American Options in an Imperfect Complete Market with Default

. We study pricing and hedging for American options in an imperfect market model with default, where the imperfections are taken into account via the nonlinearity of the wealth dynamics. The payoﬀ is given by an RCLL adapted process ( ξ t ). We deﬁne the seller’s price of the American option as the minimum of the initial capitals which allow the seller to build up a superhedging portfolio. We prove that this price coincides with the value function of an optimal stopping problem with a nonlinear expectation E g (induced by a BSDE), which corresponds to the solution of a nonlinear reﬂected BSDE with obstacle ( ξ t ). Moreover, we show the existence of a superhedging portfolio strategy. We then consider the buyer’s price of the American option, which is deﬁned as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged. We show that the buyer’s price is equal to the value function of an optimal stopping problem with a nonlinear expectation, and that it can be characterized via the solution of a reﬂected BSDE with obstacle ( ξ t ). Under the additional assumption of left upper semicontinuity along stopping times of ( ξ t ), we show the existence of a super-hedge ( τ, ϕ ) for the buyer.


Introduction
We consider an American option associated with a terminal time T and a payoff given by an RCLL adapted process (ξ t ). The case of a classical perfect market has been largely studied in the literature (see e.g. [21,25]). The seller's price (often called superhedging price or fair price in the literature), denoted by u 0 , is classically defined as the minimal initial capital which enables the seller to invest in a portfolio which covers his liability to pay to the buyer up to T no matter what the exercise time chosen by the buyer. Moreover, this price is equal to the value function of the following optimal stopping time problem where T is the set of stopping times valued in [0, T ]. Here,ξ t denotes the discounted value of ξ t , equal to e − t 0 rsds ξ t , where (r t ) is the instantaneous interest process, and E Q denotes the expectation under the unique equivalent martingale measure Q. In [12], the seller's price is characterized via a reflected BSDE with lower obstacle (ξ t ).
The aim of the present paper is to study pricing and hedging issues for American options in the case of imperfections in the market model taken into account via the nonlinearity of the dynamics of the wealth (or equivalently, of the portfolio's values), which are modeled via a nonlinear driver g. We moreover include the possibility of a default. The market model we consider here is complete, in the sense that for each European option, there exists a unique portfolio such that its value at the exercise time is equal to the payoff. A large class of imperfect market models can fit in our framework: e.g. the case of different borrowing and lending interest rates (see e.g. [4,5,22]), or the case of market prices impacted by the hedging strategy of a large seller (see e.g. [14,23]). In the framework of a market with default, the case of a large seller affecting the default probability is considered in [11].
We provide a characterization of the seller's price u 0 as the value of a corresponding optimal stopping problem with a nonlinear expectation, more precisely where E g is the g-evaluation (also called g-expectation) induced by a nonlinear BSDE with default jump solved under the primitive probability measure P with driver g. Note that in the particular case of a perfect market, the driver g is linear and (2) reduces to (1). We also show that the seller's price can be characterized via the solution of the reflected BSDE with driver g and lower obstacle (ξ t ), and we study the existence of a superhedging portfolio strategy for the seller. The proof of these results is based on the characterization of the value function of an optimal stopping problem with E g -expectation as the solution of an associated reflected BSDEs with driver g, provided in [14,Section 5.4] in the case of a continuous payoff, and in [27] when the payoff is only RCLL). We then consider the buyer's price of the American option, denoted byũ 0 , defined as the supremum of the initial prices which allow the buyer to select an exercise time τ and a portfolio strategy ϕ so that he/she is superhedged. We provide a characterization of the buyer's priceũ 0 as the value of a corresponding optimal stopping problem with nonlinear expectation. More precisely, we prove that where Eg is theg-expectation, associated with driverg(t, y, z, k) := −g(t, −y, −z, −k). Moreover, we show that the buyer's price can also be characterized via the solution of a nonlinear reflected BSDE with obstacle (ξ t ) and driverg. Under the additional assumption of left upper-semicontinuity along stopping times of (ξ t ), we show the existence of a super-hedge (τ ,φ) for the buyer. Wheng = g, which is the case when the market is perfect, the buyer's price is equal to the seller's price, that isũ 0 = u 0 . Wheng ≤ g, we obtainũ 0 ≤ u 0 . The interval [ũ 0 , u 0 ] can then be seen as a non-arbitrage interval for the price of the American option (see Section 5).
The paper is organized as follows: in Section 2, we introduce our imperfect market model with default and nonlinear wealth dynamics. In Section 3, we study pricing and superhedging of American options from the seller's point of view. In Section 4, we address the pricing and superhedging problem from the buyer's point of view. In Section 5, we discuss some arbitrage issues in our imperfect market model.

