A conditional version of the second fundamental theorem of asset pricing in discrete time
aa r X i v : . [ q -f i n . M F ] F e b A CONDITIONAL VERSION OF THE SECOND FUNDAMENTALTHEOREM OF ASSET PRICING IN DISCRETE TIME
LARS NIEMANN AND THORSTEN SCHMIDT
Abstract.
The second fundamental theorem of asset pricing characterizescompleteness of a financial market by uniqueness of prices of financial claims.The associated super- and sub-hedging dualities give upper and lower boundsof the no-arbitrage interval. In this article we provide conditional versions ofthese results in discrete time.The main tool we use are consistency properties of dynamic non-linear ex-pectations, which we apply to the super- and sub-hedging prices. The obtainedresults extend existing results in the literature, where the conditional settingis in most cases considered only on finite probability spaces. Using non-linearexpectations we also provide a new perspective on the optional decompositiontheorem.
Keywords: asset pricing, fundamental theorems, super- and sub-hedging du-ality, nonlinear expectation, risk measures, optional decomposition. Introduction
The mathematical analysis of financial markets starts with the remarkable thesisof Bachelier, submitted to the Academy of Paris in 1900. More than half a centurylater, the search for a precise theory of option valuation was continued in Samuelson(1965) and encompassed with the observation that replication is a key to pricing inthe famous works Black and Scholes (1973), and Merton (1973) (honoured by theNobel prize in Economics in 1991).The connection to martingales and martingale measures was started in the worksHarrison and Kreps (1979) and Harrison and Pliska (1981), which builds the foun-dation for what we nowadays call the fundamental theorems of asset pricing. Incontinuous time, semimartingales turned out to play the central role and the funda-mental theorems in this setting were established in a series of papers, see Delbaenand Schachermayer (1994, 1998, 2006) and references therein.While the study of arbitrage in continuous time requires subtle arguments usingsemimartingale theory, the results simplify significantly in discrete time. In thisrealm, the first fundamental theorem of asset pricing characterizes absence of arbi-trage of a financial market by the existence of an equivalent martingale measure.It is the core of modern financial mathematics.The second fundamental theorem considers the more special case when a marketis complete, i.e. when every European contingent claim can be replicated perfectly
Department of Mathematical Stochastics, University of Freiburg, Germany
E-mail addresses : [email protected],[email protected] . Date : March 1, 2021.Financial support from the DFG in project SCHM 2160/13-1 is gratefully acknowledged. by a trading strategy, see Biagini (2010) for an overview and literature. It char-acterizes completeness by uniqueness of the prices for claims. This well-knownresult dates back to Harrison and Kreps (1979), Harrison and Pliska (1981, 1983),although in the realm of continuous processes which excludes discrete time. Indiscrete time it was proven on a finite probability space in Taqqu and Willinger(1987), see also F¨ollmer and Schied (2011). These results however consider onlythe initial time point 0 and a conditional version is lacking.This is the topic of the present article. We will provide a self-contained proofof a conditional formulation of the second fundamental theorem of asset pricingtogether with a conditional version of the associated sub-hedging and super-hedgingdualities. These results were shown in Delbaen and Schachermayer (2006), howeverunder the strong simplification of a finite probability space.Consider a finite number of time points which we denote without loss of generalityby { , . . . , T } . In this setting we study a financial market which is free of arbitrage.Our first main result, Theorem 5.5, shows the following two (among four) equivalentconditions and hence is a conditional version of the second fundamental theorem ofasset pricing. Theorem 1.1.
For any t ∈ { , . . . , T } , the following statements are equivalent:(i) The market is complete at time t , i.e. every European contingent claim H ∈ L ( F T ) is attainable at time t .(ii) Every European contingent claim H ∈ L ( F T ) has a unique price at time t . Equally important is the characterization of the no-arbitrage interval and theassociated duality results. In the unconditional case, the no-arbitrage interval of acontingent claim collapses to a singleton if and only if the claim can be replicatedby a self-financing trading strategy, which is called attainability (see Section 5.1 fordetails). Here, we generalize this to the conditional setting. First, for any randomvariable H denote by π sup t ( H ) := ess sup { E Q [ H | F t ] : Q ∈ M He } , and π inf t ( H ) := ess inf { E Q [ H | F t ] : Q ∈ M He } the upper and the lower (conditional) bound of the no-arbitrage interval; here M He denotes the set of equivalent martingale measures under which H is integrable. InTheorem 5.4 we show the following. Theorem 1.2.
Let H ∈ L ( F T ) be a European contingent claim. Then H isattainable at time t if and only if π inf t ( H ) = π sup t ( H ) , i.e. when there is a unique arbitrage free price at time t . If H is not attainable attime t , then π inf t ( H ) and π sup t ( H ) are not arbitrage free prices at time t . The associated super- and sub-hedging dualities state that the upper (lower)bound of the no-arbitrage interval equals the smallest super-hedging price, or thelargest sub-hedging price, respectively. The conditional version of this result will beproven in Propositions 3.11 and 3.14. The smallest super-hedging price gives rise toa non-linear expectation, which will be the main tool for this proof. We thereforestart with a small exposition of non-linear expectations in a dynamic setting andprovide a series of useful results in this regard.
ONDITIONAL SECOND FUNDAMENTAL THEOREM 3
Non-linear dynamic expectations and their counterpart dynamic risk measureshave been studied in many places in the literature. Introduced in Peng (2005),non-linear expectations allow for a concise description of model uncertainty. Thisincludes the g -expectation, a Brownian Motion with uncertain drift parameter (seeCoquet et al. (2002)), and the G -Brownian Motion (see Peng (2007)) describing aBrownian Motion with uncertain volatility. For applications of non-linear expecta-tions in the study of financial markets in discrete time under uncertainty, we referto Nutz (2014) and Bouchard et al. (2015).Regarding the intensively studied topic of risk measures, we refer to the seminalarticle Artzner et al. (1999), to Delbaen (2002) for unconditional risk measures,and to Detlefsen and Scandolo (2005), Acciaio and Penner (2011) for conditionaland dynamic risk measures, amongst many others of course.The paper is structured as follows: in Section 2 we introduce dynamic non-linearexpectations and derive related results on sensitivity and consistency. In Section 3we study super- and sub-hedging and provide a new perspective on the optional de-composition theorem based on those. In Section 4 we show that the set of arbitrage-free prices is indeed an interval and study further properties. Section 5 is the coreof the paper provides a detailed study of attainability and the conditional versionof the second fundamental theorem of asset pricing. Section 6 concludes the paper.An appendix provides some references and results for non-linear expectations andsymmetry. 2. Nonlinear Expectations
In this section we present the appropriate notion of conditional non-linear ex-pectations and related properties. Let T ∈ N denote the final time horizon andlet (Ω , F ) be a measurable space with a filtration F = ( F t ) t ∈{ ,...,T } . We assume F T = F and that F is trivial.Denote by ca (Ω , F ) the space of probability measures (i.e. countably additivemeasures with total mass 1) on (Ω , F ) and consider a set of probability measures P ⊆ ca (Ω , F ). The classical case with one fixed probability measure is containedas special case by considering P = { P } with a probability measure P on (Ω , F ).A P -null set A ⊆ Ω is a possibly not measurable set being a subset of a mea-surable set A ′ ∈ F with P ( A ′ ) = 0. A set A ⊆ Ω is called a P -polar set , if A is a P -null set for every P ∈ P . We denote the collection of P -polar sets by Pol( P ).We say a property holds P -a.s., if it holds outside a P -polar set. If P = { P } , wewrite short Pol( P ) instead of Pol( { P } ).For two subsets of ca (Ω , F ), P and Q , we call Q absolutely continuous withrespect to P , denoted by Q ≪ P , if Pol( P ) ⊆ Pol( Q ). We write Q ∼ P , if Q ≪ P and P ≪ Q . Remark 2.1 (Absolute continuity does not imply total continuity) . Unlike thecase of singletons, absolute continuity does in general not imply total continuity:consider P = { δ n | n ∈ N } . For P := P n − n δ n we have P ∼ { P } . However, thesequence A n = { n } fulfils P ( A n ) = 2 − n → P ( A n ) := sup P ∈ P P ( A n ) = 1 . As a consequence, if Q ≪ P , there exists a well-defined mapping L ( P ) → L ( Q ).However, this mapping is in general not continuous, as absolute continuity does notimply total continuity, which was just shown. ⋄ CONDITIONAL SECOND FUNDAMENTAL THEOREM On L := L (Ω , F ) := { X : Ω → R | X F -measurable } we introduce theequivalence relation ∼ P by X ∼ P Y if and only if X = Y P -a.s.. Then we set L := L / P , for p ∈ (1 , ∞ ), L p := { X ∈ L : sup P ∈ P E P [ | X | p ] < ∞} and L ∞ := { X ∈ L : ∃ C > | X | ≤ C P -a.s. } . With this notation, Proposition 14 inDenis et al. (2011) shows that for each p ∈ [1 , ∞ ], L p is a Banach space.The space L can be equipped with a metric by d : L × L → R + , ( X, Y ) sup P ∈ P E P [ | X − Y | ∧ H ⊆ L containing all constants and set for t ∈ { , ..., T } H t := H ∩ L ( F t ) . The following definition introduces the notion of a conditional non-linear expec-tation and the associated notion of a non-linear dynamic expectation which is aset of conditional non-linear expectations. In the classical, linear case, i.e. when P = { P } we recover the notion of conditional expectation. Definition 2.2.
