Arbitrage-free pricing of American options in nonlinear markets
aa r X i v : . [ q -f i n . M F ] J u l ARBITRAGE-FREE PRICING OF AMERICAN 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, PolandJuly 17, 2018
Abstract
We re-examine and extend the findings from the recent paper by Dumitrescu et al. [12]who studied American options in a particular market model using the nonlinear arbitrage-freepricing approach developed in El Karoui and Quenez [18]. In the first part, we provide a detailedstudy of unilateral valuation problems for the two counterparties in an American-style contractwithin the framework of a general nonlinear market. We extend results from Bielecki et al. [5, 6]who examined the case of a European-style contract. In the second part, we present a BSDEapproach, which is used to establish more explicit pricing, hedging and exercising results whensolutions to reflected BSDEs have additional desirable properties.
Keywords : nonlinear market, American option, optimal stopping, 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
Non-linearArbitrage Pricing of Multi-Agent Financial Games . The work of T. Nie was supported by the National Natural ScienceFoundation of China (No. 11601285) and the Natural Science Foundation of Shandong Province (No. ZR2016AQ13). E. Kim, T. Nie and M. Rutkowski
Unlike contracts of a European style, contracts of American style are asymmetric between the twocounterparties (hereafter referred to as the issuer and the holder ) not only due to the oppositedirections of contractual cash flows, but also due to the fact that only one party, the holder ofan American option, has the right to exercise (that is, to stop and settle) an American contractbefore its expiration date, which we invariably denote as T . The issues of arbitrage-free pricing andrational exercising of American options within the framework of a linear market model (usually forthe classical Black and Scholes model, but possibly with trading constraints, such as: no borrowingof cash or no short-selling of shares) have been studied in numerous papers, to mention just a few:Bensoussan [4], El Karoui et al. [16], Jaillet et al. [25], Karatzas [27], Karatzas and Kou [28], Kallsenand K¨uhn [26], Klimsiak and Rozkosz [34], Myneni [37] and Rogers [44].The goal of this work is to re-examine and extend the findings from the recent paper by Du-mitrescu et al. [12] who studied American options within the framework of a particular imperfectmarket model with default using the nonlinear arbitrage-free pricing approach developed in El Karouiand Quenez [18]. In contrast to [12] (see also [11] for the case of game options), we place ourselveswithin the setup of a general nonlinear arbitrage-free market, as introduced in Bielecki et al. [5, 6],and we examine general properties of fair and acceptable unilateral prices. We also obtain moreexplicit results regarding pricing, hedging and break-even times for the issuer and rational exercisetimes for the holder using a BSDE approach without being specific about the dynamics of underlyingassets, but by focusing instead on pertinent general features of solutions to reflected BSDEs.Let us introduce some notation for a generic nonlinear market model. Let (Ω , G , G , P ) be afiltered probability space satisfying the usual conditions of right-continuity and completeness, wherethe filtration G = ( G t ) t ∈ [0 ,T ] models the flow of information available to all traders. For convenience,we assume that the initial σ -field G is trivial. Moreover, all processes introduced in what follows areimplicitly assumed to be G -adapted and, as usual, any semimartingale is assumed to be c`adl`ag. Forsimplicity of notation, we assume throughout that the trading conditions are identical for the issuerand the holder, although this assumption can be relaxed without any difficulty since we examineunilateral valuation and hedging problems. Let T = T [0 ,T ] stand for the class of all G -stopping timestaking values in [0 , T ]. We adopt the following definition of an American contingent claim.By convention, all cash flows of a contract are described from the perspective of the issuer. Hencewhen a cash flow is positive for the issuer, then the cash amount is paid by the holder and receivedby the issuer. Obviously, if a cash flow is negative for the issuer, then the cash amount is transferredfrom the issuer to the holder. For instance, when dealing with the classical case of an American putoption written on a stock S , we assume that the payoff to the issuer (respectively, the holder) equals X hτ = − ( K − S τ ) + (respectively, X hτ = ( K − S τ ) + ) if the option is exercised at time τ by the holder.This is formalized through the following definition of an American contingent claim.
Note that thesuperscript h in X h is used to emphasize that only the holder has the right to exercise and henceto stop the contract; this can be contrasted with the case of a game option (see, e.g., Kifer [29, 30]and the references therein) where the covenants stipulate that both parties may exercise and hencestop the contract. Definition 1.1. An American contingent claim with the G -adapted, c`adl`ag payoff process X h is acontract between the issuer and the holder where the holder has the right to exercise the contractby selecting a G -stopping time τ ∈ T [0 ,T ] . Then the issuer ‘receives’ the amount X hτ or, equivalently,‘pays’ to the holder the amount of − X hτ at time τ where the G -adapted payoff process X ht , t ∈ [0 , T ]is specified by the contract. Note that we do not make any a priori assumptions about the sign ofthe payoff process X h , so it can be either positive or negative, in general.More generally, an American contract is formally identified with a triplet C a = ( A, X h , T ) wherea G -adapted, c`adl`ag stochastic process A , which is predetermined by the contract’s clauses, rep-resents the cumulative cash flows from time 0 till the contract’s maturity date T . In the financialinterpretation, the process A is assumed to model all the cash flows of a given American contract, onlinear Pricing of American Options C a , which is exchanged atits initiation (by convention, at time 0), is not included in the process A so that we set A = 0.This convention is motivated by the fact that the contract’s price before the deal is made is yetunspecified and thus it needs to be determined through negotiations between the counterpartiesand we will argue that unilateral pricing does not yield a common value for the initial price of anAmerican contract, in general.When examining the valuation of an American contract at any time t ∈ [0 , T ], we implicitlyassume that it has not yet been exercised and thus the set of exercise times available at time t toits current holder is the class T [ t,T ] of all G -stopping times taking values in [ t, T ]. In principle, onecould consider two alternative conventions regarding the payoff upon exercise: either(A.1) the cash flow upon exercise at time t equals A t − A t − + X ht or(A.2) if a contract is exercised at time t , then the cash flow A t − A t − is waived, so the only cashflow occurring at time t is X ht .Unless explicitly stated otherwise, we work under covenant (A.1) and we acknowledge that the choiceof a particular settlement rule may result in a different price for an American contract C a , in general.Of course, this choice is immaterial when the process A is continuous or, simply, when it vanishes,so that the contract reduces to a pair ( X h , T ).In the classical linear market model, if the terminal payoff of a contract is negative (or positive),then it is easy to recognize whether the fair price for a given party should be positive (or negative)and thus it is less important to keep track of signs of cash flows. In contrast, in a nonlinear setup itmay occur, for instance, that each party would like to sell a contract for a positive respective priceand thus it is not reasonable to make any a priori assumptions about the signs of prices.The paper is organized as follows. In Section 2, we work in an abstract nonlinear setup, meaningthat we only make fairly general assumptions about the nonlinear dynamics of the wealth processof self-financing strategies. The main postulates of that kind are the monotonicity properties of thewealth (see Assumptions 2.1 and 2.2). We examine general properties of fair and profitable prices forthe two counterparties, the issuer and the holder. In particular, we built upon papers by Bielecki etal. [5, 6] where the arbitrage-free valuation of European contingent claims in nonlinear markets wasexamined. In Section 3, we re-examine and extend a BSDE approach to the valuation of Americanoptions in nonlinear market initiated by El Karoui and Quenez [18] and continued in Dumitrescu etal. [12]. Our main goal is to show that unilateral acceptable prices for an American contract C a canbe characterised in terms of solutions to reflected BSDEs driven by a multi-dimensional continuoussemimartingale S . For the sake of concreteness, we postulate in Section 3 that the wealth process V = V ( y, ξ, A ) satisfies V t = y − Z t g ( u, V u , ξ u ) du + Z t ξ u dS u + A t , (1)where y ∈ R is the initial wealth at time 0 of a dynamic portfolio ξ (recall that A = 0). However,in order to keep the notation comprehensive, we do not explicitly specify the process S or generator g but instead we make assumptions about solutions to the SDE (14) and the BSDE (17). Let M = ( B , S , Ψ) be a market model which is arbitrage-free with respect to European contractsin the sense of Bielecki et al. [5, 6]. Here Ψ stands for the class of all admissible trading strategiesand Ψ( y, D ) denotes the class of all admissible trading strategies from Ψ with 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 E. Kim, T. Nie and M. Rutkowski 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 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. An important feature of the nonlinear arbitrage-free approach is the concept of the benchmark wealth V b ( x ) with respect to which arbitrage opportunities of a given trader are quantified and assessed.As in [5, 6], a simple and natural candidate for the benchmark wealth can be given by the equality V b ( x ) = V ( x ) where, for an arbitrary initial endowment x ∈ R of a trader, we set for all t ∈ [0 , T ] V t ( x ) := xB ,lt { x ≥ } + xB ,bt { x< } where the risk-free lending (respectively, borrowing ) cash account B ,l (respectively, B ,b ) is usedfor unsecured lending (respectively, borrowing) of cash. Note that V ( x ) represents the wealthprocess of a trader who decided at time 0 to keep his initial cash endowment x in either the lending(when x ≥
0) or the borrowing (when x <
0) cash account and who is not involved in any othertrading activities between the times 0 and T . By convention, the quantities x and x representinitial endowments of the issuer and the holder and thus the processes V ( x ) and V ( x ) are theirrespective benchmark wealths. Since the idea of the benchmark wealth is immaterial for valuationin linear market models, it is rarely encountered in works on arbitrage-free valuation, although itcorresponds to the well-known economic concept of opportunity costs . It thus important to makefew comments on its relevance, scope and limitations.On the one hand, we acknowledge that the idea of unilateral pricing based on an initial portfolioof a trader can be rejected as unrealistic. It is thus worth stressing that even when we set x = x = 0 so that V t ( x ) = V t ( x ) = 0 for all t ∈ [0 , T ], meaning that the initial endowments oftraders are completely ignored, the asymmetry in their respective unilateral prices will show up asa result of nonlinearity of the wealth dynamics, so that this assumption would not give an essentialsimplification of our results and findings.On the other hand, it would also be possible to assume that each trader is endowed with aninitial portfolio of assets (including also the savings account B ,l and B ,b , which means that hecould be either a borrower or a lender of cash) with the current market value y at time 0 (ignoringbid-ask spreads and transaction costs). Then V b ( y ) could represent the wealth process of his staticportfolio. Assume, for instance, that the issuer of an American contract has an initial portfolio ofrisky assets, denoted as α , and cash, denoted as β , with the (static) terminal values V T ( α ) and V T ( β ), respectively. Suppose that the risky portfolio α is not used for issuer’s hedging purposes buta (positive or negative) amount x from the cash account β is used to establish the hedge. Then theissuer’s benchmark wealth V b ( α , β ) can be defined as V bt ( α , β ) := V t ( α )+ V t ( β ) for all t ∈ [0 , T ]so that, in particular, the initial wealth of the issuer is equal to y := V b ( α , β ) = V ( α ) + V ( β ).Then the issuer’s total wealth, inclusive of the price p for the contract C a entered at time 0 and thehedging strategy ϕ , equals V t ( α , β , x , p, ϕ, A ) = V t ( α ) + V t ( β − x ) + V t ( x + p, ϕ, A ) . where x no longer represents the initial wealth of the issuer, but rather the part of the initial cashendowment used for hedging of C a . It is clear that it would be more difficult to analyze the case of adynamic benchmark portfolio, although in principle this is possible. For the sake of conciseness, weare not going to study these cases in what follows, but we stress that all results in this work are validfor any specifications of the benchmark wealth processes for the issuer and the holder, respectively,since it suffices to assume that they are some G -adapted and c`adl`ag stochastic processes. onlinear Pricing of American Options We consider an extended market model M p ( C a ) in which an American contract C a is traded attime 0 at some initial price p where p is an arbitrary real number. We first give a preliminaryanalysis of unilateral fair valuation of an American contract by its issuer who is endowed with thepre-trading initial wealth x ∈ R and the corresponding benchmark wealth process V b ( x ). We start by introducing the following conditions. Since the process A is fixed throughout, to alleviatenotation, we will frequently write V ( x + p, ϕ ) instead of V ( x + p, ϕ, A ) when dealing with the issuer.By the same token, we will later write V ( x − p, ψ ) instead of V ( x − p, ψ, − A ) when examiningtrading strategies of the holder. Definition 2.1.
