DEFAULTABLE GAME OPTIONS IN A HAZARD PROCESS MODEL

Transcription

1 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 Firs draf: June 2, 26 Final version: March 3, 29 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 Europlace Insiue of Finance. The research of M. Jeanblanc was suppored by Io33, FIRN, and Moody s Corporaion gran The research of M. Rukowski was suppored by he ARC Discovery Projec DP88146.

2 2 Defaulable Game Opions 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 ha 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, 7], Davis and Lischka [14], Kallsen and Kühn [31], and 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 [34] 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 such as 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 3.1 and 3.2, which give convenien pricing formulae 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 is he minimal pre-defaul 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 4.3 and 4.4, in which we resor o suiable Galchouk-Kunia-Waanabe decomposiions of a soluion o he relaed doubly refleced BSDE and discouned prices of primary asses wih respec o various risk facors corresponding o sysemaic, idiosyncraic and even risks. 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, 7]. We conclude he paper by briefly commening on some alernaive approaches o hedging of defaulable game opions. 1.2 Convenions and Sanding Noaion Throughou his paper, we use he concep of he vecor sochasic inegral, denoed as H dx, as opposed o a more resriced noion of he componenwise sochasic inegral, which is defined as he sum d i=1 H i dx i of inegrals wih respec o one-dimensional inegraors X i. For a deailed exposiion of he heory of vecor sochasic inegraion, we refer o Shiryaev and Cherny [41] 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 he vecor sochasic inegral H dx allows one o ake ino accoun possible inerferences of local maringale and finie variaion componens of a scalar inegraor process, or of differen componens of a mulidimensional inegraor 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 He e al. [27, Theorem 7.7]. 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, for insance, Delbaen and Schachermayer [15] or Shiryaev and Cherny [41]. 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 [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 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 cf. [15, 41], 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 for he definiion of a sigma maringale, see Definiion 1.9 in [41]. The following well-known properies of sigma maringales will also be used in he sequel. Proposiion 2.1 [29, 4, 41] i The class of sigma maringales is a vecor space conaining all local maringales. I is sable wih respec o vecor sochasic inegraion, ha is, if Y is a sigma maringale and H is a predicable Y -inegrable process hen he vecor sochasic inegral H dy is a sigma maringale. ii Any locally bounded sigma maringale is a local maringale, and any bounded from below sigma maringale is a supermaringale. Remark 2.1 In he same vein, we recall ha sochasic inegraion of predicable, locally bounded inegrands preserves local maringales see, e.g., Proer [4]. We now inroduce he concep of a dividend paying game opion see also Kifer [34]. 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 erminal 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 erminal 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 erminal paymen a mauriy ξ made a mauriy dae T provided ha T τ p τ c. The possibly random ime τ in he hird bulle poin 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. 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 G T ; τ τ d τ τ d }, where he lifing ime of a call proecion τ belongs o G T. We are now in he posiion o sae he formal definiion of a defaulable game opion. Definiion 2.1 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 = C τ 1 { τ} + C 1 {<τ} + R τ 1 { τ}. 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. So, for any and τ p, τ c GT Ḡ T, he random variable π; τ p, τ c is G τ τd -measurable. 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 L := 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 he inequaliy, for every [, T ], β Û := 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] see Proposiion 3.1 and Theorem 4.1 herein 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 for he general heory of Dynkin games, see, for insance, Dynkin [2], Kifer [33], and Lepelier and Maingueneau [36]. The noion of an arbirage price of a game opion referred o in Theorem 2.1 is he dynamic noion of arbirage price for game opions, as defined in Kallsen and Kühn [31], and exended in [4] 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 2.1 Arbirage Price of a Defaulable Game Opion 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, meaning ha 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 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 2.2 A defaulable American opion is a defaulable game opion wih τ = T. A defaulable European opion is a defaulable American opion such ha he process β L cf. 4 aains is maximum a T, ha is, β L β T LT for every [, T ]. by In view of Theorem 2.1, 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 [34], Ma and Cvianić [37], and Hamadène [24]. One of our goals is o show ha his definiion is consisen wih he concep of arbirage valuaion of a defaulable game opion in he sense of Kallsen and Kühn [31]. 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 2.3 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. Remark 2.2 The reason why we do no assume ha G is rivial which would, of course, simplify several saemens is ha we apply our resuls in he subsequen work [7] o siuaions, where his assumpion fails o hold for insance, when sudying converible bonds wih a posiive call noice period. Definiion 2.4 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.

