Pointwise Arbitrage Pricing Theory in Discrete Time

M. Burzoni, M. Frittelli, Z. Hou, M. Maggis and J. Ob l´oj

June 5, 2018

Abstract

We develop a robust framework for pricing and hedging of derivative securities in discrete- time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain an abstract (pointwise) Fun- damental Theorem of Asset Pricing and Pricing–Hedging Duality. Our results are general and in particular include so-called model independent results of Acciaio et al. (2016); Burzoni et al. (2016) as well as seminal results of Dalang et al. (1990) in a classical probabilistic approach. Our analysis is scenario–based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

1 Introduction

The State Preference Model or Asset Pricing Model underpins most mathematical descriptions of Financial Markets. It postulates that the price of d financial assets is known at a certain initial time t0 = 0 (today), while the price at future times t>0 is unknown and is given by a certain random outcome. To formalize such a model we only need to fix a quadruple (X, , F,S), where X F is the set of scenarios, a -algebra and F := t t I a filtration such that the d-dimensional F {F } 2 ✓F process S := (St)t I is adapted. At this stage, no probability measure is required to specify the 2 Financial Market model (X, , F,S). F

One of the fundamental reasons for producing such models is to assign rational prices to contracts which are not liquid enough to have a market–determined price. Rationality here is understood via the economic principle of absence of arbitrage opportunities, stating that it should not be possible to trade in the market in a way to obtain a positive gain without taking any risk. Starting from this premise, the theory of pricing by no arbitrage has been successfully developed over the last 50 years. Its cornerstone result, known as the Fundamental Theorem of Asset Pricing (FTAP), establishes equivalence between absence of arbitrage and existence of risk neutral pricing rules. The intuition for this equivalence can be accredited to de Finetti for his work on coherence and previsions (see de Finetti (1931, 1990)). The ﬁrst systematic attempt to understand the absence of arbitrage opportunities in models of ﬁnancial assets can be found in the works of Ross (1976, 1977)

1 on capital pricing, see also Huberman (1982). The intuition underpinning the arbitrage theory for derivative pricing was developed by Samuelson (1965), Black and Scholes (1973) and Merton (1973). The rigorous theory was then formalized by Harrison and Kreps (1979) and extended in Harrison and Pliska (1981), see also Kreps (1981). Their version of FTAP, in the case of a ﬁnite set of scenarios X, can be formulated as follows. Consider X = ! ,...,! and let s =(s1,...,sd) { 1 n} be the initial prices of d assets with random outcome S(!)=(S1(!),...,Sd(!)) for any ! X. 2 Then, we have the following equivalence

d @H R such that H s 0 Q such that Q(!j) > 0 and 2 ·  9 2P (1) and H S(!) 0with> for some ! X () E [Si]=si, 1 j n, 1 i d · 2 Q 8     where is the class of probability measures on X. In particular, no reference probability measure P is needed above and impossible events are automatically excluded from the construction of the state space X. On the other hand, linear pricing rules consistent with the observed prices s1,...sd and the No Arbitrage condition, turn out to be (risk-neutral) probabilities with full support, that is, they assign positive measure to any state of the world. By introducing a reference probability measure P with full support and defining an arbitrage as a portfolio with H s 0, P (H S(!) ·  · 0) = 1 and P (H S(!) > 0) > 0, the thesis in (1) can be restated as · There is No Arbitrage Q P such that E [Si]=si i =1,...d. (2) () 9 ⇠ Q 8 The identification suggested by (2) allows non-trivial extensions of the FTAP to the case of a general space X with a fixed reference probability measure, and was proven in the celebrated work Dalang et al. (1990), by use of measurable selection arguments. It was then extended to continuous time models by Delbaen and Schachermayer (1994b,a).

The idea of introducing a reference probability measure to select scenarios proved very fruitful in the case of a general X and was instrumental for the rapid growth of the modern financial industry. It was pioneered by Samuelson (1965) and Black and Scholes (1973) who used it to formulate a continuous time financial asset model with unique rational prices for all contingent claims. Such models, with strong assumptions implying a unique derivative pricing rule, are in stark contrast to a setting with little assumptions, e.g. where the asset can follow any non-negative continuous trajectory, which are consistent with many rational pricing rules. This dichotomy was described and studied in the seminal paper of Merton (1973) who referred to the latter as “assumptions suciently weak to gain universal support1” and pointed out that it typically generates outputs which are not specific enough to be of practical use. For that reason, it was the former approach, with the reference probability measure interpreted as a probabilistic description of future asset dynamics, which became the predominant paradigm in the field of quantitative finance. The original simple models were extended, driven by the need to capture additional features observed in the increasingly complex market reality, including e.g. local or stochastic volatility. Such extensions can be seen as enlarging the set of scenarios considered in the model and usually led to plurality of rational prices.

1This setting has been often described as “model–independent” but we see it as a modelling choice with very weak assumptions.

2 More recently, and in particular in the wake of the ﬁnancial crisis, the critique of using a single reference probability measure came from considerations of the so-called Knightian uncertainty, going back to Knight (1921), and describing the model risk, as contrasted with ﬁnancial risks captured within a given model. The resulting stream of research aims at extending the probabilistic framework of Dalang et al. (1990) to a framework which allows for a set of possible priors . R✓P The class represents a collection of plausible (probabilistic) models for the market. In continuous R time models this led naturally to the theory of quasi-sure stochastic analysis as in Denis and Martini (2006); Peng (2010); Soner et al. (2011a,b) and many further contributions, see e.g. Dolinsky and Soner (2017). In discrete time, a general approach was developed by Bouchard and Nutz (2015). Under some technical conditions on the state space and the set they provide a version of the R FTAP, as well as the superhedging duality. Their framework includes the two extreme cases: the classical case when = P is a singleton and, on the other extreme, the case of full ambiguity R { } when coincides with the whole set of probability measures and the description of the model R becomes pathwise. Their setup has been used to study a series of related problems, see e.g. Bayraktar and Zhang (2016); Bayraktar and Zhou (2017).

Describing models by specifying a family of probability measures appears natural when starting R from the dominant paradigm where a reference measure P is fixed. However, it is not the only way, and possibly not the simplest one, to specify a model. Indeed, in this paper, we develop a di↵erent approach inspired by the original finite state space model used in Harrison and Pliska (1981) as well as the notion of prediction set in Mykland (2003), see also Hou and Oblój(2018). Our analysis is scenario based. More specifically, agent’s beliefs or a model are equivalent to selecting a set of admissible scenarios which we denote by ⌦ X. The selection may be formulated e.g. in terms ✓ of behaviour of some market observable quantities and may reflect both the information the agent has as well as the modelling assumptions she is prepared to make. Our approach clearly includes the “universally acceptable” case of considering all scenarios ⌦= X but we also show that it sub- sumes the probabilistic framework of Dalang et al. (1990). Importantly, as we work under minimal measurability requirement on ⌦, our models o↵er a flexible way to interpolate between the two settings. The scenario based specification of a model requires less sophistication than selection of a family of probability measures and appears particularly natural when considering (super)-hedging which is a pathwise property.

Our first main result, Theorem 2.3, establishes a Fundamental Theorem of Asset Pricing for an arbitrary specification of a model ⌦and gives equivalence between existence of a rational pricing rule (i.e. a calibrated martingale measure) and absence of, suitably defined, arbitrage opportunities. Interestingly the equivalence in (1) does not simply extend to a general setting: specification of ⌦which is inconsistent with any rational pricing rule does not imply existence of one arbitrage strategy. Ex post, this is intuitive: while all agents may agree that rational pricing is impossible they may well disagree on why this is so. This discrepancy was first observed, and illustrated with an example, by Davis and Hobson (2007). The equivalence is only recovered under strong assumptions, as shown by Riedel (2015) in a topological one-period setup and by Acciaio et al. (2016) in a general discrete time setup. A rigorous analysis of this phenomenon in the case ⌦= X

3 was subsequently given by Burzoni et al. (2016), who also showed that several notions of arbitrage can be studied within the same framework. Here, we extend their result to an arbitrary ⌦ X ✓ and to the setting with both dynamically traded assets and statically traded assets. We show that also in such cases agents’ di↵erent views on arbitrage opportunities may be aggregated in a canonical way into a pointwise arbitrage strategy in an enlarged ﬁltration. As special cases of our general FTAP, we recover results in Acciaio et al. (2016); Burzoni et al. (2016) as well as the classical Dalang-Morton-Willinger theorem Dalang et al. (1990). For the latter, we show that choosing a probability measure P on X is equivalent to ﬁxing a suitable set of scenarios ⌦P and our results then lead to probabilistic notions of arbitrage as well as the probabilistic version of the Fundamental Theorem of Asset pricing.

Our second main result, Theorem 2.5, characterises the range of rational prices for a contingent claim. Our setting is comprehensive: we make no regularity assumptions on the model specification ⌦, on the payo↵s of traded assets, both dynamic and static, or the derivative which we want to price. We establish a pricing–hedging duality result asserting that the infimum of prices of superhedging strategies is equal to the supremum of rational prices. As already observed in Burzoni et al. (2017), but also in Beiglböck et al. (2017) in the context of martingale optimal transport, it may be necessary to consider superhedging on a smaller set of scenarios than ⌦in order to avoid a duality gap between rational prices and superhedging prices. In this paper this feature is achieved through the set of ecient trajectories ⌦⇤ which only depends on ⌦and the market.

The set ⌦⇤ recollects all scenarios which are supported by some rational pricing rule. Its intrinsic and constructive characterisation is given in the FTAP, Theorem 2.3. Our duality generalizes the results of Burzoni et al. (2017) to the setting of abstract model specification ⌦as well as generic finite set of statically traded assets. The flexibility of model choice is of particular importance, as stressed above. The “universally acceptable” setting ⌦= X will typically produce wide range of rational prices which may not be of practical relevance, as already discussed by Merton (1973). However, as we shrink ⌦from X to a set ⌦P , the range of rational prices shrinks accordingly and, in case ⌦P corresponds to a complete market model, the interval reduces to a single point. This may be seen as a quantification of the impact of modelling assumptions on rational prices and gives a powerful description of model risk.

We note that pricing–hedging duality results have a long history in the field of robust pricing and hedging. First contributions focused on obtaining explicit results working in a setting with one dynamically traded risky asset and a strip of statically traded co-maturing call options with all strikes. In his pioneering work Hobson (1998) devised a methodology based on Skorkohod embedding techniques and treated the case of lookback options. His approach was then used in a series of works focusing on di↵erent classes of exotic options, see Brown et al. (2001); Cox and Oblój (2011b,a); Cox and Wang (2013); Hobson and Klimmek (2013); Hobson and Neuberger (2012); Henry-Labordère et al. (2016). More recently, it has been re-cast as an optimal transportation problem along martingale dynamics and the focus shifted to establishing abstract pricing–hedging duality, see Beiglböck et al. (2013); Davis et al. (2014); Dolinsky and Soner (2014); Hou and Oblój (2018).

4 The remainder of the paper is organised as follows. First, in Section 2, we present all the main results. We give the necessary definitions and in Section 2.1 state our two main theorems: the Fundamental Theorem of Asset Pricing, Theorem 2.3, and the pricing–hedging duality, Theorem 2.5, which we also refer to as the superheding duality. In Section 2.2, we generalize the results of Acciaio et al. (2016) for a multi-dimensional non-canonical stock process. Here, suitable continuity assumptions and presence of a statically traded option 0 with convex payo↵with superlinear growth allow to “lift” superhedging from ⌦⇤ to the whole ⌦. Finally, in Section 2.3, we recover the classical probabilistic results of Dalang et al. (1990). The rest of the paper then discusses the methodology and the proofs. Section 3 is devoted to the construction of strategy and filtration which aggregate arbitrage opportunities seen by di↵erent agents. We first treat the case without statically traded options when the so-called Arbitrage Aggregator is obtained through a conditional backwards induction. Then, when statically traded options are present, we devise a Pathspace Partition Scheme, which iteratively identifies the class of polar sets with respect to calibrated martingale measure. Section 4 contains the proofs with some technical remarks relegated to the Appendix.

