Weak and strong no-arbitrage conditions for continuous financial markets
aa r X i v : . [ q -f i n . P R ] M a y Weak and strong no-arbitrage conditionsfor continuous financial markets
Claudio Fontana
Laboratoire de Mathématiques et Modélisation, Université d’Évry Val d’Essonne23 Boulevard de France, Évry Cedex, 91037, France.E-mail: [email protected]
This version: May 13, 2014
Abstract
We propose a unified analysis of a whole spectrum of no-arbitrage conditions for financial marketmodels based on continuous semimartingales. In particular, we focus on no-arbitrage conditionsweaker than the classical notions of
No Arbitrage and
No Free Lunch with Vanishing Risk . Weprovide a complete characterisation of the considered no-arbitrage conditions, linking their validityto the characteristics of the discounted asset price process and to the existence and the propertiesof (weak) martingale deflators , and review classical as well as recent results.
Keywords: arbitrage, benchmark approach, continuous semimartingale, martingale deflator, mar-ket price of risk, arbitrage of the first kind, free lunch with vanishing risk.
MSC (2010):
Modern mathematical finance is strongly rooted on the no-arbitrage paradigm. In a nutshell, thisamounts to excluding the possibility of “making money out of nothing” by cleverly trading in thefinancial market. Since the existence of such a possibility is both unrealistic and, loosely speaking,conflicts with the existence of an economic equilibrium, any mathematical model for a realisticfinancial market is required to satisfy a suitable no-arbitrage condition, in the absence of which onecannot draw meaningful conclusions on asset prices and investors’ behavior.The search for a satisfactory no-arbitrage condition has a rather long history, grown at theborder between financial economics and mathematics. We do not attempt here a detailed overviewof the historical developments of modern no-arbitrage theory, but only mention the seminal papers[31]-[32] and refer the reader to [20] and [74] for more information. A decisive step in this historywas marked by the paper [14], where, in the case of locally bounded processes, it was proved theequivalence between the
No Free Lunch with Vanishing Risk (NFLVR) condition (a condition slightly tronger than the classical No Arbitrage (NA) condition) and the existence of an
Equivalent LocalMartingale Measure (ELMM) , i.e., a new probability measure equivalent to the original one suchthat the discounted asset price process is a local martingale under the new measure. The localboundedness assumption was then removed in the subsequent papers [19] and [43].The NFLVR condition has established itself as a golden standard and the vast majority of modelsproposed in quantitative finance satisfies it. However, financial market models that fail to satisfyNFLVR have also appeared in recent years. In particular, in the context of the
Benchmark Approach (see [62]-[65]), a new asset pricing theory has been developed without relying on the existence ofELMMs. A similar perspective is also adopted in the
Stochastic Portfolio Theory (see [24]-[25]),where the NFLVR condition is not imposed as a normative assumption and it is shown that arbitrageopportunities may naturally arise in the market. Related works that explicitly consider situationswhere NFLVR may fail are [7], [8], [13], [27], [28], [35], [47], [49], [56], [73] and also, in the morespecific case of diffusion models, [29], [53], [54] and [72] (see later in the text for more information).Somewhat surprisingly, these works have shown that the full strength of NFLVR is not necessarilyneeded in order to solve the fundamental problems of valuation, hedging and portfolio optimisation.However, the situation is made complicated by the fact that many different and alternative no-arbitrage conditions have been proposed in the literature during the last two decades.Motivated by the preceding discussion, the present paper aims at presenting a unified and clearperspective on the most significant no-arbitrage conditions in the context of general financial marketmodels based on continuous semimartingales . In particular, we study several no-arbitrage condi-tions which are weaker than the classical and strong NFLVR condition, namely the
No IncreasingProfit (NIP) , No Strong Arbitrage (NSA) and
No Arbitrage of the First Kind (NA1) conditions. Weprove the following implications:NFLVR = ⇒ NA1 = ⇒ NSA = ⇒ NIP. (1.1)By means of explicit examples and counterexamples, we illustrate these implications and discusstheir economic meaning, their relations with the Benchmark Approach as well as the connectionsto several other conditions which have appeared in the literature, thus providing a complete pictureof a whole spectrum of no-arbitrage conditions. Moreover, we prove that none of the converseimplications in (1.1) holds in general.We show that weak no-arbitrage conditions (i.e., NIP, NSA and NA1) can be completely charac-terised in terms of the semimartingale characteristics of the discounted price process, while this is ingeneral impossible for strong no-arbitrage conditions (NA and NFLVR), since the latter also dependon the structure of the underlying filtration. Moreover, we link the validity of different no-arbitrageconditions to the existence and the properties of (weak) martingale deflators , which can be thoughtof as weaker counterparts of density processes of ELMMs. In particular, we show that the weakNSA and NA1 conditions (as well as their equivalent formulations) can be directly checked by look-ing at the minimal weak martingale deflator, the properties of which are easily determined by themean-variance trade-off process of the discounted price process. Furthermore, we prove that NA1(as well as its equivalent formulations) is stable with respect to changes of numéraire (see Corollary5.6), unlike the classical NFLVR condition, and allows to recover NFLVR by means of a suitable The continuous semimartingale setting allows for a rather transparent analysis and covers many models widely usedin quantitative finance (in particular, almost all models developed in the context of Stochastic Portfolio Theory). hange of numéraire (see Corollary 6.7).Altogether, referring to Section 8 for a more detailed discussion on the economic aspects, thepresent study shows that the NIP and NSA conditions can be regarded as indispensable no-arbitragerequirements for any realistic financial market, but are not enough for the purposes of financial mod-eling. On the contrary, the NA1 condition, even though strictly weaker than the classical NFLVRcondition, is equivalent to a meaningful notion of market viability and allows for a satisfactorysolution of all typical problems of mathematical finance.To the best of our knowledge, there does not exist in the literature a similar unifying analysisof the weak no-arbitrage conditions going beyond the classical notions of NA and NFLVR. Theonly exception is contained in Chapter 1 of [34]. In comparison with the latter work, our approachputs more emphasis on the role of (weak) martingale deflators and also carefully takes into accountminimal no-arbitrage conditions that are weaker than the NUPBR condition, on which the presen-tation in [34] is focused. Moreover, besides providing different and original proofs, we also studythe NIP, NA1, NCT and NAA no-arbitrage conditions (see e.g. the table in Section 8), which arenot explicitly considered in [34], and drop the non-negativity assumption on the discounted assetprices.The paper is structured as follows. Section 2 presents the general setting and introduces themain no-arbitrage conditions which shall be studied in the following. Section 3, 4 and 5 discuss theNIP, NSA and NA1 conditions, respectively. Section 6 deals with the classical NFLVR condition anddiscusses its relations with the previous no-arbitrage conditions. Section 7 illustrates the implications(1.1) by means of several examples. Finally, Section 8 concludes by summarising the different no-arbitrage conditions studied in the present paper and commenting on their economic implications. Let (Ω , F , F , P ) be a given filtered probability space, where the filtration F = ( F t ) ≤ t ≤ T is assumedto satisfy the usual conditions of right-continuity and P -completeness and, for the sake of simplicity, T ∈ (0 , ∞ ) represents a finite time horizon (all the results we are going to present can be rathereasily adapted to the infinite horizon case). Note that the initial σ -field F is not assumed to betrivial. Let M denote the family of all uniformly integrable P -martingales and M loc the family ofall local P -martingales. Without loss of generality, we assume that all processes in M loc have càdlàgpaths and we denote by M c and M c loc the families of all processes in M and M loc , respectively,with continuous paths.We consider a financial market comprising d + 1 assets, whose prices are represented by the R d +1 -valued process e S = ( e S t ) ≤ t ≤ T , with e S t = ( e S t , e S t , . . . , e S dt ) ⊤ , with ⊤ denoting transposition. Weassume that e S t is P -a.s. strictly positive for all t ∈ [0 , T ] and, as usual in the literature, we then takeasset as numéraire and express all quantities in terms of e S . This means that the ( e S -discounted)price of asset is constant and equal to and the remaining d risky assets have ( e S -discounted)prices described by the R d -valued process S = ( S t ) ≤ t ≤ T , with S it := e S it / e S t for all t ∈ [0 , T ] and i = 1 , . . . , d . The process S is assumed to be a continuous R d -valued semimartingale on (Ω , F , F , P ) .In particular, S is a special semimartingale, admitting the unique canonical decomposition S = S + A + M , where A is a continuous R d -valued predictable process of finite variation and M is an R d -valued process in M c loc with M = A = 0 . Due to Proposition II.2.9 of [39], it holds that, for ll i, j = 1 , . . . , d , A i = Z a i dB and h S i , S j i = h M i , M j i = Z c ij dB, (2.1)where B is a continuous real-valued predictable increasing process, a = ( a , . . . , a d ) ⊤ is an R d -valuedpredictable process and c = (cid:0) ( c i ) ≤ i ≤ d , . . . , ( c id ) ≤ i ≤ d (cid:1) is a predictable process taking values in thecone of symmetric nonnegative d × d matrices. The processes a , c and B in (2.1) are not uniquein general, but our results do not depend on the specific choice we make (for instance, B can betaken as B = P di =1 (Var( A i ) + h M i i ) , with Var( · ) denoting the total variation). Note that we donot necessarily assume that S takes values in the positive orthant of R d . For every t ∈ [0 , T ] , let usdenote by c + t the Moore-Penrose pseudoinverse of the matrix c t . The proof of Proposition 2.1 of [22]shows that the process c + = ( c + t ) ≤ t ≤ T is predictable and, hence, the process a can be representedas a = c λ + ν, (2.2)where λ = ( λ t ) ≤ t ≤ T is the R d -valued predictable process defined by λ t := c + t a t , for all t ∈ [0 , T ] ,and ν = ( ν t ) ≤ t ≤ T is an R d -valued predictable process with ν t ∈ Ker ( c t ) := { x ∈ R d : c t x = 0 } , forall t ∈ [0 , T ] .We suppose that the financial market is frictionless, meaning that there are no trading restric-tions, transaction costs, liquidity effects or other market imperfections. In order to mathematicallydescribe the activity of trading, we need to introduce the notion of admissible strategy . To this effect,let L ( S ) be the set of all R d -valued S -integrable predictable processes, in the sense of [39], and, for H ∈ L ( S ) , denote by H · S the stochastic integral process (cid:0)R t H u dS u (cid:1) ≤ t ≤ T . Since S is a continuoussemimartingale, Proposition III.6.22 of [39] implies that L ( S ) = L loc ( M ) ∩ L ( A ) , where L loc ( M ) and L ( A ) are the sets of all R d -valued predictable processes H such that R T H ⊤ t d h M, M i t H t < ∞ P -a.s. and R T | H ⊤ t dA t | < ∞ P -a.s., respectively. Hence, due to (2.1), an R d -valued predictableprocess H belongs to L ( S ) if and only if Z T v ( H ) t dB t < ∞ P -a.s. where v ( H ) t := d X i,j =1 H it c ijt H jt + (cid:12)(cid:12)(cid:12) d X i =1 H it a it (cid:12)(cid:12)(cid:12) . Remark 2.1.
The set L ( S ) represents the most general class of predictable integrands with respectto S . In particular, it contains non-locally bounded integrands, as in [10]. Note that, for H ∈ L ( S ) ,we have H · M ∈ M c loc and the continuous semimartingale H · S admits the unique canonicaldecomposition H · S = H S + H · A + H · M . We also want to emphasize that H · S has to beunderstood as the vector stochastic integral of H with respect to S and is in general different fromthe sum of the componentwise stochastic integrals P di =1 R H i dS i ; see for instance [38] and [77].We are now in a position to formulate the following classical definition. Definition 2.2.
