Arbitrage-free modeling under Knightian Uncertainty
aa r X i v : . [ q -f i n . M F ] A p r Arbitrage-free modeling under Knightian Uncertainty
M. Burzoni M. Maggis April 28, 2020
Abstract
We study the Fundamental Theorem of Asset Pricing for a general financial marketunder Knightian Uncertainty. We adopt a functional analytic approach which requiresneither specific assumptions on the class of priors P nor on the structure of the state space.Several aspects of modeling under Knightian Uncertainty are considered and analyzed. Weshow the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Riskand discuss its relation to the choice of an appropriate technical filtration. In an abstractsetup, we show that absence of arbitrage is equivalent to the existence of approximate martingale measures sharing the same polar set of P . We then specialize our results to adiscrete-time financial market in order to obtain martingale measures. Keywords : Knightian Uncertainty, Arbitrage Theory, First Fundamental Theorem of AssetPricing, quasi-sure analysis.
MSC (2010): primary 91B24, 91G99, 46N10 secondary 91G80
JEL Classification:
C02, G10, G13.
The mathematical modeling of financial markets is a challenging task initiated over a centuryago by Bachelier Bachelier (1900), who firstly observed how the oscillations of the prices onstock exchanges could be represented as the trajectories of the Brownian Motion. After themajor contributions by Black and Scholes (1973) and Merton (1973), an outbreak of sophis-ticated mathematical models for Finance was observed in the last decades in the scientificliterature. For any of such various models the absence of arbitrage is a foundational principle.According to this condition, it is not possible to make a positive gain without taking anyshortfall risk. This is not only a reasonable feature of the model but also a property whichis typically satisfied by real markets. Indeed, it is widely accepted that markets are efficient :even if an arbitrage opportunity occurred, it would soon vanish as the traders willing to ex-ploit it would cause a change in the underlying prices. A cornerstone result is the so-called Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX26GG, email: [email protected]. Part of the research work has been conducted at ETH Zürich,the ETH foundation is gratefully acknowledged. Part of the research work is supported by the Hooke ResearchFellowship from the University of Oxford. Department of Mathematics, University of Milan, Via Saldini 50, 20133 Milan, email:[email protected] (Ω , F , P ) is given. The role of P is essentially to establish the class of eventswhich are irrelevant for the model, the P -nullsets. Such an assumption is very much exposedto the so-called model risk , namely, the fact that the outputs produced by the model are sen-sitive to the choice of P and a wrong choice might lead to severe consequences. It is thereforenatural to wonder whether a FTAP can be established in a more robust setting by the meanof a suitable functional analytic approach.The quasi-sure setting consists in replacing the given probability measure P with a class ofprobability measures P aiming at capturing the model ambiguity that agents are facing. In thissetting only events which are irrelevant with respect to any of the considered priors are deemedimpossible. In the seminal paper Bouchard and Nutz (2015), the authors construct a discretetime framework, inspired by dynamic programming ideas, in order to prove a quasi-sure ver-sion of the FTAP. Such a framework has become standard in the Robust Finance literature andmany results can be obtained within the same setting (see, e.g., Bayraktar and Zhang (2016);Bayraktar and Zhou (2017); Blanchard and Carassus (2019) in discrete-time and Biagini et al.(2017) in continuous-time). An alternative pathwise setting has been considered in the liter-ature (see, e.g., Acciaio et al. (2016); Bartl et al. (2020, 2019); Burzoni et al. (2019, 2016);Dolinsky and Soner (2014); Hou and Obłój (2018); Obłój and Wiesel (2018); Riedel (2015)),where instead of a probabilistic formulation of the problem the authors work directly on theset of scenarios Ω .The first main result of the paper (Theorem 6) is an abstract version of the FTAP in ageneral quasi-sure setting and it is the content of Section 2. The novelty is that its proofrelies only on functional analytic arguments, therefore, we do not require a structure which isamenable to the use of measurable selection techniques, as customary in the aforementionedliterature. As in the classical case, where a reference measure is given, the advantage is thatboth discrete and continuous time models can be attacked in the same way since the problemreduces to show the weak-closure of the cone of superreplicable claims. Moreover, thesetechniques are typically applied in a multitude of other related problems (among others thesuper-hedging duality and utility maximization). We consider a notion of arbitrage that we call P - sensitive (see Definition 4 and the subsequent discussion). The peculiarity of this conditionis that every market participant agrees on the presence of such arbitrage opportunities but theymay well disagree on which strategy will realize it. These models cannot represent equilibriumprices for the underlying securities and they should be excluded. The sensitive versions ofarbitrage are equivalent to their quasi-sure versions in the frameworks of Bouchard and Nutz(2015) and Soner et al. (2011). In Section 4 we explain how the role of the chosen filtration iscrucial in order to recover the standard notions, in relation to the problem of aggregation of P -dependent strategies. This is a technical aspect of the chosen mathematical framework and,from an economic point of view, whether the considered market is viable or not should notdepend on that. To this extent the sensitive notions of arbitrage captures market inefficienciesbeyond the necessary mathematical technicalities.2tarting from the general FTAP we show how stronger results can be obtained by allowingfor more structure. In particular, several aspects of modeling under Knightian Uncertainty arediscussed throughout the paper. In Section 3 we analyze the discrete-time setting and showthe convergence of the approximate martingale measures of Theorem 6 to true martingalemeasures. This is proven in Theorem 11 and we illustrate one application in the context ofMartingale Optimal Transport (MOT). Finally, Section 5 contains all the proofs of the mainresults.We conclude this Introduction with the frequently used notation. Let Ω be a separable metric space and F the associated sigma algebra of Borel measurableevents. We let M be the class of probability measures on (Ω , F ) endowed with the usualweak topology σ ( M , C b ) , where C b is the space of continuous and bounded functions on Ω .Given P , P ⊂ M we define P ≪ P if sup P ∈P P ( A ) = 0 implies sup P ∈P P ( A ) = 0 . We say that P dominates P ; P ≈ P if both P ≪ P and P ≪ P holds. We say that P and P are equivalent. P ≪ P if there exists a P ∈ P such that P ≪ P .For a given P ⊂ M , we introduce the vector space of countably additive signed measuresdominated by P , namely ca ( P ) (see Maggis et al. (2018)). We shall denote by N the familyof polar sets , namely, N := { A ⊂ A ′ | A ′ ∈ F and P ( A ′ ) = 0 ∀ P ∈ P} . A statement is said to hold quasi surely ( q . s . ) if it holds outside a polar set. It is possibleto identify measurable functions which are q . s . equal and L will indicate the quotient space. L ∞ is the subspace of q . s . bounded functions, which we endow with the norm k X k ∞ := inf { m ∈ R | P ( {| X | > m } ) = 0 ∀ P ∈ P} . If no confusion arises we will denote the q . s . partial ordering by ≤ (resp. ≥ and = ), meaningthat for any X, Y ∈ L , X ≤ Y if and only if P ( { X > Y } ) = 0 for every P ∈ P . ( L ∞ , k · k ∞ ) endowed with the q . s . order ≤ is a Banach lattice. Throughout the text we will be given apositive random variable W ≥ , and work with the space L := { X ∈ L | X/W ∈ L ∞ } , paired with the norm k X k := k X/W k ∞ . We finally introduce L + , L and L ∞ + , as the subsetsof q . s . non-negative functions in L , L and L ∞ respectively. Given a set A ⊂ L , cl ∞ A willdenote the closure with respect to k · k ∞ of A ∩ L ∞ . Fix a measurable space (Ω , F ) and W ∈ L with W ≥ . The financial market is described, inan abstract form, by the set of financial contracts attainable at zero cost denoted by K ⊂ L .Throughout the text K is assumed to be a convex cone.3 efinition 1 Let
K ⊂ L be a convex cone. • k ∈ K is an arbitrage opportunity if k ∈ L \ { } ; • ξ ∈ L \ { } is a free lunch with vanishing risk if there exist c n ↓ and { k n } ⊂ K suchthat c n + k n ≥ ξ ;We denote by (NA) and (NFLVR) absence of arbitrage and free lunch with vanishing riskrespectively. We let K λ := K ∩ { X ∈ L | X ≥ − λW } for λ ≥ and define C := { X ∈ L | X ≤ k for some k ∈ K} , (1) C λ := { X ∈ L | X ≤ k for some k ∈ K λ } , (2)where we recall that all inequalities are meant to hold q . s . . Remark 2
In the classical dominated case (
P ≪ P for some P ∈ M ), K is the class ofstochastic integrals of admissible strategies. The use of a random lower bound in the admis-sibility condition is not new and was used for instance in Biagini and Frittelli (2008). Analternative possible choice for W is W = 1 for which K λ is the set of contracts bounded frombelow by − λ , a typical constraint for continuous time models which excludes doubling strate-gies. In Section 3 we show that, in discrete time, a suitable choice for W identifies the class ofadmissible bounded strategies. Under uncertainty, the stochastic integral can be defined in thesame way for the discrete time case, since it amounts to a finite sum; in continuous time, itrequires a different construction (see e.g. Dolinsky and Soner (2014); Perkowski and Prömel(2016); Soner et al. (2011); Vovk (2012)). Both C and C λ are convex and monotone sets containing , in addition C is also a cone.They represent the class of claims which can be super-replicated at zero initial cost by meanof attainable payoffs in K and K λ respectively.As in the classical literature, we can reformulate the no-arbitrage conditions in terms ofthe cone C , i.e., (NA) ⇐⇒ C ∩ L ∞ + = { } (NFLVR) ⇐⇒ cl ∞ ( C ) ∩ L ∞ + = { } , where cl ∞ denotes the closure with respect to k · k ∞ of C ∩ L ∞ . In the context of KnightianUncertainty this straightforward generalization of the classical concepts might not be sufficientfor deriving a general no-arbitrage theory. Sensitivity: from dominated to non-dominated frameworks.
