Hedging portfolio loss derivatives with CDSs

Transcription

1 Hedging porfolio loss derivaives wih CDSs Areski Cousin and Monique Jeanblanc November 3, 2 Absrac In his paper, we consider he hedging of porfolio loss derivaives using single-name credi defaul swaps as hedging insrumens. The hedging issue is invesigaed in a general pure jump dynamic seing where defaul imes are assumed o admi a join densiy. In a firs sep, we compue defaul inensiies adaped o he global filraion of defauls. In paricular, we sress he impac of a defaul even on he price dynamics of non-defauled names. In a wo defauls seing, we also fully describe he hedging of a loss derivaive wih single name insrumens. The mehodology can be applied recursively o he case of a mulidefaul seing. We compleely characerize he hedging sraegies for general n-dimensional credi porfolios when defaul imes are assumed o be ordered. The compuaion of he hedging sraegies does no require any Markovian assumpion. Inroducion The hedging of loss derivaives such as CDO ranches or baske defaul swaps is a prominen risk-managemen issue especially given he recen revisions o he Basel II marke risk framework, Dec 2. Indeed, according o [], correlaion rading porfolios including ranches on sandard indexes and heir associaed liquid hedging posiions will coninue o be charged as hedge-ses under inernal VaR-based mehod. The pracice of hedging is sill recognized as a risk miigaion echnics for hese correlaion producs as far as compuaion of rading book capial requiremen is concerned. As a resul, he performance and efficiency of underlying hedging mehods will have a direc impac on he amoun of capial required for loss derivaives. Cousin and Lauren 2 discuss various issues relaed o he use of models in designing hedging sraegies for CDO ranches and back-esing or assessing hedging performance. In his paper, we consider he hedging of loss derivaives using single-name credi defaul swaps as hedging insrumens. The hedging issue is invesigaed in a general pure jump seing where defaul imes are assumed o admi a join densiy which is he only inpu of he model so ha our resuls can be considered as model independen and we compue defaul inensiies adaped o he global filraion of defauls. We check ha, if CDSs on each defaul are raded, he marke is complee. The hedging sraegies can be found by idenifying he erms associaed wih he fundamenal defaul maringales. This research is a par of CRIS programm Déparemen de Mahémaiques, Equipe analyse e probabilié, Universié d Evry Déparemen de Mahémaiques, Equipe analyse e probabilié, Universié d Evry; Insiu Europlace de Finance

2 We exend some recen resuls by Lauren, Cousin and Fermanian 2 and Cousin, Jeanblanc and Lauren 2. In paricular, we sress he impac of a defaul even on he price dynamics of non-defauled names. Moreover, in a wo defauls seing, we fully describe he hedging of a loss derivaive wih single name insrumens. The generalizaion o a mulidefaul seing can be done following he same mehodology. Furhermore, we are able o compleely characerize he hedging sraegies in single-name CDS for general n-dimensional credi porfolios when defaul imes are assumed o be ordered. The compuaion of he hedging sraegies does no require any Markovian assumpion. The paper is organized as follows. The firs secion aims a presening he general seing of he model and i recalls he predicable represenaion heorem. In he second secion, we invesigae he case where only one name is considered. In paricular, we exhibi he inensiy of he defaul ime and he dynamics of CDS prices. Secion hree is devoed o he case where he credi porfolio is composed of wo names. The exension o a mulivariae seing can be done using a recursive procedure. In paricular, we highligh he conagion effec occurring a defaul ime of one of he wo names on he CDS price dynamics of he oher name. We also compue he dynamics of he hedging sraegies a any ime in all he possible defaul siuaions. In Secion four, we consider he hedging of a loss derivaives wrien on a general n-dimensional porfolio. We sress ha when defaul imes are assumed o be ordered, i.e., CDS are kh-o-defaul swaps, he hedging sraegies can be simply characerized as he soluions of a linear sysem. Mahemaical ools: he general case In wha follows, we consider n defaul imes τ i, i =,..., n, ha is, non-negaive and finie random variables consruced on he same probabiliy space Ω, G, P. For any i =,..., n, we denoe by H i = {τi }, he i-h defaul process, and by H i = σh i s, s he naural filraion of H i afer compleion and regularizaion on righ. We inroduce H, he filraion generaed by he processes H i, i =,..., n, defined as H = H... H n, i.e., H = n i= Hi afer regularizaion on righ. We assume ha G,..., n := Pτ >,..., τ n > n is wice differeniable wih respec o,..., n and ha G and is derivaives do no vanish. Then, as we shall prove in he nex secion, for any i =,..., n, here exiss a non-negaive H-adaped process λ i, such ha he process M i := H i is an H-maringale. The process λ i is called he H-inensiy of τ i. This process vanishes afer τ i oherwise, afer τ i, he maringale M i would be coninuous and decreasing and can be wrien λ i = H i λ i for some H... H i H i+... H n -adaped process λ i. In erms of he process λ i, one has τi M i = H i λ i sds = H i λ i sds H i s λ i sds. In paricular, denoing by τ i, i =,..., n, he ranked sequence of defaul imes, he process λ is a deerminisic funcion on he ime inerval [, τ [ i.e., λ = λ where λ is deerminisic, a deerminisic funcion evaluaed a ime τ on he ime inerval [τ, τ 2 [ i.e., λ = λ,2, τ where λ,2 is deerminisic, and a deerminisic funcion evaluaed a imes τ j, j i on he ime inerval [τ i, τ i+ [. In paricular, he 2

