φ) PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 17 1 N ) [ 1 π e Q λ T ) I + r Q λ ri) 1) e rt ) r Q λ ri) 1) 1 1 N κ 1 4 1 N ) π 1 φ) ) 4T ) + π e Qλn ) 1e rn ) + L n=n [I e Q λ T ) I + r Q λ ri) 1) e rt ) + r Q λ ri) 1 1 κ 1 4T ) ) + e Qλn ) 1e rn ) + L 4 n=n where we used he fac ha 1 = π 1 = π I1. Thus, we can rewrie he payoff as Π, T; κ) Π, T; κ) = π [A, T) κb, T) 1 1 N ) ) + 1 φ)n + where A, T) and B, T) are defined as A, T) = 1 φ) [I e Q λ T ) I + r Q λ ri) 1) e rt ) + r Q λ ri) 1 and B, T) = 1 4 4T n=n e Q λ n ) e rn ). This proves 7.1). In view of he above lea and since he price of he CDS index opion C 0, T; κ) a ie 0 i.e. oday) is given by C 0, T; κ) = E [e r Π, T; κ) we herefore ge [ C 0, T; κ) = e r E π [A, T) κb, T) 1 1 N ) ) + 1 φ)n + 7.2) where A, T) and B, T) are given as in Lea 7.1. Since no closed forulas are known for he enries in he vecor π i is difficul o find analyical expressions for he forulas in he RHS of Equaion 7.2). Insead we rely on Mone Carlo siulaions of he filering probabiliies π ogeher wih he copac forula for he payoff funcion Π, T; κ) given in 7.1). However, we can sill derive lower bounds for he price C 0, T; K) in our nonlinear filering odel by using he observaion in Equaion 2.3.3). This is done in he following proposiion.

18 ALEXANDER HERBERTSSON AND RÜDIGER FREY Proposiion 7.2. Le C 0, T; κ) be he price oday of an CDS index opion wih srike κ, exercise dae and auriy T. Then, wih noaion as above, C 0, T; κ) 1 φ)e r Q [N = 1 + e r j=0 K p k, T; κ) 1 j ) 1 Q [X = k, N = j + j=0 ) + 1 φ)j Q [N = j 7.3) where [A, ) p k, T; κ) = T) κb, T) 1 k 7.4) for A, T) and B, T) defined as in Lea 7.1. Proof. Recall ha we have Π, T; κ) = Π, T; κ)1 {N<} + Π, T; κ)1 {N=} and in view of 7.1) we clearly see ha Π, T; κ)1 {N=} = 1 φ) for all κ which we also concluded in Equaion 2.3.5) Subsecion??. Thus, given his observaion we have C 0, T; κ) = e r E [ Π, T; κ)1 {N<} + 1 φ)e r Q [N = 7.5) which siply is Equaion 2.3.9) resaed. Noe ha E [ Π, T; κ)1 {N<} can be rewrien as E [ Π, T; κ)1 {N<} 1 = j=0 E [ Π, T; κ)1 {N=j} 7.6) and we nex give a lower bound for he quaniy E [ Π, T; κ)1 {N=j}. Bu firs we need soe ore noaion. For each sae k in he sae space of he underlying process X [A, defined in Secion 3, le p k, T; κ) denoe he k-h coponen in he vecor T) κb, T) ) 1, ha is [A, ) p k, T; κ) = T) κb, T) 1. 7.7) Furherore, we reind he reader ha π is a a row-vecor given by π = ) π 1,..., π K where each processes π k saisfy he K-diensional syse of SDE-s in Equaion 4.8) presened in Corollary 4.2. Hence, his observaion ogeher wih Equaion 7.1) and k

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 19 Equaion 7.7) hen iplies ha we can rewrie he quaniy E [ Π, T; κ)1 {N=j} as follows E [ [ Π, T; κ)1 {N=j} = E [π A, T) κb, T) 1 1 j ) + K = E K = E E [ K π k p k, T; κ) 1 j ) + 1 φ) j π k p k, T; κ) 1 j ) 1 {N=j} + π k p k, T; κ) 1 j ) 1 {N=j} + ) + 1 φ)j 1 {N=j} ) + 1 {N=j} ) + 1 φ)j 1 {N =j} ) + 1 φ)j 1 {N =j} 7.8) where he las inequaliy is due o Jensens inequaliy. The quaniy inside he ax expression on he las line in Equaion 7.8) can be rewrien as E [ K = π k p k, T; κ) 1 j ) 1 {N=j} + K 1 φ)j 1 {N =j} p k, T; κ) 1 j ) E [ π k 1 1 φ)j {N =j} + Q [N = j. 7.9) Furherore, since π k = Q[ X = k F M we have 1{N=j} E [ π k 1 [ [ {N =j} = E Q X = k F M = E [ Q [ X = k, N = j F M = Q [X = k, N = j 7.10) where he second equaliy follows fro he fac ha N is F M -easurable since F M = F Y F Z in view of Equaion 3.2.2). Hence, insering 7.10) in 7.9) and using 7.6) and 7.8), we rerieve he following lower bound for E [ Π, T; κ)1 {N<} E [ Π, T; κ)1 {N<} 1 K j=0 p k, T; κ) 1 j ) Q [X = k, N = j + 1 j=0 1 φ)j Q [N = j ) + 7.11)