We say that a triplet ( p, ϕ, τ ) ∈ R × Ψ( x + p, A ) × T satisfies:(AO) ⇐⇒ V τ ( x + p, ϕ ) + X hτ ≥ V bτ ( x ) and P (cid:0) V τ ( x + p, ϕ ) + X hτ > V bτ ( x ) (cid:1) > , (SH) ⇐⇒ V τ ( x + p, ϕ ) + X hτ ≥ V bτ ( x ) , (BG ε ) ⇐⇒ V τ ( x + p, ϕ ) + X hτ ≤ V bτ ( x ) + ε, (BE) ⇐⇒ V τ ( x + p, ϕ ) + X hτ = V bτ ( x ) , (NA) ⇐⇒ either V τ ( x + p, ϕ ) + X hτ = V bτ ( x ) or P (cid:0) V τ ( x + p, ϕ ) + X hτ < V bτ ( x ) (cid:1) > . Let us first explain the meaning of acronyms appearing in Definition 2.1: (AO) stands for arbi-trage opportunity , (SH) for superhedging , (BG) for bounded gain , (BE) for break-even and (NA) for no-arbitrage . For a more detailed explanation of each of these properties, see Definitions 2.2–2.6.For brevity, we write ( p, ϕ, τ ) ∈ (AO) if a triplet ( p, ϕ, τ ) satisfies condition (AO); an analogousconvention will be applied to other conditions introduced in Definition 2.1.If property (SH) is satisfied by a triplet ( p, ϕ, τ ), then we say that an issuer’s superhedging at time τ arises. Note that, from the optional section theorem, condition (SH) holds for a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) and all τ ∈ T if and only if ( p, ϕ ) is such that the inequality V t ( x + p, ϕ )+ X ht ≥ V bt ( x )is valid for every t ∈ [0 , T ]. This simple observation justifies the following definition of property (SH)for a pair ( p, ϕ ). Definition 2.2.
We say that a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) satisfies (SH) (briefly, ( p, ϕ ) ∈ (SH))if the inequality V t ( x + p, ϕ ) + X ht ≥ V bt ( x ) holds for every t ∈ [0 , T ]. Then ( p, ϕ ) is called an issuer’s superhedging strategy in the extended market M p ( C a ).Property (AO) of a triplet ( p, ϕ, τ ) is referred to as the issuer’s strict superhedging (or the issuer’sarbitrage opportunity ) at time τ . Definition 2.3.
We say that a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) satisfies condition (AO) if a triplet( p, ϕ, τ ) complies with condition (AO) for every τ ∈ T . Then we also say that ( p, ϕ ) creates an issuer’s arbitrage opportunity in the extended market M p ( C a ).The following lemma is an immediate consequence of Definition 2.3. Lemma 2.1.
If a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) is such that V t ( x + p, ϕ ) + X ht > V bt ( x ) for every t ∈ [0 , T ] , then ( p, ϕ ) fulfills (AO) and thus an issuer’s arbitrage opportunity arises in the extendedmarket M p ( C a ) . We say that no issuer’s arbitrage arises for ( p, ϕ ) at τ if ( p, ϕ, τ ) satisfies condition (NA). Itreadily seen that if a triplet ( p, ϕ, τ ) fails to satisfy (NA), then it fulfills (AO) and thus an issuer’sarbitrage opportunity arises at time τ for the issuer’s strategy ( p, ϕ ). By convention, we henceforthset inf ∅ = ∞ and sup ∅ = −∞ . Note that the superscript f stands here for fair and i for issuer . E. Kim, T. Nie and M. Rutkowski
Definition 2.4.
We say that p f,i ( x , C a ) is an issuer’s fair price for C a if no issuer’s arbitrageopportunity ( p, ϕ ) may arise in M p ( C a ) when p = p f,i ( x , C a ). Hence the set of issuer’s fair pricesequals H f,i ( x ) := (cid:8) p ∈ R | ∀ ϕ ∈ Ψ( x + p, A ) ∃ τ ∈ T : ( p, ϕ, τ ) ∈ (NA) (cid:9) and the upper bound for issuer’s fair prices is given by p f,i ( x , C a ) := sup (cid:8) p ∈ R | p is an issuer’s fair price for C a (cid:9) = sup H f,i ( x ) . (2)If the equality p f,i ( x , C a ) = max H f,i ( x ) holds (that is, whenever p f,i ( x , C a ) ∈ H f,i ( x )), then p f,i ( x , C a ) is denoted as b p f,i ( x , C a ) and called the issuer’s maximum fair price for C a .To alleviate notation, the variables ( x , C a ) will be sometimes suppressed and thus we will write p f,i , p f,i , b p f,i , etc., rather than p f,i ( x , C a ) , p f,i ( x , C a ) , b p f,i ( x , C a ) , etc. Assumption 2.1.
The forward monotonicity property holds: 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 ]. Lemma 2.2.
Let Assumption 2.1 be satisfied. If p ∈ H f,i ( x ) , then for any p ′ < p we have that p ′ ∈ H f,i ( x ) . Hence if H f,i ( x ) = ∅ , then either H f,i ( x ) = ( −∞ , p f,i ] = ( −∞ , b p f,i ] for some p f,i ∈ R or H f,i ( x ) = ( −∞ , p f,i ) where p f,i ∈ R ∪ {∞} . Proof.
We argue by contradiction. If H f,i ( x ) = ∅ , then p f,i = −∞ . Let us now consider the casewhere H f,i ( x ) = ∅ . Assume that p ∈ H f,i ( x ) and a number p ′ < p is not an issuer’s fair price.Then there exists ϕ ′ ∈ Ψ( x + p ′ , A ) such that ( p ′ , ϕ ′ , τ ) fulfills (AO) for every τ ∈ T . Consequently,by Assumption 2.1, there exists ϕ ∈ Ψ( x + p, A ) such that a triplet ( p, ϕ, τ ) complies with (AO)for every τ ∈ T . This clearly contradicts the assumption that p belongs to H f,i ( x ) and thus weconclude that the asserted properties are valid. ✷ The bounded gain condition (BG ε ) stipulates that issuer’s gains associated with a triplet ( p, ϕ, τ )are bounded from above by ε . It leads to the following definition of an issuer’s superhedging costwith negligible gain. Definition 2.5.
We say that p ∈ R is an issuer’s superhedging cost with negligible gain for C a iffor every ϕ ∈ Ψ( x + p, A ) such that condition (SH) is satisfied by ( p, ϕ ) ∈ R × Ψ( x + p, A ) andfor every ε >
0, there exists a τ ε ∈ T such that V τ ε ( x + p, ϕ ) + X hτ ε ≤ V bτ ε ( x ) + ε . The set of allissuer’s superhedging costs with negligible gain for C a is denoted by H n,i ( x ), that is, H n,i ( x ) := (cid:8) p ∈ R | ∀ ( p, ϕ ) ∈ R × Ψ( x + p, A ) satisfying (SH) ∀ ǫ > ∃ τ ε ∈ T : ( p, ϕ, τ ε ) ∈ (BG ε ) (cid:9) . Remark 2.1.
Note that the issuer’s superhedging cost with negligible gain is not necessarily unique.For example, assume that a pair ( p, ϕ ) ∈ R × Ψ( x + p, A ) satisfies (SH) so that V τ ( x + p, ϕ ) + X hτ ≥ V bτ ( x ) for all τ ∈ T and, in addition, V T − ( x + p, ϕ ) + X hT − = V bT − ( x ). It is possible to supposethat there exists a δ > ϕ ′ ∈ R × Ψ( x + p + δ, A ) such that ( p, ϕ ′ ) ∈ (SH), wehave V T − ( x + p, ϕ ) + X hT − = V T − ( x + p + δ, ϕ ′ ) + X hT − = V bT − ( x ) and V T ( x + p + δ, ϕ ′ ) + X hT >V T ( x + p, ϕ ) + X hT . Then it is obvious that both p and p + δ are issuer’s superhedging costs withnegligible gain.Let us finally introduce a stopping time related to the break-even condition (BE) introduced inDefinition 2.1. Definition 2.6.
If condition (BE) is satisfied by ( p, ϕ, τ ) ∈ R × Ψ( x + p, A ) × T , then a stoppingtime τ ∈ T is called an issuer’s break-even time for the pair ( p, ϕ ) ∈ R × Ψ( x + p, A ).Note that even when the pair ( p, ϕ ) is fixed, the uniqueness of an issuer’s break-even time τ isnot ensured, in general. Obviously, any issuer’s break-even time can be formally classified as oneof the exercise times available to the holder of C a but, as we will argue in what follows, an issuer’s onlinear Pricing of American Options rational exercise time for the holder. This is due to the factthat it may not actually always be advantageous for the holder to exercise at a stopping time thatcauses the issuer to break even or prohibits the issuer’s arbitrage opportunities. Firstly, usually theholder is not informed about the issuer’s trading strategy. Secondly, the holder should be behavingin a rational way for his own payoff and hedging abilities. A holder’s rational exercise time can betypically identified with a particular instance of a holder’s break-even time , which is introduced inDefinition 2.8. The earliest issuer’s break-even time will be later denoted as τ ∗ ,i , whereas for theearliest holder’s rational exercise time we will use the symbol τ ∗ ,h . Let us now analyze the holder’s fair pricing problem for C a . We assume that he is endowed withthe pre-trading initial wealth x ∈ R and his computation refers to the benchmark wealth process V b ( x ). Recall that we write V ( x − p, ψ ) := V ( x − p, ψ, − A ) when there is no danger of confusion. Definition 2.7.
We say that ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T satisfy:(AO ′ ) ⇐⇒ V τ ( x − p, ψ ) − X hτ ≥ V bτ ( x ) and P (cid:0) V τ ( x − p, ψ ) − X hτ > V bτ ( x ) (cid:1) > , (SH ′ ) ⇐⇒ V τ ( x − p, ψ ) − X hτ ≥ V bτ ( x ) , (BL ′ ε ) ⇐⇒ V τ ( x − p, ψ ) − X hτ ≥ V bτ ( x ) − ε, (BE ′ ) ⇐⇒ V τ ( x − p, ψ ) − X hτ = V bτ ( x ) , (NA ′ ) ⇐⇒ either V τ ( x − p, ψ ) − X hτ = V bτ ( x ) or P (cid:0) V τ ( x − p, ψ ) − X hτ < V bτ ( x ) (cid:1) > . Property (AO ′ ) (respectively, (SH ′ )) is called the strict superhedging (respectively, superhedging )condition for the holder. Note that the bounded loss property (BL ′ ε ) means that the holder’s lossesare bounded from below by − ε . Condition (BE ′ ) leads to the following definition. Definition 2.8.
If the equality V τ ′ ( x − p, ψ ) − X hτ ′ = V bτ ′ ( x ) holds, then a stopping time τ ′ ∈ T is called a holder’s break-even time for the pair ( p, ψ ) ∈ R × Ψ( x − p, − A ).The concept of a holder’s arbitrage opportunity reflects the fact that the holder has the rightto exercise an American contract, that is, to conveniently choose a stopping time τ at which thecontract is settled and terminated. Specifically, a holder’s arbitrage opportunity in M p ( C a ) is atriplet ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T satisfying condition (AO ′ ).For a triplet ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T , we say that no holder’s arbitrage arises for ( p, ψ ) at time τ if ( p, ψ, τ ) fulfills (NA ′ ). It is easily seen that a triplet ( p, ψ, τ ) ∈ R × Ψ( x − p, − A ) × T fails to satisfy (AO ′ ) if and only if it satisfies (NA ′ ). Definition 2.9.
We say that p f,h ( x , C a ) is a holder’s fair price for C a if no holder’s arbitrageopportunity ( p, ψ, τ ) may arise in the extended market M p ( C a ) when p = p f,h ( x , C a ). Hence theset of holder’s fair prices equals H f,h ( x ) := (cid:8) p ∈ R | ∀ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (NA ′ ) (cid:9) and the lower bound for the holder’s fair prices is given by p f,h ( x , C a ) := inf (cid:8) p ∈ R | p is a holder’s fair price for C a (cid:9) = inf H f,h ( x ) . (3)If the equality p f,h ( x , C a ) = min H f,h ( x ) holds, then p f,h ( x , C a ) is denoted as ˘ p f,h ( x , C a ) andcalled the holder’s minimum fair price for C a . Lemma 2.3.
