Arbitrage-Free Pricing of Game Options in Nonlinear Markets
aa r X i v : . [ q -f i n . M F ] J u l ARBITRAGE-FREE PRICING OF GAME OPTIONSIN NONLINEAR MARKETS
Edward Kim b , Tianyang Nie a ∗ and Marek Rutkowski b,ca School of Mathematics, Shandong University,Jinan, Shandong 250100, China b School of Mathematics and Statistics, University of SydneySydney, NSW 2006, Australia c Faculty of Mathematics and Information Science, Warsaw University of Technology,00-661 Warszawa, PolandNovember 6, 2018
Abstract
The goal is to re-examine and extend the findings from the recent paper by Dumitrescu etal. [11] who studied game options within the nonlinear arbitrage-free pricing approach developedin El Karoui and Quenez [16]. We consider the setup introduced in Kim et al. [26] wherecontracts of an American style were examined. We give a detailed study of unilateral pricing,hedging and exercising problems for the counterparties within a general nonlinear setup. Wealso present a BSDE approach, which is used to obtain more explicit results under suitableassumptions about solutions to doubly reflected BSDEs.
Keywords : nonlinear market, game option, doubly reflected BSDE
Mathematics Subjects Classification (2010) : 91G40, 60J28 ∗ The research of T. Nie and M. Rutkowski was supported by the DVC Research Bridging Support Grant
Pricingof American and game options in markets with frictions . The work of T. Nie was supported by the NationalNatural Science Foundation of China (No. 11601285) and the Natural Science Foundation of Shandong Province (No.ZR2016AQ13). E. Kim, T. Nie and M. Rutkowski
Unlike contracts of an American style, the so-called game options are symmetric between the twocounterparties, hereafter referred to as the hedger and the counterparty , in the sense that both partieshave the right to stop the contract before its nominal maturity date, denoted as T . We adopt here theconvention according to which a random moment when the game contract is stopped by the hedgeris called a cancellation time whereas for the corresponding moment for the counterparty is called an ⁀ exercise time. Within the framework of the classical Black-Scholes options pricing model, the notionof an abstract game option was first introduced and studied by Kifer [24] who coined the term Israelioption . His results were subsequently extended to a general (possibly incomplete) semimartingalemodel by Kallsen and K¨uhn [22]. Arbitrage-free pricing and rational cancellation/exercising of gameoptions (in particular, convertible bonds) within the framework of a linear market model have beenstudied by several authors, to mention a few: Ayache et al. [1], Bielecki et al. [4], Dolinsky andKifer [9], Hamad´ene [20], Kallsen and K¨uhn [22, 23], Matoussi et al. [33], K¨uhn et al. [29] andKyprianou [30].It is well known that valuation of game options, at least within the framework of a linear marketmodel, is closely related to the concept of the zero-sum two-person Dynkin stopping game, as firstintroduced by Dynkin [13] and later modified by Neveu [34]. From the mathematical perspective,one should also mention papers on Dynkin games and doubly reflected backward stochastic differen-tial equations (DRBSDEs) that underpin valuation of game contract, in particular, Bayraktar andYao [2], Cr´epey and Matoussi [7], Cvitani´c and Karatzas [8], Dumitrescu et al. [10], Essaky andHassani [17], Grigorova et al. [18], Grigorova and Quenez [19], Hamad`ene and Ouknine [21], andLepeltier and San Mart´ın [31]. The interested reader is referred to Kifer [25] for a recent survey ofresults on Dynkin games and their applications to game options.More recently, some authors have studied nonlinear variants of a Dynkin game and their ap-plications to game options. The aim of this work is to re-examine and extend the findings fromDumitrescu et al. [11] (see also [12] for the case of American options) who have studied game op-tions within the framework of a particular imperfect market model with default using the nonlineararbitrage-free pricing approach developed in El Karoui and Quenez [16]. In contrast to [11], we placeourselves within the setup of a general nonlinear arbitrage-free market, as introduced in Bielecki etal. [3, 5], and we examine general properties of unilateral fair prices for the two counterparties.Let us introduce the notation for a general nonlinear market model. We consider a filteredprobability space (Ω , G , G , P ) satisfying the usual conditions of right-continuity and completeness,where the filtration G = ( G t ) t ∈ [0 ,T ] models the flow of information available to all traders. Forconvenience, the inception date of a contract under consideration is set to be 0 and it is assumedthat the initial σ -field G is trivial. Moreover, all processes introduced in what follows are implicitlyassumed to be G -adapted and, as usual, any semimartingale is assumed to be c`adl`ag. For the sakeof notational convenience, we assume throughout that trading conditions are identical for the twocounterparties, although this assumption can be relaxed without any difficulty by modifying thenotation. By convention, all cash flows of a contract are described from the perspective of thehedger. Hence when a cash flow is positive for the hedger, then the cash amount is paid by thecounterparty and received by the hedger. Obviously, if a cash flow is negative for the hedger, thenthe cash amount is transferred from the hedger to the counterparty. Let us state a formal definitionof a game contract. Definition 1.1. A game contract with G -adapted, c`adl`ag processes A, X h , X c , X b is a contractbetween the two parties, called the hedger and the counterparty , where:(i) σ ∈ T is chosen by the hedger and called the cancellation time ,(ii) τ ∈ T is chosen by the counterparty and called the exercise time .The process A with A = 0 gives the cumulative cash flows of the contract up to its effective maturity σ ∧ τ ∧ T . The contract expires at time σ ∧ τ ∧ T and its terminal payoff at time σ ∧ τ ∧ T , as seenfrom the perspective of the hedger, is given by the following expression, for every σ, τ ∈ T , I ( X h , X c , X b ; σ, τ ) := X hσ { σ<τ } + X cτ { τ<σ } + X bσ { τ = σ } . (1) onlinear Pricing of Game Options A, X h , X c , X b , T ) or, simply, C g . In theexisting literature, it is common to assume that the inequalities X ht < X ct and X ht ≤ X bt ≤ X ct aresatisfied for every t ∈ [0 , T ]. In that case, the payoff process X b in fact plays only a minor role,since one can prove that the solutions to the hedging and valuation problems for the hedger and thecounterparty will depend only on the terminal value X bT and thus the values X bt for t ∈ [0 , T ) areimmaterial. We will make these assumptions in Section 3 where we analyze a BSDE approach togame options, since they are convenient when dealing with solutions to DRBSDEs. This should becontrasted with the situation studied in Section 2, where we do not need to make any assumptionsabout the ordering of the payoffs X h , X c and X b .An important element of arbitrage-free pricing in a nonlinear market is the concept of the bench-mark wealth V b ( x ), that is, the process with respect to which arbitrage opportunities of a particulartrader are quantified and assessed. As in Bielecki and Rutkowski [5], Bielecki et al. [3] and Kimet al. [26], the benchmark wealth can be given by the equality V b ( x ) = V ( x ) where, for any ini-tial endowment x ∈ R of a trader, we set V ( x ) := xB ,l { x ≥ } + xB ,b { x< } where the risk-free lending (respectively, borrowing ) cash account B ,l (respectively, B ,b ) is used for unsecured lending(respectively, borrowing) of cash. Note that V ( x ) represents the wealth process of a trader whodecided at time 0 to keep his initial cash endowment x in either the lending (when x ≥
0) or theborrowing (when x <
0) cash account and who is not involved in any other trading activities be-tween the times 0 and T . More generally, if a trader is endowed with an initial portfolio of assets(including also the savings account B ,l and B ,b , which means that he can also be a borrower or alender of cash) with the current market value x at time 0 (ignoring bid-ask spreads and transactioncosts), then V b ( x ) stands for the wealth process of the trader’s static portfolio under the postulatethat it is hold unchanged till the date T . By convention, the quantities x and x represent initialendowments of the hedger and the counterparty, respectively, and thus the processes V ( x ) and V ( x ) are their respective benchmark wealths. From the economic perspective, the notion of thebenchmark wealth can be seen as a plausible formalization of the well known idea of opportunitycosts , which is instrumental in financial decision making. Note also that, even if we set x = x = 0and V bt (0) = 0 for all t ∈ [0 , T ], then, due to nonlinear dynamics of the wealth process, unilateralprices will still be asymmetric. A particular choice for the benchmark process V b ( x ) is important inpractical applications, but it is immaterial for all results presented in this work.In Section 2, we work in an abstract nonlinear setup, meaning that we only make very generalassumptions about the nonlinear dynamics of the wealth process of self-financing strategies. Themain assumptions of that kind are the forward monotonicity and the strict forward monotonic-ity of wealth processes (see Assumptions 2.1 and 2.2, respectively), the comparison and the strictcomparison postulates for trading strategies (Assumptions 2.3 and 2.6, respectively), as well as thereplicability postulates for the hedger and the counterparty (Assumptions 2.4 and 2.7, respectively).The main result in Section 2 is Theorem 2.1 where we show that, under mild and natural assump-tions about the underlying nonlinear market model, a unilateral hedger’s acceptable price for thegame contract is unique and corresponds to the fair price obtained through replication. To this end,one needs to properly define the concepts of a fair price and replication of a game contract in anonlinear setup. Using the symmetric features of a game contract, we show in Theorem 2.2 that thesame properties of pricing are true for the counterparty as well.In Section 3, we present a BSDE approach to game options in a fairly general setup. We obtainthere more explicit results regarding pricing, hedging and rational cancellation/exercise times usinga BSDE approach without being specific about the dynamics of underlying assets, but by focusinginstead on desirable properties of solutions to DRBSDEs. In Section 3.4, we show that the hedger’sacceptable price can be characterized by a unique solution to the hedger’s DRBSDE. Next, in Section3.5, we analyze the sets of hedger’s rational cancellation times and break-even times under thepostulate that the contract is traded at time 0 at the hedger’s acceptable price. The correspondingresults for the counterparty are stated without proofs in Sections 3.4 and 3.5. It should be noted thatthe DRBSDEs for the hedger and the counterparty differ and thus, in general, unilateral acceptableprices computed by the two counterparties will not coincide. E. Kim, T. Nie and M. Rutkowski
Let M = ( B , S , Ψ) be a market model which is arbitrage-free with respect to European contracts inthe sense of Bielecki et al. [3, 5]. Here Ψ stands for the class of all admissible trading strategies andΨ( y, D ) denotes the class of all admissible trading strategies from Ψ with the initial wealth y ∈ R and with external cash flows D . For any trading strategy ϕ ∈ Ψ( y, D ), we denote by V ( y, ϕ, D ) the wealth process of ϕ . Obviously, the equality V ( y, ϕ, D ) = y holds for all y ∈ R and any strategy ϕ .It is assumed throughout that the processes D, X h , X c and the wealth process V ( y, ϕ, D ) are c`adl`agand G -adapted. We will gradually make more assumptions about the dynamics of wealth processes. We first conduct a preliminary analysis of unilateral fair valuation from the hedger’s viewpoint.Unlike in the case of options of an American style, which were studied by Kim et al. [26], thesymmetry of the game contract simplifies the analysis, in the sense that it suffices to address thevaluation and hedging problem for the hedger and, subsequently, to apply analogous arguments toestablish results pertaining to the counterparty.We consider an extended market model, denoted as M p ( C g ), in which a game contract C g =GCC ( A, X h , X c , X b , T ) is traded at time 0 at some initial price p where p can be an arbitrary realnumber. We first give a preliminary analysis of unilateral fair valuation of a game contract by thehedger. We henceforth assume that the hedger (respectively, the counterparty) is endowed with aninitial pre-trading wealth of x ∈ R (respectively, x ∈ R ) units of cash.Note that, for brevity, the variables ( A, X h , X c , X b ) are frequently omitted in what follows whenthere is no danger of confusion; in particular, the payoff I ( X h , X c , X b ; σ, τ ) will be usually denotedas I ( σ, τ ). Similarly, since the process A is fixed throughout, we will frequently write V ( x + p, ϕ )instead of V ( x + p, ϕ, A ) when dealing with the hedger. By the same token, we will later write V ( x − p, ψ ) instead of V ( x − p, ψ, − A ) when examining trading strategies of the counterparty.We introduce the following conditions regarding the presence (or absence) of unilateral hedger’sgains with respect to his predetermined benchmark V b ( x ). Definition 2.1.
A quadruplet ( p, ϕ, σ, τ ) ∈ R × Ψ( x + p, A ) × T × T is said to satisfy:(AO) ⇐⇒ V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) ≥ V bσ ∧ τ ( x )and P (cid:0) V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) > V bσ ∧ τ ( x ) (cid:1) > , (SH) ⇐⇒ V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) ≥ V bσ ∧ τ ( x ) , (BE) ⇐⇒ V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) = V bσ ∧ τ ( x ) , (NA) ⇐⇒ either V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) = V bσ ∧ τ ( x )or P (cid:0) V σ ∧ τ ( x + p, ϕ ) + I ( σ, τ ) < V bσ ∧ τ ( x ) (cid:1) > . Let us explain the meaning of acronyms appearing in Definition 2.1: (AO) stands for arbitrageopportunity , (SH) for superhedging , (BE) for break-even and (NA) for no-arbitrage . For a detailedexplanation of each property, see Definitions 2.2–2.5.
Definition 2.2.
If condition (BE) is satisfied by a quadruplet ( p, ϕ, σ, τ ), then a stopping time τ ∈ T is called a hedger’s break-even time for ( p, ϕ, σ ) ∈ R × Ψ( x + p, A ) × T .Property (SH) of ( p, ϕ, σ, τ ) is referred to as the hedger’s superhedging at time τ . In view of theoptional section theorem, it is easy to see that property (SH) holds for a given triplet ( p, ϕ, σ ) ∈ R × Ψ( x + p, A ) × T and all τ ∈ T if and only if V σ ∧ t ( x + p, ϕ ) + I ( σ, t ) ≥ V bσ ∧ t ( x ) for all t ∈ [0 , T ].This observation justifies the following definition of property (SH) for a triplet ( p, ϕ, σ ). Definition 2.3.
