No-Arbitrage Symmetries
aa r X i v : . [ q -f i n . M F ] A ug NO-ARBITRAGE SYMMETRIES
I. L. DEGANO, S.E.FERRANDO, AND A.L. GONZALEZA
BSTRACT . The no-arbitrage property is widely accepted to be a centerpiece of mod-ern financial mathematics and could be considered to be a financial law applicable toa large class of (idealized) markets. The paper addresses the following basic question:can one characterize the class of transformations that leave the law of no-arbitrage in-variant? We provide a geometric formalization of this question in a non probabilisticsetting of discrete time, the so-called trajectorial models. The paper then character-izes, in a local sense, the no-arbitrage symmetries and illustrates their meaning in adetailed example. Our context makes the result available to the stochastic setting as aspecial case.
Keywords: No arbitrage symmetry, convexity preserving maps, non-probabilistic mar-kets.2000 Math Subject Classification: 91B24; 91G10; 26B25; 49J35.1. I
NTRODUCTION
The principle of no-arbitrage plays a fundamental role in modern financial mathe-matics, see [F¨ollmer & Schied (2011)] and references therein (we mostly restrict ourcomments and developments to a discrete time setting). In plain language, the as-sumption of no-arbitrage means that risky asset models should rule out apriori thepossibility of making a profit without taking in any risk. This hypothesis implies apricing methodology based on martingale stochastic processes, this is the risk-neutralvaluation ([Bingham & Kiesel (2004)]), and as such plays the role of a financial law.Its empirical validity has been studied ([Kamara & Miller (1995)]) and even if arbi-trage opportunities may be available they are believed to be rare, short lived and hardto profit from. One could compare the notion of no-arbitrage to a physical law thatapplies in idealized conditions such as the principle of inertia and think of it as a fun-damental financial law applicable to a large class of (idealized) markets. With thispoint of view in mind, and as a preliminary step, we pose the question: can we charac-terize the class of transformations that leave the law of no-arbitrage invariant ? This ismuch akin to Galilean/Lorentz transformations leaving the class of inertial frames in-variant. Therefore, we look for no-arbitrage preserving transformations (referred alsoas no-arbitrage symmetries) mapping a set of financial events into another set of finan-cial events with the property that the no-arbitrage property holds for both classes of
Date : August 12, 2020. events. One is also interested in providing a financial interpretation to such set of sym-metries (much like Galilean/Lorentz transformations having a physical interpretation)and exploring some financial implications.The financial events mentioned in the previous paragraph have to be linked to finan-cial transactions as it is the latter that fall under the scope of the no-arbitrage principle.In most situations, each such transaction involves two goods X and Y and a price X Y ( t ) so that X = X Y ( t ) Y . X Y ( t ) is the integer number of units of asset Y required to pur-chase one unit of asset X . In terms of dimensional units [ X Y ( t )] = [ X ] / [ Y ] and thereference asset Y is called the (chosen) numeraire ([Vecer (2011)]). This discussionsuggests that an analysis of the no-arbitrage principle could be done in terms of pricesand numeraires (numeraire free approaches are also possible) and that is the way weproceed in the paper.The setting where we precisely pose and answer the above raised question is a setof sequences of multidimensional prices that evolve in discrete time. That type of setis called a trajectorial model (for a set of risky assets); in our investigation there is noneed to assume any probability structure on such a set. In this way we can work on amore general setting than the (discrete time) stochastic framework and the question westudy becomes a natural one unencumbered by unnecessary additional structure. Wealso briefly indicate how our main results, characterizing no-arbitrage transformations,apply to the stochastic setting as a special case.The mentioned trajectorial framework has been developed in [Ferrando et al (2019b)]and [Degano et al (2018)] (see also [Ferrando & Gonz´alez (2019)]) for the 1-dimensionalcase with a 0 interest rate bank account as numeraire and is here extended to the d -dimensional case and a general numeraire (but we restrict ourselves to finite time,as opposed to unbounded or infinite discrete time as in [Ferrando et al (2019b)] and[Ferrando & Gonz´alez (2019)], respectively). It then follows that the global notion ofno-arbitrage, i.e. involving several time steps, can be reduced to the one step notionof no-arbitrage. This is a classical reduction in discrete and finite time and allows toconcentrate our efforts in the local notion (i.e. involving one step into the future) ofno-arbitrage.We can now be more precise about our search for no-arbitrage preserving transfor-mations. Definition [Disperse sets] Consider a set E ⊆ R d ; E is called disperse if for each h ∈ R d : [ h · Y = ∀ Y ∈ E ] or [ inf Y ∈ E h · Y < Y ∈ E h · Y > ] , where h · Y represents the euclidean inner product.We prove in the paper (by means of Proposition 6, Proposition 3 and Definition 8)that the notion of a set being disperse is equivalent to the risky assets (one step timeevolution) obeying the no-arbitrage principle. Therefore, our original question on no-arbitrage preserving transformations becomes: characterize the set of transformationsthat leave the disperse property invariant. We are then in the context of the modernview of a geometry where we study the set of transformations that leave certain prop-erties of a space of points invariant. O-ARBITRAGE SYMMETRIES 3