20 ALEXANDER HERBERTSSON AND RÜDIGER FREY where p k, T; κ) is given by Equaion 7.7). Nex, plugging 7.11) ino 7.5) finally yields he following lower bound for he opion price C 0, T; κ), C 0, T; κ) 1 φ)e r Q [N = 1 + e r j=0 K p k, T; κ) which proves 7.3). 1 j ) 1 Q [X = k, N = j + j=0 ) + 1 φ)j Q [N = j Thus, Proposiion 7.2 esablish a lower bound for he opion price C 0, T; κ) as funcion of he probabiliies Q [X = k, N = j and Q [N = j for each sae k and j = 0, 1,...,. Furherore, we also reark ha he quaniy in he righ hand side of 7.3) does no depend on any of he noise paraeers in he filering odel inroduced in Secion 3. Tha is, he expression in he righ hand side of 7.3) is independen of he apping a : {1, 2,..., K} R l which is used in 3.2.1) o generae he noisy inforaion F Z and he corresponding noisy arke inforaion F M in 3.2.2), which in urn creaes he nonlinear filering odel inroduced in Secion 3. The following corollary o Proposiion 7.2 gives condiions for he possibiliies of having an equaliy in 7.3) insead of an inequaliy. Corollary 7.3. Le C 0, T; κ) be he price oday of an CDS index opion wih srike κ, exercise dae and auriy T. Then here exiss a consan κ such ha for κ κ i holds C 0, T; κ) = 1 φ)e r Q [N = 1 K + e r p k, T; κ) 1 j j=0 where κ is given by and ) 1 Q [X = k, N = j + e r j=0 1 φ) j Q [N = j 7.12) κ = in,...,k κ k. 7.13) κ k = inκ k, 0) for κ k = e ka, T)1 e k B, T)1. Here A, T), B, T) and p k, T; κ) are defined as in Proposiion 7.2. Proof. Firs recall he opion pricing forula 7.2) [ C 0, T; κ) = e r E π [A, T) κb, T) 1 1 N ) ) + 1 φ)n + 7.14) where A, T) and B, T) are given as in Lea 7.1. Fro Theore A.1 in Appendix A, Proposiion 5.2 and Equaion 6.2) in Proposiion 6.1 we conclude ha e k B, T)1 0

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 21 for each sae k. Siilarly, fro he Equaions 6.4) and 6.5) in Proposiion 6.1, we also conclude ha e k A, T)1 0 for every sae k. Therefore, for each k he quaniy e k A, T) κb, T) ) 1 = ek A, T)1 κe k B, T)1 7.15) is he difference of wo posiive expressions when κ 0. Consequenly, for each sae k here is a salles srike spread denoed by κ k bounded below by zero) for which he payoff in 7.15) is non-negaive for all κ κ k. More explici, κ k is given by where κ k is defined as κ k = inκ k, 0) κ k = e ka, T)1 e k B, T)1. Furherore, le κ be κ = in,...,k κ k. In view of κ k defined above, we conclude ha 7.15) is non-negaive for all saes k and all srike spreads κ κ. Consequenly ) π A, T) κb, T) 1 0 a.s. for κ κ 7.16) ha is ) Q [π A, T) κb, T) 1 0 = 1 for κ κ. By using 7.16) we conclude ha he ax expression in 7.14) is superfluous for κ κ and he opion price C 0, T; κ) can hen be rewrien as [ C 0, T; κ) = e r E π [A, T) κb, T) 1 1 N ) ) + 1 φ)n + ) = e r E [π A, T) κb, T) 1 1 N ) + 1 φ)n 7.17) ) = e r E [π A, T) κb, T) 1 1 N ) + e r1 φ) E [N. Furherore, by following siilar seps ) as in he Equaions 7.9)-7.10) we can rewrie he expression E [π A, T) κb, T) 1 ) 1 N as ) E [π A, T) κb, T) 1 1 N ) 1 K = p k, T; κ) 1 j ) Q [X = k, N = j j=0 7.18) where p k, T; κ) is defined as in 7.4). Noe ha e r1 φ) E [N = 1 φ)e r Q [N = + e r 1 j=0 1 φ)j Q [N = j 7.19)