We call a mapping E t : H → H t an F t -conditional non-linearexpectation , if(i) E t is monotone : for X, Y ∈ H the condition X ≤ Y implies E t ( X ) ≤ E t ( Y ),(ii) E t preserves measurable functions : for X t ∈ H t we have E t ( X t ) = X t .We call E = ( E t ) t ∈{ ,...,T } a non-linear dynamic expectation , if for every t ∈{ , ..., T } the mapping E t : H → H t is an F t - conditional non-linear expectation.We introduce further properties which will be of interest in the context ofdynamic non-linear expectations. First, we introduce some well-known proper-ties regarding the set H , all in an appropriate conditional formulation. Denote H + t := { X ∈ H t : X ≥ } . We call H (i) additive , if H + H ⊆ H . (ii) F t -translation-invariant , if H + H t ⊆ H .(iii) F t -convex , if for λ t ∈ H t with 0 ≤ λ t ≤ λ t H + (1 − λ t ) H ⊆ H (iv) F t -positively homogeneous , if H + t · H ⊆ H .(v) F t -local , if A H ⊆ H for every A ∈ F t .Finally, we call H translation-invariant , if it is F t -translation-invariant for every t ∈ { , ..., T } and do so in a similar fashion for the other properties.Next, we introduce well-known properties of non-linear conditional expectations,all in an appropriate conditional formulation which are frequently used for examplein the context of risk measures. An F t -conditional expectation E t is called(i) subadditive , if H is additive and E t ( X + Y ) ≤ E t ( X ) + E t ( Y ) holds for X, Y ∈ H .(ii) F t -translation-invariant , if H is F t -translation-invariant and E t ( X + X t ) = E t ( X ) + X t holds for any X ∈ H and any X t ∈ H t .(iii) F t -convex , if H is F t -convex and for λ t ∈ H t with 0 ≤ λ t ≤ X, Y ∈ H the inequality E t ( λ t X + (1 − λ t ) Y ) ≤ λ t E t ( X ) + (1 − λ t ) E t ( Y ) ONDITIONAL SECOND FUNDAMENTAL THEOREM 5 holds.(iv) F t -positively homogeneous , if H is F t -positively homogeneous and E t ( X t X ) = X t E t ( X )holds for any X ∈ H and any X t ∈ H + t .(v) F t -sublinear , if it is subadditive and F t -positively homogeneous.(vi) F t -local , if H is F t local and E t ( A X ) = A E t ( X )for any X ∈ H and any A ∈ F t .Finally, we call a dynamic expectation E = ( E t ) t ∈{ ,...,T } translation-invariant, sub-additive, convex or positively homogeneous, if for every t ∈ { , ..., T } the F t -conditional expectation E t has the corresponding property.2.1. Sensitivity and Consistency.
In contrast to a classical expectation, a non-linear expectation might contain only little information on underlying random vari-ables. Sensitivity is a property which allows at least to separate zero from positiverandom variables: we call an F t -conditional non-linear expectation E t sensitive , iffor every X ∈ H with X ≥ E t ( X ) = 0 we have X = 0. Similarly, we call thedynamic non-linear expectation E sensitive, if all E t , t = 0 , . . . , T are sensitive. Remark 2.3 (Sensitivity in the context of risk measures) . For risk-measures thereis also the notion of sensitivity. A conditional risk measure ρ t on L ∞ ( P ) is calledsensitive, if for A ∈ F with P ( A ) > δ > { ρ t ( − δ A ) > } has positive probability. Under the bijection ρ t ( X ρ t ( − X )) between riskmeasures and translation-invariant expectations, both notions coincide. Indeed, if H is local, then E t is sensitive if and only if for every A ∈ F with A / ∈ Pol( P )and δ > {E t ( δ A ) > } / ∈ Pol( P ) ⋄ Time-consistency is an important property in the context of dynamic risk mea-sures, which has been intensively studied, see Acciaio and Penner (2011) We intro-duce an appropriate generalization for dynamic non-linear expectations: we call adynamic expectation E consistent , if for s, t ∈ { , ..., T } with s ≤ t the equality E s = E s ◦ E t holds.Since F = F T , E T is the identity and hence, every expectation is consistentbetween T − T , i.e. E T − ◦ E T = E T − . Remark 2.4 (Extension of consistency to stopping times) . For simplicity, we re-strict our definition of consistency to deterministic times s, t ∈ { , . . . , T } . This caneasily be generalized: indeed, let τ be a stopping time with values in { , . . . , T } .Given ( E t ) t , we define E τ ( H ) := X s { τ = s } E s ( H ) . CONDITIONAL SECOND FUNDAMENTAL THEOREM
If ( E t ) t is consistent, then for any two such stopping times σ, τ with σ ≤ τ , theequality E σ ◦ E τ = E σ holds whenever E is translation-invariant and local. ⋄ The remarkable connection between sensitivity and consistency can already beseen from the simple observation that a consistent dynamic expectation is alreadysensitive, if E is sensitive. Remark 2.5.
Given two sets of probability measures P and Q , the non-linearexpectation E := sup Q ∈ Q E Q is well-defined on L ∞ ( P ) if and only if Q ≪ P .Moreover, sensitivity of E is equivalent to Q ∼ P . ⋄ Lemma 2.6 below generalizes the well-known result that the acceptance sets ofconsistent expectations are decreasing: if E is consistent, then {E s ≤ } ⊇ {E t ≤ } for s ≤ t . See (F¨ollmer and Schied, 2011, Lemma 11.11) for the correspondingrisk-measure result, and note that only preservation of constants is needed for theproof. Lemma 2.6.
Let E be a consistent, local non-linear dynamic expectation and t ∈{ , . . . , T } . Then, for H ∈ H the condition E t ( H ) ≤ implies E s ( A H ) ≤ forall A ∈ F t and all s ≤ t . If E is sensitive, then the converse is also true.Proof. First, locality implies that E t ( H A ) = A E t ( H ) ≤
0. Together with mono-tonicity we obtain E s ( A H ) = E s ( A E t ( H )) ≤ . Now, suppose E is sensitive. Then E s is sensitive. To show that E t ( H ) ≤
0, it thussuffices to show E s ( A E t ( H )) = 0for A := {E t ( H ) ≥ } ∈ F t . However, as above, E s ( A E t ( H )) = E s ( A H )and the latter vanishes by assumption. (cid:3) The following Lemma clarifies and generalizes previous uniqueness results, suchas (Cohen et al., 2012, Lemma 1) showing that every coherent expectation hasat most one coherent and consistent extension. Here, uniqueness is understoodas uniqueness up to a polar-set, where the set of reference measures comes fromthe representation of E in the sense of (Peng, 2010, Theorem I.2.1.). Regardingthe mentioned result, note that for any collection P of probability measures, theassociated non-linear expectation sup P ∈ P E P [ · ] is sensitive; see Remark 2.5.A space H is called symmetric, if − H = H . In Section A.2 we study thisconcept in more detail.The essential infimum appearing in the following Lemma is the infimum withrespect to the order induced by the polar sets. Hence it lacks the well-knownproperties from the dominated case as, for example, it might not exist, and it hasno countable representation. To emphasize this, we denote it by P − ess inf. SeeCohen et al. (2012) for further details. ONDITIONAL SECOND FUNDAMENTAL THEOREM 7
Lemma 2.7.