7 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 7 Definiion 2.5 Consider he game opion wih he ex-dividend cumulaive discouned cash flows βπ given by 2. i An issuer hedge wih cos process ρ 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, càdlàg 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 ii A holder hedge wih cos process ρ 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, càdlàg 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 11, we need o inser he minus sign in he righ-hand side of 12 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 12 is he value a ime τ of a sraegy wih a cos process ρ, when he players adop heir 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 he recovery a defaul. Remark 2.3 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 under Q we hus recover he noion of mean self-financing hedge, in he sense of Schweizer [42]. ii Regarding he admissibiliy of hedging sraegies see, e.g., Delbaen and Schachermayer [15], noe ha he lef-hand side 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 lef-hand side in formula 11 is bounded from below for any holder hedge wih a cos V, ζ, ρ, τ p. Obviously, he class of all hedges wih semimaringale cos processes is oo large for any pracical purposes. Therefore, we will resric our aenion o hedges wih a G-sigma maringale cos ρ under a paricular risk-neural measure Q. Assumpion 2.1 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 13 and 14.

8 8 Defaulable Game Opions Lemma 2.1 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. 13 esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G essinfv V p V, a.s. 14 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. I is easily seen from 9 and 1 ha, for any sopping ime τ p GT, β V = β τp τ c V τp τ c p τ c ζ u dβ u Xu β π; τ p, τ c In paricular, by aking τ p =, we obain ha, for any [, T ], β V = β τc V τc τc ζ u dβ u Xu β π;, τ c p τ c τc ζu dβ u Xu + β u dρ u. 15 ζu dβ u Xu + β u dρ u. 16 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, as well as he process τc ζu dβ u Xu + β u dρ u are G-sigma maringales. The laer process is bounded from below his follows from 2 4 and 16, so ha i is a bounded from below local maringale [29, p.216] and hus also a supermaringale. By aking condiional expecaions in 15, we hus obain for any sopping ime τ p G T recall ha τ c is fixed β V E Q β π; τ p, τ c G, τp G T, and hus, by he assumed posiiviy of he process β, V essinf τc Ḡ T esssup τ p G T E Q π; τp, τ c G, a.s. The required inequaliy 13 is an immediae consequence of he las formula. 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. 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 GT, we obain β V E Q β π; τ p, τ c G, a.s., τc Ḡ T, so ha and his in urn implies 14. V esssup τp G T essinf τ c Ḡ T E Q π; τp, τ c G, a.s., 3 Valuaion in a Hazard Process Se-Up In order o ge more explici pricing and hedging resuls for defaulable game opions, we will now sudy he so-called hazard process se-up.

9 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 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 he filraion F of he non-increasing process 1 H see Jeulin [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 3.1 i The assumpion ha G is coninuous implies ha τ d is a oally inaccessible G- sopping ime see, e.g., [18]. Moreover, τ d avoids F-sopping imes, in he sense ha Qτ d = τ = for any F-sopping ime τ see Coculescu and Nikeghbali [1]. ii If G coninuous, he addiional assumpion ha G has a finie variaion implies in fac 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 local F-maringales are local G-maringales see, for insance, [8], whereas τ d is an F-pseudo-sopping ime whenever all F-local maringales sopped a τ d are G-local maringales see Nikeghbali and Yor [39] and Appendix A. Some consequences of Assumpion 3.1 useful for his work are summarized in Lemma A.1. The nex definiion refers o some auxiliary resuls, which are relegaed o Appendix A. Definiion 3.1 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 addiional assumpions. Sricly speaking, hese assumpions are superfluous, in he sense ha all he resuls below are rue wihou Assumpion 3.2. Indeed, by making use of Lemmas A.2 and A.3 and Definiion 3.1, i is always possible o reduce he original problem o he case described in Assumpion 3.2. Since his would make he noaion heavier, wihou adding much value, we prefer o work under his sanding assumpion. Assumpion 3.2 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. 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.