The notion of sensitiv-ity was introduced in Maggis et al. (2018) and, as we discuss below, it should be interpretedin terms of aggregation of trading strategies with respect to the different measures in the set P . For P ≪ P , we define the linear (projection) map j P : L → L P X [ X ] P (3)where [ X ] P is the P -equivalence class of X in L P . We say that that
A ⊂ L is monotone if Y ≤ X and X ∈ A implies Y ∈ A . efinition 3 A set
A ⊂ L is called sensitive if there exists a family R ⊂ M with R ≪ P such that A = \ P ∈R j − P ( j P ( A )) . The set R will be called reduction set for A . We will typically use P itself as a reduction set. As the space L does only depend on thepolar sets N we could alternative choose any P ′ ≈ P and we will occasionally do so.To better understand the previous definition let us consider for a moment the dominated setting, namely, suppose there exists a reference probability P equivalent to the family P . Inthis case C is composed of equivalence classes with respect to P and contains, in particular, allthe indicators of P -null sets. It is well know that, in a discrete framework, (NA) is equivalentto the existence of martingale measures for S or, stated otherwise, C ∩ ( L ∞ P ) + = { } ⇐⇒ C ≈ P, (4)where the set C := { Q ∈ M | E Q [ X ] ≤ ∀ X ∈ C} is necessarily composed of measureswhich are absolutely continuous with respect to P . The no-arbitrage condition (4) could betrivially rewritten as j P ( C ) ∩ ( L ∞ P ) + = { } since, in the dominated case, the map j P is obviouslythe identity. Let us now take into account a class of non-dominated probabilities P ⊂ M .In order to embed j P ( C ) in L , for any P ∈ P , we need to consider all the elements in L whichcoincide P -a.s. with an element of j P ( C ) . More precisely, we introduce the set j − P ( j P ( C )) andthe no-arbitrage condition with respect to a single P would read as j − P ( j P ( C )) ∩ L ∞ + = { } .Since we have a whole class of non-dominated P , representing the uncertainty of the model,we need to consider the set e C = \ P ∈P j − P ( j P ( C )) . (5)We stress that any j − P ( j P ( C )) is a subset of L , i.e., it is a collection of quasi-sure equivalenceclasses of contingent claims. To understand how the set e C is related to the probabilistic modelsin P we can write, more explicitly, e C = { X ∈ L | ∀ P ∈ P , ∃ k P ∈ K s.t. X ≤ k P P -a.s. } The set e C induces the following no-arbitrage conditions. Definition 4
We say that it holds: (sNA) ˙ ⇐⇒ e C ∩ L ∞ + = { } (sNFLVR) ˙ ⇐⇒ cl ∞ ( e C ) ∩ L ∞ + = { } , where we have emphasized in the acronyms that these are the sensitive versions of the previousnotions. When (sNA) is violated there exists a non-negative contingent claim for which it is not possibleto assign a reasonable price. On the one hand, X is non-negative quasi-surely , thus, its pricecannot be less or equal than zero as it would create an obvious arbitrage. On the other hand a If P ≪ P ′ for some P ′ ∈ M , the Halmos Savage Lemma (see Halmos and Savage (1949)) implies thatthere exists a probability P which is equivalent to P . X disagrees with the output of any plausible model . Indeed, since X ∈ e C , X can be super-hedged at zero cost under any of the priors and it will not be traded at a positiveprice. The same claim X induces arbitrage opportunities in every model P . This means thatthe inefficiency of the market is identified by every market participant, nevertheless, theymight well disagree which strategy should be implemented in order to exploit an arbitrage .Arguing as in the classical case, these situations would trigger a change in the prices of theunderlying assets which will make such opportunities disappear. We stress that (sNA) doesnot imply that the classical no arbitrage condition holds under any of the P ∈ P .As pointed out above, if P is dominated, we clearly have e C = C (i.e. C is sensitive). How-ever, as we demonstrate below this is not always the case under Knightian uncertainty, unlessthe framework is chosen carefully. We show that the discrepancy e C 6 = C can be often resolvedby extending the filtration in an appropriate technical way, which allows for aggregation of P -dependent strategies (see Section 4) and for which (sNA) ⇔ (NA) . Since the aim of thepaper is to provide a general FTAP which is not tailor made to any specific underlying settingwe refrain to assume that C is sensitive and continue to work with e C . The sensitive version of the FTAP.
We first introduce the class of dual elements. Recallthat C and C λ are defined in (1) and (2) for λ ≥ . Definition 5 An approximate separating class is a sequence of probabilities Q := { Q n } n ∈ N such that there exists P ∈ P with Q ≪ P and, for any n ∈ N , E Q n [ X ] ≤ n ∀ X ∈ C n , (6) We denote by Q app the collection of approximate separating classes. We now state the main result of the section. To this end recall that N represents the classof polar sets for P and that a set A ⊂ L is Fatou-closed if for any k · k -bounded sequence { X n } n ∈ N ⊂ A , X n → X q . s . , we have X ∈ A . Theorem 6
Suppose that under (sNFLVR) C is Fatou closed. The following are equivalent:1. (sNFLVR) Q app ≈ P . Moreover, ∀ A ∈ F \ N , ∃ δ > , Q ∈ Q app such that inf Q ∈Q Q ( A ) ≥ δ . For discrete time financial market models, (sNFLVR) guarantees that C is Fatou closed (seeLemma 8 below), as a consequence, we do not need such an assumption in the subsequentTheorem 9 and 11. Whether the same result holds for continuous time models is an interestingquestion which goes beyond the scope of this paper and is left for future investigations. Inter-estingly, in the recent paper Cheridito et al. (ming), it is shown that a general MOT dualityholds only if the set of attainable payoffs is Fatou closed.We provide the proof of Theorem 6 in Section 5. One of the main technical point is toshow that the sensitive version of any C λ is closed in an appropriate weak topology, which is This situation is reminiscent of the example of the two call options with different strikes but same pricegiven in Davis and Hobson (2007). With a slight abuse of notation, Q app ≈ P means that the whole collection of probabilities belonging tosome approximate separating class is equivalent to P . C λ contains a sufficiently rich class of dynamicstrategies in the underlying process S which allows to identify martingale measures. The abovetheorem essentially says that (sNFLVR) is equivalent to the fact that for any non-polar set A ,we can find approximate martingale measures for S which assign positive probability to A (seealso (9) below). In particular this implies that, under (sNFLVR) , the class of approximatemartingale measures is non empty and equivalent to P . Remark 7
Given
Q ∈ Q app , it is possible to define a super-additive functional ψ ( · ) :=inf Q ∈Q E Q [ · ] in the spirit of Aliprantis et al. (2001) and Aliprantis and Tourky (2002), which,by (6) , is a non-linear separator of the cone C . In the context of Knightian Uncertainty, non-linearity arises also for pricing rules related to economic equilibrium and absence of arbitrage(see e.g. Beissner and Riedel (2019); Burzoni et al. (2017)). We call Theorem 6 an “abstract” version of the FTAP since it is obtained in a general setupand its implications can be strenghtened if we are willing to choose a more specific setting oradopt stronger assumptions. More precisely:1. From a technical point of view, a desirable property is e C = C which can hold underthe (NA) assumption (resp. (NFLVR) ), provided a suitable underlying structure. Bydefinition, e C = C automatically implies (NA) ⇔ (sNA) (resp. (sNFLVR) ⇔ (NFLVR) ).As discussed above, such a situation occurs when P is dominated. We will explainin Section 4 that this is related to the choice of the filtration and it holds true inthe framework of Bouchard and Nutz (2015), where, in addition, these four notions ofarbitrage are all equivalent.2. The approximate separating classes of Theorem 6 can be used to obtain linear pricingfunctionals as “true” martingale measures for a given underlying process. In Section 3 wewill show that this is possible under some further assumptions and illustrate the resultin a discrete-time MOT framework.3. When both the two points (1. and 2.) above are fulfilled, Theorem 6 in discrete timehas the more familiar form: (NA) ⇐⇒ Q mtg ≈ P , where Q mtg := { Q ≪ P | E Q [ k ] = 0 ∀ k ∈ K} .4. (On No Free Lunch) Mathematically, one could obtain separating measures using theHahn Banach Theorem, under the No Free Lunch (NFL) condition: cl σ ( C ) ∩ L ∞ + = { } , where cl σ denotes the closure with respect to σ ( L ∞ , ca ( P )) -topology of C . As in thedominated case, it is not a priori clear how limit points in cl σ ( C ) are related to thepayoffs of implementable strategies, thus, a clear economic interpretation is missing. In this section we further analyze the discrete-time setting and show how to obtain martingalemeasures from Theorem 6. Let T ∈ N , and I := { , ..., T } . The price process is given byan R d -valued stochastic process S = ( S t ) t ∈ I with S jt ∈ L for every t ∈ I, j = 1 , . . . , d , andwe also assume the existence of a numeraire asset S t = 1 for all t ∈ I . Moreover, we fix afiltration F := {F t } t ∈ I such that the process S is F -adapted. A finite set of F -measurable7ptions Φ = ( φ , . . . , φ m ) ( φ i ∈ L for every i ) is available for static trading and, withoutloss of generality, we assume their initial price to be . An admissible semi-static strategy isa couple ( H, h ) where H is an R d -valued, F -predictable stochastic process with H jt ∈ L forevery t ∈ I, j = 1 , . . . , d and h ∈ R m . The final payoff is ( H ◦ S ) T + h · Φ ∈ L where thestochastic integral is defined as ( H ◦ S ) t := t X k =1 d X j =1 H jk ( S jk − S jk − ) , t ∈ I , with ( H ◦ S ) = 0 . We denote by H the class of semi-static admissible strategies with zeroinitial cost.We choose W := 1 + T X t =1 d X j =1 | S jt | + m X i =1 | φ i | . The sets C and C λ takes the following explicit form C = { X ∈ L | X ≤ ( H ◦ S ) T + h · Φ for some ( H, h ) ∈ H} , (7) C λ = { X ∈ L | X ≤ ( H ◦ S ) T + h · Φ for some ( H, h ) ∈ H λ } , (8)where H λ := { ( H, h ) ∈ H | ( H ◦ S ) T + h · Φ ≥ − λW } for λ ≥ . Note that, in particular, ∪ λ ≥ H λ contains any bounded strategy.Recall that a set A ⊂ L is closed with respect to q . s . convergence if for any sequence { X n } n ∈ N , X n → X q . s . implies X ∈ A . Lemma 8
Under (NA) the convex set C is closed with respect ot q . s . convergence and henceboth C and C λ are Fatou closed. Proof.
