HEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT

Transcription

1 HEDGING OF CREDIT DERIVATIVES IN MODELS WITH TOTALLY UNEXPECTED DEFAULT Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France Marek Rukowski School of Mahemaics Universiy of New Souh Wales Sydney, NSW 252, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology -661 Warszawa, Poland July 1, 25 This work was compleed during our visi o he Isaac Newon Insiue for Mahemaical Sciences in Cambridge. We hank he organizers of he programme Developmens in Quaniaive Finance for he kind inviaion. The research of T.R. Bielecki was suppored by NSF Gran and Moody s Corporaion gran The research of M. Jeanblanc was suppored by Zéliade, Iô33, and Moody s Corporaion gran The research of M. Rukowski was suppored by he 25 Faculy Research Gran PS

2 2 Hedging of Credi Derivaives Conens 1 Toally Unexpeced Defaul General Se-up Inensiy of a Sopping Time F-Inensiy of a Random Time Hypohesis H Hazard Process Canonical Consrucion Change of a Probabiliy Measure Preliminary Lemma Case of he Brownian Filraion Exension o Orhogonal Maringales Semimaringale Model wih a Common Defaul Dynamics of Asse Prices Pre-defaul Values Marke Observables Recovery Schemes Risk-Neural Valuaion Price Dynamics of a Survival Claim X,, τ Price Dynamics of a Recovery Claim, Z, τ Price Dynamics of a Defaulable Claim X, Z, τ Trading Sraegies in a Semimaringale Se-up Unconsrained Sraegies Consrained Sraegies Synheic Asse Case of Coninuous Asse Prices Maringale Approach o Valuaion and Hedging Defaulable Asse wih Toal Defaul Defaul-Free Marke Arbirage-Free Propery Hedging a Survival Claim Hedging a Recovery Process Bond Marke Credi-Risk-Adjused Forward Price Vulnerable Opion on a Defaul-Free Asse Vulnerable Swapion Two Defaulable Asses wih Toal Defaul Hedging a Survival Claim Opion on a Defaulable Asse Opion o Exchange Defaulable Asses PDE Approach o Valuaion and Hedging Defaulable Asse wih Toal Defaul Hedging wih he Savings Accoun Defaulable Asse wih Non-Zero Recovery Arbirage-Free Propery Pricing PDE and Replicaing Sraegy Hedging of a Survival Claim Hedging of a Recovery Payoff Two Defaulable Asses wih Toal Defaul

3 T.R. Bielecki, M. Jeanblanc and M. Rukowski 3 Inroducion This paper presens some mehods o hedge defaulable derivaives under he assumpion ha here exis radeable asses wih dynamics allowing for eliminaion of defaul risk of derivaive securiies. We invesigae hedging sraegies in alernaive frameworks wih differen degrees of generaliy, an absrac semimaringale framework and a more specific Markovian se-up, and we use wo alernaive approaches. On he one hand, we use he sochasic calculus approach in order o esablish raher absrac characerizaion resuls for hedgeable coningen claims in a fairly general se-up. We subsequenly apply hese resuls o derive closed-form soluions for prices and replicaing sraegies in paricular models. On he oher hand, we examine he PDE approach in a Markovian seing. In his mehod, he arbirage price and he hedging sraegy for an aainable coningen claim are described in erms of soluions of a pair of coupled PDEs. Again, for some sandard examples of defaulable claims, we provide explici formulae for prices and hedging sraegies for furher examples of rading sraegies involving radeable credi derivaives, we refer o Lauren [28] or Bielecki e al. [7]. As expeced, boh mehods yield idenical resuls for some special cases considered in his work. For he sake of simpliciy, we only deal wih financial models wih no more han hree primary asses models wih an arbirary number of primary asses were sudied in Bielecki e al. [5]. Also, i is posulaed hroughou ha he defaul ime is he same for all defaulable securiies. An exension of our resuls o he case of several possibly dependen defaul imes is crucial if someone wishes o cover he so-called baske credi derivaives in his regard, see Secion 6 in Bielecki e al. [6]. Le us commen briefly on he erminology used in his work. Tradiionally, credi risk models are classified eiher as srucural models also known as value-of-he-firm models or as reduced-form models also ermed inensiy-based models. In heir original forms, he wo approaches, srucural and reduced-form, are exreme cases in he sense ha he defaul ime is modelled eiher as a predicable sopping ime he firs momen when he firm s value his some barrier, as in Black and Cox [8], or by a oally inaccessible sopping ime defined via is inensiy, as in Jarrow and Turnbull [22]. However, as argued by several auhors see, for insance, Duffie and Lando [16], Giesecke [2], Jarrow and Proer [21], or Jeanblanc and Valchev [25], probabilisic properies of defaul ime are direcly relaed o he publicly available informaion i is imporan, for insance, wheher he value of he firm and/or he defaul riggering barrier are observed by he marke wih absolue accuracy. In fac, in several srucural models he defaul ime is no longer predicable, as i was he case in classic models wih deerminisic defaul riggering barrier and full observaion of he firm value process see, Meron [29] or Black and Cox [8]. For his reason, we decided o refer o credi risk models considered in his work as models wih oally unexpeced defaul he sric mahemaical erm, oally inaccessible sopping ime, seems o be raher cumbersome for a frequen use. For a more exhausive presenaion of mahemaical heory of credi risk, we refer o Arvaniis and Gregory [1], Bielecki and Rukowski [2], Bielecki e al. [3], Cossin and Piroe [14], Duffie and Singleon [17], Lando [27], or Schönbucher [33]. Acknowledgemens Some resuls of his work were presened by Monique Jeanblanc a he Inernaional Workshop on Sochasic Processes and Applicaions o Mahemaical Finance held a Risumeikan Universiy on March 3-6, 25. She deeply hanks he paricipans for quesions and commens. The firs version of his paper was wrien during her say a Nagoya Ciy Universiy on he inviaion by Professor Yoshio Miyahara, whose he warm hospialiy is graefully acknowledged. The work was compleed during our visi o he Isaac Newon Insiue for Mahemaical Sciences in Cambridge. We hank he organizers of he programme Developmens in Quaniaive Finance for he kind inviaion.