2 Main Results

We work on a Polish space X and denote its Borel sigma-algebra and the set of all probability BX P measures on (X, ). If is a sigma algebra and P , we denote with P ( ):= N BX G✓BX 2P N G { ✓ A P (A)=0 the class of P -null sets from . We denote with A the sigma-algebra 2G| } G F generated by the analytic sets of (X, ) and with pr the sigma algebra generated by the class BX F ⇤of projective sets of (X, ). The latter is required for some of our technical arguments and BX we recall its properties in the Appendix. In particular, under a suitable choice of set theoretical axioms, it is included in the universal completion of , see Remark 5.4. As discussed in the BX introduction, we consider pointwise arguments and think of a model as a choice of universe of scenarios ⌦ X. Throughout, we assume that ⌦is an analytic set. ✓ Given a family of measures we say that a set is polar (with respect to ) if it belongs R✓P R to N A Q(A)=0 Q and a property is said to hold quasi surely ( -q.s.) if it { ✓ 2BX | 8 2R} R holds outside a polar set. For those random variables g whose positive and negative part is not Q- + integrable (Q ) we adopt the convention = when we write E [g]=E [g ] E [g]. 2P 11 1 Q Q Q Finally for any sigma-algebra we shall denote by (X, ; Rd) the space of -measurable d- G L G G dimensional random vectors. For a given set A X and f,g (X, ; R) we will often refer to ✓ 2L G f g on A whenever f(!) g(!) for every ! A (similarly for = and <).   2 We ﬁx a time horizon T N and let T := 0, 1,...,T . We assume the market includes both 2 { } liquid assets, which can be traded dynamically through time, and less liquid assets which are only available for trading at time t = 0. The prices of assets are represented by an Rd-valued stochastic process S =(St)t on (X, X ). In addition we may also consider presence of a vector 2T B d˜ of non-traded assets represented by an R -valued stochastic process Y =(Yt)t on (X, X ) 2T B with Y0 a constant, which may also be interpreted as market factors, or additional information

5 available to the agent. The prices are given in units of some fixed numeraire asset S0,which itself is thus normalized: S0 = 1 for all t T. In the presence of the additional factors Y ,welet t 2 S,Y S,Y S,0 S F := ( t )t be the natural filtration generated by S and Y (When Y 0wehaveF = F F 2T ⌘ the natural filtration generated by S). For technical reasons, we will also make use of the filtration pr pr pr S,Y F := ( t )t where t is the sigma algebra generated by the projective sets of (X, t ), F 2T F F pr 1 namely t := ((Su,Yu) (L) L ⇤,u t) (see the Appendix for further details). Clearly, F pr | 2  pr S,Y FS,Y Fpr and is “non-anticipative” in the sense that the atoms of and are the ✓ Ft Ft Ft same. Finally, we let denote the vector of payo↵s of the statically traded assets. We consider the setting when = ,..., is finite and each is A-measurable. When there are no { 1 k} 2 F statically traded assets we set = 0 which makes our notation consistent.

For any ﬁltration F, (F) is the class of F-predictable stochastic processes, with values in Rd, H which represent admissible trading strategies. Gains from investing in S, adopting a strategy H, T d j j j T are given by (H S)T := t=1 j=1 Ht (St St 1)= t=1 Ht St. In contrast, j can only be · bought or sold at time t =P 0 (withoutP loss of generalityP with zero initial cost) and held until the k k maturity T , so that trading strategies are given by ↵ R and generate payo↵ ↵ := ↵jj 2 · j=1 at time T .Welet (F) denote the set of such F-admissible trading strategies (↵,H). A P Given a ﬁltration F, universe of scenarios ⌦and set of statically traded assets , we let

⌦,(F):= Q S is an F-martingale under Q, Q(⌦) = 1 and EQ[]=0 . M { 2P| 8 2 } The support of a probability measure Q is given by supp(P ):= C C closed, P (C)=1 . { 2BX | } f We often consider measures with finite support and denote it withT a superscript , i.e. for a given f set of probability measures we put f := Q supp(Q)isfinite .Towit, (F) denotes R R { 2R| } M⌦, finitely supported martingale measures on ⌦which are calibrated to options in . Define

M M M S,Y P S,Y F := ( t )t , where t := t ( ) , (3) F 2T F F _N FT P ( S,Y ) 2M⌦\, F M S,Y and we convene is the power set of ⌦whenever ⌦,(F )= . Ft M ; Remark 2.1 In this paper we only consider ﬁltrations F which satisfy FS,Y F FM . All such ✓ ✓ ﬁltrations generate the same set of martingale measures, in the sense that any Q ⌦,(F) 2M ˆ M ˆ M uniquely extends to a measure Q ⌦,(F ) and, reciprocally, for any Q ⌦,(F ),the 2M 2M ˆ restriction Q belongs to ⌦,(F). Accordingly, with a slight abuse of notation, we will write |F M M ⌦,(F)= ⌦,(F )= ⌦,. M M M In the subsequent analysis, the set of scenarios charged by martingale measures is crucial:

f ⌦⇤ := ! ⌦ Q such that Q(!) > 0 = supp(Q). (4) 2 |9 2M⌦, Q f n o 2[M⌦, f We have by definition that for every Q its support satisfies supp(Q) ⌦⇤ . Notice that 2M⌦, ✓ f M the key elements introduced so far namely ⌦,, , F and ⌦⇤ , only depend on the four M M⌦, basic ingredients of the market: ⌦, S, Y and . Finally, in all of the above notations, we omit the subscript ⌦when ⌦= X and we omit the subscript when = 0, e.g. f denotes all finitely M supported martingale measures on X.

6 2.1 Fundamental Theorem of Asset Pricing and Superhedging Duality

We now introduce di↵erent notions of arbitrage opportunities which play a key role in the statement of the pointwise Fundamental Theorem of Asset Pricing.

Deﬁnition 2.2 Fix a ﬁltration F, ⌦ X and a set of statically traded options . ✓

A One-Point Arbitrage (1p-Arbitrage) is a strategy (↵,H) (F) such that ↵ +(H S)T • 2A · 0 on ⌦ with a strict inequality for some ! ⌦. 2

A Strong Arbitrage is a strategy (↵,H) (F) such that ↵ +(H S)T > 0 on ⌦. • 2A ·

A Uniformly Strong Arbitrage is a strategy (↵,H) (F) such that ↵ +(H S)T >" • 2A · on ⌦,forsome">0.

Clearly, the above notions are relative to the inputs and we often stress this and refer to an arbitrage in (F) and on ⌦. We are now ready to state the pathwise version of Fundamental Theorem of A Asset Pricing. It generalizes Theorem 1.3 in Burzoni et al. (2016) in two directions: we include an analytic selection of scenarios ⌦and we include static trading in options as well as dynamic trading in S.

Theorem 2.3 (Pointwise FTAP on ⌦ X) Fix ⌦ analytic and a ﬁnite set of A-measurable ✓ F statically traded options. Then, there exists a ﬁltration F which aggregates arbitrage views in that:

S,Y No Strong Arbitrage in (F) on ⌦ e ⌦,(F ) = ⌦⇤ = A ()M 6 ;() 6 ; S,Y M and F F F . Further, ⌦⇤ is analytice and there exists a trading strategy (↵⇤,H⇤) (F) ✓ ✓ 2A which is an Arbitrage Aggregator in that ↵⇤ +(H⇤ S)T 0 on ⌦ and e · e

⌦⇤ = ! ⌦ ↵⇤ (!)+(H⇤ S) (!)=0 . (5) { 2 | · T }

Moreover, one may take F and (↵⇤,H⇤) as constructed in (21) and (20) respectively.

All the proofs of the resultse of sections 2.1 and 2.2 are given in Section 4.

Remark 2.4 Several examples where Theorem 1 may fail replacing F with FS,Y ,canbefoundin Burzoni et al. (2016). We stress that = does not imply existence of a Strong Arbitrage in M⌦, ; S,Y e (F ) – this is true only under additional strong assumptions, see Theorem 2.12 and Example A 3.14 below. In general, the former corresponds to a situation when all agents agree that rational option pricing is not possible but they may disagree on why this is so. Our result shows that any such arbitrage views can be aggregated into one strategy (↵⇤,H⇤) in a ﬁltration F which does not perturb the calibrated martingale measures, see Remark 2.1 above. The proof of Theorem 2.3 relies e on an explicit — up to a measurable selection — construction of the enlarged ﬁltration F and the

Arbitrage Aggregator strategy (↵⇤,H⇤). It also puts in evidence that these objects may require the e use of projective sets, which is also explained in Remark 3.3.

We turn now to our second main result. For a given set of scenarios A X, deﬁne the superhedging ✓ price on A:

pr ⇡A,(g):= inf x R (↵,H) (F ) such that x + ↵ +(H S)T g on A . (6) { 2 |9 2A · }

7 Following the intuition in Burzoni et al. (2017), we expect to obtain pricing–hedging duality only when considering superhedging on the set of scenarios visited by martingales, i.e. we consider ⇡ (g). In Theorem 2.5 there is no need for the construction of a larger ﬁltration ˜, as explained ⌦⇤ , F above. Indeed, in Theorem 2.3 such a ﬁltration is used for aggregating arbitrage opportunities C which, in particular, yields a positive gain on the set (⌦⇤ ) . On the contrary, the aim of Theorem 2.5 is to show a pricing–hedging duality where both the primal and dual elements depend only on the set of ecient paths ⌦⇤ .

Theorem 2.5 Fix ⌦ analytic and a ﬁnite set of A-measurable statically traded options. Then, F for any A-measurable g F ⇡ (g)= sup E [g]= sup E [g] (7) ⌦⇤ , Q Q Q f Q ⌦, 2M⌦, 2M

pr and, if ﬁnite, the left hand side is attained by some strategy (↵,H) (F ). 2A For the case with no options, it was claimed in Burzoni et al. (2017) that the superhedging strategy is universally measurable. This is true under the set-theoretic axioms that guarantee that projective sets are universally measurable (see Remark 5.4), but in general Theorem 2.5 o↵ers a correction and only asserts measurability with respect to Fpr. To prove our main results, we ﬁrst deal with the case when = 0 and then extend iterating on the number k of statically traded options. The proofs are intertwined and we explain their logic at the beginning of Section 4. Further, for technical reasons, in Section 4 we need to show that some results stated here for ⌦analytic, such as Proposition 2.6 below, also extend to ⌦ ⇤, see Remark 4.5. 2 The following proposition is important as it shows that there are no One-Point arbitrage on ⌦if and only if each ! ⌦is weighted by some martingale measure Q f . 2 2M⌦, Proposition 2.6 Fix ⌦ analytic. Then there are no One-Point Arbitrages on ⌦ with respect to pr F if and only if ⌦=⌦⇤ .

Under a mild assumption, this situation has further equivalent characterisations:

Remark 2.7 Under the additional assumption that is not perfectly replicable on ⌦,thefollowing are easily shown to be equivalent:

(1) No One-Point Arbitrage on ⌦ with respect to Fpr.

k f (2) For any x R , when "x > 0 is small enough, = . 2 M⌦,+"xx 6 ; f (3) When ">0 is small enough, for any x Rk such that x <", = . 2 | | M⌦,+x 6 ; where +x = + x ,..., + x . { 1 1 n n} In particular, small uniform modiﬁcations of the statically traded options do not a↵ect the existence of calibrated martingale measures.

Remark 2.8 In Bouchard and Nutz (2015) the notion of no-Arbitrage (NA( )) depends on a P reference class of probability measures . If we choose as the set of all the probability measures P P

8 on ⌦ then NA( ) corresponds to the notion of no One-Point Arbitrage in this paper and, in this P case, Proposition 2.6 is a reformulation of the First Fundamental Theorem showed therein. Notice that such theorem in Bouchard and Nutz (2015) was proven under the assumption that ⌦ is equal to the T -fold product of a Polish space ⌦1 (where T is the time horizon). Proposition 2.6 extends this result to ⌦ equal to an analytic subset of a general Polish space.

Arbitrage de la classe . In Burzoni et al. (2016) a large variety of di↵erent notions of arbitrage S were studied, with respect of a given class of relevant measurable sets. In order to cover the present setting of statically traded options, we adapt the Deﬁnition of Arbitrage de la Classe in Burzoni S et al. (2016).