We work on a self contained framework for financial markets centered on a setof (mutidimensional) trajectories modeling a collection of assets with the possibil-ity that any of them could play the role of a numeraire asset. No probability mea-sures, filtrations, cardinality or topological assumptions are required of the trajectoryset. The approach singles out local trajectory properties that can be used to consis-tently build an associated option price theory. The paper does not pursue this latterpossibility as we focus on symmetry transformations (option pricing developmentsare in [Ferrando et al (2019b)] for the 1-dimensional case). Such trajectory propertieshave already made their appearance in the stochastic literature ([Dalang et al. (1990)],[Bender et al. (2010)], [Jarrow et al (2009)]). To relate to the well established sto-chastic approach ([F¨ollmer & Schied (2011)]) the reader could think that our paperconcentrates on financial developments that only depend on the support of a givenstochastic process independently of any possible probability distribution. Some con-nections with the stochastic setting are developed in [Ferrando et al (2019b)], the refer-ence [Ferrando & Gonz´alez (2019)] provides a first mathematical step to extend somemartingale notions from the standard setting to a trajectorial setting.The subject of the paper is the study of fundamental symmetry transformations asso-ciated to the no-arbitrage principle in a non-probabilistic setting. Results are obtainedwith minimal assumptions and, in this way, providing a wider financial context fortheir availability.We describe the contents of the paper. Section 2 introduces the setting which iscentered on a trajectory space. Section 3 studies the notions of no-arbitrage and 0-neutrality (a weakening of no-arbitrage) in trajectory based markets. Section 3.1 intro-duces local conditions (i.e. properties that are conditioned on a given state of affairsand involve one step into the future) which are necessary and sufficient to establishno-arbitrage and 0-neutrality. These local conditions play the analogue role of themartingale condition in stochastic markets. Section 4 develops purely geometric re-sults in R d , independent of any financial setting, that form the backbone to derivethe set of no-arbitrage and 0-neutral symmetries. Section 5 characterizes two classesof transformations, one preserving the local no-arbitrage property and the other classpreserving the 0-neutral property. In particular, we prove that a change of numerairebelongs to both such classes of transformations. The uncovered transformations canthen be considered to be symmetries satisfied by price relationships and we providean illustrative example. Appendix A provides proofs for results in the main body ofthe paper. Appendix B develops some results on convex analysis that we rely upon.Finally, we use the words arbitrage-free and no-arbitrage interchangeably.2. G ENERAL T RAJECTORIAL S ETTING
We introduce the mathematical setting of a dynamic financial market with a finitenumber of assets whose initial prices are known. Uncertainty of future prices is givenby a set of multidimensional sequences that we call trajectories. The trading strate-gies are given by portfolios that will be successively re-adjusted, taking into accountthe information available at each stage. The present paper is essentially self-containedWe extend work presented in [Degano et al (2018)] and [Ferrando et al (2019b)]. The
I. L. DEGANO, S.E.FERRANDO, AND A.L. GONZALEZ latter reference presents a non-probabilistic one-dimensional, discrete time, setting toprice European options. The reference [Degano et al (2018)] provides examples anda computational algorithm to evaluate price bounds for European options. A detaileddiscussion and justification of why a trajectorial modeling approach is worth study-ing is presented in [Ferrando et al (2019b)] as well as in [Degano et al (2018)]. Thepresent paper extends the setting from those two papers to the multidimensional case.There is empirical evidence suggesting that liquid markets do not allow for arbi-trage opportunities. Therefore, and from a modeling point of view, the no-arbitrageprinciple assumes that market models should not contain arbitrage strategies. Theno-arbitrage assumption allows to develop a theory constraining relative prices. Weremark in passing that under the weaker condition of 0-neutrality (see Definition 6 aswell as [Ferrando et al (2019b)]), it is possible to obtain well defined price bounds forEuropean options.More precisely, we consider a market with d + [ , T ] . The model will be discrete in the sense that the trading instances areindexed by integer numbers. Given s = ( s , s , s , . . . , s d ) ∈ R d + , as initial prices ofassets S k , we will denote by a trajectory S , a sequence taking values in R d + such that S i = ( S i , S i , S i , . . . , S di ) with S = s . A portfolio will be a sequence of functions defined on the trajectory sets which wewill denote by Φ = ( H , H ) = { ( H i , H i , . . . , H di ) } i ≥ . Each coordinate H ji , 0 ≤ j ≤ d represents the portfolio holdings at stage i , for the j -thasset with [ H ji ] = S j (a unit of asset S j ). The asset values and the invested amountscan take values in general subsets of the real numbers.The portfolio re-balancing stages may be triggered by arbitrary events of the marketwithout the need to be directly associated with time. To incorporate this greater degreeof generality we will add a new source of uncertainty to the trajectories’ coordinates(these additional coordinates are relevant when constructing specific models). We willdenote them by W = { W i } i ≥ , the W i can be vector valued and take values in arbitrarysets. In financial terms, this new variable can represent any observable value of interest,such as volume of transactions, time, quadratic variation of trajectories, etc., as in[Ferrando et al (2019a)].In case one intends to price financial derivatives in the proposed setting, we add afinite time horizon T . We will use a positive integer m , to indicate the stage at whichthe trajectory reaches the time T . This new variable plays a key role in calculating thefair price interval for options, although it does not intervene in the market properties. Definition 1 (Trajectory set) . Consider Σ = { Σ i } a given family of subsets of R d + , Ω = { Ω i } is a family of sets and Θ ⊆ N . For given s ∈ R d + and w ∈ Ω , a trajectorybased set S is a subset of S ∞ ( s , w ) ≡ { S = { S i ≡ ( S i , W i , m ) } i ≥ : S i ∈ Σ i , W i ∈ Ω i , m ∈ Θ } , such that ( S , W ) = ( s , w ) . The elements of S will be called trajectories . O-ARBITRAGE SYMMETRIES 5