Every sensitive F -conditional expectation E on a symmetric spacehas at most one translation-invariant, local, consistent extension ( E t ) t . If it exists,it is given by E t ( H ) = P − ess inf { H t ∈ H t : H − H t ∈ A t } where A t := { H ∈ H : E ( A H ) ≤ , ∀ A ∈ F t } . Proof.
Lemma 2.6 characterizes for t ≥ A t := {E t ≤ } solelyin terms of E and F . Indeed, it yields that A t = { H ∈ H : E ( A H ) ≤ ∀ A ∈ F t } . This allows to recover every translation-invariant non-linear expectation on a sym-metric space from its acceptance set through the representation E t ( H ) = P − ess inf { H t ∈ H t : H t ≥ E t ( H ) } = P − ess inf { H t ∈ H t : H − H t ∈ A t } . Summarizing, we have not only shown uniqueness of the extension, but even ob-tained an explicit expression. (cid:3)
In the appendix we show in Proposition A.4, that translation-invariance implieslocality if H ⊆ L ∞ . In particular, every dynamic risk-measure on L ∞ in the senseof (F¨ollmer and Schied, 2011, Section 11) has at most one consistent extension.Moreover, one can show that not every coherent risk measure has a consistentextension. Remark 2.8.
Consider the non-linear expectations E t ( · ) := ess sup Q E Q [ ·| F t ], t = 0 , . . . , T , for some collection of equivalent measures Q . Up to some continuityconstraints, every sublinear expectation is of this form. Then, {E t ≤ } = { H ∈ H : E Q [ H | F t ] ≤ ∀ Q ∈ Q } = { H ∈ H : E Q [ H A ] ≤ ∀ Q ∈ Q , ∀ A ∈ F t } = { H ∈ H : H A ∈ {E ≤ } ∀ A ∈ F t } However, ( E t ) t is only consistent if Q is stable under pasting. See also Artzneret al. (2007). ⋄ Super- and Sub-hedging
Before we consider the second fundamental theorem, we provide some resultsregarding super- and sub-hedging in a classical, finite-dimensional financial marketin discrete time.In this regard, fix a single probability measure P and consider a filtered prob-ability space (Ω , F , P, ( F t ) t ∈{ ,...,T } ). Furthermore, assume that F is trivial and F T = F . We directly work on discounted prices which are described through the d -dimensional discounted price process X = ( X t ) t ∈{ ,...,T } . Denote by Pred thecollection of d -dimensional predictable processes.Self-financing trading strategies can be identified with predictable processes.Hence, throughout the paper, we identify a self-financing trading strategy witha predictable process, ξ ∈ Pred. For such ξ and t ∈ { , . . . , T } , we write( ξ · X ) t = t X k =1 ξ k ( X k − X k − ) = t X k =1 ξ k ∆ X k , CONDITIONAL SECOND FUNDAMENTAL THEOREM for the gains from self-financing trading in X until t .Denote by M e := M e ( P ) the set of martingale measures equivalent to P . Weassume that the market is free of arbitrage, or equivalently, that M e = ∅ .Arbitrage-free prices will be studied in more detail in Section 4.1. In particular,as a consequence from the first fundamental theorem, the set of arbitrage-free pricesof a European contingent claim is given by its expectations under all equivalentmartingale measures.In this section we study the upper bound of the set of no-arbitrage prices inmore detail. It is given by¯ E t ( H ) := ess sup { E Q [ H | F t ] : Q ∈ M e } , (3.1)for H ∈ L ∞ ( P ). We will show soon that ¯ E is indeed a consistent, dynamic non-linear expectation. We also revisit the well-known fact that the upper bound of theno-arbitrage interval ¯ E is equal to the smallest super-hedging price, E t ( H ) := ess inf { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≥ H } , (3.2)where G t ( ξ ) := ( ξ · X ) T − ( ξ · X ) t .For a proof of this result we refer to (F¨ollmer and Schied, 2011, Corollary 7.18).While E = ¯ E is easy to show, the case t ≥ E ( H ) for the process ( E t ( H )) t ∈{ ,t...,T } anduse a similar notation for the other dynamic non-linear expectations. Lemma 3.1.
For all t = 0 , . . . , T , the super-hedging price E t defined in (3.2) is asensitive, sub-linear expectation on L ∞ .Proof. First, we show that E t ( H ) is bounded for H ∈ L ∞ ( P ). The inequality E t ( H ) ≤ k H k follows by definition. If H t ∈ L t is a superhedging price, choose ξ ∈ Pred with H t + G t ( ξ ) ≥ H . Consider the set A := { H t < − k H k} ∈ F t . Thenthere exists η ∈ Pred with G t ( η ) ≥ A ( H − H t ) ≥ P ( A ) = 0 byabsence of arbitrage. We conclude that − k H k ≤ E t ( H ) ≤ k H k .Second, one easily checks the properties of a sublinear expectation. The sensi-tivity of E t follows from the no-arbitrage assumption. (cid:3) Up to now we treated only bounded random variables, which excludes for exam-ple European calls. More generally, we are interested in
European contingent claims (claims for short) which are simply non-negative random variables, i.e. elements of L ( F T ). This however has a major downside, as the space L ( F T ) is of course notsymmetric. Symmetry allows us to associate to the dynamic expectation E anotherdynamic expectation E ∗ = ( E ∗ t ) t ∈{ ,...,T } by E ∗ t ( X ) := −E t ( − X ) . (3.3)For properties we refer to the appendix. This tool will play a major role in ouranalysis of perfect hedges.In this regard, we extend the non-linear expectation E t to the symmetric space H = L ( F T ). Note that the properties stated in Lemma 3.1 transfer to this largerspace. However, for unbounded H the non-linear expectation E t ( H ) is in general ONDITIONAL SECOND FUNDAMENTAL THEOREM 9 not real-valued. To be precise, it takes values in ( −∞ , + ∞ ]. The same holds truefor ¯ E t by Proposition 4.1.Likewise, the associated expectations E ∗ t , ( ¯ E t ) ∗ take values in [ −∞ , ∞ ). In par-ticular, if H ∈ L is symmetric with respect E t , then E t ( H ) is real-valued.3.1. Super-hedging and sub-hedging dualities.
At time t < T , an F t - mea-surable random variable π t is a superhedging price for the European claim H due attime T , if there is a self-financing trading strategy which provides always a terminalwealth greater than H . This is equivalent to the existence of a predictable process ξ , such that π t + G t ( ξ ) ≥ H. Remark 3.2.
For every H ∈ L ( F T ), and every ξ ∈ Pred one has the orthogonality E t ( H + G t ( ξ )) = E t ( H ) . (3.4)Indeed, for every H ∈ L ( F T ), and every ξ ∈ Pred, the set of superhedging pricesfor H coincides with the set of superhedging prices for H + G t ( ξ ). This follows fromlinearity of Pred ∋ ξ G t ( ξ ). Note that, in view of Lemma 5.2, this is a specialcase of Proposition A.14, as every G t ( ξ ) for ξ ∈ Pred is clearly attainable at time t . ⋄ The following result shows that E t ( H ) from Equation (3.2) is actually a super-hedging price. Lemma 3.3.
Fix t ∈ { , . . . , T } . For every H ∈ L ( F T ) , E t ( H ) is a superhedgingprice for H .Proof. The set M := { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≥ H } of superhedgingprices is directed downwards. Hence, by (F¨ollmer and Schied, 2011, Theorem A.33)there exists a sequence ( H nt ) n ⊆ M with H nt ↓ E t ( H ) a.s. By construction, we maywrite for each n ∈ N , H = H nt + G t ( ξ n ) − U n for some ξ n ∈ Pred and U n ∈ L ( F T ). As the cone { G t ( ξ ) − U : ξ ∈ Pred , U ∈ L ( F T ) } is closed due to the no-arbitrage assumption, the claim follows. (cid:3) Lemma 3.4.
