VALUATION AND HEDGING OF DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL

Transcription

1 VALUATION AND HEDGING OF DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Séphane Crépey Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France and Europlace Insiue of Finance Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, NSW 252, Ausralia and Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology -661 Warszawa, Poland December 23, 27 The research of T.R. Bielecki was suppored by NSF Gran and Moody s Corporaion gran The research of S. Crépey was suppored by Io33 and he 25 Faculy Research Gran PS6987. The research of M. Jeanblanc was suppored by Io33 and Moody s Corporaion gran The research of M. Rukowski was suppored by he 27 Faculy Research Gran PS12918.

2 2 Defaulable Game Opions in a Hazard Process Model 1 Inroducion The goal of his work is o analyze valuaion and hedging of defaulable conracs wih game opion feaures wihin a hazard process model of credi risk. Our moivaion for considering American or game clauses ogeher wih defaulable feaures of an opion is no han much a ques for generaliy, bu raher he fac ha he combinaion of early exercise feaures and defaulabiliy is an inrinsic feaure of some acively raded asses. I suffices o menion here he imporan class of converible bonds, which were sudied by, among ohers, Andersen and Buffum [2], Ayache e al. [3], Bielecki e al. [4], Davis and Lischka [15], Kallsen and Kühn [31], or Kwok and Lau [35]. In Bielecki e al. [4], we formally defined a defaulable game opion, ha is, a financial conrac ha can be seen as an inermediae case beween a general mahemaical concep of a game opion and much more specific converible bond wih credi risk. We concenraed here on developing a fairly general framework for valuing such conracs. In paricular, building on resuls of Kifer [33] and Kallsen and Kühn [31], we showed ha he sudy of an arbirage price of a defaulable game opion can be reduced o he sudy of he value process of he relaed Dynkin game under some risk-neural measure Q for he primary marke model. In his sochasic game, he issuer of a game opion plays he role of he minimizer and he holder of he maximizer. In [4], we deal wih a general marke model, which was assumed o be arbirage-free, bu no necessarily complee, so ha he uniqueness of a risk-neural or maringale measure was no posulaed. In addiion, alhough he defaul ime was inroduced, i was lef largely unspecified. An explici specificaion of he defaul ime will be an imporan componen of he model considered in his work. As is well known, here are wo main approaches o modeling of defaul risk: he srucural approach and he reduced-form approach. In he laer approach, also known as he hazard process approach, he defaul ime is modeled as an exogenous random variable wih no reference o any paricular economic background. One may objec o reduced-form models for heir lack of clear reference o economic fundamenals, such as he firm s asse-o-deb raio. However, he possibiliy of choosing various parameerizaions for he coefficiens and calibraing hese parameers o any se of CDS spreads and/or implied volailiies makes hem very versaile modeling ools, well-suied o price and hedge derivaives consisenly wih plain-vanilla insrumens. I should be acknowledged ha srucural models, wih heir sound economic background, are beer suied for inference of reliable deb informaion, such as: risk-neural defaul probabiliies or he presen value of he firm s deb, from he equiies, which are he mos liquid among all financial insrumens. Bu he srucure of hese models, as rich as i may be and which can include a lis of facors such as sock, spreads, defaul saus, credi evens, ec. is never rich enough o yield consisen prices for a full se of CDS spreads and/or implied volailiies of relaed opions. As we ulimaely aim o specify models for pricing and hedging conracs wih opional feaures in paricular, converible bonds, we favor he reduced-form approach in he sequel. 1.1 Ouline of he Paper From he mahemaical perspecive, he goal of he presen paper is wofold. Firs, we wish o specialize our previous valuaion resuls o he hazard process se-up, ha is, o a version of he reduced-form approach, which is slighly more general han he inensiy-based se-up. Hence we posulae ha filraion G modeling he informaion flow for he primary marke admis he represenaion G = H F, where he filraion H is generaed by he defaul indicaor process H = 1 { τd } and F is some reference filraion. The main ool employed in his secion is he effecive reducion of he informaion flow from he full filraion G o he reference filraion F. The main resuls in his par are Theorems 3.3 and 3.4, which give convenien pricing formulas wih respec o he reference filraion F. The second goal is o sudy he issue of hedging of a defaulable game opion in he hazard process se-up. Some previous aemps o analyze hedging sraegies for defaulable converible bonds were done by Andersen and Buffum [2] and Ayache e al. [3], who worked direcly wih

3 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 3 suiable variaional inequaliies wihin he Markovian inensiy-based se-up. Our preliminary resuls for hedging sraegies in a hazard process se-up, Proposiions 4.1 and 4.2, can be informally saed as follows: under he assumpion ha a relaed doubly refleced BSDE admis a soluion Θ, M, K under some risk-neural measure Q, for which various ses of sufficien condiions are given in he lieraure, he sae-process Θ of he soluion muliplied by he defaul indicaor process is he minimal super-hedging price up o a G, Q-sigma or local maringale cos process. More specific properies of hedging sraegies are subsequenly analyzed in Proposiions 4.4 and 4.5, in which we resor in he general se-up of his paper o suiable Galchouk-Kunia- Waanabe decomposiions of a soluion o he relaed doubly refleced BSDE and discouned prices of primary asses. I is noeworhy ha hese decomposiions, hough seemingly raher absrac in a general se-up considered here, are by no means arificial. On he conrary, hey arise naurally in he conex of paricular Markovian models ha are sudied in he follow-up paper by Bielecki e al. [6] see also [7]. We conclude he paper by menioning some alernaive approaches o hedging of game opions. 1.2 Convenions and Sanding Noaion We use hroughou his paper he vecor as opposed o componenwise sochasic inegraion, as developed in Cherny and Shiryaev [1] see also Chaelain and Sricker [9] and Jacod [28]. Given a sochasic basis saisfying he usual condiions, an R d -valued semimaringale inegraor X and an R 1 d -valued row vecor predicable inegrand H, he noion of vecor sochasic inegral H dx allows one o ake ino accoun possible inerferences of differen componens of a mulidimensional process. Well-defined vecor sochasic inegrals include, in paricular, all inegrals wih a predicable and locally bounded inegrand e.g., any inegrand of he form H = Y where Y is an adaped càdlàg process, see [27, Theorem 7.7]. Even in he one-dimensional case, he concep of vecor sochasic inegral is indeed more general han a sochasic inegral defined as he sum of inegrals of componens of H wih respec o he relaed componens of X, all supposed o be well defined in he classic sense. The usual properies of sochasic inegral, such as: lineariy, associaiviy, invariance wih respec o equivalen changes of measures and wih respec o inclusive changes of filraions, are known o hold for he vecor sochasic inegral. Moreover, unlike oher kinds of sochasic inegrals, vecor sochasic inegrals form a closed space in a suiable opology. This feaure makes hem well adaped o many problems arising in he mahemaical finance, such as Fundamenal Theorems of Asse Pricing see [1, 4] and Secion 2. By defaul, we denoe by he inegrals over, ]. Oherwise, we explicily specify he domain of inegraion as a subscrip of. Noe also ha, depending on he conex, τ will sand eiher for a generic sopping ime or i will be given as τ = τ p τ c for some specific sopping imes τ c and τ p. Finally, we consider he righ-coninuous and compleed versions of all filraions, so ha hey saisfy he so-called usual condiions. 2 Semimaringale Se-Up Afer recalling some fundamenal valuaion resuls from Bielecki e al. [4], we will examine basic feaures of hedging sraegies for defaulable game opions ha are valid in a general semimaringale se-up. The imporan special case of a hazard process framework is sudied in he nex secion. We assume hroughou ha he evoluion of he underlying primary marke is modeled in erms of sochasic processes defined on a filered probabiliy space Ω, G, P, where P denoes he saisical probabiliy measure. Specifically, we consider a primary marke composed of he savings accoun and of d risky asses, such ha, given a finie horizon dae T > : he discoun facor process β, ha is, he inverse of he savings accoun, is a G-adaped, finie