A direct application of (Bouchard and Nutz, 2015, Remark 2.1 and Theorem 2.2)guarantees that C is closed with respect to q . s . convergence. Since C is closed with respect to q . s . convergence, it is Fatou closed. Consider now a k · k -bounded sequence { X n } ⊂ C λ suchthat X n → X , q . s . for some X ∈ L . By definition of C λ , there exists ( H n , h n ) ∈ H λ such that X n ≤ ( H n ◦ S ) T + h n · Φ , from which we deduce that ( − λW ) ∨ X n ∈ C λ ⊂ C . Moreover, fromthe closure of C with respect to q . s . convergence, the limit ( − λW ) ∨ X =: ˜ X belongs to C .By definition of C , there exists ( H, h ) ∈ H such that ˜ X ≤ ( H ◦ S ) T + h · Φ and, necessarily, ( H, h ) ∈ H λ . From X ≤ ˜ X and the monotonicity of C λ , X ∈ C λ .Using Lemma 8 we can specialize Theorem 6 to the present discrete time setup withoutthe Fatou closure assumption. Theorem 9
The following are equivalent:1. (sNFLVR) Q app ≈ P . Moreover, ∀ A ∈ F \ N , ∃ δ > , Q ∈ Q app such that inf Q ∈Q Q ( A ) ≥ δ .The two conditions are further equivalent to (NA) if, in addition, C is sensitive under (NA) .
8e call measures in Q app approximate martingale measures . Indeed, for every A ∈ F k − the one-step strategy H = ( H t ) t ∈ I with H t ( ω ) = 1 A ( ω )1 { k } ( t ) for every ω ∈ Ω and t ∈ I ,satisfies ( H, ∈ H λ . Similarly (0 , ± e i ) ∈ H λ for every i = 1 , . . . , m , where { e i } mi =1 denotesthe canonical basis of R m . Their final payoffs are thus contained in C λ , from which, for every n ∈ N and { Q n } ∈ Q app , it holds | E Q n [ φ i ] | ≤ n , | E Q n [1 A ( S jk − S jk − )] | ≤ n , (9)for any i = 1 , . . . , m , j = 1 , . . . , d , A ∈ F k − , k = 1 , . . . , T .We now show that, under some additional weak assumptions, Theorem 6 implies theexistence of true martingale measures. Assumption 10 (Ω , m ) is a Polish space with respect to a metric m .(i) For any t ∈ I , S t : Ω → R d + is a continuous function ;(ii) For any P ∈ P there exists a compact set K P such that P ( K P ) = 1 .(iii) F := {F t } t ∈I is the natural filtration generated by S . Note that the previous conditions are not restrictive. If S is only Borel measurable, by(Aliprantis and Border, 2006, Theorem 4.59) there exists a Polish topology τ on Ω such thatthe Borel sigma algebra is the same and the process S is τ -continuous. Thus, assumption(i) can be made without loss of generality. Assumption (ii) can be easily fulfilled when theclass of priors P has the only scope of fixing the polar sets. For Ω a Polish space, the class R := { P ( · | K ) | K ⊂ Ω compact , P ∈ P} satisfies (ii) and R ≈ P . Indeed,
R ≪ P is trivial.If A ∈ F \ N , there exists P ∈ P such that P ( A ) > . By (Aliprantis and Border, 2006,Theorem 12.7), we find a compact set K ⊂ A such that P ( K ) > , from which P ( A | K ) > and P ≪ R .Let now Q := { Q ∈ Q | Q ∈ Q app , S is a ( Q, F ) -martingale with E Q [Φ] = 0 } , where theclosure Q is taken in the σ ( M , C b ) sense. Theorem 11
Under Assumption 10, the following are equivalent:1. (sNFLVR) ;2. Q app ≈ P and P ≪ Q . Remark 12
Theorem 11 cannot guarantee that the limiting measures Q satisfies Q ≪ P , asweak limits do not, in general, preserve absolute continuity with respect to a measure. A martingale optimal transport framework.
Set
Ω = R d × T + and let S t ( ω ) = ω t be thecanonical process. We assume that, for any of the assets S j , a certain finite number N ( j ) of call options are available for semi-static trading with payoffs ( S jT − k ji ) + for some k ji > and with prices c ji , for i = 1 , . . . N ( j ) . We assume that c ji ≥ , otherwise there is an obviousarbitrage opportunity, and we also assume that for a sufficiently large strike price the optionsare traded at zero price; we model this by setting c jN ( j ) = 0 . The corresponding set of optionswith zero prices is given by Φ = { ( S jT − k ji ) + − c j | j = 1 , . . . , d, i = 1 , . . . , N ( j ) } . FollowingDavis and Hobson (2007) we construct the support function R j as the maximal convex non-increasing function such that R j ( k ji ) ≤ c ji . As R j is λ -a.s. twice differentiable, following the More precisely is q . s . equal to a continuous function λ is the Lebesgue measure on R . µ j on R as dµ j /dλ = ( R j ) ′′ . Note that, from the assumption c jN ( j ) = 0 , it follows that R j ( x ) = 0 for all x ≥ k jN ( j ) and, therefore, µ j has compact support.Let µ := ⊗ dj =1 µ j the product measure of { µ j } dj =1 on R d . Let K be the compact supportof µ . We consider a family of probability measures P satisfying P ⊂ { P ∈ M | P T ∼ µ } , (10)where P T denotes the marginal of P on the last component of Ω . The interpretation is thefollowing. If we denote by Q the (unknown) set of measures which are used in the marketto price the options Φ , µ represents the (approximation) of the distribution of S T under any Q ∈ Q . Any probability measure equivalent to Q defines the same null-events and shouldbe considered as a plausible model. Therefore, the only constraint that we can deduce frommarket data is that the distribution of S T under P ∈ P should be equivalent to µ . Notethat, differently from the standard martingale optimal transport setup, we are not assumingto know, in addition, all the marginals at intermediate time. This case could be easily incor-porated.Denote by K T the T -fold product of the compact set K . Lemma 13
Under (NA) , P ( K T ) = 1 for every P ∈ P . Proof.
For every n ∈ N , t = 1 , . . . , T − , consider the closed-valued multifunction Ψ t,n ( ω ) := (cid:26) H ∈ R d | H · ( x − ω t ) ≥ n ∀ x ∈ K \ { ω t } (cid:27) , ω ∈ Ω . The domain of Ψ t,n is defined as dom (Ψ t,n ) := { ω ∈ Ω | Ψ t,n ( ω ) = ∅} . The compactness of K and the hyperplane separating theorem implies that ∪ n ∈ N dom (Ψ t,n ) = { ω ∈ Ω | S t ( ω ) / ∈ K } .From (Burzoni et al., 2016, Lemma A.7), Ψ t,n is F St -measurable, thus, it admits a measurableselector ψ t,n on its domain which we extend to the whole Ω by setting ψ t,n = 0 on thecomplementary set. By letting H t := P ∞ n =1 ψ t,n { ψ t,n − =0 } with ψ t, = 0 , we obtain H t · ( S T − S t ) ≥ with strict positivity on { ω ∈ Ω | S t ( ω ) / ∈ K } . If now, by contradiction, P ( K T ) < ,there exists ≤ t ≤ T − such that P ( S t ∈ K c ) > . Thus, H t as above provides an arbitrageopportunity.We could deduce the following Proposition 14
Under the assumption of this paragraph, the following are equivalent:1. (sNFLVR) ;2. Q app ≈ P and P ≪ Q . Proof.
Assumption 10 is satisfied in the framework of this subsection. The result followsdirectly from Theorem 11.
Remark 15
If, in addition, the class P is chosen with the structure of Bouchard and Nutz(2015), from Theorem 17 below, the above are further equivalent to (NA) . Remark 16
In the classical MOT framework it is well know that Strassen’s Theorem ensurethat the set of martingale measures with prescribed marginals is non empty if and only if themarginals are in convex order. This is typically taken as a no-arbitrage condition. The abovetheorem explains such a no-arbitrage condition from a different point of view. The role of filtrations in the aggregation process
In this section we depict two well known examples, borrowed from the recent literature, inwhich the cone C turns out to be sensitive under the no-arbitrage condition. We show howsensitivity is related to the possibility of obtaining an aggregation property for superhedgingstrategies. In both examples the filtration will play a crucial role and will be an opportuneenlargement of the natural filtration, which will not affect the structure of the set of martingalemeasures for the discounted price process, calibrated on liquid options. We stress that ourmain goal is to explain some significant features of the models rather than recovering thesewell established results. Nevertheless, Theorem 17 below is a direct proof of the sensitivityof C which does not rely on the results of Bouchard and Nutz (2015) and which provides newinsights on the properties of the superhedging functional in that framework. The product structure of Bouchard and Nutz (2015).