Let a ∈ R + . An element H ∈ L ( S ) is said to be an a -admissible strategy if H = 0 and ( H · S ) t ≥ − a P -a.s. for all t ∈ [0 , T ] . An element H ∈ L ( S ) is said to be an admissible strategy if it is an a -admissible strategy for some a ∈ R + . For a ∈ R + , we denote by A a the set of all a -admissible strategies and by A the set of alladmissible strategies, i.e., A = S a ∈ R + A a . As usual, H it represents the number of units of asset i held in the portfolio at time t . The condition H = 0 amounts to requiring that the initial position n the risky assets is zero and, hence, the initial endowment is entirely expressed in terms of thenuméraire asset. For H ∈ A , we define the gains from trading process G ( H ) = (cid:0) G t ( H ) (cid:1) ≤ t ≤ T by G t ( H ) := ( H · S ) t , for all t ∈ [0 , T ] . According to Definition 2.2, the process G ( H ) associated toan admissible strategy H ∈ A is uniformly bounded from below by some constant. This restrictionis needed since the set L ( S ) is too large for the purpose of modeling reasonable trading strategiesand may also contain doubling strategies. This possibility is automatically ruled out if we impose alimit to the line of credit which can be granted to every market participant. For ( x, H ) ∈ R + × A ,we define the portfolio value process V ( x, H ) = (cid:0) V t ( x, H ) (cid:1) ≤ t ≤ T by V ( x, H ) := x + G ( H ) . Thiscorresponds to considering portfolios generated by self-financing admissible strategies.We now introduce five main notions of arbitrage. Definition 2.3. (i) An element H ∈ A generates an increasing profit if the process G ( H ) is predictable and if P (cid:0) G s ( H ) ≤ G t ( H ) , for all ≤ s ≤ t ≤ T (cid:1) = 1 and P (cid:0) G T ( H ) > (cid:1) > . If there exists nosuch H ∈ A we say that the No Increasing Profit (NIP) condition holds;(ii) an element H ∈ A generates a strong arbitrage opportunity if P (cid:0) G T ( H ) > (cid:1) > . If thereexists no such H ∈ A , i.e., if (cid:8) G T ( H ) : H ∈ A (cid:9) ∩ L = { } , we say that the No StrongArbitrage (NSA) condition holds;(iii) a non-negative random variable ξ generates an arbitrage of the first kind if P ( ξ > > and for every v ∈ (0 , ∞ ) there exists an element H v ∈ A v such that V T ( v, H v ) ≥ ξ P -a.s. Ifthere exists no such random variable ξ we say that the No Arbitrage of the First Kind (NA1) condition holds;(iv) an element H ∈ A generates an arbitrage opportunity if G T ( H ) ≥ P -a.s. and P (cid:0) G T ( H ) > (cid:1) > . If there exists no such H ∈ A , i.e., if (cid:8) G T ( H ) : H ∈ A (cid:9) ∩ L = { } , we say that the No Arbitrage (NA) condition holds;(v) a sequence { H n } n ∈ N ⊂ A generates a Free Lunch with Vanishing Risk if there exist an ε > and an increasing sequence { δ n } n ∈ N with ≤ δ n ր such that P (cid:0) G T ( H n ) > − δ n (cid:1) = 1 and P (cid:0) G T ( H n ) > ε (cid:1) ≥ ε , for all n ∈ N . If there exists no such sequence we say that the NoFree Lunch with Vanishing Risk (NFLVR) condition holds.
The NIP condition is similar to the No Unbounded Increasing Profit condition introduced in [47]and represents the strongest notion of arbitrage among those listed above. The “unboundedness” inthe original definition of [47] can be explained as follows: if H ∈ A yields an increasing profit inthe sense of Definition 2.3- (i) , we have H n := nH ∈ A and G ( H n ) ≥ G ( H ) , for every n ∈ N . Thismeans that the increasing profit generated by H can be scaled to arbitrarily large levels of wealth.The NSA condition corresponds to the notion of absence of arbitrage opportunities adopted inSection 3 of [53] as well as to the NA + condition studied in [79]. The above formulation of the notionof arbitrage of the first kind has been introduced by [49]. The NA and NFLVR conditions are classicaland, in particular, go back to the seminal papers [31], [32] and [14]. Note that the NA conditioncan be equivalently formulated as C ∩ L ∞ + = { } , where C := (cid:0) { G T ( H ) : H ∈ A} − L (cid:1) ∩ L ∞ . Theabove definition of NFLVR is taken from [47] and can be shown to be equivalent to C ∩ L ∞ + = { } , The reason for requiring G ( H ) to be predictable will become clear in Theorem 3.1, which is formulated with respectto possibly discontinuous locally square-integrable semimartingales, in the sense of Definition II.2.27 of [39]. Of course, assoon as S is continuous, the predictability requirement becomes unnecessary. ith the bar denoting the closure in the norm topology of L ∞ , as in [14]. In the following sections,we shall also examine several other no-arbitrage conditions equivalent to the ones introduced inDefinition 2.3. An increasing profit represents an investment opportunity which does not require any initial in-vestment nor any line of credit and, moreover, generates an increasing wealth process, yielding anon-zero final wealth with strictly positive probability. As such, the notion of increasing profit rep-resents the most egregious form of arbitrage and, therefore, should be banned from any reasonablefinancial model. The following theorem characterises the NIP condition. At no extra cost, we stateand prove the result for general locally square-integrable semimartingales, in the sense of DefinitionII.2.27 of [39].
Theorem 3.1.
The following are equivalent, using the notation introduced in (2.1) - (2.2) :(i) the NIP condition holds;(ii) for every H ∈ L ( S ) , if H ⊤ t c t = 0 P ⊗ B -a.e. then H ⊤ t a t = 0 P ⊗ B -a.e.;(iii) ν t = 0 P ⊗ B -a.e.Proof. ( i ) ⇒ ( ii ) : Let us define the product space Ω := Ω × [0 , T ] . Suppose that NIP holds andlet H = ( H t ) ≤ t ≤ T be a process in L ( S ) such that H ⊤ t c t = 0 P ⊗ B -a.e. (so that H · M = 0 ) but P ⊗ B (cid:0) ( ω, t ) ∈ Ω : H ⊤ t ( ω ) a t ( ω ) = 0 (cid:1) > . By the Hahn-Jordan decomposition (see [16], Theorem2.1), we can write H · A = R ( D + − D − ) dV , where D + and D − are two disjoint predictable subsetsof Ω such that D + ∪ D − = Ω and V := Var( H · A ) . Let ψ := D + − D − and define the R d -valuedpredictable process ˜ H := ψH ((0 ,T ]] . Due to the associativity of the stochastic integral, it is clearthat ˜ H ∈ L ( S ) and ˜ H · M = 0 . Thus, using again the associativity of the stochastic integral, ˜ H · S = ˜ H · A = ( ψH ) · A = ψ · ( H · A ) = ψ · V = V. The process V is non-negative, increasing and predictable and satisfies P ( V T > > , since H · A issupposed to be not identically zero. Clearly, this amounts to saying that ˜ H generates an increasingprofit, thus contradicting the assumption that NIP holds. Hence, it must be H ⊤ t a t = 0 P ⊗ B -a.e. ( ii ) ⇒ ( iii ) : let H = ( H t ) ≤ t ≤ T be an R d -valued predictable process such that k H t ( ω ) k ∈ { , } forall ( ω, t ) ∈ Ω . Since H ⊤ t c t = 0 P ⊗ B -a.e. implies that H ⊤ t a t = 0 P ⊗ B -a.e., condition (iii) followsdirectly from the absolute continuity result of Theorem 2.3 of [16]. ( iii ) ⇒ ( i ) : suppose that ν t = 0 P ⊗ B -a.e. and let H ∈ A generate an increasing profit. Theprocess G ( H ) = H · S is predictable and increasing, hence of finite variation. In particular, H · S is aspecial semimartingale and, hence, due to Proposition 2 of [38], we can write H · S = H · A + H · M .This implies that H · M = H · S − H · A is also predictable and of finite variation. Theorem III.15of [66] then implies that H · M = 0 , being a predictable local martingale of finite variation. Hence: G t ( H ) = ( H · A ) t = Z t H ⊤ u a u dB u = Z t H ⊤ u c u λ u dB u = Z t d h H · M, M i u λ u = 0 P -a.s.for all t ∈ [0 , T ] . In particular, P (cid:0) G T ( H ) > (cid:1) = 0 , thus contradicting the hypothesis that H generates an increasing profit. learly, ν t = 0 P ⊗ B -a.e. means that dA ≪ d h M, M i . The latter condition is known in the lit-erature as the weak structure condition and the process λ is usually referred to as the instantaneousmarket price of risk (see e.g. [35], Section 3). We want to point out that results similar to Theorem3.1 have already appeared in the literature, albeit under stronger assumptions. In particular, Theo-rem 3.5 of [16] (see also [67], Theorem 1, and Appendix B of [48]) shows that dA ≪ d h M, M i holdsunder the classical NA condition, which is strictly stronger than NIP (see Section 6). Somewhatmore generally, [45] (see also [34], Theorem 1.13) prove that dA ≪ d h M, M i holds under the NSAcondition, which is also strictly stronger than the NIP condition (see Section 4). Theorem 3.1 showsthat the weak structure condition dA ≪ d h M, M i is exactly equivalent to the NIP condition, whichrepresents an indispensable requirement for any reasonable financial market. Remarks 3.2. 1)
As can be seen by inspecting the proof of Theorem 3.1, the NIP condition isalso equivalent to the absence of elements H ∈ A such that the gains from trading process G ( H ) is predictable and of finite variation (not necessarily increasing) and satisfies P ( G T ( H ) > > . In general, as long as the NIP condition holds, there may exist many R d -valued predictableprocesses γ = ( γ t ) ≤ t ≤ T such that dA t = d h M, M i t γ t . However, for any such process γ , elementarylinear algebra gives Π c t ( γ t ) = λ t , where we denote by Π c t ( · ) the orthogonal projection onto therange of the matrix c t , for t ∈ [0 , T ] . In turn, this implies that R T γ ⊤ t c t γ t dB t ≥ R T λ ⊤ t c t λ t dB t , thusshowing the minimality property of the process λ introduced in (2.2). A strong arbitrage opportunity consists of an investment opportunity which does not require anyinitial capital nor any line of credit and leads to a non-zero final wealth with strictly positiveprobability. Of course, this sort of strategy should be banned from any reasonable financial market,since every agent would otherwise benefit in an unlimited way from a strong arbitrage opportunity(compare also with the discussion in Section 8). According to Definition 2.3, it is evident that anincreasing profit generates a strong arbitrage opportunity. Two examples of models which satisfyNIP but allow for strong arbitrage opportunities will be presented in Section 7, thus showing thatNSA is strictly stronger than NIP.Let us now introduce another notion of arbitrage, which has been first formulated in [16] andturns out to be equivalent to the notion of strong arbitrage opportunity.
Definition 4.1.
An element H ∈ A generates an immediate arbitrage opportunity if there existsa stopping time τ such that P ( τ < T ) > and if H = H (( τ,T ]] and G t ( H ) > P -a.s. for all t ∈ ( τ, T ] . If there exists no such H ∈ A we say that the No Immediate Arbitrage (NIA) conditionholds.
We then have the following simple lemma (compare also with [16], Lemma 3.1).
Lemma 4.2.
The NSA condition and the NIA condition are equivalent.Proof.
Suppose that H ∈ A generates a strong arbitrage opportunity and define the stopping time τ := inf { t ∈ [0 , T ] : G t ( H ) > } ∧ T . Since P ( G T ( H ) > > , we have P ( τ < T ) > . For asequence { θ n } n ∈ N dense in (0 , , let us define the process ˜ H := P ∞ n =1 − n H (( τ, ( τ + θ n ) ∧ T ]] . Clearly, e have ˜ H ∈ A . Furthermore, on the event { τ < T } it holds that, for every ε > , G τ + ε ( ˜ H ) = ( ˜ H · S ) τ + ε = ∞ X n =1 n (cid:0) ( H · S ) ( τ +( ε ∧ θ n )) ∧ T − ( H · S ) τ (cid:1) = ∞ X n =1 n ( H · S ) ( τ +( ε ∧ θ n )) ∧ T > P -a.s.thus showing that ˜ H generates an immediate arbitrage opportunity at the stopping time τ . Con-versely, Definitions 2.3- (ii) and 4.1 directly imply that an immediate arbitrage opportunity is alsoa strong arbitrage opportunity.Recall that, due to Theorem 3.1, the NIP condition is equivalent to a = c λ P ⊗ B -a.e., wherethe processes a , c , λ and B are as in (2.1)-(2.2). Since NSA (or, equivalently, NIA) is stronger thanNIP, it is natural to expect that NSA will imply some additional properties of the process λ . This isconfirmed by the next theorem. As a preliminary, let us define the mean-variance trade-off process b K = ( b K t ) ≤ t ≤ T as b K t := Z t λ ⊤ u d h M, M i u λ u = Z t λ ⊤ u c u λ u dB u = Z t a ⊤ u c + u a u dB u , for t ∈ [0 , T ] . (4.1)Let also b K ts := b K t − b K s , for s, t ∈ [0 , T ] with s ≤ t . Following [52] and [79], we also define thestopping time α := inf (cid:8) t ∈ [0 , T ] : b K t + ht = ∞ , ∀ h ∈ (0 , T − t ] (cid:9) , with the usual convention inf ∅ = ∞ . The next theorem is essentially due to [79], but we opt for aslightly different proof. Theorem 4.3.