4 4 Defaulable Game Opions in a Hazard Process Model variaion, posiive, coninuous and bounded process, he risky asses are G-semimaringales wih càdlàg sample pahs. The primary risky asses, wih R d -valued price process X, pay dividends, whose cumulaive value process, denoed by D, is assumed o be a G-adaped, càdlàg and R d -valued process of finie variaion. Given he price process X, we define he cumulaive price X of primary risky asses as X = X + β 1 [,] β u dd u. 1 In he financial inerpreaion, he las erm in 1 represens he curren value a ime of all dividend paymens from he asses over he period [, ], under he assumpion ha all dividends are immediaely reinvesed in he savings accoun. We assume ha he primary marke model is free of arbirage opporuniies, hough presumably incomplee. In view of he Firs Fundamenal Theorem of Asse Pricing see [16, 1], and accouning in paricular for he dividends, his means ha here exiss a risk-neural measure Q M, where M denoes he se of probabiliy measures Q P for which β X is a sigma maringale wih respec o G under Q. Given a sandard sochasic basis, an R d -valued process Y is called a sigma maringale if here exiss an R d -valued local maringale M and an R d -valued process H such ha H i is a predicable [1, secion 3] M i -inegrable process and Y i = Y i + H i dm i for i = 1,..., d see Lemma 5.1ii in [1]. We shall use he following well-known properies of sigma maringales. Proposiion 2.1 [1, 4, 29] i The class of sigma maringales is a vecor space conaining all local maringales. I is sable wih respec o sochasic inegraion, ha is, if Y is a sigma maringale and H is a predicable Y -inegrable process hen he inegral H dy is a sigma maringale. ii Any bounded from below sigma maringale is a supermaringale and any locally bounded sigma maringale is a local maringale. Remark 2.1 In he same vein, we recall ha sochasic inegraion of predicable locally bounded inegrands preserves local maringales see, e.g., [4]. We now inroduce he concep of a dividend paying game opion see also Kifer [33]. In broad erms, a dividend paying game opion, wih he incepion dae and he mauriy dae T, is a conrac wih he following cash flows ha are paid by he issuer of he conrac and received by is holder: a dividend sream wih he cumulaive dividend a ime denoed by D, a pu paymen L made a ime = τ p if τ p τ c and τ p < T ; ime τ p is called he pu ime and is chosen by he holder, a call paymen U made a ime = τ c provided ha τ c < τ p T ; ime τ c, known as he call ime, is chosen by he issuer and may be subjec o he consrain ha τ c τ, where τ is he lifing ime of he call proecion, a paymen a mauriy ξ made a mauriy dae T provided ha T τ p τ c. Of course, here is also he iniial cash flow, namely, he purchasing price of he conrac, which is paid a he iniiaion ime by he holder and received by he issuer. Le us now be given an [, + ]-valued G-sopping ime τ d represening he defaul ime of a reference eniy, wih defaul indicaor process H = 1 {τd }. A defaulable dividend paying game opion is a dividend paying game opion such ha he conrac is erminaed a τ d, if i has no been pu or called and has no maured before. In paricular, here are no more cash flows relaed o his conrac afer he defaul ime. In his seing, he dividend sream D is assumed o include a possible recovery paymen made a he defaul ime. We are ineresed in he problem of he ime evoluion of an arbirage price of he game opion. Therefore, we formulae he problem in a dynamic way by pricing he game opion a any ime [, T ]. We wrie G T o denoe he se of all G-sopping imes wih values in [, T ] and we le Ḡ T

5 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 5 sand for he se {τ GT ; τ τ d τ τ d }, where he lifing ime of a call proecion τ belongs o GT. The sopping ime τ G T is used o model he resricion ha he issuer of a game opion may be prevened from making a call on some random ime inerval [, τ. We are now in he posiion o sae he formal definiion of a defaulable game opion. Definiion 2.2 A defaulable game opion wih lifing ime of he call proecion τ GT is a game opion wih he ex-dividend cumulaive discouned cash flows β π; τ p, τ c given by he formula, for any [, T ] and τ p, τ c GT Ḡ T, β π; τ p, τ c = β u dd u + 1 {τ<τd }β τ 1 {τ=τp<t }L τp + 1 {τ<τp}u τc + 1 {τ=t } ξ, 2 where τ = τ p τ c and he dividend process D = D [,T ] equals D = 1 H u dc u + [,] [,] R u dh u 3 for some coupon process C = C [,T ], which is a G-predicable, real-valued, càdlàg process wih bounded variaion, and some real-valued, G-predicable recovery process R = R [,T ], he pu paymen L = L [,T ] and he call paymen U = U [,T ] are G-adaped, real-valued, càdlàg processes, he inequaliy L U holds for every [τ d τ, τ d T, he paymen a mauriy ξ is a G T -measurable, real-valued random variable. I is clear ha, for any fixed, π; τ p, τ c is a G τ τd -measurable random variable. We furher assume ha R, L and ξ are bounded from below, so ha here exiss a consan c such ha, for every [, T ], β u dd u + 1 {<τd }β 1{<T } L + 1 {=T } ξ c. 4 [,] Symmerically, we shall someimes addiionally assume ha R, U and ξ are bounded from below and from above, or ha 4 is supplemened by, for every [, T ], β u dd u + 1 {<τd }β 1{<T } U + 1 {=T } ξ c. 5 [,] 2.1 Valuaion of a Defaulable Game Opion We will sae he following fundamenal pricing resul wihou proof, referring he ineresed reader o [4] for more deails. The goal is o characerize he se of arbirage ex-dividend prices of a game opion in erms of values of relaed Dynkin games [2, 34, 36]. The noion of an arbirage price of a game opion referred o in Theorem 2.2 is he dynamic noion of arbirage price for game opions, defined in Kallsen and Kühn [31], exended o he case of dividend-paying primary asses and dividend-paying game opions by resoring o he ransformaion of prices ino cumulaive prices. Noe ha in he sequel, he saemen Π [,T ] is an arbirage price for he game opion is in fac o be undersood as X, Π [,T ] is an arbirage price for he exended marke consising of he primary marke and he game opion. Theorem 2.2 Assume ha a process Π is a G-semimaringale and here exiss Q M such ha Π is he value of he Dynkin game relaed o a game opion, specifically, esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G = Π 6 = essinf τc Ḡ T esssup τ p G T E Q π; τp, τ c G, [, T ].