It is important to note that if ˜ S = { ( ˜ S i , ˜ W i , ˜ m ) } and ˆ S = { ( ˆ S i , ˆ W i , ˆ m ) } are two trajec-tories, ˜ S i could unfold at a different time than ˆ S i . That is, the index i will be associatedwith portfolio re-balances stages but they will not be necessarily associated to (uni-form) time. It is only assumed that the stage i + i . We define M : S → N as the projection on the third coordinate of S , that is: M ( S ) = m . The results and properties that appear in this section only involve the first coordinate S i nonetheless, we will continue using the notation that includes the coordinates W i forconsistency.To build an adequate market model, we are going to require that any portfolio benon-anticipative. The non-anticipativity of the portfolios expresses the fact that invest-ments must be made at the beginning of each period, so that they can not anticipatespecific future price changes. Definition 2 (Portfolio) . Let S be a trajectory set, a portfolio Φ is a sequence of(pairs of) functions Φ ≡ { ( H i , H i ) } i ≥ with H i : S → R and H i : S → R d such thatfor all S , S ′ ∈ S , with S ′ i = S i for all ≤ i ≤ k, where k < min { M ( S ) , M ( S ′ ) } , then Φ k ( S ) = Φ k ( S ′ ) . For a portfolio Φ , H ji ( S ) represents the number of units held for the j -th asset duringthe period between i and i +
1. Therefore, H ji ( S ) S ji is the value invested in the j -thasset at stage i , while H ji ( S ) S ji + is the value just before rebalancing at the end of theperiod. So, the total value of the portfolio Φ at the beginning of the period i is H i ( S ) S i + H i ( S ) · S i ≡ H i ( S ) S i + d ∑ j = H ji ( S ) S ji , and at the end of the period, the value of Φ will change to H i ( S ) S i + + H i ( S ) · S i + = H i ( S ) S i + + d ∑ j = H ji ( S ) S ji + . In the next re-balancing, the investor will invest Φ i + ; in general, H i + ( S ) S i + + H i + ( S ) · S i + may be different from H i ( S ) S i + + H i ( S ) · S i + . In this latter case, it fol-lows that some units of the assets were added or removed, without replacement, fromthe portfolio. However, this situation is precluded for many applications. For example,if the goal is to look for a “fair” price for a certain financial contract, this value shouldbe the minimum necessary to cover the obligations generated by the contract, that is,any injection or withdrawal of money will affect this property. This reasoning justifiesthe use of the following concept. Definition 3 (Self-financing portfolio) . A portfolio Φ is called self-financing if for all S ∈ S and i ≥ , (1) H i ( S ) S i + + H i ( S ) · S i + = H i + ( S ) S i + + H i + ( S ) · S i + . The self-financing property means that the portfolio is re-balanced in such a waythat its value is preserved. From this property it is clear that the accumulated gains
I. L. DEGANO, S.E.FERRANDO, AND A.L. GONZALEZ and losses resulting from price fluctuations are the only sources of variation of theportfolio:(2) H k ( S ) S k + H k ( S ) · S k = H S + H · S + k − ∑ i = (cid:0) H i ( S ) ∆ i S + H i ( S ) · ∆ i S (cid:1) , for k ≥
0, where ∆ i S = S i + − S i and ∆ i S = S i + − S i . The value H S + H · S represents the initial investment corresponding to the portfolio coordinate Φ .We will mention below some examples of strategies that will be used later. Example 1. (1)
The null portfolio Φ = , ( S ) = { ( , ) } i ≥ for all S ∈ S where is the null vector of R d . (2) Set h ∈ R and h ∈ R d , we will define by constant portfolio Φ = h by h ( S ) = { ( h , h ) } i ≥ for all S ∈ S . (3) Set Φ = { ( H i , H i ) } i ≥ a self-financing portfolio. We will denote by − Φ to thesequence of functions { ( − H i , − H i ) } i ≥ . It is easy to see that − Φ is a self-financing portfolio. (4) Set ˆ Φ = { ( ˆ H i , ˆ H i ) } i ≥ and ˜ Φ = { ( ˜ H i , ˜ H i ) } i ≥ two portfolios. We define Φ ≡ ˆ Φ + ˜ Φ to be the sequence Φ = { ˆ H i + ˜ H i , ˆ H i + ˜ H i } i ≥ . Numeraire.
To be definite, we will consider that the real numbers S ki express theprice of asset S k in a common currency, a unit of which we denote generically by $.That is, in terms of dimensions [ S ki ] = $ / S k where S k is one unit of asset S k (it is wellknown that an algebra of dimensions is available through dimensional analysis as in[Whitney (1968)]). Notice that [ H ki ] = S k . On the other hand, for financial reasons,it is important to work with an arbitrary reference asset; this is achieved by taking areference asset as num´eraire . For example, in some cases it is useful to select the valueof a bank account as num´eraire.Toward this end, we will assume from here onward that S i > i ≥
0. Thishypothesis will allow us to use S as num´eraire. For each S ∈ S , we will build asequence of relative prices X ( S ) = { ( X ( S i ) , W i , m ) } i ≥ where X : D ⊆ R d + → R d isa perspective function defined by(3) X ( s ) ≡ (cid:18) s s , . . . , s d s (cid:19) , D ≡ { s = ( s , . . . , s d ) ∈ R d + : s > } . The numerical value of X j ( S i ) (i.e. stripped from its units), is the number of units ofthe asset S , now the num´eraire, which are required to acquire one unit of the S j asset. Remark 1.