Market model with default
Let (Ω, G, P ) be a complete probability space equipped with two stochastic processes: a unidimensional standard Brownian motion W and a jump process N defined by N t = 1 ϑ≤t for all t ∈ [0, T ], where ϑ is a random variable which models a default time. We assume that this default can appear after any fixed time, that is P (ϑ ≥ t) > 0 for all t ≥ 0. We denote by G = {G t , t ≥ 0} the augmented filtration generated by W and N (in the sense of [7,). We denote by P the predictable σ-algebra. We suppose that W is a G-Brownian motion. Let (Λ t ) be the predictable compensator of the nondecreasing process (N t ). Note that (Λ t∧ϑ ) is then the predictable compensator of (N t∧ϑ ) = (N t ). By uniqueness of the predictable compensator, Λ t∧ϑ = Λ t , t ≥ 0 a.s. We assume that Λ is absolutely continuous w.r.t. Lebesgue's measure, so that there exists a nonnegative process λ, called the intensity process, such that Λ t = t 0 λ s ds, t ≥ 0. Since Λ t∧ϑ = Λ t , λ vanishes after ϑ. The compensated martingale is given by Let T > 0 be the terminal time. We define the following sets: |U t | 2 λ t dt because the intensity λ vanishes after ϑ. Note that, without loss of generality, we may assume that U vanishes after ϑ. 1 Moreover, T denotes the set of stopping times τ such that τ ∈ [0, T ] a.s. and for each S in T , T S is the set of stopping times τ such that S ≤ τ ≤ T a.s. Recall that in this setup, we have a martingale representation theorem with respect to W and M (see [24]).
We consider a complete financial market with default as in [2], which consists of one risk-free asset, with price process S 0 satisfying dS 0 t = S 0 t r t dt, and two risky assets with price processes S 1 , S 2 evolving according to the following equations: The process S 0 = (S 0 t ) 0≤t≤T corresponds to the price of a non risky asset with interest rate process r = (r t ) 0≤t≤T , S 1 = (S 1 t ) 0≤t≤T to a non defaultable risky asset, and S 2 = (S 2 t ) 0≤t≤T to a defaultable asset with total default. The price process S 2 vanishes after ϑ.
All the processes σ 1 , σ 2 , r, µ 1 , µ 2 are predictable (that is P-measurable). We suppose that the coefficients σ 1 , σ 2 > 0, and r, σ 1 , We set σ := (σ 1 , σ 2 ) , where denotes transposition. We consider an investor, endowed with an initial wealth equal to x, who can invest his wealth in the three assets of the market. At each time t < ϑ, he chooses the amount ϕ 1 t (resp. ϕ 2 t ) of wealth invested in the first (resp. second) risky asset. However, after time ϑ, he cannot invest his wealth in the defaultable asset since its price is equal to 0, and he only chooses the amount ϕ 1 t of wealth invested in the first risky asset. Note that the process ϕ 2 can be defined on the whole interval [0, T ] by setting ϕ 2 t (ω) = 0 for each (ω, t) such that t > ϑ(ω).
, P, λtdP ⊗ dt)) can be identified with U 1 t≤ϑ , since U 1 t≤ϑ is a predictable process satisfying Ut1 t≤ϑ = Ut λtdP ⊗ dt-a.s. A process ϕ . = (ϕ 1 t , ϕ 2 t ) 0≤t≤T is called a risky assets strategy if it belongs to H 2 × H 2 λ . The value of the associated portfolio (also called wealth) at time t is denoted by V x,ϕ t (or simply V t ).
The perfect market model. In the classical case of a perfect market model, the wealth process and the strategy satisfy the self financing condition: where θ 1 t := Note that (θ 2 t ) vanishes after ϑ, as the process (λ t ). We suppose below that the processes θ 1 and θ 2 √ λ are bounded.
Consider a European contingent claim with maturity T > 0 and G T -measurable payoff ξ in L 2 . The problem is to price and hedge this claim by constructing a replicating portfolio. From [11,Proposition 2], there exists a unique process (X, Z, K) ∈ S 2 × H 2 × H 2 λ solution of the following BSDE with default jump: The solution (X, Z, K) provides the replicating portfolio. More precisely, the process X corresponds to its value, and the hedging risky assets strategy ϕ in H 2 λ is given by ϕ = Φ(Z, K), where Φ is the one-to-one map defined on H 2 × H 2 λ by: Definition 2.1. Let Φ : H 2 × H 2 λ → H 2 × H 2 λ be the one-to-one map defined for each (Z, K) ∈ H 2 × H 2 λ by Φ(Z, K) := ϕ, where ϕ = (ϕ 1 , ϕ 2 ) is given by which is equivalent to K t = −ϕ 2 t ; Z t = ϕ t σ t . Note that the processes ϕ 2 and K, which belong to H 2 λ , both vanish after time ϑ. The process X coincides with V X0,ϕ , the value of the portfolio associated with initial wealth x = X 0 and portfolio strategy ϕ. Since X T = ξ a.s. , we derive that V X0,ϕ T = ξ a.s. 2 From the seller's point of view, this portfolio is a hedging portfolio. Indeed, by investing at initial time 0 the initial amount X 0 in the reference assets along the strategy ϕ, the seller can pay the amount ξ to the buyer at time T (and similarly at each initial time t with initial amount X t ). We derive that X t is the price at time t of the option, called hedging price, and denoted by X t (ξ). By the representation property of the solution of a λ-linear BSDE with default jump (see [11,Theorem 1]), the solution X of BSDE (4) is given by: where ζ t,· satisfies dζ t,s = ζ t,s − [−θ 1 s dW s − θ 2 s dM s ] with ζ t,t = 1. This defines a linear price system X: ξ → X(ξ). Suppose now that which is a usual assumption in the literature on default risk. Moreover, the price system X is increasing and corresponds to the classical arbitrage-free price system. The imperfect market model M g . From now on, we assume that there are imperfections in the market which are taken into account via the nonlinearity of the dynamics of the wealth. More precisely, the dynamics of the wealth V associated with strategy ϕ = (ϕ 1 , ϕ 2 ) can be written via a nonlinear driver, defined as follows: Definition 2.2 (Driver, λ-admissible driver). A function g is said to be a driver if g : [0, T ] × Ω × R 3 → R; (ω, t, y, z, k) → g(ω, t, y, z, k) is P ⊗ B(R 3 )− measurable, and such that g(., 0, 0, 0) ∈ H 2 . A driver g is called a λ-admissible driver if moreover there exists a constant C ≥ 0 such that dP ⊗ dt-a.s. , for each (y 1 , z 1 , k 1 ), (y 2 , z 2 , k 2 ), A nonnegative constant C which satisfies this inequality is called a λ-constant associated with driver g.