6 6 Defaulable Game Opions in a Hazard Process Model Then Π is an arbirage ex-dividend price of he game opion, called he Q-price of he game opion. The converse holds rue hus any arbirage price is a Q-price for some Q M under he following inegrabiliy assumpion esssup Q M E Q sup β u dd u + 1 {<τd }β 1{<T } L + 1 {=T } ξ G <, a.s. 7 [,T ] [,] Noe ha defaulable American or European opions can be seen as special cases of defaulable game opions. Definiion 2.3 A defaulable American opion is a defaulable game opion wih τ = T. A defaulable European opion is a defaulable American opion such ha β L β T LT for every [, T ]. by In view of Theorem 2.2, he cash flows φ of a defaulable European opion can be redefined β φ = T β u dd u + 1 {τd >T }β T ξ, [, T ] Hedging of a Defaulable Game Opion We adop he definiion of hedging game opions semming from successive developmens, saring from he hedging of American opions examined by Karazas [32], and subsequenly followed by El Karoui and Quenez [23], Kifer [33], Ma and Cvianić [37] and Hamadène [24] see also Schweizer [41]. This definiion will be laer shown o be consisen wih he concep of arbirage pricing of a defaulable game opion. Recall ha X resp. X is he price process resp. cumulaive price process of primary raded asses, as given by 1. The following definiions are sandard, accouning for he dividends on he primary marke. Definiion 2.4 By a self-financing primary rading sraegy we mean any pair V, ζ such ha: V is a G -measurable real-valued random variable represening he iniial wealh, ζ is an R 1 d -valued, β X-inegrable process represening holdings in primary risky asses. Definiion 2.5 The wealh process V of a primary rading sraegy V, ζ is given by he formula, for [, T ], β V = β V + ζ u dβ u Xu. 9 Given he wealh process V of a primary sraegy V, ζ, we uniquely specify a G-opional process ζ by seing V = ζ β 1 + ζ X, [, T ]. The process ζ represens he number of unis held in he savings accoun a ime, when we sar from he iniial wealh V and we use he sraegy ζ in he primary risky asses. Recall ha we denoe τ = τ p τ c. Definiion 2.6 i An issuer hedge wih cos process ρ for he game opion wih ex-dividend cumulaive discouned cash flows βπ cf. 2 is represened by a quadruple V, ζ, ρ, τ c such ha: V, ζ is a primary sraegy wih he wealh process V given by 9, a cos process ρ is a real-valued G-semimaringale wih ρ =, a fixed call ime τ c belongs o Ḡ T, he following inequaliy is valid, for every pu ime τ p GT, β τ V τ + β u dρ u β π; τ p, τ c, a.s. 1

7 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 7 ii A holder hedge wih cos process ρ for he game opion is a quadruple V, ζ, ρ, τ p such ha: V, ζ is a primary sraegy wih he wealh process V given by 9, a cos process ρ is a real-valued G-semimaringale wih ρ =, a fixed pu ime τ p belongs o GT, he following inequaliy is valid, for every call ime τ c Ḡ T, β τ V τ + β u dρ u β π; τ p, τ c, a.s. 11 Issuer or holder hedges a no cos ha is, wih ρ = are hus in effec issuer or holder superhedges. A more explici form of condiion 1 reads: for a fixed τ c Ḡ T and every τ p G T V τ + β 1 τ β 1 τ β u dρ u 12 β u dd u + 1 {τ<τd } 1 {τ=τp<t }L τp + 1 {τ<τp}u τc + 1 {τp=τ c=t }ξ, a.s. The lef-hand side in he las formula is he value process of a sraegy wih cos ρ, when he players adop he respecive exercise policies τ p and τ c, whereas he righ-hand side represens he payoff o be done by he issuer, including pas dividends and recovery a defaul. By he righ-coninuiy of he involved processes, condiion 12 is in urn equivalen o he following saemen: for a fixed call ime τ c Ḡ T chosen by he issuer, he inequaliy V τc + β 1 τ c τc = β 1 τ c τc β u dρ u lim π;, τ c 13 + β u dd u + 1 { τc<τ d } 1 {<τc}l + 1 {τc <T }U τc + 1 {=τc=t }ξ, is saisfied almos surely, for any [, T ] or, inerchangeably, for any [, T ], a.s.. Likewise, condiion 11 is equivalen o: for a fixed pu ime τ p GT chosen by he holder, he inequaliy V τp + β 1 τ p = β 1 τ p p p β u dρ u lim π; τ p, = π; τ p, {τp <T }L τp + 1 {<τp}u + 1 {=τc=t }ξ, β u dd u 1 {τp <τ d } holds almos surely, for any [ τ, T ] or, inerchangeably, for any [, T ], a.s.. Remark 2.7 i The process ρ is o be inerpreed as he running financing cos, ha is, he amoun of cash added o if dρ or wihdrawn from if dρ he hedging porfolio in order o ge a perfec, bu no longer self-financing, hedge. In he special case where ρ is a G-maringale we hus recover he noion of mean self-financing hedge in he sense of Schweizer [41]. ii Regarding he admissibiliy of hedging sraegies see, e.g., Delbaen and Schachermayer [16], noe ha he l.h.s. in formula 1 discouned wealh process inclusive of financing coss is bounded from below for any issuer hedge wih a cos V, ζ, ρ, τ c. Likewise, in he case of a bounded payoff π ha is, assuming 5, he l.h.s. in formula 11 is bounded from below for any holder hedge wih a cos V, ζ, ρ, τ p. The class of all hedges wih semimaringale cos processes is obviously oo large for any pracical purpose, so we will resric our aenion o hedges wih a G-sigma maringale cos ρ under a paricular risk-neural measure Q.