Notice the above definition of X singles out s but of course any othercoordinate could be used (relying on the -component simplifies the notation). In fact,and for more generality, one could replace s by a linear map B ( s ) > on s k > . We O-ARBITRAGE SYMMETRIES 7 do not pursue here this possibility but our results will apply to such numeraire by justmoving to a new trajectory market with s = B ( s ) . Given S ∈ S and k ≥
0, we will denote by V Φ k ( S ) the relative value of the portfolio Φ ∈ H given by V Φ k ( S ) ≡ H k ( S ) + H k ( S ) · X ( S k ) . Clearly V Φ k ( S ) = Φ k ( S ) · S k S k , then V Φ k ( S ) can be interpreted as the value of the portfo-lio at the beginning of the stage k expressed in units of the num´eraire. In addition, G Φ k ( S ) will denote the profits generated up to the stage k associated with Φ ∈ H for atrajectory S ∈ S , that is(4) G Φ k ( S ) ≡ k − ∑ i = H i ( S ) · ∆ i X ( S ) for k ≥ ∆ i X ( S ) = X ( S i + ) − X ( S i ) . G Φ k ( S ) reflects, in terms of the num´eraire, the net gains accumulated by the portfolio Φ at the beginning of the k -th stage.A self-financing portfolio for a path S ∈ S will also be self-financing for the X ( S ) sequence. Proposition 1.
Let S be a space of trajectories, and let Φ be a portfolio on S . Thenthe following statements are equivalent: (1) Φ is self-financing. (2) H i − ( S ) + H i − ( S ) · X ( S i ) = H i ( S ) + H i ( S ) · X ( S i ) for all S ∈ S and i ≥ . (3) V Φ k ( S ) = V Φ + G Φ k ( S ) = H + H · X ( S ) + k − ∑ i = H i ( S ) · ∆ i X ( S ) for all k ≥ .Proof. Note that Proposition 5.7 of [F¨ollmer & Schied (2011)] is valid even in caseswhere we do not have a market indexed by pre-set time stages. Therefore the sameidea used in that result applies to our setting. (cid:3)
Remark 2.
From the previous Proposition, we know that the H component of a self-financed portfolio Φ satisfies (5) H k ( S ) − H k − ( S ) = − ( H k ( S ) − H k − ( S )) · X ( S k ) . Given that (6) H = V Φ − H · X ( S ) , the sequence H is completely determined by the initial investment V Φ and H by meansof the previous equations. Remark 3.
For a given set of portfolios H , in virtue of Remark 2, and display (4)(which depends on Φ = ( H , H ) , just through H) we will set the definition (7) H S ≡ { H : ( H , H ) ∈ H } for later use. I. L. DEGANO, S.E.FERRANDO, AND A.L. GONZALEZ
Definition 4 (Trajectory market) . Given s ∈ R d + , w ∈ Ω , a trajectory based set S ⊆ S ∞ ( s , w ) and a portfolio set H , we say that M = S × H is a trajectorybased market if it satisfies the following properties: (1) For each S ∈ S , the coordinate S i > for all i ≥ . (2) All Φ ∈ H are self-financing and Φ = belongs to H . (3) For each ( S , Φ ) ∈ M there exists N Φ ( S ) ∈ N such that Φ k ( S ) = Φ N Φ ( S ) = for all k ≥ N Φ ( S ) .We will say that the market is semi-bounded if for each Φ ∈ H there is n Φ ∈ N suchthat N Φ ( S ) ≤ n Φ for all S ∈ S and it is n -bounded , for n ∈ N , if N Φ ( S ) ≤ M ( S ) ≤ nfor each pair ( S , Φ ) ∈ M . A portfolio set H obeying items ( ) and ( ) above will becalled admissible. The third property of the previous definition states that the adjustments of the port-folio Φ for a trajectory S will end at, or before, the stage N Φ ( S ) −
1, which meansthat the portfolio was liquidated on, or before, the period N Φ ( S ) . In this case, thecorresponding portfolio will be called liquidated .The above setting incorporates, as a special case, a discrete time stochastic model.Given a process Y = { Y i = ( Y i , . . . , Y di ) } i ≥ on a probability space ( Ω , P ) with filtration F = { F i } i ≥ and F trivial, Y ki : Ω → R , Y ki ∈ F i . We can then define S = { S = { ( S i ≡ Y i ( ω )) } i ≥ : for some ω ∈ Ω } . One can also define a set of trajectories S by means of a sequence of admissible stopping times τ = { τ i } i ≥ : S ∈ S then S = { ( S i ≡ Y τ i ( ω ) ( ω )) } i ≥ for some ω ∈ Ω . Another way to proceed is to use a givencollection of such sequences of stopping times, for the details we refer to Section 6 in[Ferrando et al (2019b)].3. A RBITRAGE AND
EUTRALITY
A model for common market situations should not allow for investors that are ableto generate a profit in a transaction without any risk/possibility of losing money. Suchan investment opportunity is called an arbitrage opportunity.