Let x ∈ R be the initial wealth and let ϕ = (ϕ 1 , ϕ 2 ) in H 2 × H 2 λ be a portfolio strategy. We suppose now that the associated wealth process V x,ϕ t (or simply V t ) is defined as the unique solution in S 2 of the forward stochastic differential equation: with initial condition V 0 = x.Equivalently, setting Z t = ϕ t σ t and K t = −ϕ 2 t , the dynamics (8) of the wealth process V t can be written as follows: In the following, our imperfect market model is denoted by M g . Note that in the case of a perfect market (see (3)), we have: which is a λ-admissible driver since by the assumptions on the coefficients of the model, the processes θ 1 and θ 2 √ λ are bounded.

Nonlinear pricing system E g (for the seller)
Pricing and hedging European options in the imperfect market M g leads to BSDEs with nonlinear driver g and a default jump. By [11, Proposition 2], we have Proposition 2.3. Let g be a λ-admissible driver, let ξ ∈ L 2 (G T ). There exists a unique solution (X(T, ξ), Z(T, ξ), K(T, ξ)) (denoted simply by (X, Z, K)) in S 2 × H 2 × H 2 λ of the following BSDE: Let us consider a European option with maturity T and terminal payoff ξ in L 2 (G T ) in this market model. Let x ∈ R and let ϕ ∈ H 2 × H 2 λ . The process ϕ is called a hedging risky-assets portfolio strategy for the seller (associated with initial capital x) if the value of the associated portfolio is equal to ξ at time T , that is V x,ϕ T = ξ a.s.
Let (X, Z, K) be the solution of BSDE (11). The process X is equal to the value of the portfolio associated with initial capital x = X 0 and the risky assets strategy ϕ = Φ(Z, K) (where Φ is defined in Definition 2.1), that is X = V X0,ϕ . Since X T = ξ, the process ϕ is thus a hedging risky-assets strategy for the seller associated with initial capital x = X 0 .
Remark 2.4 (Completeness property of the market M g ). By the above observations, we derive that for each ξ ∈ L 2 (G T ), there exists a unique initial capital x ∈ R and a unique risky-assets strategy ϕ such that the value of the associated portfolio is equal to ξ at time T . By analogy with the classical case of a perfect market, this property is called completeness property of the market M g .
Using the previous notation, the initial value X 0 = X 0 (T, ξ) of the hedging portfolio is thus a sensible price (at time 0) of the claim ξ for the seller since this amount is the unique initial capital which allows him/her to build a hedging risky-assets strategy. Similarly, X t = X t (T, ξ) satisfies an analogous property at time t, and is called the hedging price (for the seller) at time t. This leads to a nonlinear pricing system, first introduced in [14, Section 2.3] in a Brownian framework, then called g-evaluation and denoted by E g in [26]. For each We make the following assumption (see [11,Assumption 4 in Section 3.3]).
Remark 2.6. Assume that g(t, 0, 0, 0) = 0 dP ⊗ dt-a.s. Then for all S ∈ [0, T ], E g ·,S (0) = 0 a.s. Moreover, by the comparison theorem for BSDEs with default jump (see [11,Theorem 3]), the nonlinear pricing system E g is Note that, by the flow property of BSDEs, for each S ∈ [0, T ] and for each ξ ∈ L 2 (G S ), the g-evaluation For examples of imperfect market models of type M g , the reader is referred to [10,11,13]. Note that when the market is perfect, the prices S 0 , S 1 and S 2 are E g -martingales. 3 This property does not necessarily holds for all λ-admissible drivers g (as for example in a market model with taxes on risky investment profits, see [10] and [13]).

American option pricing from the seller's point of view
Let us consider an American option associated with horizon T > 0 and payoff given by an RCLL adapted process (ξ t , 0 ≤ t ≤ T ) in S 2 . At time 0, it consists in the selection of a stopping time τ ∈ T and the payment of the payoff ξ τ from the seller to the buyer.
The seller's price of the American option at time 0, denoted by u 0 , is classically defined as the minimal initial capital which enables the seller to invest in a portfolio which covers his liability to pay to the buyer up to T no matter what the exercise time chosen by the buyer. More precisely, for each initial wealth x, we denote by A(x) the set of all portfolio strategies ϕ ∈ H 2 such that V x,ϕ t ≥ ξ t , 0 ≤ t ≤ T a.s. The seller's price of the American option is thus defined by Note that u 0 ∈ R. We shall see below that u 0 is finite.