8 8 Defaulable Game Opions in a Hazard Process Model Assumpion 2.8 In he sequel, we work under a fixed, bu arbirary, risk-neural measure Q M. All he measure-dependen noions like local maringale, compensaor, ec., implicily refer o he probabiliy measure Q. In paricular, we define V c resp. V p as he se of iniial values V for which here exiss an issuer resp. holder hedge of he game opion wih he iniial value V resp. V and wih a G-sigma maringale cos under Q. The following resul gives some preliminary conclusions regarding he iniial cos of a hedging sraegy for he game opion under he presen, raher weak, assumpions. In Proposiion 4.2, we shall see ha, under sronger assumpions, he infima are aained and hus we obain equaliies, raher han merely inequaliies, in 15 and 16. Lemma 2.3 i We have by convenion, essinf = ii If inequaliy 5 is valid hen essinf τc Ḡ T esssup τ p G T E Q π; τp, τ c G essinfv V c V, a.s. 15 esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G essinfv V p V, a.s. 16 Proof. i Assume ha for some sopping ime τ c Ḡ T he quadruple V, ζ, ρ, τ c is an issuer hedge wih a G-sigma maringale cos ρ for he game opion. Then 9 and 1 imply ha, for any [, T ], τc τc β V = β τc V τc ζ u dβ u Xu β π;, τ c ζu dβ u Xu + β u dρ u. 17 The sochasic inegral ζ u dβ u Xu wih respec o a G-sigma maringale β X is a G-sigma maringale. Hence he sopped process τ c ζ u dβ u Xu and he process τc ζu dβ u Xu + β u dρ u are G-sigma maringales as well. The laer process is bounded from below his follows from 2 4 and 17, so ha i is a bounded from below local maringale [29, p.216] and hus a supermaringale. Moreover, for any sopping ime τ p GT, he inequaliy in formula 17 sill holds wih replaced by τ p. By aking expecaions, we obain recall ha τ c is fixed and hus, by posiiviy of β, The las inequaliy yields 15. β V E Q β π; τ p, τ c G, τp G T, V essinf τc Ḡ T esssup τ p G T E Q π; τp, τ c G, a.s. ii Le V, ζ, ρ, τ p be a holder hedge wih a G-sigma maringale cos ρ for he game opion for some sopping ime τ p GT. Then 9 and 11 imply ha, for any [ τ, T ], β V = β τp V τp τp ζ u dβ u Xu β π; τ p, τp ζu dβ u Xu + β u dρ u. 18 Under condiion 5, he sochasic inegral in he las formula is bounded from below and hus we conclude, by he same argumens as in par i, ha i is a supermaringale. Consequenly, for a fixed sopping ime τ p G T, so ha and his implies 16. β V E Q β π; τ p, τ c G, a.s., τc Ḡ T, V esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G, a.s.,

9 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 9 3 Valuaion in a Hazard Process Se-Up In order o ge more explici pricing hedging resuls, we will now sudy he so-called hazard process se-up. 3.1 Sanding Assumpions Given an [, + ]-valued G-sopping ime τ d, we assume ha G = H F, where he filraion H is generaed by he process H = 1 {τd } and F is some reference filraion. As expeced, our approach will consis in effecively reducing he informaion flow from he full filraion G o he reference filraion F. Le G sand for he process G = Qτ d > F for R +. The process G is a bounded F-supermaringale, as he opional projecion on F of he non-increasing process 1 H see [3]. In he sequel, we shall work under he following sanding assumpion. Assumpion 3.1 We assume ha he process G is sricly posiive and coninuous wih finie variaion, so ha he F-hazard process Γ = lng, R +, is well defined and coninuous wih finie variaion. Remark 3.2 i If G is coninuous hen τ d is a oally inaccessible G-sopping ime see, e.g., [19], which in paricular avoids F- even G- predicable sopping imes. More precisely, he assumpion ha he process G is coninuous lies somewhere beween assuming ha τ d avoids F-sopping imes and assuming ha τ d avoids F-predicable sopping imes. ii Assuming G coninuous, he furher assumpion ha G has a finie variaion in fac implies ha G is non-increasing see Lemma A.1i. This lies somewhere beween assuming furher he sronger H Hypohesis and assuming furher ha τ d is an F-pseudo-sopping ime see Nikeghbali and Yor [39]. Recall ha he H Hypohesis means ha all square-inegrable F-maringales are G-maringales see, e.g., [8], or, by a sandard localizaion argumen, ha all F-local maringales are G-local maringales, whereas τ d is an F-pseudo-sopping ime iff all F-local maringales sopped a τ d are G-local maringales see Nikeghbali and Yor [39] and Appendix A. More deailed consequences of Assumpion 3.1 useful for his work are summarized in Lemma A.1. The nex definiion refers o some auxiliary resuls proved in Appendix A. Definiion 3.3 The F-sopping ime τ, he F -measurable random variable χ and he F-adaped or F-predicable process Ỹ inroduced in Lemmas A.2 and A.3 are called he F-represenaives of τ, χ and Y, respecively. In he conex of credi risk, where τ d represens he defaul ime of a reference eniy, hey are also known as he pre-defaul values of τ, χ and Y. To simplify he presenaion, we find i convenien o make an addiional assumpion. Sricly speaking, Assumpion 3.4 is superfluous, in he sense ha all he resuls below are rue in general; by making use of Lemmas A.2 and A.3, one can reduce he original problem o he case described in Assumpion 3.4. Since his would make he noaion heavier, wihou adding much value, we prefer o work under his sanding assumpion. Assumpion 3.4 i The discoun facor process β is F-adaped. ii The coupon process C is F-predicable. iii The recovery process R is F-predicable. iv The payoff processes L, U are F-adaped and he random variable ξ is F T -measurable. v The call proecion τ is an F-sopping ime.

10 1 Defaulable Game Opions in a Hazard Process Model 3.2 Reducion of a Filraion The nex lemma shows ha he compuaion of he lower and upper value of he Dynkin games 6 wih respec o G-sopping imes can be reduced o he compuaion of he lower and upper value wih respec o F-sopping imes. Lemma 3.1 We have esssup essinf τp GT τ E Q c Ḡ T π; τp, τ c G = esssupτp F essinf T τ c F T E Q π; τp, τ c G and essinf esssup τc Ḡ T τ E p G Q T π; τp, τ c G = essinfτc F T esssup τ E p FT Q π; τp, τ c G. Proof. For τ p, τ c GT Ḡ T, we have π; τ p, τ c = π; τ p τ d, τ c τ d = π; τ p τ d, τ c τ d = π; τ p, τ c for some sopping imes τ p, τ c FT F T, where he middle equaliy follows from Lemma A.3, and he oher wo from he definiion of π. Since, clearly, FT G T and F T Ḡ T, we conclude ha he lemma is valid. Under our assumpions, he compuaion of condiional expecaions of cash flows π; τ p, τ c wih respec o G can be reduced o he compuaion of condiional expecaions of F-equivalen cash flows π; τ p, τ c wih respec o F. Le α := β exp Γ sand for he credi-risk adjused discoun facor. Noe ha α is bounded, like β. Lemma 3.2 For any sopping imes τ p F T and τ c F T we have ha where π; τ p, τ c is given by, wih τ = τ p τ c, α π; τ p, τ c = E Q π; τp, τ c G = 1{<τd } E Q π; τp, τ c F, 19 α u dc u + R u dγ u + α τ 1{τ=τp<T }L τp + 1 {τ<τp}u τc + 1 {τ=t } ξ. 2 Proof. Formula 19 is an immediae consequence of formula 2 and Lemma A.4. Noe ha π; τ p, τ c is an F τ -measurable random variable. A comparison of formulas 2 and 2 shows ha we have effecively moved our consideraions from he original marke subjec o he defaul risk, in which cash flows are discouned according o he discoun facor β, o he ficiious defaul-free marke, in which cash flows are discouned according o he credi risk adjused discoun facor α. Recall ha he original cash flows π; τ p, τ c are given as G τ τd -measurable random variables, whereas he F-equivalen cash flows π; τ p, τ c are manifesly F τ -measurable and hey depend of he defaul ime τ d only via he hazard process Γ. For he purpose of compuaion of ex-dividend prices of a defaulable game opion hese wo marke models are equivalen. This is shown by he following resul shows, which is obained by combining Theorem 2.2 wih Lemmas 3.1 and 3.2. Theorem 3.3 Assuming condiion 7, le Π be he arbirage ex-dividend Q-price for a game opion. Then we have, for any [, T ], Π = 1 {<τd } Π, 21 where Π saisfies esssup τp F T essinf τ c F T E Q π; τp, τ c F = Π 22 = essinf τc F T esssup τ p F T E Q π; τp, τ c F.