4 4 Hedging of Credi Derivaives 1 Toally Unexpeced Defaul In his secion, we describe briefly he fundamenal feaures of he credi risk models wih unexpeced defaul. Also, we collec here few echnical resuls ha are used in subsequen secions. 1.1 General Se-up We assume ha we are given a probabiliy space Ω, G, P and a nonnegaive random variable τ on his space. We always posulae ha τ is sricly posiive wih probabiliy 1. Noe ha he probabiliy measure P represens he hisorical probabiliy reflecing he real-life dynamics of prices of primary raded asses, raher han some maringale measure for our financial model. We firs focus on differen definiions of defaul inensiy encounered in he lieraure Inensiy of a Sopping Time Suppose ha Ω, G, P is endowed wih some filraion G such ha τ is a G-sopping ime. Le H be he defaul process, defined as H = 1 { τ} noe ha H is a bounded G-submaringale. We say ha τ admis a G-inensiy if here exiss a G-adaped, nonnegaive process λ such ha he process M = H τ λ u du = H λ u du 1 is a G-maringale he second equaliy in 1 follows from he fac ha he process H is sopped a τ. Then M is called he compensaed G-maringale of he defaul process H. In order for a G-sopping ime τ o admi a G-inensiy λ, i has o be oally inaccessible wih respec o G, so ha Pτ = θ = for any G-predicable sopping ime θ. The simples example is he momen of he firs jump a Poisson process. Noe ha he inensiy λ necessarily vanishes afer defaul. Remark 1.1 Some auhors define he inensiy as he process λ such ha H τ λ u du is a G-maringale. In ha case, he process λ is no uniquely defined afer ime τ F-Inensiy of a Random Time We change he perspecive, and we no longer assume ha he filraion G is given a priori. We assume insead ha τ is a posiive random variable on some probabiliy space Ω, G, P. Le H = H, be he naural filraion generaed by he defaul process H,, and le F = F, be some reference filraion in Ω, G, P. We assume hroughou ha he informaion available o an invesor is modeled by he filraion G = F H. Consequenly, we can reduce our sudy o he case where he defaul inensiy if i exiss is G-adaped, meaning ha he process M given by 1 is a G-maringale for some G-adaped process λ. In his seing, here exiss a process λ = λ,, called he F-inensiy of τ, which is F-adaped and equal o λ before defaul, so ha λ 1 { τ} = λ 1 { τ} for every R +. The exisence of λ and is uniqueness under some echnical condiions follows from he following resul see Dellacherie e al. [15], Page 186. Lemma 1.1 Le G = F H. Then for any G-predicable process ζ here exiss an F-predicable process ζ such ha ζ 1 { τ} = ζ 1 { τ}, R +. 2 If, in addiion, he inequaliy F := Pτ F < 1 holds for every R + hen he process ζ saisfying 2 is unique.

5 T.R. Bielecki, M. Jeanblanc and M. Rukowski 5 Of course, we have ha τ τ M = H λ u du = H λ u du. Suppose ha he reference filraion is chosen in such a way ha he defaul evens {τ } are no in F. Then he F-inensiy λ is uniquely defined afer τ and, ypically, does no vanish afer τ. 1.2 Hypohesis H In his secion, we focus on he invariance propery of he so-called hypohesis H under an equivalen change of a probabiliy measure. Definiion 1.1 We say ha filraions F and G, wih F G, saisfy he hypohesis H under P whenever any F-local maringale L follows also a G-local maringale. Remark 1.2 We emphasize ha, in general, an F-maringale may fail o follow a G-maringale. As a rivial example, consider a fixed dae T > and ake G = F T for every [, T ]. Then any F-maringale L saisfies E P L G s = L for s, and hus L is no a G-maringale, in general. I is even possible, bu more difficul, o produce an example of an F-maringale, which is no a semimaringale wih respec o G. For oher couner-examples, in paricular hose involving progressive enlargemen of filraions, we refer ineresed reader o Proer [32], or Mansuy and Yor [3]. The original formulaions of he hypohesis H refer o maringales or even square-inegrable maringales, raher han local maringales. We shall show ha in our se-up he definiion given above is equivalen o he original definiion. In fac, he hypohesis H posulaes a cerain form of condiional independence of σ-fields associaed wih F and G, raher han a specific propery of F-local maringales. In paricular he following well known resul is valid. Lemma 1.2 Assume ha G = F H, where F is an arbirary filraion and H is generaed by he process H = 1 {τ }. Then he following condiions are equivalen o he hypohesis H. i For any, h R +, we have i For any R +, we have Pτ F = Pτ F +h. 3 Pτ F = Pτ F. 4 ii For any R +, he σ-fields F and G are condiionally independen given F under P, ha is, E P ξ η F = E P ξ F E P η F for any bounded, F -measurable random variable ξ and bounded, G -measurable random variable η. iii For any R +, and any u he σ-fields F u and G are condiionally independen given F. iv For any R + and any bounded, F -measurable random variable ξ: E P ξ G = E P ξ F. v For any R +, and any bounded, G -measurable random variable η: E P η F = E P η F. Proof. The proof of equivalence of condiions i -v can be found, for insance, in Secion of Bielecki and Rukowski [2] for relaed resuls, see Ellio e al. [19]. Using monoone class heorem i can be shown ha condiions i and i are equivalen. Hence, we shall only show ha condiion iv and he hypohesis H are equivalen. Assume firs ha he hypohesis H holds. Consider any bounded, F -measurable random variable ξ. Le L = E P ξ F be he maringale associaed wih ξ. Then, H implies ha L is also a local maringale wih respec o G, and hus a G-maringale, since L is bounded recall ha any bounded local maringale is a maringale. We conclude ha L = E P ξ G and hus iv holds.

6 6 Hedging of Credi Derivaives Suppose now ha iv holds. Firs, we noe ha he sandard runcaion argumen shows ha he boundedness of ξ in iv can be replaced by he assumpion ha ξ is P-inegrable. Hence, any F-maringale L is an G-maringale, since L is clearly G-adaped and we have, for every s, L = E P L s F = E P L s G. Now, suppose ha L is an F-local maringale so ha here exiss an increasing sequence of F- sopping imes τ n such ha lim n τ n =, for any n he sopped process L τ n follows a uniformly inegrable F-maringale. Hence, L τ n is also a uniformly inegrable G-maringale, and his means ha L follows a G-local maringale Hazard Process Le τ be a random ime on a space Ω, G, P such ha he filraions F and G = F H saisfy he hypohesis H. Then, from 4, he process F = Pτ F is increasing. We make he sanding assumpion ha F < 1 for every R +, and we define he F-hazard process Γ by seing Γ = ln 1 F. Le, in addiion, he process F be absoluely coninuous wih respec o he Lebesgue measure, so ha F = f u du, R +, for some F-progressively measurable or F-predicable process f. Then he F-hazard process saisfies where he F-inensiy γ is given by Γ = γ u du, R +, γ = f 1 F, R +. 5 From now on, we ake 5 as he definiion of he F-inensiy γ, and we make he sanding assumpion ha he hypohesis H holds under P. The following auxiliary resul is sandard see, for insance, Ellio e al. [19] or Blanche-Scallie and Jeanblanc [9]. Lemma 1.3 For any P-inegrable, F T -measurable random variable X we have, for [, T ], E P X1 {T <τ} G = 1 {<τ} e Γ E P Xe Γ T F Canonical Consrucion We now describe he canonical consrucion of a random ime wih a given F-hazard process. Le Ψ be an F-adaped, increasing, nonnegaive process wih Ψ = and lim Ψ =. We define a nonnegaive random variable τ by seing τ = inf { R + : Ψ Θ}, where Θ is a random variable independen of F, wih he exponenial disribuion of parameer 1. Of course, he exisence of Θ on he original probabiliy space Ω, G, P is no guaraneed, so ha we allow for an exension of he underlying probabiliy space. We shall now find he process F = P{τ F }. Since clearly {τ > } = {Θ > Ψ }, we ge P{τ > F } = P{Θ > Ψ F } = e Ψ. Consequenly, 1 F = P{τ > F } = E P P{τ > F } F = e Ψ,