Remark 3.1. Suppose that g(t, 0, 0, 0) = 0. From Remark 2.6, we derive that if ξ · ≥ 0, the infimum in the definition of u 0 can be taken only over nonnegative initial wealths, that is, We define the g-value associated with the American option as the value function (at time 0) of the E g -optimal stopping problem associated with payoff (ξ t ), that is Proposition 3.2 (Characterization of the g-value). There exists a unique process (Y, Z, K, A) solution of the reflected BSDE (RBSDE) associated with driver g and obstacle ξ in where A c denotes the continuous part of A and A d its discontinuous part. Moreover, for each S ∈ T , we have In particular, Y 0 is equal to the g-value sup τ ∈T E g 0,τ (ξ τ ). Note that this result was shown in [14,Theorem 5.9] in a Brownian framework with a continuous obstacle, and generalized in [27,Theorem 3.3] to the case of a filtration associated with a Brownian motion and a Poisson random measure, with an RCLL obstacle.
Proof. Let us first show that there exists a unique solution of RBSDE (16). As usual, we first consider the case when the driver g(t) does not depend on the solution. By using the representation property of G-martingales (see [24]) and some results of optimal stopping theory, one can show, proceeding as in [12] (see also [17] and [27]), that there exists a unique solution of the associated RBSDE (16). The proof in the case of a general λ-admissible driver is based on a usual fixed point argument and a priori estimates for RBSDEs with default given in Appendix (see Lemma 8.1). Equality (18)   Proof. Let τ be a predictable stopping time. By (16), we have ∆A τ = (∆Y τ ) − . Using the Skorokhod conditions s. We derive that ∆A τ = 0 a.s. It follows that A is continuous.
We now provide two characterizations of the seller's price, which generalize those provided in the literature in the case of a perfect market (see [12]) to the case of an imperfect market.
Theorem 3.4 (Seller's price of the American option). The seller's price u 0 of the American option is equal to the g-value, that is Moreover, we have where (Y, Z, K, A) is the solution of the nonlinear reflected BSDE (16) and the portfolio strategy ϕ * := Φ(Z, K) (where Φ is defined in Definition 2.1) is a superhedging strategy for the seller.
Note that in the case of a perfect market, equality (19) reduces to the well-known characterization of the price of the American option as the value function of a classical optimal stopping problem, and the equality u 0 = Y 0 corresponds to the well-known characterization of this price as the solution of the linear reflected BSDE associated with the linear driver (10) (see [12]).
Proof. The proof is based on the characterization of the g-value as the solution of the reflected BSDE (16) (see Proposition 3.2). It is sufficient to show that u 0 = Y 0 and ϕ * ∈ A(u 0 ). Let H be the set of initial capitals which allow the seller to be "super-hedged", that is By (8)-(9), for almost every ω, the trajectory of the value of this portfolio t → V Y0,ϕ * t (ω) satisfies the following forward differential equation: where (16), for almost every ω, the function t → Y t (ω) satisfies: where by a comparison result for forward differential equations (see e.g. [10] in the Appendix), we derive that s. , which implies the desired property (20). It follows that Y 0 ≥ u 0 . Let us show the converse inequality. Let x ∈ H. There exists ϕ ∈ A(x) such that V x,ϕ t ≥ ξ t , 0 ≤ t ≤ T a.s. For each τ ∈ T we thus have V x,ϕ τ ≥ ξ τ a.s. By taking the E g -evaluation in this inequality, using the monotonicity of E g and the E g -martingale property of the wealth process V x,ϕ , we obtain , which holds for any x ∈ H. By taking the infimum over x ∈ H, we obtain u 0 ≥ Y 0 . We derive that u 0 = Y 0 . By (20), we thus have ϕ * ∈ A(u 0 ), which ends the proof.
Remark 3.5. In general, except when g does not depend on y, by (22), we have Remark 3.6. In [20], it is proved that the seller's price of the American option is equal to the g-value in the case of a higher interest rate for borrowing, by using a dual approach. This approach relies on the convexity properties of the driver and cannot be adapted to our case, except when g is convex with respect to (y, z, k).
We define now the seller's price of the American option at each time/stopping time S ∈ T . We first define, for each initial wealth X ∈ L 2 (G S ), a super-hedge against the American option as a portfolio strategy ϕ ∈ s. , where V S,X,ϕ denotes the wealth process associated with initial time S and initial condition X. The seller's price at time S is defined by the random variable where A S (X) is the set of all super-hedges associated with initial time S and initial wealth X. Using equality (18) and similar arguments to those used in the proof of Theorem 3.4 above, one can show the following result, which generalizes the results of Theorem 3.4 to any time S ∈ T .
where (Y, Z, K, A) is the solution of reflected BSDE (16).
In other terms, the seller's price family of random variables (u(S), S ∈ T ) can be aggregated by an RCLL adapted process, which we call the seller's price process of the American option. Moreover, this price process is both characterized as the value function process of the E g -optimal stopping problem with payoff (ξ t ), and also as the solution (Y t ) of the reflected BSDE (16).
Suppose now that the buyer has bought the American option at the selling price u 0 = Y 0 . We address the problem of the choice of her/his exercise time. We introduce the following definition.
Definition 3.8. A stopping timeτ ∈ T is called a rational exercise time for the buyer of the American option if it is optimal for Problem (19), that is, if it satisfies sup τ ∈T E g 0,τ (ξ τ ) = E g 0,τ (ξτ ). By the optimality criterium provided in [27] (see Proposition 3.5), we have: Proposition 3.9. (Characterization of rational exercise times) Let τ ∈ T . Then, τ is a a rational exercise time for the buyer if and only if Y τ = ξ τ a.s. and A τ = 0 a.s. , where (Y, Z, K, A) is the solution of the reflected BSDE (16).