We say that a triplet ( p, ϕ, σ ) ∈ R × Ψ( x + p, A ) × T satisfies condition (SH) if theinequality V σ ∧ t ( x + p, ϕ ) + I ( σ, t ) ≥ V bσ ∧ t ( x ) holds for all t ∈ [0 , T ]. In that case, a triplet ( p, ϕ, σ )is called a hedger’s superhedging strategy in the extended market M p ( C g ). onlinear Pricing of Game Options hedger’s strict superhedging condition (or the hedger’s arbi-trage opportunity ) for ( p, ϕ, σ ) at time τ . Note that if a triplet ( p, ϕ, σ ) is such that the inequality V σ ∧ t ( x + p, ϕ ) + I ( σ, t ) > V bσ ∧ t ( x ) holds for every t ∈ [0 , T ], then condition (AO) is satisfied by( p, ϕ, σ ) for every τ ∈ T , but the converse implication does not hold, in general. Definition 2.4.
We say that a triplet ( p, ϕ, σ ) ∈ R × Ψ( x + p, A ) × T satisfies condition (AO) ifa quadruplet ( p, ϕ, σ, τ ) complies with (AO) for all τ ∈ T . In that case, we also say that ( p, ϕ, σ )creates a hedger’s arbitrage opportunity in the extended market M p ( C g ). We say that no hedger’sarbitrage arises for ( p, ϕ, σ ) if there exists τ ∈ T such that the quadruplet ( p, ϕ, σ, τ ) satisfies (NA). Definition 2.5.
We say that p f,h ( x ) = p f,h ( x , X h , X c , X b ) is a hedger’s fair price for C g if nohedger’s arbitrage opportunity ( p, ϕ, σ ) may arise in the extended market M p ( C g ) when p = p f,h ( x ).The set of hedger’s fair prices equals K f,h ( x ) := (cid:8) p ∈ R | ∀ ( ϕ, σ ) ∈ Ψ( x + p, A ) × T ∃ τ ∈ T : ( p, ϕ, σ, τ ) ∈ (NA) (cid:9) and the upper bound for the hedger’s fair prices equals p f,h ( x ) := sup K f,h ( x ). If the equality p f,h ( x ) = max K f,h ( x ) is satisfied (that is, if p f,h ( x ) ∈ K f,h ( x )), then p f,h ( x ) is denoted as b p f,h ( x ) and called the hedger’s maximum fair price for C g . Assumption 2.1.
The forward monotonicity of wealth holds meaning that for all x, p ∈ R , ϕ ∈ Ψ( x + p, A ) and p ′ > p (respectively, p ′ < p ), there exists a trading strategy ϕ ′ ∈ Ψ( x + p ′ , A ) suchthat V t ( x + p ′ , ϕ ′ ) ≥ V t ( x + p, ϕ ) (respectively, V t ( x + p ′ , ϕ ′ ) ≤ V t ( x + p, ϕ )) for every t ∈ [0 , T ].Under the postulate of the forward monotonicity of wealth, we obtain the interval structure ofthe fair prices set. Lemma 2.1.
Let Assumption 2.1 be satisfied. If p ∈ K f,h ( x ) then for any p ′ < p , we have that p ′ ∈ K f,h ( x ) . Therefore, if K f,h ( x ) = ∅ , then either K f,h ( x ) = ( −∞ , p f,h ] = ( −∞ , b p f,h ] or K f,h ( x ) = ( −∞ , p f,h ) .Proof. We argue by contradiction. If K f,h ( x ) = ∅ , then p f,h = −∞ . Let us now consider thecase where K f,h ( x ) = ∅ . Assume that p ∈ K f,h ( x ) and a number p ′ such that p ′ < p is not anhedger’s fair price. Then there exists ϕ ′ ∈ Ψ( x + p ′ , A ) such that ( p ′ , ϕ ′ , τ ) satisfy (AO) for every τ ∈ T . Consequently, by Assumption 2.1, there exists ϕ ∈ Ψ( x + p, A ) such that ( p, ϕ, τ ) satisfy(AO) for every τ ∈ T . This clearly contradicts the assumption that p belongs to K f,h ( x ) and thuswe conclude that the asserted properties are valid. ✷ In the next step, we analyze the hedger’s superhedging costs for the game contract.
Definition 2.6.
The lower bound for the hedger’s strict superhedging costs equals p a,h ( x ) = p a,h ( x , X h , X c , X b ) := inf K a,h ( x )where K a,h ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, σ ) ∈ Ψ( x + p, A ) × T : ( p, ϕ, σ ) ∈ (AO) (cid:9) . If the equality p a,h ( x ) := min K a,h ( x ) holds, then p a,h ( x ) is denoted as ˘ p a,h ( x ) and called the hedger’s minimum strict superhedging cost for C g .For conciseness, the variables ( x , X h , X c , X b ) or x will be frequently suppressed when dealingwith the hedger’s valuation problem and thus we will write p a,h instead of p a,h ( x , X h , X c , X b ) or p a,h ( x ), etc. Notice that K a,h ( x ) is the complement of K f,h ( x ) and thus either K f,h ( x ) = ( −∞ , b p f,h ] and K a,h ( x ) = ( p a,h , ∞ ) (2)or K f,h ( x ) = ( −∞ , p f,h ) and K a,h ( x ) = [ ˘ p a,h , ∞ ) . (3) E. Kim, T. Nie and M. Rutkowski
Definition 2.7.
The lower bound for the hedger’s superhedging costs equals p s,h ( x ) = p s,h ( x , X h , X c , X b ) := inf K s,h ( x )where K s,h ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, σ ) ∈ Ψ( x + p, A ) × T : ( p, ϕ, σ ) ∈ (SH) (cid:9) . If the equality p s,h ( x ) := min K s,h ( x ) holds, then p s,h ( x ) is denoted as ˘ p s,h ( x ) and called the hedger’s minimum superhedging cost for C g .Since K a,h ( x ) ⊆ K s,h ( x ), we always have that p s,h ≤ p a,h but, in principle, it may occur that p s,h < p a,h = p f,h . To avoid this awkward situation, we introduce Assumption 2.2, which ensuresthat p s,h = p a,h and thus also p s,h = p f,h . Assumption 2.2.
The strict forward monotonicity of wealth holds meaning that for all x, p ∈ R , ϕ ∈ Ψ( x + p, A ) and p ′ > p (respectively, p ′ < p ), there exists a trading strategy ϕ ′ ∈ Ψ( x + p ′ , A )such that V t ( x + p ′ , ϕ ′ ) > V t ( x + p, ϕ ) (respectively, V t ( x + p ′ , ϕ ′ ) < V t ( x + p, ϕ )) for every t ∈ [0 , T ].Obviously, Assumption 2.2 of the strict forward monotonicity of wealth is stronger than Assump-tion 2.1 of forward monotonicity of wealth and thus the former encompasses the latter. Lemma 2.2.
If Assumption 2.2 is satisfied, then p f,h = p s,h = p a,h .Proof. Let us first assume that K s,h ( x ) = ∅ so that p s,h < ∞ . Since Assumption 2.2 holds, it isclear that for any p ∈ K s,h ( x ) and arbitrary ε >
0, there exists a strategy ϕ ′ ∈ Ψ( x + p + ε, A )such that condition (AO) is satisfied by the pair ( p + ε, ϕ ′ ). This means that p + ε belongs to K a,h ( x ) and thus p + ε ≥ p a,h . From the arbitrariness of p ∈ K s,h ( x ) and ε >
0, we obtain theinequality p s,h ≥ p a,h . Since p s,h ≤ p a,h , it is clear that p s,h = p a,h . Using also (2) and (3), especially K a,h ( x ) is the complement of K f,h ( x ), we obtain the equality p f,h = p a,h and thus we concludethat p f,h = p s,h = p a,h . Let us now assume that K s,h ( x ) = ∅ . Then p s,h = p a,h = ∞ and, since K f,h ( x ) is the complement of K a,h ( x ), we see that p f,h = ∞ as well. ✷ The next step is to introduce the concept of a replication cost. Let ( p, ϕ, σ ) be a hedger’s super-hedging strategy such that he breaks even at σ ∧ τ when the exercise time τ is coincidentally chosenby his counterparty. Then we say that ( p, ϕ, σ ) is a hedger’s replicating strategy for a game contract. Definition 2.8.
The lower bound for hedger’s replication costs for C g is given by the equality p r,h ( x ) = p r,h ( x , X h , X c , X b ) := inf K r,h ( x )where K r,h ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, σ, τ ) ∈ Ψ( x + p, A ) × T × T : ( p, ϕ, σ ) ∈ (SH) & ( p, ϕ, σ, τ ) ∈ (BE) (cid:9) . If the equality p r,h ( x ) := min K r,h ( x ) holds, then p r,h ( x ) is denoted as ˘ p r,h ( x ) and called the hedger’s minimum replication cost for C g . Definition 2.9.
The lower bound for hedger’s fair replication costs for C g is given by the equality p f,r,h ( x ) = p f,r,h ( x , X h , X c , X b ) := inf K f,r,h ( x )where K f,r,h ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, σ, τ ) ∈ Ψ( x + p, A ) × T × T : ( p, ϕ, σ ) ∈ (SH) & ( p, ϕ, σ, τ ) ∈ (BE); ∀ ( ϕ ′ , σ ′ ) ∈ Ψ( x + p, A ) × T , ∃ τ ′ ∈ T : ( p, ϕ ′ , σ ′ , τ ′ ) ∈ (NA) (cid:9) . If p f,r,h ( x ) := min K f,r,h ( x ), then p f,r,h ( x ) is denoted as ˘ p f,r,h ( x ) and called the hedger’s mini-mum fair replication cost for C g . onlinear Pricing of Game Options K s,h ( x ) ⊇ K r,h ( x ) ⊇ K f,r,h ( x ) = K f,h ( x ) ∩ K r,h ( x ) and thus p s,h ≤ p r,h ≤ p f,r,h . Lemma 2.3.
Let Assumption 2.2 be satisfied.(i) If K f,r,h ( x ) = ∅ , then −∞ < b p f,h = ˘ p r,h = ˘ p f,r,h = ˘ p s,h < + ∞ .(ii) If K r,h ( x ) = ∅ , then p f,h = p s,h ≤ p r,h < ∞ .(iii) If K r,h ( x ) = ∅ , then p f,h = p s,h ≤ p r,h = ∞ .Proof. We first prove part (i). It is clear that it in enough to show p f,r,h ≤ p s,h . For this purpose,we notice that K f,r,h ( x ) ⊆ K f,h ( x ). Since K f,r,h ( x ) = ∅ , it follows that p f,r,h < + ∞ and p f,r,h ≤ p f,h . Consequently, the equalities p f,h = p r,h = p f,r,h = p s,h are valid. Moreover, fromthe inclusion K f,r,h ( x ) ⊆ K f,h ( x ) and the equalities sup K f,h ( x ) = p f,h = p f,r,h = inf K f,r,h ( x ),we deduce that for arbitrary p , p ∈ K f,r,h ( x ) and p ∈ K f,h ( x ) we have that p = p ≥ p .This means that K f,r,h ( x ) is a singleton and its unique element is not less than any element of K f,h ( x ). Consequently, b p f,h and ˘ p f,r,h are well defined and satisfy −∞ < b p f,h = ˘ p f,r,h < + ∞ .Furthermore, K f,r,h ( x ) ⊆ K r,h ( x ) and thus the equality ˘ p f,r,h = p r,h implies that ˘ p r,h is welldefined and is equal to ˘ p f,r,h . We conclude that b p f,h = ˘ p r,h = ˘ p f,r,h = ˘ p s,h . This means, inparticular, that K f,h ( x ) = ( −∞ , b p f,h ] = ( −∞ , ˘ p r,h ] = ( −∞ , ˘ p f,r,h ]. Statements (ii) and (iii) areobvious consequences of Lemma 2.2. ✷ We will henceforth work under the following postulate of the (backward) comparison property fortrading strategies.
Assumption 2.3.
The comparison property for trading strategies holds meaning that for all τ ∈T , x, p, p ′ ∈ R , ϕ ∈ Ψ( x + p, A ) and ϕ ′ ∈ Ψ( x + p ′ , A ), if the inequality V τ ( x + p ′ , ϕ ′ ) ≥ V τ ( x + p, ϕ )holds, then p ′ ≥ p .Under Assumption 2.3, if for some G -adapted process X and a given stopping time θ ∈ T thereexists a pair ( x, ϕ ) ∈ R × Ψ( x, A ) satisfying V θ ( x, ϕ ) = X θ , then x is unique and it is denoted by E h ( X θ ). We set X lt := V bt ( x ) − X ct , X ut := V bt ( x ) − X ht and X mt := V bt ( x ) − X bt for every t ∈ [0 , T ]and we denote J ( x , X l , X u , X m , σ, τ ) := V bσ ∧ τ ( x ) − I ( σ, τ ) = X lτ { τ<σ } + X uσ { σ<τ } + X mσ { τ = σ } . (4)The hedger’s relative reward J ( x , X l , X u , X m , σ, τ ) will be henceforth denoted as J h ( σ, τ ). Notethat it is not postulated in Section 2 that X l ≤ X m ≤ X u , although we will need to make thisassumption later on when addressing the hedger’s valuation problem using a BSDE approach studiedin Section 3.The following assumption is used to ensure that the quantity E h ( J h ( σ, τ )) is well defined for allstopping times σ, τ ∈ T . Assumption 2.4.
We assume that the process J h is replicable for the hedger, in the sense thatfor a given x ∈ R and every σ, τ ∈ T there exists a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) such that V σ ∧ τ ( x + p, ϕ ) = J h ( σ, τ ).The following lemma is obvious and thus its proof is omitted. Lemma 2.4.
If Assumptions 2.3 and 2.4 are satisfied, then the hedger’s nonlinear evaluation ( σ, τ )
7→ E h ( J h ( σ, τ )) is well defined for all σ, τ ∈ T and is unique. Definition 2.10.