Let Assumption 2.1 be satisfied for − A . If p ∈ H f,h ( x ) , then for any p ′ > p we havethat p ′ ∈ H f,h ( x ) . Therefore, if H f,h ( x ) = ∅ , then either H f,h ( x ) = [ p f,h , ∞ ) = [ ˘ p f,h , ∞ ) or H f,h ( x ) = ( p f,h , ∞ ) . E. Kim, T. Nie and M. Rutkowski
Definition 2.10.
We say that p ∈ R is a holder’s cost with negligible loss if for every ε > ψ, τ ) ∈ Ψ( x − p, − A ) × T such that V τ ( x − p, ψ ) − X hτ ≥ V bτ ( x ) − ε . The set of allholder’s costs with negligible loss for C a is denoted by H n,h ( x ), that is, H n,h ( x ) := (cid:8) p ∈ R | ∀ ǫ > ∃ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (BL ′ ε ) (cid:9) . The concepts of a (strict) superhedging strategy and the associated cost for the issuer and theholder are fairly standard. For the issuer, they are based on conditions (SH) and (AO), respectively,whereas for the holder they hinge on conditions (SH ′ ) and (AO ′ ), respectively. This means that forthe issuer we need to impose some conditions that are valid for every τ ∈ T , whereas for the holderit suffices to postulate that the analogous conditions are satisfied for some τ ∈ T . We first introduce the notion of the lower bound for issuer’s strict superhedging costs.
Definition 2.11.
The lower bound for issuer’s strict superhedging costs for C a is given by p a,i ( x , C a ):= inf H a,i ( x ) where H a,i ( x ) := (cid:8) p ∈ R : ∃ ϕ ∈ Ψ( x + p, A ) : ( p, ϕ ) ∈ (AO) (cid:9) . If the equality p a,i ( x , C a ) = min H a,i ( x ) holds, then it is denoted as ˘ p a,i ( x , C a ) and called the issuer’s minimum strict superhedging cost for C a .It is readily seen that H a,i ( x ) is the complement of H f,i ( x ) and thus, in view of Lemma 2.2,the equality p a,i ( x , C a ) = p f,i ( x , C a ) is satisfied under Assumption 2.1. More precisely, we dealwith the following alternative: either H f,i ( x ) = ( −∞ , b p f,i ] and H a,i ( x ) = ( p a,i , ∞ ) (4)or H f,i ( x ) = ( −∞ , p f,i ) and H a,i ( x ) = [ ˘ p a,i , ∞ ) . (5) Definition 2.12.
The lower bound for issuer’s superhedging costs for C a is given by p s,i ( x , C a ):= inf H s,i ( x ) where H s,i ( x ) := (cid:8) p ∈ R : ∃ ϕ ∈ Ψ( x + p, A ) : ( p, ϕ ) ∈ (SH) (cid:9) . If the equality p s,i ( x , C a ) = min H s,i ( x ) holds, then p s,i ( x , C a ) is denoted as ˘ p s,i ( x , C a ) andcalled the issuer’s minimum superhedging cost for C a .It is clear that H a,i ( x ) ⊆ H s,i ( x ) and thus p s,i ( x , C a ) ≤ p a,i ( x , C a ). In general, it may occurthat p s,i ( x , C a ) < p a,i ( x , C a ) = p f,i ( x , C a ). To avoid this problematic situation, we introduce As-sumption 2.2, which ensures that p s,i ( x , C a ) = p a,i ( x , C a ) and thus also p s,i ( x , C a ) = p f,i ( x , C a ).Obviously, Assumption 2.2 is stronger than Assumption 2.1. Assumption 2.2.
The forward strict monotonicity property holds: 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 ]. Lemma 2.4.
If Assumption 2.2 is satisfied, then the equality p s,i ( x , C a ) = p a,i ( x , C a ) holds andthus p f,i ( x , C a ) = p s,i ( x , C a ) = p a,i ( x , C a ) . onlinear Pricing of American Options Proof.
Let us first assume that H s,i ( x ) = ∅ or, equivalently, that p s,i ( x , C a ) < ∞ (recall thatinf ∅ = ∞ ). Since Assumption 2.2 holds, it is clear that for an arbitrary p ∈ H s,i ( x ) and any ε >
0, there exists a strategy ϕ ′ ∈ Ψ( x + p + ε ) such that condition (AO) is satisfied by the pair( p + ε, ϕ ′ ). Hence p + ε belongs to H a,i ( x ) and thus p + ε ≥ p a,i ( x , C a ). From the arbitrarinessof p ∈ H s,i ( x ) and ε >
0, we infer that p s,i ( x , C a ) ≥ p a,i ( x , C a ). From the discussion afterDefinition 2.12, we get p s,i ( x , C a ) ≤ p a,i ( x , C a ) and thus p s,i ( x , C a ) = p a,i ( x , C a ). Recalling that p f,i ( x , C a ) = p a,i ( x , C a ) (see the comments after Definition 2.11), we conclude that p f,i ( x , C a ) = p s,i ( x , C a ) = p a,i ( x , C a ) if H s,i ( x ) = ∅ . Let us now assume that H s,i ( x ) = ∅ so that H a,i ( x ) = ∅ as well, since H a,i ( x ) ⊆ H s,i ( x ). Then, on the one hand, p s,i ( x , C a ) = p a,i ( x , C a ) = ∞ and, onthe other hand, p f,i ( x , C a ) = ∞ , since H f,i ( x ) = R being the complement of H a,i ( x ). Hence theasserted equalities are satisfied in that case as well. ✷ Let us now examine the holder’s strict superhedging costs.
Definition 2.13.
The upper bound for holder’s strict superhedging costs for C a is given by p a,h ( x , C a ):= sup H a,h ( x ) where H a,h ( x ) := (cid:8) p ∈ R | ∃ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (AO ′ ) (cid:9) . If the equality p a,h ( x , C a ) = max H a,h ( x ) holds, then p a,h ( x , C a ) is denoted as b p a,h ( x , C a ) andcalled the holder’s maximum strict superhedging cost for C a .It is easily seen that H a,h ( x ) is the complement of H f,h ( x ). Hence we infer from Lemma 2.3that the equality p a,h ( x , C a ) = p f,h ( x , C a ) is satisfied under Assumption 2.1 and either H a,h ( x ) = ( −∞ , p a,c ) and H f,h ( x ) = [ ˘ p f,h , ∞ ) (6)or H a,h ( x ) = ( −∞ , b p a,c ] and H f,h ( x ) = ( p f,h , ∞ ) . (7)Similarly as for the issuer, we also introduce the concept of a superhedging strategy for the holder. Definition 2.14.
The upper bound for holder’s superhedging costs for C a equals p s,h ( x , C a ) :=sup H s,h ( x ) where H s,h ( x ) := (cid:8) p ∈ R | ∃ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (SH ′ ) (cid:9) . If the equality p s,h ( x , C a ) = max H s,h ( x ) holds, then p s,h ( x , C a ) is denoted as b p s,h ( x , C a ) andcalled the holder’s maximum superhedging cost for C a .It is clear that H a,h ( x ) ⊆ H s,h ( x ) and thus the inequality p s,h ( x , C a ) ≥ p a,h ( x , C a ) is valid.Furthermore, Assumption 2.2 ensures that the equality holds. Lemma 2.5.
If Assumption 2.2 is satisfied with − A , then the equality p s,h ( x , C a ) = p a,h ( x , C a ) holds and thus p f,h ( x , C a ) = p s,h ( x , C a ) = p a,h ( x , C a ) .Proof. Let us first assume that H s,h ( x ) = ∅ or, equivalently, that p s,h ( x , C a ) > −∞ . UnderAssumption 2.2, for any p ∈ H s,h ( x ) and ε >
0, there exists a pair ( ϕ ′ , τ ) ∈ Ψ( x − ( p − ε )) × T suchthat ( p − ε, ϕ ′ , τ ) satisfy condition (AO ′ ), that is, p − ε ∈ H a,h ( x ) and thus p − ε ≤ p a,h ( x , C a ). Since p ∈ H s,h ( x ) and ε > p s,h ( x , C a ) ≤ p a,h ( x , C a ). The inequality p s,h ( x , C a ) ≤ p a,h ( x , C a ) is always satisfied and thus p s,h ( x , C a ) = p a,h ( x , C a ) = p f,h ( x , C a )where the last equality is known to hold under Assumption 2.1. Let us now assume that H s,h ( x ) = ∅ .Then, on the one hand, p s,h ( x , C a ) = p a,h ( x , C a ) = −∞ since H a,h ( x ) ⊆ H s,h ( x ) and, on theother hand, p f,h ( x , C a ) = −∞ since H f,h ( x ) is the complement of H a,h ( x ). Hence the assertedequalities are valid in that case as well. ✷ E. Kim, T. Nie and M. Rutkowski
Our next goal is to analyze the following problem: under which assumptions a suitably defined replication cost of an American contract is also its maximum (respectively, minimum) fair pricefor the issuer (respectively, the holder). The answer to this question will lead us to the cruciallyimportant concept of unilateral acceptable prices computed by the counterparties.
We will now study the concept of replication of the contract C a by the issuer. We work hereafterunder Assumption 2.2 and thus, in view of Lemma 2.4, we have that p f,i ( x , C a ) = p s,i ( x , C a ) = p a,i ( x , C a ) . (8) Definition 2.15.
The lower bound for issuer’s replication costs for C a is given by p r,i ( x , C a ):= inf H r,i ( x ) where H r,i ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, τ ) ∈ Ψ( x + p, A ) × T : ( p, ϕ ) ∈ (SH) & ( p, ϕ, τ ) ∈ (BE) (cid:9) . If the equality p r,i ( x , C a ) = min H r,i ( x ) holds, then p r,i ( x , C a ) is denoted as ˘ p r,i ( x , C a ) and calledthe issuer’s minimum replication cost for C a .Note that in Definition 2.15 we focus on a particular issuer’s superhedging strategy for which abreak-even time exists and we do not impose any restrictions on wealth processes of other tradingstrategies available to the issuer. Hence, in principle, for p ∈ H r,i ( x ), it may happen that thereexists another pair, say ( p, ψ ), which is an issuer’s strict superhedging strategy. This would meanthat the issuer’s replication cost is not an issuer’s fair price for C a . To eliminate this shortcomingof Definition 2.15, in the next definition we impose, in addition, the no-arbitrage condition (NA) onall issuer’s trading strategies associated with p . Definition 2.16.
The lower bound for issuer’s fair replication costs for C a is given by p f,r,i ( x , C a ) :=inf H f,r,i ( x ) where H f,r,i ( x ) := (cid:8) p ∈ R | ∃ ( ϕ, τ ) ∈ Ψ( x + p, A ) × T : ( p, ϕ ) ∈ (SH) & ( p, ϕ, τ ) ∈ (BE); ∀ ϕ ′ ∈ Ψ( x + p, A ) ∃ τ ′ ∈ T : ( p, ϕ ′ , τ ′ ) ∈ (NA) (cid:9) . If the equality p f,r,i ( x , C a ) = min H f,r,i ( x ) holds, then p f,r,i ( x , C a ) is denoted as ˘ p f,r,i ( x , C a )and called the issuer’s minimum fair replication cost for C a . Definition 2.17.
If the set H f,r,i ( x ) is a singleton, then its unique element is denoted as p i ( x , C a )and called the issuer’s acceptable price .Obviously, if p i ( x , C a ) is well defined, then it coincides with ˘ p f,r,i ( x , C a ). Let us examine someuseful relationships between the conditions introduced in Definition 2.1. Lemma 2.6. (i) If ( p, ϕ ) satisfy (SH) and ( p, ϕ, τ ) satisfy (NA), then (BE) is valid for ( p, ϕ, τ ) .(ii) If ( p, ϕ ) fail to satisfy (SH), then there exists τ ∈ T such that ( p, ϕ, τ ) satisfy (NA).(iii) The following conditions are equivalent:(a) for any ( p, ϕ ) satisfying (SH) there exists τ ∈ T such that ( p, ϕ, τ ) satisfy (BE),(b) for any ( p, ϕ ) , there exists τ ∈ T such that ( p, ϕ, τ ) satisfy (NA).Proof. (i) The statement is an almost immediate consequence of definitions of conditions (SH),(NA) and (BE). Indeed, suppose that a pair ( p, ϕ ) satisfies (SH) and a triplet ( p, ϕ, τ ) complies with(NA). Clearly, the inequality P ( V τ ( x + p, ϕ ) + X hτ < V bτ ( x )) > p, ϕ, τ )must satisfy (BE). onlinear Pricing of American Options p, ϕ ) satisfies (AO), then it fulfills (SH) and thus if ( p, ϕ ) fails to satisfy(SH), then condition (AO) is not met. Recalling that, in essence, the class of pairs ( p, ϕ ) satisfying(AO) is the complement of (NA), we conclude that there exists a stopping time τ ∈ T such that thetriplet ( p, ϕ, τ ) complies with (NA).(iii) We first show that (a) implies (b). We know from (ii) that if a pair ( p, ϕ ) does not satisfycondition (SH), then there exists τ ∈ T such that ( p, ϕ, τ ) complies with (NA). If ( p, ϕ ) satisfies(SH) then, from condition (a), there exists τ ∈ T such that the triplet ( p, ϕ, τ ) fulfills (BE), and thusit also satisfies (NA). Let us now assume that condition (b) is met. Then, from part (i), condition(a) holds as well. ✷ In view of (8), the proof of the following lemma is obvious and thus it is omitted.