Suppose now that the price of the American option is equal to the seller's price not only at time 0 but at all time S ∈ T , that is, the price at time S is equal to u(S) = Y S (see Proposition 3.7). Suppose that the buyer buys the American option at time 0 (at price u 0 = Y 0 ). Let us show that it is not profitable for him to exercise his option at a stopping time τ which is not a rational exercise time. First, it is not in his interest to exercise at a time t such that Y t > ξ t since he would loose a financial asset (the option) with value Y t and only receive the lower amount ξ t . Second, it is not in his interest to exercise at a stopping time τ greater thanτ , defined bȳ τ := inf{s ≥ 0, A s = 0}. Let us show that Yτ = V Y0,ϕ * τ a.s. Note that by definition ofτ , Aτ = 0 a.s. Hence, for a.e. ω, the trajectories t → Y t (ω) and t → V Y0,ϕ * t (ω) are solutions on [0,τ (ω)] of the same differential equation (with initial value Y 0 ), which implies that they are equal, by uniqueness of the solution. Hence, Yτ = V Y0,ϕ * τ a.s. Without loss of generality, we can suppose that for each ω, we have Yτ (ω) = V Y0,ϕ * τ (ω). Let τ ≥τ . Let B := {τ >τ }. Suppose that P (B) > 0. Hence, A τ > 0 a.s. on B. Then, by exercising the option at timē τ , the option holder receives the amount Yτ which he can invest in the market along the strategy ϕ * . Since Yτ = V Y0,ϕ * τ , by the flow property of the forward differential equation (8) with ϕ = ϕ * and x = Y 0 , the value of the associated portfolio is equal to V Yτ ,ϕ * Since A τ > 0 on B, by the strict comparison result for forward differential equations applied to (21) and (22) (see [10] in the Appendix), we get V Y0,ϕ * τ > Y τ a.s. on B, which implies V Yτ ,ϕ * τ = V Y0,ϕ * τ > ξ τ a.s. on B. We thus have V Yτ ,ϕ * τ ≥ ξ τ a.s. with P (V Yτ ,ϕ * τ > ξ τ ) > 0. Hence, at timeτ , it is more interesting for the buyer to exercise immediately than later. Proposition 3.10. (Existence of rational exercise times) Suppose that the payoff ξ is left u.s.c. along stopping times. Let (Y, Z, K, A) be the solution of the reflected BSDE (16). Let τ * := inf{s ≥ 0, Y s = ξ s } andτ := inf{s ≥ 0, A s = 0}. 4 The stopping time τ * (resp.τ ) is the minimal (resp. maximal) rational exercise time.
By Proposition 3.9, τ * is thus a rational exercise time and is the minimal one. From the definition ofτ , and the continuity of A, we have Aτ = 0 a.s. Also, we have a.s. for all t >τ , A t > Aτ = 0. Since A increases only on the set {Y · = ξ · }, it follows that Yτ = ξτ . By Proposition 3.9,τ is a rational exercise time and is the maximal one.
When ξ is only RCLL, there does not exist necessarily a rational exercise time for the buyer. However, we have the following result.
where K is a constant which only depends on T and the λ-constant C of g.
The proof, which relies on similar arguments as those used in the proof of Theorem 3.2 in [27], is left to the reader.
Remark 3.12. Following the terminology of classical optimal stopping theory, τ ε is a Kε-optimal stopping time for the E g -optimal stopping problem sup τ E g 0,τ (ξ τ ).

The buyer's point of view
Let us consider the pricing and hedging problem of a European option with maturity T and payoff ξ ∈ L 2 (G T ) from the buyer's point of view. Supposing the initial price of the option is z, he starts with the amount −z at time t = 0, and wants to find a risky-assets strategyφ such that the payoff that he receives at time T allows him to recover the debt he incurred at time t = 0 by buying the option, that is such that V −z,φ T + ξ = 0 a.s. or equivalently, V −z,φ T = −ξ a.s. The buyer's price of the option is thus equal to the opposite of the seller's price of the option with payoff −ξ, that is −E g 0,T (−ξ) = −X 0 , where (X,Z,K) is the solution of the BSDE associated with driver g and terminal condition −ξ. Let us specify the hedging strategy for the buyer. Suppose that the initial price of the option is z := −X 0 . The processX is equal to the value of the portfolio associated with initial value −z =X 0 and strategỹ ϕ : s. , which yields thatφ is the hedging risky-assets strategy for the buyer. Similarly, −E g t,T (−ξ) = −X t satisfies an analogous property at time t, and is called the hedging price for the buyer at time t.
Note that for all ξ ∈ L 2 (G T ), we have −E g ·,T (−ξ) = Eg ·,T (ξ). 4 Note that by Proposition 3.7, the process (Yt) is equal to the seller's price process of the American option.
This leads to the nonlinear pricing system Eg relative to the buyer (corresponding to theg-evaluation), defined for each (S, ξ) ∈ [0, T ] × L 2 (G S ) by Eg ·,S (ξ), which is equal to the solution of the BSDE with driverg, terminal time S and terminal condition ξ.