3 value of he inensiy depends no only of he number of defaul occurred in he pas, bu also on he imes where he defauls have aken place, which is more realisic. The following predicable represenaion heorem holds rue see Brémaud [5]. Theorem. Le B H T be an inegrable random variable. Then, here exis H-predicable processes ϑ i, i =,..., n such ha T B = EB + ϑ i sdms i, i= and E T ϑi s λ i sds <. Moreover, if B is square inegrable, hese processes are unique in he class of processes which saisfy E T ϑi s 2 λ i sds <. Due o he inegrabiliy assumpion and he predicable propery of he ϑ s, he processes ϑi sdm i s are H-maringales. In wha follows, for a bounded variaion lef-coninuous process A, we denoe T... da s for,t ]... da s. We shall in he firs par presen compuaions for he inensiy in erms of he densiy of τ in he case n =. Then, we shall sudy he case n = 2 and we deermine he hedging sraegy of any payoff, when he hedging insrumens are CDSs. The mehodology can be easily exended o oher hedging insrumens, as defaulable zero-coupons, digial CDSs. The mulidefaul case can be sudied along he same lines. For simpliciy, we resric our aenion o he paricular siuaion of ranked imes. 2 The single defaul case In his secion, we presen some well known resuls concerning he dynamics of a CDS wrien on a single defaul, working in he filraion H. As we shall see in he nex secion, he dynamics of a CDS wih he same recovery will be differen in he filraion aking ino accoun he knowledge of oher defauls. 2. Some imporan maringales We recall some well known resuls see Ellio [], Dellacherie [9] and Bielecki and Rukowski [2]. Here, τ is a non-negaive random variable on he probabiliy space Ω, G, P wih survival funcion G := Pτ > = Pτ = F where F is he cumulaive disribuion funcion of τ. We assume ha G >,, and ha G is differeniable, i.e., ha τ admis a densiy f, so ha G = f. The filraion is H = H. Proposiion 2. For any inegrable random variable X and for any Borelian bounded funcion h {<τ} E P X H = {<τ} G E PX {<τ} E P hτ H = {τ } hτ {<τ} G hudgu. 3

4 The process M, defined as τ fs M = H Gs ds = H H s fs Gs ds is a P, H-maringale. In oher erms, he H-inensiy of τ is H λ where λ is he deerminisic funcion λ = f G. Noe ha he survival probabiliy G can be expressed in erms of he deerminisic funcion λ: indeed we have proved ha λ = f/g = G /G. Solving his ODE wih iniial condiion G = leads o G = Pτ > = exp λu du. Noe ha λd = Pτ d τ >. The defaul inensiy can be inerpreed as he insananeous condiional defaul probabiliy given ha defaul has no ye occurred. 2.2 Price of a radiional single-name CDS We assume ha P is he pricing measure. We denoe by B he savings accoun, henceforh he price process of any radeable securiy, paying no coupons or dividends, is a P, H-maringale, when discouned by B. The ex-dividend price of an asse paying dividends is T B E Bs dd s H where D represens he cumulaive dividend. In ha case, he discouned cumulaive dividend price V cum is such ha T V cum B = E is a maringale. As usual, B is given by B = exp B s dd s H + r u du, R +, where he shor-erm ineres rae r is here a deerminisic process. Bs dd s Le us recall ha a credi defaul swap is a bilaeral conrac involving a proecion seller and a proecion buyer. We consider a CDS mauring a ime T. If a defaul even occurs a ime τ < T, hen he proecion seller delivers o he proecion buyer he unrecovered porion of he loss δτ where δ is a deerminisic funcion. As for he premium leg, we assume for simpliciy ha he fee is paid o he proecion seller in More precisely, he quaniy δτ is equal o he loss given defaul associaed wih he reference eniy imes he CDS noional amoun. 4