The NSA condition holds if and only if ν t = 0 P ⊗ B -a.e. and α = ∞ P -a.s.Proof. Suppose first that NSA holds. Since NSA implies NIP, Theorem 3.1 gives that ν t = 0 P ⊗ B -a.e. The fact that α = ∞ P -a.s. then follows from Theorem 3.6 of [16] together with Lemma 4.2(compare also with [45], Sections 3-4).Conversely, suppose that ν t = 0 P ⊗ B -a.e. and α = ∞ P -a.s. and let H ∈ A generate astrong arbitrage opportunity. Due to Lemma 4.2, we can equivalently suppose that H generatesan immediate arbitrage opportunity with respect to a stopping time τ with P ( τ < T ) > . Since P ( α = ∞ ) = 1 , we have P (cid:0) b K τ + hτ = ∞ , ∀ h ∈ (0 , T − τ ] (cid:1) = 0 . For each n ∈ N , define the stoppingtime ρ n := inf (cid:8) t > τ : b K tτ ≥ n (cid:9) ∧ T . Since b K is continuous and does not jump to infinity, it is clearthat ρ n > τ P -a.s. on the set { τ < T } , for all n ∈ N . Let us then define the predictable process λ τ,n := λ (( τ,ρ n ]] , for every n ∈ N . Then, on the set { τ < T } , Z T ( λ τ,nt ) ⊤ d h M, M i t λ τ,nt = Z T (( τ,ρ n ]] λ ⊤ t d h M, M i t λ t = b K ρ n τ ≤ n P -a.s.For every n ∈ N , we can define the stochastic exponential b Z τ,n := E ( − λ τ,n · M ) as a strictly positiveprocess in M c , due to Novikov’s condition. It is obvious that b Z τ,n = 1 on [[0 , τ ]] and b Z τ,n = b Z τ,nρ n on [[ ρ n , T ]] . We now apply the integration by parts formula to b Z τ,n ( H · S ) ρ n , where ( H · S ) ρ n denotes the process H · S stopped at ρ n , and use the fact that b Z τ,nt d ( H · A ) ρ n t = b Z τ,nt H ⊤ t dA ρ n t , since b Z τ,n ∈ L ( H · A ) (being b Z τ,n adapted and continuous, hence predictable and locally bounded), and he fact that dA = d h M, M i λ and H = H (( τ,T ]] : d (cid:0) b Z τ,nt ( H · S ) ρ n t (cid:1) = b Z τ,nt d ( H · S ) ρ n t + ( H · S ) ρ n t d b Z τ,nt + d (cid:10) b Z τ,n , ( H · S ) ρ n (cid:11) t = b Z τ,nt d ( H · M ) ρ n t + b Z τ,nt d ( H · A ) ρ n t + ( H · S ) ρ n t d b Z τ,nt − b Z τ,nt H ⊤ t d h M, M i t λ τ,nt = b Z τ,nt d ( H · M ) ρ n t + ( H · S ) ρ n t d b Z τ,nt + b Z τ,nt H ⊤ t (cid:0) dA ρ n t − d h M, M i t λ τ,nt (cid:1) = b Z τ,nt d ( H · M ) ρ n t + ( H · S ) ρ n t d b Z τ,nt . This shows that b Z τ,n ( H · S ) ρ n is a non-negative local martingale and, by Fatou’s lemma, also asupermartingale, for every n ∈ N . Since b Z τ,n ( H · S ) ρ n = 0 , the supermartingale property impliesthat b Z τ,nt ( H · S ) ρ n t = 0 for all t ∈ [0 , T ] P -a.s., meaning that H · S = 0 P -a.s. on S n ∈ N [[0 , ρ n ]] .Since ρ n > τ P -a.s. on { τ < T } and P ( τ < T ) > , this contradicts the fact that ( H · S ) t > P -a.s. for all t ∈ ( τ, T ] , thus showing that there cannot exist an immediate arbitrage opportunity.Equivalently, due to Lemma 4.2, the NSA condition holds.Theorem 4.3 shows that NSA holds as long as the mean-variance trade-off process b K does notjump to infinity (however, b K T is not guaranteed to be finite). In particular, it is important toremark that we can check whether a financial market allows for strong arbitrage opportunities bylooking only at the semimartingale characteristics of the discounted price process S .We now introduce the important concept of (weak) martingale deflator , which represents a weakercounterpart to the density process of an equivalent local martingale measure (see Remark 6.4) andcorresponds to the notion of martingale density introduced in [75]-[76]. Definition 4.4.
Let Z = ( Z t ) ≤ t ≤ T be a non-negative local martingale with Z = 1 . We say that Z is a weak martingale deflator if the product ZS i is a local martingale, for all i = 1 , . . . , d . If Z satisfies in addition Z T > P -a.s. we say that Z is a martingale deflator .A (weak) martingale deflator Z is said to be tradable if there exists a sequence { θ n } n ∈ N ⊆ A anda sequence { τ n } n ∈ N of stopping times increasing P -a.s. to τ := inf (cid:8) t ∈ [0 , T ] : Z t = 0 or Z t − = 0 (cid:9) such that /Z τ n = V (1 , θ n ) P -a.s., for every n ∈ N . Remark 4.5.
Fatou’s lemma implies that any weak martingale deflator Z is a supermartingale(and also a true martingale if and only if E [ Z T ] = 1 ). Furthermore, if Z is a martingale deflator,so that Z T > P -a.s., the minimum principle for non-negative supermartingales (see e.g. [68],Proposition II.3.4) implies that τ = ∞ P -a.s. It can be verified that a martingale deflator istradable if and only if there exists a strategy θ ∈ A such that /Z = V (1 , θ ) (indeed, it sufficesto define θ := P ∞ n =1 θ n (( τ n − ,τ n ]] , with τ := 0 ). This also explains the meaning of the terminology tradable .We denote by D weak and D the families of all weak martingale deflators and of all martingaledeflators, respectively. The next lemma shows the fundamental property of (weak) martingale defla-tors. At little extra cost, we state and prove the result for the case of general (possibly discontinuousand non-locally bounded) semimartingales (we refer to Section III.6e of [39] and to [46] for the defi-nition and the main properties of σ -martingales). The result is more or less well-known but, for theconvenience of the reader, we give a detailed proof in the Appendix. To the best of our knowledge, for a weak martingale deflator Z , the definition of tradability as in Definition 4.4 seemsto be new, but is related to condition H from [45]. emma 4.6. Let Z ∈ D weak . Then, for any H ∈ L ( S ) , the product Z ( H · S ) is a σ -martingale.If in addition H ∈ A , then Z ( H · S ) ∈ M loc . If Z ∈ D weak and H ∈ A , Lemma 4.6 implies that Z (1+ H · S ) is a non-negative local martingaleand, hence, a supermartingale. This means that Z is a P -supermartingale density , according to theterminology adopted in [5]. If we also have Z T > P -a.s., i.e., Z ∈ D , then Z is an equivalentsupermartingale deflator in the sense of Definition 4.9 of [47]. The importance of supermartingaledensities/deflators has been first recognized by [51] in the context of utility maximisation.We now show that the NSA condition ensures the existence of a tradable weak martingaledeflator. This can already be guessed by carefully examining the proof of Theorem 4.3, but, sincethe result is of interest, we prefer to give full details. Proposition 4.7.
Let τ := inf (cid:8) t ∈ [0 , T ] : b K t = ∞ (cid:9) . If the NSA condition holds then the process b Z := E ( − λ · M ) [[0 ,τ )) is a tradable weak martingale deflator. Furthermore, b ZN ∈ M loc for any N = ( N t ) ≤ t ≤ T ∈ M loc orthogonal to M (in the sense of [39], Definition I.4.11).Proof. Note first that, due to Theorem 4.3, we have τ > P -a.s. Furthermore, the sequence { τ n } n ∈ N , defined as τ n := inf { t ∈ [0 , T ] : b K t ≥ n } , n ∈ N , is an announcing sequence for τ , inthe sense of I.2.16 of [39], and we have [[0 , τ )) = S n ∈ N [[0 , τ n ]] . Since b K T ∧ τ n ≤ n P -a.s. for every n ∈ N , the process b Z := E ( − λ · M ) is well-defined as a continuous local martingale on [[0 , τ )) , inthe sense of Section 5.1 of [37]. On { τ ≤ T } , we have b K τ = ∞ and b Z τ − = 0 P -a.s. By letting b Z = b Z τ − = 0 on [[ τ, T ]] , b Z can be extended to a continuous local martingale on the whole interval [0 , T ] . Furthermore, the integration by parts formula gives that, for every i = 1 , . . . , d , d ( b ZS i ) t = b Z t dS it + S it d b Z t + d h b Z, S i i t = b Z t dM it + b Z t d h M i , M i t λ t + S it d b Z t − b Z t λ ⊤ t d h M, M i i t = b Z t dM it + S it d b Z t . Since S i and b Z are continuous, this implies that b ZS i ∈ M c loc , for every i = 1 , . . . , d . We have thusshown that b Z = E ( − λ · M ) [[0 ,τ )) ∈ D weak . To prove the tradability of b Z , note that the process / b Z is well defined on [[0 , τ )) = S n ∈ N [[0 , τ n ]] . Itô’s formula gives then the following, for every n ∈ N : d b Z τ n t = − (cid:0) b Z τ n t (cid:1) d b Z τ n t + 1 (cid:0) b Z τ n t (cid:1) d h b Z i τ n t = 1 b Z τ n t λ t dM τ n t + 1 b Z τ n t λ ⊤ t d h M, M i τ n t λ t = θ nt dS t , (4.2)where θ n := ((0 ,τ n ]] λ b Z − ∈ A , for all n ∈ N . Finally, for any N = ( N t ) ≤ t ≤ T ∈ M loc orthogonal to M : b ZN = N + b Z · N + N − · b Z + h b Z, N i = N + b Z · N + N − · b Z − b Zλ · h M, N i = N + b Z · N + N − · b Z, where we have used the continuity of b Z and the orthogonality of M and N . Since b Z and N − are predictable and locally bounded, being adapted and left-continuous, and since N, b Z ∈ M loc ,Theorem IV.29 of [66] implies that b ZN ∈ M loc . Remark 4.8 ( On the minimal martingale measure ) . The process b Z is the candidate density processof the minimal martingale measure , originally introduced in [26] and defined as a probability measure b Q ∼ P on (Ω , F ) with b Q = P on F such that S is a local b Q -martingale and every local P -martingaleorthogonal to the martingale part M in the canonical decomposition of S (with respect to P ) remainsa local b Q -martingale. However, even if NSA holds, the process b Z can fail to be a well-defined density rocess for two reasons. First, if P ( b Z T > < , the measure b Q defined by d b Q := b Z T dP fails tobe equivalent to P , being only absolutely continuous. Second, b Z may fail to be a true martingale,being instead a strict local martingale in the sense of [23], i.e., a local martingale which is not atrue martingale, so that E [ b Z T ] < E [ b Z ] = 1 . In the latter case, b Q fails to be a probability measure,since b Q (Ω) = E [ b Z T ] < . Remark 4.9.
A strong arbitrage opportunity corresponds to the notion of arbitrage adopted in thecontext of the Benchmark Approach, see e.g. Section 7 of [64] and Section 10.3 of [65]. However,we want to make the reader aware of the fact that typical applications of the Benchmark Approachrequire assumptions stronger than NSA, namely the existence of the
Growth Optimal Portfolio(GOP) . Theorem 4.12 of [47] shows that the existence of a (non-exploding) GOP is equivalent tothe
No Unbounded Profit with Bounded Risk (NUPBR) condition, which is strictly stronger thanNSA (see e.g. Example 7.6). Hence, in the context of the Benchmark Approach, not only strongarbitrage opportunities but also slightly weaker forms of arbitrage must be ruled out from the market(see also Section 8 for a related discussion). The NSA condition has been also adopted in [13] as anecessary (but not sufficient) requirement in order to construct the GOP.