Lemma 2.7.
We have that H s,i ( x ) ⊇ H r,i ( x ) ⊇ H f,r,i ( x ) = H f,i ( x ) ∩ H r,i ( x ) and thus p f,i = p s,i ≤ p r,i ≤ p f,r,i . (9)In the next result, we study the basic properties of issuer’s costs. Proposition 2.1. (i) If H f,r,i ( x ) = ∅ , then it is a singleton and the issuer’s acceptable price p i = p i ( x , C a ) satisfies − ∞ < p i = b p f,i = ˘ p r,i = ˘ p s,i < + ∞ . (10) (ii) If H r,i ( x ) = ∅ , then p f,i = p s,i ≤ p r,i < ∞ .(iii) If H r,i ( x ) = ∅ , then p f,i = p s,i ≤ p r,i = ∞ .Proof. We start by proving (i). If H f,r,i ( x ) = ∅ , then from the inclusion H f,r,i ( x ) ⊆ H f,i ( x )we obtain the inequality p f,r,i ≤ p f,i and thus, in view of (9), we have p f,i = p s,i = p r,i = p f,r,i .Moreover, H f,i ( x ) = ∅ and H s,i ( x ) = ∅ and thus p f,i > −∞ and p s,i < ∞ . We conclude that −∞ < p f,i = p r,i = p f,r,i = p s,i < ∞ . From the inclusion H f,r,i ( x ) ⊆ H f,i ( x ) and the equalitiessup H f,i ( x ) = p f,i = p f,r,i = inf H f,r,i ( x ), we deduce that for arbitrary p , p ∈ H f,r,i ( x ) and p ∈ H f,i ( x ) we have p = p ≥ p and thus H f,r,i ( x ) is a singleton and its unique elementis not less than any element of H f,i ( x ). Consequently, b p f,i and p i are well defined and satisfy −∞ < b p f,i = p i < + ∞ . Furthermore, H f,r,i ( x ) ⊆ H r,i ( x ) and thus the equality p i = p r,i implies that ˘ p r,i exists as well and coincides with p i . We conclude that (10) is valid. This means,in particular, that H f,i ( x ) = ( −∞ , b p f,i ] = ( −∞ , ˘ p r,i ] = ( −∞ , p i ]. Parts (ii) and (iii) are easyconsequences of (9). ✷ Remark 2.2.
Part (i) in Proposition 2.1 shows that if there exists a number p for which an issuer’sfair replicating strategy exists and p is the unique issuer’s acceptable price, his maximum fair price,minimum replication cost, and the infimum of (strict) superhedging costs (but, obviously, it is nota strict superhedging cost). We thus deal here with a highly desirable situation but, sadly, it is noteasy to check whether a number p with above-mentioned properties exists. As expected, to overcomethis difficulty we will impose additional assumptions on wealth processes of trading strategies. Recallthat in the complete linear market, one can show that H r,i ( x ) = ∅ but, to the best of our knowledge,the properties of the set H f,r,i ( x ) have not been studied in the existing literature. We postulate that Assumption 2.2 is satisfied with − A and thus, in view of Lemma 2.5, we havethat p f,h ( x , C a ) = p s,h ( x , C a ) = p a,h ( x , C a ) . (11)The notion of the holder’s replication cost introduced is through the following definition. Definition 2.18.
The upper bound for holder’s replication costs for C a is given by p r,h ( x , C a ):= sup H r,h ( x ) where H r,h ( x ) = (cid:8) p ∈ R | ∃ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (BE ′ ) (cid:9) . E. Kim, T. Nie and M. Rutkowski
Equivalently, p r,h ( x , C a ) := − inf (cid:8) q ∈ R | ∃ ( ψ, τ ) ∈ Ψ( x + q, − A ) × T : V τ ( x + q, ψ ) − X hτ = V bτ ( x ) (cid:9) . If the equality p r,h ( x , C a ) = max H r,h ( x ) holds, then p r,h ( x , C a ) is denoted as b p r,h ( x , C a ) andcalled the holder’s maximum replication cost for C a .To establish the existence of the holder’s acceptable price, we will use the concept of the holder’sfair replication costs. Definition 2.19.
The upper bound for holder’s fair replication costs for C a is given by p f,r,h ( x , C a ) =sup H f,r,h ( x ) where H f,r,h ( x ) := (cid:8) p ∈ R | ∃ ( ψ, τ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ, τ ) ∈ (BE ′ ) & ∀ ( ψ ′ , τ ′ ) ∈ Ψ( x − p, − A ) × T : ( p, ψ ′ , τ ′ ) ∈ (NA ′ ) (cid:9) . If p f,r,h ( x , C a ) = max H f,r,h ( x ), then p f,r,h ( x , C a ) is denoted as b p f,r,h ( x , C a ) and called the holder’s maximum fair replication cost for C a . Definition 2.20.
If the set H f,r,h ( x ) is a singleton, then its unique element is denoted as p h ( x , C a )and called the holder’s acceptable price .Notice that if p h ( x , C a ) is well defined, then it is equal to b p f,r,h ( x , C a ). The following lemma isan easy consequence of Definition 2.7. Lemma 2.8. (i) If ( p, ψ, τ ) satisfy (BE ′ ), then they satisfy (SH ′ ).(ii) If ( p, ψ, τ ) satisfy (SH ′ ) and (NA ′ ), then they satisfy (BE ′ ).(iii) If ( p, ψ, τ ) do not satisfy (SH ′ ), then they satisfy (NA ′ ). We infer from Lemma 2.8 that H s,h ( x ) ⊇ H r,h ( x ) ⊇ H f,r,h ( x ) = H r,h ( x ) ∩ H f,h ( x ) andthus, in view of (11), we have p f,h = p s,h ≥ p r,h ≥ p f,r,h . (12) Proposition 2.2. (i) If H f,r,h ( x ) = ∅ , then it is a singleton and the holder’s acceptable price p h = p h ( x , C a ) satisfies − ∞ < p h = ˘ p f,h = b p r,h = b p s,h < + ∞ . (13) (ii) If H r,h ( x ) = ∅ , then −∞ < p r,h ≤ p f,h = p s,h .(iii) If H r,h ( x ) = ∅ , then −∞ = p r,h ≤ p f,h = p s,h .Proof. It is clear that H f,r,h ( x ) ⊆ H f,h ( x ) and thus if H f,r,h ( x ) = ∅ , then p f,r,h < + ∞ and p f,r,h ≥ p f,h . Furthermore, the sets H s,h ( x ) and H r,h ( x ) are nonempty and thus we infer from(12) that −∞ < p f,h = p r,h = p f,r,h = p s,h < ∞ . From the inclusion H f,r,h ( x ) ⊆ H f,h ( x ) and theequalities inf H f,h ( x ) = p f,h = p f,r,h = sup H f,r,h ( x ), we see that for arbitrary p , p ∈ H f,r,h ( x )and p ∈ H f,h ( x ) we have p = p ≤ p and thus H f,r,h ( x ) is a singleton and its unique elementis less than any element of H f,h ( x ). Consequently, ˘ p f,h and p h are well defined and they satisfy −∞ < p h = ˘ p f,h < + ∞ . Furthermore, H f,r,h ( x ) ⊆ H r,h ( x ) and thus the equality p h = p r,h impliesthat b p r,h exists and coincides with p h . We conclude that (13) is valid. The proofs of statements (ii)and (iii) are straightforward in view of (12). ✷ The goal of this section is to re-examine and extend a BSDE approach to the valuation of Americanoptions in nonlinear market, which was initiated in the paper by Dumitrescu et al. [11]. Our maingoal is to show that unilateral acceptable prices for an American contract C a can be characterized in onlinear Pricing of American Options S .In this section, we postulate that the wealth process V = V ( y, ϕ, A ) satisfies the forward equation V t = y − Z t g ( u, V u , ξ u ) du + Z t ξ u dS u + A t , (14)where y ∈ R represents the initial wealth at time 0 of a given trading strategy ξ (recall that A = 0).By applying Lemma 3.1 with y = x + p < x + p ′ = y , f = f = g and z = ξ , it is easy to checkthat Assumption 2.2 is met when the dynamics of the wealth process are given by the SDE (14) andare uniquely specified by the initial value y and the process ξ (for instance, when g = g ( t, v, z ) isLipschitz continuous with respect to v ). Let us first describe more explicitly the main features of the mechanism of nonlinear trading, whichunderpins the wealth dynamics given by (14). We start by introducing the notation for traded assets,that is, cash accounts, risky assets, and funding accounts associated with risky assets. It should bestressed, however, that our further developments will not depend on the choice of a particular modelfor primary assets and trading arrangements and thus our general results are capable of covering abroad spectrum of market models.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 [5, 6]). Let us denote by B the collection of all cash and funding accounts available toa trader. For simplicity of presentation, we maintain our assumption that the issuer and the holderhave identical market conditions but it is clear that this assumption is not relevant for our furtherdevelopments and thus it can be easily relaxed. A trading strategy is an R d +2 -valued, G -adaptedprocess ϕ = ( ξ , . . . , ξ d ; ψ ,l , ψ ,b , . . . , ψ d,l , ψ d,b ) where the components represent all outstandingpositions in the risky assets S j , j = 1 , , . . . , d , cash accounts B ,l , B ,b , and funding accounts B j,l , B j,b , j = 1 , , . . . , d for risky assets. Definition 3.1.
We say that a trading strategy ( y, ϕ ) is self-financing for C a and we write ϕ ∈ Ψ( y, A ) if the 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 all admissible trading strategies is arbitrage-free in a suitable sense, forinstance, the market model M can be assumed to be regular , in the sense of Bielecki et al. [5].4 E. Kim, T. Nie and M. Rutkowski
Since the arbitrage-free feature of a nonlinear market was studied in El Karoui and Quenez [18]and, more recently, Bielecki et al. [5, 6], we do not elaborate on this issue here and we refer theinterested reader to these works. It is important to notice that, due to the trading constraints,differential funding costs and possibly also some additional adjustment processes, which are notexplicitly stated in Definition 3.1, the dynamics of the wealth process are nonlinear, in general. Werefer the reader to Bielecki et al. [5, 6] for more encompassing versions of the self-financing propertyof a trading strategy (see, for instance, Definition 4.5 in [6] or Definition 2.2 in [5]) and to Nie andRutkowski [38, 39, 40, 41] for explicit examples of nonlinear models with trading constraints andadjustments (in particular, the collateralization of contracts). We only observe that each particulartrading arrangement gives rise to an explicit mapping g in the dynamics (14) of the wealth, whichis sometimes quite complex, but at the same time usually fairly regular.The following elementary lemma addresses the issue of the (strict) monotonicity of the wealthprocess driven by (14). Note that since the process z is assumed to be given in the statement ofLemma 3.1, we may interpret the SDE (15) as a deterministic integral equation, which holds foralmost all ω ∈ Ω. Lemma 3.1.