Starting from the frameworkof Section 3 and letting W = 1 , we further assume the following requirements. The underlyingspace Ω = Ω T is a T -fold product of a Polish space Ω and Ω t := Ω t . For every t ∈ I , F t isthe universal completion of the Borel sigma-algebra B Ω t , defined as \ P ∈M (Ω t ) B Ω t ∨ N Pt , where N Pt = { N ⊂ A ∈ B Ω t | P ( A ) = 0 } .Fix t ∈ I , the event ω ∈ Ω t can be seen as a path observed up to time t and P t ( ω ) ⊂ M (Ω ) is a prescribed convex set of priors, on the node ( t, ω ) . It is assumed thatgraph ( P t ) = { ( ω, P ) | ω ∈ Ω t , P ∈ P t ( ω ) } is analytic, thus, it admits a universally measurable selector P t : Ω t → M (Ω ) : this allowsto introduce the set of multiperiod probabilistic models (priors) as P := { P ⊗ P ⊗ . . . ⊗ P T − | P t ( · ) ∈ P t ( · ) , t = 0 , . . . , T − } . We set Q := { Q ≪ P | S is an F -martingale under Q and E Q [Φ] = 0 } . For simplicity we assume in the following that
Φ = 0 . In this specific framework the followingFTAP was proved in Bouchard and Nutz (2015). (NA) holds if and only if P and Q share the same polar sets N .Our aim is to establish the sensitivity of the cone C by showing that the P - q . s . superhedgingprice is the supremum of the P -a.s. superhedging price with P varying in P . Theorem 17
Consider the measurable space (Ω , F T ) and the class P as described above.Then under (NA) we have C = \ P ∈P j − P ( j P ( C )) and the four notions (NA) , (sNA) , (NFLVR) and (sNFLVR) are all equivalent. Π( X ) the quasi-suresuperhedging price of an upper semianalytic function X : Ω → R (see (Bertsekas and Shreve,2007, Chapter 7.7) for more on semianalytic functions). For t = 0 , . . . , T − , let ∆ S t +1 := S t +1 − S t and for a function f : Ω t × Ω → R n , the notation f ( ω ; · ) indicates that f isconsidered as a function on Ω with the first coordinate fixed. Let Π T = X and define, bybackward iteration, Π t ( ω ) := inf { x ∈ R | ∃ H ∈ R d s.t. x + H · ∆ S t +1 ( ω ; · ) ≥ Π t +1 ( ω ; · ) P t ( ω ) - q . s . } , for ω ∈ Ω t . We refer to Π t as the conditional superhedging price of X at time t , in particular Π = Π( X ) . In the same spirit, we define π t ( ω, P ) := inf { x ∈ R | ∃ H ∈ R d s.t. x + H · ∆ S t +1 ( ω ; · ) ≥ Π t +1 ( ω ; · ) P -a.s. } , for ω ∈ Ω t . We aim at showing that under (NA) there exists a P -polar set N such that thefollowing holds:1. Π t ( ω ) = sup P ∈P t ( ω ) π t ( ω, P ) , for every ω ∈ N c ;2. Π t : Ω t → R is upper semianalytic;3. for any ε > , there exists a universally measurable kernel P ε : Ω t → M (Ω ) such that P ε ( ω ) ∈ P t ( ω ) and π t ( ω, P εt ( ω )) ≥ Π t ( ω ) − ε , for every ω ∈ N c .The first property shows that the quasi-sure superhedging price for a given X is the worstcase among all the P -a.s superhedging prices for X . The other two properties guarantee thatit is possible to construct models in P , by means of an appropriate measurable selection of P t , for which the almost-sure superhedging price is arbitrarily close to Π( X ) . Remark 18
Let
P ⊂ M and X ∈ L . For any P ∈ P , the measure dP ′ dP = c | X | , with c anormalizing constant, is equivalent to P and E P ′ [ | X | ] < ∞ . Thus, P ≈ ˜ P where ˜ P = { P ≪ P | E P [ | X | ] < ∞} . Now we show that inf { x ∈ R | x ≥ X P - q . s . } = sup P ∈ ˜ P E P [ X ] . Indeed, if the l.h.s. is infinite the inequality ≥ is trivial. Otherwise, if x is such that x ≥ X P q . s . then x ≥ E P [ X ] for any P ≪ P ′ with E P [ | X | ] < ∞ and P ′ ∈ P ; the latter exists fromthe first observation above. The inequality ≥ follows. Let M be the value of the l.h.s. above(or an arbitrary large M if it is not finite). For any ε > , there exists P ′ ∈ P such that theset A := { M − ε < X } satisfies P ′ ( A ) > . Take P ′′ ∼ P ′ such that E P ′′ [ | X | ] < ∞ as aboveand note that it still holds P ′′ ( A ) > . Define the probability P ( · ) := P ′′ ( · | A ) which satisfies P ≪ P ′′ ∼ P ′ and M − ε < E P [ X ] . From ε (and M in the infinite case) being arbitrary, theinequality ≤ follows. We start by showing the measurability properties.
Proposition 19
For any t ∈ I , Π t : Ω t → R is upper semianalytic. roof. We proceed by backward iteration. For t = T , Π T = X which is upper semianalyticby assumption. Assume the same is true up to t + 1 . Using Remark 18, we first rewrite theconditional superhedging price as follows, Π t ( ω ) = inf H ∈ R d inf { x ∈ R | x ≥ Π t +1 ( ω ; · ) − H · ∆ S t +1 ( ω ; · ) P t ( ω ) - q . s . } = inf H ∈ R d sup P ∈ ˜ P t ( ω ) E P [Π t +1 ( ω ; · ) − H · ∆ S t +1 ( ω ; · )] , where ˜ P t ( ω ) := { P ≪ P t ( ω ) | E P [ | ∆ S t +1 ( ω ; · ) | ] < ∞} and where we recall that E P [Π t +1 ( ω ; · )] = R Ω Π t +1 ( ω ; ˜ ω ) dP (˜ ω ) (similarly for the other term). By an application of the minimax theorem(see e.g. (Terkelsen, 1972, Corollary 2)), we can rewrite Π t ( ω ) = inf n ∈ N inf H ∈ B n (0) sup P ∈ ˜ P t ( ω ) E P [Π t +1 ( ω ; · ) − H · ∆ S t +1 ( ω ; · )]= inf n ∈ N sup P ∈ ˜ P t ( ω ) inf H ∈ B n (0) E P [Π t +1 ( ω ; · ) − H · ∆ S t +1 ( ω ; · )] , where B n (0) is the closed (compact) ball in R d of radius n centered in . Indeed, the abovefunction of ( H, P ) is affine in the first variable (hence convex and continuous) and linear(hence concave) in the second one. We now show that the functions f n : Ω t × M (Ω ) → R defined as f n ( ω, P ) = E P [Π t +1 ( ω ; · )] + inf H ∈ B n (0) g ( ω, P, H ) , (11) g ( ω, P, H ) := − H · E P [∆ S t +1 ( ω ; · )] , are upper semianalytic. We first observe that g is continuous in H with ( ω, P ) fixed and g is Borel measurable in ( ω, P ) with H fixed, as a consequence of (Bertsekas and Shreve,2007, Proposition 7.26 and 7.29). In other words g is a Carathéodory function, which isa particular case of normal integrand (see (Rockafellar and Wets, 1998, Definition 14.27))By (Rockafellar and Wets, 1998, Theorem 14.37), ( ω, P ) inf H ∈ B n (0) g ( ω, P, H ) is Borelmeasurable (in particular upper semianalytic). Moreover, since Π t +1 is upper semianalyticby the inductive assumption, the map ( ω, P ) E P [Π t +1 ( ω ; · )] is also upper semianalyticby (Bertsekas and Shreve, 2007, Proposition 7.26 and 7.48). We conclude that f n is uppersemianalytic as the sum of upper semianalytic functions.Define now the Borel-measurable function φ : Ω × M (Ω ) × M (Ω ) R ∪ { + ∞} as φ ( ω, P, P ′ ) := E P [ dP ′ dP ] if P ′ ≪ P and + ∞ otherwise. Consider the set D ⊂ Ω t × M (Ω ) defined by D := { ( ω, P ) | ω ∈ Ω t , P ≪ P t ( ω ) , E P [ | ∆ S t +1 ( ω ; · ) | ] < ∞} . From the definition of φ we have φ ( ω, P, P ′ ) = 1 if and only if P ′ ≪ P and therefore D is theprojection on the first and the third components of the analytic set (cid:0) graph( P t ) × M (Ω ) (cid:1) ∩ φ − (1) intersected with the Borel set of probability measures for which ∆ S t +1 is integrable.We deduce that D is also analytic (see also (Bouchard and Nutz, 2015, Lemma 4.11)).Clearly sup P ≪ P t ( ω ) f n ( ω, P ) = sup P ∈ D ω f n ( ω, P ) with D ω = { P ∈ M (Ω ) | ( ω, P ) ∈ D } and by (Bertsekas and Shreve, 2007, Proposition 7.47), ω sup P ≪ P t ( ω ) f n ( ω, P ) is uppersemianalytic. Finally, the class of upper semianalytic functions is closed under countableinfimum by (Bertsekas and Shreve, 2007, Lemma 7.30 (2)) and the result follows.In exactly the same way we can show the following.13 emma 20 The map ψ : Ω → R , defined as ψ ( ω ) = sup P ∈P t ( ω ) π t ( ω, P ) is upper semiana-lytic. Moreover, for any ε > , there exists a universally measurable kernel P εt : Ω t → M (Ω ) such that P εt ( ω ) ∈ P t ( ω ) and π t ( ω, P εt ( ω )) ≥ ψ ( ω ) − ε on ψ < ∞ and π t ( ω, P εt ( ω )) ≥ /ε on ψ = ∞ . Proof.
As in the proof of Proposition 19, the function f n in (11) is upper semianalyticfor every n ∈ N . Note that π t ( ω, P ) = inf n ∈ N sup P ′ ≪ P f n ( ω, P ) so that, π t is again uppersemianalytic. Using the fact that graph( P t ) is analytic we deduce that ψ is upper semianalyticand the existence of a universally measurable ε -optimizer from (Bertsekas and Shreve, 2007,Proposition 7.50).Given the above measurability properties we can now focus on the pointwise property 1.above. Proposition 21
Assume (NA) and X ∈ L ∞ . There exists a P -polar set N such that Π t ( ω ) =sup P ∈P t ( ω ) π t ( ω, P ) , for every t ∈ I and ω ∈ N c . Proof.