An arbitrage of the first kind amounts to a non-negative and non-zero payoff which can be super-replicated via a non-negative portfolio by every market participant, regardless of his/her initialwealth. It is evident that a strong arbitrage opportunity yields an arbitrage of the first kind.Indeed, let H ∈ A generate a strong arbitrage opportunity and define ξ := G T ( H ) . By Definition2.3- (ii) , it holds that P ( ξ ≥
0) = 1 and P ( ξ > > . Moreover, for any v ∈ (0 , ∞ ) , we also have V T ( v, H ) = v + G T ( H ) > ξ , thus showing that ξ generates an arbitrage of the first kind. A modelsatisfying NSA but allowing for arbitrages of the first kind will be presented in Example 7.6, thusshowing that NA1 is strictly stronger than NSA.We now introduce two alternative notions of arbitrage which will be shown to be equivalent toan arbitrage of the first kind (Lemma 5.2). Definition 5.1. (i) A sequence { H n } n ∈ N ⊂ A generates an unbounded profit with bounded risk if the collec-tion { G T ( H n ) } n ∈ N is unbounded in probability, i.e., if lim m →∞ sup n ∈ N P ( G T ( H n ) > m ) > .If there exists no such sequence we say that the No Unbounded Profit with Bounded Risk(NUPBR) condition holds;(ii) let { x n } n ∈ N ⊂ R + be a sequence such that x n ց as n → ∞ . A sequence { H n } n ∈ N ⊂ A with H n ∈ A x n , for all n ∈ N , generates a cheap thrill if V T ( x n , H n ) → ∞ P -a.s. as n → ∞ onsome event with strictly positive probability. If there exists no such sequence we say that the No Cheap Thrill (NCT) condition holds.
The NUPBR condition has been first introduced under that name in [47] and corresponds tocondition BK in [43] (the same condition also plays a key role in the seminal paper [14]). Note thatthere is no loss of generality in considering -admissible strategies in Definition 5.1- (i) . Indeed, wehave { G T ( H ) : H ∈ A a } = a { G T ( H ) : H ∈ A } , for any a > , and, hence, the set of all finalwealths generated by a -admissible strategies is bounded in probability if and only if the set of all nal wealths generated by -admissible strategies is bounded in probability. The notion of cheapthrill has been introduced by [53] in the context of a complete Itô process model and can easilybe shown to be equivalent to the notion of asymptotic arbitrage of the first kind (with respect tothe fixed probability measure P ) of [44], hence the name “arbitrage of the first kind” in Definition2.3- (iii) .The next lemma proves the equivalence between the notions introduced in Definition 5.1 and thenotion of arbitrage of the first kind. The proof relies on techniques similar to those used in Section3 of [14] or in Proposition 1 of [49] and does not rely on the continuity of S . Lemma 5.2.
The NA1, NUPBR and NCT conditions are all equivalent.Proof.
Let the random variable ξ generate an arbitrage of the first kind. By definition, for every n ∈ N , there exists a strategy H n ∈ A /n such that V T (1 /n, H n ) ≥ ξ P -a.s. For every n ∈ N , define e H n := nH n , so that { e H n } n ∈ N ⊂ A and G T ( e H n ) = nG T ( H n ) ≥ nξ − P -a.s. Since P ( ξ > > ,this implies that the collection { G T ( e H n ) : n ∈ N } is unbounded in probability.Let { H n } n ∈ N ⊂ A generate an unbounded profit with bounded risk, so that P (cid:0) G T ( H n ) ≥ n (cid:1) > β for all n ∈ N and for some β > . Let e H n := n − H n , for every n ∈ N , so that e H n ∈ A /n and P (cid:0) G T ( e H n ) ≥ (cid:1) > β . Let f n := n − + G T ( e H n ) ≥ P -a.s., for all n ∈ N . Due to Lemma A1.1 of [14],there exists a sequence { g n } n ∈ N , with g n ∈ conv { f n , f n +1 , . . . } , such that { g n } n ∈ N converges P -a.s.to a non-negative random variable g as n → ∞ . For all n ∈ N , let K n be the convex combinationof strategies { e H m } m ≥ n corresponding to g n . It is easy to check that K n ∈ A /n , for every n ∈ N .Furthermore, we have G T ( K n ) = g n + O (cid:0) n − (cid:1) , so that G T ( K n ) → g P -a.s. as n → ∞ . Thelast assertion of Lemma A1.1 of [14] implies that P ( g > > . By letting x n := log( n ) /n and e K n := log( n ) K n , for every n ∈ N , so that e K n ∈ A x n , we then obtain a sequence { e K n } n ∈ N whichgenerates a cheap thrill.Finally, let the sequence { H n } n ∈ N generate a cheap thrill, with respect to { x n } n ∈ N . By definition,this implies that, for each n ∈ N , the set C n := (cid:8) V T ( x m , H m ) : m ∈ N , m ≥ n (cid:9) is hereditarilyunbounded in probability on Ω u := (cid:8) ω ∈ Ω : lim n →∞ V T ( x n , H n )( ω ) = ∞ (cid:9) , in the sense of [6].Then e C n := conv C n is hereditarily unbounded in probability on Ω u as well, for all n ∈ N . Similarlyas in the proof of Proposition 1 of [49], by Lemma 2.3 of [6], for every n ∈ N there exists an element f n ∈ e C n such that P (cid:0) Ω u ∩ { f n < } (cid:1) < P (Ω u ) / n +1 . Let A := T n ∈ N { f n ≥ } and ξ := A . Then: P (Ω u \ A ) = P (cid:18) [ n ∈ N (cid:0) Ω u ∩ { f n < } (cid:1)(cid:19) ≤ X n ∈ N P (cid:0) Ω u ∩ { f n < } (cid:1) < X n ∈ N P (Ω u )2 n +1 = P (Ω u )2 , which implies P ( A ) > , thus showing that P ( ξ ≥
0) = 1 and P ( ξ > > . Note also that ξ ≤ A f n ≤ f n P -a.s., for every n ∈ N . Since f n ∈ conv (cid:8) V T ( x m , H m ) : m ∈ N , m ≥ n (cid:9) , for every n ∈ N , and x n ց as n → ∞ , this implies that ξ generates an arbitrage of the first kind. Remark 5.3.
We want to mention that the recent paper [33] provides an alternative characterisationof NA1 in terms of the equivalent
No Gratis Events (NGE) condition. In particular, the NGEcondition (and, consequently, NA1 as well) is shown to be numéraire-independent. We shall give avery simple proof of the latter property in Corollary 5.6. We want to point out that a cheap thrill is also equivalent to an approximate arbitrage in the sense of [13], as thereader can easily verify. However, we shall use the term “approximate arbitrage” with a different meaning in Section 6. he following theorem gives several equivalent characterisations of the NA1 condition (anotherequivalent and useful characterisation will be provided in the next section, see Corollary 6.7). Theorem 5.4.
The following are equivalent, using the notation introduced in (2.1) - (2.2) and (4.1) :(i) any (and, consequently, all) of the NA1, NUPBR and NCT conditions holds;(ii) ν t = 0 P ⊗ B -a.e. and b K T = R T a ⊤ t c + t a t dB t < ∞ P -a.s., i.e., λ ∈ L loc ( M ) ;(iii) there exists a tradable martingale deflator;(iv) D 6 = ∅ , i.e., there exists a martingale deflator.Proof. ( i ) ⇒ ( ii ) : due to Lemma 5.2, the NA1, NUPBR and NCT conditions are equivalent. So,let us assume that NUPBR holds. Since NUPBR implies NSA, Theorem 4.3 gives that ν t = 0 P ⊗ B -a.e. and α = inf (cid:8) t ∈ [0 , T ] : b K t + ht = ∞ , ∀ h ∈ (0 , T − t ] (cid:9) = ∞ P -a.s. It remains to show that b K T < ∞ P -a.s. Suppose on the contrary that P ( τ ≤ T ) > , where τ := inf (cid:8) t ∈ [0 , T ] : b K t = ∞ (cid:9) ,so that P ( b K T = ∞ ) = P ( b Z T = 0) > , where the process b Z is defined as in Proposition 4.7. Definethe sequence { τ n } n ∈ N of stopping times τ n := inf (cid:8) t ∈ [0 , T ] : b K t ≥ n (cid:9) , for n ∈ N . Clearly, wehave τ n ր τ P -a.s. as n → ∞ . As shown in equation (4.2), we have θ n = ((0 ,τ n ]] λ b Z − ∈ A and G T ( θ n ) = b Z − T ∧ τ n − , for every n ∈ N , so that G T ( θ n ) → b Z − T ∧ τ − P -a.s. as n → ∞ . Since b Z T ∧ τ = 0 on { τ ≤ T } and P ( τ ≤ T ) > , this shows that (cid:8) G T ( H n ) : n ∈ N (cid:9) cannot be bounded inprobability, thus contradicting the assumption that NUPBR holds. ( ii ) ⇒ ( iii ) : this follows directly from Proposition 4.7, since b K T < ∞ P -a.s. implies τ = ∞ P -a.s. ( iii ) ⇒ ( iv ) : by Definition 4.4, this implication is trivial. ( iv ) ⇒ ( i ) : let Z ∈ D and suppose that the random variable ξ generates an arbitrage of the firstkind, so that for every v ∈ (0 , ∞ ) there exists an element H v ∈ A v satisfying V T ( v, H v ) ≥ ξ P -a.s.Due to Lemma 4.6, the product Z V ( v, H v ) = Z ( v + H v · S ) is a non-negative local martingale and,hence, also a supermartingale. As a consequence, for every v ∈ (0 , ∞ ) , E (cid:2) Z T ξ (cid:3) ≤ E (cid:2) Z T V T ( v, H v ) (cid:3) ≤ E (cid:2) Z V ( v, H v ) (cid:3) = v. Since Z T > P -a.s., this contradicts the assumption that P ( ξ > > . Due to Lemma 5.2, theNUPBR and NCT conditions hold as well.Results analogous to Theorem 5.4 have already been obtained in Section 4 of [49], in Section 3of [35] and also earlier in Theorem 2.9 of [12]. However, the proof given here is rather short andemphasises the role of the tradability of the martingale deflator b Z introduced in Proposition 4.7.In particular, it shows that the event { b K T = ∞} corresponds to the explosion of the final wealthgenerated by a sequence of -admissible strategies (see also Section 6 of [13] for a related discussion). Remark 5.5 ( The numéraire portfolio ) . The NA1 condition can be shown to be equivalent to theexistence of the numéraire portfolio , defined as the strictly positive portfolio process V ∗ := V (1 , θ ∗ ) , θ ∗ ∈ A , such that V (1 , θ ) /V ∗ is a supermartingale for all θ ∈ A (see e.g. [5]). In the setting of thepresent paper, it is easy to verify that, as long as NA1 holds, the numéraire portfolio coincides withthe inverse of the tradable martingale deflator b Z , as follows from Theorem 5.4 together with Lemma4.6 and Fatou’s lemma (compare also with [35], Lemma 5). The equivalence between NUPBR andthe existence of the numéraire portfolio is proved in full generality in [47] (see also [13] for relatedresults). n important property of the NA1 condition (as well as of NUPBR and NCT), which is notshared in general by stronger no-arbitrage conditions (see e.g. [17]), is its invariance with respectto a change of numéraire , as shown in the next corollary. Corollary 5.6.
Let V := V (1 , θ ) be a P -a.s. strictly positive portfolio process, for some θ ∈ A .The NA1 condition holds (for S ) if and only if the NA1 condition holds for ( S/V, /V ) .Proof. Due to Theorem 5.4, it suffices to show that
D 6 = ∅ if and only if there exists a martingaledeflator for ( S/V, /V ) . If Z ∈ D 6 = ∅ , Lemma 4.6 implies that Z ′ := ZV is a strictly positive localmartingale with Z ′ = 1 . Since Z ′ ( S/V, /V ) = Z ( S, ∈ M loc , this shows that Z ′ is a martingaledeflator for ( S/V, /V ) . Conversely, if Z ′ is a martingale deflator for ( S/V, /V ) then Z := Z ′ /V isa strictly positive local martingale with Z = 1 and ZS = Z ′ S/V ∈ M loc , meaning that Z ∈ D .The next lemma describes the general structure of all martingale deflators. The result goes backto [12] and [76] (compare also with [1], Theorem 5), but we give a short proof in the Appendix forthe sake of completeness . Lemma 5.7.
Suppose that any (and, consequently, all) of the NA1, NUPBR and NCT conditionsholds. Then every martingale deflator Z = ( Z t ) ≤ t ≤ T admits the following representation: Z = E ( − λ · M + N ) = b Z E ( N ) , for some N = ( N t ) ≤ t ≤ T ∈ M loc with N = 0 , h N, M i = 0 and ∆ N > − P -a.s. and where theprocess b Z is defined as in Proposition 4.7. Theorem 5.4 and Lemma 5.7 show that b Z = E ( − λ · M ) can be rightfully considered as the minimal (tradable) martingale deflator (compare with part 2 of Remark 3.2 and Remark 4.8). Indeed, besidesbeing the martingale deflator with the “simplest” structure, if b Z fails to be a martingale deflator,i.e., if P ( b Z T = 0) > , then there cannot exist any other martingale deflator. Remark 5.8.