11 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 11 Hence he Dynkin game wih cos crierion E Q π; τp, τ c F on F T F T admis he value Π, which coincides wih he pre-defaul ex-dividend price a ime of he game opion under he riskneural measure Q. The following resul is he converse of Theorem 3.3. I follows immediaely by Lemmas 3.1, 3.2 and by he if par of Theorem 2.2 noing also ha Π defined by 21 is obviously a G- semimaringale if Π is a G-semimaringale. Theorem 3.4 Le Π be he value of he Dynkin game wih he cos crierion E Q π; τp, τ c F on F T F T, for any [, T ]. Then Π defined by 21 is he value of he Dynkin game wih he cos crierion E Q π; τp, τ c G on G T Ḡ T, for any [, T ]. If, in addiion, Π is a G-semimaringale hen Π is he arbirage ex-dividend Q-price for he game opion. Theorems 3.3 and 3.4 hus allow us o reduce he sudy of a game opion o he sudy of Dynkin games 22 wih respec o he reference filraion F. 3.3 Valuaion via Doubly Refleced BSDEs In his secion, we shall characerize he arbirage ex-dividend Q-price of a game opion as a soluion o a judiciously chosen doubly refleced BSDE. To his end, we firs recall some auxiliary resuls concerning he relaionship beween Dynkin games and doubly refleced BSDEs. Given an addiional F-adaped process F of finie variaion, we consider he following doubly refleced BSDE wih he daa F, ξ, L, U, τ see Cvianić and Karazas [14], Hamadène and Hassani [25], Crépey e al. [13, 12], Bielecki e al. [6, 7]: α Θ = α T ξ + α T F T α F + T α u dk u T α u dm u, [, T ], L Θ Ū, [, T ], 23 T Θ u L u dk + u = T Ūu Θ u dk u =, where he process Ū = Ū [,T ] equals, for [, T ], Ū = 1 {< τ} + 1 { τ} U. Definiion 3.5 By a Q-soluion o he doubly refleced BSDE 23, we mean a riple Θ, M, K such ha: he sae process Θ is a real-valued, F-adaped, càdlàg process, α dm is a real-valued F-maringale vanishing a ime, K is an F-adaped coninuous finie variaion process vanishing a ime, all condiions in 23 are saisfied, where in he hird line K + and K denoe he Jordan componens of K, and where he convenion ha ± = is made in he hird line. Here by Jordan decomposiion we mean he decomposiion K = K + K, where he non-decreasing coninuous processes K + and K vanish a ime and define muually singular measures. The sae process Θ in a soluion o 23 is clearly an F-semimaringale. So here are obvious hough raher arificial cases in which 23 does no admi a soluion: i suffices o ake τ = and L = U, assumed no o be an F-semimaringale. I is also clear ha a soluion would no necessarily be unique if we did no impose he condiion of a muual singulariy of he non-negaive measures defined by K + and K see, e.g., [25, Remark 4.1]. In applicaions see [6, 13, 12, 7], he inpu process F is ypically given as a Lebesgue inegral αf = αf du and he componen M of a soluion o 23 is usually searched for in he form M = Z dn +dn for some R q -valued and real-valued square-inegrable F-maringales N and n see Secion

12 12 Defaulable Game Opions in a Hazard Process Model 4.3. For concree including Markovian specificaions of he presen se-up and sufficien condiions for he exisence and uniqueness of a soluion o 23, we refer he reader o, e.g., [13, 25, 12, 7, 14]. Basically, in any model endowed wih he maringale represenaion propery, he exisence and uniqueness of a soluion o 23 supplemened by suiable inegrabiliy condiions on he daa and he soluion is equivalen o he so-called Mokobodski condiion, namely, he exisence of a quasimaringale Z such ha L Z U on [, T ] see, in paricular, Crépey and Maoussi [13], Hamadène and Hassani [25, Theorem 4.1], and previous works in his direcion, saring wih [14]. I is hus saisfied when one of he barriers is a quasimaringale and, in paricular, when one of he barriers is given as S l where S is an Iô-Lévy process S wih square-inegrable special semimaringale decomposiion componens see [13] and l is a consan in R { }. This covers, for insance, he call payoff of a converible bond, see Bielecki e al. [4, 7]. Remark 3.6 i Since K, and hus K + and K, are coninuous, he minimaliy condiions hird line in 23 are equivalen o T Θ u L u dk + u = T Ūu Θ u dk u = 24 Indeed he relaed inegrands here and in he hird line of 23 differ on an a mos counable se whereas he inegraors define aomless measures on [, T ]; see, e.g., [13]. In he preprin version [5] of his paper we defined more general noions of ε-hedges, ha were peraining in he case where here may be jumps in he process K. Since in all exising works on doubly refleced BSDEs he process K is acually found o be a coninuous process see e.g. [13, 25, 7, 14], we impose in his paper he coninuiy of K in Definiion 3.5 and we only consider hedges, no ε-hedges. Noe, however, ha essenially all he resuls of his paper can be exended o possible jumps in K, using he generalized noion of ε-hedge defined in [5], and wih he minimaliy condiions saed as 24 insead of he hird line of 23 in Definiion 3.5. ii Since F is a given process, he BSDE 23 can be rewrien as α Θ = α T ξ + T α u dk u T α u dm u, [, T ], L Θ Û, [, T ], 25 T Θ u L u dk + u = T Ûu Θ u dk u =, where Θ = Θ + F, ξ = ξ + F T, L = L + F and Û = Ū + F. This shows ha he problem of solving 23 can be formally reduced o he case of F = wih suiably modified reflecing barriers L, Û and erminal condiion ξ. Noe ha, in spie of his formal reducion, he freedom o choose he driver of a relaed BSDE associaed wih a game opion is imporan from he poin of view of applicaions his is apparen in he follow-up papers [6, 7]; see also [5]. iii In he special case where all F-maringales are coninuous and where he F-semimaringale F and he barriers L and U are coninuous see [14, 26, 7], i is naural o look for a coninuous soluion of 23, ha is, a soluion of 23 given by a riple of coninuous processes Θ, M, K. iv In he conex of Markovian se-ups, he probabilisic BSDE approach may be complemened by a relaed analyic variaional inequaliy approach. This issue is deal wih in Bielecki e al. [6] see also [7]. Noe, however, ha he variaional inequaliy approach srongly relies on he BSDE approach. Moreover, he BSDE-based simulaion mehod is he only efficien way of numerically solving he pricing problem if he dimension of he problem number of model facors is greaer han hree or four. Indeed in such case he compuaional cos of deerminisic numerical schemes based on he variaional inequaliy approach becomes prohibiive. In order o esablish a link beween a soluion o he relaed doubly refleced BSDE and he arbirage ex-dividend Q-price of he defaulable game opion, we firs recall he general relaionship beween doubly refleced BSDEs and Dynkin games wih purely erminal cos, before applying his resul o dividend-paying game opions in he ficiious defaul-free marke in Proposiion 3.5.