Definition 5 (Arbitrage opportunity) . Given a trajectory based market M = S × H , Φ ∈ H is an arbitrage opportunity if: • ∀ S ∈ S , V Φ N Φ ( S ) ≥ V Φ . • ∃ S ∗ ∈ S such that V Φ N Φ ( S ∗ ) > V Φ .We say that M is arbitrage-free if H does not contain arbitrage opportunities. The particular case S i =
1, for all i , gives X ( S i ) = ( S i , . . . , S di ) ; i.e. the originalcurrency is the asset S and is being used as numeraire and so [ S i ] = $ / $. Currency, ifincluded as a traded asset and in the presence of a riskless bank account with non-zerointerest rates, will lead to an arbitrage as per Definition 5. That is, currency, underthe mentioned conditions will be banned as a traded asset whenever we assume a noarbitrage market (as well as a 0-neutral market). Notice the relevant discussion in[Vecer (2011)] about arbitrage and non-arbitrage assets, currency being an arbitrageasset (as contrasted to an interest bearing money market account). O-ARBITRAGE SYMMETRIES 9
Our use of an arbitrary value for V Φ in the definition of an arbitrage opportunityis nonstandard, textbook definitions require V Φ ≤ Φ with V ˜ Φ = et al (2019b)] and [Degano et al (2018)]. Definition 6 (0-neutral market) . Let M = S × H be a trajectory based market. Wesay that M is if sup Φ ∈ H (cid:26) inf S ∈ S G Φ N Φ ( S ) (cid:27) = sup Φ ∈ H ( inf S ∈ S " N Φ ( S ) − ∑ i = H i ( S ) · ∆ i X ( S ) = . In [Ferrando et al (2019b)] it is shown that this property is also sufficient to obtain apricing interval for financial derivatives. The next Proposition shows that 0-neutralityis weaker than arbitrage-free.
Proposition 2.
Let M = S × H be an arbitrage-free trajectory based market. Then, M is -neutral.Proof. We are going to prove it by contraposition. Note that if M = S × H is atrajectory based market, 0 ∈ H , then it is always true thatsup Φ ∈ H { inf S ∈ S G Φ N Φ ( S ) } ≥ . That is, if M is not 0-neutral, there exist a portfolio Φ such thatinf S ∈ S G Φ N Φ ( S ) > . Thus G Φ N Φ ( S ) > S ∈ S . Then V Φ N Φ ( S ) = V Φ + G Φ N Φ ( S ) > V Φ for all S ∈ S . Then Φ is an arbitrage porfolio. (cid:3) It is clear how to generate simple examples of 0-neutral markets which contain arbi-trage (see [Ferrando et al (2019b)] and [Degano et al (2018)]). Following [Pliska (1997),Section 1.2] it is possible to define another properties of the market, closely related to0-neutral and arbitarge-free, namely dominant portfolios and law of one price . Under the appropriate hypotheses, the following chain of implications for a trajectory basedmarket holds:Arbitrage-free ⇒ No dominant portfolios ⇒ ⇒ Law of one price.At this point, we have introduced enough properties of multidimensional trajectorymarkets in order to address our goal of characterizing no-arbitrage symmetry transfor-mations.3.1.
Relationships Between Local and Global Properties.
From the definitions, itis not clear how to construct arbitrage-free or 0-neutral markets. For the case of semi-bounded markets, one can obtain necessary and sufficient conditions, only involvinglocal properties of the trajectory set, implying trajectorial markets that are arbitrage-free (or 0-neutral). Such characterizations play an analogous role to the equivalenceof no arbitrage stochastic markets and the possibility to equivalently modify the sto-chastic process into a martingale process. In fact, in the arbitrage-free case the localtrajectorial conditions correspond to a probability free notion of a martingale sequence(see [Ferrando & Gonz´alez (2019)]). We will use these characterizations to pose andanswer our opening question on no-arbitrage preserving transformations.At the k -th stage, the information about the future available to investors is that S isan element of the set S ( S , k ) ≡ (cid:8) S ′ ∈ S : S ′ i = S i , ≤ i ≤ k and M ( S ′ ) > k (cid:9) ⊆ S . We will call the pair ( S , k ) a node and will refer to the set S ( S , k ) as trajectory setconditioned at the node ( S , k ) . The future information contained in ˜ S ∈ S ( S , k ) dependson the past only through S , . . . , S k . The multiple number of trajectories emanatingfrom a node reflects the non-deterministic nature of the assets’ time evolution. Astrajectories unfold more coordinates become available and so the investor increaseshis knowledge about possible future scenarios. This is expressed mathematically as S ( S , k ′ ) ⊆ S ( S , k ) for k ′ > k . The following notation will also be used(8) ∆ X ( S ( S , k ) ) ≡ { ∆ k X ( S ′ ) : S ′ ∈ S ( S , k ) } ⊆ R d , where ∆ k X ( S ′ ) = X ( S ′ k + ) − X ( S ′ k ) has been introduced before.We will refer as local to any property relative to a node ( S , k ) and only involvingelements of ∆ X ( S ( S , k ) ) .The definitions below, are the local counterpart of those of arbitrage-free and 0-neutral for the whole market. We are then going to derive the global properties fromthe local ones. Definition 7 (Local notions) . Given a trajectory based market M = S × H , let S ∈ S and k ≥ . (1) ( S , k ) is called an arbitrage-free node with respect to H if [ H k ( S ) · ∆ k X ( S ′ ) = ∀ S ′ ∈ S ( S , k ) ] or [ inf S ′ ∈ S ( S , k ) H k ( S ) · ∆ k X ( S ′ ) < ] , O-ARBITRAGE SYMMETRIES 11 for all H ∈ H S (the latter as in (7)). (2) ( S , k ) is called a -neutral node with respect to H if, for all H ∈ H S :inf S ′ ∈ S ( S , k ) H k ( S ) · ∆ k X ( S ′ ) ≤ . M is called locally arbitrage-free ( -neutral) if each ( S , k ) is an arbitrage-free ( -neutral) node w.r.t. H . A node that is not arbitrage-free w.r.t. H , will be called anarbitrage node w.r.t. H . Notice that an arbitrage-free node w.r.t. H is always 0-neutral w.r.t. H . Clearly,there are natural examples of nodes which are 0-neutral w.r.t. H but no arbitrage-freew.r.t. H (hence these are arbitrage nodes). It is then of interest to indicate that thereare results ([Ferrando et al (2019b)]) that justify option prices obtained for general 0-neutral markets (in particular these markets may contain 0-neutral nodes which arearbitrage nodes w.r.t. H ).Admittedly, attaching the qualifier “w.r.t. H ” to some of the above notions doesnot play a substantial role in the paper. In fact, Proposition 3 below provides sufficientconditions on trajectory nodes that imply that those nodes are arbitrage-free (0-neutral)w.r.t. any (admissible) H .The conclusions in Proposition 3 below are consequences of characterizations givenby Propositions 6 and 7 in Section 4.1. Proposition 3.