For every bounded European contingent claim H ∈ L ∞ ( P ) , the pro-cess of super-hedging prices E ( H ) is an M e -supermartingale.Proof. Fix t ∈ { , ..., T − } . Using Lemma 3.3, we may choose ξ ∈ Pred with E t ( H ) + G t ( ξ ) ≥ H. Next, we apply E t +1 to this inequality. Note that G t ( ξ ) = G t +1 ( ξ ) + ξ t +1 ∆ X t +1 and E t +1 ( E t ( H )) = E t ( H ). From Equation (3.4) it follows that E t +1 ( G t ( ξ )) = ξ t +1 ∆ X t +1 . Hence, E t ( H ) + ξ t +1 ∆ X t +1 ≥ E t +1 ( H ) . For the ξ chosen above, ξ t +1 ∆ X t +1 is bounded from below. Thus, its conditionalexpectation under Q vanishes for all Q ∈ M e . We obtain the inequality E t ( H ) ≥ E Q [ E t +1 ( H ) | F t ] , for each Q ∈ M e and each t ∈ { , ..., T − } which implies the claim. (cid:3) Corollary 5.3 proved below will allow to study the martingale property of E ( H ).It will establish that for any t ∈ { , . . . , T } , the process ( E s ( H )) s ≥ t is an M e -martingale, if and only if H is attainable at time t and therefore allows to charac-terize the martingale property in terms of attainability.For the following result, recall from Equation (3.2) that the dynamic non-linearexpectation on L ∞ ( P ) given by the smallest super-hedging price was denoted by E t ( H ) := ess inf { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≥ H } , with t ∈ { , . . . , T } and H ∈ L ∞ ( P ). Theorem 3.5.
The dynamic non-linear expectation E = ( E t ) t =0 ,...,T , defined inEquation (3.2) is consistent on L ∞ .Proof. First, applying Lemma 3.3 to the European contingent claims E t +1 ( H ) and H allows to choose strategies ξ, η ∈ Pred such that E t ( E t +1 ( H )) + G t ( ξ ) ≥ E t +1 ( H )and E t +1 ( H ) + G t +1 ( η ) ≥ H. Combining both inequalities yields E t ( E t +1 ( H )) + G t ( ξ ) + G t +1 ( η ) ≥ H. Note that the sum G t ( ξ ) + G t +1 ( η ) can be seen as the gains process of a strategyfrom time t on. Hence, the claim H can be super-replicated at time t at price E t ( E t +1 ( H )). As E t ( H ) is by definition the smallest super-hedging price for claim H at time t , we obtain E t ( E t +1 ( H )) ≥ E t ( H ) . Second, analogously to the second inequality above, we obtain from Lemma 3.3the existence of θ ∈ Pred, such that E t ( H ) + G t ( θ ) ≥ H. Applying E t +1 to this inequality gives E t ( H ) + E t +1 (G t ( θ )) ≥ E t +1 ( H ) . Again using the orthogonality relation as introduced in Remark 3.2, in particularEquation (3.4), we obtain that E t +1 (G t ( θ )) = θ t +1 ∆ X t +1 + E t +1 (G t +1 ( θ )) = θ t +1 ∆ X t +1 . This implies E t ( H ) + θ t +1 ∆ X t +1 ≥ E t +1 ( H ) . Similarly as above we conclude that E t ( E t +1 ( H )) ≤ E t ( H )and the result is proven. (cid:3) Remark 3.6 (Consistency of ¯ E ) . Consistency of ¯ E is related to the stability of M e . A set of equivalent probability measures M is said to be stable under pasting ,if for any two measures Q , Q , and every t ∈ { , . . . , T } , the pasting Q ⊙ t Q of Q and Q in t , defined by Q ⊙ t Q ( A ) := E Q [ Q ( A | F t )] ( A ∈ F T )(3.5) ONDITIONAL SECOND FUNDAMENTAL THEOREM 11 is contained in M . In case of M e this is easily verified, see Proposition 6.42 inF¨ollmer and Schied (2011), as for any s, t ∈ { , . . . , T } one has E Q ⊙ t Q [ ·| F s ] = E Q [ E Q [ ·| F s ∨ t ] | F s ] , by (F¨ollmer and Schied, 2011, Lemma 6.41). (F¨ollmer and Schied, 2011, Theorem11.22) now shows that the expectation ¯ E is consistent. See Section 3.2 for a relateddiscussion. ⋄ Corollary 3.7.
For every t ∈ { , . . . , T } , and every H ∈ L ∞ ( P ) , the superhedging-duality E t ( H ) = ¯ E t ( H )(3.6) holds.Proof. In view of Theorem 3.5 and Remark 3.6, Lemma 2.7 already implies theclaim. (cid:3)
Remark 3.8.
It is worth to recall how the equality E = ¯ E is usually established,as it allows for a non-linear interpretation. To ease the argument, assume, for themoment, that the price process is (locally) bounded. Denote by K := { G ( ξ ) : ξ ∈ Pred } the set of claims attainable at price 0, and by C := ( K − L ) ∩ L ∞ ( P ) thecorresponding cone. By the very definition of C , a measure Q ∼ P is contained in M e if and only if E Q [ H ] ≤ H ∈ C . The Bipolar Theorem then impliesthat H ∈ C if and only if E Q [ H ] ≤ Q ∈ M e . This can also be read as:the acceptance set { ¯ E ≤ } of ¯ E coincides with C . One readily verifies that byconstruction C = {E ≤ } , and translation-invariance of both E and ¯ E implies E = ¯ E . ⋄ Corollary 3.9.
For every H ∈ L ∞ ( P ) the subhedging-duality ess sup { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≤ H } = ess inf { E Q [ H | F t ] : Q ∈ M e } , (3.7) holds.Proof. This follows from Corollary 3.7, as E ∗ t ( H ) = ess sup { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≤ H } (3.8)and ( ¯ E t ) ∗ ( H ) = ess inf { E Q [ H | F t ] : M e } . (cid:3) Up to now, super- and subhedging dualities were only established for boundedEuropean claims. The next result shows continuity from below of the non-linearconditional expectations E and ¯ E , which allows to extend the dualities to Europeanclaims bounded from below (or bounded from above). Lemma 3.10.
The expectations E and ¯ E are continuous from below on L ( F T ) .Proof. Let ( H n ) ⊆ L ( F T ) with H n ↑ H ∈ L . We start with ¯ E . Choose asequence ( Q m ) ⊆ M e withsup m E Q m [ H | F t ] = ess sup M e E Q [ H | F t ] . Monotone convergence then impliessup n ¯ E t ( H n ) = sup n sup m E Q m [ H n | F t ]= sup m E Q m [ H | F t ] = ¯ E t ( H ) . For E , one proceeds similar as in the proof of Lemma 3.3, to obtain sup n E t ( H n ) ≥E t ( H ). (cid:3) Proposition 3.11.
The superhedging-duality (3.6) , and consistency of E extendsto claims, i.e. to L ( F T ) .Proof. In view of Corollary 3.7, and Theorem 3.5 this follows from Lemma 3.10. (cid:3)
In the context of Remark 3.8, Proposition 3.11 allows us to state the followingconditional version of the Bipolar relationship: a claim H ∈ L ( F T ) can be super-replicated at time t at price zero, if and only if E Q [ H | F t ] ≤ Q ∈ M e .It is worth to recall the classical way to prove the superhedging-duality. Usually,one proves the M e -supermartingale property of ¯ E . Then, one relies on the optionaldecomposition, to show the difficult inequality ¯ E t ≥ E t .In continuous time, the optional decomposition theorem, and the resulting con-struction of superhedging strategies were first obtained in El Karoui and Quenez(1995) in the setting of continuous processes. The extension to general locallybounded semimartingales was achieved in Kramkov (1996). Later, the assumptionof local boundedness was removed, see F¨ollmer and Kabanov (1997), Delbaen andSchachermayer (1999) and references therein. For the statement in discrete time, werefer to F¨ollmer and Kabanov (1997), Theorem 2, and F¨ollmer and Schied (2011),Theorem 7.5. The case of a finite probability space in discrete time is treated inDelbaen and Schachermayer (2006), Theorem 2.6.1.While it is possible to derive consistency of the super-hedging prices E using theoptional decomposition, by taking advantage of the M e -supermartingale propertyof E established in Lemma 3.4, let us mention that the superhedging-duality inProposition 3.11 allows us to prove the optional decomposition with little effort.We record this in Theorem 3.12 below, whose proof is similar to the proof of Delbaenand Schachermayer (2006), Theorem 2.6.1. Theorem 3.12.