5 coninuous ime, i.e., he amoun κd is paid by he proecion buyer during he ime inerval d, ill ime τ T. The ime- marke value of a CDS wih paymen a defaul δ and wih a conracual spread κ is equal o V κ = D κp, where D and P, he defaul leg and he premium leg, are given by D = B E Bτ δτ {<τ T } H T τ P = B E Bu du H τ and he cumulaive dividend price is V cum κ = B E In he case of a zero ineres rae, Bτ δτ {τ T } κ T τ B u du H. V κ = {<τ} Eδτ {τ T } κt τ H, V cum κ = Eδτ {τ T } κt τ H. I is worhwhile o noe ha he ex-dividend price is no a maringale under he pricing measure, despie he fac ha he ineres rae is null. However, he cumulaive dividend price is a maringale, his will be useful laer on. In wha follows, we resric our aenion o he case of nil ineres rae. We recall a well know resul see, e.g., [4]. Proposiion 2.2 The price a ime [, T ] of a credi defaul swap wih spread κ is V κ = {<τ} Ṽ κ, [, T ], where Ṽκ is a deerminisic funcion associaed wih he pre-defaul value of he CDS and equals Ṽ κ = T T δu dgu κ Gu du. G Proof. From Proposiion 2., we have, on he se { < τ}, V κ = = T G δu dgu κ G T T u dgu + T GT G T δu dgu κ T GT G u dgu. where, in he las equaliy, we have used an inegraion by pars o obain T Gu du = T GT G T u dgu. 5

6 2.3 Dynamics of CDS Prices in a single defaul seing Here, we compue he dynamics of he CDS s price. I is useful see [3] o obain he hedging sraegy of a defaulable claim based on CDS and savings accoun. Proposiion 2.3 The dynamics of he ex-dividend price V κ on [, T ] are dv κ = V κ dm + H κ δ λ d, where he P, H-maringale M is given in Proposiion 2.. Proof. I suffices o noe ha V κ = H Ṽκ wih Ṽ κ given in Proposiion 2.2, so ha, using inegraion by pars formula, dv κ = H dṽκ Ṽ κ dh. Using he explici expression of Ṽκ, we find easily ha we have dṽκ = λṽκ d + κ δ λ d. The SDE for V follows. Commen 2. I is well known ha he risk neural dynamics of a dividend paying asse is ds = dm δ d, where m is a maringale and δ is he dividend rae. Here, he premium κ is similar o a dividend o be paid up o ime, hence he quaniy κ H d appears. The δ can be inerpreed as a dividend o be received, a ime, wih probabiliy λd. A defaul ime, he price jumps from V τ κ o, as can be seen in he righ-hand side of he dynamics. Corollary 2. The dynamics of he cum-dividend price V cum on [, T ] are dv cum κ = δ V κ dm. 2 Proof. The cumdividend price is The resul follows. V cum κ = V κ + {<τ} δτ κ τ = V κ + δsdh s κ H κsdh s. 3 Two defaul imes Le us firs sudy he case wih wo random imes τ, τ 2. For i =, 2, we denoe by H i, he defaul process associaed wih τ i. The filraion generaed by he process H i is denoed H i and he filraion generaed by he wo processes H, H 2 is H = H H 2. Noe ha, since H i = στ i, an H H 2 -measurable random variable is 6

