Game options in an imperfect market with default

Transcription

1 Game opions in an imperfec marke wih defaul Roxana Dumirescu Marie-Claire Quenez Agnès Sulem arxiv: v2 [q-fin.mf] 2 Jul 217 July 4, 217 Absrac We sudy pricing and erhedging sraegies for game opions in an imperfec marke wih defaul. We exend he resuls obained by Kifer in [23] in he case of a perfec marke model o he case of an imperfec marke wih defaul, when he imperfecions are aken ino accoun via he nonlineariy of he wealh dynamics. We inroduce he seller s price of he game opion as he infimum of he iniial wealhs which allow he seller o be erhedged. We prove ha his price coincides wih he value funcion of an associaed generalized Dynkin game, recenly inroduced in [14], expressed wih a nonlinear expecaion induced by a nonlinear BSDE wih defaul jump. We moreover sudy he exisence of erhedging sraegies. We hen address he case of ambiguiy on he model, - for example ambiguiy on he defaul probabiliy - and characerize he robus seller s price of a game opion as he value funcion of a mixed generalized Dynkin game. We sudy he exisence of a cancellaion ime and a rading sraegy which allow he seller o be er-hedged, whaever he model is. Key-words: Game opions, imperfec markes, generalized Dynkin games, nonlinear expecaions, backward sochasic differenial equaions, nonlinear pricing, er-hedging price, doubly refleced backward sochasic differenial equaions. 1 Inroducion Game opions, which have been inroduced by Kifer (2) [23], are derivaive conracs ha can be erminaed by boh counerparies a any ime before a mauriy dae T. More precisely, a game opion allows he seller o cancel i and he buyer o exercise i a any sopping ime smaller han T. If he buyer exercises a ime τ before he seller cancels, hen Deparmen of Mahemaics, King s College London, Unied Kingdom, roxana.dumirescu@kcl.ac.uk LPMA, Universié Paris 7 Denis Didero, Boie courrier 712, Paris cedex 5, France, quenez@mah.univ-paris-didero.fr INRIA Paris, 3 rue Simone Iff, CS 42112, Paris Cedex 12, France, agnes.sulem@inria.fr The auhors hank an anonymous referee for insighful commens which led o a significanly improved version of he paper. 1

2 he seller pays he buyer he amoun ξ τ, bu if he seller cancels before he buyer exercises, hen he pays he amoun ζ σ ξ τ o he buyer a he cancellaion ime σ. The difference ζ σ ξ σ is inerpreed as a penaly ha he seller pays o he buyer for he cancellaion of he conrac. In shor, if he buyer selecs an exercise ime τ and he seller selecs a cancellaion ime σ, hen he laer pays o he former he payoff ξ τ 1 τ σ +ζ σ 1 τ>σ a ime τ σ. In he case of classical perfec markes, Kifer inroduces he fair price of he game opion, defined as he minimum iniial wealh needed for he seller o cover his liabiliy o pay he payoff o he buyer unil a cancellaion ime, whaever is he exercise ime chosen by he buyer. He shows boh in he CCR discree-ime model and in he Black and Scholes model ha his price is equal o he value funcion of he following Dynkin game: inf E Q[ ξ τ 1 τ σ + ζ σ 1 τ>σ ] = inf τ σ σ τ E Q [ ξ τ 1 τ σ + ζ σ 1 τ>σ ], (1.1) where ξ and ζ are he discouned values of ξ and ζ, equal o e r ξ and e r ζ respecively in he Black and Scholes model, where r is he insananeous ineres rae. Here, E Q denoes he expecaion under he unique maringale probabiliy measure Q of he marke model. Furher research on he pricing of game opions and on more sophisicaed gameype financial conracs includes in paricular papers by Dolinsky and Kifer (27) [12] and Dolinsky and al. (211) [11] in he discree ime case, and by Hamadène (26) [18] in a coninuous ime perfec marke model wih coninuous payoffs ξ and ζ. We also menion he paper by Bielecki and al. (29) [3] which sudies he pricing of game opions in a marke model wih defaul. Noe ha in [22], Kallsen and Kuhn (24) sudy game opions in an incomplee marke. They consider anoher ype of pricing called neural valuaion via uiliy maximizaion. The aim of he presen paper is o sudy pricing and hedging issues for game opions in he case of imperfecions in he marke model aken ino accoun via he nonlineariy of he wealh dynamics, modeled via a nonlinear driver g. We moreover include he possibiliy of a defaul. A large class of imperfec marke models can fi in our framework, like differen borrowing and lending ineres raes, or axes on he profis from risky invesmens. Our model also includes he case when he seller of he opion is a large rader whose hedging sraegy may affec he marke prices and he defaul probabiliy. Here, we pose ha he payoffs ξ and ζ associaed wih he game opion are righconinuous lef-limied (RCLL) only and hey saisfy Mokobodzki s condiion. We call seller s price of he game opion, he infimum (denoed by u ) of he iniial wealhs such ha here exiss a cancellaion ime σ and a porfolio sraegy which allow he seller o pay ξ τ (a ime τ) o he buyer if he buyer exercises a any ime τ σ, and ζ σ (a ime σ) if he buyer has sill no exercise a ime σ. Noe ha his infimum is no necessarily aained. We provide a characerizaion of he seller s price u of he game opion as he (common) value of a corresponding generalized Dynkin game (recenly inroduced in [14]). More precisely, we show ha u = inf τ σ Eg [ξ τ 1 τ σ +ζ σ 1 τ>σ ] = inf σ τ E g [ξ τ 1 τ σ +ζ σ 1 τ>σ ], (1.2) 2

3 where E g is a nonlinear expecaion/evaluaion induced by a nonlinear BSDE wih defaul jump solved under he primiive probabiliy measure P wih driver g. Noe ha in he paricular case of a perfec marke, he driver g is linear and one can show by using an acualizaion procedure and a change of probabiliy measure ha (1.2) corresponds o (1.1). We prove ha, under an addiional lef-regulariy assumpion on ζ (bu no on ξ), here exis a cancellaion ime and a rading sraegy which allow he seller o be er-hedged. In his case, he infimum in he definiion of he seller s price u is aained. When ζ is only RCLL, he infimum is no necessarily aained. However, we show ha for each ε >, he amoun u allows he seller o be er-hedged up o ε unil a well chosen cancellaion ime. The proofs of hese resuls rely on he links beween generalized Dynkin games and nonlinear doubly refleced BSDEs wih defaul jump. The second main quesion we sudy is he pricing and erhedging problem of game opions in he case of uncerainy on he (imperfec) marke model. To he bes of our knowledge, his problem has no been sudied in he lieraure excep by Dolinsky (214) in [1] in a discree ime framework. In paricular, our model can ake ino accoun an ambiguiy on he defaul probabiliy as illusraed in Secion 4.3. We prove ha he robus seller s price of he game opion under uncerainy, defined as he infimum of he iniial wealhs wih allow he seller o be erhedged whaever he model is, coincides wih he value funcion of a mixed generalized Dynkin game. We also sudy he exisence of robus erhedging sraegies. The paper is organized as follows: in Secion 2, we inroduce our imperfec marke model wih defaul and nonlinear wealh dynamics. In Secion 3, we sudy pricing and erhedging of game opions and heir links wih generalized Dynkin games. In Secion 4, we address he case of an imperfec marke wih model ambiguiy. Secion 5 provides some complemenary resuls concerning he buyer s poin of view and he case wih dividends. Some resuls on doubly refleced BSDEs wih defaul jumps and a useful lemma of analysis are given in Appendix. 2 Imperfec marke model wih defaul 2.1 Marke model wih defaul Le (Ω,G,P) be a complee probabiliy space equipped wih wo sochasic processes: a unidimensional sandard Brownian moion W and a jump process N defined by N = 1 ϑ for any [,T], where ϑ is a random variable which models a defaul ime. We assume ha his defaul can appear a any ime ha is P(ϑ ) > for any. We denoe by G = {G, } he augmened filraion ha is generaed by W and N (in he sense of [9, IV-48]). We pose ha W is a G-Brownian moion. We denoe by P he G-predicable σ-algebra. Le (Λ ) be he predicable compensaor of he nondecreasing process (N ). Noe ha (Λ ϑ ) is hen he predicable compensaor of (N ϑ ) = (N ). By uniqueness of he predicable compensaor, Λ ϑ = Λ, a.s. We assume ha Λ is absoluely coninuous w.r.. Lebesgue s measure, so ha here exiss a nonnegaive process λ, called he inensiy 3