10 1 Defaulable Game Opions 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, similarly o β, he process α is bounded. Lemma 3.2 For any sopping imes τ p F T and τ c F T we have ha E Q π; τp, τ c G = 1{<τd } E Q π; τp, τ c F, 17 where π; τ p, τ c is given by, wih τ = τ p τ c, α π; τ p, τ c = 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. 18 Proof. Formula 17 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 formulae 2 and 18 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 on he defaul ime τ d only via he hazard process Γ. For he purpose of compuaion of he ex-dividend price of a defaulable game opion hese wo marke models are in fac equivalen. This follows from he nex resul, which is obained by combining Theorem 2.1 wih Lemmas 3.1 and 3.2. Theorem 3.1 Pre-defaul Price of a Defaulable Game Opion Assuming condiion 7, le Π be he arbirage ex-dividend Q-price for a game opion. Then we have, for any [, T ], where Π saisfies Π = 1 {<τd } Π, 19 esssup τp F T essinf τ c F T E Q π; τp, τ c F = Π 2 = essinf τc F T esssup τ p F T E Q π; τp, τ c F. Hence he Dynkin game wih cos crierion E Q π; τ p, 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 risk-neural measure Q.

11 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 11 The following resul is he converse of Theorem 3.1. I is an immediae consequence of Lemmas 3.1, 3.2 and he if par of Theorem 2.1 noing also ha Π defined by 19 is obviously a G- semimaringale if Π is a G-semimaringale. Theorem 3.2 Le Π be he value of he Dynkin game wih he cos crierion E Q π; τ p, τ c F on F T F T, for any [, T ]. Then Π defined by 19 is he value of he Dynkin game wih he cos crierion E Q π; τ p, 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 3.1 and 3.2 hus allow us o reduce he sudy of a game opion o he sudy of Dynkin games 2 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 an associaed 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 α, F, ξ, L, U, τ see Cvianić and Karazas [13], Hamadène and Hassani [25], Crépey [11], Crépey and Maoussi [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 ], 21 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 3.2 By a Q-soluion o he doubly refleced BSDE 21, 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 21 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. By he Jordan decomposiion, we mean he decomposiion K = K + K, where he nondecreasing coninuous processes K + and K vanish a ime and define muually singular measures. The sae process Θ in a soluion o 21 is clearly an F-semimaringale. So here are obvious hough raher arificial cases in which 21 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]. Remark 3.2 In applicaions see [6, 7, 11, 12], he inpu process F is ypically given in he form of he Lebesgue inegral αf = αf du and he componen M of a soluion o 21 is usually searched for in he form M = Z dn + n for some R q -valued and real-valued square-inegrable F-maringales N and n see also Assumpion 4.2 in Secion 4.3. For more explici in paricular, Markovian specificaions of he presen se-up and sufficien condiions for he exisence and uniqueness of a soluion o 21, he ineresed reader is referred o, e.g., [7, 11, 12, 13, 25].