The inequality ≥ is trivial so we only need to show the converse. From (NA) , thelocal condition N A ( P t ( ω )) holds for any ω outside a polar set N (see (Bouchard and Nutz,2015, Lemma 4.6)) and we can assume Π t ( ω ) < ∞ as X is q . s . bounded. Fix ω ∈ N c and t ∈ I . If Π t ( ω ) = −∞ the equality follows so we also assume Π t ( ω ) > −∞ .Without loss of generality we assume that the random vector S t +1 is composed only ofnon-redundant asset. Let ˇΠ t ( ω ) = sup { x ∈ R | ∃ H ∈ R d s.t. x + H · ∆ S t +1 ( ω ; · ) ≤ Π t +1 ( ω ; · ) P t ( ω ) - q . s . } , for ω ∈ Ω t , be the subhedging price. We can distinguish two cases: ˇΠ t ( ω ) < Π t ( ω ) or ˇΠ t ( ω ) = Π t ( ω ) . In the first case, for any x ∈ ( ˇΠ t ( ω ) , Π t ( ω )) we have that the local no arbitragecondition holds for the extended one period market Y = 0 , Y := [∆ S t +1 ; − Π t +1 + x ] . Indeed,suppose [ H, h ] ∈ R d +1 satisfies H · ∆ S t +1 + h ( − Π t +1 + x ) ≥ . If h = 0 , N A ( P t ( ω )) andthe non-redundancy imply H = 0 . If h = 0 , dividing both sides by h we see that H/h is either a sub or a super-hedging strategy which is not possible by the choice of x . Since N A ( P t ( ω )) holds for the one-period market, (Bayraktar and Zhou, 2017, Lemma 2.7) impliesthat there exists P ′ ∈ P t ( ω ) such that N A ( P ′ ) holds and for which the assets are still nonredundant. For the same reason as above, a P ′ -a.s. superhedging strategy with initial price x would be a P -a.s. arbitrage in the extended market, which is excluded. Thus, π t ( ω, P ′ ) ≥ x and sup P ∈P t ( ω ) π t ( ω, P ) ≥ x . By taking the supremum for x ∈ ( ˇΠ t ( ω ) , Π t ( ω )) , the desiredinequality follows.Finally, when ˇΠ t ( ω ) = Π t ( ω ) < ∞ , we have that for some H, ˇ H ∈ R d , Π t ( ω ) + H · ∆ S t +1 ≥ Π t +1 ≥ ˇΠ t +1 ≥ Π t ( ω ) + ˇ H · ∆ S t +1 , P t ( ω ) - q . s .. By the local no arbitrage condition H = ˇ H and Π t +1 is P t ( ω ) - q . s . replicable. Similarly asbefore the local no arbitrage condition holds for the extended one period market ( Y , Y ) .Applying again (Bayraktar and Zhou, 2017, Lemma 2.7) we deduce that there exists P ′ ∈P t ( ω ) such that π t ( ω, P ′ ) = Π t ( ω ) . This implies the desired inequality.We can now conclude the proof of the sensitivity of C . Proof of Theorem 17.
Let X ∈ ˜ C , namely, the P -a.s. superhedging price Π P ( X ) isnon-positive, for any P ∈ P . For any ε > , consider the probability measure P ε := P ε ⊗ · · · ⊗ P εt ⊗ · · · ⊗ P εT , { P εt } from Lemma20. By construction P ε ∈ P and, by Proposition 21, Π( X ) ≤ Π P ε ( X ) + εT . As X ∈ ˜ C , Π P ε ( X ) ≤ and since ε is arbitrary, we conclude Π( X ) ≤ . Remark 22
The previous results reads as follows: fix X ∈ L ∞ and assume that for every P ∈ P we find H P ∈ H such that g ≤ ( H P ◦ S ) T , P -a.s where g is a representative of j P ( X ) . The strategy depends on P but not on the representative g ∈ j P ( X ) . The equality C = T P ∈P j − P ◦ j P ( C ) guarantees that in this case there exists a strategy H ∈ H which isindependent on P ∈ P such that g ≤ ( H ◦ S ) T , P -a.s for any P ∈ P , where g is anyrepresentative of X . The use of the universal filtration is crucial and guarantees the rightmeasurability framework for the proof of these results. Remark 23 (Pointwise framework)
Theorem 17 can be obtained in the pointiwise setupproposed by Burzoni et al. (2019) using the superhedging duality. Indeed once the superhedgingduality is obtained we can automatically deduce that sensitivity of the cone C . Also in this caseone needs to extend the natural filtration in an opportune way in order to obtain an aggregationresult for superhedging strategies. Quasi-sure aggregation in continuous time.
The second case is an example of non-dominated volatility uncertainty (see e.g. Denis and Martini (2006); Soner et al. (2011);Beissner and Denis (2018)) which we briefly outline. For the sake of exposition, we restrictour attention to (Soner et al., 2011, Example 4.5). We set C ([0 , T ]) the space of continuousfunctions on [0 , T ] taking values in R . Let P be the Wiener measure on Ω = { ω ∈ C ([0 , T ]) | ω (0) = 0 } . Let B := { B t } t ∈ [0 ,T ] be the canonical process, i.e. B t ( ω ) = ω t , ≤ t ≤ T .The process B is a standard Brownian motion under P with respect to the rough filtration F = {F t } ≤ t ≤ T := { σ ( B s | ≤ s ≤ t ) } ≤ t ≤ T and F + = ( F + t ) ≤ t ≤ T its right continuous ver-sion. Recall that from Karandikar (1995) the quadratic variation can be defined pathwise andis given by the F adapted process ( h B i t ) t ∈ [0 ,T ] . Following (Soner et al., 2011, Example 4.5) weconsider a class of piecewise constant diffusion coefficients V defined by a = P ∞ n =0 a n [ τ n ,τ n +1 ) ,where { τ n } n ∈ N is any non-decreasing sequence of F stopping times, with τ = 0 , τ n ≤ T and a n being a positive valued F τ n measurable random variable. Let P := {P a } a ∈V be the familycomposed by the measures P a = P ◦ ( X a ) − , where X a is the unique strong solution of dX t = a t ( X ) dB t P -a.s.The existence of a strong solution for such a class is proved in (Soner et al., 2011, Appendix).In particular, we have h B i t = R t a u du P a -a.s. for every t ∈ [0 , T ] (see (4.10) in Soner et al.(2011)). For any probability P we set N Pt = { A ⊂ B | B ∈ F t and P ( B ) = 0 } and introducethe enlarged filtration F V given by F V t = \ a ∈V F + t ∨ N P a t . (12)Recall that that any P a uniquely extends to F V t for any t ∈ [0 , T ] and the filtration is stillright continuous (see Soner et al. (2011)). Example 24
First we provide an example where
C 6 = e C unless we choose, in the discrete timemodel, the right continuous version of F . Whereas in continuous time the use of the right ontinuous filtration is customary in discrete time is not. Consider a one period model bychoosing two deterministic stopping times τ < τ = 1 . Suppose that V = [ σ, σ ] for some σ < σ non-negative, meaning that any plausible density of the quadratic variation process isconstant and bounded in a given interval. The class of corresponding probabilities is denotedby P := { P a } a ∈ [ σ,σ ] . Let X := B {h B i =ˆ a } for some ˆ a ∈ [ σ, σ ] . We consider first the rawfiltration F which implies that F is trivial. We can easily see that X ∈ e C . Indeed, for any a ∈ [ σ, σ ] with a = ˆ a we have X = 0 P a -a.s. with ∈ C . Moreover, X = B P ˆ a -a.s. with B ∈ C as it corresponds to the buy and hold strategy of one unit of risky asset. We deduce X ∈ \ P ∈P j − P ◦ j P ( C ) = e C . On the other hand it is not possible to find a trading strategy H ∈ R such that HB ≥ X q . s . .Indeed HB should be P a non-negative for any σ = ˆ a , nevertheless, the P a distribution of B is equal to the P distribution of aB (see (Soner et al., 2011, Section 8)). This implies X / ∈ C and consequently
C 6 = e C . Remark 25