We say that a triplet ( v h ( x ) , σ ∗ ,h , τ ∗ ,h ) ∈ R × T × T solves the hedger’s optimalreplication problem for C g if v h ( x ) = E h ( J h ( σ ∗ ,h , τ ∗ ,h )) − x where the stopping times σ ∗ ,h and τ ∗ ,h are such that E h ( J h ( σ ∗ ,h , τ ∗ ,h )) = min σ ∈T max τ ∈T E h ( J h ( σ, τ )) . (5) E. Kim, T. Nie and M. Rutkowski
Assumption 2.5.
The hedger’s optimal replication problem for C g has a solution ( v h ( x ) , σ ∗ ,h , τ ∗ ,h ).Furthermore, for p ∗ ,h = v h ( x ) there exists a trading strategy ϕ ∗ ∈ Ψ( x + p ∗ ,h , A ) such that thetriplet ( p ∗ ,h , ϕ ∗ , σ ∗ ,h ) satisfies condition (SH) and the quadruplet ( p ∗ ,h , ϕ ∗ , σ ∗ ,h , τ ∗ ,h ) complies withcondition (BE), so that the set K r,h ( x ) is nonempty. Remark 2.1.
Notice that the hedger’s optimal replication problem does not hinge on the existenceof a saddle point for the nonlinear Dynkin game associated with E h and J h . To be more specific,we do not assume that the corresponding nonlinear Dynkin game has the value , in the sense thatinf σ ∈T sup τ ∈T E h ( J h ( σ, τ )) = sup τ ∈T inf σ ∈T E h ( J h ( σ, τ )) . Lemma 2.5.
If Assumptions 2.3–2.5 are met, then p s,h ( x ) ≥ v h ( x ) .Proof. The assertion is trivial when K s,h ( x ) = ∅ since then p s,h ( x ) = ∞ . Let thus p belong to K s,h ( x ) = ∅ . Then there exists a pair ( ϕ, σ ) ∈ Ψ( x + p, A ) × T such that ( p, ϕ, σ ) satisfies (SH)so that V σ ∧ t ( x + p, ϕ ) ≥ J h ( σ, t ) for all t ∈ [0 , T ]. Consequently, for all τ ∈ T , we have that V σ ∧ τ ( x + p, ϕ ) ≥ J h ( σ, τ ) and, due to Assumption 2.3, this implies that for all τ ∈ T , we have x + p ≥ E h ( J h ( σ, τ )). It is now easy to see x + p ≥ sup τ ∈T E h ( J h ( σ, τ )) ≥ inf σ ∈T sup τ ∈T E h ( J h ( σ, τ )) . Using equation (5) and Assumption 2.5, we conclude that p ≥ v h ( x ). Since p ∈ K s,h ( x ) wasarbitrary, this yields the desired inequality. ✷ Lemma 2.6.
If Assumptions 2.2–2.5 are met, then p f,h ( x ) ≤ v h ( x ) .Proof. We argue by contradiction. Suppose that p f,h ( x ) > v h ( x ). From Assumption 2.5, for p ∗ ,h = v h ( x ) there exists a trading strategy ϕ ∗ ∈ Ψ( x + p ∗ ,h , A ) such that V σ ∗ ,h ∧ τ ∗ ,h ( x + p ∗ ,h , ϕ ∗ ) = J ( σ ∗ ,h , τ ∗ ,h ) and the triplet ( p ∗ ,h , ϕ ∗ , σ ∗ ,h ) satisfies (SH). Consequently, in view of Assumption 2.2,for any p ′ > v h ( x ) there exists ϕ ′ ∈ Ψ( x + p ′ , A ) such that ( p ′ , ϕ ′ , σ ∗ ,h ) is a hedger’s strictsuperhedging strategy. This contradicts the assumption that p f,h ( x , X c , X h , X b ) > v h ( x ), sincewe have shown that if p belongs to the interval ( v h ( x ) , p f,h ( x )), then p ′ ∈ K f,h ( x ) ∩ K a,h ( x ) = ∅ (recall that K a,h ( x ) is the complement of K f,h ( x )). ✷ The next assumption, which is manifestly stronger than Assumption 2.3, is crucial to ensure thatthe solution v h ( x ) to the hedger’s optimal replication problem yields the hedger’s maximum fairprice b p f,h ( x ). Assumption 2.6.
For all τ ∈ T , x, p, p ′ ∈ R , ϕ ∈ Ψ( x + p, A ) and ϕ ′ ∈ Ψ( x + p ′ , A ), if the inequality V τ ( x + p ′ , ϕ ′ ) ≥ V τ ( x + p, ϕ ) holds, then p ′ ≥ p . If, in addition, V τ ( x + p ′ , ϕ ′ ) = V τ ( x + p, ϕ ), then p ′ > p .The next result shows that, under Assumptions 2.2 and 2.4–2.6, the hedger’s maximum fairprice coincides with the solution v h ( x ) to the hedger’s optimal replication problem introduced inDefinition 2.10. It is useful to observe that E h ( J h ( σ, τ )) − x = (cid:8) p ∈ R | ∃ ϕ ∈ Ψ( x + p, A ) : V σ ∧ τ ( x + p, ϕ ) = J h ( σ, τ ) (cid:9) , (6)where, in view of Assumption 2.6, the set in the right-hand side has only one element. Theorem 2.1.
Let Assumptions 2.2 and 2.4–2.6 be valid. If p ∈ K a,h ( x ) , then the inequality p > v h ( x ) holds and thus b p f,h ( x ) = v h ( x ) . Moreover, K f,r,h ( x ) = ∅ and v h ( x ) = b p f,h ( x ) = ˘ p f,r,h ( x ) = ˘ p s,h ( x ) . (7) The stopping time τ ∗ ,h is a hedger’s break-even time for the triplet (˘ p f,r,h ( x ) , ϕ f,r,h , σ ∗ ,h ) where atrading strategy ϕ = ϕ f,r,h is implicitly given by equation (6) with ( p, σ, τ ) = (˘ p f,r,h ( x ) , σ ∗ ,h , τ ∗ ,h ) .More explicitly, a trading strategy ϕ f,r,h belongs to Ψ( x + ˘ p f,r,h ( x ) , A ) and is such that V σ ∗ ,h ∧ τ ∗ ,h (cid:0) x + ˘ p f,r,h ( x ) , ϕ f,r,h (cid:1) = J h (cid:0) σ ∗ ,h , τ ∗ ,h (cid:1) . onlinear Pricing of Game Options Proof.
We denote (noticing Assumption 2.3, the right side of the following equation is singleton) p r,h ( x , σ, τ ∗ ,h ) := (cid:8) p ∈ R | ∃ ϕ ∈ Ψ( x + p, A ) : ( p, ϕ, σ, τ ∗ ,h ) ∈ (BE) (cid:9) . (8)Let p be an arbitrary number from K a,h ( x ). Then there exists a pair ( ϕ, σ ) ∈ Ψ( x + p, A ) × T suchthat for every τ ∈ T we have V σ ∧ τ ( x + p, ϕ ) ≥ J h ( σ, τ ) and P (cid:0) V σ ∧ τ ( x + p, ϕ ) > J h ( σ, τ ) (cid:1) > τ = τ ∗ ,h V σ ∧ τ ∗ ,h ( x + p, ϕ ) ≥ J h ( σ, τ ∗ ,h ) , P (cid:0) V σ ∧ τ ∗ ,h ( x + p, ϕ ) > J h ( σ, τ ∗ ,h ) (cid:1) > . From equation (8) and Assumption 2.6, we get V ( x + p, ϕ ) = x + p > x + p r,h ( x , σ, τ ∗ ,h ) , but, in view of Assumption 2.5, we also have that x + v h ( x ) = E h ( J h ( σ ∗ ,h , τ ∗ ,h )) ≤ E h ( J h ( σ, τ ∗ ,h )) = x + p r,h ( x , σ, τ ∗ ,h )and thus p > v h ( x ). From equation (2) and Lemma 2.6, we deduce that v h ( x ) = b p f,h ( x ).Moreover, in view of Assumption 2.5, we also have that v h ( x ) ∈ K f,r,h ( x ) and thus K f,r,h ( x ) = ∅ .Hence to establish (7) it suffices to make use of Lemma 2.3. The last part of the statement is animmediate consequence of Definitions 2.2 and 2.10. ✷ The following definition hinges on Theorem 2.1.
Definition 2.11.
If the set K f,r,h ( x ) is a singleton, then its unique element is denoted as p h ( x )and called the hedger’s acceptable price for C g . Remark 2.2.
From Theorem 2.1 and the proof of Lemma 2.3, we know that under Assumptions2.2 and 2.4–2.6, the hedger’s acceptable price for C g is well defined. Due to symmetric features of a game contract, to address the pricing problem for the counterparty,it suffices to make appropriate modifications in the statements of results from Section 2.4. Note, inparticular, that the hedger is receiving the initial price p , whereas the counterparty is paying the priceprovided, of course, if p is nonnegative. More formally, a number p is added to the initial endowmentof the hedger, but it is subtracted from the initial endowment of the counterparty. For this reason,the counterparty searches for maximum superhedging and replication costs and minimum fair prices,which are denoted as b p s,c ( x ) , b p r,c ( x ) and ˘ p f,c ( x ), respectively. Definition 2.12.
A quadruplet ( p, ψ, σ, τ ) ∈ R × Ψ( x − p, − A ) × T × T is said to satisfy(AO ′ ) ⇐⇒ V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) ≥ V bσ ∧ τ ( x )and P (cid:0) V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) > V bσ ∧ τ ( x ) (cid:1) > , (SH ′ ) ⇐⇒ V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) ≥ V bσ ∧ τ ( x ) , (BE ′ ) ⇐⇒ V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) = V bσ ∧ τ ( x ) , (NA ′ ) ⇐⇒ either V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) = V bσ ∧ τ ( x )or P (cid:0) V σ ∧ τ ( x − p, ψ ) − I ( σ, τ ) < V bσ ∧ τ ( x ) (cid:1) > . Of course, all other definitions and results formulated and established in Sections 2.1-2.4 for thehedger have analogous (although not identical) versions for the counterparty. Since there is no needto present all of them here, we only state the following important definitions.
Definition 2.13.
If condition (BE ′ ) is satisfied by ( p, ψ, σ, τ ), then a stopping time σ ∈ T is calleda counterparty’s break-even time for ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ).0 E. Kim, T. Nie and M. Rutkowski
Definition 2.14.
We say that a triplet ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T satisfies condition (SH ′ )if the inequality V τ ∧ t ( x − p, ψ ) − I ( τ, t ) ≥ V bτ ∧ t ( x ) holds for all t ∈ [0 , T ]. In that case, a triplet( p, ψ, τ ) is called a counterparty’s superhedging strategy in the extended market M p ( C g ).We will identify conditions under which the counterparty’s maximum fair replication cost, de-noted as b p f,r,c ( x ), is well defined and coincides with a solution to the counterparty’s optimal replica-tion problem given by Definition 2.15. It is convenient to define x ut := X ct + V bt ( x ) , x lt := X ht + V bt ( x )and x mt := X bt + V bt ( x ) for every t ∈ [0 , T ] and we denote e J ( x , x l , x u , x m , σ, τ ) := I ( σ, τ ) + V bσ ∧ τ ( x ) = J ( x , x u , x l , x m , σ, τ )= x lσ { σ<τ } + x uτ { τ<σ } + x mσ { τ = σ } . (9)For brevity, the counterparty’s relative reward e J ( x , x l , x u , x m , σ, τ ) is also denoted as J c ( σ, τ ).To formulate the counterparty’s optimal replication problem, we postulate that Assumptions 2.2and 2.3 hold but with the process A replaced by − A . For a given process Y and a fixed stoppingtime θ ∈ T , we denote by E c ( Y θ ) the unique y ∈ R such that there exists a strategy ψ ∈ Ψ( y, − A )satisfying V θ ( y, ψ ) = Y θ . It is also easy to check that x − E c ( Y τ ) = (cid:8) p ∈ R | ∃ ψ ∈ Ψ( x − p, − A ) : V τ ( x − p, ψ ) = Y τ (cid:9) . (10)As for the hedger (see Assumption 2.4), we postulate that the game contract is replicable for thecounterparty. Assumption 2.7.
We assume that the process J c is replicable for the counterparty, in the sensethat for a given x ∈ R and every σ, τ ∈ T there exists a pair ( q, ψ ) ∈ R × Ψ( x + q, − A ) such that V θ ( x + q, ψ ) = J c ( σ, τ ).For the counterparty, we have the following definition of the optimal replication, which corre-sponds to Definition 2.10 for the hedger. Definition 2.15.
We say that a triplet ( v c ( x ) , σ ∗ ,c , τ ∗ ,c ) ∈ R × T × T is a solution to the coun-terparty’s optimal replication problem for the game contract C g if v c ( x ) = x − E c ( J c ( σ ∗ ,c , τ ∗ ,c ))where the stopping times σ ∗ ,c and τ ∗ ,c are such that E c ( J c ( σ ∗ ,c , τ ∗ ,c )) = min τ ∈T max σ ∈T E c ( J c ( σ, τ )) . (11) Assumption 2.8.
The counterparty’s optimal replication problem for the game contract C g has asolution ( v c ( x ) , σ ∗ ,c , τ ∗ ,c ). Furthermore, for p ∗ ,c = v c ( x ) there exists ψ ∗ ∈ Ψ( x − p ∗ ,c , − A ) suchthat the triplet ( p ∗ ,c , ψ ∗ , τ ∗ ,c ) satisfies (SH ′ ) and the quadruplet ( p ∗ ,s , ψ ∗ , σ ∗ ,c , τ ∗ ,c ) satisfies (BE ′ ),so that K r,c ( x ) = ∅ .In view of Definition 2.14, the triplet ( p ∗ ,c , ψ ∗ , σ ∗ ,c ) satisfies the inequality V σ ∗ ,c ∧ t ( x − p ∗ ,c , ψ ∗ ) ≥ J c ( σ ∗ ,c , t ) for all t ∈ [0 , T ]. The proof of Lemma 2.7 is omitted since it is analogous to the proofs ofLemmas 2.5 and 2.6. Lemma 2.7.