12 12 Defaulable Game Opions Basically, in any model endowed wih he maringale represenaion propery, he exisence and uniqueness of a soluion o 21 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 [12], Hamadène and Hassani [25, Theorem 4.1] and previous works in his direcion, saring wih Cvianić and Karazas [13]. 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 [12] and l is a consan in R { }. This framework covers, for insance, he payoff a call of a converible bond examined in [4, 7]. Remark 3.3 i Since K, and hus K + and K, are coninuous, he minimaliy condiions hird line in 21 are equivalen o T Θ u L u dk + u = T Ūu Θ u dk u =. 22 Indeed he relaed inegrands here and in he hird line of 21 differ on an a mos counable se whereas he inegraors define aomless measures on [, T ]; see, e.g., [12]. In he preprin version [5] of his work, 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 [7, 12, 13, 25], we decided o impose here he coninuiy of K in Definiion 3.2 and we only consider hedges, as opposed o 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 22 insead of he hird line in condiion 21 of Definiion 3.2. ii Since F is a given process, he BSDE 21 can be rewrien as α Θ = α T ξ + T α u dk u T α u dm u, [, T ], L Θ Û, [, T ], 23 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 21 can be formally reduced o he case of F = wih suiably modified reflecing barriers L, Û and erminal condiion ξ. However, 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 [7, 13, 26], i is naural o look for a coninuous soluion of 21, ha is, a soluion of 21 given by a riple of coninuous processes Θ, M, K. iv In he conex of a Markovian se-up, he probabilisic BSDE approach may be complemened by a relaed analyic variaional inequaliy approach; his issue is deal wih in he follow-up papers [6, 7]. Noe, however, ha he variaional inequaliy approach srongly relies on he BSDE approach. Moreover, a simulaion mehod based on he BSDE is he only efficien way of numerically solving he pricing problem whenever he problem dimension number of model facors is greaer han hree or four. Indeed, in ha case he compuaional cos of deerminisic numerical schemes based on he variaional inequaliy approach becomes prohibiive. In order o esablish a relaionship 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 3.1. Observe ha if Θ, M, K solves 21 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. 24

13 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 13 Proposiion 3.1 Verificaion Principle for a Dynkin Game Le Θ, M, K be a soluion o 21 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. 25 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 25 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, 24 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 25. 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 3.1 o a defaulable game opion. To his end, we firs rewrie 18 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. 26 [,] [,] Le us denoe by Ē equaion 23 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 3.3 The doubly refleced BSDE Ē admis a soluion Θ, M, K. Le us sress ha Assumpion 3.3, 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. he commens following Definiion 3.2.

14 14 Defaulable Game Opions We denoe, for [, T ], Π = Θ F, Π = 1 {<τd } Π, Π = Π + β 1 [,] β u dd u 27 and τd m = β Π + β u dk u. 28 The following lemma is crucial in wha follows Lemma 3.3i is acually he key of he proof of Proposiion 4.1 below. Lemma 3.3 i The process m given by 28 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 21 wih F given by F in 26. 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. 29 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 27, 28 and par i, he process Π is clearly a G-semimaringale. iii By 28, we have ha τd β Π = m β u dk u, 3 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. Remark 3.4 In view of 3 and since K is coninuous, he process m given by 28 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 ]. Therefore, 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 useful connecion beween Θ, M, K and he arbirage exdividend Q-price of he defaulable game opion. Proposiion 3.2 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. Proof. In view of 18, he presen assumpions imply ha Π is he value of he Dynkin game 2, by Proposiion 3.1, 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 3.3ii. To conclude he proof, i suffices o make use of he las saemen in Theorem 3.2.

15 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 15 4 Hedging in a Hazard Process Se-up In he remaining par of his work, we examine in some deail he exisence and basic properies of hedging sraegies for defaulable game opions in a hazard process se-up. 4.1 Cos Process of a Hedging Sraegy From now on, we shall work under Assumpion 3.3. Le hus Θ, M, K denoe a soluion o Ē and le Π and Π be defined by 27. In paricular, Π is he arbirage Q-price for he game opion by Proposiion 3.2 and he lef-hand sides in 13 and 14 are equal o Π. Finally, recall ha he G-maringale m is defined by 28. Le us sress ha some of he key argumens underlying he following resul are classical, and hey are already conained in Lepelier and Maingueneau [36] see, in paricular, Theorem 11 herein. Proposiion 4.1 can hus be seen as a naural exension of heir resuls o he defaulable case, in which wo filraions are involved. I is noable ha our assumpions are made relaive o he filraion F, whereas conclusions are drawn relaive o he filraion G. Proposiion 4.1 Hedging wih a Local Maringale Cos Le ζ be an arbirary R 1 d -valued and β X-inegrable process. Then he following saemens are valid. i Le he process ρζ be given by ρ ζ = and β dρ ζ = dm ζ dβ X. 31 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 he process ρζ be given by ρ ζ = and β d ρ ζ = dm ζ dβ X. 32 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. Recall ha, according o our convenion see Secion 1.2, he β X-inegrabiliy of an R 1 d -valued sochasic process ζ implies is G-predicabiliy. Noe also 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 3.3i, 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 31, we obain V = Π and, for every [, T ], and hus β V + dβ V = ζ dβ X = dm β dρ β u dρ u = m + β Π Π τd = β Π + β u dk u + β Π Π, 33 where he second equaliy follows from 28. Recall ha he sopping ime τc F T is given by see Proposiion 3.2 τc = inf { [ τ, T ] ; Π } U T.