13 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 13 Observe ha if Θ, M, K solves 23 hen we have, for any sopping ime τ F T, α Θ = α τ Θ τ + α τ F τ α F + α u dk u α u dm u. 26 Proposiion 3.5 Verificaion Principle for a Dynkin Game Le Θ, M, K be a soluion o 23 wih F =. Then Θ is he value of he Dynkin game wih cos crierion E Q θ; τp, τ c F on F T F T, where θ; τ p, τ c is he F τ -measurable random variable defined by α θ; τ p, τ c = α τ 1{τ=τp<T }L τp + 1 {τ=τc<τ p}u τc + 1 {τ=t } ξ, where τ = τ p τ c. Moreover, for any [, T ], he pair of sopping imes τ p, τ c F T F T given by τ p = inf { u [, T ] ; Θ u L u } T, τ c = inf { u [ τ, T ] ; Θ u U u } T, is a saddle-poin of his Dynkin game, in he sense ha we have, for any τ p, τ c F T F T, E Q θ; τp, τ c F Θ E Q θ; τ p, τ c F. 27 Proof. Excep for he presence of τ, he resul is sandard see, e.g., Lepelier and Maingueneau [36]. Le us firs check ha he righ-hand side inequaliy in 27 is valid for any τ c F T. Le τ denoe τp τ c. By definiion of τp and coninuiy of K +, we see ha K + equals on [, τ ]. Since K is non-decreasing, 26 applied o τ yields α Θ α τ Θ τ α u dm u. Taking condiional expecaions recall ha α udm u is an F-maringale, and using also he facs ha Θ τ p L τ p if τp < T, Θ τ p = ξ if τp = T and Θ τc U τc recall ha τ c F T, so ha τ c τ and Ūτ c = U τc, we obain α Θ E Q α τ Θ τ F E Q α τ 1{τ =τ p <T } L τ p + 1 {τ =τ c<τ p } U τc + 1 {τ =T }ξ F. We conclude ha Θ E Q θ; τ p, τ c F for any τc F T. This complees he proof of he righhand side inequaliy in 27. The lef-hand side inequaliy can be shown similarly. I is in fac sandard, since i does no involve τ, and hus he deails are lef o he reader. Le us now apply Proposiion 3.5 o a defaulable game opion. To his end, we firs rewrie 2 as follows where α π; τ p, τ c = α τ Fτ α F + α τ 1{τ=τp<T }L τp + 1 {τ<τp}u τc + 1 {τ=t } ξ, F := α 1 α u d D u wih D := dc u + R u dγ u. 28 [,] [,] Le us denoe by Ē equaion 25 wih F = F, ha is, T α Θ = α T ξ + α u dk u T α u dm u, [, T ], L Θ Û, [, T ], T Θ u L u dk u + = T Ûu Θ u dku =, wih ξ = ξ + F T, L = L + F and Û = Ū + F. Ē Assumpion 3.7 The doubly refleced BSDE Ē admis a soluion Θ, M, K.

14 14 Defaulable Game Opions in a Hazard Process Model Le us sress ha Assumpion 3.7, heroic as i may seem in he general hazard process se-up, is in fac a plausible assumpion in any reasonable applicaion one may hink of cf. commens following Definiion 3.5. We denoe, for [, T ], Π = Θ F, Π = 1 {<τd } Π, Π = Π + β 1 [,] β u dd u 29 and τd m = β Π + β u dk u. 3 The following lemma is crucial in wha follows Lemma 3.6i is acually he key of he proof of Proposiion 4.1 below. Lemma 3.6 i The process m given by 3 is a G-maringale sopped a τ d. ii The process Π is a G-semimaringale. iii The process β Π is a special G-semimaringale. Proof. i The riple Π, M, K saisfies 23 wih F given by F in 28. Therefore, for every [, T ], T T T α Π = α T ξ + α u d D u + α u dk u α u dm u and hus α u dm u = α Π α Π + α u dk u + Using Lemma A.4, i is easy o check ha we have, for any u T, u 1 {<τd }e Γ E Q α v dm v F = E Q m u m G. α u d D u. 31 Since he inegral α v dm v is an F-maringale, he process m is a G-maringale. I is also clear ha i is sopped a τ d. ii In view of 29, 3 and par i, he process Π is clearly a G-semimaringale. iii By 3, we have ha τd β Π = m β u dk u, 32 where m is a G-maringale, by i, and where he second erm in he righ-hand side is a G-adaped and coninuous hence G-predicable processes of finie variaion. In view of 32, and since K is coninuous, he process m given by 3 can equivalenly be redefined as he canonical G-local maringale componen of he discouned cumulaive Q-value process β Π. The processes m and β Π are easily seen o coincide on he random inerval [, τ c τ p τ d T ] and hus boh m and β Π can be inerpreed on his inerval as he discouned cumulaive Q-value of a defaulable game opion. The following resul esablishes a link beween Θ, M, K and he arbirage ex-dividend Q-price of he defaulable game opion. Proposiion 3.7 Verificaion Principle for a Defaulable Game Opion The process Π is he arbirage ex-dividend Q-price for he game opion. Moreover, for any [, T ], he saddle-poin τp, τc FT F T for he relaed Dynkin game 6 on G T Ḡ T is given by τ p = inf { u [, T ] ; Π u L u } T, τ c = inf { u [ τ, T ] ; Π u U u } T.

15 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 15 Proof. In view of 2, he presen assumpions imply ha Π is he value of he Dynkin game 22, by Proposiion 3.5, wih saddle-poin τp, τc. Therefore, by Lemmas 3.1 and 3.2, Π is he value of he Dynkin game associaed wih he game opion on GT Ḡ T, wih saddle-poin τ p, τc. Moreover, Π is a G-semimaringale, by Lemma 3.6ii. We conclude by making use of he las par in Theorem Hedging in a Hazard Process Se-up In his secion, which is he cenral par of his work, we examine in some deail he exisence and properies of hedging sraegies for defaulable game opions in a hazard process se-up. 4.1 Preliminary Resuls From now on, we shall work under Assumpion 3.7. Le hus Θ, M, K denoe a soluion o Ē and le Π and Π be defined by 29. In paricular, Π is he arbirage Q-price for he game opion by Proposiion 3.7 and he lef-hand sides in 15 and 16 are equal o Π. Finally, recall ha he G-maringale m is defined by 3. Proposiion 4.1 i Le ζ be an arbirary R 1 d -valued, β X-inegrable process and le he process ρζ be given by ρ ζ = and β dρ ζ = dm ζ dβ X. 33 Then Π, ζ, ρζ, τ c is an issuer hedge wih G-sigma local, in case β X and ζ are locally bounded maringale cos. ii Le ζ be an arbirary R 1 d -valued, β X-inegrable process and le he process ρζ be given by ρ ζ = and β d ρ ζ = dm ζ dβ X. 34 Then Π, ζ, ρζ, τ p is a holder hedge wih a G-sigma maringale local maringale, when β X and ζ are locally bounded cos process. Noe ha he equaliy ρ ζ = ρζ is valid for any process ζ, since β dρ ζ = dm + ζ dβ X = dm ζ dβ X. Proof of Proposiion 4.1. The argumens for a holder are essenially symmerical o hose for an issuer; we hus only prove par i. By Lemma 3.6i, he process ρζ is a G-sigma maringale, and a G-local maringale if β X and ζ are locally bounded processes. For he ease of noaion, we wrie ρ = ρζ. Le V denoe he wealh process of he primary sraegy Π, ζ. By combining 9 wih 33, we obain V = Π and, for [, T ] : and hus noe ha m = β Π V = β 1 m dβ V = ζ dβ X = dm β dρ β u dρ u = Π τd + β 1 β u dk u β 1 β u dρ u, 35 where he second equaliy follows from 3. Recall ha he sopping ime τc F T is given by see Proposiion 3.7 { τc = inf [ τ, T ] ; Π } U T,