The equivalence ( i ) ⇔ ( iv ) in Theorem 5.4 has been recently established for generalsemimartingale models in the papers [50], in the one-dimensional case, and [81], in the R d -valuedcase (see also [78] for an alternative proof). We also want to mention that in [49] it is shown that D 6 = ∅ is equivalent to the existence of a finitely additive measure Q on (Ω , F ) , weakly equivalent to P and locally countably additive, under which S has a kind of local martingale behavior (see alsoSection 5 of [7] for related results). The goal of this section consists in studying the NFLVR condition, on which the classical no-arbitrage pricing theory is based (we refer the reader to [20] for a complete account thereof), andthe relations with the weak no-arbitrage conditions discussed so far. As can be seen from Definition2.3, the NA (and, hence, the NFLVR) condition excludes arbitrage possibilities that may requireaccess to a finite line of credit and, hence, cannot be realized in an unlimited way be every marketparticipant. Let us begin this section by introducing another (last) notion of arbitrage. Actually, in the one-dimensional case, an analogous result can already be found in [82] (see also [37], Theorem 6.11). efinition 6.1. A sequence { H n } n ∈ N ⊂ A c , for some c > , generates an Approximate Arbitrage if P ( G T ( H n ) ≥ → as n → ∞ and there exists a constant δ > such that P ( G T ( H n ) > δ ) > δ ,for all n ∈ N . If there exists no such sequence we say that the No Approximate Arbitrage (NAA) condition holds.
The notion of approximate arbitrage has been first introduced in [52] in the context of a completeItô-process model and turns out to be equivalent to the notion of free lunch with vanishing riskintroduced in Definition 2.3- (v) , as shown in the next lemma, the proof of which combines severaltechniques already employed in [14] and [52]. Recall that C = (cid:0) { G T ( H ) : H ∈ A} − L (cid:1) ∩ L ∞ ,according to the notation introduced at the end of Section 2. Lemma 6.2.
The NFLVR condition and the NAA condition are equivalent.Proof.
Suppose that NFLVR fails to hold. Then, as in Proposition 3.6 of [14], there exists either anarbitrage opportunity or a cheap thrill. Clearly, if there exists an arbitrage opportunity then therealso exists an approximate arbitrage. We now show that the existence of a cheap thrill yields anapproximate arbitrage, thus proving that NAA implies NFLVR. Due to Lemma 5.2 together withTheorem 5.4, the existence of a cheap thrill is equivalent to < P ( b K T = ∞ ) =: δ . For a fixed κ > δ and for every n ∈ N , define the stopping times σ n := inf (cid:8) t ∈ [0 , T ] : b Z t = 1 /n (cid:9) ∧ T and ̺ n := inf (cid:8) t ∈ [ σ n , T ] : b Z t = b Z σ n /κ (cid:9) ∧ T, where b Z is as in Proposition 4.7. For every n ∈ N , define Y n := b Z σ n / b Z ̺ n and H n := [[ σ n ,̺ n ]] ( b Z/ b Z σ n ) − λ .Itô’s formula implies then the following, for all t ∈ [0 , T ] : Y nt = 1 + Z t b Z σ n b Z u { σ n ≤ u ≤ ̺ n } λ u dM u + Z t b Z σ n b Z u { σ n ≤ u ≤ ̺ n } λ ⊤ u d h M, M i u λ u = 1 + G t ( H n ) , thus showing that { H n } n ∈ N ⊂ A . Furthermore: { b K T = ∞} = { b Z T = 0 } ⊆ { σ n < ̺ n < T } ⊆ { G T ( H n ) = b Z σ n / b Z ̺ n − } = { G T ( H n ) = κ − } and, hence, P ( G T ( H n ) ≥ κ − ≥ δ , ∀ n ∈ N . Since we have { G T ( H n ) < } ⊆ { σ n < T }∩{ b K T < ∞} and P ( σ n < T, b K T < ∞ ) → as n → ∞ , we also get P ( G T ( H n ) ≥
0) = 1 − P ( G T ( H n ) < ≥ − P ( σ n < T, b K T < ∞ ) → as n → ∞ , thus showing that the sequence { H n } n ∈ N ⊂ A yields an approximate arbitrage.Conversely, suppose that the sequence { H n } n ∈ N ⊂ A c generates an approximate arbitrage. Bydefinition, for every ε > , we have P ( G T ( H n ) − > ε ) ≤ P ( G T ( H n ) < → as n → ∞ . Thismeans that G T ( H n ) − → in probability as n → ∞ and, passing to a subsequence, we can assumethat the convergence takes place P -a.s. For every n ∈ N , let f n := G T ( H n ) ∧ δ ∈ C , so that P ( f n = δ ) > δ and f − n → P -a.s. as n → ∞ . Due to Lemma A1.1 (and the subsequent Remark 2)of [14], there exists a sequence { g n } n ∈ N , with g n ∈ conv { f n , f n +1 , . . . } , such that g n → g P -a.s. as n → ∞ for some random variable g : Ω → [0 , δ ] . More precisely, due to the bounded convergencetheorem (since − c ≤ g n ≤ δ P -a.s. for all n ∈ N ), δ P ( g > ≥ E [ g { g> } ] = E [ g ] = lim n →∞ E [ g n ] ≥ δ , eaning that β := P ( g > ≥ δ > . Egorov’s theorem gives that g n converges to g as n → ∞ uniformly on a set Ω ′ with P (Ω ′ ) ≥ − β/ . For every n ∈ N , define h n := g n ∧ δ Ω ′ , so that { h n } n ∈ N ∈ C and h n → g Ω ′ in the norm topology of L ∞ , i.e., g Ω ′ ∈ C ∩ L ∞ + . Since P ( g Ω ′ >
0) = 1 − P (cid:0) { g = 0 } ∪ Ω ′ c (cid:1) ≥ P (Ω ′ ) − P ( g = 0) ≥ − β/ − (1 − β ) = β/ > , this shows that NFLVR fails to hold, thus proving that NFLVR implies NAA.Before formulating the next theorem, which essentially corresponds to the main result of [14],we need to recall the classical and well-known notion of Equivalent Local Martingale Measure . Definition 6.3.
A probability measure Q on (Ω , F ) with Q ∼ P is said to be an Equivalent LocalMartingale Measure (ELMM) for S if S is a local Q -martingale. Remark 6.4 ( On martingale deflators and ELMMs ) . Suppose that there exists an ELMM Q for S . Letting Z Q = ( Z Qt ) ≤ t ≤ T be its density process, Bayes’ rule implies that Z Q S ∈ M loc , meaningthat Z Q /Z Q ∈ D . Conversely, in view of Remark 4.5, an element Z ∈ D can be taken as the densityprocess of an ELMM if and only if E [ Z T ] = 1 . Theorem 6.5.
The following are equivalent, using the notation introduced in (2.1) - (2.2) and (4.1) :(i) the NFLVR condition holds;(ii) there exists an ELMM for S ;(iii) ν t = 0 P ⊗ B -a.e., b K T = R T a ⊤ t c + t a t dB t < ∞ P -a.s. and there exists N = ( N t ) ≤ t ≤ T ∈ M loc with N = 0 , h N, M i = 0 and ∆ N > − P -a.s. such that b Z E ( N ) ∈ M ;(iv) the conditions NA1 (or, equivalently, NUPBR/NCT) and NA both hold;(v) the NAA condition holds.Proof. (i) ⇔ (ii) : this is the content of Corollary 1.2 of [14], recalling that S is a continuous (and,hence, locally bounded) semimartingale. (ii) ⇔ (iii) : this equivalence follows from Theorem 5.4 and Lemma 5.7 together with Remark 6.4. (ii) ⇒ (iv) : the existence of an ELMM for S implies that D 6 = ∅ . Hence, due to Theorem 5.4, theNA1 condition (as well as NUPBR and NCT) holds. Let H ∈ A yield an arbitrage opportunity.Lemma 4.6 and Bayes’ rule imply that the process G ( H ) is a local Q -martingale uniformly boundedfrom below. Due to Fatou’s lemma, it is also a Q -supermartingale and, hence, E Q [ G T ( H )] ≤ .Since Q ∼ P , this clearly contradicts the assumption that H yields an arbitrage opportunity. (iv) ⇔ (v) ⇔ (i) : these equivalences follow from Lemma 6.2 together with Proposition 3.6 of [14]. Remarks 6.6. 1)
As can be seen from part (iii) of Theorem 6.5, the NFLVR condition, unlikethe weak no-arbitrage conditions discussed in the previous sections, does not only depend on thecharacteristics of S but also on the structure of the underlying filtration F . In particular, thismeans that in general one cannot prove the existence of arbitrage opportunities by relying on thecharacteristics of the discounted price process only (to this effect, compare also [47], Example 4.7) . We want to mention that, in some special cases, it is possible to check the NFLVR condition in terms of the charac-teristics of the discounted price process S . For instance, in the case when S is a continuous exponential semimartingaleand one can take dB t = dt in (2.1), a probabilistic characterisation of the absence of arbitrage opportunities in terms ofthe characteristics of S has been obtained in the recent paper [55]. In the case of non-negative one-dimensional Markoviandiffusions, necessary and sufficient conditions for the validity of NFLVR are provided in [59]. ) We want to warn the reader that NFLVR does not ensure that the measure b Q defined by d b Q/dP := b Z T is an ELMM for S , since NFLVR fails to imply in general that E [ b Z T ] = 1 . In view ofRemark 4.8, this amounts to saying that NFLVR does not guarantee the existence of the minimalmartingale measure (a counterexample is provided in [18]). In other words, the NFLVR conditioncannot be checked by looking only at the properties of the minimal (weak) martingale deflator b Z ,unlike weaker no-arbitrage conditions. There is no general implication between NA1 and NA. On the one hand, as shown in example7.7, it might well be that NA1 holds but nevertheless there exist arbitrage opportunities. On theother hand, it is possible to construct models that admit no arbitrage opportunities but do notsatisfy NA1 (an explicit example can be found in Section 4 of [52]; see also [34], Example 1.37).For models based on continuous semimartingales, without assuming a priori the validity of NA1, acharacterisation of NA is given in Theorem 9 of [45] and in Theorem 2.1 of [79].The following corollary gives an interesting alternative characterisation of NA1, thus comple-menting Theorem 5.4.
Corollary 6.7.
The NA1 condition holds if and only if there exists a P -a.s. strictly positive portfolioprocess V := V (1 , θ ) , for some θ ∈ A , such that the NFLVR condition holds for ( S/V, /V ) .Proof. Due to Theorem 5.4, the NA1 condition implies the existence of a tradable martingale deflator Z , so that /Z = V (1 , θ ) for some θ ∈ A (see also Remark 4.5). By letting V := V (1 , θ ) , this meansthat /V ∈ M loc and S/V ∈ M loc and so P is an ELMM for ( S/V, /V ) . Due to Theorem 6.5, thisimplies that ( S/V, /V ) satisfies NFLVR. Conversely, if NFLVR holds for ( S/V, /V ) , Theorem 6.5gives the existence of an ELMM Q for ( S/V, /V ) , with density process Z Q . By Bayes’ rule, wehave Z Q /V ∈ M loc and Z Q S/V ∈ M loc . This means that Z := Z Q / ( Z Q V ) ∈ D . Theorem 5.4then implies that NA1 holds for S .In particular, the above corollary shows that, as long as NA1 holds, we can find a suitablenuméraire V such that the classical and stronger NFLVR condition holds in the V -discountedfinancial market ( S/V, /V ) , regardless of the validity of NFLVR for the original financial market.In particular, if NA1 holds, the process b Z is a tradable martingale deflator and, hence, letting b V := 1 / b Z , the NFLVR condition holds for ( S/ b V , / b V ) . This suggests that, even in the absence ofan ELMM for S , the financial market ( S/ b V , / b V ) can be regarded as a natural setting for solvingpricing and portfolio optimisation problems, as it is indeed proposed in the context of the BenchmarkApproach (see e.g. [65], Chapter 10). In the present section, we provide several examples and counterexamples for the different no-arbitrage conditions discussed so far. In particular, we aim at illustrating the relationships (1.1).
Example 7.1.