7 a consan on he se { < τ τ 2 }, a στ τ 2 -measurable random variable on he se {τ τ 2 < τ τ 2 }, i.e., a στ -measurable random variable on he se {τ < τ 2 }, and a στ 2 -measurable random variable on he se {τ 2 < τ }. We recall ha a στ -measurable random variable is a Borel funcion of τ. a στ, τ 2 -measurable random variable i.e., a Borel funcion hτ, τ 2 on he se {τ τ 2 }. To summarize, for fixed, any H H 2 -measurable random variable Z admis a represenaion as Z = h {<τ τ 2} + h τ {τ <τ 2} + h 2 τ 2 {τ2 <τ } + hτ, τ 2 {τ τ 2 }. We denoe by G, s = Pτ >, τ 2 > s he survival probabiliy of he pair τ, τ 2 and we assume ha his funcion is wice differeniable. We denoe by i G, he parial derivaive of G wih respec o he i-h variable, i =, 2 and by,2 G, he second order parial derivaive of G. The densiy of he pair τ, τ 2 is denoed by f. Simulaneous defauls are precluded in his framework, i.e., Pτ = τ 2 =. Even if he case of wo defaul imes is more involved, closed form expressions for he inensiies are available. I is imporan o ake ino accoun ha he choice of he filraion is very imporan. Indeed, in general, an H maringale is no an H H 2 maringale. We shall illusrae his imporan fac below. 3. Inensiies We presen he compuaion of maringales associaed wih defaul imes τ i, i =, 2, in differen filraions. In paricular, we shall obain he compuaion of he inensiies in various filraions. 3.. Filraion H i We sudy, for any fixed i, he Doob-Meyer decomposiion of he sub-maringale H i in he filraion H i. In oher erms, we compue he H i -compensaor of H i. From Proposiion 2., he process M i τi := H i f i s ds 3 G i s is an H i -maringale. Here, G i s = F i s = Pτ i s = s f iudu. Hence, he process H i fi G i is he H i -inensiy of τ i. Noe ha, hanks o Theorem., any H i -maringale can be wrien as a sochasic inegral wih respec o M i Filraion H We recall a resul proved in Bielecki e al. [4]. Proposiion 3. The process M defined as τ τ 2 M := H Gs, s τ Gs, s ds,2gs, τ 2 τ τ 2 2 Gs, τ 2 ds 7

8 is an H-maringale. The process M 2 defined as is an H-maringale. M 2 := H 2 τ τ 2 2 Gs, s τ2 Gs, s ds,2gτ, s τ τ 2 Gτ, s ds Proof. The proof relies on some Iô s calculus o obain he Doob-Meyer decomposiion of Qτ > H 2. We refer he reader o [4] for deails. This means ha he H-inensiy of τ akes ino accoun he knowledge of τ 2 and is equal o he deerminisic funcion G, G, on he se < τ τ 2 and o he random quaniy ϕ, τ 2 where ϕ, s =,2G,s 2G,s on he se τ 2 < τ. In a closed form, he processes H i λi sds, i =, 2, are H-maringales, where λ = H H 2 G, H 2,2 G, τ 2 G, 2 G, τ 2 Here = H H 2 λ + H H 2 λ 2 τ 2, λ 2 = H 2 H 2G, H,2 Gτ, G, Gτ, = H H 2 λ 2 + H H 2 λ 2 τ. λ i = ig, G,, 4 λ 2 s =,2G, s f, s = 2 G, s 2 G, s, 5 λ 2 s =,2Gs, fs, = Gs, Gs,. Noe ha he minus signs in he value of he inensiy are due o he fac ha G is decreasing wih respec o is componens, hence he firs derivaives are non-posiive and he second order derivaive,2 G equal o he densiy of he pair τ, τ 2 is non-negaive. The quaniy λ d is equal o Pτ d τ τ 2 >, ha is he probabiliy ha τ occurs in he ime inerval [, + d], knowing ha neiher τ nor τ 2 have occurred before ime. The quaniy 2 λ s = f,s evaluaed a s = τ 2G,s 2, represens he value of he defaul inensiy process of 2 τ wih respec o he filraion H on he even {τ 2 < }. This quaniy λ sd is also he probabiliy ha τ occurs in he ime inerval [, + d], knowing ha τ has no occurred before and ha s = τ 2. Since we are working in he filraion 2 H H 2 for boh maringales M i, he compensaed maringale of he couning process H = H + H 2 = 2 i= {τ i } is M := H λ sds where λ = λ + λ 2 2 The sum of wo maringales in he same filraion is a maringale. 8