7 T.R. Bielecki, M. Jeanblanc and M. Rukowski 7 and so F is an F-adaped, coninuous, increasing process. We conclude ha for every R + F = 1 e Ψ = P{τ F } = P{τ F }, 6 and hus Ψ coincides wih he F-hazard process Γ of τ and he hypohesis H is valid. I is also no difficul o show ha he process M = H Γ τ = H Ψ τ follows a G-maringale. The following resul shows ha under he hypohesis H, for any random ime τ wih coninuous hazard process Γ, he auxiliary random variable Θ can be consruced on he original probabiliy space, using τ and Γ see El Karoui [18] or Blanche-Scallie and Jeanblanc [9]. Lemma 1.4 Le τ be a random ime on a probabiliy space Ω, G, P such ha he F-hazard process Γ of τ under P is coninuous and he hypohesis H holds. Then here exiss a random variable Θ on Ω, G, P, independen of F and wih he exponenial disribuion of parameer 1, such ha τ = inf { R + : Γ Θ}. 7 Proof. I suffices o check ha he random variable Θ = Γ τ has he desired properies. Indeed, we have, for every R +, PΘ F = PΓ τ F = Pτ A F = exp Γ A = e, where A is he lef inverse of Γ, so ha Γ A = for every R Change of a Probabiliy Measure Kusuoka [26] shows, by means of a couner-example, ha he hypohesis H is no invarian wih respec o an equivalen change of he underlying probabiliy measure, in general. I is worh noing ha his couner-example is based on wo filraions, H 1 and H 2, generaed by he wo random imes τ 1 and τ 2, and he chooses H 1 o play he role of he reference filraion F. We shall argue ha in he case where F is generaed by a Brownian moion or, more generally, by some maringale orhogonal o M under P, he above-menioned invariance propery is valid under mild echnical assumpions Preliminary Lemma Le us firs examine a general se-up in which G = F H, where F is an arbirary filraion and H is generaed by he defaul process H. We say ha Q is locally equivalen o P if Q is equivalen o P on Ω, G for every R +. Then here exiss he Radon-Nikodým densiy process η such ha dq G = η dp G, R +. 8 Par i in he nex lemma is well known see Jamshidian [24]. We assume ha he hypohesis H holds under P. Lemma 1.5 i Le Q be a probabiliy measure equivalen o P on Ω, G for every R +, wih he associaed Radon-Nikodým densiy process η. If he densiy process η is F-adaped hen we have Qτ F = Pτ F for every R +. Hence, he hypohesis H is also valid under Q and he F-inensiies of τ under Q and under P coincide. ii Assume ha Q is equivalen o P on Ω, G and dq = η dp, so ha η = E P η G. Then he hypohesis H is valid under Q whenever we have, for every R +, E P η H F E P η F = E Pη H F E P η F. 9

8 8 Hedging of Credi Derivaives Proof. To prove i, assume ha he densiy process η is F-adaped. We have for each, h R + Qτ F = E Pη 1 {τ } F E P η F = Pτ F = Pτ F +h = Qτ F +h, where he las equaliy follows by anoher applicaion of he Bayes formula. follows from par i in Lemma 1.2. To prove par ii, i suffices o esablish he equaliy The asserion now F := Qτ F = Qτ F, R +. 1 Noe ha since he random variables η 1 {τ } and η are P-inegrable and G -measurable, using he Bayes formula, par v in Lemma 1.2, and assumed equaliy 9, we obain he following chain of equaliies Qτ F = E Pη 1 {τ } F E P η F = E Pη 1 {τ } F E P η F = E Pη 1 {τ } F E P η F = Qτ F. We conclude ha he hypohesis H holds under Q if and only if 9 is valid. Unforunaely, sraighforward verificaion of condiion 9 is raher cumbersome. For his reason, we shall provide alernaive sufficien condiions for he preservaion of he hypohesis H under a locally equivalen probabiliy measure Case of he Brownian Filraion Le W be a Brownian moion under P wih respec o is naural filraion F. Since we work under he hypohesis H, he process W is also a G-maringale, where G = F H. Hence, W is a Brownian moion wih respec o G under P. Our goal is o show ha he hypohesis H is sill valid under Q Q for a large class Q of locally equivalen probabiliy measures on Ω, G. Le Q be an arbirary probabiliy measure locally equivalen o P on Ω, G. Kusuoka [26] see also Secion in Bielecki and Rukowski [2] proved ha, under he hypohesis H, any G-maringale under P can be represened as he sum of sochasic inegrals wih respec o he Brownian moion W and he jump maringale M. In our se-up, Kusuoka s represenaion heorem implies ha here exis G-predicable processes θ and ζ > 1, such ha he Radon-Nikodým densiy η of Q wih respec o P saisfies he following SDE dη = η θ dw + ζ dm 11 wih he iniial value η = 1. More explicily, he process η equals η = E θ u dw u E ζ u dm u = η 1 η 2, 12 where we wrie η 1 = E θ u dw u = exp θ u dw u 1 2 θu 2 du, 13 and τ η 2 = E ζ u dm u = exp ln1 + ζ u dh u ζ u γ u du. 14 Moreover, by virue of a suiable version of Girsanov s heorem, he following processes Ŵ and M are G-maringales under Q Ŵ = W θ u du, M = M 1 {u<τ} γ u ζ u du. 15