4 process, such ha Λ = λ sds,. Since Λ ϑ = Λ, λ vanishes afer ϑ. We denoe by M he compensaed maringale which saisfies M = N λ s ds. Le T > be he finie horizon. We inroduce he following ses: S 2 is he se of G-adaped RCLL processes ϕ such ha E[ T ϕ 2 ] < +. A 2 is he se of real-valued non decreasing RCLL predicable processes A wih A = and E(A 2 T ) <. [ ] H 2 is he se of G-predicable processes Z such ha Z 2 T := E Z 2 d <. [ ] H 2 λ := T L2 (Ω [,T],P,λ d),equippedwihhescalarproduc U,V λ := E U V λ d, [ ] for all U,V in H 2 λ. For each U H2 λ, we se U 2 λ := E T U 2 λ d <. [ ] Noe ha for each U H 2 λ, we have U 2 λ = E T ϑ U 2 λ d because he G-inensiy λ vanishes afer ϑ. Moreover, we can pose ha for each U in H 2 λ = L2 (Ω [,T],P,λ d), U (or is represenan in L 2 (Ω [,T],P,λ d) sill denoed by U) vanishes afer ϑ. Moreover, T denoes he se of sopping imes τ such ha τ [,T] a.s. and for each S in T, T S is he se of sopping imes τ such ha S τ T a.s. We recall he maringale represenaion heorem (see e.g. [19]): Lemma 2.1. Any G-local maringale m = (m ) T has he represenaion m = m + z s dw s + l s dm s, [,T] a.s., (2.1) where z = (z ) T and l = (l ) T are predicable such ha he wo above sochasic inegrals are well defined. If m is a square inegrable maringale, hen z H 2 and l H 2 λ. We consider now a financial marke wih hree asses wih price process S = (S,S 1,S 2 ) governed by he equaion: ds = Sr d ds 1 = S1 [µ1 d+σ1 dw ] ds 2 = S2 [µ 2 d+σ2 dw dm ]. The process S = (S ) T corresponds o he price of a non risky asse wih ineres rae process r = (r ) T, S 1 = (S 1 ) T o a non defaulable risky asse, and S 2 = (S 2 ) T o a defaulable asse wih oal defaul. The price process S 2 vanishes afer ϑ. All he processes σ 1,σ 2, r,µ 1,µ 2 are predicable (ha is P-measurable). We se σ = (σ 1,σ 2 ). We make he following assumpions: 4

5 Assumpion 2.2. The coefficiens σ 1,σ 2 >, and r, σ 1,σ 2, µ 1,µ 2,λ, λ 1,(σ 1 ) 1, (σ 2 ) 1 are bounded. We consider an invesor, endowed wih an iniial wealh equal o x, who can inves his wealh in he hree asses of he marke. A each ime < ϑ, he chooses he amoun ϕ 1 (resp. ϕ 2 ) of wealh invesed in he firs (resp. second) risky asse. However, afer ime ϑ, he invesor canno inves his wealh in he defaulable asse since is price is equal o, and he only chooses he amoun ϕ 1 of wealh invesed in he firs risky asse. Noe ha he process ϕ 2 can be defined on he whole inerval [,T] by seing ϕ 2 = for each ϑ. A process ϕ. = (ϕ 1,ϕ2 ) T is called a risky asses saegy if i belongs o H2 H 2 λ. We denoe by V x,ϕ (or simply V ) he wealh, or equivalenly he value of he porfolio, a ime. The amoun invesed in he non risky asse a ime is hen given by V (ϕ 1 +ϕ2 ). The perfec marke model. In he classical case of a perfec marke model, he wealh process and he sraegy saisfy he self financing condiion: dv = (r V +ϕ 1 (µ 1 r )+ϕ 2 (µ 2 r ))d+(ϕ 1 σ 1 +ϕ 2 σ 2 )dw ϕ 2 dm. (2.2) Seing K := ϕ 2, and Z := ϕ 1 σ1 +ϕ2 σ2, which implies ha ϕ1 = (Z +σ 2 K )(σ 1 ) 1, we ge dv = r V +(Z +σ 2 K )(µ 1 r )(σ 1 ) 1 K (µ 2 r )d+z dw +K dm = (r V +Z θ 1 +K θ 2 λ )d+z dw +K dm, where θ 1 := µ1 r σ 1 and θ 2 := σ2 θ1 µ2 +r λ 1 { ϑ}. Consider a European coningen claim wih mauriy T > and payoff ξ which is G T measurable, belonging o L 2. The problem is o price and hedge his claim by consrucing a replicaing porfolio. From [15, Proposiion 2.6 ], here exiss an unique process (X,Z,K) S 2 H 2 H 2 λ soluion of he following BSDE wih defaul jump: dx = (r X +Z θ 1 +K θ 2 λ )d Z dw K dm ; X T = ξ. (2.3) The soluion (X, Z, K) provides he replicaing porfolio. More precisely, he process X correspondsoisvalue, andhehedgingriskyassessaegyϕ H 2 λ isgivenbyϕ = Φ(Z,K), where Φ is he one o one map defined on H 2 H 2 λ by: Definiion 2.3. Le Φ be he funcional defined by where ϕ = (ϕ 1,ϕ 2 ) is given by Φ : H 2 H 2 λ H2 H 2 λ ;(Z,K) Φ(Z,K) := ϕ, ϕ 2 = K ; ϕ 1 = Z +σ 2 K σ 1 which is equivalen o K = ϕ 2 ; Z = ϕ 1 σ1 +ϕ2 σ2 = ϕ σ. 5,

6 Noe ha he processes ϕ 2 and K, which belong o H 2 λ, boh vanish afer ime ϑ. The process X coincides wih V X,ϕ, he value of he porfolio associaed wih iniial wealh x = X and porfolio sraegy ϕ. From he seller s poin of view, his porfolio is a hedging porfolio. Indeed, by invesing he iniial amoun X in he reference asses along he sraegy ϕ, he seller can pay he amoun ξ o he buyer a ime T (and similarly a each iniial ime ). We derive ha X is he price a ime of he opion, called hedging price, and denoed by X (ξ). By he represenaion propery of he soluion of a λ-linear BSDE wih defaul jump (see [15, Theorem 2.13]), we have ha he soluion X of BSDE (2.3) can be wrien as follows: X (ξ) = E[e T r sds ζ,t ξ G ], (2.4) where ζ, saisfies dζ,s = ζ,s [ θ 1 sdw s θ 2 sdm s ]; ζ, = 1. (2.5) This defines a linear price sysem X: ξ X(ξ). Suppose now ha θ 2 < 1, ϑ d dp a.s. (2.6) Then ζ, >. Le Q be he probabiliy measure which admis ζ,t as densiy on G T. Using Girsanov s heorem, i can be shown ha Q is he unique maringale probabiliy measure.in his case, he price sysem X is increasing and corresponds o he classical arbirage free price sysem (see [19, 4, 3]). Remark 2.4. We have presened above he case of a defaulable asse wih oal defaul. A differen model for he asse price S 2 (see e.g. [19, Chaper 7, Secion 9.3]) could be considered: ds 2 = S 2 [µ2 d+σ 2 dw +β dm ], where β and β > 1, wih β, β 1 afer he defaul ime ϑ. We pose ha bounded. In his case, he price does no vanish µ 1 r σ 1 1 {>ϑ} = µ2 r σ 2 1 {>ϑ} d dp a.s. (2.7) Le ζ, be defined by (2.5) wih θ 1 = µ1 r ; θ 2 σ 1 = µ2 σ2 θ1 r 1 { ϑ}. Assume ha β λ θ 2 < 1, ϑ d dp -a.s. The assumpion (2.7) ensures ha he probabiliy measure Q wih ζ,t as densiy on G T is he unique maringale probabiliy measure. The arbirage free price of he coningen claim ξ is given by (2.4) and saisfies BSDE (2.3); moreover, he hedging sraegy ϕ = (ϕ 1,ϕ 2 ) is given by: ϕ 2 = K β and ϕ 1 = Z ϕ2 σ2. σ 1 The imperfec marke model M g. From now on, we assume ha here are imperfecions in he marke which are aken ino accoun via he nonlineariy of he dynamics of he wealh. More precisely, he dynamics of he wealh V associaed wih sraegy ϕ = (ϕ 1,ϕ 2 ) can be wrien via a nonlinear driver, defined as follows: 6