22 ALEXANDER HERBERTSSON AND RÜDIGER FREY and his observaion ogeher wih 7.18) in 7.17) for κ κ hen yields ha C 0, T; κ) = 1 φ)e r Q [N = 1 K + e r p k, T; κ) 1 j j=0 which proves 7.14). ) 1 Q [X = k, N = j + e r j=0 1 φ) j Q [N = j Noe ha Equaion 7.12) in Corollary 7.3 is siply Equaion 7.3) wih a sric equaliy and wihou a ax funcion for he second expression in he righ hand side of 7.3). We herefore conclude ha he inequaliy in 7.3) urns ino a sric equaliy when he srike spread κ saisfies κ κ wih κ defined as in I is rearkable ha we in he noise odel inroduced in Secion 3 for srike spreads κ such ha κ κ have derived an sei-explici expression for he opion price C 0, T; κ) given by 7.12), which is independen of he apping a : {1, 2,..., K} R l used in 3.2.1) o generae he noisy inforaion F Z. Equaion 7.3) in Proposiion 7.2 provides a lower bound for he opion price C 0, T; κ) as funcion of he probabiliies Q [X = k, N = j and Q [N = j for each sae k and j = 0, 1,...,. Furherore, 7.12) in Corollary 7.3 gives an exac expression for opion price C 0, T; κ) when κ κ as funcion of Q [X = k, N = j and Q [N = j. Therefore we will below ouline how o copue he probabiliies Q [X = k, N = j and Q [N = j in our odel presened in Secion 3. Consider a bivariae Markov process H on a sae space S H defined as S H = {1,..., K} {0, 1,..., } 7.20) where S H = K + 1). So each sae j S H can be wrien as a pair j = k, j) where k and j are inegers such ha 1 k K and 0 j. The firs coponen of H belongs o {1,...,K} while he second coponen of H is defined on {1,..., }. The inuiive idea behind he bivariae Markov process H is of course ha he firs coponen of H should iic he facor process X defined in Secion 3.1 while he second coponen of H should represen N, i.e. he nuber of defauled obligors in he porfolio a ie, as defined in previous secions. More specific, for any pair k, j) S H and for any ie poin 0, we wan ha he evens {H = k, j)} and {X = k, N = j} should have he sae probabiliy under he risk-neural easure Q, ha is Q [H = k, j) = Q [X = k, N = j where k, j) S H and 0. 7.21) In view of he above descripion of he bivariae Markov process H we now specify he generaor Q H for H on S H. Firs, we can for a fixed value k, say, of he firs coponen of H consider he second coponen of H as a pure deah process on {0, 1,..., }, i.e. a process which couns he nuber of defauled obligors in he porfolio given ha he underlying econoy is in sae k, ha is X = k. Therefore, for any j = 0, 1,..., 1 he process H can jup fro k, j) o k, j +1) wih inensiy j)λk) where he apping λ is defined as in Equaion??). Recall ha λk) is he individual defaul inensiy when he facor process is in sae k, i.e. X = k. Nex, for a fixed value j, say, of he second

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 23 coponen of H i.e. he nuber of defauled obligors a ie are j) consider wo disinc saes k and k in {0, 1,..., }. Then inspired by he consrucion of he underlying facor process X wih generaor Q, we le he bivariae process H jup fro k, j) o k, j) wih inensiy Q k,k where k k. These are he only allowed ransiions for H. Hence, he generaor Q H for H is hen given by Q H ) k,j),k,j+1) = j)λk) 0 j 1, 1 k K Q H ) k,j),k,j) = Q k,k 0 j 1, 1 k, k K k k Q H ) k,),k,) = 0 1 k, k K and for each pair k, j we also have ha Q H ) k,j),k,j) = k,j ) S H,k k,j j 7.22) Q H ) k,j),k,j ). 7.23) where he oher enries in Q H are zero. In view of his consrucion i is easy o see ha i us hold Q [H = k, j) = Q [X = k, N = j where k, j) S H and 0. 7.24) Le α R K+1) be he iniial disribuion of he Markov process H on he sae space S H wih generaor Q H and consider j S H. Fro Markov heory we know ha Q [H = j = αe Q H e j, 7.25) where e j R K+1) is a colun vecor where he enry a posiion j is 1 and he oher enries are zero. Furherore, e Q H is he arix exponenial which has a closed for expression in ers of he eigenvalue decoposiion of Q H. Thus, in view of 7.24) and 7.25) we have for any j = k, j) S H and ha Q [X = k, N = j = αe Q H e k,j). 