Definition 2.9 Let be a class of measurable subsets of ⌦ such that / .Fixafiltration S✓BX ; 2S F and a set of statically traded options . An Arbitrage de la classe on ⌦ is a strategy (↵,H) S 2 (F) such that ↵ +(H S)T 0 on ⌦ and ! ⌦ ↵ +(H S)T > 0 contains a set in . A · { 2 | · } S Notice that: (i) when = ⌦ then the Arbitrage de la Classe coincides with the notion of S { } S Strong Arbitrage; (ii) when the class consists of all non-empty subsets of ⌦the Arbitrage de la S Classe coincides with the notion of 1p-Arbitrage. S We now apply our Theorem 2.3 to characterize No Arbitrage de la classe in terms of the structure S of the set of martingale measures. In this way we generalize Burzoni et al. (2016) to the case of semi-static trading. Define M := A ⌦ Q(A)=0 Q f . N { ✓ | 8 2M⌦,}

Corollary 2.10 (FTAP for the class ) Fix ⌦ analytic and a ﬁnite set of A-measurable S F statically traded options. Then, there exists a ﬁltration F such that:

M No Arbitrage de la classe in (F) on ⌦ e ⌦, = and = S A ()M 6 ; N \S ; and FS,Y F FM . e ✓ ✓ e 2.2 Pointwise FTAP for arbitrary many options in the spirit of Acciaio et al. (2016)

In this section, we want to recover and extend the main results in Acciaio et al. (2016). A similar result can be also found in Cheridito et al. (2017) under slightly di↵erent assumptions. We work in the same setup as above except that we can allow for a larger, possible uncountable, set of pr statically traded options = i : i I . Trading strategies (↵,H) (F ) correspond to { 2 } 2A dynamic trading in S using H (Fpr) combined with a static position in a ﬁnite number of 2H options in .

d (T +1) Assumption 2.11 In this section, we assume that S takes values in R+⇥ and all the options are continuous derivatives on the underlying assets S, more precisely 2 d (T +1) i = gi S for some continuous gi : R ⇥ R, i I. + ! 8 2

9 In addition, we assume 0 I and = g (S ) for a strictly convex super-linear function g on 2 0 0 T 0 Rd, such that other options have a slower growth at inﬁnity:

T gi(x) lim =0, i I/ 0 , where m(x0,...,xT ):= g0(xt). x m(x) 8 2 { } | |!1 t=0 X

The option 0 can be only bought at time t =0. Therefore admissible trading strategies (F) A consider only positive values for the static position in 0.

The presence of 0 has the e↵ect of restricting non-trivial considerations to a compact set of values for S and then the continuity of gi allows to aggregate di↵erent arbitrages without enlarging the ﬁltration. This results in the following special case of the pathwise Fundamental Theorem of Asset

Pricing. Denote by ⌦, := Q ⌦, EQ[0] 0 . M { 2M \{ 0} |  }

Theorem 2.12 Consider ⌦ analytic and such that ⌦=⌦⇤, ⇡ ( ) > 0 and there exists !⇤ ⌦ f ⌦⇤ 0 2 such that S0(!⇤)=S1(!⇤)=...= ST (!⇤). Under Assumption 2.11, the following are equivalent:

pr (1) There is no Uniformly Strong Arbitrage on ⌦ in (F ); A pr (2) There is no Strong Arbitrage on ⌦ in (F ); A (3) = . M⌦, 6 ; d (T +1) Moreover, when any of these holds, for any upper semi-continuous g : R ⇥ R that satisﬁes f + ! g+(x) lim =0, (8) x m(x) | |!1 the following pricing–hedging duality holds:

⇡⌦,(g(S)) = sup EQ[g(S)]. (9) Q 2M⌦, Remark 2.13 We show in Remark 3.15 below thatf the pricing–hedging duality may fail in general when super-replicating on the whole set ⌦ as in (9). This conﬁrms the intuition that the existence of an option 0 which satisﬁes the hypothesis of Theorem 2.12 is crucial. However as shown in

Burzoni et al. (2017) Section 4, the presence of such 0 is not sucient. In fact the pricing hedging duality (9) may fail if g is not upper semicontinuous.

2.3 Classical Model Speciﬁc setting and its selection of scenarios

In this section we are interested in the relation of our results with the classical Dalang, Morton and Willinger approach from Dalang et al. (1990). For simplicity, and ease of comparison, throughout this section we restrict to dynamic trading only: = 0 and (FP)= (FP). For any ﬁltration F, A H we let FP be the P-completion of F. Recall that a (F, P)–arbitrage is a strategy H (FP)such 2A that (H S)T 0 P-a.s. and P((H S)T > 0) > 0, which is the classical notion of arbitrage.

Proposition 2.14 Consider a probability measure P and let ⌧P := Q Q P . 2P M { 2M| ⌧ } S,Y M There exists a set of scenarios ⌦P A and a ﬁltration F such that F F F and 2F ✓ ✓

No Strong Arbitrage in (F) on ⌦eP ⌧P = . e A ()M 6 ; e 10 Further, S,Y No (F , P)–arbitrage P (⌦P)⇤ =1 ⇠P = , () ()M 6 ; P where ⇠ := Q Q P . M { 2M| ⇠ } S,Y Proof. For 1 t T , we denote t 1 the random set from (40) with ⇠ =St and = t 1   G G F (see Appendix 5.1 for further details). Consider now the set

U := ! X St(!) t 1(!) { 2 | 2 } t=1 \ and note that, by Lemma 5.5 in the Appendix, U X and P(U) = 1. Consider now the set U ⇤ 2B deﬁned as in (4) (using U in the place of ⌦and for = 0) and deﬁne

U if P(U ⇤) > 0 ⌦P := , 8 U U ⇤ if P(U ⇤)=0 < \ which satisﬁes ⌦P A and P(⌦P) = 1.: 2F

In both the proofs of suciency and necessity the existence of the technical ﬁltration is consequence of Theorem 2.3. To prove suciency let Q ⌧P and observe that, since Q(⌦P) = 1, we have 2M P ⌦P = . Since necessarily Q(U ⇤) > 0wehaveP(U ⇤) > 0 and hence ⌦ = U X . From M 6 ; 2B Theorem 2.3 we have No (⌦P, F˜) Strong Arbitrage. To prove necessity observe ﬁrst that (⌦P)⇤ is either equal to U ⇤,ifP(U ⇤) > 0, or to the empty set otherwise. In the latter case, Theorem 2.3 with ⌦= U would contradict No (⌦P, F˜) Strong P P ˜ Arbitrage. Thus, (⌦ )⇤ = and P((⌦ )⇤)=P(U ⇤) > 0. Note now that by considering P( ):=P( 6 ; · ·| P ˜ (⌦ )⇤) we have, by construction 0 ri(t 1) P a.s. for every 1 t T ,whereri( ) denotes the 2   · relative interior of a set. By Rokhlin (2008) we conclude that P˜ admits an equivalent martingale measure and hence the thesis. The last statement then also follows.

3 Construction of the Arbitrage Aggregator and its ﬁltra- tion

3.1 The case without statically traded options

The following Lemma is an empowered version of Lemma 4.4 in Burzoni et al. (2016), which relies on measurable selections arguments, instead of a pathwise explicit construction. In the following ! we will set St = St St 1 and ⌃t 1 the level set of the trajectory ! up to time t 1 of both traded and non-traded assets, i.e.

! ⌃t 1 = ! X S0:t 1(!)=S0:t 1(!) and Y0:t 1(!)=Y0:t 1(!) , (10) { 2 | } where S0:t 1 := (S0,...,Se t 1)andY0:et 1 := (Y0,...,Yt 1) . Moreover,e by recalling that ⇤= 1 pr,n 1 1 n ⌃n (see the Appendix), we deﬁne t := ((Su,Yu) (L) L ⌃n,u t). [ 2N F | 2 

11 Lemma 3.1 Fix any t 1,...,T and ⇤. There exist n N, an index 0,...,d , 2{ } 2 2 2{ } 1 pr,n d pr,n 0 random vectors H ,...,H (X, t 1 ; R ), t -measurable sets E ,...,E such that the sets 2L F F Bi := Ei ,i=0,...,,formapartitionof satisfying: \ 1. if >0 and i =1,..., then: Bi = ; Hi S (!) > 0 for all ! Bi and Hi S (!) 0 6 ; · t 2 · t for all ! Bj B0. 2[j=i [ pr,n d 0 0 2. H (X, t 1 ; R ) such that H St 0 on B we have H St =0on B . 8 2L F · · Remark 3.2 Clearly if =0then B0 =(which include the trivial case = ). Notice also that ; for any ⇤ and t = 1,...,T we have that Hi = Hi,,Bi = Bi,, = depend explicitly on 2 { } t t t t and .

Remark 3.3 To appreciate why the use of projective sets is necessary, consider a market with d 2 assets, T 2 trading periods periods and a Borel selection of paths ⌦ .Fora 2BX given t>0 and a realized price St, an investor willing to exploit an arbitrage opportunity (if it exists) needs to analyze all the possible evolution of the price St+1 given the realized value St.

Mathematically, this conditional set is the projection of St+1 at time t and it is provided by Lemma 3.1. Such a projection does not preserve Borel measurability and therefore the arbitrage vector H1 (if it exists) is an analytically measurable strategy. Now if B1 = H1S > 0 = , agents considering B1 significant will call H1 an arbitrage { t+1 }6 ; 1 1 opportunity. On the other hand, even on the restricted set A := ⌦ B A there might \ 2F 1 be ineciencies. Note that, in general, A is neither Borel nor analytic but only in A.Its F projection, obtained by a second application of Lemma 3.1, does not need to be analytic measurability and the potential arbitrage strategy H2 will be, in general, projective. Agents considering B2 = H2S > 0 = significant will call H an arbitrage strategy. As shown by Lemma 3.1 { t+1 }6 ; 2 the geometry of the finite dimensional space Rd imposes that the procedure terminates in a finite number of steps.

Proof. Fix t 1,...,T and consider, for an arbitrary ⇤, the multifunction 2{ } 2 ! d t, : ! X St(!)1(!) ! ⌃t 1 R (11) 2 7! | 2 ✓

! 1 where ⌃t 1 is defined in (10). By definition ofe ⇤, theree e exists ` N such that ⌃` .Wefirst 2 2 pr,`+1 d show that t, is an t 1 -measurable multifunction. Note that for any open set O R F ✓ 1 ! X t,(!) O = =(S0:t 1,Y0:t 1) ((S0:t 1,Y0:t 1)(B)) , { 2 | \ 6 ;}

1 pr,` pr,` where B =(S 1 ) (O). First S 1 is an -measurable random vector then B , t t F 2F the sigma-algebra generated by the `-projective sets of X. Second Su,Yu are Borel measurable functions for any 0 u t 1 so that, from Lemma 5.3, we have that (S0:t 1,Y0:t 1)(B) belongs   to the sigma-algebra generated by the (` + 1)-projective sets of Mat((d + d˜) t; R) (the space ⇥ of (d + d˜) t matrices with real entries) endowed with its Borel sigma-algebra. Applying again ⇥ 1 pr,`+1 Lemma 5.3 we deduce that (S0:t 1,Y0:t 1) ((S0:t 1,Y0:t 1)(B)) t 1 and hence the desired 2F measurability for t,.

12 Let Sd be the unit sphere in Rd, by preservation of measurability (see Rockafellar and Wets (1998), pr,`+1 Chapter 14-B) the following multifunction is closed valued and t 1 -measurable F d ⇤ (!):= H S H y 0 y t,(!) . t, 2 | · 8 2 It follows that it admits a Castaing representation (see Theorem 14.5 in Rockafellar and Wets n pr,`+1 d (1998)), that is, there exists a countable collection of measurable functions ⇠t, n N (X, t 1 ; R ) { } 2 ✓L F n n such that ⇤ (!)= ⇠ (!) n N for every ! such that ⇤ (!) = and ⇠ (!) = 0 for ev- t, { t, | 2 } t, 6 ; t, n ery ! such that ⇤ (!)= . Recall that every ⇠ is a measurable selector of ⇤ and hence, t, ; t, t, ⇠n S 0 on . Note moreover that, t, · t d d n ! X, y R ⇠ y>0 = y R ⇠ (!) y>0 (12) 8 2 2 | · 2 | t, · ⇠ ⇤ (!) n N 2 [t, [2 The inclusion ( ) is clear, for the converse note that if y satisﬁes ⇠n (!) y 0 for every n N ◆ t, ·  2 then by continuity ⇠ y 0 for every ⇠ ⇤ (!). ·  2 t, We now deﬁne the the conditional standard separator as

1 1 ⇠ := ⇠n (13) t, 2n t, n=1 X pr,`+1 which is t 1 -measurable and, from (12), satisﬁes the following maximality property: ! X F { 2 | ⇠(!) S (!) > 0 ! X ⇠ (!) S (!) > 0 for any ⇠ measurable selector of ⇤ . · t }✓{ 2 | t, · t } t,

0 Step 0: We take A := and consider the multifunction t,A⇤ 0 and the conditional standard d separator ⇠t,A0 in (13). If ⇤ 0 (!) is a linear subspace of R (i.e. H ⇤ 0 (!)implies t,A 2 t,A 0 0 0 necessarily H ⇤ 0 (!) ) for any ! A then set = 0 and A = B (In this case 2 t,A 2 obviously E0 = X).