7 Definiion 2.5 (Driver, λ-admissible driver). A funcion g is said o be a driver if g : [,T] Ω R 3 R; (ω,,y,z,k) g(ω,,y,z,k) which is P B(R 3 ) measurable, and such ha g(.,,,) H 2. A driver g is called a λ-admissible driver if moreover here exiss a consan C such ha dp d-a.s., for each (y 1,z 1,k 1 ), (y 2,z 2,k 2 ), g(ω,,y 1,z 1,k 1 ) g(ω,,y 2,z 2,k 2 ) C( y 1 y 2 + z 1 z 2 + λ k 1 k 2 ). (2.8) The posiive real C is called he λ-consan associaed wih driver g. Noe ha condiion (2.8) implies ha for each > ϑ, since λ =, g does no depend on k. In oher erms, for each (y,z,k), we have: g(,y,z,k) = g(,y,z,), > ϑ dp d-a.s. Le x R be he iniial wealh and le ϕ = (ϕ 1,ϕ 2 ) in H 2 H 2 λ be a porfolio sraegy. We pose ha he associaed wealh process V x,ϕ (or simply V ) saisfies he following dynamics: dv = g(,v,ϕ σ, ϕ 2 )d ϕ σ dw +ϕ 2 dm, (2.9) wih V = x. Since g is lipschiz wih respec o y, his formulaion makes sense. Indeed, seing f 1 := ϕ σ dw s + ϕ 2 dm s, for each ω, he deerminisic funcion (V Y,ϕ (ω)) is defined as he unique soluion of he following deerminisic differenial equaion: V x,ϕ (ω) = x g(ω,s,v x,ϕ s (ω),ϕ s σ s (ω), ϕ 2 s(ω))ds+f 1 (ω), T. (2.1) Noe ha, equivalenly, seing Z = ϕ σ and K = ϕ 2, he dynamics (2.9) of he wealh process V can be wrien as follows: dv = g(,v,z,k )d Z dw K dm. (2.11) In he following, our imperfec marke model is denoed by M g. Noe ha in he case of a perfec marke (see (2.3)), we have: which is a λ-admissible driver by Assumpion A nonlinear pricing sysem g(,y,z,k) = r y θ 1 z θ2 kλ, (2.12) Pricing and hedging European opions in he imperfec marke M g leads o BSDEs wih nonlinear driver g and a defaul jump. By [15, Proposiion 2.6], we have Proposiion 2.6. Le g be a λ-admissible driver, le ξ L 2 (G T ). There exiss an unique soluion (X(T,ξ),Z(T,ξ),K(T,ξ)) (denoed simply by (X,Z,K)) in S 2 H 2 H 2 λ of he following BSDE: dx = g(,x,z,k )d Z dw K dm ; X T = ξ. (2.13) 7

8 Le us consider a European opion wih mauriy T and erminal payoff ξ L 2 (G T ) in his marke model. Le (X,Z,K) be he soluion of BSDE (2.13). The process X is equal o he wealh process associaed wih iniial value x = X, sraegy ϕ = Φ(Z,K) (where Φ is defined in Definiion 2.3) ha is X = V X,ϕ. Is iniial value X = X (T,ξ) is hus a sensible price (a ime ) of he claim ξ for he seller since his amoun allows him/her o consruc a rading sraegy ϕ H 2 H 2 λ, called hedging sraegy (for he seller), such ha he value of he associaed porfolio is equal o ξ a ime T. Moreover, by he uniqueness of he soluion of BSDE (2.13), i is he unique price (a ime ) which saisfies his hedging propery. Similarly, X = X (T,ξ) saisfies an analogous propery a ime, and is called he hedging price a ime. This leads o a nonlinear pricing sysem, firs inroduced by El Karoui-Quenez([17]) in a Brownian framework(laer called g-evaluaion in[24]) and denoed by E g. For each S [,T], for each ξ L 2 (G S ) he associaed g-evaluaion is defined by E g,s (ξ) := X (S,ξ) for each [,S]. In order o ensure he(sric) monooniciy and he no arbirage propery of he nonlinear pricing sysem E g, we make he following assumpion (see [15, Secion 3.3]). Assumpion 2.7. Assume ha here exiss a bounded map γ : [,T] Ω R 4 R; (ω,,y,z,k 1,k 2 ) γ y,z,k 1,k 2 (ω) P B(R 4 )-measurable and saisfying dp d-a.s., for each (y,z,k 1,k 2 ) R 4, g(,y,z,k 1 ) g(,y,z,k 2 ) γ y,z,k 1,k 2 (k 1 k 2 )λ, (2.14) and P-a.s., for each (y,z,k 1,k 2 ) R 4, γ y,z,k 1,k 2 > 1. This assumpion is saisfied e.g. when g(, ) is non decreasing wih respec o k, or if g is C 1 in k wih k g(, ) > λ on { ϑ}. In he special case of a perfec marke, g is given by (2.12), which implies ha k g(, ) = θλ 2. In his case, Assumpion 2.7 is hus equivalen o θ 2 < 1, which corresponds o he usual assumpion (2.6) made in he lieraure on defaul risk. Remark 2.8. Suppose ha g(,,,) = dp d-a.s. Then he price of an opion wih a null payoff is equal o, ha is, for each S [,T], E g,s () = a.s. Moreover, by he comparison heorem for BSDEs wih defaul jump (see [15, Theorem 2.17]), i follows ha he nonlinear pricing sysem E g is nonnegaive, ha is, for each S [,T], for all ξ L 2 (G S ), if ξ a.s., hen E g,s (ξ) a.s. Definiion 2.9. Le Y S 2. The process (Y ) is said o be a srong E-ermaringale (resp. maringale) if E σ,τ (Y τ ) Y σ (resp. = Y σ ) a.s. on σ τ, for all σ,τ T. Proposiion 2.1. For each S [,T] and for each ξ L 2 (G S ), he associaed price (or g- evaluaion) E g,s (ξ) is an Eg -maringale. Moreover, for each x R and each porfolio sraegy ϕ H 2 H 2 λ, he associaed wealh process V x,ϕ is an E g -maringale. 8