9 T.R. Bielecki, M. Jeanblanc and M. Rukowski 9 Proposiion 1.1 Assume ha he hypohesis H holds under P. Le Q be a probabiliy measure locally equivalen o P wih he associaed Radon-Nikodým densiy process η given by formula 12. If he process θ is F-adaped hen he hypohesis H is valid under Q and he F-inensiy of τ under Q equals γ = 1 + ζ γ, where ζ is he unique F-predicable process such ha he equaliy ζ 1 { τ} = ζ 1 { τ} holds for every R +. Proof. Le P be he probabiliy measure locally equivalen o P on Ω, G, given by d P G = E ζ u dm u dp G = η 2 dp G. 16 We claim ha he hypohesis H holds under P. From Girsanov s heorem, he process W follows a Brownian moion under P wih respec o boh F and G. Moreover, from he predicable represenaion propery of W under P, we deduce ha any F-local maringale L under P can be wrien as a sochasic inegral wih respec o W. Specifically, here exiss an F-predicable process ξ such ha L = L + ξ u dw u. This shows ha L is also a G-local maringale, and hus he hypohesis H holds under P. Since dq G = E θ u dw u d P G, by virue of par i in Lemma 1.5, he hypohesis H is valid under Q as well. The las claim in he saemen of he lemma can be deduced from he fac ha he hypohesis H holds under Q and, by Girsanov s heorem, he process M = M is a Q-maringale. 1 {u<τ} γ u ζ u du = H 1 {u<τ} 1 + ζ u γ u du We claim ha he equaliy P = P holds on he filraion F. Indeed, we have d P F = η dp F, where we wrie η = E P η 2 F, and E P η 2 F = E P E ζ u dm u F = 1, R +, 17 where he firs equaliy follows from par v in Lemma 1.2. To esablish he second equaliy in 17, we firs noe ha since he process M is sopped a τ, we may assume, wihou loss of generaliy, ha ζ = ζ where he process ζ is F-predicable see Lemma 1.1. Moreover, in view of 7 he condiional cumulaive disribuion funcion of τ given F has he form 1 exp Γ ω. Hence, for arbirarily seleced sample pahs of processes ζ and Γ, he claimed equaliy can be seen as a consequence of he maringale propery of he Doléans exponenial. Formally, i can be proved by following elemenary calculaions, where he firs equaliy is a consequence of 14, ζ u dm u F E P E = E P = E P 1 τ = E P + 1{ τ} ζτ exp u 1 + 1{ u} ζu exp 1 + ζu γu exp u ζ v γ v dv γ u e R u 1 + ζ v γ v dv du F γv dv du ζ u γ u du F F

10 1 Hedging of Credi Derivaives + E P exp = = 1 exp 1 + ζu γu exp ζ v γ v dv γ u e R u u 1 + ζ v γ v dv γ v dv du F 1 + ζ v γ v dv du + exp + exp ζ v γ v dv exp ζ v γ v dv where he second las equaliy follows by an applicaion of he chain rule. γ v dv = 1, γ u e R u γ v dv du Exension o Orhogonal Maringales Equaliy 17 suggess ha Proposiion 1.1 can be exended o he case of arbirary orhogonal local maringales. Such a generalizaion is convenien, if we wish o cover he siuaion considered in Kusuoka s counerexample. Le N be a local maringale under P wih respec o he filraion F. I is also a G-local maringale, since we mainain he assumpion ha he hypohesis H holds under P. Le Q be an arbirary probabiliy measure locally equivalen o P on Ω, G. We assume ha he Radon-Nikodým densiy process η of Q wih respec o P equals dη = η θ dn + ζ dm 18 for some G-predicable processes θ and ζ > 1 he properies of he process θ depend, of course, on he choice of he local maringale N. The nex resul covers he case where N and M are orhogonal G-local maringales under P, so ha he produc M N follows a G-local maringale. Proposiion 1.2 Assume ha he following condiions hold: a N and M are orhogonal G-local maringales under P, b N has he predicable represenaion propery under P wih respec o F, in he sense ha any F-local maringale L under P can be wrien as L = L + ξ u dn u, R +, for some F-predicable process ξ, c P is a probabiliy measure on Ω, G such ha 16 holds. Then we have: i he hypohesis H is valid under P, ii if he process θ is F-adaped hen he hypohesis H is valid under Q. The proof of he proposiion hinges on he following simple lemma. Lemma 1.6 Under he assumpions of Proposiion 1.2, we have: i N is a G-local maringale under P, ii N has he predicable represenaion propery for F-local maringales under P. Proof. In view of c, we have d P G = η 2 dp G, where he densiy process η 2 is given by 14, so ha dη 2 = η ζ 2 dm. From he assumed orhogonaliy of N and M, i follows ha N and η 2 are orhogonal G-local maringales under P, and hus Nη 2 is a G-local maringale under P as well. This means ha N is a G-local maringale under P, so ha i holds. To esablish par ii in he lemma, we firs define he auxiliary process η by seing η = E P η 2 F. Then manifesly d P F = η dp F, and hus in order o show ha any F-local maringale under P follows an F-local maringale under P, i suffices o check ha η = 1 for every R +, so ha P = P on F. To his end, we noe ha E P η 2 F = E P E ζ u dm u F = 1, R +,

11 T.R. Bielecki, M. Jeanblanc and M. Rukowski 11 where he firs equaliy follows from par v in Lemma 1.2, and he second one can esablished similarly as he second equaliy in 17. We are in a posiion o prove ii. Le L be an F-local maringale under P. Then i follows also an F-local maringale under P and hus, by virue of b, i admis an inegral represenaion wih respec o N under P and P. This shows ha N has he predicable represenaion propery wih respec o F under P. We now proceed o he proof of Proposiion 1.2. Proof of Proposiion 1.2. We shall argue along he similar lines as in he proof of Proposiion 1.1. To prove i, noe ha by par ii in Lemma 1.6 we know ha any F-local maringale under P admis he inegral represenaion wih respec o N. Bu, by par i in Lemma 1.6, N is a G-local maringale under P. We conclude ha L is a G-local maringale under P, and hus he hypohesis H is valid under P. Asserion ii now follows from par i in Lemma 1.5. Remark 1.3 I should be sressed ha Proposiion 1.2 is no direcly employed in wha follows. We decided o presen i here, since i sheds some ligh on specific echnical problems arising in he conex of modelling dependen defaul imes hrough an equivalen change of a probabiliy measure see Kusuoka [26]. Example 1.1 Kusuoka [26] presens a couner-example based on he wo independen random imes τ 1 and τ 2 given on some probabiliy space Ω, G, P. We wrie M i = H i τ i γ i u du, where H i = 1 { τi } and γ i is he deerminisic inensiy funcion of τ i under P. Le us se dq G = η dp G, where η = η 1 η 2 and, for i = 1, 2 and every R +, η i = 1 + ηu ζ i u i dmu i = E ζu i dmu i for some G-predicable processes ζ i, i = 1, 2, where G = H 1 H 2. We se F = H 1 and H = H 2. Manifesly, he hypohesis H holds under P. Moreover, in view of Proposiion 1.2, i is sill valid under he equivalen probabiliy measure P given by d P G = E ζu 2 dmu 2 dp G. I is clear ha P = P on F, since E P η 2 F = E P E ζu 2 dmu 2 H 1 = 1, R +. However, he hypohesis H is no necessarily valid under Q if he process ζ 1 fails o be F-adaped. In Kusuoka s couner-example, he process ζ 1 was chosen o be explicily dependen on boh random imes, and i was shown ha he hypohesis H does no hold under Q. For an alernaive approach o Kusuoka s example, hrough an absoluely coninuous change of a probabiliy measure, he ineresed reader may consul Collin-Dufresne e al. [12]. 2 Semimaringale Model wih a Common Defaul In wha follows, we fix a finie horizon dae T >. For he purpose of his work, i is enough o formally define a generic defaulable claim hrough he following definiion. Definiion 2.1 A defaulable claim wih mauriy dae T is represened by a riple X, Z, τ, where:

12 12 Hedging of Credi Derivaives i he defaul ime τ specifies he random ime of defaul, and hus also he defaul evens {τ } for every [, T ], ii he promised payoff X F T represens he random payoff received by he owner of he claim a ime T, provided ha here was no defaul prior o or a ime T ; he acual payoff a ime T associaed wih X hus equals X1 {T <τ}, iii he F-adaped recovery process Z specifies he recovery payoff Z τ received by he owner of a claim a ime of defaul or a mauriy, provided ha he defaul occurred prior o or a mauriy dae T. In pracice, hedging of a credi derivaive afer defaul ime is usually of minor ineres. Also, in a model wih a single defaul ime, hedging afer defaul reduces o replicaion of a non-defaulable claim. I is hus naural o define he replicaion of a defaulable claim in he following way. Definiion 2.2 We say ha a self-financing sraegy φ replicaes a defaulable claim X, Z, τ if is wealh process V φ saisfies V T φ1 {T <τ} = X1 {T <τ} and V τ φ1 {T τ} = Z τ 1 {T τ}. When dealing wih replicaing sraegies, in he sense of Definiion 2.2, we will always assume, wihou loss of generaliy, ha he componens of he process φ are F-predicable processes. 2.1 Dynamics of Asse Prices We assume ha we are given a probabiliy space Ω, G, P endowed wih a possibly muli-dimensional sandard Brownian moion W and a random ime τ admiing an F-inensiy γ under P, where F is he filraion generaed by W. In addiion, we assume ha τ saisfies 4, so ha he hypohesis H is valid under P for filraions F and G = F H. Since he defaul ime admis an F-inensiy, i is no an F-sopping ime. Indeed, any sopping ime wih respec o a Brownian filraion is known o be predicable. We inerpre τ as he common defaul ime for all defaulable asses in our model. For simpliciy, we assume ha only hree primary asses are raded in he marke, and he dynamics under he hisorical probabiliy P of heir prices are, for i = 1, 2, 3 and [, T ], or equivalenly, dy i dy i = Y i µi, d + σ i, dw + κ i, dm, 19 = Y i µi, κ i, γ 1 { τ} d + σ i, dw + κ i, dh. 2 The processes µ i, σ i, κ i = µ i,, σ i,, κ i,,, i = 1, 2, 3, are assumed o be G-adaped, where G = F H. In addiion, we assume ha κ i 1 for any i = 1, 2, 3, so ha Y i are nonnegaive processes, and hey are sricly posiive prior o τ. Noe ha, according o Definiion 2.2, replicaion refers o he behavior of he wealh process V φ on he random inerval [[, τ T ]] only. Hence, for he purpose of replicaion of defaulable claims of he form X, Z, τ, i is sufficien o consider prices of primary asses sopped a τ T. This implies ha insead of dealing wih G-adaped coefficiens in 19, i suffices o focus on F-adaped coefficiens of sopped price processes. However, for he sake of compleeness, we shall also deal wih T -mauriy claims of he form Y = GY 1 T, Y 2 T, Y 3 T, H T see Secion 5 below Pre-defaul Values As will become clear in wha follows, when dealing wih defaulable claims of he form X, Z, τ, we will be mainly concerned wih he so-called pre-defaul prices. The pre-defaul price Ỹ i of he ih asse is an F-adaped, coninuous process, given by he equaion, for i = 1, 2, 3 and [, T ], dỹ i = Ỹ i µi, κ i, γ d + σ i, dw 21

13 T.R. Bielecki, M. Jeanblanc and M. Rukowski 13 wih Ỹ i = Y i. Pu anoher way, Ỹ i is he unique F-predicable process such ha see Lemma 1.1 Ỹ i 1 { τ} = Y i 1 { τ} for R +. When dealing wih he pre-defaul prices, we may and do assume, wihou loss of generaliy, ha he processes µ i, σ i and κ i are F-predicable. I is worh sressing ha he hisorically observed drif coefficien equals µ i, κ i, γ, raher han µ i,. The drif coefficien denoed by µ i, is already credi-risk adjused in he sense of our model, and i is no direcly observed. This convenion was chosen here for he sake of simpliciy of noaion. I also lends iself o he following inuiive inerpreaion: if φ i is he number of unis of he ih asse held in our porfolio a ime hen he gains/losses from rades in his asse, prior o defaul ime, can be represened by he differenial φ i dỹ i = φ i Ỹ i µi, d + σ i, dw φ i Ỹ i κ i, γ d. The las erm may be here separaed, and formally reaed as an effec of coninuously paid dividends a he dividend rae κ i, γ. However, his inerpreaion may be misleading, since his quaniy is no direcly observed. In fac, he mere esimaion of he drif coefficien in dynamics 21 is no pracical. Sill, if his formal inerpreaion is adoped, i is someimes possible make use of he sandard resuls concerning he valuaion of derivaives of dividend-paying asses. I is, of course, a delicae issue how o separae in pracice boh componens of he drif coefficien. We shall argue below ha alhough he dividend-based approach is formally correc, a more perinen and simpler way of dealing wih hedging relies on he assumpion ha only he effecive drif µ i, κ i, γ is observable. In pracical approach o hedging, he values of drif coefficiens in dynamics of asse prices play no essenial role, so ha hey are considered as marke observables Marke Observables To summarize, we assume hroughou ha he marke observables are: he pre-defaul marke prices of primary asses, heir volailiies and correlaions, as well as he jump coefficiens κ i, he financial inerpreaion of jump coefficiens is examined in he nex subsecion. To summarize we posulae ha under he saisical probabiliy P we have dy i = Y i µi, d + σ i, dw + κ i, dh, 22 where he drif erms µ i, are no observable, bu we can observe he volailiies σ i, and hus he asses correlaions, and we have an a priori assessmen of jump coefficiens κ i,. In his general se-up, he mos naural assumpion is ha he dimension of a driving Brownian moion W equals he number of radable asses. However, for he sake of simpliciy of presenaion, we shall frequenly assume ha W is one-dimensional. One of our goals will be o derive closed-form soluions for replicaing sraegies for derivaive securiies in erms of marke observables only whenever replicaion of a given claim is acually feasible. To achieve his goal, we shall combine a general heory of hedging defaulable claims wihin a coninuous semimaringale se-up, wih a judicious specificaion of paricular models wih deerminisic volailiies and correlaions Recovery Schemes I is clear ha he sample pahs of price processes Y i are coninuous, excep for a possible disconinuiy a ime τ. Specifically, we have ha Y i τ := Y i τ Y i τ = κ i,τ Y i τ, so ha Yτ i = Yτ 1 i + κ i,τ = Ỹ τ 1 i + κ i,τ. A primary asse Y i is ermed a defaul-free asse defaulable asse, respecively if κ i = κ i, respecively. In he special case when κ i = 1, we say ha a defaulable asse Y i is subjec o a oal defaul, since is price drops o zero a ime τ and says here forever. Such an asse ceases o