If Assumptions 2.2–2.3 and 2.7–2.8 are met, then we have p s,c ( x ) ≤ v c ( x ) and p f,c ( x ) ≥ v c ( x ) . We are in a position to state the counterparty’s version of Theorem 2.1. The proof of Theorem2.2 is based on similar arguments as the proof of Theorem 2.1 and thus it is not presented here.
Theorem 2.2.
Let Assumptions 2.2–2.3 and 2.7–2.8 be satisfied. If p ∈ K a,c ( x ) , then the inequality p < v c ( x ) holds and thus ˘ p f,c ( x ) = v c ( x ) . Moreover, K f,r,c ( x ) = ∅ and v c ( x ) = ˘ p f,c ( x ) = b p f,r,c ( x ) = b p s,c ( x ) . (12) onlinear Pricing of Game Options The stopping time σ ∗ ,c is a counterparty’s break-even time for the triplet ( b p f,r,c ( x ) , ψ f,r,c , τ ∗ ,c ) wherea trading strategy ψ = ψ f,r,c is implicitly given by equation (10) with ( p, σ, τ ) = ( b p f,r,c ( x ) , σ ∗ ,c , τ ∗ ,c ) .More explicitly, a trading strategy ψ f,r,c belongs to Ψ( x − b p f,r,c ( x ) , − A ) and is such that V σ ∗ ,c ∧ τ ∗ ,c (cid:0) x − b p f,r,c ( x ) , ψ f,r,c (cid:1) = J h (cid:0) σ ∗ ,c , τ ∗ ,c (cid:1) . As for the hedger, we can now define the counterparty’s acceptable price for C g . Definition 2.16.
If the set K f,r,c ( x ) is a singleton, then its unique element is denoted as p c ( x )and called the counterparty’s acceptable price for C g . Remark 2.3.
Similarly to Remark 2.2, under Assumptions 2.2–2.3 and 2.7–2.8, one can show thatthe counterparty’s acceptable price for the game contract C g is well defined. Observe that unilateralacceptable prices for C g , as computed by the hedger and counterparty, will not coincide, in general. In this section, we re-examine and extend a BSDE approach to the valuation of game options innonlinear market, which was initiated by Dumitrescu et al. [11]. Our main goal is to show thatunilateral acceptable prices for a game contract C g can be characterized in terms of solutions toDRBSDEs driven by a multi-dimensional continuous semimartingale S . In this section, we postulatethat the wealth process V = V ( y, ϕ, A ) satisfies the SDE V t = y − Z t g ( u, V u , ξ u ) du + Z t ξ u dS u + A t , (13)where y ∈ R and a process ξ are given (recall also that A = 0). By applying Lemma 3.1 in [26]with y = x + p < x + p ′ = y , f = f = g and z = ξ , one can check that the condition of strictmonotonicity of wealth (see Assumption 2.2) is met when the dynamics of the wealth process aregiven by (13), for instance, the mapping g = g ( t, v, z ) is Lipschitz continuous with respect to v . We will briefly describe the main features of the mechanism of nonlinear trading, which generatesthe wealth dynamics given by (13). We first introduce the notation for traded assets in our marketmodel, which are: cash accounts, risky assets, and funding accounts associated with risky assets. Itshould be stressed that results in this section do not depend on the choice of a particular model forprimary assets and trading arrangements.Let S = ( S , . . . , S d ) stand for the collection of prices of a family of d risky assets where theprocesses S , . . . , S d are continuous semimartingales. Continuous processes of finite variation, de-noted as B ,l and B ,b , represent the lending borrowing unsecured cash accounts, respectively. Forevery j = 1 , , . . . , d , we denote by B j,l (respectively, B j,b ) the lending (respectively, borrowing ) funding account associated with the i th risky asset, and also assumed to be continuous processesof finite variation. The financial interpretation of these accounts varies from case to case (for moredetails, see [3, 5]). Let us denote by B the collection of all cash and funding accounts availableto a trader. For simplicity of presentation, we maintain our assumption that the hedger and thecounterparty have identical market conditions but it is clear that this assumption is not relevantfor our further developments and thus it can be easily relaxed. A trading strategy is an R d +2 -valued, G -adapted process ϕ = ( ξ , . . . , ξ d ; ψ ,l , ψ ,b , . . . , ψ d,l , ψ d,b ) where the components representall outstanding positions in the risky assets S j , j = 1 , , . . . , d , cash accounts B ,l , B ,b , and fundingaccounts B j,l , B j,b , j = 1 , , . . . , d for risky assets.2 E. Kim, T. Nie and M. Rutkowski
Definition 3.1.
We say trading strategy ( y, ϕ ) is self-financing for C g and we write ϕ ∈ Ψ( y, A ) ifthe wealth process V ( y, ϕ, A ), which is given by V t ( y, ϕ, A ) = d X j =1 ξ jt S jt + d X j =0 ψ j,lt B j,lt + d X j =0 ψ j,bt B j,bt , satisfies, for every t ∈ [0 , T ], V t ( y, ϕ, A ) = y + d X j =1 Z t ξ ju dS ju + d X j =0 Z t ψ j,lu dB j,lu + d X j =0 Z t ψ j,bu dB j,bu + A t subject to additional constraints imposed on the components of ϕ . In particular, we postulate that ψ j,lt ≥ , ψ j,bt ≤ ψ j,lt ψ j,bt = 0 for all j = 0 , , . . . , d and t ∈ [0 , T ].Due to additional trading constraints, which depend on the particular trading mechanism, thechoice of an initial value y and a process ξ is known to uniquely specify the wealth process of a self-financing strategy ϕ ∈ Ψ( y, A ). In addition, one needs also to introduce some form of admissibility of trading strategies and to postulate that the market model M = ( B , S , Ψ( A )) where the classΨ( A ) = ∪ y ∈ R Ψ( y, A ) of admissible trading strategies is arbitrage-free in a suitable sense, for instance,the market model M can be assumed to be regular , in the sense of Bielecki et al. [3]. It is importantto notice that, due to the trading constraints, differential funding costs and possibly also someadditional adjustment processes, which are not explicitly stated in Definition 3.1, the dynamics of thewealth process are nonlinear, in general. We refer the reader to Bielecki et al. [3, 5] for more detailson the self-financing property of a nonlinear trading strategy and to Nie and Rutkowski [35, 36, 37]for explicit examples of nonlinear markets. Our goal in this section is to examine the valuation and hedging of game contracts for the specialcase where the wealth dynamics are driven by (13). Therefore, we henceforth postulate that thewealth process V = V ( y, ϕ, A ) is governed by the SDE (13) where ξ = ( ξ , . . . , ξ d ) is given and themapping g satisfies some additional assumptions. To keep the presentation succinct and coveringseveral alternative nonlinear market models, we will directly postulate that the associated (possiblydoubly reflected) BSDEs enjoy desirable properties, such as: the existence, uniqueness, and the strictcomparison property for solutions to BSDEs, which are known to hold under various circumstances.As was already mentioned, a particular instance of a market model given by (13) has been studiedin detail by Dumitrescu et al. [11].We will use the standard terminology related to nonlinear evaluations generated by solutions toBSDEs (see, e.g., Chapter 3 in Peng [38] or Section 4 in Peng [39]). Consider the following BSDEon [0 , s ] Y t = ζ s + Z st g ( u, Y u , Z u ) du − Z st Z u dM u − ( H s − H t ) , (14)where ζ s ∈ L ( G s ), M is a d -dimensional martingale, the process H is real-valued and G -adapted,and the generator g : Ω × [0 , T ] × R × R d → R is P ⊗ B ( R ) ⊗ B ( R d ) / B ( R )-measurable where P is the σ -field of predictable sets on Ω × [0 , T ]. Assume that the BSDE (14) has a unique solution( Y, Z ) in a suitable space of stochastic processes (see, e.g., [6, 35]). For every 0 ≤ t ≤ s ≤ T and ζ s ∈ L ( G s ), we denote E g,Ht,s ( ζ s ) = Y t where ( Y, Z ) solves the BSDE (14) with Y s = ζ s . Then thesystem of operators E g,Ht,s : L ( G s ) → L ( G t ) is called the E g,H -evaluation . It is worth noting thata deterministic dates t ≤ s appearing in the BSDE (14) can be replaced by arbitrary G -stoppingtimes τ ≤ σ from T and thus the notion of the E g,H -evaluation can be extended to stopping times E g,Hτ,σ : L ( G σ ) → L ( G τ ). The concept of the (strict) comparison property is of great importance inthe theory of BSDEs and nonlinear evaluations. onlinear Pricing of Game Options Definition 3.2.
We say that the comparison property of E g,H holds if for every stopping time τ ∈ T and random variables ζ τ , ζ τ ∈ L ( G τ ), the following property is valid: if ζ τ ≥ ζ τ then E g,H ,τ ( ζ τ ) ≥ E g,H ,τ ( ζ τ ). We say that the strict comparison property of E g,H holds if for every τ ∈ T and ζ τ , ζ τ ∈ L ( G τ ) if ζ τ ≥ ζ τ and ζ τ = ζ τ then E g,H ,τ ( ζ τ ) > E g,H ,τ ( ζ τ ).The nonlinear evaluation given by solutions to the BSDE with M = S and H = A is interpretedas the hedger’s nonlinear evaluation. Note that here S is the process of (possibly discounted) assetsprices and thus it is assumed to be a d -dimensional continuous martingale after a change of aprobability measure. Definition 3.3.
The nonlinear evaluation E g,A associated with the BSDE Y t = ζ s + Z st g ( u, Y u , Z u ) du − Z st Z u dS u − ( A s − A t ) (15)is denoted by E g,h and called the hedger’s g -evaluation for A .In Section 3.4 and 3.5, we address the hedger’s pricing problem and we work under the followingstanding assumption. Assumption 3.1.
We postulate that:(i) the wealth process V = V ( y, ϕ, A ) of any trading strategy ϕ ∈ Ψ( y, A ) satisfies (13),(ii) for any fixed y ∈ R and any process ξ such that the stochastic integral in (13) is well defined,the SDE (13) has a unique strong solution,(iii) the strict monotonicity property holds for the wealth V ( y, ϕ, A ) (see Assumption 2.2),(iv) for every ( s, ζ s ) ∈ [0 , T ] × L ( G s ) the BSDE (15) has a unique solution ( Y, Z ) on [0 , s ],(v) the inequalities X ht < X ct and X ht ≤ X bt ≤ X ct hold for all t ∈ [0 , T ].In Sections 3.7 and 3.8, when studying the counterparty’s unilateral valuation and exercisingproblems, we will use a slight modification of Assumption 3.1 with the process A replaced by − A and with suitably modified assumptions about BSDEs and the counterparty’s g -evaluation E g,c . Remark 3.1.
In view of Lemma 3.1 in Kim et al. [26] and condition (ii) in Assumption 3.1,condition (iii) in Assumption 3.1 is not restrictive, since it is valid for every generator g . For explicitassumptions about the generator g ensuring that the BSDE (15), where S is a multi-dimensional,continuous, square-integrable martingale enjoying the predictable representation property, has aunique solution and the strict comparison property of the hedger’s g -evaluation E g,h holds, thereader is referred to Theorems 3.2 and 3.3 in Nie and Rutkowski [35]. We first introduce the general notation for a doubly reflected BSDE (DRBSDE). Let ζ l and ζ u bethe two G -adapted, c`adl`ag processes such that ζ lt < ζ ut for all t ∈ [0 , T ]. Also, let ζ mT ∈ L ( G T )be a random variable such that ζ lT ≤ ζ mT ≤ ζ uT . We work under Assumption 3.1 regarding thedynamics of the wealth process and the properties of solutions to BSDE ( g, ζ mT ). In addition, we willmake additional assumptions about the existence and uniqueness of solutions to a DRBSDE for thehedger and counterparty. They can be justified by results established in numerous papers; see, forinstance, Bayraktar and Yao [2], Cvitani´c and Karatzas [8], Cr´epey and Matoussi [7], Essaky andHassani [17], Dumitrescu et al. [10], Grigorova et al. [18], Hamad`ene and Ouknine [21], Klimsiak[27, 28] or Lepeltier and Xu [32] and the references therein. In particular, Assumption 3.2 is knownto be satisfied by several stochastic market models that were studied in papers on the so-calledvaluation adjustments.The definition of a solution to the DRBSDE with obstacles ζ l and ζ u , as stated in Assumption3.2, is a minor modification of Definition 2.4 in Dumitrescu et al. [10]. For alternative versions ofthe minimality (Skorokhod) conditions in the definition of a solution to DRBSDE with progressively4 E. Kim, T. Nie and M. Rutkowski measurable (or G -optional) obstacles, we refer to Klimsiak [27, 28] and Grigorova et al. [18]. Notethat we deliberately do not specify particular spaces of stochastic processes in which the components Y and Z are searched for, since our further results do not depend on the choice of these spaces. Onlythe properties of the processes L and U in a unique solution ( Y, Z, L, U ) to the hedger’s DRBSDE(20) (and, analogously, of the processes ℓ and u in a unique solution ( y, z, ℓ, u ) to the counterparty’sRBSDE (31)) will play an essential role and thus they are stated explicitly and analyzed in detail. Assumption 3.2.