9 = H H 2 G, + 2 G, G, H H 2,2 G, τ 2 2 G, τ 2 H2 H,2 Gτ, 2 Gτ,. 3.2 Dynamics of prices of defaul coningen claims In his secion, our aim is o find he dynamics of he price of a coningen claim wih payoff hτ, τ 2. This conains in paricular he case of firs or second o defaul claim, wih payoff associaed wih hu, v = {u<v} ϕu or hu, v = {u<v<t } ψv. The goal is o find he dynamics of Z := Ehτ, τ 2 H. wih The firs sep is o prove ha Z = hτ, τ 2 H H 2 + ψ, τ, H H 2 + ψ,, τ 2 H 2 H + H H 2 ψ, ψ, u, := ψ,, v := ψ, := Gu, 2 G, v G, du hu, vfu, vdv, hu, vfu, vdu, dvhu, vfu, v. The proof follows from ieraive condiioning and use of Proposiion 2.. We leave he deails o he reader. One noes ha, on he one hand, for any funcion φ, H H 2 φτ, τ 2 = H φτ, = dhu φu, dh u, dh 2 v φu, v, so ha, using inegraion by pars formula and re-arranging he erms h, dz = τ2 ψ,, τ 2 H 2 + ψ,, ψ, H 2 hτ +, ψ, τ, H + ψ,, ψ, H + H 2 2 ψ, u, dh u + H + H H 2 d d ψ, d. On he oher hand, one checks ha, wih easy compuaion, ha ψ,, v = λ 2, v ψ,, v h, v, ψ,, vdh 2 v dh dh 2 9

10 2 ψ, u, = λ 2 u, ψ, u, hu,, d d ψ, = λ + λ 2 ψ, + G, G, ψ,, + 2 G, ψ,,. I follows ha dz = h, τ2 ψ,, τ 2 H 2 + ψ,, ψ, H 2 dm hτ +, ψ, τ, H + ψ,, ψ, H dm Dynamics of CDS prices Le us now examine he valuaion of a single-name CDS wrien on name, in he case of null ineres rae. Our aim is o show ha he dynamics of his CDS will be affeced by he informaion on τ 2 : when τ 2 occurs, he inensiy of τ changes, and his will change he parameers of he price dynamics. We reproduce some resuls appearing in Bielecki e al. [4]. We consider a CDS wih a consan spread κ which delivers δτ a ime τ if τ < T, where δ is a deerminisic funcion. The value of he CDS akes he form V κ = Ṽκ {<τ τ 2} + V κ {τ τ 2 <τ }. Firs, we resric our aenion o he case < τ τ 2. Proposiion 3.2 On he se { < τ τ 2 }, he value of he CDS is Ṽ κ = T T δu Gu, du κ Gu, du. G, Proof. The value V κ of his CDS, compued in he filraion H, i.e., aking care on he informaion on he second defaul conained in ha filraion, is V κ = {<τ}e δτ {τ T } κ T τ H Le us denoe by τ = τ τ 2 he firs defaul ime. Then, {<τ} V κ = {<τ} Ṽ κ, where Ṽ κ = = = Pτ > E δτ {τ T } {<τ} κ T τ {<τ} G, E δτ {τ T } {<τ} κ T τ {<τ} T δupτ du, τ 2 > G,

11 T κ u Pτ du, τ 2 > T κ Pτ du, τ 2 >. In oher erms, using inegraion by pars formula, we end up wih Ṽ κ = T T δu Gu, du κ Gu, du. G, T Proposiion 3.3 On he even {τ 2 < τ }, he CDS price is given by V κ = V 2 V 2 s = T T δufu, s du κ 2 Gu, s du 2 G, s. τ 2 where Proof. One has V κ = V κ = E δτ {τ T } κ T τ στ 2 T T = δufu, τ 2 du κ 2 Gu, τ 2 du. 2 G, τ 2 In he financial inerpreaion, V 2 s is he marke price a ime of a CDS on he firs credi name, under he assumpion ha he defaul τ 2 occurs a ime s and he firs name has no ye defauled recall ha simulaneous defauls are excluded, since we have assumed ha G is differeniable. The price of a CDS is V = V κ {<τ2 τ } + V κ {τ2 τ <τ }. Differeniaing he deerminisic funcion which gives he value of he CDS, we obain dṽκ = λ + λ 2 Ṽ κ + κ λ δ λ 2 V 2 d, where for i =, 2 he funcion λ i is he deerminisic pre-defaul inensiy of τ i given in 4 and λ 2 d V κ = τ 2 V κ δ + κ d where λ 2 u is given in 5. Proposiion 3.4 The price of a CDS follows dv κ = H H 2 κ δ λ d + H H 2 κ δ λ 2 τ 2 d V κ dm + H V 2 V κ dm 2. 6