0 d Step 1: If there exists an ! A such that ⇤ 0 (!) is not a linear subspace of R then we set 2 t,A 1 1 0 1 H = ⇠ 0 , E = ! X H S > 0 , B = ! A H S > 0 = E and 1 t,A { 2 | 1 t } { 2 | 1 t } \ 1 0 1 0 d A = A B = ! A H1St =0 .Ifnow ⇤ 1 (!) is a linear subspace of R for any \ { 2 | } t,A ! A1 then we set = 1 and A1 = B0. If this is not the case we proceed iterating this 2 scheme.

1 d Step 2: notice that for every ! A we have St(!) R1(!):= y R H1(!) y =0 which 2 2 { 2 | · } can be embedded in a subspace of Rd whose dimension is d 1. We consider the case in 1 which there exists one ! A such that ⇤ 1 (!) is not a linear subspace of R (!): we set 2 t,A 1 2 2 0 2 H = ⇠ 1 , E = ! X H S > 0 , B = ! A H S > 0 = E and 2 t,A { 2 | 2 t } { 2 | 2 t } \ 2 1 2 1 A = A B = ! A H S =0 .Ifnow ⇤ 2 (!) is a linear subspace of R (!) for \ { 2 | 2 t } t,A 1 any ! A2 then we set = 2 and A2 = B0. If this is not the case we proceed iterating this 2 scheme.

The scheme can be iterated and ends at most within d Steps, so that, there exists n ` +2d  yielding the desired measurability.

13 Deﬁne, for ⌦ ⇤, 2

⌦T := ⌦

t i ⌦t 1 := ⌦t Bt,t 1,...,T , (14) \ 2{ } i=1 [ i i, where Bt := Bt , t := t are the sets and index constructed in Lemma 3.1 with =⌦t, for 1 t T . Note that we can iteratively apply Lemma 3.1 at time t 1since=⌦ ⇤.   t 2 Corollary 3.4 For any t 1,...,T , ⌦ analytic and Q we have t Bi is a subset of a 2{ } 2M⌦ [i=1 t Q-nullset. In particular t Bi is an polar set. [i=1 t M⌦

Proof. Let =⌦. First observe that the map T, in (11) is TA 1-measurable. Indeed the set F 1 1 1 B =(S 1 ) (O) is analytic since it is equal to S (O) if0/ O or S (O) if0 O. t t \ 2 t [ 2 1 1 The measurability of T, follows from Lemma 5.2. As a consequence, H and B from Lemma 1 d 1 1 1 3.1 satisfy: H (X, TA 1; R ) and B = H ST > 0 A. Suppose Q(B ) > 0. The 2L F { · }2F 1 strategy Hu := H 1T 1(u) satisﬁes: Q Q S Q H is F -predictable, where F = t ( X ) t 0,...,T . • {F _N B } 2{ } (H S) 0 Q-a.s. and (H S) > 0 on B1 which has positive probability. • · T · T Thus, H is an arbitrage in the classical probabilistic sense, which leads to a contradiction. Since B1 is a Q-nullset, there exists B˜1 such that B1 B˜1 and Q(B˜1) = 0. Consider now the Borel- 2BX ✓ ˜ measurable version of ST given by ST = ST 1X B˜1 + ST 11B˜1 . We iterate the above procedure \ replacing S with S˜ at each step up to time t. As in Lemma 3.1, the procedure ends in a ﬁnite ˜ ˜ number of step yielding a collection B˜i t such that t Bi t B˜i with Q( t B˜i) = 0. { t}i=1 [i=1 t ✓[i=1 t [i=1 t

Corollary 3.5 Let B0 the set provided by Lemma 3.1 for =⌦. For every ! B0 there exists t t 2 t f S Q with Q( ! ) > 0 such that EQ[St t 1](!)=St 1(!). 2P { } |F

0 ! ! 0 d Proof. Fix ! Bt and let ⌃t 1 be given as in (10). We consider D := St(⌃t 1 Bt ) R and 2 \ ✓ C := v v conv(D), R+ where conv(D) denotes the convex hull of D. Denote by ri(C) { | 2 2 } ! 0 the relative interior of C. From Lemma 3.1 item 2 we have H St(!) 0 for all ! ⌃t 1 Bt · 2 \ ! 0 implies H St(!) = 0 for all ! ⌃t 1 Bt , which is equivalent to 0 ri(C). From Remark 4.8 · 2 \ 2 in Burzoni et al. (2016) we have that for every x D there exists a ﬁnitee collectione x m D 2 { j}j=1 ✓ and m+1 withe 0 < 1,e m+1 = 1, such that { j}j=1 j  j=1 j P m 0= jxj + m+1x. (15) j=1 X ! 0 Choose now x := St(!) and note that for every j =1,...m there exists !j ⌃t 1 Bt 2 \ f S such that St(!j)=xj. Choose now Q with conditional probability Q( t 1)(!):= m 2P ·|F j=1 j!j + m+1!,where! denotes the Dirac measure with mass point in !. From (15), we have the thesis. P e e

14 Lemma 3.6 For ⌦ ⇤, the set ⌦⇤, defined in (4) with =0, coincides with ⌦ defined in (14), 2 0 and therefore ⌦⇤ ⇤. Moreover, if ⌦ is analytic then ⌦⇤ is analytic and we have the following 2 f ⌦⇤ = = = . 6 ;()M⌦ 6 ;()M⌦ 6 ; Proof. The proof is analogous to that of Proposition 4.18 in Burzoni et al. (2016), but we give here a self-contained argument. Notice that ⌦⇤ ⌦0 follows from the definitions and Corollary ✓ f 3.4. For the reverse inclusion, it suces to show that for ! ⌦0 there exists a Q such that ⇤ 2 2M⌦ Q( ! ) > 0, i.e. ! ⌦⇤. From Corollary 3.5, for any 1 t T , there exists a finite number of { ⇤} ⇤ 2   ! 0 elements of ⌃t 1 Bt named Ct(!):= !,!1,...,!m , such that \ { } m

St 1(!)=t(!)St(!)+ t(!j)St(!j) (16) j=1 X m where t(!) > 0 and t(!)+ j=1 t(!j) = 1. T Fix now ! ⌦0. We iteratively build a set ⌦f which is suitable for being the finite support of a ⇤ 2 P discrete martingale measure (and contains ! ). ⇤ 1 t 1 t Start with ⌦f = C1(! ) which satisfies (16) for t = 1. For any t>1, given ⌦ ,define⌦f := ⇤ f t 1 T C (!) ! ⌦ .Once⌦ is settled, it is easy to construct a martingale measure via (16): t | 2 f f n o T Q( ! )= (!) ! ⌦T { } t 8 2 f t=1 Y f Since, by construction, t(! ) > 0 for any 1 t T ,wehaveQ( ! ) > 0 and Q . ⇤   { ⇤} 2M⌦ For the last assertion, suppose ⌦is analytic. From Remark 5.6 in Burzoni et al. (2017),⌦ ⇤ is also analytic. In particular, if = then, from Corollary 3.4, Q(⌦⇤) = 1 for any Q .This M⌦ 6 ; 2M⌦ implies ⌦⇤ = . The converse implication is trivial. 6 ;

Lemma 3.7 Suppose =0. Then, no One-Point Arbitrage ⌦⇤ =⌦. , Proof. “ ”If(H S) 0 on ⌦then (H S) =0Q-a.s. for every Q f . From the hypothesis ( T T 2M⌦ we have supp(Q) Q f =⌦from which the thesis follows. “ ”. Let 1 t T and [{ | 2M⌦} )   1 =⌦t. Note that if t from Lemma 3.1 is strictly positive then H is a One-Point Arbitrage. We thus have = 0 for any 1 t T and hence ⌦ =⌦. From Lemma 3.6 we have ⌦⇤ =⌦. t   0 Deﬁnition 3.8 We call Arbitrage Aggregator the process

t i,⌦t Ht⇤(!):= Ht (!)1 i,⌦t (!) (17) Bt i=1 X for t 1,...,T , where Hi,⌦t ,Bi,⌦t , are provided by Lemma 3.1 with =⌦. 2{ } t t t t

Remark 3.9 Observe that from Lemma 3.1 item 1, (H⇤ S) (!) 0 for all ! ⌦ and from T 2 Lemma 3.6, (H⇤ S) (!) > 0 for all ! ⌦ ⌦⇤. T 2 \ pr,n Remark 3.10 By construction we have that H⇤ is -measurable for every t 1,...,T ,for t F 2{ } i,⌦t pr,n some n N. Moreover any B is the intersection of an -measurable set with ⌦t.Asa 2 t Ft d pr,n consequence we have (Ht⇤) ⌦t :⌦t R is ( t ) ⌦t -measurable. | ! F |

15 Remark 3.11 In case there are no options to be statically traded, =0, the enlarged ﬁltration F required in Theorem 2.3 is given by e S,Y := (H⇤,...,H⇤ ),t 0,...,T 1 (18) Ft Ft _ 1 t+1 2{ } S,Y := (H⇤,...,H⇤ ). (19) FeT FT _ 1 T so that the Arbitrage Aggregatore from (17) is predictable with respect to F = t t . {F } 2T e e 3.2 The case with a ﬁnite number of statically traded options

Throughout this section we consider the case of a finite set of options . As in the previous section pr,n 1 1 pr,n pr,n we consider t := ((Su,Yu) (L) L ⌃n,u t) and F := ( t )t . F | 2  F 2T Definition 3.12 A pathspace partition scheme (↵?,H?) of ⌦ is a collection of trading strategies R H1,...,H (Fpr,n),forsomen N, ↵1,...,↵ Rk and arbitrage aggregators H˜ 0,...,H˜ , 2H 2 2 for some 1 k, such that   (i) ↵i, 1 i , are linearly independent,   (ii) for any i ,  i i (H S)T + ↵ 0 on Ai⇤ 1, · i i where A0 =⌦, Ai := (H S)T + ↵ =0 Ai⇤ 1 and Ai⇤ is the set ⌦⇤ in (4) with ⌦=Ai { · }\ and =0for 1 i ,   (iii) for any i =0,...,, H˜ i is the Arbitrage Aggregator, as defined in (17) substituting ⌦ with

Ai,

k 1 (iv) if

(H S) + ↵ 0 on A⇤ . T ·

We note that as deﬁned in (ii) above, each A ⇤so that A⇤ ⇤by Lemma 3.6. The purpose i 2 i 2 of a pathspace partition scheme is to iteratively split the pathspace ⌦in subsets on which a Strong Arbitrage strategy can be identiﬁed. For the existence of calibrated martingale measure it will be crucial to see whether this procedure exhausts the pathspace or not. Note that on Ai we can perfectly replicate i linearly independent combinations of options ↵j , 1 j i.In ·   consequence, we make at most k such iterations, k, and if = k then all statically traded  options are perfectly replicated on A⇤ which reduces here to the setting without statically traded options.

? ? Deﬁnition 3.13 A pathspace partition scheme (↵ ,H ) is successful if A⇤ = . R 6 ; We illustrate now the construction of a successful pathspace partition scheme.