Let g j : Ω × [0 , T ] × R × R d → R , j = 1 , be P ⊗ B ( R ) ⊗ B ( R d ) / B ( R ) -measurable.Consider the SDEs for j = 1 , v jt = y j − Z t g j ( u, v ju , z u ) du + k jt + Z t z u dS u + H t , (15) where z is an R d -valued, G -progressively measurable stochastic process, the G -adapted process k = k − k is nondecreasing with k = 0 , S is an R d -valued semimartingale, and H is a c`adl`ag, G -adaptedprocess. Assume that (15) has a unique solution v j for j = 1 , . If g ( t, v t , z t ) ≤ g ( t, v t , z t ) , dt ⊗ d P -a.e. and y ≥ y (respectively, y > y ), then v t ≥ v t (respectively, v t > v t ) for all t ∈ [0 , T ] .Proof. If we set ¯ v t := v t − v t for all t ∈ [0 , T ], then ¯ v is easily seen to satisfy¯ v t = y − y + Z t (cid:0) g ( u, v u , z u ) − g ( u, v u , z u ) (cid:1) du + k t = y − y + Z t (cid:0) λ u ¯ v u + g ( u, v u , z u ) − g ( u, v u , z u ) (cid:1) du + k t where λ u := (cid:0) g ( u, v u , z u ) − g ( u, v u , z u ) (cid:1) (¯ v u ) − { v u = v u } . Thus d (cid:16) e − R t λ u du ¯ v t (cid:17) = e − R t λ u du (cid:0) ( g ( t, v t , z t ) − g ( t, v t , z t )) dt + dk t (cid:1) . After integrating both sides from 0 to t , and noticing that the process k = k − k is nondecreasingand e − R t λ u du >
0, we obtain the inequality¯ v t ≥ ¯ v e R t λ u du + Z t e R ts λ u du (cid:0) g ( s, v s , z s ) − g ( s, v s , z s ) (cid:1) ds from which the assertion of the lemma follows. ✷ We henceforth postulate that the wealth process V = V ( y, ϕ, A ) is governed by the SDE (14) where ξ = ( ξ , . . . , ξ d ) is given and the mapping g satisfies some additional assumptions. This will allow usto refer to a large body of existing literature on the theory of BSDEs and, in particular, to exploit thelink between reflected BSDEs and solutions to nonlinear optimal stopping problems (see, for instance,Cvitani´c and Karatzas [9], Dumitrescu et al. [10], El Karoui et al. [15], Grigorova and Quenez [22],Grigorova et al. [20, 21], and Quenez and Sulem [45]). Our goal in this section is to examine onlinear Pricing of American Options , s ] Y t = ζ s + Z st g ( u, Y u , Z u ) du − Z st Z u dM u − ( H s − H t ) , (16)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 (16) has a unique solution( Y, Z ) in a suitable space of stochastic processes (see, e.g., [7, 39]). 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 (16) 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 (16) 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. 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 ).In view of its financial interpretation, 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 ) (17)is henceforth denoted by E g,i and called the issuer’s g -evaluation . In Section 3.3, we address theissuer’s pricing problem and we work under the following standing assumption. Assumption 3.1.
We postulate that:(i) the wealth process V = V ( y, ϕ, A ) of a trading strategy ϕ ∈ Ψ( y, A ) satisfies (14),(ii) for any fixed y ∈ R and any process ξ such that the stochastic integral in (14) is well defined,the SDE (14) 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 (17) has a unique solution ( Y, Z ) on [0 , s ] . Note that in Section 3.5, where we study the holder’s pricing problem, we will use a modifiedversion of Assumption 3.1 in which the process A is replaced by − A . Remark 3.1.
In view of Lemma 3.1 and condition (ii) in Assumption 3.1, condition (iii) in As-sumption 3.1 is not restrictive, since it is satisfied for every generator g . For explicit assumptionsabout g ensuring that the BSDE (17), where S is a multi-dimensional, continuous, square-integrablemartingale enjoying the predictable representation property, has a unique solution and the strictcomparison property of the issuer’s g -evaluation E g,i holds, see Theorems 3.2 and 3.3 in Nie andRutkowski [39].6 E. Kim, T. Nie and M. Rutkowski
We denote by X := V b ( x ) − X h the issuer’s relative reward and we assume that the process X issquare-integrable. Then, by Assumption 3.1(iv), the following BSDE on [0 , T ] Y t = X T + Z Tt g ( u, Y u , Z u ) du − Z Tt Z u dS u − ( A T − A t ) (18)has a unique solution ( Y, Z ) in a suitable space of stochastic processes. To study the issuer’s pricingproblem for an American contract C a , we make the following assumption (see, e.g., Definition 2.4 inQuenez and Sulem [45]). Assumption 3.2.
The reflected BSDE with the lower obstacle X ( dY t = − g ( t, Y t , Z t ) dt + Z t dS t + dA t − dK t ,Y T = X T , Y t ≥ X t , R T ( Y t − X t ) dK ct = 0 , ∆ K dt = − ∆( Y t − A t ) { Y t − = X t − } , (19)has a unique solution ( Y, Z, K ) where K is a G -predictable, c`adl`ag, nondecreasing process such that K = 0 and K = K c + K d is its unique decomposition into continuous and jump components.Reflected BSDEs were introduced in seminal papers by El Karoui et al. [16, 15] where theywere applied to solutions of optimal stopping problems and pricing of American options. Theywere subsequently studied by several authors who dealt with various frameworks, to mention a few:Aazizi and Ouknine [1], Baadi and Ouknine [2], Cr´epey and Matoussi [8], Essaky [19], Hamad`ene [23],Hamad`ene and Ouknine [24], Grigorova et al. [20, 21], Klimsiak [32, 33], Klimsiak et al. [35] andQuenez and Sulem [45]. They have shown that Assumption 3.2 is met in many instances of ourinterest but, due to space limitations, we are not going to quote any particular result here.It is worth stressing that we deliberately do not specify particular spaces of stochastic processesin which the components Y and Z are searched for, since our further results do not depend on thechoice of these spaces. Only the properties of the process K in a unique solution ( Y, Z, K ) to theissuer’s RBSDE (19) (and, by the same token, of the process k in a unique solution ( y, z, k ) to theholder’s RBSDE (30)) are essential in what follows and thus they are stated explicitly and analyzedin some detail. Definition 3.3.
We say that v i ( C a ) ∈ R is the value of the issuer’s optimal stopping problem for C a if v i ( C a ) = sup τ ∈T E g,i ,τ ( X τ ) . A stopping time τ ∗ ∈ T is called a solution to the issuer’s optimal stopping problem if v i ( C a ) = b v i ( C a )where b v i ( C a ) := E g,i ,τ ∗ ( X τ ∗ ) = max τ ∈T E g,i ,τ ( X τ ) . (20) Assumption 3.3.
The value v i ( C a ) to the issuer’s optimal stopping problem exists and satisfies v i ( C a ) = Y . Assumption 3.4.
The stopping time τ i := inf { t ∈ [0 , T ] | Y t = X t } is a (not necessarily unique)solution to the issuer’s optimal stopping problem so that b v i ( C a ) = E g,i ,τ i ( X τ i ). Remark 3.2.
Various results pertaining to Assumptions 3.2–3.4 were obtained under alternativeassumptions on the generator g and the processes X and S in (19) by, among others, El Karoui etal. [15], Grigorova et al. [20] and Quenez and Sulem [45]. Although it is common to set S = W and A = 0, an extension to a more general situation is also feasible when the generator g satisfies asuitable Lipschitz-type condition (see [39] for the case of a BSDE without reflection).The first main result in this section is the following theorem. onlinear Pricing of American Options Theorem 3.1.
Let Assumptions 3.1–3.4 be satisfied and let ( Y, Z, K ) be the unique solution to theissuer’s reflected BSDE (19) . Then the following assertions are valid:(i) if E g,i has the strict comparison property, then ˘ p r,i ( x , C a ) = ˘ p s,i ( x , C a ) = Y − x = E g,i ,τ i ( X τ i ) − x = b v i ( C a ) − x (21) and ( p ′ , ϕ ′ , τ ′ ) = ( Y − x , Z, τ i ) is an issuer’s replicating strategy for C a ,(ii) if E g,i has the strict comparison property, then the issuer’s acceptable price for C a equals p i ( x , C a ) = Y − x = b v i ( C a ) − x .Proof. The proof of Theorem 3.1 is split into three steps, which are formulated as Propositions 3.1,3.2 and 3.3. ✷ We start by analyzing the issuer’s minimum superhedging cost (note that Assumption 3.4 is notpostulated in Proposition 3.1).
Proposition 3.1.
If Assumptions 3.1–3.3 are satisfied and E g,i has the comparison property, thenthe issuer’s minimum superhedging cost is well defined and satisfies ˘ p s,i ( x , C a ) = v i ( C a ) − x = Y − x , (22) where ( Y, Z, K ) is the unique solution to the reflected BSDE (19) .Proof. We first prove that p s,i ≤ Y − x . It suffices to show that for the initial value p ′ := Y − x ,where Y is obtained from the reflected BSDE (19), we can find an issuer’s superhedging strategy,that is, there exists a trading strategy ϕ ′ ∈ Ψ( x + p ′ , A ) such that V t ( x + p ′ , ϕ ′ ) ≥ X t for all t ∈ [0 , T ]. To this end, we set ( p ′ , ϕ ′ ) = ( Y − x , Z ) where ( Y, Z, K ) is the unique solution to thereflected BSDE (19). Then, on the one hand, the value process V = V ( x + p ′ , ϕ ′ ) is a uniquestrong solution (by Assumption 3.1(ii)) to the following SDE where the initial value V = Y andthe process Z are fixed dV t = − g ( t, V t , Z t ) dt + Z t dS t + dA t . (23)On the other hand, if ( Y, Z, K ) solves the reflected BSDE (19), then the process e Y = Y can also beseen as a unique strong solution to the following SDE d e Y t = − g ( t, e Y t , Z t ) dt + Z t dS t + dA t − dK t , (24)where, once again, the initial value e Y = Y and the processes Z and K are given. Therefore, fromLemma 3.1 with g = g = g we infer that V t ≥ e Y t = Y t for all t ∈ [0 , T ]. Since Y t ≥ X t for all t ∈ [0 , T ], we conclude that V t ≥ X t for all t ∈ [0 , T ]. Consequently, ( x + p ′ , ϕ ′ ) = ( Y , Z ) is anissuer’s superhedging strategy and thus p s,i ≤ Y − x .We will now show that p s,i ≥ Y − x . Let us consider an arbitrary p ∈ R for which there exists ϕ ∈ Ψ( x + p, A ) such that ( p, ϕ ) satisfy (SH). If we can show that x + p ≥ Y , then the inequality p s,i ≥ Y − x will hold by the definition of the lower bound p s,i . To this end, we observe that V τ ( x + p, ϕ ) ≥ X τ for every τ ∈ T since, by Definition 2.2, we have that V t ( x + p, ϕ ) ≥ X t forall t ∈ [0 , T ]. Consequently, by applying the mapping E g,i to both sides and using the comparisonproperty of E g,i , we obtain x + p = E g,i ,τ ( V τ ( x + p, ϕ )) ≥ E g,i ,τ ( X τ ) . Since τ ∈ T is arbitrary, we conclude that x + p ≥ sup τ ∈T E g,i ,τ ( X τ ) = v i ( C a ) = Y where the secondequality follows from Assumption 3.3. Hence p s,i ≥ Y − x and thus we conclude that the equality p s,i = Y − x is valid.Finally, from the first part of the proof, we know that for p ′ = Y − x there exists a tradingstrategy ϕ ′ = Z ∈ Ψ( x + p ′ , A ) such that V t ( x + p ′ , ϕ ′ ) ≥ X t for all t ∈ [0 , T ] so that Y − x ∈H s,i ( x ). Consequently, we have that p s,i ( x , C a ) = ˘ p s,i ( x , C a ) = Y − x . ✷ Consider the solution (
Y, Z, K ) and the pair ( Y , Z ) introduced in the proof of Proposition 3.1.In the next result, we examine the existence of an issuer’s replicating strategy for C a .8 E. Kim, T. Nie and M. Rutkowski
Proposition 3.2.
If Assumptions 3.1–3.4 are satisfied and E g,i has the strict comparison property,then the following assertions are valid:(i) the pair ( Y − x , Z ) is an issuer’s replicating strategy for C a and τ i is an issuer’s break-eventime for the pair ( Y − x , Z ) ,(ii) the issuer’s minimum superhedging and replication costs satisfy ˘ p r,i ( x , C a ) = ˘ p s,i ( x , C a ) = Y − x = E g,i ,τ i ( X τ i ) − x = b v i ( C a ) − x . (25) Proof.