12 Proof. obains Using inegraion by pars formula for V κ = Ṽκ H H 2 + V κ H H 2, one dv κ = H H 2 dṽκ + H H 2 d V κ V κ dh + H V 2 Ṽκ dh 2, which leads o he resul afer ligh compuaions. Commen 3. As for a single name CDS, he quaniy δ λ corresponds o he dividend δ o be paid a ime wih probabiliy λ d on he se < τ τ 2 and δ λ 2 corresponds o he dividend δ o be 2 paid a ime wih probabiliy λ d on he se τ 2 < < τ. The quaniy V 2 Ṽ represens he jump in he value of he CDS, when defaul τ 2 occurs a ime. The cumulaive dividend price of he CDS is I follows ha hence, since he cumulaive price is a maringale V cum κ = Eδτ {τ T } κ T τ H. dv cum κ = dv κ + δdh κ H d, dv κ = dm δλ d + κ H d, where dm = dv cum κ δdm. This is an easy way o obain he drif erm in Equaion 6. I urns ou ha he cum-dividend CDS price process has he following dynamics dv cum κ = H H 2 κ δ λ d 3.4 CDSs as hedging asses + H H 2 κ δ λ 2 d V κ dm + H V 2 V κ dm 2 + δdh κ H d = δ V κ dm + H V 2 V κ dm 2. Assume now ha a CDS wrien on τ 2 is also raded in he marke. We denoe by δ i, i =, 2 he recovery assumes o be deerminisic and V i, i =, 2 he prices of he wo CDSs wih spreads κ i. We assume ha hese CDSs are raded in he marke. Since he CDS are paying dividends, a self financing sraegy consising in ϑ i shares of CDS s has value X = ϑ + ϑ V + ϑ 2 V 2 and dynamics dx = ϑ dv,cum + ϑ 2 dv 2,cum = ϑ δ V dm + H V 2 Ṽ dm 2 +ϑ 2 δ 2 V dm H 2 V 2 Ṽ 2 = ϑ δ V + ϑ 2 H 2 V 2 Ṽ 2 dm + ϑ H V 2 Ṽ + ϑ 2 δ 2 V 2 dm dm 2 2

13 The posiion ϑ in he savings accoun which is worh a consan in his zero ineres-rae se-up is necessary o make he sraegy self-financing. Noe ha, due o liquidiy issue, one needs o use rolling CDS-s in pracice so o consruc marke feasible hedging porfolio. We refer he reader o [3] for more deails on rolling-cds. Mahemaically, here is lile difference beween porfolios consising of CDS-s 3, and porfolios consising of rolling CDS-s, so porfolio consising of CDS-s is chosen for illusraion purpose. Le A H T be a erminal payoff wih price A = EA H, hen from Theorem. here exis predicable processes π and π 2 such ha A = EA + π sdm s + In order o hedge ha claim, i remains o solve he linear sysem π 2 sdm 2 s. Hence, on he se < τ τ 2, noing ha V i ϑ δ V + ϑ 2 H 2 V 2 Ṽ 2 = π, ϑ H V 2 Ṽ + ϑ 2 δ 2 V 2 = π 2. = Ṽ i on ha se, ϑ = ϑ 2 = π δ 2 Ṽ 2 π 2 V 2 Ṽ 2 δ Ṽ δ 2 Ṽ 2 V 2 Ṽ V 2 Ṽ 2, π 2 δ Ṽ π V 2 Ṽ δ Ṽ δ 2 Ṽ 2 V 2 Ṽ V 2 Ṽ 2. On he se τ < < τ 2 ϑ = π δ 2 V 2 π 2 V 2 V 2 δ V δ 2 V 2, ϑ 2 = π 2 δ 2 V 2. On he se τ 2 < < τ On he se τ τ 2 < ϑ = π δ V ϑ =, ϑ 2 = π2 δ V π V 2 V 2 δ V δ 2 V 2 π δ V, ϑ 2 = π 2 δ 2 V 2. As we saw above, for he case A = hτ, τ 2, one has a closed form for he coefficiens π: π = h, τ 2 ψ,, τ 2 H 2 + ψ,, ψ, H 2, π 2 = hτ, ψ, τ, H + ψ,, ψ, H. 3 A rolling-cds has a fixed mauriy T and a ime-dependen conracual spread equal o he curren CDS marke spread.. 3