16 16 Defaulable Game Opions 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 FT, wih τ = τ p τc cf. 12, β τ V τ + β u dρ u dd u 1 {τ<τd }β τ 1 {τ=τp<t }L τp + 1 {τ<τp}u τ c + 1 {τp=τc =T } ξ. 34 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, 33 hus yields β τ V τ + β u dρ u dd u = β τ Π τ + β u dk u β τ Π τ = 1 {τ<τd }β τ Πτ, [,τ τ d ] where, by Ē, we have Πτ 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 34 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 risks associaed wih a game opion, for insance, she may be willing o ake some direcional bes regarding specific risks. For his reason, we decided no o posulae a priori ha ρ should be minimized in some sense as, for insance, in Schweizer [42]. ii I is possible o inroduce he issuer rivial hedge Π,, ρ, τ c resp. he holder rivial hedge Π,, ρ, τ p wih he G-local maringale cos ρ = β 1 u dm u, [, T ]. Obviously, his hedge is of no pracical ineres, since i implicily assumes ha one is no ineresed in hedging any risks. 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 2.1 and Proposiion 4.1. In he conex of specific Cox Ross Rubinsein or Black Scholes, say models, analogous resuls can be found in Kifer [34]. Our main conribuion here is an exension of hese resuls o he presen se-up involving a reducion of filraion, as well as o a fairly general class of semimaringale models. We use here he noaion essmin insead of a more common symbol essinf in order o emphasize ha he respecive bounds are in fac aained. Proposiion 4.2 Under he assumpions of Proposiion 4.1, he following saemens are valid. i The equaliy Π = essmin V c holds, so ha 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 = essmin V 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 3.2. Therefore, he infimum is aained and we have equaliy, raher han inequaliy, in Lemma 2.1i.

17 T.R. Bielecki, S. Crépey, M. Jeanblanc and M. Rukowski 17 ii In view of 5 and Lemma 2.1ii, he second claim can be proven in he same way as par i. iii This follows immediaely from pars 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 an iniial wealh Π and a G-sigma maringale cos or a local maringale cos, in suiable cases is by no means surprising. The minimaliy saemen esablishes a connecion beween arbirage prices and hedging in a general incomplee marke. Le us conclude his secion by menioning ha one could sae analogous definiions and resuls regarding hedging sraegies for a defaulable game opion saring a any dae [, T ]. 4.2 Risk Facors of a Defaulable Game Opion Le N d = H Γ τd sand for he compensaed defaul process. Under our sanding assumpion ha he F-hazard process Γ of τ d is a coninuous and non-decreasing process cf. Remark 3.1ii, he process N d is known o be a G-maringale. Recall also ha he avoidance propery holds, in he sense ha Qτ d = τ = for any F-sopping ime τ cf. Remark 3.1i. An analysis of hedging sraegies in he nex secion hinges on he following lemma, which yields he risk decomposiion of he discouned cumulaive value process of a defaulable game opion. More formally, he maringale componen m cf. Remark 3.4 is represened in erms of he pure jump maringale N d and a real-valued F-maringale M, which arise as he second componen of a soluion o he doubly refleced BSDE 21. Inuiively, he process M models he pre-defaul risk associaed wih a defaulable game opion, as opposed o he even risk, which is due o an unexpeced occurrence of he defaul even, and which is modeled hrough he jump maringale N d. Lemma 4.1 The G-maringale m defined by 28 saisfies dm = 1 { τd }β dm + Y dn d, 35 where he F-predicable process Y equals Y = R Π. 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. 27 and 28 dm = dβ Π + 1 { τd }β dk = de α Π + 1 { τd }β dk + β dd. 36 I may happen ha he F-semimaringale α Π fails o be also a G-semimaringale, so a direc applicaion of he G-inegraion by pars formula o Eα Π is no possible. 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 he avoidance propery of Remark 3.1i, formula 29, and he assumpions ha he coupon process C is F-predicable and he hazard process Γ is coninuous so ha C τd = Γ τd =, we obain he equaliy Π τd =. Using 29, we nex deduce from 36 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.