Let V = ( V t ) t ∈{ ,...,T } be a non-negative M e -supermartingale.Then there exists an adapted increasing process C with C = 0 , and a strategy ξ ∈ Pred such that V t = V + ( ξ · X ) t − C t . Proof.
For every t ∈ { , . . . , T } , and every Q ∈ M e the inequality E Q [ V t | F t − ] ≤ V t − holds. Equivalently, this can be written in terms of the non-linear expectation ¯ E as¯ E t − ( V t ) ≤ V t − . Applying the superhedging-duality Proposition 3.11 to the claim V t ∈ L ( F t ), we conclude that E t − (∆ V t ) ≤ . Hence, for t ∈ { , . . . , T } there exists a strategy ξ ( t ) ∈ Pred such that∆ V t ≤ G t − ( ξ ( t ) ) . ONDITIONAL SECOND FUNDAMENTAL THEOREM 13
Using the orthogonality relation as introduced in Remark 3.2, in particular Equation(3.4), an application of E t yields ∆ V t ≤ ξ ( t ) t ∆ X t . Summing over t ∈ { , . . . , T } , we obtain ξ ∈ Pred such that ( ξ · X ) − V is increasing. (cid:3) Lemma 3.13.
For every claim H ∈ L ( F T ) , E ∗ t ( H ) defined in (3.8) is a subhedg-ing price for H .Proof. Similar to Lemma 3.3. (cid:3)
Proposition 3.14.
Consider the claim H ∈ L ( F T ) . If E t ( H ) is finite, then thesubhedging-duality (3.7) holds at time t .Proof. Pick ξ ∈ Pred with E t ( H ) + G t ( ξ ) ≥ H , and consider the claim ˜ H := E t ( H ) + G t ( ξ ) − H . Applying the superhedging-duality in form of Proposition 3.11to ˜ H yields E t ( E t ( H ) + G t ( ξ ) − H ) = ¯ E t ( E t ( H ) + G t ( ξ ) − H ) . Using translation in conjunction with Remark 3.2, in particular Equation (3.4),gives E t ( H ) + E t ( − H ) = E t ( H ) + ¯ E t ( − H )and the claim follows. (cid:3) We briefly collect the associated symmetric statements of Lemma 3.10, Proposi-tion 3.11 and Proposition 3.14.
Proposition 3.15.
The following statements hold true:(i) The expectations E ∗ and ( ¯ E ) ∗ are continuous from above on L − ( F T ) .(ii) The subhedging-duality (3.7) , and consistency of ( E ∗ t ) extends to L − ( F T ) .(iii) Consider H ∈ L − ( F T ) . If E ∗ t ( H ) is finite, then the superhedging-duality (3.6) holds at time t . Implications of the super-hedging duality.
We end this section discussingthe implications and limitations of the super-hedging-duality. Recall from Remark3.2 that E t (G t ( ξ )) = 0 for every trading strategy ξ ∈ Pred, simply by linearity of ξ G t ( ξ ).By duality, this implies for every claim H = H t + G t ( ξ ) attainable at time t that E Q [G t ( ξ ) | F t ] = 0 for every Q ∈ M e , and every replicating strategy ξ .Here, the conditional setting fundamentally differs from the unconditional one.Indeed, a claim H attainable at time zero can always be replicated by a boundedinitial investment since F is trivial. This is no longer the case for t >
0. Here,one has to pay a finite, but not necessarily bounded, price. In particular, the gainsG t ( ξ ) = H − H t might be unbounded from below and expectations might no longerexist. This of course relates to the observation that for arbitrary ξ ∈ Pred thestochastic integral (( ξ · X ) t ) t ∈{ ,...,T } is in general only a local martingale under Q ∈ M e due to lacking integrability.This is of course well known, the link between integrability and the martingaleproperty is discussed in Jacod and Shiryaev (1998), Meyer (2006) in detail. Asmentioned above, the superhedging-duality implies the desired integrability of G t ( ξ )for replicating strategies of claims. This can also be checked directly. Indeed, if 0 ≤ H = H t + G t ( ξ ), set for n ∈ N , A n := { H t ≤ n } , reducing the problem to the bounded case. By monotoneconvergence we obtain E Q [ H | F t ] = H t for every Q ∈ M e . However, note that thishinges on the boundedness of H from below; this indicates that the superhedging-duality does not hold in L .Finally, we mention that this is in contrast to robust versions of the superhedging-duality, as in Bouchard et al. (2015) or Burzoni et al. (2017), where the superhedging-duality is known to hold on L . In these robust approaches, the duality gap is usu-ally closed by enlarging the set of pricing measures. We refer to Obloj and Wiesel(2018) for a detailed discussion.4. The conditional no-arbitrage interval
This section studies the conditional no-arbitrage interval and we begin with someresults on pricing conditional on past information. It seems to be worthwhile point-ing out that in case of a finite probability space, conditional pricing can be reducedto the unconditional setting by exploiting the structure of the set of equivalentmartingale measures.The no-arbitrage interval has been intensively studied, but in the unconditionalcase with t = 0. We refer once more to F¨ollmer and Schied (2011), Section 5 for adetailed treatment of this case. In the following, we extend the notions and resultstherein to arbitrary t ∈ { , . . . , T } .4.1. Structure of arbitrage-free prices.
The main goal in computing arbitrage-free prices relying on the fundamental theorem of asset pricing is to obtain a priceprocess for a new security which can be added to the market without violatingabsence of arbitrage.In this spirit, a random variable π t ∈ L ( F t ) is called arbitrage-free price at time t of a European contingent claim H ∈ L ( F T ) if there exists an adapted process X d +1 such that X d +1 t = π t , X d +1 T = H and the market ( X, X d +1 ) extended with X d +1 is free of arbitrage. Denote by Π t ( H ) the collection of arbitrage free prices.For every claim H ∈ L ( F T ) we define M He := { Q ∈ M e : H ∈ L ( Q ) } . Note that for an attainable claim H , M He = M e . The fundamental theorem ofasset pricing implies immediately that the set of arbitrage-free prices is given byexpectations under the risk-neutral measures, which we state here for clarity It isan immediate consequence from the first fundamental theorem of asset pricing.
Proposition 4.1.
For every claim H ∈ L ( F T ) , Π t ( H ) = { E Q [ H | F t ] : Q ∈ M He } , and the set of arbitrage-free prices is non-empty. The following lemma shows that already when a risk-neutral conditional expec-tation has finite values, it is an arbitrage-free price.
Lemma 4.2.
Let H ∈ L ( F T ) and Q ∈ M e . If E Q [ H | F t ] is finite-valued, thenit is an arbitrage-free price. The unconditional version of this result is, for example, given in Theorem 5.29 in F¨ollmer andSchied (2011).
ONDITIONAL SECOND FUNDAMENTAL THEOREM 15
Proof. If E Q [ H | F t ] is finite, it is in L ( F T ), and by Proposition 4.1 there exists˜ Q ∈ M e such that E Q [ H | F t ] is integrable with respect to ˜ Q . We now paste Q and ˜ Q . As mentioned in Remark 3.6, the pasting ˜ Q ⊙ t Q of ˜ Q with Q in F t givenby (3.5) is again an equivalent martingale measure. By construction, we even have˜ Q ⊙ t Q ∈ M He . Moreover, it follows that E ˜ Q ⊙ t Q [ H | F t ] = E Q [ H | F t ] . This is an arbitrage-free price by Proposition 4.1. (cid:3)
Denote the upper and the lower bound of the no-arbitrage set at time t by π sup t ( H ) := ess sup Π t ( H ) , and π inf t ( H ) := ess inf Π t ( H ) . At the moment it is unclear if Π t ( H ) is actually an interval: indeed, since P E P [ H | G ] is in general not linear when G is not trivial, convexity of Π t ( H ) is nolonger an immediate consequence. However, we will prove (conditional) convexityin Corollary 4.8. An explicit description of Π t ( H ) in terms of π sup t ( H ) and π inf t ( H )will be given in Theorem 5.8. Remark 4.3.