14 3.5 Mulidefaul seing The same mehodology can be applied in a mulidefaul seing. The only difficuly is ha one has o rank he defauls and o disinguish he various possibiliies. For example, for hree defauls, he dynamics of he cumdividend price of he CDS wrien on he defaul τ is dv cum = δ V dm + H H 3 V 2 + H 3 V 32, τ 3 V dm 2 + H H 2 V 3 + H 2 V 23 τ 2, V dm 3, where V is he ex-dividend price of he CDS a ime, V 2 u is he price of he CDS on he se {τ 2 = u} for u < < τ 3, V 3 v is he price of he CDS on he se {τ 3 = v} for v < < τ 2, V 23 u, v is he price of he CDS, on he se {τ 2 = u, τ 3 = v} for u < v < and V 32 u, v is he price of he CDS, on he se {τ 2 = u, τ 3 = v} for v < u <. 4 Paricular case: ordered defauls In his secion, we consider he paricular case where defaul imes are ordered, i.e., τ < τ 2 <... < τ n. Recall ha simulaneous defauls are precluded in our seing. Then, single-name CDS k offers credi proecion agains he kh defaul occurring in he porfolio so ha i can be viewed as a kh-o-defaul swap. We firs consider a seing wih wo names only, hen we invesigae he hedging of loss derivaives wrien on a mulivariae n-dimensional credi porfolio. 4. Dynamics of CDS prices in a wo defauls seing Le us now assume ha τ < τ 2, a.s. In ha case, G, s = G, for s, hence he maringale M defined in Proposiion 3. simplifies: τ M = H Gs, s τ Gs, s ds = f s H G s ds where G s = Pτ > s = Gs, s = s f udu = s Gu, udu. The H-maringale M is H -adaped, hence is an H -maringale. From Theorem. applied o he case n =, any H -maringale is a sochasic inegral w.r.. M, hence is a H-maringale. Furhermore, he inensiy of τ 2 vanishes on he se < τ and M 2 = H 2 τ2 τ fτ, s Gτ, s ds = H2 {τ<s<τ 2} fτ, s Gτ, s ds. Proposiion 4. Le V i, i =, 2 be he price of a CDS on name i, wih conracual spread κ i and paymen a defaul given by a deerminisic funcion δ i. The H-dynamics of V is dv = V dm + H κ δ λ d 7 4

15 wih λ = f G. The H-dynamics of V 2 is dv 2 = V 2 dm H 2 κ 2 d H 2 H δ 2 λ 2 τ d + V 2 V 2 dm. Proof. Apply Proposiion 2.3 or 6 o obain 7, and 6 o obain Mulidefaul seing Le G be he survival funcion of he join defauls, assumed o be differeniable and G j be he survival funcion of he j-firs defauls G,..., n = Pτ >,..., τ n > n G j,..., j = Pτ >,..., τ j > j. We shall denoe by f he densiy of he n-uple τ i, i n and by f j he densiy of he j-uple τ i, i j. Since he defauls are ordered, seing j =,..., j one has G j j := Pτ >,..., τ j > j = Pτ >,..., τ j > j, τ j+ > j,..., τ n > j = G j, j,..., j. From an immediae exension of Proposiion 3., noing ha he densiy of he defauls is null ouside he se { 2... n, he fundamenal maringales are τj M j = H j λj j s τ j τ,..., τ j ds where and,j =... j. λ j j j =,jg j j,,j G j j, Proposiion 4.2 If V i is he price process of a CDS wih mauriy T, wrien on he i-h defaul, wih spread κ i and paymen a defaul given by a deerminisic funcion δ i, hen dv i = V dm i i HH i i δ i λ i i τ,..., τ i d i + V i j j= τ,..., τ j V i dm j + H i κ i d, 9 where V i j j = T δ i uf j+ j, u du κ T i,j G j+ j, u du,j G j j. 5