Example 3.14 Let X = R2. Consider a ﬁnancial market with one dynamically traded asset S and two options available for static trading :=(1,2). Let S be the canonical process i.e. St(x)=xt for t =1, 2 with initial price S =2. Moreover, let := g (S ,S ) c for i =1, 2 with c>0, 0 i i 1 2

16 + gi := (x2 Ki) 1[0,b](x1)+c1(b, )(x1) and K1 >K2. Namely, i is a knock in call option on S 1 with maturity T =2, strike price K , knock in value b 0 and cost c>0, but the cost is recovered i if the option is not knocked in. Consider the pathspace selection ⌦=[0, 4] [0, 4]. ⇥

Start with A0 :=⌦and suppose 0

˜ 0 ˜ 0 1. the process (H1 , H2 ):=(0, 1 0 (S1) 1 4 (S1)) is an arbitrage aggregator: when S1 hits { } { } the values 0, 4 , the price process does not decrease, or increase, respectively. It is easily { } seen that there are no more arbitrages on A0 from dynamic trading only. Thus, A0⇤ = (0, 4) [0, 4] 0 0 4 4 . { ⇥ }[{{ }⇥{ }}[{{ }⇥{ }} 2. Suppose now we have a semi-static strategy (H1,↵1) such that

1 1 (H S) + ↵ 0 on A⇤. T · 0 1 2 where ↵ =( ↵,↵) R for some ↵>0(sinceK1 >K2 and 1,2 have the same cost). 2 Moreover since = 0 if S / [0,b], we can choose (H1,H1)=(0, 0). A positive gain is 1 2 1 2 obtained on B := [0,b] (K , 4] (see Figure 1a). Thus, A := A⇤ B . 1 ⇥ 2 1 0 \ 1 3. (H˜ 1, H˜ 1):=(0, 1 (S )) is an arbitrage aggregator on A : the price process does not 1 2 [K2,b] 1 1 increase and (H˜ 1, H˜ 1) yields a positive gain on B := K S b K K (see 1 2 2 { 2  1  } \ {{ 2}⇥{ 2}} Figure 1b). Thus, A⇤ = A B . 1 1 \ 2 4. The null process H2 and the vector ↵2 := ( 1, 0) R2 satisﬁes 2 2 2 (H S) + ↵ 0 on A⇤. T · 1 A positive gain is obtained on [0,b] [0,K ]. Thus A := (b, 4) [0, 4] 4 4 . ⇥ 2 2 ⇥ [{{ }⇥{ }}

Obviously there are no more semi-static 1p-arbitrage opportunities and A2 = A2⇤, = 2. We set H˜ 2 0 and the partition scheme is successful with arbitrage aggregators H˜ 0, H˜ 1, H˜ 2, and ⌘ semi-static strategies (Hj,↵j), for j =1, 2, as above.

Remark 3.15 The previous example also shows that ⇡ (g)=⇡ (g) is a rather exceptional ⌦⇤ , ⌦, case if we do not assume the existence of an option with dominating payo↵as in Theorem 2.12.

Consider indeed the market of example 3.14 with b := 4, 0

Remark 3.16 Note that if a partition scheme is successful then there are no One-Point Arbitrages on A⇤ . When

B1 2 2

s s K2 K2 B2

t t 0 1 2 0 1 2

a) b)

Figure 1: Some steps of the pathspace partition scheme with b =1.5, K2 = 1. On the left, the 1 strategy (H1,↵ ), deﬁned on A0⇤, has positive gain on B1. As a consequence the pathspace reduces to A = A⇤ B . On the right, for S K the arbitrage aggregator H˜ , deﬁned on A , has 1 0 \ 1 1 2 1 1 positive gain on B2.

k i i i where Hˆ := H H . Since, by construction (H S) + ↵ =0on A⇤ for any i=1 i T · ˆ i =1,...,, we obtainP that H is a One-Point Arbitrage with =0on A⇤ . From Lemma 3.7 we have a contradiction.

Remark 3.17 As we shall see, Lemma 4.3 implies relative uniqueness of (↵?,H?) in the sense R ? ? ? ? that either every (↵ ,H ) is not successful or all (↵ ,H ) are successful and then A⇤ =⌦⇤ . R R Deﬁnition 3.18 Given a pathspace partition scheme we deﬁne the Arbitrage Aggregator as

? ? i i ˜ i (↵ ,H )= ↵ 1A⇤ , H 1A⇤ + H 1A A , (20) i i i\ i⇤ i=1 i=1 i=1 ! X X X ? ? ˜ 0 with (↵ ,H )=(0, H 1⌦ ⌦ ) if =0. \ ⇤ To make the above arbitrage aggregator predictable we need to enlarge the ﬁltration. We therefore introduce the arbitrage aggregating ﬁltration F given by

S,Y ˜ 0 ˜ ˜ 0 ˜ t =Ft A0,A0⇤,...,A,A⇤ e (H1 ,...,H1 ,...,Ht+1,...,Ht+1), F _{ }_ (21) S,Y ˜ 0 ˜ ˜ 0 ˜ T =F A0,A⇤,...,A,A⇤ (H ,...,H ,...,H ,...,H ). Fe T _{ 0 }_ 1 1 T T It will follow,e as a consequence of Lemma 4.2, that M for any t =0,...,T and in particular, Ft ✓FT S as observed before, any Q ⌦,(F ) extends uniquely to a measure in ⌦,(F). 2M e M e 4Proofs

We ﬁrst describe the logical ﬂow of our proofs and we point out that we need to show some of the results for ⌦ ⇤(and not only analytic). In particular, showing ⌦⇤ ⇤is involved. First, 2 2 Theorem 2.3 and then Theorem 2.5 are established when = 0. Then, we show Theorem 2.5

18 under the further assumption that ⌦⇤ ⇤for all 1 i k,wherei = 1,...,i . Note that i 2   { } in the case with no statically traded options (= 0) and for⌦ ⇤, the property ⌦⇤ =⌦⇤ ⇤ 2 2 follows from the construction and is shown in Lemma 3.6. This allows us to prove Proposition 2.6 for which we use Theorem 2.5 only when ⌦ =⌦, which belongs to ⇤by assumption. Proposition ⇤ i 2.6 in turn allows us to establish Lemma 4.3 which implies that in all cases ⌦⇤ ⇤. This then i 2 completes the proofs of Theorem 2.3 and Theorem 2.5 in the general setting.

4.1 Proof of the FTAP and pricing hedging duality when no options are statically traded

Proof of Theorem 2.3, when no options are statically traded. In this case we consider

⌦ ⇤, the technical ﬁltration as described in Remark 3.11 and the Arbitrage Aggregator H⇤ 2 deﬁned by (17). We prove that

f Strong Arbitrage on ⌦in (F) = . 9 H ,M⌦ ; f Notice that if H (F) satisﬁes (H S)T (!) > 0 ! ⌦then,ifthereexistse Q we would get 2H 8 2 2M⌦ 0 < EQ[(H S)T ] = 0 which is a contradiction. For the opposite implication, let H⇤ be the Arbitrage e c Aggregator from (17) and note that (H⇤ S) (!) 0 ! ⌦and ! (H⇤ S) (!) > 0 =(⌦⇤) . T 8 2 { | T } f c If = then, by Lemma 3.6, (⌦⇤) =⌦and H⇤ is therefore a Strong Arbitrage on ⌦in (F). M⌦ ; H The last assertion, namely ⌦⇤ = ! ⌦ (H⇤ S)T (!)=0 , follows straightforwardly from the { 2 | } e deﬁnition of H⇤.

Proposition 4.1 (Superhedging on ⌦ X without options) Let ⌦ ⇤. We have that for ✓ 2 any g (X, A; R) 2L F

⇡⌦⇤ (g)= sup EQ[g], (22) Q f 2M⌦ pr with ⇡⌦ (g)=inf x R H (F ) such that x +(H S)T g on ⌦⇤ . In particular, the ⇤ { 2 |9 2H } left hand side of (22) is attained by some strategy H (Fpr). 2H f Proof. Note that by its definition in (4), ⌦⇤ = if and only if = and in this case both sides ; M⌦ ; in (22) are equal to . We assume now that ⌦⇤ = and recall from Lemma 3.6 we have ⌦⇤ ⇤. 1 6 ; 2 1 By definition, there exists n N such that ⌦⇤ ⌃ . The second part of the statement follows 2 2 n with the same procedure proposed in Burzoni et al. (2017) proof of Theorem 1.1. The reason can be easily understood recalling the following construction, which appears in Step 1 of the proof. For any ` N, D pr,`,1 t T , G (X, pr,`), we define the multifunction 2 2F   2L F ! d+2 t,G,D : ! [St(!); 1; G(!)] 1D(!) ! ⌃t 1 R 7! | 2 ✓ 1 d ! where [St; 1; G] 1D = St 1D,...,Set 1D, 1De,G1D eande ⌃t is given as in (10). We show that pr,`+1 d 2 t,G,D is an t 1 -measurable⇥ multifunction. Let O ⇤ R R be an open set and observe that F ✓ ⇥ 1 ! X t,G,D(!) O = =(S0:t 1,Y0:t 1) ((S0:t 1,Y0:t 1)(B)) , { 2 | \ 6 ;} 1 pr,` where B =([S ; 1; G] 1 ) (O). First [S , 1,G] 1 is an -measurable random vector then t D t D F B pr,`, the sigma-algebra generated by the `-projective sets of X. Second (S ,Y ) is a Borel 2F u u

19 measurable function for any 0 u t 1 so that we have, as a consequence of Lemma 5.3,   that (S0:t 1,Y0:t 1)(B) belongs to the sigma-algebra generated by the (` + 1)-projective sets in Mat((d + d˜) t; R) (the space of (d + d˜) t matrices with real entries) endowed with its Borel ⇥ ⇥ 1 sigma-algebra. Applying again Lemma 5.3 we deduce that (S0:t 1,Y0:t 1) ((S0:t 1,Y0:t 1)(B)) pr,`+1 2 t 1 and hence the desired measurability for t,G,D. F The remaining of Step 1, Step 2, Step 3, Step 4 and Step 5 follows replicating the argument in Burzoni et al. (2017).

4.2 Proof of the FTAP and pricing hedging duality with statically traded options

We ﬁrst extend the results from Lemma 3.6 to the present case of non-trivial .

Lemma 4.2 Let ⌦ be analytic. For any Q we have Q(⌦⇤ )=1. In particular, = 2M⌦, M⌦, 6 ; if and only if f = . M⌦, 6 ;

Proof. Recall that ⌦analytic implies that⌦ ⇤ is analytic, from Remark 5.6 in Burzoni et al. (2017). Let Q˜ and consider the extended market (S, S˜)withS˜j equal to a Borel-measurable version 2M⌦, t S of E ˜ [j ] for any j =1,...,k and t T (see Lemma 7.27 in Bertsekas and Shreve (2007)). Q |Ft 2 In particular Q˜ ˜ , the set of martingale measure for (S, S˜) which are concentrated on ⌦. 2 M⌦ Denote by ⌦˜ ⇤ the set of scenarios charged by martingale measures for (S, S˜) as deﬁned in (4). f f From Corollary 3.4 and Lemma 3.6 we deduce that Q˜(⌦˜ ⇤) = 1. Since, obviously, ˜ we M⌦ ✓M⌦, also have ⌦˜ ⇤ ⌦⇤ . Since the former has full probability the claim follows. ✓

As highlighted above, we start with the proof of Theorem 2.5 under the further assumption that

⌦⇤ ⇤for all 1 n k,wheren = 1,...,n . This assumption is then shown to hold at n 2   { } the end of this subsection.

Proof of Theorem 2.5 under the assumption ⌦⇤ ⇤ for all n k. Similarly to the n 2  proof of Proposition 4.1 we note that the statement is clear when f = so we may assume M⌦, ; the contrary. For any A-measurable g, standard arguments implies, F sup E [g] ⇡ (g), Q ⌦⇤ , Q f  2M⌦, so that it remains to show the converse inequality. We prove the statement by induction on the number of static options used for superhedging. For this we consider the superhedging problem with additional options on ⌦ and denote its superhedging cost by ⇡ (g)whichisdeﬁned n ⇤ ⌦⇤ ,n as in (6) but with n replacing .

Assume that ⌦⇤ ⇤ for all n k. The case n = 0 corresponds to the super-hedging problem n 2  on ⌦⇤ when only dynamic trading is possible. Since by assumption ⌦⇤ ⇤, the pricing–hedging 2 duality and the attainment of the inﬁmum follow from Proposition 4.1. Now assume that for some n

20 We show that the same statement holds for n + 1. Note that the attainment property is always f satisfied. Indeed using the notation of Bouchard and Nutz (2015), we have NA( ⌦ , ). As a M ⇤ n consequence of Theorem 2.3 in Bouchard and Nutz (2015), which holds also in the setup of this paper, the infimum is attained whenever is finite.

The proof proceeds in three steps.

Step 1. First observe that if n+1 is replicable on ⌦⇤ by semi-static portfolios with the static hedging part restricted to n,i.e. x + h n(!)+(H S)T (!)=n+1(!), for any ! ⌦⇤ , f · 2 then necessarily x = 0 (otherwise ⌦, = ). Moreover since any such portfolio has zero ex- f M ; f pectation under measures in we have that EQ[n+1]=0 Q . In particular ⌦⇤ ,n ⌦⇤ ,n f f M 8 2M ⌦ , = ⌦ , and (23) holds for n + 1. M ⇤ n M ⇤ n+1

Step 2. We now look at the more interesting case, that is n+1 is not replicable. In this case, we show that:

sup EQ[n+1] > 0 and inf EQ[n+1] < 0. f Q f (24) Q ⌦ , ⌦⇤ ,n 2M ⇤ n 2M

f Inequalities and are obvious from the assumption ⌦ , = . From the inductive hypothesis  M ⇤ 6 ; we only need to show that ⇡ ( ) is always strictly positive (analogous argument applies ⌦⇤ ,n n+1 to ⇡⌦ , ( n+1)). Suppose, by contradiction, ⇡⌦ , (n+1) = 0. Since the inﬁmum is attained, ⇤ n ⇤ n there exists some (↵,H) Rn (Fpr) such that 2 ⇥H

↵ (!)+(H S) (!) (!) ! ⌦⇤ . · n T n+1 8 2

Since is not replicable the above inequality is strict for some! ˜ ⌦⇤ . Then, by taking n+1 2 ˜ f ˜ expectation under Q ⌦ , such that Q( !˜ ) > 0, we obtain 2M ⇤ { }

0=E ˜ [↵ +(H S) ] >E˜ [ ]=0. (25) Q · n T Q n+1 which is clearly a contradiction.