Given a trajectory set S , consider a node ( S , k ) . (1) If: (9) 0 ∈ ri (cid:0) co (cid:0) ∆ X ( S ( S , k ) ) (cid:1)(cid:1) . then ( S , k ) is an arbitrage-free node w.r.t. any (admissible) H . (2) If: (10) 0 ∈ cl (cid:0) co (cid:0) ∆ X ( S ( S , k ) ) (cid:1)(cid:1) . then ( S , k ) is a -neutral node w.r.t. any (admissible) H . According with these results we introduce the following notions which will play acrucial role for the remaining of the paper.
Definition 8 ( H -Independent local properties) . A node ( S , k ) is called arbitrage-free if (9) is satisfied; it is called if (10) is satisfied. We call S locally arbitrage-free (locally -neutral), if every node ( S , k ) is arbitrage-free ( -neutral). Remark 4.
The above definitions rely on a numeraire (through the perspective func-tion X ). We will show in Section 5 that once the properties hold for one numeraire,they hold for any numeraire.
Therefore, if S is locally arbitrage-free (locally 0-neutral), then M = S × H islocally arbitrage-free (locally 0-neutral) for any (admissible) H . Remark 5.
Condition (9) appears in the stochastic literature as equivalent to one steparbitrage-free markets [Cutland & Roux (2012), Lemma 3.42] , [Elliot & Kopp (2005),Prop 3.3.4] , [F¨ollmer & Schied (2011), Cor 1.50] , [Jacod & Shirayev (1990) ] . The local notions in Definition 8 allow us to ensure global conditions on a trajec-tory based market. In particular, the results in the rest of this section characterize anarbitrage-free market (0-neutral) by means of arbitrage-free (0-neutral) nodes w.r.t. H . Theorem 1 (No arbitrage: local implies global) . If M = S × H is locally arbitrage-free (as per Definition 7) and semi-bounded, then M is arbitrage-free (as per Defini-tion 5). See proof in Appendix A.In order to establish a converse to Theorem 1, consider ξ ∈ R d , S ∈ S and k ≥ ξ ( S , k ) i : S → R d , for any i ≥
0, by ξ ( S , k ) i ( S ′ ) = (cid:26) ξ if i = k and S ′ ∈ S ( S , k ) . V , we can obtain from the equations (5) and (6) a sequence of functions { ξ i } i ≥ in such a way that the sequence(11) Ξ ( S , k ) = { ( ξ i , ξ ( S , k ) i ) } i ≥ be self-financing. Also, defining N Ξ ( S , k ) ( S ′ ) = k + S ′ ∈ S , it is easy to see that Ξ ( S , k ) is a portfolio. We will call this type of portfolios as restricted portfolios at thenode ( S , k ) . Proposition 4 (No arbitrage: global implies Local) . If M = S × H is arbitrage freeand the restricted portfolios belong to H then, S is locally arbitrage-free (as perDefinition 8). In particular M is locally arbitrage-free. See proof in Appendix A.We now carry out a similar analysis for the notion of 0-neutral. The followingTheorem shows that a trajectory based market will be 0-neutral if it is locally 0-neutral.
Theorem 2 (0-neutral: local implies global) . Let M = S × H be a semi-boundedtrajectory market. Then if M is locally -neutral (as per Definition 7) then, M is -neutral (as per Definition 6). See proof in Appendix A.
Proposition 5 (0-neutral: global implies local) . Let M = S × H be a -neutraltrajectory market such that the restricted portfolios belong to H . Then, any node ( S , k ) is a -neutral node (in particular, ( S , k ) is -neutral with respect to H ). See proof in Appendix A.4. G
EOMETRIC C HARACTERIZATIONS
We develop geometric characterizations for the local notions introduced in the pre-vious section. Definition 9 below is a stronger version of Definition 7 that dispensesof the qualifier “w.r.t. H ” present in the latter definition. O-ARBITRAGE SYMMETRIES 13
Local Geometric Characterizations.Definition 9 (Disperse and 0-neutral sets) . Consider a set E ⊆ R d ; E is called disperseif for each h ∈ R d : (12) [ h · y = ∀ y ∈ E ] or [ inf y ∈ E h · y < y ∈ E h · y > ] . E is called -neutral if for each h ∈ R d : (13) [ inf y ∈ E h · y ≤ y ∈ E h · y ≥ ] . Notice that (13) is equivalent to just requiring the validity of one of the two inequali-ties appearing in the conjunction in (13). Similarly, (12) is equivalent to [ h · y = ∀ y ∈ E ] or [ inf y ∈ E h · y < ] (this later inequality could be replaced by [ sup y ∈ E h · y > ] ). Wehave written Definition 9 in its present form for emphasis.Results and notions from convex analysis that we will rely upon in this section aredetailed in Appendix B. Proposition 6.