The DRBSDE with parameters ( g, ζ l , ζ u , ζ mT ) − dY t = g ( t, Y t , Z t ) dt − Z t dS t − dA t + dL t − dU t , Y T = ζ mT ,ζ lt ≤ Y t ≤ ζ ut , R T ( Y t − ζ lt ) dL ct = R T ( ζ ut − Y t ) dU ct = 0 , ∆ L d = − ∆( Y − A ) { Y − = ζ l − } , ∆ U d = ∆( Y − A ) { Y − = ζ u − } , (16)has a unique solution ( Y, Z, L, U ) where the processes
L, U are G -predictable, c`adl`ag, nondecreasingand such that L = U = 0. Moreover, the equalities L = L c + L d and U = U c + U d give theirunique decompositions into continuous and jump components.Dumitrescu et al. [10] provide a thorough examination of a particular instance of a DRBSDEdriven by a Brownian motion and a compensated random measure. They postulate, using thepresent setup and notation, that the conditions ∆ L dτ = − ∆( Y τ − A τ ) { Y τ − = ζ lτ − } and ∆ U dτ =∆( Y τ − A τ ) { Y τ − = ζ uτ − } are satisfied for every G -predictable stopping time τ . Since L and U are G -predictable, nondecreasing processes, these conditions are equivalent to ∆ L d = − ∆( Y − A ) { Y − = ζ l − } and ∆ U d = ∆( Y − A ) { Y − = ζ u − } where the equality means that the processes in the right- and left-hand sides are indistinguishable. Remark 3.2.
Using the arguments from the proof of Theorem 3.7 of Dumitrescu et al. [10], it iseasy to show that if the process ζ l − A (respectively, − ζ u + A ) is left-upper-semicontinuous, thenthe process L (respectively, U ) is continuous. This important observation motivates Assumption 3.4for the hedger, as well as the analogous Assumption 3.6 for the counterparty.Let ζ m be a G -adapted, c`adl`ag process such that ζ lt ≤ ζ mt ≤ ζ ut for all t ∈ [0 , T ] and let the payoff J ( ζ l , ζ u , ζ m , σ, τ ) of a stochastic stopping game be given by J ( ζ l , ζ u , ζ m , σ, τ ) := ζ lτ { τ<σ } + ζ uσ { σ<τ } + ζ mτ { τ = σ } . (17)Then the upper and lower values of the nonlinear Dynkin game associated with the hedger’s g -evaluation E g,h and with the payoff J ( ζ l , ζ u , ζ m , σ, τ ) are defined by the following expressions V h ( ζ l , ζ u , ζ m ) := inf σ ∈T sup τ ∈T E g,h ,σ ∧ τ (cid:0) J ( ζ l , ζ u , ζ m , σ, τ ) (cid:1) , V h ( ζ l , ζ u , ζ m ) := sup τ ∈T inf σ ∈T E g,h ,σ ∧ τ (cid:0) J ( ζ l , ζ u , ζ m , σ, τ ) (cid:1) , so that, manifestly, the inequality V h ( ζ l , ζ u , ζ m ) ≥ V h ( ζ l , ζ u , ζ m ) is always true.In view of the existing results on linear and nonlinear Dynkin games (see, in particular, Theorems3.5, 3.7, 4.8 and 4.10 in Dumitrescu et al. [10])), it would be tempting to postulate that the above-mentioned nonlinear Dynkin game has the value , in the sense that the equality V h ( ζ l , ζ u , ζ m ) = V h ( ζ l , ζ u , ζ m ) holds and, moreover, that the value of the game coincides with the initial value Y ofa solution to the DRBSDE (16), that is, the equality Y = V h ( ζ l , ζ u , ζ m ) holds where V h ( ζ l , ζ u , ζ m ) := V h ( ζ l , ζ u , ζ m ) = V h ( ζ l , ζ u , ζ m ) . This relationship between the nonlinear Dynkin game with the payoff given by (17) and a solutionto the DRBSDE (16) is known to be satisfied in some setups and thus it could be postulated as well. onlinear Pricing of Game Options Y = inf σ ∈T sup τ ∈T E g,h ,σ ∧ τ ( J ( ζ l , ζ u , ζ m , σ, τ )) is valid for the hedger’s and counterparty’s nonlinearDynkin games.Put another way, it is not necessary to postulate that the nonlinear Dynkin game associated withthe game contract has the value, since only the upper value V h ( X l , X u , X m ) matters for the hedger.Similarly, the counterparty’s price will be expressed in terms of the upper value V c ( x l , x u , x m ) where,in general, the processes x l , x u and x m do not coincide with X l , X u and X m , respectively.We conclude that it order to cover both the case of the hedger and that of the counterparty, itis convenient to introduce the following assumption, which is supported by results in Dumitrescu etal. [10] and Grigorova et al. [18]. Assumption 3.3.
Let ζ m be a G -adapted, c`adl`ag process such that ζ lt ≤ ζ mt ≤ ζ ut for all t ∈ [0 , T ].We postulate that the following equalities hold Y = V h ( ζ l , ζ u , ζ m ) := inf σ ∈T sup τ ∈T E g,h ,σ ∧ τ (cid:0) J ( ζ l , ζ u , ζ m , σ, τ ) (cid:1) = E g,h ,σ ∗ ∧ τ ∗ (cid:0) J ( ζ l , ζ u , ζ m , σ ∗ , τ ∗ ) (cid:1) where σ ∗ := inf { t ∈ [0 , T ] | Y t = ζ ut } and τ ∗ := inf { t ∈ [0 , T ] | Y t = ζ lt } . Remark 3.3.
In the special case when S = W is a Brownian motion, one may also refer to Bayraktarand Yao [2] (see Assumptions (H.1)–(H.5) and Theorem 2.1 in [2]) for explicit assumptions ensuringthat a particular nonlinear Dynkin game has the value, a unique solution ( Y, Z, L, U ) exists in asuitable space of stochastic processes, and the equality Y = V h ( ζ l , ζ u , ζ m ) holds (in fact, they showthat Y = V h ( ζ l , ζ u , ζ m )). In Sections 3.4 and 3.5, it is postulated throughout that Assumptions 3.1–3.3 are satisfied by thehedger’s wealth and the hedger’s DRBSDE with parameters ( g, X l , X u , X mT ) where the processes X l , X u and the random variable X mT are given in the statement of Proposition 3.1 (see also Section2.4). Note that the obstacles X l and X u are c`adl`ag processes, since the processes X h , X c and V b ( x ) are assumed to be c`adl`ag. Finally, recall that the hedger’s relative reward J h ( σ, τ ) = J ( x , X l , X u , X m , σ, τ ) is given by (see equation (4)) J h ( σ, τ ) := V bσ ∧ τ ( x ) − I ( σ, τ ) = X lτ { τ<σ } + X uσ { σ<τ } + X mσ { τ = σ } . (18)We start by analyzing the lower bound for the hedger’s superhedging costs. The following resultcorresponds to Proposition 3.4 in Dumitrescu et al. [11] where, however, the nonlinear Dynkin gameassociated with a particular game option is also shown to have the value. Proposition 3.1.
If the comparison property of the hedger’s g -evaluation holds and the process − X u + A is left-upper-semicontinuous, then the lower bound for the hedger’s superhedging costssatisfies p s,h ( x ) = Y − x = inf σ ∈T sup τ ∈T E g,h ,σ ∧ τ ( J h ( σ, τ )) − x , (19) where ( Y, Z, L, U ) is the unique solution to the hedger’s DRBSDE with parameters ( g, X l , X u , X mT ) − dY t = g ( t, Y t , Z t ) dt − Z t dS t − dA t + dL t − dU t , Y T = X mT ,X lt ≤ Y t ≤ X ut , R T ( Y t − X lt ) dL ct = R T ( X ut − Y t ) dU ct = 0 , ∆ L d = − ∆( Y − A ) { Y − = X l − } , ∆ U d = ∆( Y − A ) { Y − = X u − } , (20) where X lt := V bt ( x ) − X ct < X ut := V bt ( x ) − X ht for all t ∈ [0 , T ] and X lT ≤ X mT := V bT ( x ) − X bT ≤ X uT . E. Kim, T. Nie and M. Rutkowski
Proof.
We fix x ∈ R and we first prove that p s,h ( x ) ≤ Y − x . It suffices to show that for the initialvalue p ′ = Y − x , where Y is obtained from the DRBSDE (20), we can find a hedger’s superhedgingstrategy. Our goal is thus to show that there exists a strategy ( ϕ ′ , σ ′ ) ∈ Ψ( x + p ′ , A ) × T such thatthe triplet ( p ′ , ϕ ′ , σ ′ ) fulfills condition (SH). To this end, we define the strategy ( p ′ , ϕ ′ ) = ( Y − x , Z )where ( Y, Z, L, U ) is a unique solution to the DRBSDE (20). Then, for a given process Z , the wealth V = V ( x + p ′ , ϕ ′ ) satisfies the forward SDE ( dV t = − g ( t, V t , Z t ) dt + Z t dS t + dA t ,V = Y . (21)Furthermore, if we set σ h := inf { t ∈ [0 , T ] | Y t = X ut } , then using the continuity of U (see Remark3.2), we obtain U σ h = 0, and thus, U = 0 on the stochastic interval [0 , σ h ] , the DRBSDE (20) canbe viewed as a forward SDE for Y with given processes Z and L dY t = − g ( t, Y t , Z t ) dt + Z t dS t + dA t − dL t ,Y = Y , X lt ≤ Y t ≤ X ut , R T ( Y t − X lt ) dL ct = 0 , ∆ L d = − ∆( Y − A ) { Y − = X l − } . (22)From Lemma 3.1 in [26], we infer that V t ≥ Y t for all t ∈ [0 , σ h ] and thus, since Y t ≥ X lt for all t ∈ [0 , T ], we conclude that V t ≥ X lt for all t ∈ [0 , σ h ]. Moreover, V σ h ≥ Y σ h = X uσ h ≥ X mσ h .Consequently, for all t ∈ [0 , T ], V σ h ∧ t ≥ X lt { t<σ h } + X uσ h { σ h
Assumption 3.4.
The processes X l − A and − X u + A are left-upper-semicontinuous so that theprocesses L and U in the solution ( Y, Z, L, U ) to the hedger’s DRBSDE (20) are continuous.In the next result, we identify a hedger’s replicating strategy for C g . Recall that the concept of ahedger’s replicating strategy for C g was introduced in Section 2.3. Note that when defining stoppingtimes, we henceforth adopt the convention that inf ∅ = T . Proposition 3.2.
If Assumption 3.4 is satisfied and the comparison property of the hedger’s g -evaluation holds, then the following assertions are valid:(i) the triplet ( Y − x , Z, σ h ) is a hedger’s replicating strategy for C g , where ( Y, Z, L, U ) is the unique onlinear Pricing of Game Options solution to the DRBSDE (20) and the stopping time σ h := inf { t ∈ [0 , T ] | Y t = X ut } ,(ii) the hedger’s minimum superhedging and replication costs satisfy ˘ p r,h ( x ) = ˘ p s,h ( x ) = Y − x , (23) (iii) we have ˘ p r,h ( x ) = E g,h ,σ h ∧ τ h ( J h ( σ h , τ h )) − x , where τ h := inf { t ∈ [0 , T ] | Y t = X lt } . Proof.
We already know that Y − x = p s,h ( x ) ≤ p r,h ( x ) and thus to prove parts (i) and (ii), itsuffices to show that ( p ′ , ϕ ′ , σ ′ ) = ( Y − x , Z, σ h ) is a hedger’s replicating strategy for C g . Noticingthat the right-continuity of the processes Y and X u gives Y σ h = X uσ h and the continuity of L and U ensures that L τ h = 0 and U σ h = 0, hence the DRBSDE (20) can be viewed on [0 , σ h ∧ τ h ] as theSDE (22) with a predetermined process Z and the process L satisfying L t = 0 for all t ∈ [0 , τ h ].Consequently, from the uniqueness of a solution to the SDE (22), we obtain the equality V t ( Y , Z ) = Y t on [0 , σ h ∧ τ h ]. In particular, P ( τ h = σ h ) = 0 (since X l < X u ) and V σ h ∧ τ h ( Y , Z ) = Y σ h ∧ τ h = X lτ h { τ h <σ h } + X uσ h { τ h >σ h } = J h ( σ h , τ h ) , which means that ( Y − x , Z, σ h ) is a hedger’s replicating strategy for C g . Assertion (iii) is now animmediate consequence of Assumption 3.3 applied to the hedger’s DRBSDE (20). ✷ The last step in solving the hedger’s valuation hinges on showing that the hedger’s minimumreplication cost is also his maximum fair price and to give alternative representations for the hedger’sacceptable price. The proof of the next result is fairly similar to the proof of Theorem 2.1; it isnevertheless given here for the sake of completeness.
Theorem 3.1.
If Assumption 3.4 is satisfied and the strict comparison property of the hedger’s g -evaluation holds, then the hedger’s acceptable price p h ( x ) satisfies p h ( x ) = ˘ p f,r,h ( x ) = b p f,h ( x ) = ˘ p r,h ( x ) = Y − x = V h ( X l , X u , X m ) − x . Proof.