Step 3. Given (24), we now show that (23) holds for n + 1, also in the case that n+1 is not replicable. We ﬁrst use a variational argument to deduce the following equalities:

⇡⌦ ,n+1 (g)= inf⇡⌦ ,n (g ln+1) (26) ⇤ l ⇤ 2R =inf sup EQ[g ln+1] l R f 2 Q ⌦ , 2M ⇤ n

=inf inf sup EQ[g ln+1] N l N f | | Q ⌦ , 2M ⇤ n

=inf sup inf EQ[g ln+1], N f l N Q ⌦ , | | 2M ⇤ n

=inf sup (EQ[g] N EQ[n+1] ) N f | | Q ⌦ , 2M ⇤ n

21 The ﬁrst equality follows by deﬁnition, the second from the inductive hypothesis, the fourth is obtained with an application of min–max theorem (see Corollary 2 in Terkelsen (1972)) and the last one follows from an easy calculation.

f f We also observe that there exist Qsup ⌦ , and Qinf ⌦ , such that 2M ⇤ n 2M ⇤ n 1 1 EQ [n+1] ⇡⌦ , (n+1) 1 and EQ [n+1] ⇡⌦ , ( n+1) 1 . sup 2 ⇤ n ^ inf 2 ⇤ n ^ From (24) and the inductive hypothesis EQinf [n+1] < 0

EQm [g] lim =+ and lim EQ [g]=+ . (27) m E [ ] 1 m m 1 !1 | Qm n+1 | !1

Given Qm such that (27) is satisﬁed, we can construct a sequence of calibrated measures Qˆm f { } { }✓ ⌦ , , as described above, so that M ⇤ n+1

E ˆ [n+1]=mEQ [n+1]+(1 m)E ˜ [n+1]=0, Qm m Qm for some [0, 1]. We stress that Q˜ can only be equal to Q or Q , which do not depend { m}✓ m inf sup on m. A simple calculation shows

E [ ] Q˜m n+1 m = . E ˜ [n+1] EQ [n+1] Qm m From

E ˆ [g]=m(EQ [g] E ˜ [g]) + E ˜ [g] Qm m Qm Qm we have two cases: either m a>0 and from EQ [g] + we deduce E ˆ [g] + ; or ! m ! 1 Qm ! 1 0 which happens when E [ ] . Nevertheless in such a case, from (27) we obtain m ! | Qm n+1 |!1 again E [g] + as m . Therefore, =sup f E [g] ⇡ (g) and hence Qˆ Q ⌦⇤ ,n+1 m Q ⌦ , ! 1 !1 1 2M ⇤ n+1  the duality.

Case 2 We are only left with the case where (27) is not satisfied. For any N N,wedefine 2 m the decreasing sequence sN := sup f (EQ[g] N EQ[n+1] ) and let QN m N asequence Q ⌦ , 2M ⇤ n | | { } 2 m realizing the supremum. If there exists a subsequence sNj such that EQ [n+1] = 0 for m> | Nj | m¯ (Nj), then the duality follows directly from (26). Suppose this is not the case. We claim that f we can find a sequence QN ⌦ , such that { }✓M ⇤ n

lim (EQ [g] N EQ [n+1] )= lim sN and lim EQ [n+1] =0. (28) N N | N | N N | N | !1 !1 !1

22 Let indeed c(N):=limsupm EQm [g]/ EQm [n+1] .IfsupN c(N)= , since (27) is not !1 N | N | 2N 1 satisﬁed, there exists m = m(N) such that EQm(N) [n+1] converges to 0 as N , from which | N | !1 the claim easily follows. Suppose now supN c(N) < . Then (by taking subsequences if needed), 2N 1

sN =lim(EQm [g] N EQm [n+1] ) (c(N) N)lim EQm [n+1] , m N N m N !1 | |  !1 | |

Note that aN := limm EQm [n+1] satisﬁes limN NaN < otherwise, from (26), ⇡⌦ , (g)= !1 N !1 ⇤ n+1 | | f 1 which is not possible as, for any Q ⌦ ,, we have that ⇡⌦⇤ ,n+1 (g) EQ[g] > .In 1 2M ⇤ 1 particular, limN aN = 0 and the claim easily follows. !1 Given a sequence as in (28) we now conclude the proof. It follows from (26) that

⇡ (g)= inf sup inf E [g l ] ⌦⇤ ,n+1 Q n+1 N f l N Q ⌦ , | | 2M ⇤ n

=limEQ [g] N EQ [n+1] N N | N | !1 lim EQ [g].  N N !1 The calibrating procedure described above yields [0, 1] such that Qˆ = Q +(1 )Q˜ N 2 N N N N N 2 f ˜ ⌦ , . Moreover, as EQN [n+1] 0 and QN can only be either Qinf or Qsup,theseN satisfy M ⇤ n+1 | |! N 1. This implies, E ˆ [g] EQ [g] 0 as N from which it follows ! QN N ! !1 ⇡ (g) lim E [g]= lim E [g] sup E [g]. ⌦⇤ ,n+1 QN QˆN Q  N N  f !1 !1 Q ⌦ , 2M ⇤ n+1 The converse inequality follows from standard arguments and hence we obtain ⇡ (g)= ⌦⇤ ,n+1 sup f EQ[g] as required. Q ⌦ , 2M ⇤ n+1

We now prove Proposition 2.6 for the more general case of ⌦ ⇤. We use Theorem 2.5 only when 2 ⌦⇤ =⌦, which belongs to ⇤by assumption.

pr Proof of Proposition 2.6. “ ” is clear since, if a strategy (↵,H) (F ) satisﬁes ↵ + ( 2A · (H S) 0 on ⌦then, by deﬁnition in (4), for any ! ⌦⇤ =⌦, we can take a calibrated T 2 martingale measure which assigns a positive probability to !,whichimplies↵ +(H S) = 0 on · T !.Since! is arbitrary we obtain the thesis. We prove “ ” by iteration on number of options used ) for static trading. No One-Point Arbitrage using dynamic trading and in particular means that there is no One-Point Arbitrage using only dynamic trading. From Lemma 3.7 we have ⌦⇤ =⌦ and hence for any ! ⌦thereexistsQ f such that Q( ! ) > 0. 2 2M⌦ { } Note that if, for some j k, is replicable on ⌦⇤ by dynamic trading in S then there exist  j pr,n n N and (x, H) R (F ) such that x +(H S)T = j on ⌦⇤. No One-Point Arbitrage 2 2 ⇥H implies x = 0 and hence E [ ] = 0 for every Q f . With no loss of generality we assume that Q j 2M⌦ ( ,..., ) is a vector of non-replicable options on ⌦⇤ with k k. We now apply Theorem 2.5 1 k1 1  in the case with = 0 to 1 and argue that

m := min ⇡ ( ),⇡ ( ) > 0. 1 { ⌦⇤ 1 ⌦⇤ 1 }

Indeed, if m1 < 0 then we would have a Strong Arbitrage and if m1 = 0, since the superhedging pr price is attained, there exists H (F ) such that, for example, 1 (H S)T on ⌦. In order to 2H 

23 avoid One-Point Arbitrage we have to have 1 =(H S)T on ⌦which is a contradiction since 1 is f not replicable. This shows that m1 > 0whichinturnimpliesthereexistQ1,Q2 ⌦ such that f 2M EQ1 [1] > 0 and EQ2 [1] < 0. Then, for any Q ⌦,thereexist↵,, [0, 1], ↵ + + =1 2M f 2 and E↵Q +Q +Q[1] = 0. Thus, for any ! ⌦⇤ there exists Q such that Q( ! ) > 0. 1 2 2 2M⌦,1 { } In particular, ⌦⇤ =⌦and we may apply Theorem 2.5 with ⌦and = 1 (Indeed ⌦ ⇤and 1 { } 2 we can therefore apply the version of Theorem 2.5 proved in this section). Deﬁne now

m := min ⇡ ( ),⇡ ( ) j =2,...,k . 1,j { ⌦⇤,1, j ⌦⇤,1 j }8 1 By absence of Strong Arbitrage we necessarily have m 0 for every j =2,...,k . Let j I = 1,j 1 2 2 j =2,...,k1 m1,j =0, by No One-Point Arbitrage, we have perfect replication of j using { | } f semistatic strategies with 1 on ⌦and in consequence for any Q we have EQ[j] = 0 for 2M⌦,1 all j I . We may discard these options and, up to re-numbering, assume that ( ,..., )isa 2 2 2 k2 vector of the remaining options, non-replicable on ⌦with semistatic trading in ,withk k . 1 2  1 If k2 2, m1,2 > 0 by Theorem 2.5 and absence of One-Point Arbitrage using arguments as above. f Hence, there exist Q1,Q2 such that EQ [2] > 0 and EQ [2] < 0. As above, this implies 2M⌦,1 1 2 that ⌦ =⌦ =⌦. We can iterate the above arguments and the procedure ends after at ⇤ , ⇤ 1 { 1 2} most k steps showing ⌦⇤ =⌦as required.

The following Lemma shows that the outcome of a successful partition scheme is the set ⌦⇤ .

? ? Lemma 4.3 Recall the deﬁnition of ⌦⇤ in (4).Forany (↵ ,H ), Ai⇤ =⌦⇤↵j :j i for any R { ·  } ? ? i . Moreover, if (↵ ,H ) is successful, then A⇤ =⌦⇤ .  R

Proof. If ⌦⇤ = , then the claim holds trivial. We now assume ⌦⇤ = , fix a partition scheme ; 6 ; ? ? (↵ ,H ) and prove the claim by induction on i. For simplicity of notation, let ⌦i⇤ := ⌦⇤↵j :j i R { ·  } with ⌦0 =⌦. By definition of A0 we have A0⇤ =⌦⇤ =⌦0⇤. Suppose now Ai⇤ 1 =⌦i⇤ 1 for some i i . Then, by definition of ⌦i we have, ⌦i⇤ ⌦i⇤ 1 = Ai⇤ 1.Further,since(Hi S)T +↵ 0 on  ✓ f · Ai⇤ 1 with strict inequality on Ai⇤ 1 Ai, it follows that ⌦i⇤ Ai. Finally, from j ⌦i⇤, ↵ :j i f f f \ ✓ M { ·  } ✓ j = we also have ⌦⇤ A⇤. For the reverse inclusion consider ! A⇤. Ai, ↵ :j i Ai Ai⇤ i i i M { ·  } ✓M M ✓ f 2 By definition of Ai⇤ and Lemma 3.6, there exists Q A with Q( ! ) > 0. Since on Ai⇤, all 2M i⇤ { } options ↵j , 1 j i, are perfectly replicated by the dynamic strategies Hj, it follows that f ·   Q A , ↵j :j i so that ! ⌦i⇤. 2M i⇤ { ·  } 2 Suppose now (↵?,H?) is successful. In the case = k,since↵i form a basis of Rk we have f f R = j and hence ⌦ = A from the above. Suppose

Remark 4.4 It follows from Lemma 3.6 that Ai⇤ ⌦⇤↵j :j i introduced in Lemma 4.3, belongs ⌘ { ·  } to ⇤ for any i . In particular ⌦⇤ in (4) is in ⇤ (see also the discussion after Deﬁnition 3.12). 

Remark 4.5 Observe that, in the proof of Lemma 4.3, we apply Proposition 2.6 with ⌦=A⇤ which is only known to belong to ⇤. For this reason we need Proposition 2.6 to hold for a generic set in ⇤ (and not only analytic).