9 Proof. By he flow propery of BSDEs, he soluion of a BSDE wih driver g is an E g - maringale. The firs asserion follows. The second one is obained by noing ha V x,ϕ is he soluion of he BSDE wih driver g, erminal ime T and erminal condiion V x,ϕ T. Example 2.11 (Examples of marke imperfecions). Differen borrowing and lending ineres raes R and r, wih R r : he driver g is hen of he form g(,v,ϕ σ, ϕ 2 ) = r V ϕ 1 (µ 1 r ) ϕ 2 (µ 2 r )+(R r )(V ϕ 1 ϕ 2 ), where ϕ 2 vanishes afer ϑ and (see e.g. [7]). Large invesor seller: Suppose ha he seller of he opion is a large rader whose hedging sraegy ϕ and is associaed cos V may influence he marke prices (see e.g. [6, 2]). Taking ino accoun he possible feedback effecs in he marke model, he large rader-seller may pose ha he coefficiens are of he form σ (ω) = σ(ω,,v,ϕ ) where σ : Ω [,T] R 3 ; (ω,,x,z,k) σ(ω,,x,z,k) is a P B(R 3 )-measurable map, and similarly for he oher coefficiens r, µ 1, µ 2. The driver is hus of he form: g(,v,ϕ σ (,V,ϕ ), ϕ 2 ) = r(,v,ϕ )V ϕ 1 ( µ1 r )(,V,ϕ ) ϕ 2 ( µ2 r )(,V,ϕ ). Here, he map Ψ : (ω,,y,ϕ) (z,k) wih z = ϕ σ (ω,,y,ϕ) and k = ϕ 2 is assumed o be one o one wih respec o ϕ, and such ha is inverse Ψ 1 ϕ is P B(R 3 )- measurable. Taxes on risky invesmens profis: Le ρ ], 1[ represens an insananeous ax coefficien (see e.g. [16]). The driver is hen given by: g(,v,ϕ σ,ϕ 2 β ) = r V ϕ 1 (µ1 r ) ϕ 2 (µ2 r )+ρ(ϕ 1 +ϕ2 )+. 3 Pricing and hedging of game opions in he imperfec marke M g Le T > be he erminal ime. Le ξ and ζ be adaped RCLL processes in S 2 wih ζ T = ξ T a.s.and ξ ζ, T a.s. We pose ha Mokobodzki s condiion is saisfied, ha is here exis wo nonnegaive RCLL ermaringales H and H in S 2 such ha: ξ H H ζ T a.s. The game opion consiss for he seller o selec a cancellaion ime σ T and for he buyer o choose an exercise ime τ T, so ha he seller pays o he buyer a ime τ σ he amoun I(τ,σ) := ξ τ 1 τ σ +ζ σ 1 σ<τ. 9

10 We now inroduce he seller s price of he game opion, denoed by u, defined as he infimum of he iniial wealhs which enable he seller o choose a cancellaion ime σ and o consruc a porfolio which will cover his liabiliy o pay he payoff o he buyer up o σ no maer he exercise ime chosen by he buyer. Definiion 3.1. For each iniial wealh x, a er-hedge agains he game opion is a pair (σ,ϕ) of a sopping ime σ T and a porfolio sraegy ϕ H 2 H 2 1 λ such ha V x,ϕ ξ, σ a.s. and V x,ϕ σ ζ σ a.s. (3.1) We denoe by S(x) = S ξ,ζ (x) he se of all er-hedges associaed wih iniial wealh x. We define he seller s price as u := inf{x R, (σ,ϕ) S(x)}. (3.2) When he infimum in (3.2) is aained, he amoun u allows he seller o be er-hedged, and is called he erhedging price. Remark 3.2. We have (,) S(ζ ) since V ζ, = ζ and ζ ξ. By (3.2), we hus ge u ζ. Moreover, when g(,,,) = dp d-a.s. and ζ, hen we can resric ourselves o nonnegaive iniial wealhs, ha is u = inf{x, (σ,ϕ) S(x)}. Indeed, le x R be such ha here exiss (σ,ϕ) S(x). Then, Vσ x,ϕ ζ σ a.s. Now, by Proposiion 2.1 he wealh process V x,ϕ is an E g -maringale. We hus have x = E,σ(V g σ x,ϕ ). Since he pricing sysem E g is nonnegaive (see Remark 2.8), i follows ha x = E,σ(V g σ x,ϕ ). We now provide a dual formulaion of he seller s price, expressed in erms of he nonlinear pricing sysem E g.we inroduce he following definiion: Definiion 3.3. We define he g-value of he game opion as inf E g,τ σ [I(τ,σ)]. (3.3) Our aim is o show ha he seller s price u of he game opion is equal o is g-value. To his purpose, we firs give he following characerizaion of he g-value. Proposiion 3.4. (Characerizaion of he g-value of he game opion) Suppose ha he payoffs ξ and ζ are (only) RCLL. The g-value of he game opion saisfies: inf E g,τ σ[i(τ,σ)] = τ T 1 Noe ha condiion (3.1) is equivalen o V x,ϕ σ I(,σ), T a.s. inf σ T Eg,τ σ[i(τ,σ)] = Y, (3.4) 1

11 where (Y,Z,K,A,A ) is he unique soluion in S 2 H 2 H 2 λ A2 A 2 of he doubly refleced BSDE (DRBSDE) associaed wih driver g and barriers ξ,ζ, ha is dy = g(,y,z,k )d+da da Z dw K dm ; Y T = ξ T, (3.5) wih (i) ξ Y ζ, T a.s., (ii) da da (i.e. he measures da and da are muually singular) T T (iii) (Y ξ )da c = a.s. and (ζ Y )da c = a.s. A d τ = Ad τ 1 {Y τ =ξ τ } and A d τ = A d τ 1 {Yτ =ζ τ } a.s. τ T predicable. Using he erminology inroduced in [14], he firs equaliy in (3.4) means ha he generalized Dynkin game associaed wih he crierium E g,τ σ[i(τ,σ)] is fair. When g is linear and when here is no defaul, his corresponds o a well-known resul on classical Dynkin games and linear DRBSDEs (see e.g. [8, 18]). Proof. The proof of exisence and uniqueness of a soluion (Y,Z,K,A,A ) of he DRBSDE (3.5) is given in appendix. Proceeding as in he proof of [14, Theorem 4.9] which was given in he framework of a random Poisson measure, we can prove ha for each S T, Y S = essinf σ TS ess τ TS E g S,τ σ [I(τ,σ)] = ess τ T S essinf σ TS E g S,τ σ [I(τ,σ)] a.s. The resuls of he proposiion hen follow by aking S =. Proposiion 3.5. Le (Y,Z,K,A,A ) be he unique soluion of he DRBSDE (3.5). When ξ (resp. ζ) is lef-u.s.c. along sopping imes, hen A (resp. A ) is coninuous. Proof. Noe firs ha for each predicable sopping ime τ, by (3.5), we have ( Y τ ) + = A τ a.s. and ( Y τ ) = A τ a.s. Suppose ha now ha ζ is lef-u.s.c. along sopping ime. Le τ be a predicable sopping ime. Using he equaliy A τ = ( Y τ) + ogeher wih he Skorokhod condiions saisfied by A, we ge A τ = 1 {Y τ =ζ τ }(Y τ Y τ ) + = 1 {Yτ =ζ τ }(Y τ ζ τ ) +. (3.6) Now, since ζ is lef-u.s.c. along sopping imes, we have Y τ ζ τ Y τ ζ τ a.s., where he las equaliy follows from he inequaliy Y ζ. Using (3.6), we derive ha A τ = a.s. I follows ha A is coninuous. By similar argumens, one can show ha if ξ is lef-u.s.c. along sopping imes, hen A is coninuous. Using he above proposiions, we can now show he dual formulaion for he seller s price. We firs consider he simpler case when ζ is lef lower-semiconinuous (or equivalenly ζ is lef-u.s.c.) along sopping imes. In his case, we prove below ha he seller s price is equal o he g-value and ha he infimum in (3.2) is aained. This implies ha he seller s price is he er-hedging price. Moreover, a er-hedge sraegy is provided via he soluion of he associaed DRBSDE. 11