16 16 Defaulable Game Opions in a Hazard Process Model In order o prove ha he quadruple Π, ζ, ρ, τc is an issuer hedge for he game opion, i is enough o show ha we have for any τ p GT, wih τ = τ p τc cf. 12: V τ + βτ 1 β u dρ u dd u 1 {τ<τd } 1 {τ=τp<t }L τp + 1 {τ<τp}u τ c + 1 {τp=τc =T } ξ. 36 From he definiion of τc, he minimaliy condiions in Ē and he coninuiy of K i follows ha K = and hus K on [, τc ]. Since τ τc, 35 hus yields V τ + βτ 1 β u dρ u dd u = Π τ + βτ 1 β u dk u Π τ = 1 Π {τ<τd } τ, where, by Ē, we have Πτ [,τ τ d ] 1 {τ<t }L τ + 1 {τ=t }ξ. In addiion, by he definiion of τc, we have ha Π τ c U τ c on he even {τc < T }. I is now easy o see ha 36 is saisfied and hus V, ζ, ρ, τc is indeed an issuer hedge. Remark 4.1 i The siuaion where ρ can be made equal o zero by he choice of a suiable sraegy ζ in Proposiion 4.1 corresponds o a paricular form of hedgeabiliy of a game opion in which an issuer and a holder are able o hedge all risks embedded in a defaulable game opion. The case where ρ corresponds eiher o non-hedgeabiliy of a game opion or o he siuaion in which an issuer or a holder is able o hedge, bu she prefers no o hedge all he risks embedded in he opion, for insance, she may be willing o ake some bes regarding specific risk direcions. Tha is why we do no posulae a priori ha ρ should be minimized in some sense as, for insance, in Schweizer [41]. ii I is possible o inroduce he issuer rivial hedge Π,, ρ, τc resp., he holder rivial hedge Π,, ρ, τp wih he G-local maringale cos ρ = β 1 u dm u, [, T ]. Obviously, his hedge is of a minor pracical ineres, since i implicily assumes one is no ineresed in hedging. The rivial hedge or, more precisely, he exisence of any hedge is used in he proof of Proposiion 4.2, however. Le us now draw some conclusions from Lemma 2.3 and Proposiion 4.1. Proposiion 4.2 Under he assumpions of Proposiion 4.1, we have ha: i Π = essminv, c so Π is he minimum of iniial wealhs of an issuer hedge wih a G-sigma maringale cos. ii We have ha Π V p. If, in addiion, 5 holds hen Π = essminv p and Π is he minimum of iniial wealhs of a holder hedge wih a G-sigma maringale cos. iii The above saemens are also valid wih local maringale insead of sigma maringale herein. Proof. i By applying Proposiion 4.1 o he rivial hedge of Remark 4.1ii, we ge, in paricular, ha Π V, c where Π is also equal o he Q-value of he relaed Dynkin game, by Proposiion 3.7. Thus, he infimum is aained and we have equaliy, raher han inequaliy, in Lemma 2.3i. ii The second claim can be proven as par i, assuming 5. iii This follows immediaely of i and ii, since he cos ρ of he rivial hedge is a G-local maringale. Given our definiion of hedging wih a cos and he definiion of Π, he fac ha here exiss a hedge wih iniial wealh Π and G-sigma or local, in suiable cases maringale cos is by no means surprising. The minimaliy saemen esablishes a connecion beween arbirage prices and hedging in a general, incomplee marke. I is also easy o see ha one could sae analogous definiions and resuls regarding hedging a defaulable game opion, saring a any dae [, T ].

17 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski Discouned Cumulaive Value Dynamics For he nex resul, we will need he following echnical assumpion. Assumpion 4.2 The process M does no jump a τ d, ha is, M τd := M τd M τd =. Assumpion 4.2 is noably saisfied when he defaul ime τ d avoids F-sopping imes, in he sense ha Qτ d = τ = for any F-sopping ime τ. This indeed implies ha an F-adaped càdlàg process e.g., M does no jump a τ d. This avoidance assumpion is sandard in he lieraure on he progressive enlargemen of filraion see, e.g., [1, 18]. I holds, for insance, in he case where τ d is consruced by he canonical consrucion cf. [8]. Anoher common siuaion in which Assumpion 4.2 is saisfied is of course when all F-maringales are coninuous cf. Remark 3.6iii. Le N d = H Γ τd sand for he compensaed jump-o-defaul process. Under our sanding assumpion ha he F-hazard process Γ of τ d is a coninuous and non-decreasing process cf. Remark 3.2ii, he process N d is known o be a G-maringale. An analysis of hedging sraegies in he nex secion will rely on he following lemma, which yields he dynamics of he discouned cumulaive value process of a game opion or, more precisely, of is maringale componen m see he commens following he proof of Lemma 3.6. Lemma 4.3 Under Assumpion 4.2, he G-maringale m defined by 3 saisfies dm = 1 { τd }β dm + R Π dn d. 37 Proof. Le us inroduce he Doléans-Dade maringale see, e.g., [8] E = 1 {<τd }e Γ = 1 E u dn d u, so ha α E = β 1 {<τd } and α E = β 1 { τd }. Then cf. 29 and 3 dm = dβ Π + 1 { τd }β dk = de α Π + 1 { τd }β dk + β dd. 38 I may happen ha he F-semimaringale α Π is no a G-semimaringale, so a direc applicaion of he G-inegraion by pars formula o Eα Π is precluded. However, by Lemma A.1iv, he process α Π sopped a τ d is a G-semimaringale. I is also clear ha Eα Π = Eα τd Π τd. Hence, by applying he inegraion by pars formula o Eα τd Π τd, we obain de α τd Π τd = E d α τd Π τd α Π dn d + d[e, α τd Π τd ], where, in addiion, we have ha [E, α τd Π τd ] = e Γτ d α τd Π τd H. Using Assumpion 4.2, formula 31 and he facs ha he coupon process C is F-predicable and he hazard process Γ is coninuous, so ha C τd = Γ τd =, we check ha Π τd =. Using 31, we hen deduce from 38 ha dm = E d α τd Π τd α Π dn d + 1 { τd }β dk + β dd = 1 { τd }β dk dc R dγ + dm Π dn d + 1{ τd }β dk + β dd = 1 { τd }β dc R dγ + dm Π dn d + β dd. Using 3 and he equaliy C τd =, we finally arrive a he equaliy dm = 1 { τd }β dm + R Π dn d, which is he required resul.