14 14 Hedging of Credi Derivaives exis afer defaul, in he sense ha i is no longer raded afer defaul. This feaure makes he case of a oal defaul quie differen from oher cases, as we shall see in our sudy below. In marke pracice, i is common for a credi derivaive o deliver a posiive recovery for insance, a proecion paymen in case of defaul. Formally, he value of his recovery a defaul is deermined as he value of some underlying process, ha is, i is equal o he value a ime τ of some F-adaped recovery process Z. For example, he process Z can be equal o δ, where δ is a consan, or o g, δy where g is a deerminisic funcion and Y, is he price process of some defaul-free asse. Typically, he recovery is paid a defaul ime, bu i may also happen ha i is posponed o he mauriy dae. Le us observe ha he case where a defaulable asse Y i pays a pre-deermined recovery a defaul is covered by our se-up defined in 19. For insance, he case of a consan recovery payoff δ i a defaul ime τ corresponds o he process κ i, = δ i Y i 1 1. Under his convenion, he price Y i is governed under P by he SDE dy i = Y i µi, d + σ i, dw + δ i Y i 1 1 dm. 23 If he recovery is proporional o he pre-defaul value Y i τ, and is paid a defaul ime τ his scheme is known as he fracional recovery of marke value, we have κ i, = δ i 1 and dy i = Y i µi, d + σ i, dw + δ i 1 dm Risk-Neural Valuaion To provide a parial jusificaion for he posulaed dynamics of he price of a defaulable asse delivering a recovery, le us sudy a oy example wih wo asses: a savings accoun wih consan ineres rae r and a defaulable asse Y represened by a defaulable claim X, Z, τ. In his oy model, he only source of noise is he defaul ime, hence, he only relevan filraion is H in oher words, he reference filraion F is rivial. We assume ha by choosing oday s prices of a large family liquidly raded defaulable asses, he marke implicily specifies a maringale measure Q, equivalen o he hisorical probabiliy P. More precisely, he probabiliy disribuion of τ under an equivalen maringale measure e.m.m. Q can be inferred from marke daa. We are hus ineresed in he dynamics of he price process of X, Z, τ under Q. I is worh noing ha in his subsecion we adop a oally differen perspecive han in he res of he presen paper. In fac, no aemp o replicae a defaulable claim is done in his secion. We assume insead ha he risk-neural defaul inensiy can be uniquely deermined from prices of raded asses, and we posulae ha he price of X, Z, τ is defined by he sandard risk-neural valuaion formula. The argumen ha formally jusifies he use of his pricing rule is ha we obain in his way an arbirage-free marke model in which Q is a maringale measure, and a defaulable claim can be considered o be an addiional raded asse. Since we do no assume here ha a defaulable claim is aainable, is spo price ha is, he price expressed in unis of cash depends explicily on he risk-neural defaul inensiy. As was menioned above, he arbirage price of a defaulable claim, when expressed in erms of radeable asses used for is replicaion, will be shown o no depend direcly on real-world or risk-neural defaul inensiy. To conclude, he raionale for he calculaions given below, is ha we srive here o jusify he dynamics of prices of primary asses in our model. The risk-neural valuaion considered in his subsecion is no suppored by replicaion-based argumens, and hus i is no surprising ha i exhibis specific feaures ha are no presen in he replicaion-based valuaion. We make he sanding assumpion ha τ admis a coninuous cumulaive disribuion funcion F under Q. Hence, he hazard funcion Γ is also coninuous, and he process M = H Γ τ is an H-maringale under Q. The following resul is sandard see, e.g., Proposiion in Bielecki and Rukowski [2].

15 T.R. Bielecki, M. Jeanblanc and M. Rukowski 15 Proposiion 2.1 Assume ha he cumulaive disribuion funcion F of τ is coninuous. Le M h be an H-maringale given by M h = E Q hτ H for some Borel measurable funcion h : R + R such ha he random variable hτ is Q-inegrable. Then where we wrie M h = M h + hu gu d M u = M h + g = e b Γ E Q 1{<τ} hτ. hu M h u d M u, 25 Remark 2.1 Using he above proposiion, i can be easily shown ha on Ω, G T we have dp = E T ζu d M u dq, for some H-predicable process ζ Price Dynamics of a Survival Claim X,, τ. In wha follows, we shall refer o a defaulable claim of he form X,, τ as a survival claim. By virue of he risk-neural valuaion formula, he price of he payoff 1 {T <τ} X ha seles a ime T equals, for every [, T ], Y = e r E Q 1 {T <τ} e rt X H. Noe ha X is F T -measurable, and hus consan since he σ-field F T is rivial. To find he dynamics of he price process, i suffices o apply Proposiion 2.1 o he funcion hu = 1 {u>t } e rt X. For he Q-maringale M h = e r Y, we hus ge, for every [, T ], e r Y = Y e ru Y u d M u. Suppose ha Γ = γu du. Then an applicaion of Iô s formula yields dy = ry d Y d M = r + 1 {<τ} γ Y d Y dh. 26 We deal here wih an example of a defaulable asse ha is subjec o he oal defaul, meaning ha is price vanishes a and afer defaul Price Dynamics of a Recovery Claim, Z, τ. Recall ha our sandard convenion sipulaes ha he recovery Z is paid a he ime of defaul. Hence, he price process Y of, Z, τ is given by he expression Y = e r E Q 1 {T τ} e rτ Zτ H. We now have hu = 1 {u T } e ru Zu. Consequenly, e r Y = Y + e ru Zu e ru Y u d Mu. By applying Iô s formula, we conclude ha ha he dynamics under Q of an asse ha delivers Zτ a ime τ are dy = ry d + Z Y d M = r + 1 {<τ} γ Y d 1 {<τ} Z γ d + Z Y dh.