12 Theorem 3.6 (Seller s/er-hedging price and er-hedge of he game opion). Suppose ha ζ is lef lower-semiconinuous along sopping imes (and ξ is only RCLL). The seller s price (3.2) of he game opion coincides wih he g-value of he game opion, ha is u = inf E g,τ σ [I(τ,σ)] = τ T inf σ T Eg,τ σ [I(τ,σ)]. (3.7) Le (Y,Z,K,A,A ) is he soluion of he DRBSDE associaed wih driver g and barriers ξ,ζ. The seller s price is equal o Y, ha is u = Y. Moreover, he infimum in (3.2) is aained. The seller s price is hus he er-hedging price and here exiss a er-hedge sraegy (σ,ϕ ) associaed wih he iniial amoun u, given by σ := inf{, Y = ζ } and ϕ := Φ(Z,K), (3.8) where Φ is defined in Definiion 2.3. Remark 3.7. In he special case of a perfec marke model, our resul gives ha u is characerized as he value funcion of a classical Dynkin game problem, which is shown in he lieraure (see e.g. [23, 18]) under an addiional regulariy assumpion on ξ, by using an acualizaion procedure, a change of probabiliy measure, and some resuls on classical Dynkin games. Moreover, in his paricular case, he characerizaion of u and of he er-hedge via he soluion of a linear doubly refleced BSDE are shown in [18] by using he links beween linear DRBSDEs and classical Dynkin games (firs provided in [8]). To solve he problem in he case of an imperfec marke model, when g is nonlinear, we need o use oher argumens, in paricular some properies of he nonlinear g-evaluaion E g, comparison heorems for backward SDEs and for forward differenial equaions, and he links beween nonlinear doubly refleced BSDEs and generalized Dynkin games (firs provided in [14]). Proof. By Proposiion 3.4, he g-value of he game opion is equal Y. Noe ha u = infh, where H is he se of iniial capials which allow he seller o be er-hedged, ha is H = {x R : (σ,ϕ) S(x)}. Le us show ha Y u. I is sufficien o prove ha here exiss (σ,ϕ ) S(Y ). By Proposiion 3.5, since ζ is lef-u.s.c. along sopping imes, he process A is coninuous. Le σ be defined as in (3.8). We have a.s. ha Y < ζ for each [,σ [. Since Y is soluion of he DRBSDE (3.5), he process A is hus consan on [,σ [ a.s. and even on [,σ ] by coninuiy. Hence, A σ = A = a.s. For almos every ω, we hus have Y (ω) = Y g(s,ω,y s (ω),z s (ω),k s (ω))ds+f (ω) A (ω), σ (ω). (3.9) 12

13 where f := Z sdw s + K sdm s. Now, he wealh V Y,ϕ., associaed wih he iniial capial Y and he financial sraegy ϕ := Φ(Z,K) saisfies for almos every ω he forward deerminisic differenial equaion: V Y,ϕ (ω) = Y g(s,v Y,ϕ s (ω),z s (ω),k s (ω))ds+f (ω), T. (3.1) Since A is non decreasing, by applying he classical comparison resul on [,σ (ω)] (see e.g. Lemma 6.2) for he wo forward differenial equaions (3.9) and (3.1), wih he same coefficien (s,x) g(s,ω,x,z s (ω),k s (ω)), we ge V Y,ϕ Y ξ, σ a.s., where he las inequaliy follows from he inequaliy Y ξ. We also have V Y,ϕ σ Y σ = ζ σ a.s., where he las equaliy follows from he definiion of he sopping ime σ and he righconinuiy of Y and ζ. Hence, (σ,ϕ ) S(Y ), (3.11) which implies ha Y H. We hus ge he inequaliy Y u. I remains o show ha u Y. Since Y = inf σ T τ T E g,t [I(τ,σ)] (by Proposiion 3.4), i is sufficien o show ha u inf E g,t [I(τ,σ)]. (3.12) Le x H. There exiss (σ,ϕ) S(x), ha is a pair (σ,ϕ) of a sopping ime σ T and a porfolio sraegy ϕ H 2 H 2 x,ϕ λ such ha V ξ, σ a.s. and Vσ x,ϕ ζ σ a.s., which implies ha for all τ T we have Vτ σ x,ϕ I(τ,σ) a.s. By aking he E g -evaluaion in he above inequaliy and hen he remum on τ T, using he monooniciy of he E g -evaluaion and he E g -maringale propery of he wealh process V x,ϕ (see Proposiion 2.1), we obain x = E,τ σ[v g τ σ] x,ϕ E,τ σ[i(τ,σ)], g for each τ T. By aking he remum over τ T, and hen he infimum over σ T, we ge x inf E g,τ σ [I(τ,σ)]. This inequaliy holds for any x H. By aking he infimum over x H, we obain he inequaliy (3.12), which yields ha u Y. Since Y u, we ge Y = u. Moreover, his equaliy ogeher wih (3.11) implies ha (σ,ϕ ) S(u ). The proof is hus complee. 13

14 Remark 3.8. Le ˆσ be a sopping ime such ha A ˆσ = a.s. and Yˆσ = ζˆσ a.s. By he above proof, he pair (ˆσ,ϕ ) is a er-hedge for he iniial amoun u, ha is (ˆσ,ϕ ) S(u ). For example, under he assumpion of Theorem 3.6 (ha is, he lef-u.s.c. propery along sopping imes of ζ), he sopping ime σ := inf{ : A > } saisfies hese wo equaliies. Noe ha σ σ. In general, he equaliy does no hold. Remark 3.9. Noe ha under he assumpion of Theorem 3.6, here does no necessarily exis a saddle poin for he generalized Dynkin game (3.4). However, if we pose addiionally ha ξ is lef-u.s.c. along sopping ime, here exiss a saddle poin. More precisely, in his case, by [14, Theorem 4.7], he pair (τ,σ ), wih σ defined in (3.8) and τ := inf{ : Y = ξ }, is a saddle poin for he generalized Dynkin game (3.4), ha is, for all (τ,σ) T 2 we have E g,τ σ [I(τ,σ )] Y = E g,τ σ [I(τ,σ )] E g,τ σ [I(τ,σ)], which implies ha τ is opimal for he opimal sopping problem τ T E g [I(τ,σ )]. The same properies also hold for he pair ( τ, σ) where τ := inf{ : A > }. We consider now he general case when ζ is only RCLL (as ξ). In his case, he seller s price u is sill equal o he g-value bu i does no necessarily allow he seller o build a er-hedge agains he opion. We inroduce he definiion of ε-er-hedges: Definiion 3.1. For each iniial wealh x and for each ε >, an ε-er-hedge agains he game opion is a pair (σ,ϕ) of a sopping ime σ T and a risky-asses sraegy ϕ H 2 H 2 λ such ha V x,ϕ ξ, σ a.s. and Vσ x,ϕ ζ σ ε a.s. In oher erms, by invesing he iniial capial amoun x in he marke following he riskyasses sraegy ϕ, he seller is compleely hedged before σ, and a he cancellaion ime σ, he is hedged up o an amoun of ε. We prove below ha when ζ and ξ are only RCLL, he seller s price u is equal o he g-value and ha here exis an ε-er-hedge for he game opion. Theorem 3.11 (Seller s price and ε-er-hedge of he game opion). Suppose ha he process ζ and ξ are only RCLL. The seller s price (3.2) of he game opion coincides wih he g-value of he game opion, ha is u = inf E g,τ σ [I(τ,σ)] = τ T inf σ T Eg,τ σ [I(τ,σ)]. Le (Y,Z,K,A,A ) be he soluion of he DRBSDE associaed wih driver g and barriers ξ,ζ. The seller s price is equal o Y, ha is u = Y. (3.13) The infimum in (3.2) is no nessarily aained. Le ϕ := Φ(Z,K) and for each ε >, le σ ε := inf{ : Y ζ ε}. (3.14) The pair (σ ε,ϕ ) is an ε-er-hedge for he iniial capial u. 14