Let E ⊆ R d .E is disperse iff ∈ ri ( co ( E )) . Proof.
Assume first that E is disperse. In order to proceed to deduce a contradiction,we assume that 0 / ∈ ri ( co ( E )) ; by the separation Theorem 8, there exists ξ ∈ R d suchthat: • ξ · x ≥ x ∈ ri ( co ( E )) , and • ξ · x ∗ > x ∗ ∈ ri ( co ( E )) . Then, by means of Proposition 11, it follows that for all x ∈ ri ( co ( E ))( α x + ( − α ) y ) ∈ ri ( co ( E ) for all y ∈ E and α ∈ ( , ] . Therefore, ξ · ( α x + ( − α ) y ) = αξ · x + ( − α ) ξ · y ≥ y ∈ E and α ∈ ( , ] . It then follows that ξ · y ≥ y ∈ E . This contradicts the fact that E is disperse.Conversely, assume that 0 ∈ ri ( co ( E )) . We may assume there exists ˆ y ∈ E suchthat h · ˆ y =
0. It is enough to establish that inf y ∈ E h · y <
0, we may then assume thereexists y ∗ ∈ E such that h · y ∗ >
0. As y ∗ ∈ co ( E ) and 0 ∈ ri ( co ( E )) , it follows fromProposition 13 in Appendix B that there exists ε > − ε y ∗ ∈ co ( E ) . Then, it follows from Theorem 9 that there exists y ( ) , . . . , y ( d + ) ∈ E such that − ε y ∗ = λ y ( ) + · · · + λ d + y ( d + ) with d + ∑ i = λ i = , λ i ≥ . Then 0 > − ε ( h · y ∗ ) = d + ∑ i = λ i (cid:16) h · y ( i ) (cid:17) . Therefore, there must be some 1 ≤ j ≤ d + h · y ( j ) <
0, and theninf y ∈ E h · y < . (cid:3) Similarly to Proposition 6, the following result characterizes the 0-neutral property of E . Proposition 7.
Let E ⊆ R d .E is − neutral iff ∈ cl ( co ( E )) . Proof.
Assume E is 0-neutral and 0 / ∈ cl ( co ( E )) . Since the closure of a convex set isa convex set (Proposition 12) and closed, by Theorem 8 in Appendix B, it follows thatthere exists ξ ∈ R d such that inf ξ · y > y ∈ cl ( co ( E )) .Thus inf y ∈ E ξ · y > , which contradicts our hypothesis.Assume now 0 ∈ cl ( co ( E )) . It is enough to show that inf y ∈ E h · y ≤ h ∈ R d . To proceed by contradiction, assume there exists h ∈ R d , and ε > ε < inf y ∈ E h · y ≤ h · y for all y ∈ E (otherwise we are done). From our hypothesis,there exists a sequence { x j } ∞ j = ⊆ co ( D ) such that x j → j → ∞ . By Theorem 9 inAppendix B, for each x j there exists y ( , j ) , . . . , y ( d + , j ) ∈ E such that x j = λ j y ( , j ) + · · · + λ jd + y ( d + , j ) with d + ∑ i = λ ji = , λ ji ≥ . Then 0 = h · = h · (cid:18) lim j → ∞ x j (cid:19) = lim j → ∞ (cid:0) h · x j (cid:1) == lim j → ∞ d + ∑ i = λ ji (cid:16) h · y ( i , j ) (cid:17) ≥ lim j → ∞ d + ∑ i = λ ji ε = ε . which is a contradiction, thus, we conclude. (cid:3) Lemma 1 below uses the following notation, for E ⊂ R d , and x ∈ R d , let E − x ≡ { x − x : x ∈ E } ⊂ R d . Since the translation t x : R d → R d , given by t x ( x ) = x − x , is of the form (30) inAppendix B, and an homeomorphism, the lemma below follows. Lemma 1. ∈ ri ( co ( E − x )) iff x ∈ ri ( co ( E )) . ∈ cl ( co ( E − x )) iff x ∈ cl ( co ( E )) . O-ARBITRAGE SYMMETRIES 15
Convexity Preserving Maps.
In order to identify transformations that preserveno-arbitrage (0-neutrality) and in view of Proposition 6 (Proposition 7) and Lemma 1we first look for transformations F : R d → R d ′ preserving relative interiors or closuresof convex sets in R d .The notions introduced below are expanded in Appendix B where we also providedue references and introduce related definitions and further results. Definition 10 (Strict inversely convexity preserving) . Let V and V ′ be real linearspaces, and C ⊂ V a nonempty convex subset. A map g : C → V ′ is called strict in-versely convexity preserving if (14) g (( x , y )) ⊆ ( g ( x ) , g ( y )) for all x , y ∈ C , where ( x , y ) = { tx + ( − t ) y : 0 < t < } (with a similar definition for [ x , y ] , see Ap-pendix B). Moreover, g is said to preserve segments strictly if equality holds in (14). The next lemma provides necessary and sufficient conditions on a transformation F in order to preserve ri ( co ( E )) , for E ⊂ R d . Lemma 2.