It suffices to show that ˘ p r,h ( x ) belongs to the set K f,h ( x ) or, equivalently, that the inequality˘ p r,h ( x ) < p holds for every p ∈ K a,h ( x ). To this end, we will argue by contradiction. Let us denote p ′ = ˘ p r,h ( x ). Assume that p ′ ∈ K a,h ( x ) so that there exists a strategy ( ϕ ′ , σ ′ ) ∈ Ψ( x + p ′ , A ) × T such that the triplet ( p ′ , ϕ ′ , σ ′ ) fulfills condition (AO). Then we have that, for every τ ∈ T , P (cid:0) V σ ′ ∧ τ ( x + p ′ , ϕ ′ ) ≥ J h ( σ ′ , τ ) (cid:1) = 1 , P (cid:0) V σ ′ ∧ τ ( x + p ′ , ϕ ′ ) > J h ( σ ′ , τ ) (cid:1) > . Setting τ = τ h and using the strict comparison property of the hedger’s g -evaluation, we obtain x + p ′ = E g,h ,σ ′ ∧ τ h (cid:0) V σ ′ ∧ τ h ( x + p ′ , ϕ ′ ) (cid:1) > E g,h ,σ ′ ∧ τ h ( J h ( σ ′ , τ h )) . However, in view of Assumption 3.3 and Proposition 3.2 (iii), we have that E g,h ,σ ′ ∧ τ h ( J h ( σ ′ , τ h )) ≥ E g,h ,σ h ∧ τ h ( J h ( σ h , τ h )) = x + p ′ and thus we obtain x + p ′ = E g,h ,σ ′ ∧ τ h (cid:0) V σ ′ ∧ τ h ( x + p ′ , ϕ ′ ) (cid:1) > E g,h ,σ ′ ∧ τ h ( J h ( σ ′ , τ h )) ≥ E g,h ,σ h ∧ τ h ( J h ( σ h , τ h )) = x + p ′ , which is a clear contradiction. Consequently, b p r,h ( x ) / ∈ K a,h ( x ) and thus b p r,h ( x ) ∈ K f,h ( x ),which means that the set K f,r,h ( x ) is nonempty. In view of Lemma 2.3, we conclude that b p f,h ( x ) =˘ p r,h ( x ) = ˘ p f,r,h ( x ) = ˘ p s,h ( x ). The assertion now follows from Propositions 3.1 and 3.2. ✷ E. Kim, T. Nie and M. Rutkowski
Our next goal is to provide some useful characterizations of the class all possible hedger’s rationalcancellation times for the game contract C g . Definition 3.4.
We say that σ ′ ∈ T is a hedger’s rational cancellation time for C g if the contractis traded at the hedger’s acceptable price p h ( x ) at time 0 and there exists a trading strategy ϕ ∈ Ψ( x + p h ( x ) , A ) such that V σ ′ ∧ τ ( x + p h ( x ) , ϕ ) ≥ J h ( σ ′ , τ ) for every τ ∈ T .In view of the regularity of the wealth process and the process J h ( σ ′ , · ), the condition appearingin Definition 3.4 can be restated as follows: the inequality V σ ′ ∧ t ( x + p h ( x ) , ϕ ) ≥ J h ( σ ′ , t ) holdsfor all t ∈ [0 , T ].Assume that the game contract C g is traded at time 0 at the hedger’s acceptable price p h ( x ).Then, from Theorem 3.1 and Propositions 3.1 and 3.2, we know that there exists a triplet ( ϕ ′ , σ ′ , τ ′ ) ∈ Ψ( x + p h ( x ) , A ) × T × T such that, on the one hand, ( p h ( x ) , ϕ ′ , σ ′ ) ∈ (SH) and, on the otherhand, ( p h ( x ) , ϕ ′ , σ ′ , τ ′ ) ∈ (BE). We thus see that the classes of all hedger’s rational cancellationand break-even times (see Definition 3.5) are non-empty. We first aim to characterize the classof all hedger’s rational cancellation times. To this end, we will need the following version of thecomparison property for solutions to BSDEs with generator g . Assumption 3.5.
The following extended comparison property for solutions to BSDEs holds: if for j = 1 , ( − dY js = g j ( s, Y js , Z js ) ds − Z js dS s + dH js ,Y jτ = ζ j , where τ ∈ T , ζ ≥ ζ , g ( s, Y s , Z s ) ≥ g ( s, Y s , Z s ) for all s ∈ [0 , τ ] and the process H − H isnondecreasing, then Y s ≥ Y s for every s ∈ [0 , τ ]. Lemma 3.1.
Let Assumption 3.5 be satisfied and the strict comparison property of the hedger’s g -evaluation hold. Assume that ( Y, Z, L, U ) is the unique solution to the DRBSDE (20) . If σ ′ ∈ T is a hedger’s rational cancellation time and τ ′ ∈ T is such that L τ ′ = 0 and Y τ ′ = X lτ ′ , then we have U σ ′ ∧ τ ′ = 0 and Y σ ′ ∧ τ ′ = J h ( σ ′ , τ ′ ) .Proof. Assume that σ ′ ∈ T is a hedger’s rational cancellation time for C g . Then there exists ahedger’s trading strategy ϕ ∈ Ψ( x + p h ( x ) , A ) such that V σ ′ ∧ τ ( x + p h ( x ) , ϕ ) ≥ J h ( σ ′ , τ ) forevery counterparty’s exercise time τ ∈ T . The comparison property of the hedger’s g -evaluationyields x + p h ( x ) = E g,h ,σ ′ ∧ τ (cid:0) V σ ′ ∧ τ ( x + p h ( x ) , ϕ ) (cid:1) ≥ E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ ))and thus also x + p h ( x ) ≥ sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ )) . In particular, if τ ′ ∈ T is such that Y τ ′ = X lτ ′ , then x + p h ( x ) ≥ sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ )) ≥ E g,h ,σ ′ ∧ τ ′ ( J h ( σ ′ , τ ′ )) ≥ E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) , (24)where the last inequality holds because it is possible to show that J h ( σ ′ , τ ′ ) ≥ Y σ ′ ∧ τ ′ . Indeed,since the quadruplet ( Y, Z, L, U ) solves the DRBSDE (20), we have Y σ ′ ∧ τ ′ = Y τ ′ = X lτ ′ if τ ′ ≤ σ ′ , Y σ ′ ∧ τ ′ = Y σ ′ ≤ X uσ ′ if σ ′ < τ ′ ≤ T and Y σ ′ ∧ τ ′ = X mT if τ ′ = σ ′ = T . Since J h ( σ ′ , τ ′ ) = X lτ ′ { τ ′ <σ ′ } + X uσ { σ ′ <τ ′ } + X mσ { τ ′ = σ ′ } ,X l < X u and X l ≤ X m ≤ X u , it is now easy to see that J h ( σ ′ , τ ′ ) ≥ Y σ ′ ∧ τ ′ . Recalling that p h ( x ) = Y − x , we obtain from (24) Y ≥ E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) . (25) onlinear Pricing of Game Options τ ′ , we also have that L τ ′ = 0 holds and thus the DRBSDE (20) for 0 ≤ r ≤ t ≤ τ ′ can be written as ( − dY r = g ( r, Y r , Z r ) dr − Z r dS r − dA t − dU r , Y t = Y t ,X lr ≤ Y r ≤ X ur , R t ( X ur − Y r ) dU cr = 0 , ∆ U d = ∆( Y − A ) { Y − = X u − } . Using Assumption 3.5, we obtain the inequality E g,hs,t ( Y t ) ≥ Y s for 0 ≤ s ≤ t ≤ τ ′ , which means thatthe process Y is an E g,h -submartingale on [0 , τ ′ ]. From (25) and the postulated strict comparisonproperty of the hedger’s g -evaluation, we will now deduce that for every 0 ≤ s ≤ σ ′ ∧ τ ′ E g,hs,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = Y s . (26)Suppose, on the contrary, that equality (26) is not true. Then the strict comparison property of thehedger’s g -evaluation would yield E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = E g,h ,s ( E g,hs,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ )) > E g,h ,s ( Y s ) ≥ Y , which would contradict (25). From (26), we have E g,ht,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = Y t and thus, for all 0 ≤ s ≤ t ≤ σ ′ ∧ τ ′ , E g,hs,t ( Y t ) = E g,hs,t ( E g,ht,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ )) = E g,hs,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = Y s , where the last equality also comes from (26). We have thus shown that Y is an E g,h -martingale on[0 , σ ′ ∧ τ ′ ] so that U σ ′ ∧ τ ′ = 0 and U σ ′ ∧ t = 0 for all t ∈ [0 , τ ′ ]. In particular, we have E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = Y . (27)By combining (24) with (27), we get Y = x + p h ( x ) = E g,h ,σ ′ ∧ τ ′ (cid:0) V σ ′ ∧ τ ′ ( x + p, ϕ ) (cid:1) = sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ ))= E g,h ,σ ′ ∧ τ ′ ( J h ( σ ′ , τ ′ )) = E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) . Since we have shown that J h ( σ ′ , τ ′ ) ≥ Y σ ′ ∧ τ ′ , using the strict comparison property of the hedger’s g -evaluation, we obtain the desired equality J h ( σ ′ , τ ′ ) = Y σ ′ ∧ τ ′ . ✷ We will study two possible exercise times for the counterparty τ h := inf { t ∈ [0 , T ] | Y t = X lt } , ¯ τ h := inf { t ∈ [0 , T ] | L t > } , which are of a special interest for the hedger’s valuation and rational cancellation problems, asemphasized by the superscript h . Proposition 3.3.
Let Assumptions 3.4–3.5 be satisfied and the strict comparison property of thehedger’s g -evaluation hold. Then the following assertions are true:(i) if a stopping time σ ′ ∈ T is such that (a) U σ ′ = 0 and (b) Y σ ′ = X uσ ′ , then σ ′ is a hedger’srational cancellation time,(ii) if σ ′ is a hedger’s rational cancellation time, then U σ ′ ∧ τ h = U σ ′ ∧ ¯ τ h = 0 , Y σ ′ ∧ τ h = J h ( σ ′ , τ h ) and Y σ ′ ∧ ¯ τ h = J h ( σ ′ , ¯ τ h ) .Proof. (i) If condition (a) is satisfied then, arguing as in the proof of Proposition 3.1, we obtainthe existence of a hedger’s trading strategy ϕ ∈ Ψ( x + p h ( x ) , A ) such that the wealth process V = V ( x + p h ( x ) , ϕ, A ) satisfies V t ≥ Y t ≥ X lt for all t ∈ [0 , σ ′ ]. Moreover, since (b) holds, we get V σ ′ ≥ Y σ ′ = X uσ ′ ≥ X mσ ′ and, for all t ∈ [0 , T ], V σ ′ ∧ t ≥ X lt { t<σ ′ } + X uσ ′ { σ ′
0, there exists a δ ∈ [0 , ε ] such that L ¯ τ h + δ > R T ( Y t − X lt ) dL t = 0, there exists a δ ∈ [0 , δ ] such that Y ¯ τ h + δ = X l ¯ τ h + δ . From theright-continuity of Y and X l and the fact that ε and 0 ≤ δ ≤ δ ≤ ε are arbitrary, we deduce that Y ¯ τ h = X l ¯ τ h . Moreover, since L t = 0 for all t ∈ [0 , ¯ τ h ), in view of the continuity of L , we also havethat L ¯ τ h = 0. ✷ Corollary 3.1.
Let Assumptions 3.4–3.5 be satisfied and the strict comparison property of thehedger’s g -evaluation hold. Then the stopping times σ h and ¯ σ h , which are given by σ h := inf { t ∈ [0 , T ] | Y t = X ut } , ¯ σ h := inf { t ∈ [0 , T ] | U t > } , are hedger’s rational cancellation times.Proof. We first consider the stopping time σ h . From the right-continuity of Y and X u , we obtainthe equality Y σ h = X uσ h . Moreover, from the definition of σ h , we have Y t < X ut for t ∈ [0 , σ h ) andthus U t = 0 for all t ∈ [0 , σ h ] observing that U is continuous. Using part (i) in Proposition 3.3, weconclude that σ h is a hedger’s rational cancellation time. Let us now focus on the stopping time¯ σ h . Then, similarly to the proof of Proposition 3.3, using the right-continuity of Y and X u , we canshow that Y ¯ σ h = X u ¯ σ h and U t = 0 for all t ∈ [0 , ¯ σ h ]. Thus, using again part (i) in Proposition 3.3,we conclude that ¯ σ h is also one of hedger’s rational cancellation times. ✷ In the case of an American option and a nonlinear market, it is easy to characterize the earliestand latest holder’s rational exercise times for the holder of the option (see, for instance, Section3.6 in Kim et al. [26]). In contrast, since a game contract can be stopped by either of the twoparties at any instance, it is much harder to provide a full characterization of the earliest and latestrational cancellation times for the hedger and, by the same token, the earliest and latest rationalexercise times for the counterparty. Nevertheless, it is possible to show that the hedger’s rationalcancellation times σ h and ¯ σ h enjoy the property of being, at least in some special circumstances,the earliest and the latest among all hedger’s rational cancellation times. Corollary 3.2.
Let Assumptions 3.4–3.5 be satisfied and the strict comparison property of thehedger’s g -evaluation hold. Then the following assertions are true:(i) if σ ′ is a hedger’s rational cancellation time such that σ ′ ≤ σ h on the event E h := { σ h ≤ ¯ τ h } ,then σ ′ = σ h on E h ,(ii) if σ ′ is a hedger’s rational cancellation time such that σ ′ ≥ ¯ σ h on the event ¯ E h := { ¯ σ h < ¯ τ h } ,then σ ′ = ¯ σ h on ¯ E h .Proof. (i) We argue by contradiction. Let σ ′ be a stopping time such that σ ′ ≤ σ h on the event E h and P ( { σ ′ < σ h } ∩ E h ) >
0. In view of the assumption that ¯ τ h ≥ σ h on E h , we have σ ′ ∧ ¯ τ h = σ ′ on E h . Due to the definition of the stopping time σ h , we have that Y σ ′ < X uσ ′ on { σ ′ < σ h } . Thus Y σ ′ ∧ ¯ τ h = Y σ ′ < X uσ ′ = X uσ ′ ∧ ¯ τ h = J h ( σ ′ , ¯ τ h ) on { σ ′ < σ h } ∩ E h where the last equality followsfrom (18). In view of part (ii) in Proposition 3.3, we conclude that σ ′ cannot be a hedger’s rationalcancellation time.(ii) Once again we argue by contradiction. Let σ ′ be a stopping time such that σ ′ ≥ ¯ σ h on the event¯ E h and P ( { σ ′ > ¯ σ h } ∩ ¯ E h ) >
0. Since manifestly ¯ τ h > ¯ σ h on ¯ E h , we have that P ( σ ′ ∧ ¯ τ h > ¯ σ h ) > σ h , this means that P ( U σ ′ ∧ ¯ τ h > > σ ′ cannot be a hedger’s rational cancellation time. ✷ From Corollary 3.2 it follows, in particular, that if the inequality ¯ τ h ≥ σ h holds, then σ h isthe earliest among all hedger’s rational cancellation times, that is, if σ ′ is any hedger’s rationalcancellation time such that σ ′ ≤ σ h , then σ ′ = σ h . Similarly, if ¯ τ h > ¯ σ h , then ¯ σ h is the latest amongall hedger’s rational cancellation times, that is, if σ ′ is any hedger’s rational cancellation time suchthat σ ′ ≥ ¯ σ h , then the equality σ ′ = ¯ σ h holds. onlinear Pricing of Game Options In the next step we will examine the set of hedger’s break-even times , which are introduced inDefinition 3.5. Obviously, a hedger’s break-even time is in fact a particular exercise time selected bythe counterparty, but its name is aimed to emphasize that the choice of that time by the counterpartyaffects in a very special way the hedger’s financial outcome. In other words, it is a particularexercise time related to the hedger’s unilateral valuation problem, but it is usually meaningless forthe counterparty’s unilateral valuation and exercising problems in a general nonlinear framework.