16 16 Hedging of Credi Derivaives Price Dynamics of a Defaulable Claim X, Z, τ. By combining he formula above wih 26, and using Remark 2.1 ogeher wih Girsanov s heorem, we arrive a he following resul. Proposiion 2.2 The price process Y of a defaulable claim X, Z, τ saisfies under Q wih he iniial condiion dy = ry d + Z Y d M Y = E Q 1{T <τ} e rt X + 1 {T τ} e rτ Zτ = e rt +b ΓT X + Under he saisical probabiliy P, he price process Y saisfies T dy = ry + 1 {<τ} Z Y γζ d + Z Y dm, where he G-maringale M under P equals M = M + 1 {u<τ} γuζu du. Zu γue b Γu du. Remark 2.2 Proposiion 2.2 can be exended o he case when he recovery is random, and is given in he feedback form as Z = g, Y for some funcion g, y, which is Lipschiz coninuous wih respec o y. Assume, for insance, ha he claim is subjec o he fracional recovery of marke value, so ha Z = δy for some consan δ. If, in addiion, ζ and γ are consan, hen we obain cf. 24 dy = Y r + 1 {<τ} δ 1 γζ d + δ 1 dm. Noe ha here he drif coefficien µ = r +1 {<τ} δ 1 γζ in dynamics of Y follows a G-predicable process, bu i is no F-predicable. However, he drif of he pre-defaul value Ỹ is simply r. 3 Trading Sraegies in a Semimaringale Se-up We consider rading wihin he ime inerval [, T ] for some finie horizon dae T >. For he sake of exposiional clariy, we resric our aenion o he case where only hree primary asses are raded. The general case of k raded asses was examined by Bielecki e al. [4]. We firs recall some general properies, which do no depend on he choice of specific dynamics of asse prices. In his secion, we consider a fairly general se-up. In paricular, processes Y i, i = 1, 2, 3, are assumed o be nonnegaive semi-maringales on a probabiliy space Ω, G, P endowed wih some filraion G. We assume ha hey represen spo prices of raded asses in our model of he financial marke. Neiher he exisence of a savings accoun, nor he marke compleeness are assumed, in general. Our goal is o characerize coningen claims which are hedgeable, in he sense ha hey can be replicaed by coninuously rebalanced porfolios consising of primary asses. Here, by a coningen claim we mean an arbirary G T -measurable random variable. We work under he sandard assumpions of a fricionless marke. 3.1 Unconsrained Sraegies Le φ = φ 1, φ 2, φ 3 be a rading sraegy; in paricular, each process φ i is predicable wih respec o he filraion G. The wealh of φ equals V φ = φ i Y i, [, T ],

17 T.R. Bielecki, M. Jeanblanc and M. Rukowski 17 and a rading sraegy φ is said o be self-financing if V φ = V φ + φ i u dy i u, [, T ]. Le Φ sand for he class of all self-financing rading sraegies. We shall firs prove ha a selffinancing sraegy is deermined by is iniial wealh and he wo componens φ 2, φ 3. To his end, we posulae ha he price of Y 1 follows a sricly posiive process, and we choose Y 1 as a numéraire asse. We shall now analyze he relaive values: Lemma 3.1 i For any φ Φ, we have V 1 φ := V φy 1 1, Y i,1 := Y i Y 1 1. V 1 φ = V 1 φ + i=2 φ i u dy i,1 u, [, T ]. ii Conversely, le X be a G T -measurable random variable, and le us assume ha here exiss x R and G-predicable processes φ i, i = 2, 3 such ha T X = YT 1 x + φ i u dyu i,1. 27 i=2 Then here exiss a G-predicable process φ 1 such ha he sraegy φ = φ 1, φ 2, φ 3 is self-financing and replicaes X. Moreover, he wealh process of φ i.e. he ime- price of X saisfies V φ = V 1 Y 1, where V 1 = x + φ i u dyu i,1, [, T ]. 28 i=2 Proof. The proof of par i is given, for insance, in Proer [31]. In he case of coninuous semimaringales, his is a well-known resul; for disconinuous processes, he proof is no much differen. We reproduce i here for he reader s convenience. Le us firs inroduce some noaion. As usual, [X, Y ] sands for he quadraic covariaion of he wo semi-maringales X and Y, as defined by he inegraion by pars formula: X Y = X Y + X u dy u + Y u dx u + [X, Y ]. For any càdlàg i.e., RCLL process Y, we denoe by Y = Y Y he size of he jump a ime. Le V = V φ be he value of a self-financing sraegy, and le V 1 = V 1 φ = V φy 1 1 be is value relaive o he numéraire Y 1. The inegraion by pars formula yields dv 1 = V dy Y 1 1 dv + d[y 1 1, V ]. From he self-financing condiion, we have dv = 3 φi dy i. Hence, using elemenary rules o compue he quadraic covariaion [X, Y ] of he wo semi-maringales X, Y, we obain dv 1 = φ 1 Y 1 dy φ 2 Y 2 dy φ 3 Y 3 dy Y 1 φ 1 dy 1 + Y 1 φ 2 dy 1 + Y 1 φ 3 dy 1 + φ 1 d[y 1 1, Y 1 ] + φ 2 d[y 1 1, Y 2 ] + φ 3 d[y 1 1, Y 1 ] = φ 1 Y 1 dy Y 1 dy 1 + d[y 1 1, Y 1 ] + φ 2 Y 2 dy Y 1 dy 1 + d[y 1 1, Y 2 ] + φ 3 Y 3 dy Y 1 dy 1 + d[y 1 1, Y 3 ].