15 Proof. By Proposiion 3.4, he g-value is equal Y. Le ε >. We have Y. ζ. ε on [,σ ε [. Since A saisfies he Skorohod condiion (iii), i follows ha almos surely, A is consan on [,σ ε [. Also, Y (σ ε ) ζ (τ ε ) ε a.s., which implies ha A σ ε = a.s. Hence, A σ = ε a.s. I follows ha for almos every ω, he deerminisic funcion Y. (ω) is he soluion of he forward deerminisic differenial equaion (3.9) on [,σ ε (ω)]. Now, for almos every ω, he wealh V Y,ϕ. (ω) is he soluion of he deerminisic differenial equaion (3.1). By applying he classical comparison resul on differenial equaions (Lemma 6.2), we derive ha V Y,ϕ Y ξ, σ ε a.s. Moreover, we have V Y,ϕ σ ε Y σε ζ σε ε, where he las inequaliy follows from definiion of he sopping ime σ ε and he righ-coninuiy of Y and ζ. Hence, (σ ε,ϕ ) is an ε-er-hedge for he iniial capial amoun Y. I remains o show ha Y = u. The proof of he inequaliy u Y, which uses Proposiion 3.4, has been done in he second par of he proof of Theorem 3.6 and does no require he coninuiy of A. Le us show he converse inequaliy. Le ε >. Le (Y,Z,K ) be he soluion of he BSDE associaed wih erminal ime σ ε and erminal. Now (V Y,ϕ,Z,K) is he soluion of he BSDE associaed wih erminal ime σ ε and erminal condiion V Y,ϕ σ ε. By an a priori esimae on BSDEs wih defaul jump (see [15, Proposiion 2.4]), since V Y,ϕ σ ε ζ σε V Y,ϕ σ ε ε a.s., we derive ha V Y,ϕ = Y Y Kε a.s., where K is a consan which only depends on T and he λ-consan C. By he comparison heorem for BSDEs, Y V Y,ϕ ξ. We derive ha he amoun Y ( Y +Kε) allows he seller o be er-hedged, and he associaed er-hedge is given by σ ε and ϕ := Φ(Z,K ). By definiion of u, we derive ha u Y Y +Kε, for each ε >. Hence, u Y. Since u Y, we ge u = Y. condiion ζ σε V Y,ϕ σ ε 4 Pricing and hedging of game opions wih model uncerainy We sudy now game opions wih uncerainy on he model, which includes in paricular he case of uncerainy on he defaul probabiliy (see Example 4.3 below). 4.1 Marke model wih ambiguiy In his secion, we need o use a measurable selecion heorem, which requires o work on an appropriae probabiliy space. We consider a Cox process model, which is a ypical example of defaul model. We work on he canonical space consruced as follows: le Ω W be he Wiener space defined by Ω W := C(R + ), ha is hese of coninuous funcions ω fromr + ino R such ha ω() =. Recall ha Ω W is a Polish space for he norm. The space Ω W is equipped wih he σ-algebra F W generaed by he coordinae process (W ) (which is equal oisborelianσ-algebra). Le P W beheprobabiliyunder which(w ) isasandard Brownian moion. Le Ω Θ := R, equipped wih is Borelian σ-algebra F Θ = B(R), and he probabiliy P Θ such ha he ideniy map Θ admis an exponenial law wih parameer 1. We consider he produc space Ω := Ω W Ω Θ, which is a Polish space. I is equipped wih 15

16 he σ-algebra F W F Θ, and he probabiliy P := P W P Θ. Le G be he σ-algebra F W F Θ compleed wih respec o P. Le F = (F, ) be he filraion F W compleed wih respec o G and P (in he sense of [2, p.3] or [9, IV]). Le ( λ ) be a bounded posiive F-predicable process. We inroduce he following random variable, which represens he defaul ime: ϑ := inf{, λ s ds Θ}. We have P(ϑ > F ) = P(ϑ > F ) = exp( λ s ds), which corresponds o he so-called condiion (H) (see e.g. [19]). We now define he defaul process: N := 1 {ϑ },. We denoe by G = (G, ) he filraion generaed by W and N augmened wih respec o G and P (in he sense of [9, IV-48]). By classical resuls, since Condiion (H) holds, we derive ha W is a G-Brownian moion. Moreover, he process M defined by M := N ϑ λ s ds,, a.s. is a G-maringale. For each, le λ := λ 1 { ϑ}. The process λ, usually called he G-inensiy of ϑ, hus vanishes afer ϑ. Le T be a given erminal ime. The ses P, S 2, H 2, H 2 λ and A2 are defined as before. LeU beanonempyclosedsubseofr. Leg : [,T] Ω R 3 U R; (,ω,y,z,k,α) g(,ω,y,z,k,α), be a given P B(R 3 ) B(U)-measurable funcion. Suppose g(,α) is uniformly λ- admissible wih respec o(y, z, k), ha saisfies he inequaliy(2.8) wih a consan C which does no depend on α. We also assume ha g(,α) is coninuous wih respec o α, and such ha α U g(,.,,,,α) H 2. Suppose also ha g(,y,z,k 1,α) g(,y,z,k 2,α) θ y,z,k 1,k 2 (k 1 k 2 )λ, (4.1) whereθ y,z,k 1,k 2 saisfieshecondiionsofassumpion2.7,inparicularheinequaliyθ y,z,k 1,k 2 > 1. Le U be he se of U-valued predicable processes. For each α U, o simplify noaion, we inroduce he map g α defined by g α (,ω,y,z,k) := g(,ω,y,z,k,α (ω)). (4.2) Noe ha hese maps g α, α U, are all λ-admissible drivers wih he same λ-consan C. The conrol α represens he ambiguiy parameer of he model. To each ambiguiy parameer α, corresponds a marke model M α where he wealh process V α,x,ϕ associaed wih an iniial wealh x and a risky asses saegy ϕ H 2 H 2 λ saisfies dv α,x,ϕ = g(,v α,x,ϕ,ϕ σ, ϕ 2,α )d ϕ σ dw +ϕ 2 dm ; V α,x,ϕ = x. (4.3) In he marke model M α, he nonlinear pricing sysem is given by E gα := {E gα,s, S [,T], [,S]}, also called g α -evaluaion. 16

17 4.2 Robus erhedging of game opions In our framework wih ambiguiy, he seller s robus price of he game opion denoed by u is defined as he infimum of he iniial wealhs which enable he seller o be erhedged for any ambiguiy parameer α U. Definiion 4.1. For an iniial wealh x R, a robus er-hedge agains he game opion is a pair (σ,ϕ) of a sopping ime σ T and a porfolio sraegy ϕ H 2 H 2 2 λ such ha V α,x,ϕ ξ, σ a.s. and V α,x,ϕ σ ζ σ a.s., α U. (4.4) We denoe by S r (x) he se of all robus er-hedges associaed wih iniial wealh x. The seller s robus price is defined as 3 u := inf{x R, (σ,ϕ) S r (x)}. (4.5) When he infimum is reached, u is called he robus erhedging price. Le α U. By Theorem 3.11, he seller s price of he game opion in he marke M α is characerized as is g α -value. Moreover, i is equal o Y α, where (Y α,z α,k α,a α,a α ) is he unique soluion in S 2 H 2 H 2 λ A2 A 2 of he DRBSDE associaed wih driver g α and barriers ξ and ζ. We now inroduce an associaed dual problem. Definiion 4.2. The dual problem associaed o he seller s er-hedging problem is v := Y α. (4.6) α U By Theorem 3.11, he seller s price Y α of he game opion in he marke M α is equal o he common value funcion of he generalized Dynkin game associaed wih driver g α, ha is, Y α = inf E,τ σ[i(τ, gα σ)] = inf,τ σ[i(τ,σ)]. τ T σ T Egα Hence, he value funcion v of he dual problem is equal o he value funcion of a mixed generalized Dynkin game, ha is v = Y α = α U α U inf E gα,τ σ [I(τ,σ)] = α U inf τ T σ T Egα,τ σ Remark 4.3. We shall see below (see Proposiion 4.8) ha v is also equal o: inf σ T α U τ T E gα,τ σ[i(τ,σ)]. [I(τ,σ)]. (4.7) 2 Condiion (4.4) is equivalen o V α,x,ϕ σ I(,σ), T a.s. for all α U. 3 Remark 3.2 also holds for he seller s robus price, ha is, u ζ. Moreover, when g(,,,) = and ζ, hen u = inf{x, (σ,ϕ) S r (x)}. 17