24 Proof of Theorem 2.3. We now prove the pointwise Fundamental Theorem of Asset Pricing, when semistatic trading strategies in a ﬁnite number of options are allowed. Let ⌦be analytic, (↵?,H?) be a pathspace partition scheme and F be given by (21). We ﬁrst show that the following R are equivalent: e 1. (↵?,H?)issuccessful. R 2. f = . M⌦, 6 ; 3. = . M⌦, 6 ; 4. No Strong Arbitrage with respect to F.

(1) (2) follows from Remark 3.17 and thee deﬁnition of ⌦⇤ in (4), since A⇤ =⌦⇤ . (2) (3) ) , follows from Lemma 4.2. To show (2) (4), observe that under Q f the expectation of any ) 2M⌦, admissible semi-static trading strategy is zero which excludes the possibility of existence of a Strong i 1 Arbitrage. For the implication (4) (1) note that, for 1 i ,wehave(H˜ S) > 0 on )   T i i Ai 1 Ai⇤ 1 from the properties of the Arbitrage Aggregator (see Remark 3.9) and (H S)T +↵ > \ · 0 on Ai⇤ 1 Ai by construction, so that a positive gain is realized on Ai 1 Ai. Finally, from \ \ ˜ ⌦=A0 =( i=1Ai 1 Ai) A and (H S)T > 0 on A A⇤ , we get [ \ [ \ i i i C (H S) + ↵ + (H˜ S) > 0 on (A⇤ ) , (29) T · T i=1 i=0 X X and equal to 0 otherwise. The hypothesis (4) implies therefore that A⇤ is non-empty and hence the pathspace partition scheme is successful. The existence of the technical ﬁltration and Arbitrage Aggregator are provided explicitly by (21) and (20). Moreover, from Lemma 4.3, ⌦⇤ = A⇤ . Finally, equation (5) follows from (29).

Proof of Corollary 2.10. Let F given by (21). We prove that

f M an Arbitrage de la classee in (F) = or contains sets of . 9 S A ()M ⌦, ; N S

( ): let (↵,H) (F) be an Arbitrage dee la classe . By deﬁnition, ↵ +(H S)T 0 on ) 2A S · ⌦and there exists A such that A ! ⌦ ↵ +(H S)T > 0 . Note now that for any 2eS ✓{ 2 | · } Q f we have E [↵ +(H S) ]=0whichimpliesQ( ! ⌦ ↵ +(H S) > 0 ) = 0. 2M⌦, Q · T { 2 | · T } Thus, if ! ⌦ ↵ +(H S) > 0 =⌦then f = , otherwise, A M . { 2 | · T } M⌦, ; 2N \S ( ): Consider the Arbitrage Aggregator (↵⇤,H⇤) as constructed in (21) which is predictable with ( respect to F given by (21). Let A M then, from (5) in Theorem 2.3, A ! ⌦ 2N \S ✓{ 2 | ↵⇤ +(H⇤ S)T > 0 which implies the thesis. · e }

Proof of Theorem 2.5 – justiﬁcation of the assumption previously made

As a consequence of Lemma 4.3, see also Remark 4.4, we obtain: ⌦⇤ ⇤for all n k. Therefore n 2  the assumption made in the proof of Theorem 2.5 at the beginning of this subsection is always satisﬁed. Moreover, for ⌦analytic, the equality between the suprema over and over f M⌦, M⌦, may be deduced following the same arguments as in the proof of Theorem 1.1, Step 2, in Burzoni et al. (2017). The proof is complete.

25 4.3 Proof of Theorem 2.12

We recall that the option 0 can be only bought at time t = 0 and the notations are as follows: pr pr 0 (F ):= (↵,H) R+ (F ) and ⌦,0 := Q ⌦ EQ[0] 0 . A { 2 ⇥H } M { 2M |  } We ﬁrst extend the results of Theorem 2.5 to the case where only 0 is available for static trading. e f f Lemma 4.6 Suppose = , ⇡⌦ (0) > 0 and ⌦⇤ A. Then, for any A-measurable g, M⌦,0 6 ; ⇤ 0 2F F

f ⇡⌦⇤ ,0 (g)= sup EQ[g]= sup EQ[g]. 0 Q f Q f 2M⌦,0 2M⌦,0 f f Proof. The assumption ⇡⌦⇤ (0) > 0 automatically implies supQ EQ[0] > 0. Moreover, by 2M⌦ f assumption, = from which inf f E [ ] 0. ⌦,0 Q Q 0 M 6 ; 2M⌦  The idea of the proof is the same of Theorem 2.5. Suppose ﬁrst that f inf EQ[0] < 0 . (30) Q f 2M⌦ Then it is easy to see that ⌦ =⌦. We use a variational argument to deduce the following ⇤ 0 ⇤ equality:

⇡⌦⇤ ,0 (g)=⇡⌦ ,0 (g)=inf sup (EQ[g] N EQ[0] ) 0 ⇤ N Q f | | 2M⌦ obtained with an application of min–max theorem (see Corollary 2 in Terkelsen (1972)). The last Step of the proof of Theorem 2.5 is only based on this variational equality and the analogous of

(30) joint with sup f E [ ] > 0. By repeating the same argument we obtain ⇡ (g)= Q Q 0 ⌦⇤ ,0 2M⌦ 0 supQ f EQ[g]. Since obviously 2M⌦,0

⇡⌦⇤ ,0 (g)= sup EQ[g] sup EQ[g] ⇡⌦⇤ ,0 (g) (31) 0 0 Q f  Q f  2M⌦,0 2M⌦,0 we have the thesis. f

f Suppose now infQ EQ[0] = 0. From Proposition 4.1, ⇡⌦⇤ ( 0) = 0 and there exists a strategy 2M⌦ ¯ pr ¯ H (F ) such that (H S)T 0 on ⌦⇤. We claim that the inequality is actually an equality 2H on ⌦ (which is non-empty by assumption). If indeed for some ! ⌦ the inequality is strict ⇤ 0 ⇤ 0 f 2 then, any Q such that Q( ! ) > 0, satisﬁes EQ[0] > 0, which contradicts ! ⌦⇤ .This 2M⌦ { } 2 0 implies that is replicable on ⌦ and thus, f = f = f . In such a case, 0 ⇤ 0 ⌦,0 ⌦,0 ⌦⇤ M M M 0

⇡⌦⇤ ,0 (g) ⇡⌦⇤ (g)= supf EQ[g]= sup EQ[g], 0 0  Q f Q f ⌦ ⌦,0 2M ⇤0 2M where, the ﬁrst equality follows from Proposition 4.1, since ⌦⇤ A ⇤by assumption. The 0 2F ✓ thesis now follows from standard arguments as above.

Proposition 4.7 Assume that ⌦ satisﬁes that there exists an !⇤ such that S0(!⇤)=S1(!⇤)=

...= ST (!⇤), ⌦=⌦⇤ and ⇡⌦⇤ (0) > 0. Then the following are equivalent:

pr (1) There is no Uniformly Strong Arbitrage on ⌦ in 0 (F ); A pr (2) There is no Strong Arbitrage on ⌦ in 0 (F ); e A e 26 (3) = . M⌦,0 6 ; f (4) f = . M⌦,0 6 ; d (T +1) Moreover,f when any of these holds, for any upper semi-continuous g : R ⇥ R such that + ! g+(x) lim =0, (32) x m(x) | |!1 T where m(x0,...,xT ):= t=0 g0(xt), we have the following pricing–hedging duality: P ⇡⌦⇤,0 (g(S)) = sup EQ[g(S)] = sup EQ[g(S)]. (33) Q Q ⌦,0 2M⌦,0 2M f Remark 4.8 We observe that the assumption ⇡⌦⇤ (0) > 0 is not binding and can be removed. In fact if ⇡ ( ) 0,(1) (3) is obviously satisfied since = = . The di↵erence is ⌦⇤ 0  ) M⌦,0 M⌦ 6 ; that the pricing–hedging duality (33) is (trivially) satisfied only in the first equation. f Proof of Proposition 4.7. (3) (2) and (2) (1) are obvious. (4) (3) is an easy consequence of Theorem 2.3. To show ) ) , pr (1) (4), we suppose there is no Uniformly Strong Arbitrage on ⌦in 0 (F ). ) A We first show that the interesting case is ⇡⌦ (0) > 0 and ⇡⌦ ( 0) = 0. The other cases follows ⇤ ⇤ e trivially from Proposition 4.1 and Lemma 4.6:

pr If ⇡⌦ ( 0) < 0, since the superhedging price is attained and ⌦=⌦⇤,thereexistH F • ⇤ 2 and x<0 such that (!)+(H S) (!) x>0, ! ⌦ 0 T 8 2 which is clearly a Uniform Strong Arbitrage on ⌦.

If ⇡⌦⇤ ( 0) > 0 and ⇡⌦⇤ (0) > 0, we have that 0 is in the interior of the price interval • f f formed by inf f E [ ] and sup f E [ ]. Thus, = and, it is Q Q 0 Q Q 0 ⌦,0 ⌦,0 2M⌦ 2M⌦ M ◆M 6 ; straightforward to see that ⌦⇤ =⌦⇤. 0 f

Note that in all these cases ⌦⇤ =⌦⇤ =⌦ A and hence (33) follows from Lemma 4.6. 0 2F

The remaining case is ⇡ ( ) > 0 and ⇡ ( ) = 0. In this case, by considering the !⇤ ⌦⇤ 0 ⌦⇤ 0 such that s = S (!⇤)=S (!⇤)=... = S (!⇤), we observe that the super-replication of 0 0 1 T 0 necessarily requires an initial capital of, at least, g (s ). From ⇡ ( ) = 0 we can rule out 0 0 ⌦⇤ 0 the possibility that g0(s0) < 0. Note now that, by the convexity of g0, for any l 0,...,T 1, T 2 g0(ST (!)) g0(Si 1(!)) Si(!) Si 1(!) g0(Sl(!)) for any ! ⌦. In particular, when i=l+1 2 l = 0, P ⇣ ⌘ T

g0(ST (!)) g0(Si 1(!)) Si(!) Si 1(!) g0(s0) ! ⌦. (34) 8 2 i=1 X ⇣ ⌘ Denote by H¯ the dynamic strategy in (34). If g0(s0) > 0, (1, H¯ ) is a Uniformly Strong Arbitrage on ⌦and hence, a contradiction to our assumption. Thus, g0(s0) = 0. In this case, it is obvious f f that the Dirac measure ! which is therefore non-empty. ⇤ 2M⌦,0 ✓ M⌦,0 f

27 f Moreover, since ! ⌦, , ⇤ 2M 0

sup EQ[g(S)] sup EQ[g(S)] g(s0,...,s0). Q f Q f 2M⌦,0 2M⌦,0 Case 1. Suppose g is boundedf from above. We show that it is possible to super-replicate g with any initial capital larger than g(s0,...,s0). To see this recall that, from the strict convexity of d (T +1) g0, the inequality in (34) is strict for any s R ⇥ such that s is not a constant path, i.e., 2 + s = s for some i 0,...,T . In fact, it is bounded away from 0 outside any small ball of i 6 0 2{ } (s0,...,s0). Hence, due to the upper semi-continuity and boundedness of g, for any ">0, there exists a suciently large K such that

g(s0,...,s0)+" + K g0(ST (!)) g0(Si 1(!)) Si(!) Si 1(!) g(S(!)) ! ⌦⇤ 8 2 i=1 n X o

f f Therefore, ⇡⌦⇤,0 (g) g(s0,...,s0) supQ EQ[g(S)] supQ EQ[g(S)]. The con-   2M⌦,0  2M⌦,0 verse inequality is easy and hence we obtain ⇡⌦ , (g)=sup f EQ[g(S)] = sup f EQ[g] ⇤ 0 Q f Q 2M⌦,0 2M⌦,0 as required. f Case 2. It remains to argue that the duality still holds true for any g that is upper semi-continuous d (T +1) and satisfies (32). We first argue that any upper semi-continuous g : R ⇥ R, satisfying + ! (32), can be super-replicated on ⌦⇤ by a strategy involving dynamic trading in S, static hedging d (T +1) in g0 and cash. Define a synthetic option with payo↵ m : R ⇥ R by + ! T T e m(x0,...,xT )= g0(xT ) g0(xi 1)(xi xi 1) . (35) Xl=0 n i=Xl+1 o e By convexity of g0, we know that