From Lemma 2.7 and Proposition 3.1, we already know that Y − x = p s,i = ˇ p s,i ≤ p r,i andthus to establish the equality ˘ p r,i ( x , C a ) = ˘ p s,i ( x , C a ), it is enough to show that the trading strategy( p ′ , ϕ ′ ) = ( Y − x , Z ), which is already known to be an issuer’s superhedging strategy, is also anissuer’s replicating strategy for C a . We first note that the definition of τ i and the right-continuityof the processes X and Y yield the equality X τ i = Y τ i . Consequently, we have that Y = b v i ( C a ) = E g,i ,τ i ( X τ i ) = E g,i ,τ i ( Y τ i ) , where the first two equalities follow from Assumptions 3.3 and 3.4. We will now show that K τ i = 0.Since ( Y, Z, K ) solves the reflected BSDE (19), we also know that Y = Y τ i + Z τ i g ( u, Y u , Z u ) du − Z τ i Z u dS u − A τ i + K τ i . Therefore, Y = E g,i ,τ i ( Y τ i + K τ i ) and thus E g,i ,τ i ( Y τ i ) = E g,i ,τ i ( Y τ i + K τ i ). From the strict comparisonproperty of E g,i , we conclude that K τ i = 0 and thus, for all t ∈ [0 , τ i ], Y t = Y − Z t g ( u, Y u , Z u ) du + Z t Z u dS u + A t . Finally, using the equality V ( Y , Z ) = Y and the postulated uniqueness of a solution to the SDE(23) (see Assumption 3.1(ii)), we obtain the equality V t ( Y , Z ) = Y t on [0 , τ i ] and thus, in particular, V τ i ( Y , Z ) = Y τ i = X τ i . We have thus shown that τ i is an issuer’s break-even time for the pair( Y − x , Z ). We have thus shown that the pair ( p ′ , ϕ ′ ) = ( Y , Z ) is an issuer’s replicating strategyfor C a . ✷ The final step in establishing Theorem 3.1 and hence providing a solution to the issuer’s valuationproblem hinges on showing that the issuer’s acceptable price is also the issuer’s maximum fair price.
Proposition 3.3.
If Assumptions 3.1–3.4 are satisfied and E g,i has the strict comparison property,then the issuer’s acceptable price p i ( x , C a ) is well defined and p i ( x , C a ) = b p f,i ( x , C a ) = ˘ p r,i ( x , C a ) = ˘ p s,i ( x , C a ) . (26) Proof.
It suffices to show that ˘ p r,i belongs to H f,i ( x ) or, equivalently, that ˘ p r,i < p for every p ∈ H a,i ( x ) (recall that H a,i ( x ) is the complement of H f,i ( x )). To this end, we will argue bycontradiction. Assume that ˘ p r,i ∈ H a,i ( x ) so that there exists a strategy ˘ ϕ ∈ Ψ( x + ˘ p r,i ) such that(˘ p r,i , ˘ ϕ ) satisfy (AO). Then we have, for every τ ∈ T , P (cid:0) V τ ( x + ˘ p r,i , ˘ ϕ ) ≥ X τ (cid:1) = 1 and P (cid:0) V τ ( x + ˘ p r,i , ˘ ϕ ) > X τ (cid:1) > . Let us now take τ = τ i . By applying the mapping E g,i to both sides, we obtain x + ˘ p r,i = E g,i ,τ i (cid:0) V τ i ( x + ˘ p r,i , ˘ ϕ ) (cid:1) > E g,i ,τ i ( X τ i ) = x + ˘ p r,i , where the last equality comes from Proposition 3.2. This is a contradiction and thus we have shownthat ˘ p r,i is not in H a,i ( x ). Recall that either H a,i ( x ) = [ p a,i , ∞ ) or H a,i ( x ) = ( p a,i , ∞ ) and weclaim that in fact the latter case is true. Indeed, from Assumption 3.1, Lemma 2.4 and Proposition3.2, we have ˘ p r,i = ˘ p s,i = p a,i and, since ˘ p r,i is not in H a,i ( x ), we have that H a,i ( x ) = ( p a,i , ∞ ).Obviously, ˘ p r,i < p for every p ∈ H a,i ( x ) and thus ˘ p r,i belongs to H f,i ( x ) meaning that the set H f,r,i ( x ) is nonempty. All equalities in (26) now follow from Proposition 2.1. ✷ onlinear Pricing of American Options In Section 3.4, we postulate that the assumptions of Theorem 3.1(ii) are satisfied and the contract C a is traded at time 0 at the issuer’s acceptable price p i = p i ( x , C a ). From Definition 2.15 andPropositions 3.2 and 3.3, we know that there exists a pair ( ϕ ′ , τ i ) ∈ Ψ( x + p i , A ) ×T such that ( p i , ϕ ′ )satisfy (SH) and ( p i , ϕ ′ , τ i ) satisfy (BE), specifically, ϕ ′ = Z and p i = Y − x , where ( Y, Z, K ) is theunique solution to the reflected BSDE (19). Our next goal is to provide a detailed characterizationof all issuer’s break-even times associated with an issuer’s replicating strategy ( p i , ϕ ′ ). To establishTheorem 3.2, which is the second main result in Section 3.3, we will need the following additionalassumption. 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 = ξ j , where τ ∈ T , ξ ≥ ξ , 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 ]. Remark 3.3.
Suitable versions of the comparison theorem for BSDEs are known and thus Assump-tion 3.5 can be checked to be met in several nonlinear market models (see, for instance, explicitexamples analyzed in Nie and Rutkowski [39]).Before stating the main result in Section 3.3, let us recall the following well-known definitionrelated to nonlinear evaluations (see, e.g., Peng [42, 43]).
Definition 3.4.
We say that a G -adapted, c`adl`ag process Y is an E g,i - supermartingale (respectively,an E g,i - martingale ) on [0 , T ] if Y s ≥ E g,is,t ( Y t ) (respectively, Y s = E g,is,t ( Y t )) for 0 ≤ s ≤ t ≤ T . Theorem 3.2.
Let Assumptions 3.1–3.5 be satisfied and the strict comparison property of E g,i hold.Then for the process ϕ ′ = Z ∈ Ψ( x + p i , A ) and an arbitrary τ ′ ∈ T the following assertions areequivalent:(i) τ ′ is an issuer’s break-even time for the pair ( p i , ϕ ′ ) ∈ R × Ψ( x + p i , A ) ,(ii) the triplet ( p i , ϕ ′ , τ ′ ) satisfies (NA) ,(iii) the equality V τ ′ ( x + p i , ϕ ′ ) = X τ ′ holds,(iv) X τ ′ = Y τ ′ and K τ ′ = 0 and thus Y is an E g,i -martingale on [0 , τ ′ ] ,(v) τ ′ is a solution to the issuer’s optimal stopping problem, that is, E g,i ,τ ′ ( X τ ′ ) = b v i ( C a ) .The stopping time τ i = inf { t ∈ [0 , T ] | Y t = X t } is the earliest issuer’s break-even time for ( p i , ϕ ′ ) .Proof. Recall that if ϕ ′ = Z , then the pair ( p i , ϕ ′ ) is an issuer’s superhedging strategy for C a (seethe proof of Proposition 3.1). It is thus is clear that assertions (i), (ii) and (iii) are equivalent.(iii) ⇒ (iv). From the proof of Proposition 3.1, we already know that V t ( x + p i , ϕ ′ ) ≥ Y t ≥ X t forall t ∈ [0 , T ] and thus, in particular, the inequality V τ ( x + p i , ϕ ′ ) ≥ Y τ ≥ X τ holds for every τ ∈ T .Since we assumed that (iii) holds, we have V τ ′ ( x + p i , ϕ ′ ) = X τ ′ and thus V τ ′ ( Y , ϕ ′ ) = Y τ ′ = X τ ′ (recall from Theorem 3.1(ii) that p i = Y − x ). It thus remains to show that K τ ′ = 0. Sincethe process V = V ( Y , ϕ ′ ) satisfies the SDE (23), it is an E g,i -martingale and thus we obtain thefollowing equalities E g,i ,τ ′ ( Y τ ′ ) = E g,i ,τ ′ (cid:0) V τ ′ ( Y , ϕ ′ ) (cid:1) = Y . (27)On the one hand, for any fixed t ∈ (0 , T ] the process ¯ Y s := E g,is,t ( Y t ) , s ∈ [0 , t ] solves the followingBSDE ( d ¯ Y s = − g ( s, ¯ Y s , ¯ Z s ) ds + ¯ Z s dS s + dA s , ¯ Y t = Y t . E. Kim, T. Nie and M. Rutkowski
On the other hand, if (
Y, Z, K ) solves the reflected BSDE (19), then for any fixed [0 , t ], the pair( e Y , e Z ) = ( Y, Z ) is a unique solution to the BSDE ( d e Y s = − g ( s, e Y s , e Z s ) ds + e Z s dS s + dA s − dK s , e Y t = Y t , where K is a predetermined increasing process. Therefore, in view of the extended comparisonproperty of solutions to BSDEs (see Assumption 3.5), the inequality E g,is,t ( Y t ) ≤ Y t holds for all s ∈ [0 , t ] and thus Y is an E g,i -supermartingale. Moreover, similarly to the above discussion, byusing the extended comparison property of solutions to BSDEs, one can show that, for any θ ∈ T ,the inequality E g,is,θ ( Y θ ) ≤ Y s holds for all s ∈ [0 , θ ]. We will now show that for every 0 ≤ s ≤ τ ′ E g,is,τ ′ ( Y τ ′ ) = Y s . (28)Let us assume, on the contrary, that equality (28) fails to hold. Then, using the strict comparisonproperty of E g,i , we obtain E g,i ,τ ′ ( Y τ ′ ) = E g,i ,s ( E g,is,τ ′ ( Y τ ′ )) < E g,i ,s ( Y s ) ≤ Y . This manifestly contradicts(27) and thus (28) is valid. For every 0 ≤ s ≤ t ≤ τ ′ , from (28), we have E g,it,τ ′ ( Y τ ′ ) = Y t and then E g,is,t ( Y t ) = E g,is,t ( E g,it,τ ′ ( Y τ ′ )) = E g,is,τ ′ ( Y τ ′ ) = Y s , where the last equality also comes from (28). We conclude that Y is an E g,i -martingale on [0 , τ ′ ]and thus the equality K τ ′ = 0 follows.(iv) ⇒ (iii). By assumption, Y τ ′ = X τ ′ and K τ ′ = 0 and thus the reflected BSDE (19) reduces tothe following BSDE on [0 , τ ′ ] ( dY t = − g ( t, Y t , Z t ) dt + Z t dS t + dA t ,Y τ ′ = X τ ′ . The above BSDE can also be represented in the forward manner, for t ∈ [0 , τ ′ ], dY t = − g ( t, Y t , Z t ) dt + Z t dS t + dA t , where the initial value Y and the process Z are given. Similarly, the wealth process V := V ( x + p i , ϕ ′ ) = V ( Y , Z ) solves the following SDE, for t ∈ [0 , T ], dV t = − g ( t, V t , Z t ) dt + Z t dS t + dA t , with initial condition V = Y . From the uniqueness of a solution to the above SDE, we infer that V t = Y t for t ∈ [0 , τ ′ ]. In particular, V τ ′ ( x + p i , ϕ ′ ) = Y τ ′ = X τ ′ , as was required to show.(iv) ⇒ (v). From the E g,i -martingale property of Y on [0 , τ ′ ], we get E g,i ,τ ′ ( Y τ ′ ) = Y . In view ofAssumption 3.3, we have that Y = v i ( C a ) and thus the equalities E g,i ,τ ′ ( X τ ′ ) = v i ( C a ) = b v i ( C a ) hold,which means that τ ′ is a solution to the issuer’s optimal stopping problem.(v) ⇒ (iv). From condition (v) and Assumption 3.3, we obtain the equality Y = E g,i ,τ ′ ( X τ ′ ). Wewill now use similar arguments as in the proof of the implication (iii) ⇒ (iv). First, the process¯ X s := E g,is,t ( X τ ′ ) solves the following BSDE ( d ¯ X s = − g ( s, ¯ X s , ¯ Z s ) ds + ¯ Z s dS s + dA s , ¯ X τ ′ = X τ ′ , and it also satisfies ¯ X = Y . Second, if ( Y, Z, K ) solves the reflected BSDE (19), then the pair( e Y , e Z ) = ( Y, Z ) is a unique solution to the BSDE ( d e Y s = − g ( s, e Y s , e Z s ) ds + e Z s dS s + dA s − dK s , e Y τ ′ = Y τ ′ ≥ X τ ′ , onlinear Pricing of American Options K is a predetermined increasing process and, obviously, e Y = Y . The extended comparisonproperty of solutions to BSDEs yields the inequality E g,it,τ ′ ( Y τ ′ ) ≤ Y t for all t ∈ [0 , τ ′ ]. Therefore, if Y τ ′ ≥ X τ ′ and Y τ ′ = X τ ′ , then the strict comparison property of E g,i gives Y ≥ E g,i ,τ ′ ( Y τ ′ ) > E g,i ,τ ′ ( X τ ′ ) = Y , which is a contradiction. This shows that Y τ ′ = X τ ′ . As in the proof of the implication (iii) ⇒ (iv),we argue that Y is an E g,i -martingale on [0 , τ ′ ] and thus the equality K τ ′ = 0 is satisfied.It remains to show that the last assertion is valid. In view of Assumption 3.4, τ i is a solutionto the issuer’s optimal stopping problem and thus, from part (v), τ i is an issuer’s break-even timefor ( p i , ϕ ′ ). In view of (iv), for any break-even time for for ( p i , ϕ ′ ) we have that X τ ′ = Y τ ′ . Thedefinition of τ i now shows that it is the earliest issuer’s break-even time for C a . ✷ We now address the pricing, hedging and exercising problems from the perspective of the holder.The corresponding nonlinear evaluation E g, − A , which is defined through solutions to the holder’sBSDE y t = ζ s + Z st g ( u, y u , z u ) du − Z st z u dS u + A s − A t , (29)is henceforth denoted by E g,h and called the holder’s g -evaluation . For brevity, we denote by x := V b ( x ) + X h the holder’s relative reward and we assume that the process x is square-integrable. Assumption 3.6.