18 18 Defaulable Game Opions Using 3 and he equaliy C τd =, we finally arrive a he formula dm = 1 { τd }β dm + R Π dn d, which is he required resul. 4.3 Hedging of Risk Facors In order o sudy non-rivial cases of hedging sraegies for a defaulable game opion in he general se-up of his paper, we need o impose more assumpions on prices of primary raded asses. Since we are working in a fairly general framework, we will be able o provide only general resuls concerning hedging sraegies. The ineresed reader is referred o he follow-up papers [6, 7] for a more deailed analysis of assumpions made in his secion and paricular examples. Firs, we recall ha he ex-dividend price X of primary risky asses saisfies X = 1 H X, for every [, T ], where he R d -valued, F-adaped process X formally represens he pre-defaul value of X. We hus assume, by convenion, ha any residual value of he primary asse a τ d is embedded in he recovery par of he dividend process for X. We denoe by R an R d -valued and F-predicable process, which is aimed o represen he recovery processes of primary risky asses. Inspired by decomposiion 35 of Lemma 4.1, we make also he following naural posulae regarding he behavior of he cumulaive price process X sopped a τ d T. Assumpion 4.1 The dynamics under Q of he cumulaive price process X of primary risky asses are, for every [, T τ d ], dβ X = β d M + Ŷ dn d 37 for some R d -valued F-maringale M, where he R d -valued, F-predicable process Ŷ is given by he equaliy Ŷ = R X for every [, T ]. By insering 35 and 37 ino 31, we obain, for every [, T τ d ], dρ ζ = dm ζ d M + Y ζ Ŷ dn d. 38 A his sage, we were only able o separae he wo principal componens of he cos process ha correspond o pre-defaul and defaul even risks, respecively, where he pre-defaul risk is now modeled by he F-maringales M and M associaed wih a game opion and primary raded asses, respecively. Remark 4.2 In wha follows, we will only be ineresed in hedging on he random inerval [, τ d T ]. Therefore, wihou loss of generaliy, we may and do assume ha ζ is F-predicable see Lemma A.2ii. This means ha he reducion of filraion mehod can also be applied o hedging of a defaulable game opion, and no only o is valuaion as was already shown in Secion 3.2. Wihin he presen framework, he even risk facor is common for all raded primary and derivaive asses. Therefore, in he nex sep, we are going o ge a closer look on pre-defaul risks of raded and derivaive asses. To his end, we make a furher sanding assumpion, in which he concep of he sysemaic risk facor also known as he marke risk facor is inroduced. Assumpion 4.2 We are given an R q -valued F-maringale, denoed by N, which is aimed o represen he sysemaic risk facor for he underlying marke model. We posulae ha he F-maringales M and M of equaions 35 and 37 saisfy he following decomposiions, for every [, T ], dm = Z dn + dn, d M = Ẑ dn + d n, 39 where Z resp. Ẑ is some F-adaped, R 1 q -valued resp. R d q -valued, N-inegrable processes and n resp. n is a real-valued resp. R d -valued F-maringale.