It is worth pointing out that if one considers the P -completion of the right-continuous version of F t , the sets {h B i = a } ∈ F V for every a ∈ [ σ, σ ] . This implies X ∈ C ,if C is defined with respect to the filtration F V . We consider again the set of processes V defined by a = P ∞ n =0 a n [ τ n ,τ n +1 ) , where { τ n } n ∈ N is a non-decreasing sequence of F stopping times, with τ = 0 , τ n ≤ T and a n being a positive F τ n measurable random variable. Admissible strategies on the underlying process B are givenby the set of stochastic processes H ( F V ) = (cid:26) H is F V -adapted and Z T | H t | d h B i t < + ∞ (cid:27) . Here we are assuming the existence of a safe numeraire asset which pays at any time andwe recall that the value of a portfolio is given by V t = h t + H t B t where h t (respectively H t )is the number of shares at time t on the riskless (respectively risky) asset.For any H ∈ H ( F V ) the assumption of (Soner et al., 2011, Theorem 6.4) are satisfied andthere exists an F V -adapted process M such that M t = R t H u dB u , P a -almost surely for all a ∈ V and therefore the following is well defined K ( F V ) := (cid:26)Z T H u dB u | H ∈ H ( F V ) , Z t H u dB u is a P a -supermtg ∀ a ∈ V (cid:27) , and under the standard self-financing condition we have dV t = H t dB t . With a slight abuse ofnotation we identify any k ∈ K ( F V ) with the equivalence class [ k ] ∈ L it generates. In thisway we can consider K ( F V ) as a subset of L .As customary, we define C ( F V ) := { X ∈ L ∞ | X ≤ k for some k ∈ K ( F V ) } and in the following we omit the dependence on the filtration. Proposition 26
Consider the measurable space ( C [0 , T ] , F V T ) and the class P := { P a } a ∈V .Then, the cone C is sensitive, i.e., C = \ P ∈P j − P ( j P ( C )) . roof. As usual we denote by e C the right hand side. Since the inclusion ⊂ is trivial weonly need to show the opposite. Fix X ∈ e C and, for later use, let ˆ X := X + k X k ∞ which is q . s . non-negative. Let T be the class of F V -stopping times. For a, ν ∈ V we set θ a,ν := inf n t ≥ | R t a u du = R t ν u du o and, for τ ∈ T and a ∈ V , V aτ := { ν ∈ V | θ a,ν > τ or θ a,ν = τ = T } . For a ∈ V arbitrary, but fixed, we can thus consider the family { E P ν [ ˆ X |F V τ ] | ν ∈ V aτ } whichis composed by P a essentially bounded elements. Set V P a τ := sup ν ∈V aτ E P ν [ ˆ X |F V τ ] , (13)where the supremum is computed as a P a -essential supremum. For every a ∈ V the family { V P a τ } τ ∈T is uniformly integrable since | V P a τ | ≤ k X k ∞ P a -a.s. for every τ ∈ T . Thereforethe assumptions of (Soner et al., 2011, Theorem 7.1) are met and we deduce, Π( ˆ X ) = sup a ∈V k V P a k ∞ , (14)where Π( X ) = inf { x ∈ R | X − x ∈ C} denotes the superhedging functional and where V P a is as in (13) with τ = 0 . Recall now that the definition of X ∈ e C reads as j P a ( X ) ≤ j P a ( k a ) P a -a.s. for some k a ∈ K ( F V ) , for any a ∈ V . Thus, X ∈ e C guarantees that X ≤ R T H at dB t P a -a.s. for every a ∈ V which implies ˆ X ≤k X k ∞ + R T H at dB t P a -a.s. for every a ∈ V . The process R t H au dB u is a P a -supermartingale.Hence, we can conclude from E P a [ R T H au dB u |F V ] ≤ for every a ∈ V that ≤ V P a = sup ν ∈V a E P ν [ ˆ X | F V ] ≤ k X k ∞ + sup ν ∈V a E P ν (cid:20)Z T H νu dB u | F V (cid:21) ≤ k X k ∞ . By (14), Π( X + k X k ∞ ) = sup a ∈V k V P a k ∞ ≤ k X k ∞ , whence Π( X ) ≤ . From (Soner et al.,2011, Theorem 7.1) and Π( X + k X k ∞ ) < ∞ we know there exists a strategy D ∈ H ( F V ) suchthat Π( X ) + k X k ∞ + Z T D t dB t ≥ ˆ X P a -a.s. , a ∈ V , and R t D u dB u is a P a -martingale for every a ∈ V . Finally k X k ∞ + Z T D t dB t ≥ ρ C ( X ) + k X k ∞ + Z T D t dB t ≥ X + k X k ∞ P a -a.s. , a ∈ V , and X ≤ R T D t dB t , P a -a.s. follows for every a ∈ V .17 Proofs of the main results
Recall that k X k = k X/W k ∞ , for any X ∈ L and W ≥ . Consider the set P W := { P ≪ P | E P [ W ] < + ∞} . It is important to notice that P W ≈ P . Indeed, for any P ∈ P , W is integrable with respectto P W ∼ P defined by dP W dP = cW , where c := 1 /E P [ W − ] is the normalizing constant.For any P ∈ P W , we have | E P [ X ] | ≤ E P h | X | W · W i ≤ ˜ c k X k for ˜ c = E P [ W ] , so that the linearfunctional X E P [ X ] is continuous on ( L , k · k ) for any P ∈ P W . Let lin( P W ) ⊂ L ∗ be thespan of the set of linear functional generated by P W and L ∗ be the topological dual of L .We redefine the projection map j P in order to map L on L ∞ P as j P : L → L ∞ P X (cid:2) XW (cid:3) P The definition slightly differs from the one given in (3), but simple inspections show that thischange does not affect the set e C . In particular e C = \ P ∈P j − P ( j P ( C )) = \ P ∈P W j − P ( j P ( C )) and similarly for e C λ . The map j P is easily shown to be continuous from ( L , σ ( L , lin( P W )) to ( L ∞ P , σ ( L ∞ P , L P )) if P ≪ P . Lemma 27 e C , e C λ and b C = ∪ λ ≥ e C λ are monotone convex sets. In addition e C and b C are cones. Proof.
We only show the monotonicity of e C , the other properties can be proven similarly.Suppose Y ≤ X q . s . with X ∈ e C . By definition of e C , for any P ∈ P , there exists X P ∈ C suchthat X = X P P - a.s. Take Y P = Y { X = X P } + X P { X = X P } and observe that Y P ≤ X P q . s . From the monotonicity of C we deduce Y P ∈ C . Moreover, from Y = Y P P - a.s. and from P ∈ P being arbitrary, Y ∈ e C . In this subsection we consider the sets C , C λ defined by equations (1) and (2) respectively, andwe will assume in Proposition 28, Lemma 29 that C is Fatou closed, which implies that C λ isalso Fatou closed. Proposition 28
For every P ≪ P we have j P ( C λ ) is σ ( L ∞ P , L P ) -closed and therefore the set T P ∈P ′ j − P ( j P ( C λ )) is σ ( L , lin( P W )) -closed for any P ′ ⊂ { P ≪ P} . The proof is based on the next two Lemmata. For λ, K > define the set C λ,K := C λ ∩ { X ∈ L | k X k ≤ K } , Lemma 29
For any probability P ≪ P and for any K ≥ λ the set j P ( C λ,K ) is σ ( L ∞ P , L P ) -closed. roof. Consider the continuous inclusion i : ( L ∞ P , σ ( L ∞ P , L P )) → ( L P , σ ( L P , L ∞ P )) . In a first step we show that C ( P ) := i ◦ j P ( C λ,K ) is closed in L P endowed with the usual norm k · k L P := E P [ | · | ] . To this end let ( Y n ) n ∈ N ⊂ C ( P ) and Y ∈ L P such that k Y n − Y k L P → ,and without loss of generality we may also assume that Y n → Y P -a.s. (by passing toa subsequence). Note that | Y | is necessarily P -a.s. bounded by KW . Choose an arbitrary X n ∈ C λ,K such that Y n = j P ( X n ) for all n ∈ N and an arbitrary X ∈ L such that Y = j P ( X ) .Consider the set F = { ω ∈ Ω | X n ( ω ) → X ( ω ) } which satisfies P ( F ) = 1 . Define e X n := X n F − KW F c ∈ C λ,K for n ∈ N . By monotonicityof C λ , e X n for all n ∈ N , and e X n → X F − KW F c =: e X . Since C λ is Fatou closed, the sameholds for C λ,K . As a consequence, e X ∈ C λ,K . From P ( F ) = 1 and the arbitrary choice of therepresentatives X n and X , we have Y = j P ( X ) = j P ( e X ) ∈ C ( P ) . Hence, C ( P ) := i ◦ j P ( C λ,K ) is k · k L P -closed in L P . As C ( P ) is convex it then follows that C ( P ) is σ ( L P , L ∞ P ) -closed andtherefore j P ( C λ,K ) is σ ( L ∞ P , L P ) -closed by continuity of i . Lemma 30
For any probability P ≪ P we have the following representation j P ( C λ ) = [ K ≥ λ j P ( C λ,K ) Proof.
Notice that by definition j P ( C λ ) ⊃ j P ( C λ,K ) for any K ≥ λ and hence j P ( C λ ) ⊃ S K ≥ λ j P ( C λ,K ) . For the converse inclusion consider Y ∈ j P ( C λ ) : there exists X ∈ C λ suchthat j P ( X ) = Y . Let ¯ K = k X k then Y ∈ j P ( C λ, ¯ K ) . Proof of Proposition 28.
We first show that, for any K ≥ λ , j P ( C λ ) ∩ { Y ∈ L ∞ P | k Y k P, ∞ ≤ K } = j P ( C λ,K ) (15)The inclusion ⊃ is clear from Lemma 30. To show the equality, let Y ∈ j P ( C λ ) with k Y k P, ∞ ≤ K . There exists X ∈ C λ with j P ( X ) = Y . Let k ∈ K λ such that X ≤ k and notice that ( − KW ) ∨ X ∧ KW ≤ k and j P (( − KW ) ∨ X ∧ KW ) = Y .From Lemma 29 the sets in (15) are σ ( L ∞ P , L P ) -closed for every K ≥ λ . The Krein-SmulianTheorem implies that j P ( C λ ) is σ ( L ∞ P , L P ) -closed and therefore j − P ◦ j P ( C λ ) is σ ( L , lin( P W )) -closed. The last assertion follows by the intersection of closed sets. In this section we prove Theorem 6 and its discrete-time version Theorem 9.