The essential supremum in the definition of π sup t ( H ) can be takenin either F t or F T . As Π t ( H ) ⊆ L ( F t ) they both coincide, since the essentialsupremum has a countable representation. ⋄ To achieve countable convexity of the set of equivalent martingale measures weexploit nonnegativity of the price process and triviality of the initial σ -algebra F in the following lemma. Lemma 4.4. M e is countably convex.Proof. Let ( Q n ) ⊆ M e and ( λ n ) n ⊆ R + with P n λ n = 1. Set Q ∗ := P n λ n Q n .Obviously, Q ∗ ∼ P . For every t ∈ { , . . . , T } we have by monotone convergence E Q ∗ [ X t ] = X n λ n E Q n [ X t ] = X n λ n X = X < ∞ and hence X T ∈ L ( Q ∗ ). Similarly, for A ∈ F t − , E Q ∗ [ X t A ] = X n λ n E Q n [ X t A ] = X n λ n E Q n [ X t − A ] = E Q ∗ [ X t − A ]and therefore E ∗ [ X t | F t − ] = X t − . (cid:3) It is important to acknowledge that, in the notation of Lemma 4.4, we typicallydo not have E Q ∗ [ H | F t ] = X n λ n E Q n [ H | F t ]for bounded H , while we have it, as just shown, for H replaced by X iT , i = 1 , . . . , d due to the Q n -martingale property of X . In general, this equality holds only ifall measures Q n agree on F t . We will exploit this observation in the proof ofProposition 4.5. Proposition 4.5.
For every t ∈ { , . . . , T } , and for every H ∈ L ( F T ) the set (cid:8) E Q [ H | F t ] : Q ∈ M e (cid:9) is F t -countably convex. Proof.
Let ( Q n ) ⊆ M e . By pasting we may assume that all Q n agree on F t . Thiswill be used throughout. Set Q ∗ := P n − n Q n . By Lemma 4.4, Q ∗ ∈ M e . Denote by Z n := dQ n /dQ ∗ the associated densities. As Q ∗ = Q n on F t for each n ∈ N , we have Z nt = E Q ∗ [ Z n | F t ] = 1 . Since H ≥
0, monotone convergence implies for a sequence ( λ nt ) ∈ L ( F t ) with P n λ t = 1, that X n λ nt E Q n [ H | F t ] = E Q ∗ h H X n λ nt Z n | F t i . Set Z := P n λ nt Z n >
0. Note that E Q ∗ h X n λ nt Z n | F t i = X n λ nt = 1and we may therefore define the measure Q by dQ/dQ ∗ := Z. Then, E Q ∗ h H X n λ nt Z n | F t i = E Q [ H | F t ] . It remains to verify that Q is indeed a martingale measure after t (recall that Q and Q ∗ agree on F t ). As the price process is nonnegative, its conditional expecta-tion is well-defined, and we obtain by monotone convergence for s ≥ tE Q [ X s +1 | F s ] = E Q ∗ h ZZ s X s +1 | F s i = 1 Z s X n λ nt E Q ∗ [ Z n X s +1 | F s ]= 1 Z s X n λ nt Z ns E Q n [ X s +1 | F s ] = X s . This allows to conclude Q ∈ M e . (cid:3) Remark 4.6.
The F t -convexity is easier to establish, since for finitely many mea-sures Q , . . . , Q n ∈ M e there exists Q ∗ ∈ M e such that the densities dQ k /dQ ∗ areuniformly bounded. In particular, it is not needed that the Q k agree on F t . ⋄ Note that, due to the integrability conditions, Π t ( H ) is not F t -countably convexfor every claim H . Even in the unconditional case this fails. Corollary 4.7.
Consider a claim H ∈ L ( F T ) , let ( Q n ) ⊆ M He and ( λ nt ) ⊆ L ( F t ) with P n λ nt = 1 . If P n λ nt E Q n [ H | F t ] is finite-valued, it is contained in Π t ( H ) .Proof. Due to Proposition 4.5, there exists Q ∈ M e with X n λ nt E Q n [ H | F t ] = E Q [ H | F t ] . Now the claim follows by Lemma 4.2. (cid:3)
Corollary 4.8.
Consider t ∈ { , . . . , T } . Then Note that this always holds for t = 0. ONDITIONAL SECOND FUNDAMENTAL THEOREM 17 (i) Π t ( H ) is F t -convex for every claim H ∈ L ( F T ) ,(ii) Π t ( H ) is directed upward for every claim H ∈ L ( F T ) ,(iii) Π t ( H ) is F t -countably convex for every bounded claim H ∈ L ∞ ( P ) , and,(iv) for H ∈ L ( F T ) , any partition ( A n ) ⊆ F t , and any sequence ( Q n ) ⊆M He ( P ) , X n A n E Q n [ H | F t ] ∈ Π t ( H ) . This observation will imply that the no-arbitrage set Π t ( H ) is indeed an (ofcourse random) interval for any t ∈ { , . . . , T } . We will give a precise proof inTheorem 5.8. Lemma 4.9.
For t ∈ { , . . . , T } , A ∈ F t and H ∈ L ( F T ) , it holds that Π t ( A H ) = A Π t ( H ) . Proof.
Let Q ∈ M e such that H A is integrable with respect to Q . By Proposition4.1, we may pick ˜ Q ∈ M e such that H A c is integrable with respect to ˜ Q . Byconstruction E Q [ H | F t ] A + E ˜ Q [ H | F t ] A c is finite, and by Corollary 4.8 andLemma 4.2 there exists Q ∗ ∈ M He with E Q ∗ [ H | F t ] = E Q [ H | F t ] A + E ˜ Q [ H | F t ] A c and therefore E Q ∗ [ H | F t ] A = E Q [ H A | F t ] . (cid:3) The next Proposition shows that, for every claim H , the non-linear expectation¯ E ( H ) = ess sup { E Q [ H | F t ] : Q ∈ M e } can be computed by considering a subsetof M e only: one can restrict to the set of martingale measure M He ( P ) under which H is integrable. In particular, for every claim H , the non-linear expectation ¯ E ( H )agrees with the upper bound of the no-arbitrage interval π sup t ( H ). This links thesuperhedging-duality Proposition 3.11 with the pricing in financial markets. Proposition 4.10.
For every H ∈ L ( F T ) we have the equalities ess sup { E Q [ H | F t ] : Q ∈ M e } = ess sup { E Q [ H | F t ] : Q ∈ M He } and ess inf { E Q [ H | F t ] : Q ∈ M e } = ess inf { E Q [ H | F t ] : Q ∈ M He } Proof.
Due to Proposition 4.1, it remains to show the first equality. Using Lemma4.9 and Lemma 4.2, it suffices to show the following: if there exits Q ∈ M e with E Q [ H | F t ] = + ∞ , then π sup t ( H ) = + ∞ .In this regard, consider Q ∈ M e with E Q [ H | F t ] = + ∞ and some Y t ∈ L t .Since E Q [ H | F t ] = + ∞ , there exists n ∈ N such that { Y t ≤ E Q [ H ∧ n | F t ] } haspositive probability. By the fundamental theorem of asset pricing and Proposition4.1 we find π t ∈ Π t ( H ) such that { Y t ≤ π t } has positive probability.Since Y t was arbitrary, it follows that π sup t = + ∞ with positive probability. Nowset A := { π sup t < + ∞} . Using Lemma 4.9, and arguing as above for the claim H A , we deduce that P ( A ) = 0. (cid:3) In view of Lemma 2.7, Proposition 4.10 is not an immediate consequence ofthe unconditional case, as the set M He is not stable. Moreover, as L ( F T ) is notsymmetric, one has to show both equalities in Proposition 4.10. A conditional version of the second fundamental theorem
Equipped with efficient tools for non-linear expectations we now prove a con-ditional version of the second fundamental theorem of asset pricing. We begin bystudying those contingent claims which are attainable, i.e. claims which can beperfectly replicated by a hedging strategy.5.1.