7.26) So 7.26) provides us wih an efficien way o copue he probabiliies Q [X = k, N = j for any 0 and any pair j = k, j) where k and j are inegers such ha 1 k K and 0 j. Noe ha here exis over 20 differen ways o copue he arix exponenial, for ore on his see e.g in Moeler & Loan 1978) and Moeler & Loan 2003). Since Q [N = j = K Q [X = k, N = j we rerieve ha K Q [N = j = αe Q H e k,j) = αe Q H e,j) 7.27) where e,j) R K+1) is a colun vecor defined as e,j) = K e k,j). Finally, le us specify he he iniial disribuion α R K+1) of he Markov process H on he sae space S H, defined as in 7.20). Firs, le α be he iniial disribuion of he process X defined in Secion 3.1. Then α k = Q [X 0 = k bu we also know ha π0 k = Q [ X 0 = k F0 M = Q [X0 = k = α k 7.28)

24 ALEXANDER HERBERTSSON AND RÜDIGER FREY which gives a relaion beween he values π k 0 and α k. Nex, le us now he iniial disribuion α R K+1). We assue ha all obligors in he porfolio are alive non-defauled) a ie = 0, i.e. oday, which iplies ha he second coponen us be zero for all saes of he econoy background process odelled by he firs coponen of he bivariae Markov process. Hence, i us hold ha K α k,0) = 1 and α k,j) = 0 for j = 1, 2,..., 7.29) which in urn guaranees ha he su of he enries in α are one. By using he forulas 7.26), 7.27) and 7.29) in 7.3) we can efficienly copue nuerical values for he lower bounds of he opion price C 0, T; κ). Siilarly, using 7.26), 7.27) and 7.29) in 7.12) will render exac values of he opion price C 0, T; κ) when κ κ where κ is defined as in 7.13). 8. Calibraing he filering odels In his secion we very briefly discuss how o esiae or calibrae he paraeers in he filering odel inroduced in Secion 3. One can esiae he paraeers θ = Q, λ, a) in several differen ways. In he paper Herbersson & Frey 2014) he auhors ouline a novel approach for esiaing he paraeers θ = Q, λ, a) in he filering odels by using ieseries daa on a CDS index and classical axiu-likelihood algorihs. This calibraionapproach naurally incorporaes he Kushner-Sraonovich SDE for he dynaics of he filering probabiliies. The copuaionally convenien forulas for he forward saring CDS index are a prerequisie for our esiaion algorih. One can easily exend he ehodology presened in his secion o also include arke quoes on sandardized CDO ranches. In paricular, Herbersson & Frey 2014) uses one source of randoness, i.e l = 1 in he noise and hen chooses he apping a : {1, 2,..., K} R l in 3.2.1) o be on he for ak) = c ln λk) for a consan c. Thus, he paraeers θ o be esiaed are hen given by θ = Q, λ, c). Furherore, he paper Herbersson & Frey 2014) also oulines alernaive ehods o esiae, such as using quadraic prograing. We refer he reader o Herbersson & Frey 2014) for ore deails on all of he above opics. There exiss oher papers using MLE echniques in credi risk odelling and we refer o Herbersson & Frey 2014) for ore references. In his paper we will no discuss esiaion and calibraion of he paraeers. Insead we will use focus on nuerical sudies of he price C 0, T; κ) for an CDS index opion in he filering odel presened in Secion 3. In pariculars we will use Mone-Carlos siulaions sudy C 0, T; κ) as funcions of he paraeers θ = Q, λ, c). Furherore, we will also invesigae how far away he siulaed prices will deviae fro he lower bound of he price derived in Proposiion 7.2. A nuerical sudy of when he inequaliy in Proposiion 7.2 urns o he equaliy in given by Corollary 7.3 will also be perfored,.