Definition 3.5.
A stopping time τ ∈ T is called a hedger’s break-even time for the triplet ( p, ϕ, σ ) ∈ R × Ψ( x + p, A ) × T if condition (BE) is satisfied by the quadruplet ( p, ϕ, σ, τ ). Remark 3.4.
Using the symmetry of the problem, we will also analyze in Section 3.8 the counter-party’s rational exercise time, which is chosen by the counterparty, and the counterparty’s break-eventime, which is selected by the hedger, but is relevant for the financial outcome of the counterparty.In the next result, we provide several alternative characterizations of all hedger’s break-eventimes associated with the hedger’s replicating strategy ( p h ( x ) , ϕ ′ , σ ′ ) = ( Y − x , Z, σ h ). Note thatthe stopping time σ ′ = σ h is a hedger’s rational cancellation time and ϕ ′ = Z where the quadruplet( Y, Z, L, U ) is a unique solution to the DRBSDE (20). Recall that the stopping time τ h is given by τ h := inf { t ∈ [0 , T ] | Y t = X lt } . Proposition 3.4.
Let Assumption 3.5 be satisfied and ( Y, Z, L, U ) be the unique solution to thehedger’s DRBSDE (20) where the process − X u + A is assumed to be left-upper-semicontinuous. For ( p h ( x ) , ϕ ′ , σ ′ ) = ( Y − x , Z, σ h ) , the following assertions are equivalent:(i) a stopping time τ ′ ∈ T is a hedger’s break-even time for the triplet ( p h ( x ) , ϕ ′ , σ ′ ) ∈ R × Ψ( x + p h ( x ) , A ) × T ,(ii) the quadruplet ( p h ( x ) , ϕ ′ , σ ′ , τ ′ ) ∈ R × Ψ( x + p h ( x ) , A ) × T × T satisfies condition (NA),(iii) the equality V σ ′ ∧ τ ′ ( x + p h ( x ) , ϕ ′ ) = J h ( σ ′ , τ ′ ) holds,(iv) the equalities Y σ ′ ∧ τ ′ = J h ( σ ′ ∧ τ ′ ) and L τ ′ ∧ σ ′ = U σ ′ = 0 hold and thus the process Y is an E g,h -martingale on [0 , σ ′ ∧ τ ′ ] ,(v) a stopping time τ ′ ∈ T is a solution to the following nonlinear optimal stopping problem: find τ ′ ∈ T such that E g,h ,σ ′ ∧ τ ′ (cid:0) J h ( σ ′ , τ ′ ) (cid:1) = sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ )) . Moreover, if the process X l − A is left-upper-semicontinuous, then τ h is a hedger’s break-even timefor the triplet ( p h ( x ) , ϕ ′ , σ ′ ) . If, in addition, the inequality σ ′ ≥ τ h holds, then τ h is the earliesthedger’s break-even time for the triplet ( p h ( x ) , ϕ ′ , σ ′ ) .Proof. From results established in Section 2.3, we note that since ( p h ( x ) , ϕ ′ , σ ′ ) is a hedger’sreplicating strategy (see Proposition 3.2), it is also his superhedging strategy, and thus it is clearthat assertions (i), (ii) and (iii) are indeed equivalent.(iii) ⇒ (iv). From the proof of Proposition 3.1, we know that V t ( x + p h ( x ) , ϕ ′ ) ≥ Y t ≥ X lt forall t ∈ [0 , σ ′ ] and thus, in particular, V σ ′ ∧ τ ′ ( x + p h ( x ) , ϕ ′ ) ≥ Y σ ′ ∧ τ ′ . Moreover, since ( Y, Z, L, U )solves the DRBSDE (20), we have X l ≤ Y ≤ X u and Y σ ′ = X uσ ′ ≥ X mσ ′ . Therefore, V σ ′ ∧ τ ′ ( x + p h ( x ) , ϕ ′ ) ≥ Y σ ′ ∧ τ ′ ≥ X lτ ′ { τ ′ <σ ′ } + X uσ ′ { σ ′ <τ ′ } + X mσ ′ { σ ′ = τ ′ } = J h ( σ ′ , τ ′ ) . (28)From (iii), V σ ′ ∧ τ ′ ( x + p h ( x ) , ϕ ′ ) = Y σ ′ ∧ τ ′ = J h ( σ ′ , τ ′ ), which is V σ ′ ∧ τ ′ ( Y , ϕ ′ ) = Y σ ′ ∧ τ ′ = J h ( σ ′ , τ ′ )by recalling that p h ( x ) = Y − x . Moreover, since the process V = V ( Y , ϕ ′ ) satisfies the SDE(21), it is easy to see that the process V ( Y , ϕ ′ ) is an E g,h -martingale. We thus have E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = E g,h ,σ ′ ∧ τ ′ (cid:0) V σ ′ ∧ τ ′ ( Y , ϕ ′ ) (cid:1) = Y . (29)2 E. Kim, T. Nie and M. Rutkowski
The process Y solves the DRBSDE (20), which can be written on 0 ≤ r ≤ t ≤ σ ′ in the followingway (recall that U σ ′ = 0) dY r = − g ( r, Y r , Z r ) dr + Z r dS r + dA t − dL r ,Y t = Y t , X lr ≤ Y r ≤ X ur , R t ( Y r − X lr ) dL cr = 0 , ∆ L d = − ∆( Y − A ) { Y − = X l − } . Using Assumption 3.5, we obtain E g,hs ∧ σ ′ ,t ∧ σ ′ ( Y t ∧ σ ′ ) ≤ Y s ∧ σ ′ for all 0 ≤ s ≤ t ≤ T , so that Y is an E g,h -supermartingale on the stochastic interval [0 , σ ′ ]. We now claim that, for all 0 ≤ s ≤ τ ′ , E g,hs ∧ σ ′ ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ ) = Y s ∧ σ ′ . (30)Suppose, on the contrary, that the equality (30) fails to hold. In that case, the strict comparisonproperty of the hedger’s g -evaluation would yield E g,h ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ ) = E g,h ,s ∧ σ ′ ( E g,hs ∧ σ ′ ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ )) < E g,h ,s ∧ σ ′ ( Y s ∧ σ ′ ) ≤ Y , which manifestly contradicts (29). Next, we claim that E g,hs ∧ σ ′ ,t ∧ σ ′ ( Y t ∧ σ ′ ) = Y s ∧ σ ′ for 0 ≤ s ≤ t ≤ T ,which means that Y is an E g,h -martingale on [0 , τ ′ ∧ σ ′ ] and thus L τ ′ ∧ σ ′ = 0. To establish thisproperty, we note that (30) gives E g,ht ∧ σ ′ ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ ) = Y t ∧ σ ′ and thus, for all 0 ≤ s ≤ t ≤ T , E g,hs ∧ σ ′ ,t ∧ σ ′ ( Y t ∧ σ ′ ) = E g,hs ∧ σ ′ ,t ∧ σ ′ ( E g,ht ∧ σ ′ ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ )) = E g,hs ∧ σ ′ ,τ ′ ∧ σ ′ ( Y τ ′ ∧ σ ′ ) = Y s ∧ σ ′ , where the last equality comes from (30).(iv) ⇒ (iii). We observe that Y σ ′ ∧ τ ′ = J h ( σ ′ ∧ τ ′ ), L τ ′ ∧ σ ′ = U σ ′ = 0 and ( Y, Z, L, U ) solves theDRBSDE (20), which thus reduces to the following BSDE on the stochastic interval [0 , σ ′ ∧ τ ′ ] ( − dY r = g ( r, Y r , Z r ) dr − Z r dS r − dA t ,Y σ ′ ∧ τ ′ = J h ( σ ′ ∧ τ ′ ) . Given the process Z , the above BSDE can be formally rewritten as the (forward) SDE, for all r ∈ [0 , τ ′ ], ( dY r = − g ( r, Y r , Z r ) dr + Z r dS r + dA t ,Y = Y . Recall that V ( x + p h ( x ) , ϕ ′ ) = V ( Y , Z ) satisfies the following SDE, for all r ∈ [0 , T ], ( dV r = − g ( r, V r , Z r ) dr + Z r dS r + dA t ,V = Y . From the uniqueness of a solution to the SDE, we deduce that V ( Y , Z ) and Y coincide on [0 , σ ′ ∧ τ ′ ].In particular, V σ ′ ∧ τ ′ ( x + p h ( x ) , ϕ ′ ) = Y σ ′ ∧ τ ′ = J h ( σ ′ ∧ τ ′ ).(iv) ⇔ (v). First, from (iv) we know that E g,h ,σ ′ ∧ τ ′ ( J h ( σ ′ ∧ τ ′ )) = E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ ) = Y . Observe thatAssumption 3.3 yields Y = sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ ∧ τ )) and thus E g,h ,σ ′ ∧ τ ′ ( J h ( σ ′ ∧ τ ′ )) = sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ ∧ τ )) . Conversely, if (v) holds, then Assumption 3.3 gives Y = E g,h ,σ ′ ∧ τ ′ ( J h ( σ ′ ∧ τ ′ )) ≤ E g,h ,σ ′ ∧ τ ′ ( Y σ ′ ∧ τ ′ )where the last inequality is a consequence of (28). Thus, similarly as for the implication (iii) ⇒ (iv),one can show (iv) holds. This completes the proof of the equivalence (iv) ⇔ (v). onlinear Pricing of Game Options τ h is a hedger’s break-even time. From the right-continuityof Y and X , we deduce that Y τ h = X lτ h . Moreover, from the definition of τ h , we have Y t > X lt for t ∈ [0 , τ h ) and thus L t = 0 for all t ∈ [0 , τ h ]. Therefore, we have Y τ h = X lτ h and L τ h = 0, that is,(iv) holds with τ ′ = τ h . Hence, from the above equivalences (in particular, (iv) ⇔ (i)), we infer that τ h is a hedger’s break-even time for the triplet ( p h ( x ) , ϕ ′ , σ ′ ) ∈ R × Ψ( x + p h ( x ) , A ) × T . Onecan also provide another simple argument: from Assumption 3.3, we have that τ h is a solution tothe following nonlinear optimal stopping problem: find τ ′ ∈ T such that E g,h ,σ ′ ∧ τ ′ (cid:0) J h ( σ ′ , τ h ) (cid:1) = sup τ ∈T E g,h ,σ ′ ∧ τ ( J h ( σ ′ , τ ))and, from the equivalence (v) ⇔ (i), we infer that τ h is a hedger’s break-even time for the triplet( p h ( x ) , ϕ ′ , σ ′ ).To complete the proof of the proposition, it remains to show that if, in addition, the inequality σ ′ ≥ τ h holds, then τ h is the earliest hedger’s break-even time for the triplet ( p h ( x ) , ϕ ′ , σ ′ ). Weargue by contradiction. Let τ ′ be any hedger’s break-even time for the triplet ( p h ( x ) , ϕ ′ , σ ′ ) suchthat τ ′ ≤ τ h and P ( { τ ′ < τ h } ) >
0. Then, from (iv) and σ ′ ≥ τ h ≥ τ ′ , it holds that Y τ ′ = X lτ ′ on { τ ′ < τ h } , which clearly contradicts the definition of τ h . ✷ In Sections 3.7 and 3.8, it is postulated without further mentioning that Assumptions 3.1–3.3 aresatisfied by the counterparty’s wealth processes associated with the process − A and the counter-party’s DRBSDE (31). Observe that the the counterparty’s DRBSDE is specified in the manneranalogous to the hedger’s equation (20) and the counterparty’s g -evaluation E g,c is defined in thesame way as the hedger’s g -evaluation E g,h but with the process A replaced by the process − A inthe BSDE (15).In particular, the lower and upper obstacles in the counterparty’s DRBSDE are c`adl`ag processes x l and x u given by x lt := X ht + V bt ( x ) < x ut := X ct + V bt ( x ) for all t ∈ [0 , T ] and the terminalvalue x mT is such that x lT ≤ x mT := X bT + V bT ( x ) ≤ x uT . An application of Assumption 3.2 to thecounterparty’s DRBSDE ensures the existence of a unique solution ( y, z, ℓ, u ) to the DRBSDE (31)with parameters ( g, x l , x u , x mT ) − dy t = g ( t, y t , z t ) dt − z t dS t + dA t + dℓ t − du t , y T = x mT ,x lt ≤ y t ≤ x ut , R T ( y t − x lt ) dℓ ct = R T ( x ut − y t ) du ct = 0 , ∆ ℓ d = − ∆( y + A ) { y − = ζ l − } , ∆ u d = ∆( y + A ) { y − = ζ u − } , (31)where, in particular, ℓ and u are G -predictable, c`adl`ag, nondecreasing processes such that ℓ = u =0. Furthermore, ℓ = ℓ c + ℓ d and u = u c + u d give their unique decompositions into continuous andjump components.Recall also that the counterparty’s relative reward J c ( σ, τ ) = e J ( x , x l , x u , x m , σ, τ ) is given byequation (9), that is, J c ( σ, τ ) = I ( σ, τ ) + V bσ ∧ τ ( x ) = x lσ { σ<τ } + x uτ { τ<σ } + x mσ { τ = σ } . In Sections 3.4 and 3.5, we have established several results for the hedger. Since the two uni-lateral valuation and hedging problems are in some sense symmetric, it is clear that the analogousresults for the counterparty should be valid as well and thus it suffices to give their statementswithout proofs. We first state the counterparty’s version of Proposition 3.1, which furnishes the linkbetween the counterparty’s superhedging costs and the unique solution ( y, z, ℓ, u ) to the counter-party’s DRBSDE (31).4
E. Kim, T. Nie and M. Rutkowski
Proposition 3.5.