Remark 4.1. If we suppose that −g(t, −y, −z, −k) ≤ g(t, y, z, k) (which is satisfied when for example g is convex with respect to (y, z, k) and g(t, 0, 0, 0) = 0), then, by the comparison theorem for BSDEs, we have In other terms, the hedging price of a European option for the buyer is always smaller than the the seller's one.
Moreover, when −g(t, −y, −z, −k) = g(t, y, z, k) (which is satisfied when for example g is linear with respect to (y, z, k), as in the perfect market case), then the pricing system for the buyer is equal the the seller's one, that is, Eg = E g .
Note that by the flow property of BSDEs, the first coordinate of the solution of a BSDE with driverg is an Eg-martingale. Moreover, the opposite of a wealth process is an Eg-martingale, that is For each x ∈ R and each portfolio strategy ϕ ∈ H 2 × H 2 λ , the process (−V x,ϕ t ) is an Egmartingale.
Proof. Since g(t, y, z, k) := −g(t, −y, −z, −k), we get that the wealth process (V x,ϕ t ) satisfies: where (Z, K) = φ −1 (ϕ). Hence, (−V x,ϕ , −Z, −K) is the solution of the BSDE associated with driverg, terminal time T and terminal conditon −V x,ϕ T . The result follows. Let us consider the case of the American option from the point of view of the buyer. Supposing the initial price of the American option is z, he starts with the amount −z at time t = 0, and wants to find a super-hedge, that is an exercise time τ and a risky-assets strategy ϕ, such that the payoff that he receives allows him to recover the debt he incurred at time t = 0 by buying the American option. This notion of super-hedge for the buyer can be defined more precisely as follows.

Definition 4.3.
A super-hedge for the buyer against the American option with initial price z ∈ R is a pair (τ, ϕ) of a stopping time τ ∈ T and a risky-assets strategy ϕ ∈ H 2 × H 2 λ such that We denote by B(z) the set of all super-hedges for the buyer associated with initial price z ∈ R. We now define the buyer's priceũ 0 of the American option as the supremum of the initial prices which allow the buyer to be super-hedged, that isũ Note thatũ 0 ∈ R. We shall see below thatũ 0 is finite.
We first consider the simpler case when ξ is left-u.s.c. along stopping times. In this case, we prove below that the infimum in (26) is attained, which implies that the buyer's priceũ 0 allows the buyer to build a super-hedge. Moreover, a super-hedge strategy is provided via the solution of a reflected BSDE associated with driverg. More precisely, we have Moreover, we haveũ where (Ỹ ,Z,K,Ã) is the solution of the reflected BSDE associated with driverg(t, y, z, k) := −g(t, −y, −z, −k) and lower obstacle ξ, that is, where the non decreasing processÃ is continuous.
Proof. Note first that, by Remark 3.3 applied to the above reflected BSDE (29), since the obstacle (ξ t ) is supposed to be left upper semicontinuous along stopping times, the processÃ is continuous. Now, by Proposition 3.2 (applied to the driverg instead of g), we have sup τ Eg 0,τ (ξ τ ) =Ỹ 0 . In order to show equality (27), it is thus sufficient to show thatũ 0 =Ỹ 0 .
We now consider the general case when ξ is only RCLL. In this case, the characterizations (27) and (28) of the buyer's priceũ 0 still hold. However, the supremum in (26) is not necessarily attained; in other words, the priceũ 0 does not necessarily allow the buyer to build a super-hedge against the American option.
We introduce the definition of an ε-super-hedge for the buyer: Definition 4.6. For each initial price z and for each ε > 0, an ε-super-hedge for the buyer against the American option is a pair (τ, ϕ) of a stopping time τ ∈ T and a risky-assets strategy ϕ ∈ H 2 × H 2 λ such that where (Ỹ ,Z,K,Ã) is the solution of the reflected BSDE associated with driverg(t, y, z, k) := −g(t, −y, −z, −k) and lower obstacle ξ, that is, Letφ := Φ(−Z, −K) and for each ε > 0, let The pair (τ ε ,φ) is an ε-super-hedge for the buyer (associated with the initial priceũ 0 ).
Let us now show thatỸ 0 =ũ 0 . First, by Proposition 3.2, we have sup τ Eg 0,τ (ξ τ ) =Ỹ 0 . Using this property together with the same arguments as those used in the second part of the proof of Theorem 4.5 (which do not require the continuity ofÃ), we obtain thatũ 0 ≤Ỹ 0 .

Links with arbitrage issues
We address now arbitrage issues associated to the problem of pricing European and American options in our imperfect market model M g . Related issues are studied in [19] and [20] in markets with convex portfolio constraints.

European option case
Consider the market M g with a European option with maturity T and payoff ξ ∈ L 2 (G T ). Recall that E g 0,T (ξ) is the hedging price for the seller (see Section 2.2).
Definition 5.1. Let x ∈ R. Let y ∈ R and ϕ in H 2 × H 2 λ . We say that (y, ϕ) is an arbitrage opportunity for the seller of the European option with initial price x if This definition means that the seller sells the European option at the price x strictly greater than the amount y which is enough to be hedged (by using the strategy ϕ). He thus makes the profit x − y > 0 at time 0. Proof. Let φ := Φ(Z, K), where (X, Z, K) is the solution of BSDE associated with driver g and terminal condition ξ. Note that X 0 = E g 0,T (ξ). Suppose that X 0 < x. Since V X0,φ T = ξ a.s. , (X 0 , φ ) is an arbitrage opportunity for the seller.