Attainability of Claims.
A European contingent claim H ∈ L ( F T ) iscalled attainable (replicable) at time t if there exists ξ ∈ Pred and H t ∈ L ( F t )such that H = H t + G t ( ξ ). In this case, we say that the strategy ξ replicates theclaim H . Remark 5.1.
Some immediate observations are due.(i) If H ∈ L ( F T ) is attainable at time s , then H is attainable at time t forevery t ≥ s .(ii) The converse is not true. Using consistency of π sup established in Theorem3.5 and Proposition 4.10, we have π sup s ( H ) = π sup s ( π sup t ( H ))and π inf s ( H ) = π inf s ( π inf t ( H ))so, by Theorem 5.4, established below, H attainable at time t is attainable attime s ≤ t if and only if π sup t ( H ) = π inf t ( H ) is attainable at time s . ⋄ Recall that H ∈ H is called symmetric with respect to E t , if E t ( H ) = −E t ( − H ) . In the following we show that for an attainable claim the non-linear expectationfor − H is well defined and coincides with −E t ( H ). Lemma 5.2.
Let H ∈ L ( F T ) . Then H is symmetric with respect to E t if andonly if H is attainable at time t .Proof. Recall that by definition (3.3) E ∗ t ( · ) = −E t ( −· ) and hence, as already estab-lished in Equation (3.8), E ∗ t ( H ) = ess sup { H t ∈ L t : ∃ ξ ∈ Pred : H t + G t ( ξ ) ≤ H } is the largest sub-hedging price. Due to Lemma 3.13, E ∗ t ( H ) is itself a sub-hedgingprice.Now, suppose first that H is attainable at time t , i.e. H = H t + G t ( ξ ) for some ξ ∈ Pred. By Remark 3.2, we obtain E ∗ t ( H ) = E ∗ t ( H t ) + E ∗ t ( G t ( ξ )) = H t = E t ( H t + G t ( ξ )) = E t ( H ) . Second, let ξ, η ∈ Pred such that E ∗ t ( H ) + G t ( ξ ) ≤ H ≤ E t ( H ) + G t ( η ) . (5.1)If H is symmetric, then E t ( H ) = E ∗ t ( H ) is finite and this implies0 ≤ H − E t ( H ) − G t ( ξ ) ≤ G t ( η − ξ ) . The no-arbitrage assumption yields G t ( η ) = G t ( ξ ). Since E t ( H ) = E ∗ t ( H ), Equation(5.1) yields that H = E t ( H ) + G t ( H ) and therefore H is attainable. (cid:3) ONDITIONAL SECOND FUNDAMENTAL THEOREM 19
Corollary 5.3.
Consider H ∈ L ( F T ) . H is attainable at time t if and only if M e ∋ Q E Q [ H | F t ] is constant.Proof. Using the dualities established in Proposition 3.11 and Proposition 3.14,Lemma 5.2 shows that H is attainable at time t if and only if H is symmetric withrespect to E πt . Now, ( E πt ) ∗ ( H ) = ess inf { E Q [ H | F t ] : Q ∈ M e } and symmetry( E πt ) ∗ ( H ) = E πt ( H ) is therefore equivalent to the constancy of E Q [ H | F t ] over Q ∈ M e . (cid:3) In view of Proposition 4.1, Corollary 5.3 extends the well-known result, that aclaim is attainable if and only if it has a unique arbitrage free price, to a conditionalsetting. See (F¨ollmer and Schied, 2011, Theorem 5.32) for the classical case.
Theorem 5.4.
Let H ∈ L ( F T ) be a European contingent claim. Then H isattainable at time t if and only if π inf t ( H ) = π sup t ( H ) , i.e. when there is a unique arbitrage free price at time t . If H is not attainable attime t , then π inf t ( H ) and π sup t ( H ) are not arbitrage free prices at time t .Proof. The first part, as just remarked, follows from Corollary 5.3. The secondpart follows from the dualities Proposition 3.11 and Proposition 3.14. (cid:3)
We obtain the following conditional version of the second fundamental theoremof asset pricing. To do so, we introduce the notation M e ⊙ t Q ∗ := { Q ⊙ t Q ∗ : Q ∈ M e } . Theorem 5.5.
The following statements are equivalent:(i) The market is complete at time t , i.e. every European contingent claim H ∈ L ( F T ) is attainable at time t .(ii) Every European contingent claim H ∈ L ( F T ) has a unique price at time t .(iii) There exists Q ∗ ∈ M e such that M e = M e ⊙ t Q ∗ .(iv) For every Q ∗ ∈ M e ( P ) the equality M e = M e ⊙ t Q ∗ holds.Proof. Theorem 5.4 provides the equivalence between market completeness at time t , and uniqueness of arbitrage-free prices at time t .As already mentioned in the Remark 3.6, the set of equivalent martingale mea-sures is stable. This implies the inclusion M e ⊙ t Q ∗ ⊆ M e for every Q ∗ ∈ M e .Now, let us first assume that every European contingent claim H ∈ L ( F T ) hasa unique price at time t , and pick Q ∗ ∈ M e . We show the inclusion M e ⊆ M e ⊙ t Q ∗ .Given Q ∈ M e , we define the measure ˜ Q ∈ M e ⊙ t Q ∗ by ˜ Q := Q ⊙ t Q ∗ . UsingRemark 3.6, we compute, for every A ∈ F T , E ˜ Q [ A ] = E Q (cid:2) E Q ∗ [ A | F t ] (cid:3) . By assumption, the claim A ∈ L ( F T ) has a unique arbitrage-free price at time t . In particular, the prices E Q ∗ [ A | F t ] and E Q [ A | F t ] do agree. Hence, E ˜ Q [ A ] = E Q (cid:2) E Q [ A | F t ] (cid:3) = E Q [ A ]and thus Q = ˜ Q ∈ M e ⊙ t Q ∗ , as desired. As Q ∗ ∈ M e was arbitrary, we concludethat ( iv ) holds. Second, suppose that the equality M e = M e ⊙ t Q ∗ holds for some Q ∗ ∈ M e ( P ).We finish the proof by showing that every European contingent claim H ∈ L ( F T )has a unique price at time t , i.e. constancy of M e ∋ Q E Q [ H | F t ]. To thisend, let ˜ Q ∈ M e . By assumption, there exists Q ∈ M e ( P ) such that ˜ Q = Q ⊙ t Q ∗ .Using Remark 3.6 again, we obtain, for every H ∈ L ( F T ), E ˜ Q [ H | F t ] = E Q ∗ [ H | F t ] , and thus Π t ( H ) is indeed a singleton. (cid:3) Remark 5.6. (i) Note that M e = M e ⊙ T Q ∗ trivially holds, as Q ⊙ T Q = Q for anytwo measures Q , Q . This is in line with the observation that every claim H ∈ L ( F T ) is attainable at time T .(ii) As Q ⊙ Q = Q for any two measures Q , Q , we recover the classical versionof the second fundamental theorem of asset pricing: market completeness isequivalent to the existence of exactly one equivalent martingale measure. ⋄ Lemma 5.7.
Let H ∈ L ( F T ) be a claim. If Y ∈ L ( F t ) , and Y A < π sup t ( H ) A for some F t -measurable subset A with P ( A ) > . Then there exists a F t -measurableset B ⊆ A with P ( B ) > and π t ∈ π t ( H ) such that Y B < π t B Proof.
First, by assumption, Y A < A ess sup Π t ( H ) = ess sup A Π t ( H ) . By definition of the essential supremum, there exists π t ∈ Π t ( H ) and a measurablesubset B ⊆ A with positive probability such that Y B < π t B and the claim follows. (cid:3) As an intuition for the above proof, it is interesting to note that π sup t inheritslocality from Π t , which was shown in Lemma 4.9, i.e. π sup t ( H ) A = ess sup Π t ( H ) A = ess sup Π t ( H A ) = π sup t ( H A ) . Theorem 5.8.