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 25 9. Nuerical sudies In his we will copue he price C 0, T; κ) for an CDS index opion in he filering odel presened in Secion 3. We will use Mone-Carlos siulaions sudy C 0, T; κ) as funcions of he paraeers θ = Q, λ, c). Furherore, we will also invesigae how far away he siulaed prices will deviae fro he lower bound of he price derived in Proposiion 7.2. A nuerical sudy of when he inequaliy in Proposiion 7.2 urns o he equaliy in given by Corollary 7.3 will also be perfored. The nuerical paraeers θ = Q, λ, c) will be exogenously given and we will no discuss esiaion and calibraion of he paraeers. This is insead sudied in he paper Herbersson & Frey 2014). We will however use soe CDS index opion prices generaed fro he Black-Scholes fraework by using he buil in rouine CDS opions in Blooberg. These prices will in urn be uilized o calibrae our lower bounds in order o ge soewha realisic paraeers for Q and λ. 9.1. Copuing prices of opions on he CDS index. In his subsecion we copue he price of an opion on he Iraxx Europe CDS index wih paraeers. By using our paraeers in he forula 7.1) in Lea 7.1 we can perfor Mone Carlo siulaions in order o find proxies for he price of opions on a CDS index. 0.038 =9 onhs, T =5 years, S0,5)=90 bp, π 0 1 =83%, θ MLE as in previous slides, 10 3 MC siulaions CDS index call opion prices 0.036 CDS index call opion prices 0.034 0.032 0.03 0.028 0.026 60 80 100 120 140 160 180 200 Srike spreads in bps) Figure 2. Prices of opions on CDS index wih differen srikes.

26 ALEXANDER HERBERTSSON AND RÜDIGER FREY Figure 2 displays he price of an opion on he Iraxx Europe CDS index wih paraeers given by. The ie o auriy is = 9 onhs, he auriy is T = 5 years, he spo spread S0, 5) is given by S0, 5) = 90 bp and he nuber of MC-siulaions are 10 3. The srike spread K runs fro 60 bp o 200 bp in seps of 10 bp. References Bielecki, T. R. & Rukowski, M. 2001), Credi Risk: Modeling, Valuaion and Hedging., Springer, Berlin. Bréaud, P. 1981), Poin Processes and Queues. Maringale Dynaics, Springer-Verlag, Berlin. Capponi, A. & Cvianic, J. 2009), Credi risk odeling wih isreporing and incoplee inforaion, Inernaional Journal of Theoreical and Applied Finance 121), 83 112. Davis, M. H. A. & Marcus, S. I. 1981), An inroducion o nonlinear filering. M. Hazewinkel and J.C. Willes, eds, Sochasic Syses: The Maheaics of Filering and Idenificaions and Applicaions, Reidel Publishing Copany, pp.53-75. Folland, G. B. 1999), Real Analysis. Modern Techniques and Their Applicaions, John Wiley & Sons, New York. Fonana, C. & Runggaldier, W. J. 2010), Credi risk and incoplee inforaion: filering and e paraeer esiaion, Inernaional Journal of Theoreical and Applied Finance 135), 683 715. Frey, R. & Runggaldier, W. 2010), Pricing credi derivaives under incoplee inforaion: a nonlinearfilering approach, Finance and Sochasics 144), 495 526. Frey, R. & Schid, T. 2012), Pricing and hedging of credi derivaives via he innovaions approach o nonlinear filering, Finance and Sochasics 161), 105 133. Graziano, G. D. & Rogers, L. C. G. 2009), A dynaic approach o he odeling of correlaion credi derivaives using Markov chains, Inernaional Journal of Theoreical and Applied Finance 