18 In order o show ha u = v, we will firs prove ha v can be characerized as he soluion of a doubly refleced BSDE. Now, by definiion, we have v = α Y α, where Y α is he soluion of he doubly refleced BSDE associaed wih barriers ξ and ζ, and wih driver g(,α ). We will show ha v coincides wih he soluion of he doubly refleced BSDE associaed wih he same barriers ξ and ζ, and wih he driver α g(,α). More precisely, le G be he map defined for each (,ω,z,k) by G(,ω,y,z,k) := g(,ω,y,z,k,α). (4.8) α U Lemma 4.4. The map G is a λ-admissible driver and saisfies Assumpion 2.7. Proof. Since U is a closed subse of a Polish space, here exiss a numerable subse D of U, dense in U. Since g is coninuous wih respec o u, he remum in (4.8) can be aken in D. I follows ha G is P B(R 3 ) measurable. Le us show ha G saisfies Assumpion (2.7). By definiion of G(,y,z,k 1 ) and by Assumpion (4.1), we have for all α U: G(,y,z,k 1 ) g(,y,z,k 2,α) g(,y,z,k 1,α) g(,y,z,k 2,α) θ y,z,k 1,k 2 (k 1 k 2 )λ. Taking he infimum on α U in his inequaliy, and using he definiion of G(,y,z,k 2 ), we derive ha G(,y,z,k 1 ) G(,y,z,k 2 ) θ y,z,k 1,k 2 (k 1 k 2 )λ, which gives he desired resul. The proof of condiion (2.8) relies on similar argumens and is lef o he reader. Hence, G is a λ-admissible driver. We now prove ha he dual funcion v is characerized as he soluion of he doubly refleced BSDE associaed wih driver G and barriers ξ and ζ. Theorem 4.5. (Characerizaion of he dual value funcion v ) Le v be defined by (4.6). We have v = Y, where (Y,Z,K,A,A ) be he soluion of he DRBSDE associaed wih driver G and barriers ξ and ζ. If U is compac, here exiss ᾱ U such ha v = Y ᾱ, which means ha he dual value funcion v is equal o he gᾱ-value of he game opion in he marke model Mᾱ. Proof. By definiion of G (see (4.8)), for each (,ω,y,z,k) [,T] Ω R 3 U, we have G(,ω,y,z,k) g(,ω,y,z,k,α (ω)). By he comparison heorem for DRBSDEs (see Theorem 5.1 in [14]), we hus have Y Y α a.s. for each α U. I follows ha Y α Y α. Le ε >. By definiion of G as a remum, for each (,ω,y,z,l) Ω [,T] R 2 R, here exiss α ε U such ha G(,ω,y,z,k) ε g(,ω,y,z,k,α ε ). Now, he se {(,ω,α) [,T] Ω U : G(,ω,Y (ω),z (ω),k (ω)) ε g(,ω,y (ω),z (ω),k (ω),α)} 18

19 belongs o P B(U). Hence, since he canonical space Ω is a Polish space, by applying a measurable selecion heorem (see e.g. [9, Secion 81, Appendix of Ch. III]) and [5, Lemma 1.2] (or [13, Lemma 26]), here exiss an U-valued predicable process (α ε ) such ha G(,Y,Z,K ) ε g(,ω,y,z,k,α ε ), T, d dp a.s. By using he esimae (6.1) ondrbsdes wih defaul jump, wih η = 1 and β = 3C 2 +2C, C 2 we derive ha here exiss a consan K, which depends only on C and T, such ha, for each ε >, Y Kε Y αε. Since Y α Y α, we hus ge Y = α Y α = v. Le us show he second asserion. If U is compac, for each (,ω,y,z,l) [,T] Ω R 2 L 2 λ, here exiss ᾱ U such ha he remum in (4.8) is aained a ᾱ. By he measurable selecion heorem of [9] and [5, Lemma 1.2], here exiss an U-valued predicable process (ᾱ ) such ha G(,Y,Z,K ) = g(,y,z,k,ᾱ ), T, d dp a.s. I follows ha Y and Y ᾱ are boh soluions of he DRBSDE associaed wih driver gᾱ. Hence, by he uniqueness of he soluion of a DRBSDE, Y = Y ᾱ. Using his resul, we now provide he following heorem: Theorem 4.6. (Seller s robus price and er-hedge) Suppose ha ζ is lef-lower semiconinuous along sopping imes (and ξ is only RCLL). The seller s robus price of he game opion defined by (4.5) is equal he dual value funcion v defined by (4.6), ha is u = v. Le (Y,Z,K,A,A ) be he soluion of he DRBSDE associaed wih driver G defined by (4.8) and barriers ξ and ζ. The seller s robus price is equal o Y, ha is u = Y. Moreover, he infimum in (4.6) is aained. The robus seller s price is hus he robus erhedging price of he game opion. Le σ := inf{, Y = ζ } and ϕ := Φ(Z,K). The pair (σ,ϕ ) is a robus er-hedge for he iniial capial u. If U is compac, here exiss ᾱ U such ha he robus erhedging price of he game opion is equal o he erhedging price in he marke model Mᾱ, ha is u = Y ᾱ. The ambiguiy parameer ᾱ corresponds o a wors case scenario among all he possible ambiguiy parameers α U. Proof. By Theorem 4.5, v = Y. Le H r be he se of iniial capials which allow he seller o be er-hedged, ha is H r = {x R : (σ,ϕ) S r (x)}. Noe ha u = infh r. Le us show ha Y u. I is sufficien o show ha here exiss (σ,ϕ ) S r (Y ). By Proposiion 3.5, since ζ is lef-u.s.c. along sopping imes, he process A is coninuous. By 19

20 definiion of σ, he process A is consan on [,σ [ a.s. and even on [,σ ] by coninuiy. Hence, A σ = A = a.s. We hus have Y = Y G(s,Y s,z s,k s )ds+ Z s dw s + K s dm s A, σ Le α U. In he marke model M α, he wealh process V α,y,ϕ. associaed wih he iniial capial Y and he financial sraegy ϕ := Φ(Z,K) saisfies V α,y,ϕ = Y g(s,v α,y,ϕ s,z s,k s,α s )ds+ Z s dw s + K s dm s. By definiion of G (see (4.8)), we have g(,ω,y,z,k,α (ω)) G(,ω,y,z,k). Hence, since A is a non decreasing process, by he comparison propery for deerminisic differenial equaions (see Lemma 6.2) applied o he wo above forward equaions, we derive ha V α,y,ϕ Y ξ, σ a.s., where he las inequaliy follows from he inequaliy Y ξ. Moreover, we have V α,y,ϕ σ Y σ = ζ σ a.s., and his holds for any α U. Hence (σ,ϕ ) S r (Y ), which implies Y H r. Thus, Y u. Le us now show ha u Y. Le x H r. There exiss (σ,ϕ) S r (x), ha is a pair (σ,ϕ) of a sopping ime σ T and a porfolio sraegy ϕ H 2 H 2 λ such ha for each α U, we have V α,x,ϕ σ I(,σ), T a.s. By he same argumens as in he proof of Theorem 3.6, we derive ha for each α U, x inf E gα,τ σ [I(τ,σ)]. By aking he remum over α U in his inequaliy, we obain x α U inf E gα,τ σ [I(τ,σ)] = v, where he las equaliy follows from he fac ha v is equal o he value funcion of he mixed generalized Dynkin game (4.7). By aking he infimum over x H r, we obain u v = Y. Since Y u, we hus ge Y = u. Since (σ,ϕ ) S r (Y ), we derive ha (σ,ϕ ) S r (u ). The las asserion of he heorem follows from Theorem 4.5. When ζ is only RCLL, by using similar argumens o hose used in he above proof and in he proof of Theorem 3.11, one can show he following resul. Theorem 4.7. [seller s robus price and ε-er-hedge] Suppose ha he process ζ and ξ are only RCLL. The seller s robus price of he game opion is equal he dual value funcion, ha is u = v. We also have u = Y, where (Y,Z,K,A,A ) is he soluion of he DRBSDE associaed wih driver G defined by (4.8) and barriers ξ and ζ. Moreover, he infimum in (4.6) is no necessarily aained. For each ε >, le σ ε := inf{ : Y ζ ε}. The pair (σ ε,ϕ ), where ϕ := Φ(Z,K), is an ε-robus er-hedge for he seller, in he sense ha V α,u,ϕ ξ, σ ε a.s. and V α,u,ϕ σ ε ζ σε ε a.s. α U. 2 a.s.