We start by giving an explicit example of a model allowing for increasing profits.Let N = ( N t ) ≤ t ≤ T ∈ M c loc and S := | N | . Tanaka’s formula (see [68], Theorem VI.1.2) gives thefollowing canonical decomposition: S t = | N | + Z t sign( N u ) dN u + L t , for all t ∈ [0 , T ] , here the process L = (cid:0) L t (cid:1) ≤ t ≤ T is the local time of N at the level . Using the notation introducedin Section 2, we have A = L and M = sign( N ) · N . We now show that dA ≪ d h M, M i does nothold, where h M, M i = h N i . In fact, Proposition VI.1.3 of [68] shows that, for almost all ω ∈ Ω , themeasure (in t ) dL t ( ω ) is carried by the set { t : N t ( ω ) = 0 } . However, for almost all ω ∈ Ω , the set { t : N t ( ω ) = 0 } has zero measure with respect to d h N i t ( ω ) . This means that L induces a measurewhich is singular with respect to the measure induced by h N i . Theorem 3.1 then implies that NIPfails.In the present context, we can also explicitly construct a trading strategy generating an increasingprofit. For simplicity, let us suppose that N = 0 P -a.s. and define the process H = ( H t ) ≤ t ≤ T by H := { N =0 }∩ ((0 ,T ]] . Clearly, H is a bounded predictable process and so H ∈ L ( S ) . Furthermore, ( H · M ) t = R t H u sign( N u ) dN u = 0 P -a.s. for all t ∈ [0 , T ] . Note also that R HdL = L , since dL t ( ω ) is carried by the set { t : N t ( ω ) = 0 } for almost all ω ∈ Ω . Hence: ( H · S ) t = Z t H u sign( N u ) dN u + Z t H u dL u = L t = sup s ≤ t (cid:18) − Z s sign( N u ) dN u (cid:19) , for all t ∈ [0 , T ] , where the last equality follows from Skorohod’s lemma (see [68], Lemma VI.2.1). This shows thatthe gains from trading process G ( H ) = H · S starts from and is non-decreasing. In particu-lar, H ∈ A . Finally, if we assume that the local martingale N is not identically zero, we alsohave P (cid:0) G T ( H ) > (cid:1) > . Indeed, suppose on the contrary that P (cid:0) G T ( H ) > (cid:1) = 0 , so that sup s ≤ T (cid:0) − R s sign( N u ) dN u (cid:1) = 0 P -a.s. and, hence, R t sign( N u ) dN u ≥ P -a.s. for all t ∈ [0 , T ] .By Fatou’s lemma, this implies that sign( N ) · N is a non-negative supermartingale, being a non-negative continuous local martingale. Since (sign( N ) · N ) = 0 , the supermartingale property gives sign( N ) · N = 0 , which in turn implies that h N i = h sign( N ) · N i = 0 , thus contradicting theassumption that N is not trivial. Remark 7.2.
An interesting interpretation of the arbitrage possibilities arising from local times canbe found in [40], where it is shown that the existence of large traders (whose orders affect marketprices) can introduce “hidden” arbitrage opportunities for the small traders, who act as price-takers.These arbitrage profits are “hidden” since they occur on time sets of Lebesgue measure zero, beingrelated to the local time of Brownian motion. Other examples of arbitrage profits arising from localtime can be found in [60] and [71]. Furthermore, in the recent paper [41] it is shown that condition (iii) of Theorem 3.1 can be violated when projecting an asset price process onto a subfiltration ifthere is a bubble in the original (larger) filtration.We now present two examples of financial market models that satisfy NIP but allow for strongarbitrage opportunities. In view of Theorem 4.3, the two following examples satisfy ν t = 0 P ⊗ B -a.e.but P ( α < T ) > , meaning that the mean-variance trade-off process is allowed to jump to infinitywith positive probability. Example 7.3.
Let M = ( M t ) ≤ t ≤ T ∈ M c loc with M = 0 and let τ be a stopping time such that P ( τ < T ) > . Define the discounted price process S = ( S t ) ≤ t ≤ T of a single risky asset as follows: S = M + h M i β ·∧ τ + (cid:0) h M i ·∨ τ − h M i τ (cid:1) γ , for some γ ≤ / < β . Then, due to Itô’s formula: dS t = dM t + (cid:16) β { t ≤ τ } h M i β − t + γ { t>τ } (cid:0) h M i t − h M i τ (cid:1) γ − (cid:17) d h M i t . heorem 3.1 implies that NIP holds. However, on { τ < T } we have that, for every ε > : b K τ + ετ = γ Z τ + ετ (cid:0) h M i t − h M i τ (cid:1) γ − d h M i t = γ log (cid:0) h M i τ + t − h M i τ (cid:1)(cid:12)(cid:12)(cid:12) εt =0 if γ = 1 / γ γ − (cid:0) h M i τ + t − h M i τ (cid:1) γ − (cid:12)(cid:12)(cid:12) εt =0 if γ < / ∞ . This shows that in the present example we have α = τ P -a.s. Hence, due to Theorem 4.3, the NSAcondition fails to hold. By letting M be a standard Brownian motion on (Ω , F , F , P ) , γ = 1 / and τ = 0 , we recover the situation considered in Example 3.4 of [16]. Example 7.4.
Let W = ( W t ) ≤ t ≤ T be a standard Brownian motion on (Ω , F , F , P ) and define S as follows, for all t ∈ [0 , T ] : S t = W t + Z t W u u du. Clearly, Theorem 3.1 implies that NIP holds. However, due to Corollary 3.2 of [42], we have R ε ( W u /u ) du = ∞ P -a.s. for every ε > , meaning that α = 0 P -a.s. Theorem 4.3 then shows thatNSA fails to hold (compare also with [70], Section 3.4). Remark 7.5.
Strong arbitrage opportunities may also arise when considering insider trading mod-els, where the original filtration F is progressively enlarged with an honest time τ which is not an F -stopping time. More specifically, as shown in [36] (see also [28], Section 6), immediate arbitrageopportunities or, equivalently, strong arbitrage opportunities (see Lemma 4.2), can be achieved inthe enlarged filtration by trading as soon as τ occurs.Let us continue by exhibiting a simple model which satisfies NSA but for which NA1 fails tohold (an analogous example can be found in [53], Section 3.1). Example 7.6.
Let W = ( W t ) ≤ t ≤ T be a standard Brownian motion on (Ω , F , F , P ) and define theprocess X = ( X t ) ≤ t ≤ T as the solution to the following SDE, for some fixed K > : dX t = K − X t T − t dt + dW t , X = 0 . The process X is a Brownian bridge (see [68], Exercise IX.2.12) starting at the level and endingat the level K > . Let us define the discounted price process S = ( S t ) ≤ t ≤ T of a single risky assetas S t := exp( X t ) , for t ∈ [0 , T ] . Then, due to Itô’s formula: dS t = S t (cid:18) K − log( S t ) T − t + 12 (cid:19) dt + S t dW t , S = 1 . It is easy to see that the condition of Theorem 4.3 is satisfied and, hence, there are no strongarbitrage opportunities, since the process b K = R · (cid:0) K − log( S u ) T − u + (cid:1) du does not jump to infinity.However, we have b K t < ∞ P -a.s. for all t ∈ [0 , T ) but b K T = ∞ P -a.s. (compare also with [9]).Theorem 5.4 then implies that NA1 fails to hold.Financial models satisfying NA1 but not NFLVR can typically be found in the context of Stochas-tic Portfolio Theory (see e.g. [25], Sections 5-6) and within the Benchmark Approach (see e.g. [62],[65], Chapters 12-13, and [34], Chapter 5). Moreover, models satisfying NA1 but not NFLVR can beconstructed in a general way by means of absolutely continuous but not equivalent changes of mea-sure (see e.g. [15], [61] and, more recently, [8], [27] and [73]) and by means of filtration enlargements see e.g. [28]). We close this section with the following example, which in particular generalizes theclassical example based on a three-dimensional Bessel process (see [17], Corollary 2.10). Example 7.7.
Let W = ( W t ) ≤ t ≤ T be a standard Brownian motion on the filtered probabilityspace (Ω , F WT , F W , P ) , with F W = ( F Wt ) ≤ t ≤ T denoting the P -augmented natural filtration of W ,and take a continuous function σ : (0 , ∞ ) → (0 , ∞ ) such that the following SDE admits a uniquestrong solution: dS t = S t σ ( S t ) dt + S t σ ( S t ) dW t , S = 1 . (7.1)Assume furthermore that R ∞ x yσ (1 /y ) dy < ∞ for some x ∈ (0 , ∞ ) . According to the notation intro-duced in Section 2, we have A t = R t S u σ ( S u ) du and M t = R t S u σ ( S u ) dW u , for all t ∈ [0 , T ] , and λ = 1 /S . Since S is locally bounded and σ ( · ) is continuous, this implies that b K T = R T λ t d h M i t < ∞ P -a.s., thus showing that NA1 holds (see Theorem 5.4). Since W enjoys the martingale represen-tation property in the filtration F W , Lemma 5.7 implies that D = { b Z } , where b Z = E ( − λ · M ) = E ( − R σ ( S ) dW ) = 1 /S . However, since R ∞ x yσ (1 /y ) dy < ∞ , for some x ∈ (0 , ∞ ) , Corollary 4.3 of[58] implies that b Z is a strict local martingale in the sense of [23], i.e., it is a local martingale whichfails to be a true martingale, so that E [ b Z T ] < . Due to Theorem 6.5, this shows that NFLVR failsfor the model (7.1).In the context of the present example, it is easy to construct explicitly an arbitrage opportunity.Indeed, let us define the process L = ( L t ) ≤ t ≤ T by L t := E [ b Z T |F t ] , for all t ∈ [0 , T ] . Then, due tothe martingale representation property, there exists an F W -predictable process θ = ( θ t ) ≤ t ≤ T with R T θ t dt < ∞ P -a.s such that L = E [ b Z T ]+ θ · W P -a.s. Let us also define the process V := L/ b Z = LS .A simple application of the product rule gives dV t = L t dS t + S t dL t + d h L, S i t = L t S t (cid:0) σ ( S t ) dt + σ ( S t ) dW t (cid:1) + S t θ t dW t + S t σ ( S t ) θ t dt = ϕ t dS t , where the process ϕ = ( ϕ t ) ≤ t ≤ T is defined as ϕ t := L t + θ t /σ ( S t ) , for t ∈ [0 , T ] . The continuity of L , S and of the function σ ( · ) implies that ϕ ∈ L ( S ) . Noting that G ( ϕ ) = V − V ≥ − E [ b Z T ] > − P -a.s., we also have ϕ ∈ A . Since G T ( ϕ ) = V T − V = 1 − E [ b Z T ] > P -a.s., this means that ϕ yieldsan arbitrage opportunity. We have thus shown that the model (7.1) allows for the possibility ofreplicating a risk-free zero-coupon bond of unitary nominal value starting from an initial investmentwhich is strictly less than one. However, not every market participant can profit from this arbitrageopportunity in an unlimited way, since the strategy ϕ ∈ A requires a non-negligible line of credit.In particular, any function of the form σ ( x ) = x µ , for µ < , satisfies the integrability condition R ∞ x yσ (1 /y ) dy < ∞ for any x ∈ (0 , ∞ ) . In the special case µ = − , it can be shown that the process S is a three-dimensional Bessel process (see [68], Chapter XI), the classical example of a financialmodel for which NA (and, hence, NFLVR as well) fails, as shown already in [15], in Corollary 2.10of [17] and in Example 4.6 of [47] (see also[21] for related results). In the present paper, we have provided a unified account of several no-arbitrage conditions proposedin the literature in the context of financial market models based on continuous semimartingales. Wehave focused on the probabilistic characterisations of weak and strong no-arbitrage conditions aswell as on the study of their relationships and of their equivalent formulations, illustrating the eneral theory by means of explicit examples and counterexamples. We now conclude the paper bycommenting on the main economic and financial implications of the different no-arbitrage conditionsconsidered so far.In economic theory, a first and fundamental issue is represented by the relation between no-arbitrage and market viability , in the sense of solvability of the portfolio optimisation problem forsome hypothetical agent who prefers more to less. In that sense, a viable model for a financial marketis a potential model of a competitive equilibrium. The relations between no-arbitrage conditionsand market viability, which go back to the seminal paper [31], clarify to what extent the existenceof