Suppose now that there exists an arbitrage opportunity (y, ϕ) for the seller. Then y < x and V y,ϕ T −ξ ≥ 0 a .s. Since E g is monotonous and V y,ϕ is an E g -martingale, we get y = E g 0,T (V y,ϕ T ) ≥ E g 0,T (ξ). Hence x > E g 0,T (ξ).
Consider now the buyer. Recall that Eg 0,T (ξ) is the buyer's hedging price (see Section 4).
Definition 5.3. Let x ∈ R. Let y ∈ R and ϕ in H 2 × H 2 λ . We say that (y, ϕ) is an arbitrage opportunity for the buyer of the European option with initial price x if y > x and V −y,ϕ This definition means that the buyer buys the European option at the price x, stricly smaller than the amount y , which, borrowed at time 0, allows him to recover his debt at time T (by using the strategy ϕ). He thus makes the profit y − x > 0 at time 0.
Using the same arguments as above, we get By Propositions 5.2 and 5.4, we get Proposition 5.6. If E g 0,T (ξ) < Eg 0,T (ξ), there does not exist arbitrage-free prices of the European option. If E g 0,T (ξ) ≥ Eg 0,T (ξ), the interval [Eg 0,T (ξ), E g 0,T (ξ)] is the set of all arbitrage-free prices. We call it the arbitrage-free interval for the European option.
Remark 5.7. Note that by similar arguments as above, one can easily show that E g 0,T (ξ) = inf{x ∈ R, ∃ϕ ∈ Remark 5.8. Suppose that ξ ≥ 0 a.s. and that g(t, 0, 0, 0) = 0. Then, by Remark 2.6, we can restrict ourselves to nonnegative prices x. In this case, we take x, y ∈ R + in the definitions above, and the propositions remain true (with x ∈ R + ).
Remark 5.9. Note that it is possible that E g 0,T (ξ) < Eg 0,T (ξ), in which case there is no arbitrage-free price. A simple example is given by taking g(t, y, z, k) = −|y| and ξ = 1. In this case, E g 0,T (1) = e −T and Eg 0,T (1) = e T .

American option case
Consider now the market M g with an American option with maturity T and payoff (ξ t ) in S 2 . Let u 0 be the superhedging price for the seller defined by (14).
Definition 5. 10. Let x ∈ R. Let y ∈ R, and let ϕ in H 2 × H 2 λ . We say that (y, ϕ) is an arbitrage opportunity for the seller of the American option with initial price x if y < x and V y,ϕ τ − ξ τ ≥ 0 a.s. for all τ ∈ T . Proof. Suppose that there exists an arbitrage opportunity (y, ϕ) for the seller of the American option with price x. Then y < x and V y,ϕ τ − ξ τ ≥ 0 a.s. for all τ ∈ T . Hence, ϕ is a superhedging strategy for the seller associated with initial wealth y. By definition of u 0 as an infimum (see (14)), we get y ≥ u 0 . Since x > y, it follows that x > u 0 . Suppose now that x > u 0 . Then, by definition of u 0 as an infimum, we derive that there exists x such that x > x ≥ u 0 and ϕ ∈ H 2 × H 2 λ such that ϕ is a superhedging strategy for the seller associated with initial wealth x , that is, such that V x ,ϕ τ ≥ ξ τ a.s. for all τ ∈ T . Since x > x, it follows that (x , ϕ) is an arbitrage opportunity for the seller of the American option with price x.
Consider now the buyer's point of view. Letũ 0 be the superhedging price for the buyer defined in (26).
Definition 5.12. Let x ∈ R. Let y ∈ R, let τ ∈ T and let ϕ ∈ H 2 × H 2 λ . We say that (y, τ, ϕ) is an arbitrage opportunity for the buyer of the American option with initial price x, if This definition means that the buyer buys the American option at the price x strictly smaller than the amount y that, borrowed at time 0, allows him to recover his debt at the exercise time τ (by using the strategy ϕ). He thus makes the profit y − x > 0 at time 0. Proof. Suppose that there exists an arbitrage opportunity (y, τ, ϕ) for the buyer of the American option with price x. We thus have y > x and V −y,ϕ τ + ξ τ ≥ 0 a.s. Hence, (τ, ϕ) is a super-hedge for the buyer associated with initial wealth y. By definition ofũ 0 as a supremum (see (26)), we get y ≤ũ 0 . Since x < y, we get x <ũ 0 .
Suppose now that x <ũ 0 . Then, by definition ofũ 0 as a supremum, we derive that there exists z such that x < z ≤ũ 0 , τ ∈ T and ϕ ∈ H 2 × H 2 λ such that (τ, ϕ) is a super-hedge for the buyer associated with z, that is, such that V −z,ϕ τ + ξ τ ≥ 0 a.s. Since z > x, it follows that (z, τ, ϕ) is an arbitrage opportunity for the buyer of the American option with price x.
Definition 5.14. A real number x is called an arbitrage-free price for the American option if there exists no arbitrage opportunity, neither for the seller nor for the buyer of the American option.
By Propositions 5.11 and 5.13, we derive the following result.
Proposition 5.15. If u 0 <ũ 0 , there does not exist arbitrage-free price for the American option.