21 We will now show ha he infimum over σ and he remum over α can be inerchanged in he expression of he dual value funcion v (see (4.7)), which, since u = v, can be wrien as follows. Proposiion 4.8. The seller s robus price u of he game opion saisfies: u = α U inf E gα,τ σ[i(τ, σ)] = inf σ T α U τ T E gα,τ σ[i(τ,σ)]. (4.9) Proof. The firs equaliy in (4.9) holds by he above heorem. Le us prove he second one. By he above heorem, we have u = Y, where (Y,Z,K,A,A ) is he soluion of he DRBSDE associaed wih driver G defined by (4.8) and barriers ξ and ζ. To obain he desired resul, i is hus sufficien o prove ha Y = inf σ T α U τ T E gα,τ σ [I(τ,σ)]. (4.1) Since by definiion (4.8), G = α U g(,α), by using similar argumens o hose used in he proof of Theorem 4.5 (in paricular a measurable selecion heorem), one can show ha he soluion of he BSDE associaed wih driver G and erminal condiion I(τ,σ) is equal o he remum over α of he soluions of he BSDEs associaed wih drivers g(,α) and he same erminal condiion, ha is E,τ σ[i(τ,σ)] G = E,τ σ[i(τ,σ)]. gα (4.11) α U On he oher hand, applying Proposiion 3.4 o he generalized Dynkin game associaed wih driver G, we obain he equaliy Y = inf Combining (4.11) and (4.12), we obain he desired equaliy (4.1). E G,τ σ[i(τ,σ)]. (4.12) 4.3 Applicaion o he case of ambiguiy on he defaul probabiliy We consider a family of a priori probabiliy measures paramerized by α U. More precisely, foreachα U, leq α beheprobabiliymeasureequivalenop,whichadmiszt α asdensiy wih respec o P, where (Z α ) is he soluion of he following SDE: dz α = Z α ν(,α )dm ; Z α = 1, where ν : (ω,, α) ν(, ω, α) is a bounded P B(U)-measurable funcion defined on Ω [,T] U wih ν(,α) > C 1 > 1. By Girsanov s heorem, we derive ha under Q α, W is a G-Brownian moion and M α := N λ s(1+ν(s,α s ))dsisag-maringale. Hence, under Q α, heg-defaul inensiy isequal o λ (1+ν(,α )). The process ν(,α ) represens he uncerainy on he defaul inensiy. 21

22 To each α U, corresponds a marke model M α associaed wih he a priori probabiliy measure Q α. In he marke M α, he dynamics of he wealh process V α,x,ϕ associaed wih an iniial wealh x and a risky asses saegy ϕ H 2 H 2 λ are posed o saisfy dv α,x,ϕ = f(,v α,x,ϕ,ϕ σ, ϕ 2,α )d ϕ σ dw +ϕ 2 dmα ; V α,x,ϕ = x, (4.13) where f : (,ω,y,z,k,α) f(,ω,y,z,k,α) is a map posed o be uniformly λ- admissible wih respec o (y,z,k), saisfying (4.1) wih θ,y,z,k 1,k 2 > ( 1 C 1 ) ( 1) and α U f(,.,,,,α) H p, for some p > 2. For example, f can be given as in (2.12) in he case of a perfec marke, or as in Examples 2.11 of marke imperfecions, wih coefficiens which may depend on α. By [15, Proposiion A.3], here is a maringale represenaion heorem for G-maringales under Q α wih respec o W and M α. Le ξ L p (G T ), where p > 2. By [15, Proposiion 2.11], he densiy ZT α of Qα wih respec o P belongs o L q for all q 2. Le p ]2,p[. Applying Hölder s inequaliy, we derive ha E Q α(ξ p ) < +. Similarly, since by assumpion f(,,,,α ) H p, wederive haf(,,,,α ) H p Qα. By[15, CorollaryA.4], hereexiss an unique soluion (X α,z α,k α ) in S p Q α Hp Q α Hp Q α,λ of he following Qα -BSDE: dx α = f(,x α,zα,kα,α )d Z α dw K α dmα ; Xα T = ξ. (4.14) As in he previous secion, o simplify noaion, for each α U, we denoe by f α he driver f α (,y,z,k) = f(,y,z,k,α ). The nonlinear price sysem in he marke model M α, denoed by E fα Q α, is hus he f α -evaluaion under he a priori probabiliy measure Q α, defined on L p. The robus er-hedges are defined as in Definiion 4.1 and he seller s robus price u is defined by (4.5). Since M α = M λ sν(s,α s )ds, he dynamics (4.13) of he wealh process V α,x,ϕ in he marke model M α can be wrien as follows: dv α,x,ϕ = λ ν(,α )ϕ 2 d+f(,v α,x,ϕ,ϕ σ, ϕ 2,α )d ϕ σ dw +ϕ 2 dm. This example hus corresponds o he model wih ambiguiy defined in Secion 4.1 wih g(,α) defined by g(,ω,y,z,k,α):= λ (ω)ν(,ω,α)k+f(,ω,y,z,k,α). By he assumpions on f, he map g saisfies he required condiions, in paricular inequaliy (4.1). Theorems 4.6 and 4.7 as well as Proposiion 4.8 hold. In paricular, he seller s robus price u of he game opion admis he following dual represenaion: u = α U inf E gα,τ σ[i(τ, σ)] = inf σ T α U τ T E gα,τ σ[i(τ,σ)]. (4.15) We now show ha for each α U, E gα is equal o he nonlinear price sysem E fα Q α relaive o he marke model M α. Firs, we have (Z α T ) 1 L q for all q 1. Indeed, The process (Z α ) 1 saisfies he following Q α -SDE: d(z α ) 1 = (Z α ) 1 ν(,α )dm α, wih (Zα ) 1 = 1. 22

23 By [15, Proposiion 2.11], (ZT α) 1 belongs o L q Q for all α q 1, which implies ha (ZT α) 1 L q for all q 1. Since p > 2, by Hölder s inequaliy, we derive ha (X α,z α,k α ) (soluion of (4.14)) belongs o S 2 H 2 H 2 λ and is hus he unique soluion in S2 H 2 H 2 λ of he P-BSDE: dx α = g α (,X α,zα,kα )d Zα dw K α dm ; XT α = ξ. Hence, for each mauriy S and each payoff η L p (G S ), we have E fα Q α,,s(η) = Egα,S (η), which gives ha E gα is equal o he nonlinear price sysem E fα Qα relaive o he marke model M α. Using his propery ogeher wih equaliies (4.15) and Theorem 4.7, we derive he following resul. Proposiion 4.9. (Seller s robus price) The seller s robus price of he game opion in his model admis he following dual represenaion: u = α U inf E fα Q α,,τ σ[i(τ,σ)] = inf Le G be he map defined for each (,ω,z,k) by σ T α U τ T E fα Q α,,τ σ[i(τ,σ)]. (4.16) G(,ω,y,z,k) := (λ (ω)ν(,ω,α)k+f(,ω,y,z,k,α)). (4.17) α U We have u = Y, where Y is he soluion of he P-DRBSDE associaed wih driver G and barriers ξ and ζ. 5 Complemenary resuls 5.1 Pricing of European opions from he buyer s poin of view Le us consider he pricing and hedging problem of a European opion wih mauriy T and payoff ξ L 2 (G T ) from he buyer s poin of view. Supposing he iniial price of he opion is z, he sars wih he amoun z a ime =, and looks o find a risky-asses sraegy ϕ such ha he payoff ha he receives a ime T allows him o recover he deb he incurred a ime = by buying he opion, ha is such ha V z, ϕ T = ξ a.s. V z, ϕ T + ξ = a.s. or equivalenly, The buyer s price of he opion is hus equal o he opposie of he seller s price of he opion wih payoff ξ, ha is E g,t ( ξ) = X, where ( X, Z, K) is he soluion of he BSDE associaed wih driver g and erminal condiion ξ. Le us specify he hedging sraegy for he buyer. Suppose ha he iniial price of he opion is z := X. The process X is equal o he value of he porfolio associaed wih iniial value z = X and sraegy ϕ := Φ( Z, K) (where Φ is defined in Definiion 2.3) ha is X = V X, ϕ = V z, ϕ. Hence, V z, ϕ T = X T = ξ a.s., which yields ha ϕ is he hedging risky-asses sraegy for he buyer. 23