Definition 31
For any set
A ⊂ L and X ∈ L , we define ρ A ( X ) := inf { x ∈ R | X − x ∈ A} . Note that for a monotone set A with ∈ A , ρ A ( X ) < ∞ for any X ∈ L ∞ . Lemma 32
The following are equivalent: ) (sNFLVR) ;ii) ρ e C ( ξ ) > for any ξ ∈ L ∞ + \ { } ;iii) ρ e C (1 A ) > for any A ∈ F \ N . Proof. i ) ⇔ ii ) : from Lemma 27, e C is monotone and ρ e C = ρ e C∩ L ∞ on L ∞ . Thus, ρ cl ∞ ( e C ) = ρ e C∩ L ∞ = ρ e C on L ∞ . The rest follows from cl ∞ ( e C ) = { ρ cl ∞ ( e C ) ≤ } = { ρ e C ≤ } . ii ) ⇒ iii ) : it follows from A ∈ L ∞ + \ { } . iii ) ⇒ ii ) : for any ξ ∈ L ∞ + \ { } we can find n ∈ N such that A := { ξ > /n } ∈ F \ N .From Lemma 27, ρ e C is monotone and positive homogeneous. We deduce, < n − ρ e C (1 A ) = ρ e C ( n − { ξ> /n } ) ≤ ρ e C ( ξ ) . Lemma 33
Consider now the conditions:a) ρ e C (1 A ) > for any A ∈ F \ N ;b) for A ∈ F \ N we can find δ ∈ (0 , and Q = { Q n } n ∈ N ⊂ P W , such that Q ≪ ¯ P for some ¯ P ∈ P and Q n ( A ) ≥ δ and E Q n ( X ) ≤ n ∀ X ∈ C n , for any n ∈ N .Then b ) ⇒ a ) . If, in addition, C is Fatou closed a ) ⇒ b ) . Proof. b) ⇒ a). Suppose that there exists A ∈ F \N such that ρ e C (1 A ) ≤ . Let δ ∈ (0 , , Q and ¯ P as in b). From ρ e C (1 A ) ≤ , it follows A − δ/ ∈ e C . In particular, by definition of e C , we have A − δ/ ∈ j − P ( j ¯ P ( C )) . Thus, there exists X ∈ C such that X = 1 A − δ/ P -a.s.More precisely, since C = ∪ λ ≥ C λ and C λ is an increasing collection of sets, there exists ¯ λ ≥ such that X ∈ C λ for every λ ≥ ¯ λ . Moreover, since { Q n } ≪ ¯ P , X = 1 A − δ/ Q n -a.s. for any n ∈ N . Using b), we deduce E Q n [1 A − δ/
4] = E Q n [ X ] ≤ n , n ≥ ¯ λ. For n ≥ ¯ λ ∨ δ − , we have Q n ( A ) ≤ δ/ which contradicts Q n ( A ) ≥ δ .a) ⇒ b). Let A ∈ F \ N and < δ < ρ e C (1 A ) . From ρ e C (1 A − δ ) > , we deduce that A − δ / ∈ e C . Therefore we can find ¯ P ∈ P such that A − δ / ∈ j − P ◦ j ¯ P ( C ) and, in particular, A − δ / ∈ j − P ◦ j ¯ P ( C n ) for any n ∈ N . We note now that the same is true for α (1 A − δ ) with α ∈ (0 , . Indeed, α (1 A − δ ) ∈ j − P ◦ j ¯ P ( C n ) ⇒ A − δ ∈ j − P ◦ j ¯ P ( C ⌈ n/α ⌉ ) which would be a contradiction. All these considerations hold true by an equivalent change ofmeasure, thus, we may assume ¯ P ∈ P W . Therefore for any α ∈ (0 , , and n ∈ N α (1 A − δ ) / ∈ − P ◦ j ¯ P ( C n ) , which is σ ( L , lin( P W )) closed by Proposition 28.Consider the σ ( L , lin( P W )) -compact and convex set A n = { α (1 A − δ ) | α ∈ [1 /n, } . Fromthe previous observation A n ∩ j − P ◦ j ¯ P ( C n ) = ∅ . For any n ∈ N , there exists µ n ∈ lin( P W ) such that sup X ∈ j − P ◦ j ¯ P ( C n ) µ n ( X ) < µ n (1 A ) − δn . (16)We observe that µ n is positive and µ n (1 Ω ) = 1 : indeed suppose that there exists ξ ∈ L + suchthat µ n ( ξ ) < . From − L + ⊂ C n , − aξ ∈ C n for any a > , from which lim a →∞ µ n ( − aξ ) =lim a →∞ − aµ n ( ξ ) = ∞ contradicts (16). Similarly, a (1 Ω − ∈ C n for any a ∈ R , which implies µ n (1 Ω ) = 1 . We deduce that µ n is the linear functional induced by some Q n ∈ P W . Moreover,for any n ∈ N , we have • Q n ≪ ¯ P . Otherwise let B ∈ F such that ¯ P ( B ) = 0 and Q n ( B ) > . From B = 0 ¯ P -a.s.we have a B ∈ j − P ◦ j ¯ P ( C n ) for every a > and sup a> E Q n [ a B ] = + ∞ contradicts(16). • Q n ( A ) ≥ δ . It follows from ∈ C n , which implies that the supremum in (16) is non-negative. • sup X ∈C n E Q n [ X ] ≤ sup X ∈ j − P ◦ j ¯ P ( C n ) E Q n [ X ] ≤ n follows again by (16).Since ¯ P is the same for every n , Q = { Q n } n ∈ N ≪ ¯ P ∈ P , which concludes the proof of b). proof of Theorem 6. It follows from Lemma 32 and Lemma 33. proof of Theorem 9.
Clearly (sNFLVR) implies (NA) and from Lemma 8 C is Fatouclosed so that the conclusion of Lemma 33 is an equivalence. The first statement follows asin the proof of Theorem 6.If we now assume that e C = C we have that (sNFLVR) is equivalent to (NFLVR) , whichis further equivalent to (NA) since, from Lemma 8, C ∩ L ∞ is k · k ∞ -closed. The implication( ⇒ ) follows directly from the first part of the Theorem. For the converse implication, let V T ( H, h ) := ( H ◦ S ) T + h · Φ . Suppose that, for some ( H, h ) ∈ H , V T ( H, h ) ≥ . s . (andhence ( H, h ) ∈ H λ for every λ ≥ ). If there exists P ∈ P such that P ( { V T ( H, h ) > } ) > then for some a > the set A = { V T ( H, h ) ≥ a } ∈ F \ N . By assumption, there exist δ > and Q ∈ Q app such that inf Q ∈Q Q ( A ) = δ > . Consider k ∈ N and define the strategy ( ˆ H k , ˆ h k ) := ka ( H, h ) . Notice that since ≤ V T ( ˆ H k , ˆ h k ) ∧ K ≤ V T ( ˆ H k , ˆ h k ) and C n is monotone, V T ( ˆ H k , ˆ h k ) ∧ K belongs to C n for every n ∈ N . From Lemma 33 b), for an arbitrarily fixed Q ∈ Q , ≤ sup k ∈ N E Q [ V T ( ˆ H k , ˆ h k ) ∧ K ] ≤ for every K > . By monotone convergence theorem E Q [ V T ( ˆ H k , ˆ h k )] ≤ for any k ∈ N . Since E Q [1 A V T ( ˆ H k , ˆ h k )] = ka E Q (cid:2) { V T ( H,h ) ≥ a } V T ( H, h ) (cid:3) ≥ kδ we have that sup k ∈ N E Q [ V T ( ˆ H k , ˆ h k )] = ∞ which is a contradiction. (sNABR) When C = e C , both sets are also equal to b C := S λ ≥ e C λ . In the general case we could define sensitive No Arbitrage with Bounded Risk (sNABR) as b C ∩ L ∞ + = { } and the the following21elations are satisfied: (sNFLVR) ⇒ (sNA) ⇒ (sNABR) ⇒ (NA) A similar characterization holds for this condition.
Theorem 34
Let C defined in (1) be Fatou closed. The following are equivalent:1. (sNABR)
2. For every non polar set A and n ∈ N , there exists < δ n ≤ n and Q n ∈ P W such that Q n ( A ) > nδ n and E Q n [ X ] < δ n ∀ X ∈ e C n , for every n ∈ N . From the equivalent formulation, it is clear that the two notions (sNFLVR) and (sNABR) arevery close. One difference is that in Theorem 6, the lower bound for Q n ( A ) is uniform for thecollection { Q n } n ∈ N . Moreover, in Theorem 6 it also holds Q ≪ ¯ P for some ¯ P ∈ P . Lemma 35
Under the same assumptions of Theorem 34 the following are equivalent:a) S λ ≥ e C λ ∩ L ∞ + = { } ;b) ρ e C λ ( ξ ) > for any ξ ∈ L ∞ + \ { } and λ ≥ ;c) ρ e C λ (1 A ) > for any A ∈ F \ N and λ ≥ . Proof.
For ease of notation, denote ρ λ = ρ e C λ . a ) ⇔ b ) : the proof follows from e C λ being σ ( L , lin( P W )) closed and monotone. b ) ⇒ c ) : it follows from A ∈ L ∞ + \ { } . c ) ⇒ b ) : We first show that ρ λ ( n A ) > for any n ∈ N and A ∈ F \ N .The inequality ≥ is clear by monotonicity. Suppose, by contradiction, there exists ¯ n such that ρ λ ( n A ) = 0 for every n > ¯ n . Since e C λ is σ ( L , lin( P W )) closed by Proposition 28, we inferthat n A ∈ e C λ for every n > ¯ n . By definition of e C λ , for every P ∈ P there exists k P ∈ K λ such that n A ≤ k P P -a.s. which implies A ∈ e C nλ i.e. ρ nλ (1 A ) = 0 . This would contradict c ) .Now for any ξ ∈ L ∞ + \ { } we can find n ∈ N such that A := { ξ > /n } ∈ F \ N . FromLemma 27, ρ λ is monotone, from which, < ρ λ ( n − A ) ≤ ρ λ ( ξ ) . Proof of Theorem 34.