If u 0 ≥ũ 0 , the interval [ũ 0 , u 0 ] is the set of all arbitrage-free prices. We call it the arbitrage-free interval for the American option.
Remark 5.16. Wheng = g, that is g(t, y, z, k) = −g(t, −y, −z, −k) (which is satisfied for example when g is linear with respect to (y, z, k)), then u 0 =ũ 0 is the unique arbitrage-free price for the American option.
Note that it is possible that u 0 <ũ 0 , and hence, that there does not exist an arbitrage-free price for the American option. A simple example is given by g(t, y, z, k) = −|y| and ξ t = 1 for all t. In this case, we have Remark 5.17. Suppose that ξ 0 ≥ 0 and that g(t, 0, 0, 0) = 0. Then, by Remarks 2.6, 3.1, and 4.4, we can restrict ourselves to nonnegative prices x. In this case, we take x, y ∈ R + in the definitions above, and the propositions remain true (with x ∈ R + ). Then, Proposition 5.15 corresponds to Theorem 4.3 of [20].

American options with intermediate cashflows
Suppose that a European option pays a terminal payoff ξ at terminal time S and intermediate cashflows. The cumulative intermediate cashflows generated by the option is modeled by a finite variational RCLL adapted process (D t ) with square integrable total variation (with D 0 = 0). By [11,Proposition 2], there exists a unique solution (X, Z, K) in S 2 × H 2 × H 2 λ of the following BSDE: The process X, denoted also by X ·,S (ξ, D), is the wealth process associated with initial value x = X 0 and strategy ϕ = Φ(Z, K). Here, D t represents the cumulative cash amount at time t withdrawn from the portfolio in order to pay the intermediate cashflows to the buyer. Hence, the amount X 0 allows the seller to be perfectly hedged against the option, in the sense that it allows him/her to pay the intermediate cashflows and the terminal payoff to the buyer, by investing the amount X 0 along the strategy ϕ in the market. The price for the seller (at time 0) of this option is thus given by X 0 = X 0,S (ξ, D) and the associated hedging strategy is equal to ϕ. Note that the driver of BSDE (37) is given by the λ-admissible "generalized" driver g(t, X t , Z t , K t )dt + dD t . This leads to the following nonlinear pricing system for the seller (see [ . If D = 0, E g,D corresponds to the g-conditional expectation E g . Concerning the pricing of the American option, the results of Section 3 can be shown by using a similar approach to the one used in Section 3, replacing the driver g by the "generalized" driver g(·)dt + dD t , and E g by E g,D .

Conclusion and perspectives
This paper deals with the pricing and hedging problem of American options in a complete imperfect market model with default. First, we have shown that the seller's price, defined as the minimal initial wealth which allows the seller to be superhedged, admits a dual representation as the supremum sup τ E g 0,τ (ξ τ ). Moreover, we have obtained an infinitesimal characterization of the price process as the first coordinate of the solution of a nonlinear reflected BSDE, and also provided a superhedging portfolio strategy, defined in terms of the solution of this reflected BSDE. Our proof relies on the characterization of the value function of an E g -optimal stopping problem as the solution of a nonlinear reflected BSDE (see [14] and [27]).
We have also considered the buyer's point of view : we have shown a dual representation of the buyer's price, and an infinitesimal characterization of the price process as the first coordinate of the solution of a reflected BSDE. Moreover, under an additional regularity assumption on the payoff, we have obtained a super-hedge pair, consisting of a suitable portfolio strategy and an appropriate exercise time, defined in terms of the solution of this reflected BSDE.
We study the problem of game options pricing in an imperfect market with default in [10]. We provide dual representations of the seller's and the buyer's price as well as infinitesimal characterizations of these prices in terms of solutions of nonlinear doubly reflected BSDEs. Our proof relies on a result we provided in [9,Section 4], which allows us to characterize the common value of an E g -optimal stopping game problem as the solution of a nonlinear doubly reflected BSDE.
Note that the characterizations of the buyer's price of the American option cannot be directly derived from those of the seller's price, contrary to the case of game options. Indeed, there is in a way a symmetry between the buyer and the seller of game options (see [10, Section 5.2]), which is not the case for American options.
In a future work, we address the pricing and hedging problem of American options when only one of the two risky assets is tradable. In this case, the market model is no longer complete.

Appendix
We give here some a priori estimates for reflected BSDEs with default jump. Lemma 8.1 (A priori estimate for RBSDEs). Let f 1 and f 2 be two λ-admissible drivers. Let C be a λ-constant associated with f 1 . Let ξ be an adapted RCLL processes. For i = 1, 2, let (Y i , Z i , K i , A i ) be a solution of the RBSDE associated with terminal time T , driver f i and obstacle ξ. Let η, β > 0 be such that β ≥ 3 η + 2C and η ≤ 1 C 2 . Letf (s) := f 1 (s, Y 2 s , Z 2 s , K 2 s ) − f 2 (s, Y 2 s , Z 2 s , K 2 s ). For each t ∈ [0, T ], we then have e βsf (s) 2 ds | G t ] a .s.
Moreover, Ȳ 2 β ≤ T η f 2 β , and if η < 1 C 2 , we then have Z 2 The proof is similar to the one given for doubly reflected SBDEs in the same framework with default (see the proof of Proposition 6.1 in the Appendix in [10]), and left to the reader.