The set of arbitrage-free prices Π t ( H ) of a claim H ∈ L ( F T ) isan F t -measurable random interval, i.e. Π t ( H ) = { λ t π sup t ( H ) + (1 − λ t ) π inf t ( H ) : λ t ∈ L ( F t ) , < λ t < } . In the case where H is attainable, the (random) interval of arbitrage-free priceshence collapses to the singleton { E Q [ H | F t ] } with any Q ∈ M e ( P ). When H is notattainable, this interval is given by ( π inf t , π sup t ). Proof.
We start with the inclusionΠ t ( H ) ⊆ { λ t π sup t ( H ) + (1 − λ t ) π inf t ( H ) : λ t ∈ L ( F t ) , < λ t < } . Let π t ∈ Π t ( H ). Theorem 5.4 implies together with Lemma 4.9 the equalities { π t = π sup t ( H ) } = { π inf t ( H ) = π t = π sup t ( H ) } , { π t < π sup t ( H ) } = { π inf t ( H ) < π t = π sup t ( H ) } . ONDITIONAL SECOND FUNDAMENTAL THEOREM 21
Thus we find 0 < λ t <
1, where λ t := π t − π inf t ( H ) π sup t ( H ) − π inf t ( H ) { π sup t ( H ) >π t } + δ { π sup t ( H )= π t } for any δ ∈ (0 , π t = λ t π sup t ( H ) + (1 − λ t ) π inf t ( H ) . To show the other inclusion we may assume, by Theorem 5.4 that H is notattainable. Using Lemma 4.9 and Corollary 4.8, we may even assume that π inf t ( H ) <π sup t ( H ), see also the remark following this proof. Next, let π inf t ( H ) < Y < π sup t ( H ).We claim that there exists π t ∈ Π t ( H ) such that Y ≤ π t . If this is shown, thensimilarly ˜ π t ≤ Y ≤ π t for ˜ π t ∈ Π t ( H ) and conditional convexity of Π t ( H ) implies Y ∈ Π t ( H ).To show the existence of π t , we use a Halmos-Savage argument. Set α :=sup { P ( { π t ≥ Y } ) : π t ∈ Π t ( H ) } . By Lemma 5.7, α >
0. Pick a sequence π nt such that P ( { π nt ≥ Y } ) converges to α , and set A n := { π nt ≥ Y } . For n ∈ N , define B n := A n \ ∪ k
Let H ∈ L ( F t ) be a European contingent claim, and let π t be an F t -measurable random variable. (i) If { π t > π sup t ( H ) } has positive probability, then π t / ∈ Π t ( H ) .(ii) If { π t = π sup t ( H ) } ∩ { π inf t ( H ) < π sup t ( H ) } has positive probability, then π t / ∈ Π t ( H ) . Conclusion
In this article we provided a conditional version of the second fundamental the-orem in discrete time. We have shown that at any time t , completeness of thefinancial market is characterized by uniqueness of the prices of all financial claims.In this regard, we provided a detailed study of the arbitrage-free price interval at t , showing conditional convexity together with the associated super-hedging andsub-hedging dualities. The latter give upper and lower bounds of the no-arbitrageinterval, and are, as usual, not arbitrage-free prices if the interval is a true interval.We also provided a new perspective on the optional decomposition theorem basedon non-linear expectations. Appendix A. Non-linear expectations
For the convenience of the reader we collect several well-known properties ofnon-linear expectations.A.1.
Basic properties.Lemma A.1.
Every subadditive conditional expectation E t : H → H t is F t -translation-invariant.Proof. Note that H + H t ⊆ H + H ⊆ H holds. For X ∈ H and X t ∈ H t ,subadditivity yields E t ( X + X t ) ≤ E t ( X ) + E t ( X t ) = E t ( X ) + X t As X = X + X t − X t we also have E t ( X ) + X t ≤ ( E t ( X + X t ) − X t ) + X t = E t ( X + X t ) . (cid:3) Lemma A.2 (Detlefsen and Scandolo (2005)) . Let H be additive and F t -local.For a F t -conditional expectation E t : H → H t the following are equivalent:(i) E t is F t -local.(ii) For X ∈ H and A ∈ F t the equality E t ( A X ) = A E t ( X ) holds.(iii) For X, Y ∈ H and A ∈ F t the equality E t ( A X + A c Y ) = A E t ( X ) + A c E t ( Y ) holds. Proposition A.3.
Every convex expectation is local.Proof.
Let E = ( E t ) t be convex. Note that H is local as A H = A H + A c ⊆ H . Now proceed as in (Detlefsen and Scandolo, 2005, Proposition 2). (cid:3) Proposition A.4.
Every translation-invariant expectation on a local set H ⊆ L ∞ is local.Proof. Let E t be translation-invariant and A ∈ F t . The inequality A X − A c k X k ≤ X ≤ A X + A c k X k yields E t ( X ) ≥ E t ( A X − A c k X k ) ONDITIONAL SECOND FUNDAMENTAL THEOREM 23 and additionally E t ( X ) ≤ E t ( A X + A c k X k )Note that R ⊆ H . Multiplying both inequalities with A gives A E t ( X ) = A E t ( A X )and thus E t ( A X ) = A E t ( A X ) + A c E t ( A X )= A E t ( X ) + A A c E t ( A X )= A E t ( X ) (cid:3) Remark A.5.
Proposition A.4 sharpens (Detlefsen and Scandolo, 2005, Propo-sition 2). Every conditional risk measure on L ∞ is local; there is no need forconvexity. ⋄ Lemma A.6.
Every translation-invariant expectation on H ⊆ L ∞ is 1-Lipschitzwith respect to. k·k ∞ . Lemma A.7.
Let E be a dynamic expectation on H . The following are equivalent:(i) E is consistent(ii) For every t ∈ { , ..., T − } we have E t = E t ◦ E t +1 .(iii) For every X, Y ∈ H and s ≤ t the condition E t ( X ) ≤ E t ( Y ) implies E s ( X ) ≤E s ( Y ) .(iv) For every X, Y ∈ H and s ≤ t the condition E t ( X ) = E t ( Y ) implies E s ( X ) = E s ( Y ) .Proof. As in (F¨ollmer and Schied, 2011, Lemma 11.11). Note that no translation-invariance is needed. (cid:3)
A.2.
Symmetry.
A space H is called symmetric, if − H = H . Proposition A.8.
Let ( E t ) t be a dynamic expectation on the symmetric set H .We define another dynamic expectation ( E ∗ t ) t by E ∗ t ( X ) := −E t ( − X ) Remark A.9. If E t is a subadditive expectation on symmetric H , then E t ≥ E ∗ t Indeed, for X ∈ H we have0 = E t (0) = E t ( X − X ) ≤ E t ( X ) + E t ( − X ) ⋄ Remark A.10. If H is symmetric, the mapping E 7→ E ∗ is a bijection on thecollection of dynamic expectations on H . For every dynamic expectation we have E = E ∗∗ . A dynamic expectation E with E = E ∗ is called symmetric. ⋄ Proposition A.11.
A dynamic expectation is linear if and only if it is symmetricand sublinear.
Definition A.12 (Definition 4.3 in Cohen et al. (2011)) . Let H be symmetric andlet E t be a F t -conditional expectation. We call X ∈ H symmetric (with respectto E t ), if E t ( X ) = E ∗ t ( X )holds. This extends as usual to dynamic expectations. Lemma A.13. (i) X ∈ H is symmetric if and only if ( − X ) is symmetric.(ii) X ∈ H is symmetric with respect to E if and only if it is symmetric withrespect to E ∗ . Proposition A.14 (Lemma 2.2 in Cohen et al. (2011)) . Let H be symmetricand additive, and let E t be a positive homogeneous F t -conditional expectation. Let Y ∈ H be symmetric and m t ∈ H t with m t Y ∈ H . Then E t ( m t Y ) = m t E t ( Y ) If E t is additionally sublinear, then for X ∈ H E t ( X + m t Y ) = E t ( X ) + m t E t ( Y ) Corollary A.15 (Lemma 4.8 in Cohen et al. (2011)) . Let H be symmetric and let E t be a sublinear F t -conditional expectation. The collection of symmetric elementsforms a R -linear space. References
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