Let the process − x u − A be left-upper-semicontinuous. If the comparison propertyof the counterparty’s g -evaluation holds, then the upper bound for the counterparty’s superhedgingcosts satisfies p s,c ( x ) = x − y = x − inf τ ∈T sup σ ∈T E g,c ,σ ∧ τ ( J c ( σ, τ )) . In the next result, which corresponds to Proposition 3.2 and gives a counterparty’s replicatingstrategy for C g , we will need the following analogue of Assumption 3.4. Assumption 3.6.
The processes x l + A and − x u − A are left-upper-semicontinuous so that theprocesses ℓ and u in the solution ( y, z, ℓ, u ) to the counterparty’s DRBSDE (31) are continuous.We define the hedger’s cancellation time σ c and the counterparty’s exercise time τ c by setting τ c := inf { t ∈ [0 , T ] | y t = x ut } , σ c := inf { t ∈ [0 , T ] | y t = x lt } . Proposition 3.6.
Let Assumption 3.6 be satisfied. If the comparison property of the counterparty’s g -evaluation holds, then the following assertions are valid:(i) ( x − y , z, τ c ) is a counterparty’s replicating strategy for C g ,(ii) the counterparty’s maximum superhedging and replication costs satisfy b p r,c ( x ) = b p s,c ( x ) = x − y = x − E g,c ,σ c ∧ τ c ( J c ( σ c , τ c )) . Let V c ( x l , x u , x m ) stand for the upper value for the counterparty’s nonlinear Dynkin game withthe payoff J c ( σ, τ ), that is, V c ( x l , x u , x m ) := inf τ ∈T sup σ ∈T E g,c ,σ ∧ τ ( J c ( σ, τ )) . The following result, which is a straightforward consequence of Theorem 3.1, shows that the coun-terparty’s maximum replication cost coincides with his minimum fair price and furnishes alternativerepresentations for the counterparty’s acceptable price p c ( x ). Theorem 3.2.
Let Assumption 3.6 be satisfied. If the strict comparison property of the counter-party’s g -evaluation holds, then the unique counterparty’s acceptable price p c ( x ) satisfies p c ( x ) = b p f,r,c ( x ) = b p r,c ( x ) = ˘ p f,c ( x ) = x − y = x − V c ( x l , x u , x m ) . Let us state the counterparty’s versions of Definition 3.4 and Proposition 3.3.
Definition 3.6.
A stopping time τ ′ ∈ T is a counterparty’s rational exercise time for C g if thecontract is traded at the counterparty’s acceptable price p c ( x ) at time 0 and there exists a tradingstrategy ψ ∈ Ψ( x − p c ( x ) , − A ) such that V τ ′ ( x − p c ( x ) , ψ ) ≥ J c ( σ, τ ′ ) for every σ ∈ T .The following hedger’s cancellation times σ c := inf { t ∈ [0 , T ] | y t = x lt } , ¯ σ c := inf { t ∈ [0 , T ] | ℓ t > } , play an important role in the counterparty’s valuation problem and thus they are designated by thesuperscript c . Proposition 3.7.
Let Assumption 3.5–3.6 be satisfied. If the strict comparison property of thecounterparty’s g -evaluation holds, then the following assertions are true:(i) if a stopping time τ ′ ∈ T is such that (a) u τ ′ = 0 and (b) y τ ′ = x uτ ′ , then τ ′ is a counterparty’srational exercise time,(ii) if τ ′ ∈ T is a counterparty’s rational exercise time, then u σ c ∧ τ ′ = u ¯ σ c ∧ τ ′ = 0 , Y σ c ∧ τ ′ = J c ( σ c , τ ′ ) and Y ¯ σ c ∧ τ ′ = J c (¯ σ c , τ ′ ) . onlinear Pricing of Game Options Corollary 3.3.
Let Assumptions 3.5–3.6 be satisfied. If the strict comparison property of the coun-terparty’s g -evaluation holds, then the stopping times τ c and ¯ τ c given by τ c := inf { t ∈ [0 , T ] | y t = x ut } , ¯ τ c := inf { t ∈ [0 , T ] | u t > } , are counterparty’s rational exercise times. Corollary 3.4.
Let Assumptions 3.5–3.6 be satisfied and the strict comparison property of thecounterparty’s g -evaluation hold. Then the following assertions are true:(i) if τ ′ is a counterparty’s rational exercise time such that τ ′ ≤ τ c on the event E c := { τ c ≤ ¯ σ c } ,then τ ′ = τ c on E c ,(ii) if τ ′ is a counterparty’s rational exercise time such that τ ′ ≥ ¯ τ c on the event ¯ E c := { ¯ τ c < ¯ σ c } ,then τ ′ = ¯ τ c on ¯ E c . In particular, if ¯ σ c ≥ τ c , then τ c is the earliest among all counterparty’s rational exercise times,that is, if τ ′ is any counterparty’s rational exercise time such that τ ′ ≤ τ c , then τ ′ = τ c . Moreover,if ¯ σ c > ¯ τ c , then ¯ τ c is the latest among all counterparty’s rational exercise times, that is, if τ ′ is anycounterparty’s rational exercise time such that τ ′ ≥ ¯ τ c , then τ ′ = ¯ τ c .As expected, the definition of the counterparty’s break-even time mimics the one for the hedger.Observe that a counterparty’s break-even time is associated with a solution to the counterparty’sunilateral valuation problem (but, of course, not with the hedger’s valuation problem, in general,unless we deal with a linear market model). Definition 3.7.
If condition (BE ′ ) is satisfied by a quadruplet ( p, ψ, σ, τ ), then σ ∈ T is called a counterparty’s break-even time for the triplet ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T .We conclude this work by considering the counterparty’s replicating strategy ( p c ( x ) , ψ ′ , τ ′ ) =( x − y , z, τ c ), where the quadruplet ( y, z, ℓ, u ) is a solution to the DRBSDE (31) and the stoppingtime τ ′ = τ c is a counterparty’s rational exercise time. The proof of Proposition 3.8, which providesseveral alternative characterizations of counterparty’s break-even times associated with the triplet( x − y , z, τ c ), is exactly the same as the proof of Proposition 3.4 and thus it is omitted. Proposition 3.8.
Let ( y, z, ℓ, u ) be the unique solution to the counterparty’s DRBSDE (31) wherethe process − x u − A is assumed to be left-upper-semicontinuous. For ( p c ( x ) , ψ ′ , τ ′ ) = ( x − y , z, τ c ) ,the following assertions are equivalent:(i) a stopping time σ ′ ∈ T is a counterparty’s break-even time for the triplet ( p c ( x ) , ψ ′ , τ ′ ) ∈ R × Ψ( x − p c ( x ) , − A ) × T , that is, V σ ′ ∧ τ ′ ( x − p c ( x ) , ϕ ′ ) = J c ( σ ′ , τ ′ ) ,(ii) the quadruplet ( p c ( x ) , ψ ′ , σ ′ , τ ′ ) ∈ R × Ψ( x − p c ( x ) , − A ) × T × T fulfills condition (NA ′ ),(iii) the equality y σ ′ ∧ τ ′ = J c ( σ ′ ∧ τ ′ ) holds,(iv) the equalities y σ ′ ∧ τ ′ = J c ( σ ′ ∧ τ ′ ) and ℓ σ ′ ∧ τ ′ = u τ ′ = 0 hold and thus the process y is an E g,c -martingale on [0 , σ ′ ∧ τ ′ ] ,(v) a stopping time σ ′ ∈ T is a solution to the following nonlinear optimal stopping problem: find σ ′ ∈ T such that E g,c ,σ ′ ∧ τ ′ (cid:0) J c ( σ ′ , τ ′ ) (cid:1) = sup σ ∈T E g,c ,σ ∧ τ ′ (cid:0) J c ( σ, τ ′ ) (cid:1) . Furthermore, if the process x l + A is left-upper-semicontinuous, then σ c is a counterparty’s break-even time for the triplet ( p c ( x ) , ψ ′ , τ ′ ) . If, in addition, the inequality τ ′ ≥ σ c holds, then σ c is theearliest counterparty’s break-even time for the triplet ( p c ( x ) , ψ ′ , τ ′ ) . Acknowledgments
The research of T. Nie and M. Rutkowski was supported by the DVC Research Bridging SupportGrant
Pricing of American and game options in markets with frictions . The work of T. Nie wassupported by the National Natural Science Foundation of China (No. 11601285) and the NaturalScience Foundation of Shandong Province (No. ZR2016AQ13).6
E. Kim, T. Nie and M. Rutkowski
References [1] Ayache, E., Forsyth, P. and Vetzal, K.: Valuation of convertible bonds with credit risk.
Journalof Derivatives
11 (2003), 9–29.[2] Bayraktar, E. and Yao, S.: Doubly reflected BSDEs with integrable parameters and relatedDynkin games.
Stochastic Processes and their Applications
125 (2015), 4489–4542.[3] Bielecki, T. R., Cialenco, I., and Rutkowski, M.: Arbitrage-free pricing of derivatives innonlinear market models.
Probability, Uncertainty and Quantitative Risk
Quantitative Finance
SIAM Journal on Financial Mathematics
Theory of Probability and its Applications
The Annals of Probability
The Annals of Probability
Stochastics:Int. J. Probab. Stoch. Process.
79 (2007), 169–195.[10] Dumitrescu, R., Quenez, M. C., and Sulem, A.: Generalized Dynkin games and doubly reflectedBSDEs with jumps.
Electronic Journal of Probability
64 (2016), 1–32.[11] Dumitrescu, R., Quenez, M. C., and Sulem, A.: Game options in an imperfect market withdefault.
SIAM Journal on Financial Mathematics
ESAIM: Proceedings and Surveys .[13] Dynkin, E. B.: Game variant of a problem on optimal stopping.
Soviet Mathematics Doklady
10 (1969), 270–274.[14] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., and Quenez, M. C.: Reflected solutionsof backward SDE’s, and related obstacle problems for PDE’s.
The Annals of Probability
Mathematical Finance
Lecture Notes in Mathematics 1656 , B. Biais et al. (Eds.), Springer, Berlin, 1997,pp. 191–246.[17] Essaky, E. H. and Hassani, M.: Generalized BSDE with 2-reflecting barriers and stochasticquadratic growth.
Journal of Differential Equations
254 (2013), 1500–1528.[18] Grigorova, M., Imkeller, P., Ouknine, Y., and Quenez, M. C.: Doubly reflected BSDEs and E f -Dynkin games: beyond the right-continuous case. Working paper, 2017 (hal-01497914).[19] Grigorova, M. and Quenez, M. C.: Optimal stopping and a non-zero-sum Dynkin game indiscrete time with risk measures induced by BSDEs. Stochastics: An International Journal ofProbability and Stochastic Processes
89 (2017), 259–279. onlinear Pricing of Game Options
SIAMJournal on Control and Optimization
45 (2) (2007), 496–518.[21] Hamad`ene, S. and Ouknine, Y.: Reflected backward SDEs with general jumps.
Theory ofProbability and its Applications
60 (2016), 263–280.[22] Kallsen, J. and K¨uhn, C.: Pricing derivatives of American and game type in incomplete markets.
Finance and Stochastics
ExoticOption Pricing and Advanced L´evy Models , Wiley, Chichester New York, 2005, pp. 277–291.[24] Kifer, Y.: Game options.
Finance and Stochastics
ISRN Probability and Statistics (2013), ID856458,17 pages.[26] Kim, E., Nie, T., and Rutkowski, M.: Valuation and hedging of American options in nonlinearmodels. Working paper, 2018.[27] Klimsiak, T.: BSDEs with monotone generator and two irregular reflecting barriers.
Bulletindes Sciences Math´ematiques
137 (2013), 268–321.[28] Klimsiak, T.: Reflected BSDEs on filtered probability spaces.
Stochastic Processes and theirApplications
125 (2015), 4204–4241.[29] K¨uhn, C., Kyprianou, A. E. and van Schaik, K.: Pricing Israeli options: a pathwise approach.
Stochastics: Int. J. Probab. Stoch. Process.
79 (2006), 117–137.[30] Kyprianou, A. E.: Some calculations for Israeli options.
Finance and Stochastics
Journal of Applied Probability
41 (2004), 162–175.[32] Lepeltier, J. P. and Xu, M.: Reflected backward stochastic differential equations with two rcllbarriers.
ESAIM: Probability and Statistics
11 (2007), 3–22.[33] Matoussi, A., Piozin, L., and Possama¨ı, D.: Second-order BSDEs with general reflection andgame options under uncertainty.
Stochastic Processes and their Applications
124 (2014), 2281–2321.[34] Neveu, J.:
Discrete-Parameter Martingales.
North-Holland Publishing Company, Amsterdam,1975.[35] Nie, T. and Rutkowski, M.: BSDEs driven by multidimensional martingales and their appli-cations to markets with funding costs.
Theory of Probability and its Applications
60 (2016),604–630.[36] Nie, T. and Rutkowski, M.: A BSDE approach to fair bilateral pricing under endogenouscollateralization.
Finance and Stochastics
20 (2016), 855–900.[37] Nie, T. and Rutkowski, M.: Fair bilateral prices under funding costs and exogenous collateral-ization.
Mathematical Finance
28 (2018), 621–655.[38] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In