T T T

m(x0,...,xT )= g0(xT ) g0(xi 1)(xi xi 1) g0(xl)=m(x0,...,xT ). Xl=0 n i=Xl+1 o Xl=0 e Since we assume there is no Uniform Strong Arbitrage, it is clear that ⇡⌦⇤,0 (m(S)) = 0. From (32) it follows that g(S) m(S) is bounded from above. By sublinearity of ⇡ ( ), we ⌦⇤,0 · have e e ⇡ (g) ⇡ (g(S) m(S)) + ⇡ (m(S)) ⌦⇤,0  ⌦⇤,0 ⌦⇤,0 =supEQ[g(S) m(S)] + 0 Q f e e 2M⌦,0 e =supEQ[g(S)] sup EQ[g(S)], Q f  Q f 2M⌦,0 2M⌦,0 where the ﬁrst equality follows from the pricing–hedging dualityf for claims bounded from above, that we established in Case 1, and the fact that ⇡ (m(S)) = 0. Moreover, for Q , ⌦⇤,0 2M⌦,0 EQ[m(S)] = 0 from which the second equality follows. e The converse inequality follows from standard arguments and hence we have obtained ⇡⌦⇤,0 (g)= e supQ f EQ[g]=supQ f EQ[g]. The equality with the supremum over ⌦,0 and ⌦,0 2M⌦,0 2M⌦,0 M M follows from the same argument for the proof of Theorem 1.1, Step 2, in Burzoni et al. (2017). f f

28 Proof of Theorem 2.12. (3) (2) and (2) (1) are obvious. ) ) Step 1. To show that (1) implies (3). Suppose there is no Uniformly Strong Arbitrage on ⌦in

(F). We first know from Proposition 4.7 that ⌦,0 = . We can use a variational argument A M 6 ; d (T +1) to deduce the following equalities: fix an arbitrary K>0 and let m : R+⇥ R defined as in e ! (35) e

⇡⌦⇤,(g(S)) = inf ⇡⌦⇤,0 (g(S) X) X Lin(/ 0 ) 2 { }

=inf sup EQ[g(S) X] (36) X Lin(/ 0 ) Q 2 { } 2M⌦,0

=inf sup EQ[g(S) X Km(S)] (37) X Lin(/ 0 ) Q 2 { } 2M⌦,0 e where the second equality follows from Proposition 4.7. Denote by (respectively ) the set of Q Q law of S under the measures Q ⌦,0 (respectively Q ⌦,0 ) and write Lin( gi i I/ 0 ) for 2M 2 M { e} 2 { } the set of ﬁnite linear combinations of elements in gi i I/ 0 . Observe that from Step 1 of the { } 2 {f} proof of Theorem 1.3 in Acciaio et al. (2016) is weakly compact and hence the same is true for Q (the weak closure of ). Q Q e By a change of variable we have

inf sup EQ[g(S) X Km(S)] = inf sup EQ[g(S)], X Lin(/ 0 ) Q G Lin( gi i I/ 0 ) 2 { } 2M⌦,0 2 { } 2 { } Q2Q

T d e(T +1) e where S := (St) is the canonical process on R ⇥ and g = g G Km. We aim at applying t=0 + min–max theorem (see Corollary 2 in Terkelsen (1972)) to the compact convex set , the convex e e Q set Lin( gi i I/ 0 ), and the function { } 2 { }

f( ,G)= g(s0,...,sT ) G(s0,...,sT ) Km(s0,...,sT ) d (s0,...,sn). Q d (T +1) Q ⇥ ZR+ ⇣ ⌘ Clearly f is ane in each of the variables. Furthermore, wee show that f( ,G)isuppersemi- · continuous on .Toseethis,ﬁxG Lin( gi i I/ 0 ). By deﬁnition of f we have that Q 2 { } 2 { }

f(Q,G)=EQ[g(S)]. (38)

It follows from Assumption 2.11 and (8) that g is boundede from above. Hence, for every sequence of Qn n with Qn Q as n for some Q weakly, we have { } 2 Q ! !1 e + + lim E n [g (S)] E [g (S)] n Q Q !1  by Portmanteau theorem, and e e

lim inf E n [g(S)] E [g(S)] n Q Q !1 + + + by Fatou’s lemma, where g := g g with eg := max g,e0 , g := ( g) .Then { }

lim supef(Qne,G)=limsupe eEQn [g(S)] e EQ[ge(S)] = f(eQ,G). n n  !1 !1 e e

29 Therefore, the assumptions of Corollary 2 in Terkelsen (1972) are satisﬁed and we have, by recalling and equation (37), Q✓Q

⇡⌦⇤,(g(S)) = inf sup EQ[g(S) X Km(S)] X Lin(/ 0 ) Q 2 { } 2M⌦,0

=infsup EQ[g(S) G(S) Kem(S)] G Lin( gi i I/ 0 ) 2 { } 2 { } Q2Q

inf sup EQ[g(S) G(S) Kme (S)]  G Lin( gi i I/ 0 ) 2 { } 2 { } Q2Q

=sup inf EQ[g(S) G(S) Kme (S)] G Lin( gi i I/ 0 ) Q2Q 2 { } 2 { }

sup inf EQ[g(S) G(S) Kme (S)]  G Lin( gi i I/ 0 ) 2 { } 2 { } Q2Q =supe inf EQ[g(S) X Kme(S)]. (39) X Lin(/ 0 ) Q ⌦, 2 { } 2M 0 f e Take g = 0. If = ,then M⌦, ;

f sup inf EQ[ X Km(S)] = , X Lin(/ 0 ) 1 Q ⌦, 2 { } 2M 0 and hence ⇡ (0) = ,f which contradicts the no arbitragee assumption. Therefore we have (1) ⌦⇤, 1 implies (3). Step 2. To show the pricing–hedging duality, suppose now = .IfQ ,then M⌦, 6 ; 62 M⌦,

inf EQ[g(S) X Km(Sf)] = . f X Lin(/ 0 ) 1 2 { } Therefore, in (39), it suces to look at measures in e= only, and hence we obtain M⌦, 6 ;

⇡⌦⇤,(g(S)) sup EQ[g(S) Kfm(S)]  Q 2M⌦, supf EQ[g(S)] + Ke sup EQ[ m(S)]  Q Q 2M⌦, 2M⌦, =supf EQ[g(S)] + K(T +f 1) sup eEQ[ g0(S)] Q Q 2M⌦, 2M⌦, f f Since g0 is bounded from above, the quantity supQ EQ[ g0(S)] is ﬁnite and, by recalling 2M⌦, that K>0 is arbitrary, we get the thesis for K 0. # f

5 Appendix

Let X be a Polish space. The so-called projective hierarchy (see Kechris (1995) Chapter V) is 1 constructed as follows. The ﬁrst level is composed of the analytic sets ⌃1 (projections of closed subsets of X NN), the co-analytic sets ⇧1 (complementary of analytic sets), and the Borel sets ⇥ 1 1 =⌃1 ⇧1. The subsequent level are deﬁned iteratively through the operations of projection 1 1 \ 1 and complementation. Namely,

1 1 ⌃ = projections of ⇧ subsets of X NN, n+1 n ⇥ 1 1 ⇧n+1 = complementary of sets in ⌃n+1, 1 =⌃1 ⇧1 . n+1 n+1 \ n+1

30 From the deﬁnition it is clear that ⌃1 ⌃1 , for any n N, and analogous inclusions hold for ⇧1 n ✓ n+1 2 n 1 and n. Sets in the union of the projective classes (also called Lusin classes) are called projective 1 1 1 sets, which we denoted by ⇤:= n1=1 n = n1=1 ⌃n = n1=1 ⇧n.

S1 1 S S Remark 5.1 We observe that ⌃ ⇧ is a sigma algebra which actually coincides with A. 1 [ 1 F 1 1 1 Moreover ⌃ ⇧ = A . 1 [ 1 F ✓ 2

We ﬁrst recall the following result from Kechris (1995) (see Exercise 37.3).

Lemma 5.2 Let f : X Rk be Borel measurable. For any n N, 7! 2 1 1 1 1. f (⌃ ) ⌃ ; n ✓ n 2. f(⌃1 ) ⌃1 . n ✓ n The following is a consequence of the previous Lemma.

Lemma 5.3 Let f : X Rk be Borel measurable. For any n N, 7! 2 1 1 1 1. f ((⌃ )) (⌃ ); n ✓ n 2. f((⌃1 )) ⌃1 . n ✓ n+1 1 Proof. From Lemma 5.2 the ﬁrst claim hold for ⌃n which generates the sigma-algebra. In 1 particular, ⌃n is contained in

1 1 1 1 A (⌃ ) f (A) (⌃ ) (⌃ ). 2 n | 2 n ✓ n 1 Since the above set is a sigma-algebra, it also contains (⌃n), from which the claim follows. For the second assertion, we recall that 1 is a sigma-algebra for any n N (see Proposition 37.1 in n 2 Kechris (1995)). In particular,

(⌃1 ) (⌃1 ⇧1 ) 1 ⌃1 . n ✓ n [ n ✓ n+1 ✓ n+1 Since, from Lemma 5.2, f(⌃1 ) ⌃1 , the thesis follows. n+1 ✓ n+1 Remark 5.4 We recall that under the axiom of Projective Determinacy the class ⇤, and hence also pr, is included in the universal completion of (see Theorem 38.17 in Kechris (1995)). F BX This axiom has been thoroughly studied in set theory and it is implied, for example, by the existence of inﬁnitely many Woodin cardinals (see e.g. Martin and Steel (1989)).

5.1 Remark on conditional supports

Let be a countably generated sub -algebra of . Then there exists a proper regular G✓BX BX conditional probability, i.e. a function P ( , ):(X, X ) [0, 1] such that: G · · B 7!

a) for all ! ⌦, P (!, ) is a probability measure on X ; 2 G · B

b) for each (ﬁxed) B X , the function P ( ,B)is -measurable and a version of E [1B ]( ) 2B G · G P |G · (here the null set where they di↵er depends on B);

31 c) N with P(N) = 0 such that P (!,B)=1B(!) for ! X N and B (here the 9 2G G 2 \ 2G null set where they di↵er does not depend on B); moreover for all ! X N,wehave 2 \ P (!,A!)=1whereA! = A : ! A, A . G { 2 2G}2G T d) For every A and B X we have P(A B)= P (!,B)P(d!) 2G 2B \ A G d R We now consider a measurable ⇠ : X R and P⇠ : X d [0, 1] deﬁned by ! ⇥BR !

P⇠(!,B):=P (!, !˜ A! ⇠(˜!) B ), G { 2 | 2 } and observe that from a) and with N as in c), for any ! X N, P (!, ) is a probability measure 2 \ ⇠ · d on (R , d ). Finally, we let B"(x) denote the ball of radius " with center in x, and we introduce BR the closed valued random set

d ! (!):= x R P⇠(!,B"(x)) > 0 ">0 , (40) ! G { 2 | 8 } for ! X N and Rd otherwise. is -measurable since, for any open set O Rd,wehave 2 \ G G ✓ 1 ! X (!) O = = N ! X N P⇠(!,O) > 0 = N ! X N P (!,⇠ (O) A!) > 0 , { 2 | G \ 6 ;} [{ 2 \ | } [{ 2 \ | G \ } with the latter belonging to from b) and c) above. By deﬁnition (!) is the support of P⇠(!, ) G G · and therefore, for every ! X, P⇠(!, (!)) = 1. Notice that since the map is -measurable 2 G G G then for ! X we have (!)= (˜!) for all! ˜ A!. 2 G G 2

Lemma 5.5 Under the previous assumption we have ! X ⇠(!) (!) X and P( ! { 2 | 2 G }2B { 2 X ⇠(!) (!) )=1. | 2 G }

Proof. Set B := ! X ⇠(!) (!) . B X follows from the measurability of ⇠ and . { 2 | 2 G } 2B G From the properties of regular conditional probability we have

P(B)=P(B X)= P (!,B)P(d!). \ G ZX Consider the atom A! = A : ! A, A . From property c) we have P (!,A!) = 1 for any \{ 2 2G} G ! X N. Therefore for every ! X N we deduce 2 \ 2 \

P (!,B)=P (!,B A!)=P (!, !˜ A! ⇠(˜!) (˜!) ) G G \ G { 2 | 2 G } = P (!, !˜ A! ⇠(˜!) (!) )=P⇠(!, (!)) = 1. G { 2 | 2 G } G Therefore

P(B)= P (!,B)P(d!)= 1X N (!)P(d!)=1. G \ ZX ZX

Acknowledgments.

Zhaoxu Hou gratefully acknowledges the support of the Oxford-Man Institute of Quantitative Fi- nance. Jan Oblójgratefully acknowledges funding received from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agree- ment no. 335421. Jan Oblójis also thankful to the Oxford-Man Institute of Quantitative Finance and St John’s College in Oxford for their financial support.

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