18 18 Defaulable Game Opions in a Hazard Process Model 4.3 Hedging via Orhogonal Decomposiions In order o sudy non-rivial hedging sraegies for a defaulable game opion in he general se-up of his paper, we resor o suiable Galchouk-Kunia-Waanabe decomposiions of a soluion o he relaed doubly refleced BSDE. Noe ha in a more specific Markovian se-up, a shor-cu o ge such decomposiion will consis in using suiable versions of he Iô formula see Bielecki e al. [6] and [7]. We assume here ha a reference R q -valued F-semimaringale, denoed by N, is given a priori. In any paricular applicaion, he choice of his process will depend on he problem a hand see [6]. By a decomposiion of M, we mean he equaliy dm = Z dn + dn, [, T ], 39 where Z is an F-adaped, R 1 q -valued, N-inegrable process and n is a real-valued F-semimaringale. As i will become apparen in he sequel, n is expeced o be orhogonal o N in some sense, [ which ] N explains, for insance, why we canno simply ake Z = in 39. Le us denoe N = N d. A decomposiion of M combined wih 37, yields, for every [, T τ d ], dm = β Z dn + β R Π dn d + β dn = β [Z, Y ] dn + β dn, 4 where we wrie Y = R Π and where [Z, Y ] sands for he concaenaion of Z and Y. Le us now focus on he discouned price process β X. By a decomposiion of β X, we mean he equaliy, for every [, T τ d ], dβ X = β Ẑ dn + β Ŷ dn d + β d n = β [Ẑ, Ŷ] dn + β d n 41 for some G-adaped, R d q+1 -valued, N τd -inegrable process β[ẑ, Ŷ ] and some G-adaped, Rd - valued process n. Proposiion 4.4 Assume ha we are given decomposiions 39 and 41. Under Assumpion 4.2, for any R 1 d -valued, β X-inegrable process ζ, he relaed cos ρ = ρζ in Proposiion 4.1 saisfies, for every [, T τ d ], dρ = [Z, Y ] ζ [Ẑ, Ŷ] dn + dn ζ d n. 42 i Assume, in addiion, ha [Ẑ, Ŷ ] is lef-inverible on [, T τ d] wih he lef inverse Λ and define he sraegy ζ by he formula, for [, T τ d ], Then he cos ρ = ρ ζ saisfies, for [, T τ d ], ζ = [Z, Y ] Λ. 43 d ρ = dn ζ d n. 44 ii Alernaively o i, le us assume addiionally ha Ŷ = and ha Ẑ is lef-inverible on [, T τ d] wih he lef inverse Λ and le us define he sraegy ζ by he formula, for [, T τ d ], Then he cos ρ = ρ ζ saisfies, for [, T τ d ], ζ = Z Λ. 45 d ρ = dn ζ d n + Y dn d. 46

19 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 19 Proof. All formulas follow from definiion 33 of he cos process and formulas 4, 43 and 45. In relaion wih Remark 4.1ii, noe ha he siuaion of Proposiion 4.4i corresponds o he hedgeable case, where he cos ρ vanishes for a sraegy ζ. The siuaion of Proposiion 4.4ii corresponds o he case of unhedgeable defaul risk. Pracically useful decomposiions of M and β X will depend on a paricular model for he primary marke, as well as on he game opion under consideraion. In an absrac se-up hey follow from maringale represenaion heorems wih orhogonal componens for complemenary resuls in he Markovian se-up we refer he ineresed reader o Bielecki e al. [6]. Le hus H 2 F resp. H 2 G denoe he class of real-valued F resp. G-maringales wih inegrable quadraic variaion over [, T ], or, by a sligh abuse of noaion, he class of vecor-valued processes wih muually srongly orhogonal componens in H 2 F resp. H 2 G. I is worh noing ha an F-maringale sopped a τ d is a G-local maringale, by Lemma A.1iii. Assumpion 4.3 The processes M and N belong o H 2 F, he process N sopped a τ d belongs o H 2 G and he process β X belongs o H 2 G. The Galchouk-Kunia-Waanabe GKW decomposiion of M wih respec o N in F see, e.g., Proer [4, IV.3, Corollary 1] hus yields a decomposiion 39 of M wih n srongly orhogonal o N in H 2 F. The GKW decomposiion of β X wih respec o N τd in G yields likewise a decomposiion 41 for some R d -valued process β d n srongly orhogonal o N τd in H 2 G. Alernaively, one may consider he GKW decomposiion heorem of β X wih respec o N τd in G, which yields a decomposiion of he form 41 wih Ŷ =, and for some Rd -valued process β d n srongly orhogonal o N τd in H 2 G. The following proposiion jusifies he informal saemen ha he sraegy ζ resp. ζ hedges he risk source N resp. N. In his resul and is proof, he symbols [, ] and [, ] F denoe he square brackes wih respec o filraions G and F, respecively. Proposiion 4.5 Le 39 be given as he GKW decomposiion of M wih respec o N. i In he siuaion of Proposiion 4.4i wih 41 given as he GKW decomposiion of β X wih respec o N τd, hen he processes ρ and N τd are orhogonal in G, in he sense ha [ ρ, N τd ] is a G-sigma maringale and a G-local maringale if ζ is locally bounded. ii In he siuaion of Proposiion 4.4ii wih 41 given as he GKW decomposiion of β X wih respec o N τd, hen he processes ρ and N τd are orhogonal in G, in he sense ha [ ρ, N τd ] is a G-sigma maringale and a G-local maringale if ζ and R are locally bounded processes. Proof. Observe firs ha n τd and N τd are G-local maringales, by Lemma A.1iii. Since n is srongly orhogonal o N in H 2 F, he process [n τd, N τd ] = [n, N] F τ d is a G-local maringale, as an F-local maringale sopped a τ d cf. Lemma A.1iii. Furhermore, by Lemma A.5, [n τd, N d ] is a G-local maringale. We conclude ha [n τd, N τd ] is a G-local maringale. In case i, so is also [ n, N τd ], since he inegral β d n is srongly orhogonal o N τd in H 2 G. Using 44, we conclude ha [ ρ, N τd ] is a G-sigma maringale and hus i follows a G-local maringale if ζ is a locally bounded process. In case ii, he inegral β d n is srongly orhogonal o N τd in H 2 G, so he process [ n, N τd ] is a G-local maringale. Furhermore, by Lemma A.5, [N τd, N d ] is a G-local maringale. In view of 46, we conclude ha [ ρ, N τd ] is a G-sigma maringale and hus i follows a G-local maringale if ζ and R are locally bounded processes.