121), 45 62. Herbersson, A. & Frey, R. 2014), Paraeer esiaion in credi odels under incoplee inforaion. forhcoing in Counicaions in Saisics - Theory and Mehods. Jackson, A. 2005), A new ehod for pricing index defaul swapions. Working paper. Lando, D. 1998), On cox processes and credi risky securiies., Review of Derivaives Research 2, 99 120. Marin, R. J. 2012), A cds opion iscellany. working paper. McNeil, A. J., Frey, R. & Ebrechs, P. 2005), Quaniaive Risk Manageen, Princeon Universiy Press, Oxford. Moeler, C. & Loan, C. V. 1978), Nineeen dubioius ways o copue he exponenial of a arix., SIAM Reviw 204), 801 836. Moeler, C. & Loan, C. V. 2003), Nineeen dubioius ways o copue he exponenial of a arix, wenyfive years laer, SIAM Reviw 451), 3 49. Morini, M. & Brigo, D. 2011), No-arageddon easure for arbirage-free pricing of index opions in a credi crises, Maheaical Finance 214), 573 593. Pedersen, C. M. 2003), Valuaion of porfolio credi defaul swapions. Quaniaive Credi Research Lehan Brohers). Rogers, L. C. G. & Willias, D. 2000), Diffusions, Markov Processes and Maringales. Volue 1. Foundaions. Second Ediion, Cabridge Universiy Press, Cabridge. Rukowski, M. & Arsrong, A. 2009), Valuaion of credi defaul swapions and credi defaul index swapions, Inernaional Journal of Theoreical and Applied Finance 127), 1027 1053. Appendix A. Feyan-Kac forulas for finie-sae Markov chains The purpose of his secion is o inroduce soe useful forulas, ha will be used hroughou his paper. We firs inroduce soe noaion. Le S X = {1, 2,..., K} and

PRICING CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION 27 consider a funcions fx) : S X R, hen we denoe f a R K -colun vecor wih f j = fj) for j S X. Nex, le X be a finie sae Markov jup process on S X = {1, 2,..., K} wih generaor Q. Then, for a funcion λx) : S X R we denoe he arix Q λ = Q I λ where I λ is a diagonal-arix such ha I λ ) k,k = λk) and e x is a row vecor in R K where he enry a posiion x is 1 and he ohers enries are zero. We now sae wih he following resul. Theore A.1. Le X be a finie sae Markov jup process on S X = {1, 2,..., K} wih generaor Q. Consider funcions λx), fx) : S X R. Then, wih noaion as above E [e 0 λxs)ds fx ) X 0 = x = e x e Q λ T ) f. A.1) A proof of Proposiion A.1 can be found on pp.273-274 in Rogers & Willias 2000). I is easy o exend Theore A.1 o yield he following equaliy, for T E [e T λx T )ds fx T ) X = x = e x e Q λ T ) f A.2) where he res of he noaion are as in Theore A.1. The ain poin in Theore A.1 is ha given he arix Q λ, hen he lef-hand side in Equaion A.1) and Equaion A.2) is sraighforward o ipleen using sandard aheaical sofware. We noe ha Theore A.1 does no hold if he funcions λ, f also depend on ie, i.e λ, x), f, x) : [0, ) S X R. In such cases, [ one generally has o rely on nuerical ODE ehod in order o find he quaniy E e 0 λs,xs)ds f, X ) X 0 = x. Alexander Herbersson), Cenre For Finance, Deparen of Econoics, School of Business, Econoics and Law, Universiy of Gohenburg, P.O Box 640, SE-405 30 Göeborg, Sweden E-ail address: Alexander.Herbersson@econoics.gu.se Rüdiger Frey), Insiue for Saisics and Maheaics, Vienna Universiy of Econoics and Business, Augasse 2-6, A-1090 Vienna, Ausria E-ail address: ruediger.frey@wu.ac.a