The reflected BSDE with the upper obstacle x ( dy t = − g ( t, y t , z t ) dt + z t dS t − dA t + dk t ,y T = x T , y t ≤ x t , R T ( x t − y t ) dk ct = 0 , ∆ k dt = ∆( y t + A t ) { y t − = x t − } , (30)has a unique solution ( y, z, k ) where k is a G -predictable, nondecreasing process such that k = 0and k = k c + k d is its unique decomposition into continuous and jump components. Definition 3.5.
We say that v h ( C a ) ∈ R is the value of the holder’s optimal stopping problem for C a if v h ( C a ) = inf τ ∈T E g,h ,τ ( x τ ) . A stopping time τ ∗ ∈ T is called a solution to the holder’s optimal stopping problem if v h ( C a ) =˘ v h ( C a ) where ˘ v h ( C a ) := E g,h ,τ ∗ ( x τ ∗ ) = min τ ∈T E g,h ,τ ( x τ ) . (31)The following assumptions can be justified by an independent analysis of a nonlinear optimalstopping problem. Although we take here the properties stated in Assumptions 3.7 and 3.8 forgranted, it is worth to mention that they are supported by recent results in Grigorova et al. [20, 21]. Assumption 3.7.
The value v h ( C a ) to the holder’s optimal stopping problem exists and satisfies v h ( C a ) = y . Assumption 3.8.
The stopping time τ h := inf { t ∈ [0 , T ] | y t = x t } is a solution to the holder’soptimal stopping problem.A salient feature of an American contract is a holder’s rational exercise time , which in ourframework is defined as follows. Definition 3.6.
We say that τ ∈ T is a rational exercise time for the holder of C a if the contractis traded at the holder’s maximum superhedging cost b p s,h = b p s,h ( x , C a ) at time 0 and there existsa strategy ψ ∈ Ψ( x − b p s,h , − A ) such that V τ ( x − b p s,h , ψ ) ≥ x τ .2 E. Kim, T. Nie and M. Rutkowski
In fact, we will use Definition 3.6 within the setup where the equality b p r,h ( x , C a ) = b p s,h ( x , C a )holds. If, in addition, the strict comparison property for the BSDE with the driver g is satisfied,then the inequality V τ ( x − b p r,h , ψ ) ≥ x τ can be replaced by the equality V τ ( x − b p r,h , ψ ) = x τ sothat a rational exercise time is also a holder’s break-even time. The following theorem describes theproperties of a solution to the holder’s pricing problem for an American contract C a . Theorem 3.3.
Let Assumption 3.1 and 3.6–3.8 be satisfied and let ( y, z, k ) be a unique solution tothe holder’s reflected BSDE (30) . Then the following assertions are valid.(i) If E g,h has the strict comparison property, then b p r,h ( x , C a ) = b p s,h ( x , C a ) = x − y = x − E g,h ,τ h ( x τ h ) = x − ˘ v h ( C a ) . (32) A holder’s replicating strategy for C a is given by the triplet ( p ′ , ψ ′ , τ ′ ) = ( x − y , z, τ h ) and τ h is aholder’s rational exercise time.(ii) If E g,h has the strict comparison property, then the holder’s acceptable price for C a equals p h ( x , C a ) = x − y = x − ˘ v h ( C a ) .Proof. As in the case of the issuer, we split the proof into three steps, which are formulated asPropositions 3.4, 3.5 and 3.6. They can be seen as the holder’s counterparts of Propositions 3.1, 3.2and 3.3, although their statements and proofs differ from the issuer’s case. ✷ Note the Assumption 3.8 is not postulated in the next result. It is worth stressing that theequality p s,h ( x , C a ) = x − y does not necessarily hold under the assumptions of Proposition 3.4and the holder’s maximum superhedging cost b p s,h ( x , C a ) is not necessarily well defined. Proposition 3.4.
If Assumptions 3.1 and 3.6–3.7 are satisfied and E g,h has the comparison property,then p s,h ( x , C a ) ≤ x − v h ( C a ) = x − y , (33) where ( y, z, k ) is the unique solution to the reflected BSDE (30) .Proof. We will show that p s,h ≤ x − y . By the definition of the supremum, it is enough toshow that x − y ≥ p for all p ∈ H s,h ( x ). From the definition of H s,h ( x ), we know that for any p ∈ H s,h ( x ), there exists a pair ( ψ, τ ) ∈ Ψ( x − p, − A ) × T such that V τ ( x − p, ψ ) ≥ x τ . Thecomparison property of E g,h gives x − p = E g,h ,τ (cid:0) V τ ( x − p, ψ ) (cid:1) ≥ E g,h ,τ ( x τ )and thus x − p ≥ inf τ ∈T E g,h ,τ (cid:0) V τ ( x − p, ψ ) (cid:1) ≥ inf τ ∈T E g,h ,τ ( x τ ) = v h ( C a ) = y , where the last equality follows from Assumption 3.7. We have thus shown that p s,h ( x , C a ) ≤ x − y = x − v h ( C a ). ✷ We will now give conditions under which, in particular, the holder’s maximum superhedging andreplication costs are well defined and we show that they are equal, that is, b p r,h ( x , C a ) = b p s,h ( x , C a ). Proposition 3.5.
If Assumptions 3.1 and 3.6–3.8 are satisfied and E g,h has the strict comparisonproperty, then:(i) ( x − y , z, τ h ) is a holder’s replicating strategy for C a ,(ii) the holder’s maximum replication cost is well defined and satisfies b p r,h ( x , C a ) = b p s,h ( x , C a ) = x − y = x − ˘ v h ( C a ) = x − E g,h ,τ h ( x τ h ) . Proof.
We already know that x − y ≥ p s,h ≥ p r,h (see (12) and (33)). Hence to prove parts (i)and (ii), it suffices to show that if ( y, z, k ) is the unique solution to the reflected BSDE (30), then( p ′ , ψ ′ , τ ′ ) = ( x − y , z, τ h ) is a holder’s replicating strategy. The wealth process V = V ( x − p ′ , ψ ′ )satisfies the SDE dV t = − g ( t, V t , z t ) dt + z t dS t − dA t , (34) onlinear Pricing of American Options V = y and the process z are given. The definition of τ h and the right-continuity of the processes x and y ensure that x τ h = y τ h so that y = ˘ v h ( C a ) = E g,h ,τ h ( x τ h ) = E g,h ,τ h ( y τ h ) , where the second equality is a consequence of Assumption 3.8, and thus we see that y = E g,h ,τ h ( y τ h ).Therefore, using the strict comparison property of E g,h and simple arguments analogous to thoseused in the derivation of the equality K τ i = 0 in the proof of Proposition 3.2, we obtain the equality k τ h = 0. Since k t = 0 on [0 , τ h ], the reflected BSDE (30) can be seen on [0 , τ h ] as the forward SDE dy t = − g ( t, y t , z t ) dt + z t dS t − dA t , (35)where the initial value y = V and the process z are given. From the uniqueness of a solutionto the SDE (34), it follows that V = y on [0 , τ h ]. Hence V τ h = y τ h = x τ h and thus the triplet( p ′ , ψ ′ , τ ′ ) = ( x − y , z, τ h ) is indeed a holder’s replicating strategy. ✷ To complete the proof of Theorem 3.4, we need to examine the existence of the holder’s acceptableprice p h ( x , C a ). This will be done in the proof of the following proposition. Proposition 3.6.
If Assumptions 3.1 and 3.6–3.8 are satisfied and E g,h has the strict comparisonproperty, then the holder’s acceptable price p h ( x , C a ) is well defined and p h ( x , C a ) = ˘ p f,h ( x , C a ) = b p r,h ( x , C a ) = b p s,h ( x , C a ) . Proof.
We will first show that b p r,h ∈ H f,h ( x ). In view of (6) and (7), it is enough to prove that b p r,h ( x , C a ) > p for every p ∈ H a,h ( x ). To this end, we argue by contradiction. Let us write b p = b p r,h ( x , C a ). Assume that b p ∈ H a,h ( x ) so that there exists ( b ϕ, b τ ) ∈ Ψ( x − b p ) × T such that( b p, b ϕ, b τ ) satisfies (AO ′ ), that is, P (cid:0) V b τ ( x − b p, b ϕ ) ≥ x b τ (cid:1) = 1 and P (cid:0) V b τ ( x − b p, b ϕ ) > x b τ (cid:1) > . By applying the mapping E g,h , we obtain x − b p = E g,h , b τ (cid:0) V b τ ( x − b p r,h , b ϕ ) (cid:1) > E g,h , b τ ( x b τ ) ≥ inf τ ∈T E g,h ,τ ( x τ ) = E g,h ,τ h ( x τ h ) = x − b p, where the last equality comes from Proposition 3.5. This is a contradiction and thus b p r,h ( x , C a ) / ∈H a,h ( x ). In general, either H a,h ( x ) = ( −∞ , p a,h ( x , C a )] or H a,h ( x ) = ( −∞ , p a,h ( x , C a )) and weargue that the latter situation occurs. Indeed, from Assumption 3.1, Lemma 2.5 and Propositions3.4 and 3.5, we have b p r,h ( x , C a ) = b p s,h ( x , C a ) = p a,h ( x , C a ) and thus, since b p r,h ( x , C a ) is not in H a,h ( x ), we conclude that H a,h ( x ) = ( −∞ , p a,h ( x , C a )). It is also clear that b p r,h ( x , C a ) > p forevery p ∈ H a,h ( x ) and thus b p r,h ( x , C a ) belongs to H f,h ( x ) so that H f,r,h ( x ) = ∅ . We completethe proof by making use of Proposition 2.2. ✷ We conclude the paper by an analysis of the properties of holder’s rational exercise times. Note thatin Theorem 3.4 we work under the assertions of Theorem 3.3. It is thus known that the equality b p r,h ( x , C a ) = b p s,h ( x , C a ) holds and thus a stopping time τ ∈ T is a holder’s rational exercise timeif the contract is traded at the holder’s maximum replication cost b p r,h = b p r,h ( x , C a ) at time 0 andthere exists a strategy ψ ∈ Ψ( x − b p r,h , − A ) such that V τ ( x − b p r,h , ψ ) = x τ . We thus deal herewith a natural extension of the classical concept of a rational exercise time for the holder of anAmerican option when the underlying market model is linear. Let us notice that in any completelinear market, albeit not in a general nonlinear market, any holder’s rational exercise time is also abreak-even time for the issuer (in particular, the equality τ h = τ i is satisfied).Recall that G -adapted, c`adl`ag process Y is an E g,h -submartingale (respectively, an E g,h -martingale)on [0 , T ] if Y s ≤ E g,hs,t ( Y t ) (respectively, Y s = E g,hs,t ( Y t )) for all 0 ≤ s ≤ t ≤ T .4 E. Kim, T. Nie and M. Rutkowski
The following result gives a characterisation of all holder’s rational exercise times and describesthe earliest and the latest rational exercise times. Of course, results of this kind are well knownfrom the existing literature on a classical optimal stopping problem (see, for instance, Kobylanskiand Quenez [36]). For nonlinear optimal stopping problems, the interested reader is also referred toDumitrescu [11] and Grigorova et al. [20, 21].