arbitrage profits is incompatible with the possibility of a competitive equilibrium.It is easy to see that the minimal NIP requirement does not suffice to ensure any form of marketviability. Indeed, Examples 7.3 and 7.4 show that NIP fails to exclude strong arbitrage opportunities.In the presence of a strong arbitrage opportunity, any agent with non-satiated strictly increasingpreferences would invest in it in an unlimited way, because a strong arbitrage opportunity does notrequire any initial investment nor any amount of credit and, at the same time, yields a positiveprofit at the final time T (see Definition 2.3- (ii) ). Of course, such a possibility would contrast withthe solvability of portfolio optimisation problems as well as with the existence of an equilibrium,because any agent could always improve the performance of his portfolio at zero cost without risk.In view of the preceding discussion, the NSA condition represents a necessary requirement formarket viability. In this regard, the result of Theorem 1 of [53] is of particular interest, since itproves that the NSA condition is actually equivalent to market viability. More specifically, in thecontext of a complete Itô process model (and considering utility from intermediate consumption aswell), [53] show that, if NSA holds, then there exists an optimal portfolio for an agent who prefersmore to less. However, the agent constructed in [53] exhibits strictly increasing but rather irregular(and discontinuous) preferences and, most importantly, has no capacity at all for undertaking a nettrade requiring access to a credit line.It is therefore of interest to study which no-arbitrage condition is equivalent to market viability,defined for a wide class of regular preferences (as it is also the case in the seminal paper [31]). Inthis regard, the paper [53] again provides an interesting result. Indeed, Theorem 2 of [53] showsthat there are no cheap thrills (see Definition 5.1), or, equivalently, NA1 holds (see Lemma 5.2), ifand only if there exists an optimal portfolio for an agent with regular preferences. This result isconfirmed and generalised in Proposition 4.19 of [47], which shows that, in a general semimartingalesetting, the failure of NA1 precludes the solvability of any portfolio optimisation problem (for anystrictly increasing concave utility function U : (0 , ∞ ) → R ). Moreover, in the recent paper [11] ithas been proved that NA1 is exactly equivalent to the solvability of portfolio optimisation problems(for any strictly increasing concave utility function U : (0 , ∞ ) → R satisfying the Inada conditionsand the asymptotic elasticity condition of [51]), up to an equivalent change of measure. Summingup, these results make clear that NA1 can be regarded as the minimal condition in order to ensurea meaningful form of market viability.Moreover, the fundamental problems of valuation and hedging can be successfully addressed aslong as the NA1 condition holds, provided that one replaces ELMMs with martingale deflators (seeDefinition 4.4). In particular, most of the classical results on the hedging and pricing of contingentclaims and on market completeness can also be obtained in terms of martingale deflators, see e.g.[25], [29], [72] and, in a general semimartingale setting, [4] and [80] (the super-hedging dualitycan also be extended to financial markets satisfying NA1, see Section 4.7 of [47]). Furthermore, s shown in Corollary 6.7, if NA1 holds then we can recover the classical NFLVR condition bymeans of a change of numéraire. This is also related to the Benchmark Approach proposed byEckhard Platen and collaborators (see e.g. [62]-[65]), which provides a coherent framework forvaluing contingent claims without relying on the existence of risk-neutral measures by consideringthe numéraire portfolio-discounted financial market.Altogether, the above discussion suggests that the NIP and NSA conditions can be regarded asindispensable “sanity checks” and are only meant to exclude almost pathological notions of arbitrage.On the other hand, the NA1 condition, while strictly weaker than the classical NFLVR condition, isequivalent to an economically sound notion of market viability and allows to successfully solve thefundamental problems of portfolio optimisation, pricing and hedging.We close the paper with the following table, which summarises the no-arbitrage conditionsintroduced in Definition 2.3 and studied so far, together with their characterisations (see Theorems3.1, 4.3, 5.4 and 6.5) and their equivalent formulations (see Lemmata 4.2, 5.2 and 6.2). Condition ProbabilisticCharacterisation Equivalent formulation
No Increasing Profit (NIP) ν t = 0 P ⊗ B -a.e. – No Strong Arbitrage (NSA) ν t = 0 P ⊗ B -a.e. and α = ∞ P -a.s. (i.e., b K does notjump to infinity) No Immediate Arbitrage (NIA)No Arbitrage of the First Kind (NA1) ν t = 0 P ⊗ B -a.e. and b K T < ∞ P -a.s. No Unbounded Profit withBounded Risk (NUPBR)No Cheap Thrill (NCT)No Free Lunch with Vanishing Risk(NFLVR) ν t = 0 P ⊗ B -a.e. and b K T < ∞ P -a.s. and ∃ N ∈ M loc with N = 0 , h N, M i = 0 , ∆ N > − P -a.s.such that b Z E ( N ) ∈ M . No Approximate Arbitrage (NAA)
Acknowledgements:
The author is thankful to Monique Jeanblanc, Wolfgang J. Runggaldier and MarekRutkowski for many useful remarks and discussions on the topic of the present paper. This research wassupported by a Marie Curie Intra European Fellowship within the 7th European Community FrameworkProgramme under grant agreement PIEF-GA-2012-332345.
A Appendix
Proof of Lemma 4.6.
The first part of the proof relies on arguments similar to those used in the proofs of Proposition3.2 of [30] and Proposition 8 of [69]. Let Z = ( Z t ) ≤ t ≤ T ∈ D weak and H ∈ L ( S ) . Define the R d +1 -valued local martingale Y = ( Y t ) ≤ t ≤ T by Y t := ( Z t S t , . . . , Z t S dt , Z t ) ⊤ and let L ( Y ) be the setof all R d +1 -valued predictable Y -integrable processes, in the sense of Definition III.6.17 of [39]. Forall n ∈ N , define also H ( n ) := H {k H k≤ n } . Then, using twice the integration by parts formula and he associativity of the stochastic integral: Z (cid:0) H ( n ) · S (cid:1) = Z − · (cid:0) H ( n ) · S (cid:1) + (cid:0) H ( n ) · S (cid:1) − · Z + (cid:2) Z, H ( n ) · S (cid:3) = (cid:0) Z − H ( n ) (cid:1) · S + (cid:0) H ( n ) · S (cid:1) − · Z + H ( n ) · [ S, Z ]= H ( n ) · ( Z − · S ) + (cid:0) H ( n ) · S (cid:1) − · Z + H ( n ) · [ S, Z ]= H ( n ) · ( ZS − S − · Z ) + (cid:0) H ( n ) · S (cid:1) − · Z = H ( n ) · ( ZS ) + (cid:16)(cid:0) H ( n ) · S (cid:1) − − H ( n ) ⊤ S − (cid:17) · Z = K ( n ) · Y where, for every n ∈ N , the R d +1 -valued predictable process K ( n ) is defined as K ( n ) i := H ( n ) i ,for all i = 1 , . . . , d , and K ( n ) d +1 := (cid:0) H ( n ) · S (cid:1) − − H ( n ) ⊤ S − . Clearly, we have K ( n ) ∈ L ( Y ) ,since K ( n ) is predictable and locally bounded, for every n ∈ N . Define the R d +1 -valued predictableprocess K by K i := H i , for all i = 1 , . . . , d , and K d +1 := ( H · S ) − − H ⊤ S − . Since H ∈ L ( S ) , H ( n ) · S converges to H · S in the Emery topology of semimartingales as n → ∞ . This implies that K ( n ) · Y = Z (cid:0) H ( n ) · S (cid:1) also converges in Emery’s topology, since the multiplication with Z is acontinuous operation. Since the space (cid:8) K · Y : K ∈ L ( Y ) (cid:9) is closed in Emery’s topology (see [39],Proposition III.6.26), we can conclude that Z ( H · S ) = ¯ K · Y for some ¯ K ∈ L ( Y ) . But since K ( n ) converges to K ( P -a.s. uniformly in t , at least along a subsequence) as n → ∞ , we can concludethat ¯ K = K (see [57]). This shows that K ∈ L ( Y ) . Since Y ∈ M loc and K ∈ L ( Y ) , PropositionIII.6.42 of [39] implies that Z ( H · S ) = K · Y is a σ -martingale. To prove the second assertion of thelemma, suppose that H ∈ A , i.e., there exists a positive constant a such that ( H · S ) t ≥ − a P -a.s.for all t ∈ [0 , T ] . The process Z ( a + H · S ) is a σ -martingale, being the sum of a local martingale anda σ -martingale. Furthermore, Proposition 3.1 and Corollary 3.1 of [46] imply that Z ( a + H · S ) is asupermartingale, being a non-negative σ -martingale, and, hence, also a local martingale (comparealso with [3], Corollary 3.5). In turn, this implies that Z ( H · S ) ∈ M loc , being the difference of twolocal martingales. Proof of Lemma 5.7.
Let Z = ( Z t ) ≤ t ≤ T ∈ D . By Definition 4.4 and Remark 4.5, the process Z ∈ M loc satisfies P (cid:0) Z t > and Z t − > for all t ∈ [0 , T ] (cid:1) = 1 . In view of Theorem II.8.3 of [39], the stochasticlogarithm L := Z − · Z is well-defined as a local martingale with L = 0 and satisfies Z = E ( L ) .Since M ∈ M c loc , the process L admits a Galtchouk-Kunita-Watanabe decomposition with respectto M , see [2]. So, we can write L = ψ · M + N for some R d -valued predictable process ψ =( ψ t ) ≤ t ≤ T ∈ L loc ( M ) and for some N = ( N t ) ≤ t ≤ T ∈ M loc with N = 0 and h N, M i = 0 . Then, forall i = 1 , . . . , d : ZS i = Z − · S i + S i · Z + h Z, S i i = Z − · A i + Z − · M i + S i · Z + h Z, M i i = Z − · (cid:16)Z d h M i , M i λ (cid:17) + Z − · M i + S i · Z + Z − · (cid:10) ψ · M + N, M i (cid:11) = Z − · (cid:16)Z d h M i , M i ( λ + ψ ) (cid:17) + Z − · M i + S i − · Z By Theorem IV.29 of [66], we have Z − · M i ∈ M c loc and S i − · Z ∈ M loc . In turn, this implies that Z − · (cid:0)R d h M i , M i ( λ + ψ ) (cid:1) ∈ M c loc , for all i = 1 , . . . , d . Since Z − > P -a.s., Theorem III.15 of [66]allows to conclude that R d h M i , M i ( λ + ψ ) = 0 for all i = 1 , . . . , d , which in turn implies that the tochastic integral ψ · M is indistinguishable from − λ · M , thus yielding the following representation: Z = E ( L ) = E ( ψ · M + N ) = E ( − λ · M + N ) = b Z E ( N ) where the last equality follows by Yor’s formula (see e.g. [66], Theorem II.38) and Proposition 4.7.Since Z > and b Z > P -a.s., we also have E ( N ) > P -a.s., meaning that ∆ N > − P -a.s. References [1]
Ansel, J.P. and Stricker, C. (1992). Lois de martingale, densités et décomposition deFöllmer Schweizer.
Annales de l’Institut Henri Poincaré , 375–392.[2] Ansel, J.P. and Stricker, C. (1993). Décomposition de Kunita-Watanabe.
Séminaire deProbabilités
XXVII , 30–32, Lecture Notes in Mathematics vol. 1557, Springer.[3]
Ansel, J.P. and Stricker, C. (1994). Couverture des actifs contingents et prix maximum.
Annales de l’Institut Henri Poincaré , 303–315.[4] Bayraktar, E., Kardaras, C. and Xing, H. (2012). Strict local martingale deflators andpricing American call-type options.
Financ. Stoch. , 275–291.[5] Becherer, D. (2001). The numeraire portfolio for unbounded semimartingales.
Financ. Stoch. , 327-341.[6] Brannath, W. and Schachermayer, W. (1999). A bipolar theorem for L (Ω , F , P ) . Sémi-naire de Probabilités
XXXIII , 349–354, Lecture Notes in Mathematics vol. 1709, Springer.[7]
Cassese, G. (2005). A note on asset bubbles in continuous-time.
Int. J. Theor. Appl. Financ. , 523–536.[8] Chau, N.H. and Tankov, P. (2014). Market models with optimal arbitrage. Preprint (avail-able at http://arxiv.org/abs/1312.4979 ).[9]
Cheng, S.T. (1991). On the feasibility of arbitrage-based option pricing when stochastic bondprice processes are involved.
J. Econ. Theory , 185–198.[10] Chou, C.S., Meyer, P.A. and Stricker, C. (1980). Sur les intégrales stochastiques deprocessus prévisibles non bornés.