Suppose (sNABR) holds. By Lemma 35, for any λ ≥ , ρ e C λ (1 A ) > and any A ∈ F \ N . Since e C λ is σ ( L , lin( P W )) closed by Proposition 28, A / ∈ e C λ for any λ ≥ and the same is true for α A for α ∈ (0 , . Indeed, α A ∈ e C λ would imply A ∈ e C λ/α , a contradiction. Thus, for any n ∈ N the σ ( L , lin( P W )) closed and convex set e C n σ ( L , lin( P W )) -compact and convex set A n = { α A | α ∈ [1 /n, } are disjoint. Forany n ∈ N , there exists Q n ∈ P W and δ n ∈ [0 , , such that sup X ∈ e C n E Q n [ X ] < δ n < Q n ( A ) n . (17)From ∈ C n , δ n > and the thesis follows.For the converse implication suppose that there exists A ∈ F \ N such that ρ e C λ (1 A ) ≤ for some λ ≥ . Since e C λ is σ ( L , lin( P W )) closed, A ∈ e C λ . Let n ∈ N with n ≥ λ . Take Q n ∈ P W such that Q n ( A ) > nδ n and E Q n [ X ] < δ n for any X ∈ e C n . From A ∈ e C n , wededuce nδ n < Q n ( A ) < δ n which yields the contradiction n < . We only need to show that (sNFLVR) implies Q = ∅ and P ≪ Q , the rest follows fromTheorem 9. Assume (sNFLVR) and introduce the notation S t = ( S , . . . , S t ) . Theorem 6ensures that for A ∈ F \ N and n ∈ N , we can find a collection Q = { Q n } n ∈ N of probabilitymeasures such that Q n ( A ) ≥ δ and E Q n [ X ] ≤ n ∀ X ∈ C n . (18)Moreover, there exists P ∈ P such that Q ≪ P . By Assumption 10, P has support on somecompact set K P , hence, the collection Q is tight and, by Prokhorov Theorem, is relativelycompact. From Lemma 33, Q ⊂ P W and therefore we can define { P n } n ∈ N by dP n dQ n = WE Qn [ W ] .Since P n ∼ Q n and Q n ( K P ) = 1 with K P independent of n , the collection { P n } n ∈ N is also tight(and hence relatively compact). Moreover, for X ∈ C n we have E Q n [ X ] = E Q n [ W ] E P n (cid:2) XW (cid:3) We deduce that there exists a convergent subsequence of { P n } n ∈ N whose limit is denoted by ¯ P ∈ M . Finally we define the measure Q by dQd ¯ P = cW where c := 1 /E ¯ P [ W − ] .Note that, since F is the natural filtration of S , any F t -measurable random variable H can bewritten as h ( S , . . . , S t ) , for some Borel-measurable function h : R d × t → R . Consider now thesets Y := (cid:26) f ( S t ) : Ω → R | f ∈ C b ( R d × t ) (cid:27) , X := (cid:26) h ( S t ) : Ω → R | h ∈ B b ( R d × t ) , E Q [ h ( S t )( S jt +1 − S jt )] = 0 ∀ j (cid:27) , where B b ( R d × t ) denotes the space of bounded measurable functions on R d × t and C b ( R d × t ) thosewhich are, in addition, continuous. We aim at using a Monotone Class Theorem to deducethat X contains all bounded F t -measurable function. First note that Y is a multiplicativeclass, namely, if Y , Y ∈ Y then Y Y ∈ Y . Next, we show that σ ( Y ) = F t . To see thislet Γ n be a sequence of compacts such that, R d × t = ∪ n ∈ N Γ n By Urysohn’s Lemma, for any n, m ∈ N there exists f n,m ∈ C b ( R d × t ) such that f n,m ( x ) = x on Γ n and f n,m ( x ) = 0 onthe complementary of Γ n + B ◦ m (0) , where B ◦ m (0) denotes the open ball with center in andradius /m . Take now an open set O ⊂ R d × t + and note that S − t ( O ) = ∪ n ∈ N S − t ( O ∩ Γ n ) . By23onstruction of f n,m , S − t ( O ∩ Γ n ) = \ m ∈ N { ω ∈ Ω | f n,m ( S t ( ω )) ∈ O ∩ Γ n } ∈ σ ( Y ) From O and n arbitrary we deduce F t ⊂ σ ( Y ) . The opposite inclusion is trivial.The next step is to show that Y ⊂ X . Let f ∈ C b ( R d × t ) and define X j := f ( S t )( S jt +1 − S jt ) for j = 1 , . . . , d . By the choice of W and f bounded, we deduce that X j /W ∈ C b (Ω) andthere exists ¯ n ∈ N such that X j ∈ C n for any n ≥ ¯ n . From (18) and W ≥ we have n ≥ E Q n [ X j ] = E Q n [ W ] E P n (cid:20) X j W (cid:21) implies E P n (cid:20) X j W (cid:21) ≤ n . Using the weak convergence of P n to ¯ P , we deduce E ¯ P [ X j /W ] ≤ . By repeating the sameargument for − X j , we obtain E ¯ P [ X j /W ] = 0 and hence E Q [ X j ] = 0 .We now note that X is a vector space and Ω ∈ Y ⊂ X . Moreover, for an increasingsequence { H n } n ∈ N ⊂ X with lim n →∞ H n = H bounded, we have that H ∈ X by dominatedconvergence. From (Protter, 2005, Theorem I.8) and t ∈ I arbitrary, we conclude that S is a ( Q, F ) -martingale. The fact that Q is calibrated to the prices of the options Φ follows from φ i /W ∈ e C n for every i = 1 , . . . , m , n ∈ N and a similar weak convergence argument.Finally we show that P ≪ Q . Let A ∈ F \ N , there exists P ∈ P such that P ( A ) > .By (Aliprantis and Border, 2006, Theorem 12.5), there exists a closed set F ⊂ A such that P ( F ) > . From W ≥ we have E P n [1 F ] ≥ E P n (cid:20) F W (cid:21) = E Q n [ W ] E Q n [1 F ] ≥ δ Due to Portemanteau’s Theorem (Aliprantis and Border, 2006, Theorem 15.3), ¯ P ( F ) ≥ lim sup P n ( F ) ≥ εδ . Thus, again by (Aliprantis and Border, 2006, Theorem 12.5), ¯ P ( A ) ≥ ¯ P ( F ) > and since Q is equivalent to ¯ P , we also have Q ( A ) > . This concludes the proof. References
Acciaio, B., Beiglböck, M., Penkner, F., and Schachermayer, W. (2016). A model-free versionof the fundamental theorem of asset pricing and the super-replication theorem.
Math.Finance , 26(2):233–251.Aliprantis, C. D. and Border, K. C. (2006).
Infinite Dimensional Analysis: a Hitchhiker’sGuide . Springer, Berlin; London.Aliprantis, C. D. and Tourky, R. (2002). The Super Order Dual of an Ordered Vector Spaceand the Riesz-Kantorovich Formula.
Transactions of the American Mathematical Society ,354(5):2055–2077.Aliprantis, C. D., Tourky, R., and Yannelis, N. C. (2001). A theory of value with non-linearprices: Equilibrium analysis beyond vector lattices.
Journal of Economic Theory , 100(1):22– 72.Bachelier, L. (1900). Théorie de la spéculation.
Annales scientifiques de l’École NormaleSupérieure , 3e série, 17:21–86. 24artl, D., Kupper, M., and Neufeld, A. (2020). Pathwise superhedging on prediction sets.
Finance and Stochastics , 24:215–248.Bartl, D., Kupper, M., Prömel, D. J., and Tangpi, L. (2019). Duality for pathwise superhedgingin continuous time.
Finance and Stochastics , 23(3):697–728.Bayraktar, E. and Zhang, Y. (2016). Fundamental theorem of asset pricing under transactioncosts and model uncertainty.
Mathematics of Operations Research , 41(3):1039–1054.Bayraktar, E. and Zhou, Z. (2017). On arbitrage and duality under model uncertainty andportfolio constraints.
Mathematical Finance , 27(4):988–1012.Beissner, P. and Denis, L. (2018). Duality and general equilibrium theory under knightianuncertainty.
SIAM Journal on Financial Mathematics , 9(1):381–400.Beissner, P. and Riedel, F. (2019). Equilibria under knightian price uncertainty.
Econometrica ,87(1):37–64.Bertsekas, D. P. and Shreve, S. E. (2007).
Stochastic Optimal Control: The Discrete-TimeCase . Athena Scientific.Biagini, S., Bouchard, B., Kardaras, C., and Nutz, M. (2017). Robust fundamental theoremfor continuous processes.
Mathematical Finance , 27(4):963–987.Biagini, S. and Frittelli, M. (2008). A unified framework for utility maximization problems:an Orlicz space approach.
Ann. Appl. Probab. , 18(3):929–966.Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities.
Journal ofPolitical Economy , 81(3):637–654.Blanchard, R. and Carassus, L. (2019). No-arbitrage with multiple-priors in discrete time. arXiv e-prints , page arXiv:1904.08780.Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-timemodels.
Ann. Appl. Probab. , 25(2):823–859.Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit inoption prices.
Jour. Bus. , pages 625–651.Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., and ObÅĆÃşj, J. (2019). Pointwise arbitragepricing theory in discrete time.
Mathematics of Operations Research , 44(3):1034–1057.Burzoni, M., Frittelli, M., and Maggis, M. (2016). Universal arbitrage aggregator in discrete-time markets under uncertainty.
Finance Stoch. , 20(1):1–50.Burzoni, M., Riedel, F., and Mete Soner, H. (2017). Viability and Arbitrage under KnightianUncertainty. arXiv e-prints , page arXiv:1707.03335.Cheridito, P., Kiiski, M., Proemel, D., and Soner, H. M. (Forthcoming). Martingale optimaltransport duality.
Mathematische Annalen .Dalang, R. C., Morton, A., and Willinger, W. (1990). Equivalent martingale measures andno-arbitrage in stochastic securities market models.
Stochastics Stochastics Rep. , 29(2):185–201. 25avis, M. H. A. and Hobson, D. G. (2007). The range of traded option prices.
Math. Finance ,17(1):1–14.Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem ofasset pricing.
Mathematische Annalen , 300(1):463–520.Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claimsin the presence of model uncertainty.
Ann. Appl. Probab. , 16(2):827–852.Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging incontinuous time.
Probab. Theory Related Fields , 160(1-2):391–427.Halmos, P. R. and Savage, L. J. (1949). Application of the radon-nikodym theorem to thetheory of sufficient statistics.
Ann. Math. Statist. , 20(2):225–241.Hou, Z. and Obłój, J. (2018). Robust pricing–hedging dualities in continuous time.
Financeand Stochastics , 22(3):511–567.Karandikar, R. L. (1995). On pathwise stochastic integration.
Stochastic Processes and theirApplications , 57(1):11 – 18.Maggis, M., Meyer-Brandis, T., and Svindland, G. (2018). Fatou closedness under modeluncertainty.
Positivity , 22(5):1325–1343.Merton, R. C. (1973). Theory of rational option pricing.
Bell Journal of Economics , 4(1):141–183.Obłój, J. and Wiesel, J. (2018). A unified Framework for Robust Modelling of FinancialMarkets in discrete time. arXiv e-prints , page arXiv:1808.06430.Perkowski, N. and Prömel, D. J. (2016). Pathwise stochastic integrals for model free finance.
Bernoulli , 22(4):2486–2520.Protter, P. E. (2005).
Stochastic Integration and Differential Equations . Springer BerlinHeidelberg.Riedel, F. (2015). Financial economics without probabilistic prior assumptions.
Decis. Econ.Finance , 38(1):75–91.Rockafellar, R. T. and Wets, R. J.-B. (1998).
Variational analysis , volume 317 of
Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] .Springer-Verlag, Berlin.Soner, H. M., Touzi, N., and Zhang, J. (2011). Quasi-sure stochastic analysis through aggre-gation.
Electron. J. Probab. , 16:no. 67, 1844–1879.Terkelsen, F. (1972). Some minimax theorems.
Math. Scand. , 31:405–413 (1973).Vovk, V. (2012). Continuous-time trading and the emergence of probability.