Theorem 3.4.
Let Assumptions 3.1 and 3.5–3.8 be satisfied and the strict comparison property of E g,h hold. In particular, let ( y, z, k ) be the unique solution to the reflected BSDE (30) . Then astopping time τ ′ ∈ T is a holder’s rational exercise time if and only if the following conditions aremet:(i) y is a E g,h -martingale on [0 , τ ′ ] , that is, k τ ′ = 0 ,(ii) the equality y τ ′ = x τ ′ holds.The earliest holder’s rational exercise time equals τ h := inf { t ∈ [0 , T ] | y t = x t } . If, in addition,the process k is continuous, then ¯ τ h := inf { t ∈ [0 , T ] | k t > } is the latest holder’s rational exercisetime.Proof. Let τ ′ ∈ T be any stopping time such that conditions (i) and (ii) are met. Since y τ ′ = x τ ′ and k τ ′ = 0, we see that the triplet ( y, z, k ) solves the following BSDE on [0 , τ ′ ] ( dy t = − g ( t, y t , z t ) dt + z t dS t − dA t ,y τ ′ = x τ ′ , which can also be written in the forward manner, for all t ∈ [0 , τ ′ ], dy t = − g ( t, y t , z t ) dt + z t dS t − dA t , where initial condition y and the process z are given. Now we take ψ = z and we recall that b p r,h ( x , C a ) = x − y (see Proposition 3.6). Hence the wealth process V = V ( x − b p r,h ( x , C a ) , ψ )satisfies the following SDE for all t ∈ [0 , T ] dV t = − g ( t, V t , z t ) dt + z t dS t − dA t , with initial condition V = y . From the uniqueness of a solution to the above SDE, we infer that V t = y t ≤ x t for every t ∈ [0 , τ ′ ]. In particular, V τ ′ = y τ ′ = x τ ′ and thus τ ′ is a rational exercisetime for the holder of C a .Let us now assume that τ ′ is a rational exercise time for the holder of C a . From Definition 3.6,it follows that for p = b p r,h ( x , C a ) = x − y there exists a strategy ψ ∈ Ψ( x − p, − A ) such that V τ ′ ( x − p, ψ ) ≥ x τ ′ . The comparison property of E g,h yields y = x − p = E g,h ,τ ′ (cid:0) V τ ′ ( x − p, ψ ) (cid:1) ≥ E g,h ,τ ′ ( x τ ′ ) ≥ E g,h ,τ ′ ( y τ ′ ) , (36)where the last inequality is valid since x τ ′ ≥ y τ ′ . For any fixed t ∈ (0 , T ], the process ¯ y s := E g,hs,t ( y t )solves the following BSDE on [0 , t ] ( d ¯ y s = − g ( s, ¯ y s , ¯ z s ) ds + ¯ z s dS s − dA s , ¯ y t = y t , and, if ( y, z, k ) solves the reflected BSDE (30), then y satisfies the following BSDE on [0 , t ] ( dy s = − g ( s, y s , z s ) ds + z s dS s − dA s + dk s ,y t = y t . Using the extended comparison property postulated in Assumption 3.5, we get y s ≤ ¯ y s = E g,hs,t ( y t )for all s ∈ [0 , t ] and thus y is an E g,h -submartingale. Furthermore, using the extended comparisonproperty of solutions to BSDEs, one can show that for any θ ∈ T we have E g,hs,θ ( y θ ) ≥ y s for all onlinear Pricing of American Options s ∈ [0 , θ ]. In view of (36) and the strict comparison property of E g,h , we deduce that for every0 ≤ s ≤ τ ′ E g,hs,τ ′ ( y τ ′ ) = y s . (37)Indeed, suppose that this is not true. Then the strict comparison property of E g,h would yield E g,h ,τ ′ ( y τ ′ ) = E g,h ,s ( E g,hs,τ ′ ( y τ ′ )) > E g,h ,s ( y s ) ≥ y , which would contradict (36).We now claim that for 0 ≤ s ≤ t ≤ τ ′ , we have that E g,hs,t ( y t ) = y s . To show this, we observe that(37) yields E g,ht,τ ′ ( y τ ′ ) = y t and thus E g,hs,t ( y t ) = E g,hs,t ( E g,ht,τ ′ ( y τ ′ )) = E g,hs,τ ′ ( y τ ′ ) = y s , where the last equality also comes from (37). We thus see that y is an E g,h -martingale on [0 , τ ′ ] andthus k τ ′ = 0. In particular, we have E g,h ,τ ′ ( y τ ′ ) = y and thus, using (36), we obtain y = E g,h ,τ ′ ( x τ ′ ) = E g,h ,τ ′ ( y τ ′ ) = E g,h ,τ ′ (cid:0) V τ ′ ( x − p, ψ ) (cid:1) . (38)By combining this equality with the inequality y τ ′ ≤ x τ ′ and the strict comparison property of E g,h ,we conclude that y τ ′ = x τ ′ . We have thus shown that if τ ′ is a rational exercise time, then conditions(i)–(ii) are valid.Let us show that τ h is a rational exercise time. From the definition of τ h and the right-continuityof y and x , we infer that y τ h = x τ h . Equality k τ h = 0 has been already established in Proposition3.5. Hence τ h satisfies conditions (i)–(ii) and thus it is one of the holder’s rational exercise timesand it is the earliest one, since y t < x t for all t ∈ [0 , τ h ).It remains to prove that ¯ τ h is the latest rational exercise time under an additional assumptionthat the process k is continuous so that k = k c . We need to show that y ¯ τ h = x ¯ τ h . For an arbitrary ε >
0, there exists δ ∈ [0 , ε ] such that k ¯ τ h + δ >
0. Since R T ( x t − y t ) dk t = 0, from the right-continuityof processes y and x and the inequality x ≥ y , we obtain the equality y ¯ τ h = x ¯ τ h . Since, obviously, k t = 0 for t ∈ [0 , ¯ τ h ), we also have k ¯ τ h = 0. This shows that ¯ τ h is one of the holder’s rationalexercise times. Moreover, it is the latest one since, if τ ′ ∈ T is such that P ( τ ′ > ¯ τ h ) >
0, then P ( k τ ′ > > k τ ′ = 0 cannot hold. Note, however, that if the continuity of k is not postulated, then it may happen that k ¯ τ h = 0 in which case ¯ τ h fails to be a rational exercisetime (for instance, such properties are always true if k = k d ). ✷ Remark 3.4.
From the proof of Theorem 3.4 (see, in particular, equation (38)), the inequality V τ ′ ( x − p, ψ ) ≥ x τ ′ and the strict comparison property of E g,h , we deduce that when the equality b p r,h ( x , C a ) = b p s,h ( x , C a ) holds, then for any rational exercise time given by Definition 3.6 we havethat V τ ( x − b p r,h , ψ ) = x τ , meaning that a rational exercise time is also a holder’s break-even time.It is also obvious that a holder’s break-even time is a rational exercise time. Thus when the equality b p r,h ( x , C a ) = b p s,h ( x , C a ) holds, then the inequality V τ ( x − b p r,h , ψ ) ≥ x τ in Definition 3.6 can bereplaced by the equality V τ ( x − b p r,h , ψ ) = x τ . Once again, it is worth noting that this observation isfully consistent with the standard definition of a rational exercise time for the holder of an Americanoption in a classical complete, linear market model, such as the Black and Scholes model. Acknowledgments
The research of T. Nie and M. Rutkowski was supported by the DVC Research Bridging SupportGrant
Non-linear Arbitrage Pricing of Multi-Agent Financial Games . 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] Aazizi, S. and Ouknine, Y.: Strong envelope and strong supermartingale: Application to re-flected backward stochastic differential equation. Working paper, 2016 (arXiv:1112.0255v2).[2] Baadi, B. and Ouknine, Y.: Reflected BSDEs when the obstacle is not right-continuous in ageneral filtration.
ALEA – Latin American Journal of Probability and Mathematical Statistics
14 (2017), 201–218.[3] Bayraktar, E. and Yao, S.: Quadratic reflected BSDEs with unbounded obstacles.
StochasticProcesses and their Applications
122 (2012), 1155–1203.[4] Bensoussan, A.: On the theory of option pricing.
Acta Applicandae Mathematicae
Probability, Uncertainty and Quantitative Risk
SIAM Journal on Financial Mathematics
Theory of Probability and its Applications
The Annals of Probability
The Annals of Probability
Electronic Journal of Probability
SIAM Journal on Financial Mathematics
ESAIM: Proceedings and Surveys (2018).[13] El Karoui, N.: Les aspects probabilistes du contrˆole stochastique. In
Lecture Notes in Mathemat-ics 876, Ecole d’Et´e de Probabilit´es de Saint-Flour IX, 1979 , P.-L. Hennequin (Ed.), Springer,Berlin, 1981, pp. 73–238.[14] El Karoui, N. and Huang, S. J.: A general result of existence and uniqueness of backwardstochastic differential equations. In
Backward Stochastic Differential Equations, Pitman Re-search Notes in Mathematics Series 364 , N. El Karoui and L. Mazliak (Eds.), Addison WesleyLongman Ltd, Harlow, Essex, 1997, pp. 27–36.[15] 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
Numerical Methods in Finance , L. C. G. Rogers and D. Talay (Eds.), CambridgeUniversity Press, Cambridge, 1997, pp. 215–231.[17] El Karoui, N., Peng, S., and Quenez, M. C.: Backward stochastic differential equations infinance.
Mathematical Finance onlinear Pricing of American Options
Lecture Notes in Mathematics 1656 , B. Biais et al. (Eds.), Springer, Berlin, 1997,pp. 191–246.[19] Essaky, E. H.: Reflected backward stochastic differential equation with jumps and RCLL ob-stacle.
Bulletin des Sciences Math´ematiques
The Annals of Applied Probability
27 (2017), 3153–3188.[21] Grigorova, M., Imkeller, P., Ouknine, Y., and Quenez, M. C.: Optimal stopping with f -expectations: the irregular case. Working paper, 2017 (hal-01403616v2).[22] 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.[23] Hamad`ene, S.: Reflected BSDEs with discontinuous barrier and application.
Stochastics andStochastics Reports
Theory ofProbability and its Applications
60 (2016), 263–280.[25] Jaillet, P., Lamberton, D., and Lapeyre, B.: Variational inequalities and the pricing of Americanoptions.
Acta Applicandae Mathematicae
Finance and Stochastics
Applied Mathematics and Optimization
Finance and Stochastics
Finance and Stochastics
ISRN Probability and Statistics (2013), ID856458,17 pages.[31] Kim, E., Nie, T., and Rutkowski, M.: Valuation and hedging of game options in nonlinearmodels. Working paper, 2018.[32] Klimsiak, T.: Reflected BSDEs with monotone generator.
Electronic Journal of Probability
Stochastic Processes and theirApplications
125 (2015), 4204–4241.[34] Klimsiak, T. and Rozkosz, A.: The early exercise premium representation for American optionson multiply assets.
Applied Mathematics and Optimization
73 (2016), 99-114.[35] Klimsiak, T., Rzymowski, M., and S lomi´nski, L.: Reflected BSDEs with regulated trajectories.Working paper, 2016 (arXiv:1608.08926v1).[36] Kobylanski, M. and Quenez, M. C.: Optimal stopping time problem in a general framework.
Electronic Journal of Probability
The Annals of Applied Probability
E. Kim, T. Nie and M. Rutkowski [38] Nie, T. and Rutkowski, M.: Fair bilateral prices in Bergman’s model with exogenous collater-alization.
International Journal of Theoretical and Applied Finance
18 (2015), 1550048.[39] 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.[40] Nie, T. and Rutkowski, M.: A BSDE approach to fair bilateral pricing under endogenouscollateralization.
Finance and Stochastics
20 (2016), 855–900.[41] Nie, T. and Rutkowski, M.: Fair bilateral prices under funding costs and exogenous collateral-ization.
Mathematical Finance
28 (2018), 621–655.[42] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In
Lecture Notes inMathematics 1856 , M. Frittelli and W. Runggaldier (Eds.), Springer, Berlin, 2004, pp. 165–253.[43] Peng, S.: Dynamically consistent nonlinear evaluations and expectations. Working paper, 2004(arXiv:0501415v1).[44] Rogers, L. C. G.: Monte Carlo valuation of American options.
Mathematical Finance