Séminaire de Probabilités
XIV , 128–139, Lecture Notes inMathematics vol. 784, Springer.[11]
Choulli, T., Deng, J. and Ma, J. (2012). The fundamental theorem of utility maximizationand numéraire portfolio. Preprint (available at http://arxiv.org/abs/1211.4598 ).[12]
Choulli, T. and Stricker, C. (1996). Deux applications de la décomposition de Galtchouk-Kunita-Watanabe.
Séminaire de Probabilités
XXX , 12–23, Lecture Notes in Mathematics vol.1626, Springer.[13]
Christensen, M.M. and Larsen, K. (2007). No arbitrage and the growth optimal portfolio.
Stoch. Anal. Appl. , 255–280.[14] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theo-rem of asset pricing.
Math. Ann. , 463–520.[15]
Delbaen, F. and Schachermayer, W. (1995a). Arbitrage possibilities in Bessel processesand their relations to local martingales.
Prob. Theory Rel. Fields , 357–366.[16]
Delbaen, F. and Schachermayer, W. (1995b). The existence of absolutely continuouslocal martingale measures.
Ann. Appl. Prob. , 926–945. Delbaen, F. and Schachermayer, W. (1995c). The no-arbitrage property under a changeof numéraire.
Stochastics and Stoch. Rep. , 213–226.[18] Delbaen, F. and Schachermayer, W. (1998a). A simple counterexample to several prob-lems in the theory of asset pricing.
Math. Financ. , 1–12.[19] Delbaen, F. and Schachermayer, W. (1998b). The fundamental theorem of asset pricingfor unbounded stochastic processes.
Math. Ann. , 215–250.[20]
Delbaen, F. and Schachermayer, W. (2006).
The Mathematics of Arbitrage . Springer,Berlin - Heidelberg - New York.[21]
Delbaen, F. and Shirakawa, H. (2002). No arbitrage condition for positive diffusion priceprocesses.
Asia Pac. Financ. Markets : 159–168.[22] Dzhaparidze, K. and Spreij, P. (1993). On correlation calculus for multivariate martingales.
Stoch. Proc. Appl. , 283–299.[23] Elworthy, K.D., Li, X.M. and Yor, M. (1999). The importance of strictly local mar-tingales; applications to radial Ornstein-Uhlenbeck processes.
Prob. Theory Rel. Fields ,325–355.[24]
Fernholz, R. (2002).
Stochastic Portfolio Theory . Springer, New York.[25]
Fernholz, R. and Karatzas, I. (2009). Stochastic portfolio theory: an overview. In
Math-ematical Modeling and Numerical Methods in Finance , eds. A. Bensoussan and Q. Zhang.Handbook of Numerical Analysis XV , 89–167, North-Holland, Oxford.[26] Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incompleteinformation. In
Applied Stochastic Analysis , eds. M.H.A. Davis and R.J. Elliott. Stochasticmonographs , 389–414, Gordon and Breach, London - New York.[27] Fontana, C. (2014). No-arbitrage conditions and absolutely continuous changes of measure. In
Arbitrage, Credit and Informational Risks , eds. C. Hillairet, M. Jeanblanc and Y. Jiao. PekingUniversity Series in Mathematics , 3–18. World Scientific, Singapore.[28] Fontana, C., Jeanblanc, M. and Song, S. (2014). On arbitrages arising with honesttimes. To appear in:
Financ. Stoch. , doi: 10.1007/s00780-014-0231-1.[29]
Fontana, C. and Runggaldier, W.J. (2013). Diffusion-based models for financial marketswithout martingale measures. In
Risk Measures and Attitudes , eds. F. Biagini, A. Richter andH. Schlesinger, 45–81. EAA Series, Springer, London.[30]
Gourieroux, C., Laurent, J.P. and Pham, H. (1998). Mean-variance hedging andnuméraire.
Math. Financ. , 179–200.[31] Harrison, J.M. and Kreps, D.M. (1979). Martingales and arbitrage in multiperiod securi-ties markets.
J. Econ. Theory , 381–408.[32] Harrison, J.M. and Pliska, S.R. (1981). Martingales and stochastic integrals in the theoryof continuous trading.
Stoch. Proc. Appl. , 215–260.[33] Herdegen, M. (2012). No-arbitrage in a numéraire independent modelling framework. NCCRFINRISK working paper no. 775.[34]
Hulley, H. (2009).
Strict Local Martingales in Continuous Financial Market Models . PhDthesis, University of Technology Sydney.[35]
Hulley, H. and Schweizer, M. (2010). M - on minimal market models and minimalmartingale measures. In Contemporary Quantitative Finance: Essays in Honour of EckhardPlaten , eds. C. Chiarella and A. Novikov, 35–51. Springer, Berlin - Heidelberg. Imkeller, P. (2002). Random times at which insiders can have free lunches.
Stoch. Stoch.Rep. , 465–487.[37] Jacod, J. (1979).
Calcul Stochastique et Problèmes de Martingales . Lecture Notes in Mathe-matics vol. 714, Springer, Berlin - Heidelberg - New York.[38]
Jacod, J. (1980). Intégrales stochastiques par rapport à une semi-martingale vectorielle etchangements de filtration.
Séminaire de Probabilités
XIV , 161–172, Lecture Notes in Mathe-matics vol. 784, Springer.[39]
Jacod, J. and Shiryaev, A.N. (2003).
Limit Theorems for Stochastic Processes , 2nd edn.Springer, Berlin - Heidelberg - New York.[40]
Jarrow, R. and Protter, P. (2005). Large traders, hidden arbitrage, and complete markets.
J. Bank. Financ. , 2803–2820.[41] Jarrow, R. and Protter, P. (2013). Positive alphas, abnormal performance, and illusoryarbitrage.
Math. Financ. , 39–56.[42] Jeulin, T. and Yor, M. (1979). Inégalité de Hardy, semimartingales, et faux-amis.
Séminairede Probabilités
XIII , 332–359, Lecture Notes in Mathematics vol. 721, Springer.[43]
Kabanov, Y. (1997). On the FTAP of Kreps-Delbaen-Schachermayer. In
Statistics and Con-trol of Stochastic Processes: The Liptser Festschrift , eds. Y. Kabanov, B.L. Rozovskii, A.N.Shiryaev, 191–203. World Scientific, Singapore.[44]
Kabanov, Y. and Kramkov, D. (1994). Large financial markets: asymptotic arbitrage andcontiguity.
Theor. Probab. Appl. , 182–187.[45] Kabanov, Y. and Stricker, C. (2005). Remarks on the true no-arbitrage property.
Sémi-naire de Probabilités
XXXVIII , 186–194, Lecture Notes in Mathematics vol. 1857, Springer.[46]
Kallsen, J. (2004). σ -localization and σ -martingales. Theory Prob. Appl. , 152–163.[47] Karatzas, I. and Kardaras, K. (2007). The numeraire portfolio in semimartingale financialmodels.
Financ. Stoch. , 447–493.[48] Karatzas, I. and Shreve, S.E. (1998).
Methods of Mathematical Finance . Springer, NewYork.[49]
Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of assetpricing. In
Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen , eds. C.Chiarella and A. Novikov, 19–34. Springer, Berlin - Heidelberg.[50]
Kardaras, C. (2012). Market viability via absence of arbitrage of the first kind.
Financ.Stoch. , 651–667.[51] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility func-tions and optimal investment in incomplete markets.
Ann. Appl. Prob. , 904–950.[52] Levental, S. and Skorohod, A.S. (1995). A necessary and sufficient condition for absenceof arbitrage with tame portfolios.
Ann. Appl. Prob. , 906–925.[53] Loewenstein, M. and Willard, G.A. (2000). Local martingales, arbitrage, and viability.
Econ. Theory , 135–161.[54] Londono, J.A. (2004). State tameness: a new approach for credit constraints.
Electron. Com-mun. Prob. , 1–13.[55] Lyasoff, A. (2012). The two fundamental theorems of asset pricing for a class of continuous-time financial markets. To appear in:
Math. Financ. , doi: 10.1111/j.1467-9965.2012.00530.x. Mancin, J. and Runggaldier, W.J. (2014). On the existence of martingale measures injump diffusion market models. In
Arbitrage, Credit and Informational Risks , eds. C. Hillairet,M. Jeanblanc and Y. Jiao. Peking University Series in Mathematics , 29–51. World Scientific,Singapore.[57] Mémin, J. (1980). Espaces des semimartingales et changement de probabilitè.
Z. Wahrschein-lichkeit. , 9-39.[58] Mijatović, A. and Urusov, M. (2012a). On the martingale property of certain local mar-tingales.
Prob. Theory Rel. Fields. , 1–30.[59]
Mijatović, A. and Urusov, M. (2012b). Deterministic criteria for the absence of arbitragein one-dimensional diffusion models.
Financ. Stoch. , 225–247.[60] Nilsen, W. and Sayit, H. (2011). No arbitrage in markets with bounces and sinks.
Int. Rev.Appl. Finan. Issues Econ. , 696–699.[61] Osterrieder, J.R. and Rheinländer, T. (2006). Arbitrage opportunities in diverse mar-kets via non-equivalent measure changes.
Ann. Financ. , 287–301.[62] Platen, E. (2002). Arbitrage in continuous complete markets.
Adv. Appl. Prob. , 540–558.[63] Platen, E. (2006). A benchmark approach to finance.
Math. Financ. , 131–151.[64] Platen, E. (2011). A benchmark approach to investing and pricing. In
The Kelly CapitalGrowth Investment Criterion , eds. L.C. MacLean, E.O. Thorp and W.T. Ziemba, 409–427.World Scientific, Singapore.[65]
Platen, E. and Heath, D. (2006).
A Benchmark Approach to Quantitative Finance .Springer, Berlin - Heidelberg.[66]
Protter, P. (2005).
Stochastic Integration and Differential Equations , 2nd edn. (v. 2.1).Springer, Berlin - Heidelberg - New York.[67]
Protter, P. and Shimbo, K. (2008). No arbitrage and general semimartingales. In
MarkovProcesses and Related Topics: A Festschrift for Thomas G. Kurtz , eds. S.N. Ethier, J. Fengand R.H. Stockbridge, 267–283. Institute of mathematical statistics, Beachwood (OH).[68]
Revuz, D. and Yor, M. (1999).
Continuous Martingales and Brownian Motion , 3rd edn.Springer, Berlin - Heidelberg.[69]
Rheinländer, T. and Schweizer, M. (1997). On L -projections on a space of stochasticintegrals. Ann. Prob. , 1810–1831.[70] Rheinländer, T. and Sexton, J. (2011).
Hedging Derivatives , World Scientific, Singapore.[71]
Rossello, D. (2012). Arbitrage in skew Brownian motion models.
Insur. Math. Econ. ,50–56.[72] Ruf, J. (2013). Hedging under arbitrage.
Math. Financ. , 297–317.[73] Ruf, J. and Runggaldier, W.J. (2014). A systematic approach to constructing market mod-els with arbitrage. In
Arbitrage, Credit and Informational Risks , eds. C. Hillairet, M. Jeanblancand Y. Jiao. Peking University Series in Mathematics , 19–28. World Scientific, Singapore.[74] Schachermayer, W. (2010). The fundamental theorem of asset pricing. In
Encyclopedia ofQuantitative Finance , ed. R. Cont, 792–801. Wiley, Chichester.[75]
Schweizer, M. (1992). Martingale densities for general asset prices.
J. Math. Econ. , 363–378.[76] Schweizer, M. (1995). On the minimal martingale measure and the Föllmer-Schweizer de-composition.
Stoch. Anal. Appl. , 573–599. Shiryaev, A.N. and Cherny, A. (2002). Vector stochastic integrals and the fundamentaltheorems of asset pricing. In
Stochastic Financial Mathematics , collected papers, Tr. Mat. Inst.Steklova , 12–56. Nauka, Moscow.[78]
Song, S. (2013). An alternative proof of a result of Takaoka. Preprint (available at http://arxiv.org/abs/1306.1062 ).[79]
Strasser, E. (2005). Characterization of arbitrage-free markets.
Ann. Appl. Prob. , 116–124.[80] Stricker, C. and Yan, J.-A. (1998). Some remarks on the optional decomposition theorem.
Séminaire de Probabilités
XXXII , 56–66, Lecture Notes in Mathematics vol. 1686, Springer.[81]
Takaoka, K. and Schweizer, M. (2014). A note on the condition of no unbounded profitwith bounded risk.
Financ. Stoch. , 393–405.[82] Yoeurp, C. and Yor, M. (1977). Espace orthogonal à une semimartingale et applications.Unpublished preprint.(1977). Espace orthogonal à une semimartingale et applications.Unpublished preprint.