18 18 Hedging of Credi Derivaives We now observe ha and Consequenly, Y 1 dy Y 1 1 dy 1 + d[y 1 1, Y 1 ] = dy 1 Y 1 1 = Y i dy Y 1 1 dy i + d[y 1 1, Y i ] = dy 1 1 Y i. dv 1 = φ 2 dy 2,1 + φ 3 dy 3,1, as was claimed in par i. We now proceed o he proof of par ii. We assume ha 27 holds for some consan x and processes φ 2, φ 3, and we define he process V 1 by seing cf. 28 V 1 = x + Nex, we define he process φ 1 as follows: φ 1 = V 1 where V = V 1 Y 1. Since dv 1 = 3 From he equaliy i follows ha dv = V 1 dy 1 + i=2 i=2 i=2 φi dy i,1 φ i u dy i,1 u, [, T ]. φ i Y i,1 = Y 1 1 V, we obain dv = dv 1 Y 1 = V dy Y dv 1 1 = V 1 dy 1 + i=2 i=2 φ i Y i, + d[y 1, V 1 ] φ i Y 1 dy i,1 + d[y 1, Y i,1 ]. dy i = dy i,1 Y 1 = Y i,1 dy 1 + Y 1 dy i,1 + d[y 1, Y i,1 ], i=2 φ i dy i Y i,1 dy 1 = V 1 i=2 φ i Y i,1 dy 1 + and our aim is o prove ha dv = 3 φi dy i. The las equaliy holds if φ 1 = V 1 i=2 φ i Y i,1 = V 1 i=2 i=2 φ i dy i, φ i Y i,1, 29 i.e., if V 1 = 3 i=2 φi Y i,1, which is he case from he definiion 28 of V 1. Noe also ha from he second equaliy in 29 i follows ha he process φ 1 is indeed G-predicable. Finally, he wealh process of φ saisfies V φ = V 1 Y 1 for every [, T ], and hus V T φ = X. We say ha a self-financing sraegy φ replicaes a claim X G T if or equivalenly, X = φ i T YT i = V T φ, X = V φ + T φ i dy i. Suppose ha here exiss an e.m.m. for some choice of a numéraire asse, and le us resric our aenion o he class of all admissible rading sraegies, so ha our model is arbirage-free.

19 T.R. Bielecki, M. Jeanblanc and M. Rukowski 19 Assume ha a claim X can be replicaed by some admissible rading sraegy, so ha i is aainable or hedgeable. Then, by definiion, he arbirage price a ime of X, denoed as π X, equals V φ for any admissible rading sraegy φ ha replicaes X. In he conex of Lemma 3.1, i is naural o choose as an e.m.m. a probabiliy measure Q 1 equivalen o P on Ω, G T and such ha he prices Y i,1, i = 2, 3, are G-maringales under Q 1. If a coningen claim X is hedgeable, hen is arbirage price saisfies π X = Y 1 E Q 1XY 1 T 1 G. We emphasize ha even if an e.m.m. Q 1 is no unique, he price of any hedgeable claim X is given by his condiional expecaion. Tha is o say, in case of a hedgeable claim hese condiional expecaions under various equivalen maringale measures coincide. In he special case where Y 1 = B, T is he price of a defaul-free zero-coupon bond wih mauriy T abbreviaed as ZC-bond in wha follows, Q 1 is called T -forward maringale measure, and i is denoed by Q T. Since BT, T = 1, he price of any hedgeable claim X now equals π X = B, T E QT X G. 3.2 Consrained Sraegies In his secion, we make an addiional assumpion ha he price process Y 3 is sricly posiive. Le φ = φ 1, φ 2, φ 3 be a self-financing rading sraegy saisfying he following consrain: 2 φ i Y i = Z, [, T ], 3 for a predeermined, G-predicable process Z. In he financial inerpreaion, equaliy 3 means ha a porfolio φ is rebalanced in such a way ha he oal wealh invesed in asses Y 1, Y 2 maches a predeermined sochasic process Z. For his reason, he consrain given by 3 is referred o as he balance condiion. Our firs goal is o exend par i in Lemma 3.1 o he case of consrained sraegies. Le ΦZ sand for he class of all admissible self-financing rading sraegies saisfying he balance condiion 3. They will be someimes referred o as consrained sraegies. Since any sraegy φ ΦZ is self-financing, from dv φ = 3 φi dy i, we obain V φ = φ i Y i By combining his equaliy wih 3, we deduce ha Le us wrie Y i,3 V φ = = V φ φ i Y i = Z + φ 3 Y i. φ i Y i. = Y i Y 3 1, Z 3 = Z Y 3 1. The following resul exends Lemma 1.7 in Bielecki e al. [3] from he case of coninuous semi-maringales o he general case see also [4]. I is apparen from Proposiion 3.1 ha he wealh process V φ of a sraegy φ ΦZ depends only on a single componen of φ, namely, φ 2. Proposiion 3.1 The relaive wealh V 3 φ = V φy 3 1 of any rading sraegy φ ΦZ saisfies V 3 φ = V 3 φ + φ 2 u u Y 2,3 u dy 2,3 Y 1,3 u dy 1,3 u + Zu 3 Y 1,3 dyu 1,3. 31 u

20 2 Hedging of Credi Derivaives Proof. Le us consider discouned values of price processes Y 1, Y 2, Y 3, wih Y 3 aken as a numéraire asse. By virue of par i in Lemma 3.1, we hus have The balance condiion 3 implies ha and hus V 3 φ = V 3 φ φ i Y i,3 = Z 3, By insering 33 ino 32, we arrive a he desired formula 31. φ i u dy i,3 u. 32 φ 1 = Y 1,3 1 Z 3 φ 2 Y 2,3. 33 The nex resul will prove paricularly useful for deriving replicaing sraegies for defaulable claims. Proposiion 3.2 Le a G T -measurable random variable X represen a coningen claim ha seles a ime T. Assume ha here exiss a G-predicable process φ 2, such ha T T X = YT 3 x + φ 2 dy Z 3 + Y 1,3 dy 1,3. 34 Then here exis G-predicable processes φ 1 and φ 3 such ha he sraegy φ = φ 1, φ 2, φ 3 belongs o ΦZ and replicaes X. The wealh process of φ equals, for every [, T ], V φ = Y 3 x + φ 2 u dyu Zu 3 + Y 1,3 dyu 1,3. 35 u Proof. As expeced, we firs se noe ha he process φ 1 is a G-predicable process φ 1 = 1 Y 1 Z φ 2 Y 2 and V 3 = x + φ 2 u dyu Zu 3 + Y 1,3 dyu 1,3. u Arguing along he same lines as in he proof of Proposiion 3.1, we obain Now, we define φ 3 = V 3 V 3 = V φ i u dy i,3 u. φ i Y i,3 = Y 3 1 V where V = V 3 Y 3. As in he proof of Lemma 3.1, we check ha φ 3 = V 3 2 φ i Y i,3, 2 φ i Y i, and hus he process φ 3 is G-predicable. I is clear ha he sraegy φ = φ 1, φ 2, φ 3 is self-financing and is wealh process saisfies V φ = V for every [, T ]. In paricular, V T φ = X, so ha φ replicaes X. Finally, equaliy 36 implies 3, and hus φ belongs o he class ΦZ. Noe ha equaliy 34 is a necessary by Lemma 3.1 and sufficien by Proposiion 3.2 condiion for he exisence of a consrained sraegy ha replicaes a given coningen claim X. 36