16 4.3 Hedging of a loss In order o hedge he payoff B, one proceeds in wo seps. The firs sep is o compue he maringale represenaion of EB H, i.e., idenify he predicable processes π such ha EB H = EB + j= π j sdm j s. We denoe by D i he dividend par associaed wih he CDS wrien on τ i. A self-financing sraegy wih value V = ϑ + ϑ i V i saisfies dv = = = i= i= j= ϑ i dv i + dd i ϑ i dm j i= i δ i V dm i i + i=j ϑ i V i j V i j j= V dm i j V i where we se V i i = δ i. I remains o solve he linear sysem wih unknown ϑ i=j ϑ i V i j V i = π j, j =,..., n. As an example, we now compue he condiional law of he loss, i.e., EfL T H where L T = n k= {τ k T }. Le B k T = E {T <τk } H or simply B k be he price of a defaulable zero-coupon wrien on he k-h defaul, wih mauriy T, hen Obviously, seing τ = EfL T H = k= fk B k+ B k n = fn + B k fk fk. k= B k = = k {τj <τ j}e {T <τk } H j= k j= PT < τ k, < τ j H j {τj <τ j} P < τ j H j Now, on he se τ j PT < τ k, < τ j H j = Φ k,j τ,..., τ j,, T 6

17 where, for j k Φ k,j,..., j,, T = P < τ j, T < τ k, τ d,..., τ j d j Pτ d,..., τ j d j On he se τ k for j = k =,j G k j,,...,, T,j G k j,..., j. Since PT < τ k, < τ k H k P < τ k H k = PT < τ k H k P < τ k H k k db k = B dm k k + ν j,k dm j = j= =,k G k k, T,k G k k,. k j= ν j,k dm j where, for j < k, we have se ν j,k = Φ k,jτ,...,τ j,,,t Φ k,j τ Φ k,j τ,...,τ j,,t,...,τ j,,, Φ k,j τ,...,τ j,, ha, seing X = EfL T H, one has dx = dm j j= k=j fk fkν i,k and ν k,k = B k. I follows and he hedging sraegy for he coningen claim fl T is he soluion ϑ of he riangular sysem Conclusion i=j ϑ i V i j V i = fk fkν j,k, j =,..., n. k=j We invesigae a quie general pure jump seing where he densiy of join defaul is known. We compue he defaul inensiies in he filraion of all he defaul imes. In paricular, a each insan when a defaul even occurs, defaul inensiies of non-defauled names are dynamically updaed. This leads o a dependence srucure among defaul imes which is regularly updaed as defauls arrive. We have seen ha he hedging of loss derivaives such as CDO ranches or baske defaul swaps can be fully described in his framework wih no Markovian assumpion. The hedging sraegies wih respec o single-name CDS can be derived analyically in a wo-defauls seing. Even if similar ideas can be exploied in higher dimension, he consrucion of dynamic hedging sraegies would involve very cumbersome compuaions if one wans o consider all possible defaul scenarios. Ineresingly, in he paricular case of ranked defaul imes, he hedging issue can be solved explicily. All hese resuls admi an exension o he case where here exiss a reference filraion, leading o inensiy processes ha depend on a facor process driven by a Brownian moion. References [] Basel Commiee on Banking Supervision Dec 2 Revisions o he Basel II marke risk framework. 7

18 [2] Bielecki, T.R. and Rukowski, M. 2, Credi risk: Modelling Valuaion and Hedging, Springer Verlag, Berlin. [3] Bielecki, T.R., Jeanblanc, M., and Rukowski, M. 28 Pricing and rading credi defaul swaps in a hazard process model. Annals of Applied Probabiliy 8, [4] Bielecki, T.R., Jeanblanc, M., and Rukowski, M. 29 Credi Risk Modelling, Lecures Noes, Osaka Universiy. [5] Brémaud, P. 98 Poin Processes and Queues: Maringale Dynamics, Springer Verlag. [6] Cousin, A. and Lauren, J.-P. 2 Dynamic hedging of synheic CDO ranches: bridging he gap beween heory and pracice, in Credi Risk Froniers: Subprime crisis, Pricing and Hedging, CVA, MBS, Raings and Liquidiy, Wiley. [7] Cousin, A., Jeanblanc, M. and Lauren, J.-P. 2 Hedging CDO ranches in a Markovian environmen, in Paris-Princeon Lecures in Mahemaical Finance, Lecure Noes in Mahemaics, Springer. [8] Lauren, J.P., Cousin, A. and Fermanian, J.D 2 Hedging defaul risk for CDOs in Markovian conagion models, Quaniaive Finance. [9] Dellacherie, C. 972 Capaciés e processus sochasiques, Springer. [] Ellio, R.J. 982 Sochasic Calculus and Applicaions, Springer, Berlin [] Kusuoka, S. 999, A remark on defaul risk models, Adv. Mah. Econ.,