Let C ⊆ R d be a convex set, x ∈ C and E ⊂ C.F : C → R d ′ is a strict inversely convexity preserving map if and only ifx ∈ ri ( co ( E )) implies F ( x ) ∈ ri ( co ( F ( E ))) . Proof.
Let x ∈ ri ( co ( E )) and b ′ ∈ co ( F ( E )) . Assume first that F ( C ) is contained ina straight line. Fix b ∈ co ( E ) , from Corollary 4, there exists a ∈ co ( E ) such that x ∈ ( a , b ) . Then by hypothesis on F and Proposition 14, F ( a ) , F ( b ) ∈ F ( co ( E )) ⊂ co ( F ( E )) , and F ( x ) ∈ F (( a , b )) ⊂ ( F ( a ) , F ( b )) . Now, since co ( F ( E )) is a segment, because it is contained in a straight line, it followsthat:If b ′ ∈ ( F ( x ) , F ( b )) , or F ( b ) ∈ ( F ( x ) , b ′ ) then F ( x ) ∈ ( F ( a ) , b ′ ) ⊂ co ( F ( E )) . On the other hand F ( x ) ∈ ( F ( b ) , b ′ ) ⊂ co ( F ( E )) . Thus in any case by Corollary 4 F ( x ) ∈ ri ( co ( F ( E ))) . If F ( C ) is not contained in a straight line, by Theorem 7, F preserves segmentsstrictly, then co ( F ( E )) = F ( co ( E )) , so b ′ = F ( b ) with b ∈ co ( E ) .As before, there exists a ∈ co ( E ) such that x ∈ ( a , b ) . Then F ( a ) ∈ co ( F ( E )) and F ( x ) ∈ F (( a , b )) = ( F ( a ) , F ( b )) , which also leads to F ( x ) ∈ ri ( co ( F ( E ))) .To establish the converse, consider the case E = { a , b } then ( a , b ) = ri ( co ( E )) and ( F ( a ) , F ( b )) = ri ( co ( F ( E ))) . Now from our hypothesis,(15) F ( x ) ∈ ( F ( a ) , F ( b )) , for any x ∈ ( a , b ) , therefore, F is strict inversely convexity preserving. (cid:3) Observe that, from Lemma 2, Proposition 6 and Lemma 1, F is strict inverselyconvexity preserving if it preserves disperse sets.In a similar way than before, F preserves cl ( co ( E )) , if for x ∈ [ a , b ] = cl ( co ( { a , b } )) it holds that F ( x ) ∈ cl ( co ( { F ( a ) , F ( b ) } )) = [ F ( a ) , F ( b )] . That is, F need to be inversely convexity preserving (see Proposition 14 in AppendixB). However this condition on its own is not sufficient (see next Lemma and Example2.) Lemma 3.
Let C ⊆ R d a convex set, E ⊂ C and F : C → R d ′ a continuous inverselyconvexity preserving map. If x ∈ cl ( co ( E )) , thenF ( x ) ∈ cl ( co ( F ( E ))) . Proof.
By continuity of F , for all E ⊆ C holds F ( cl ( E )) ⊆ cl ( F ( E )) . Furthermore,since F is a inversely convexity preserving map, F ( co ( E )) ⊂ co ( F ( E )) . Thus, since x ∈ cl ( co ( E )) , F ( x ) ∈ F ( cl ( co ( F ( E ))) ⊆ cl ( F ( co ( E )) ⊆ cl ( co ( F ( E ))) . (cid:3) Example 2.
The hypothesis of continuity in Lemma 3 can not be removed, to see thisconsider F : R → R given byF ( x ) = (cid:26) x if x ≤ x + if x > , is inversely convexity preserving, but not continuous and ∈ cl (( , ]) , butF ( ) = / ∈ [ , ] = cl ( co ( F (( , ]))) . Induced Transformations.
As indicated in Section 2.1, we have taken a stan-dard view in which the original sequency S i is given in a currency numeraire and thenthe sequence X ( S i ) is given in another (arbitrary) numeraire. Since we look for trans-formations between trajectories of financial markets that preserve their local propertieswe will be dealing with two associated functions, f and F the former acting on S i andthe latter on X ( S i ) . So we will have R d + f −→ R d ′ + and R d F −→ R d ′ . One could proceeddifferently and develop an approach which abstracts away this multiplicity; nonethe-less, we have decided to proceed the way we do as in practice that is how data is usuallypresented. This decision makes our results more readily applicable albeit at the priceof some complications.Let X and X ′ be the perspective functions over R d + and R d ′ + respectively, asdefined in (3). Since local properties are based on properties of discounted values, the O-ARBITRAGE SYMMETRIES 17 function f should induce a function F : R d → R d ′ , in such a way that the followingdiagram commutes,(16) dom X f / / X (cid:15) (cid:15) dom X ′ X ′ (cid:15) (cid:15) Im X ⊂ R d F / / Im X ′ ⊂ R d ′ that is, for s ∈ dom X , F ( X ( s )) ≡ X ′ ( f ( s )) . Therefore, if X ( s ) = X ( ˜ s ) for s , ˜ s ∈ dom X we will require that X ′ ( f ( s )) = X ′ ( f ( ˜ s )) , which gives a condition on f as we describenext.Assume s = ( s , . . . , s d ) , ˜ s = ( ˜ s , . . . , ˜ s d ) , then (cid:18) s s , . . . , s d s (cid:19) = X ( s ) = X ( ˜ s ) = (cid:18) ˜ s ˜ s , . . . , ˜ s d ˜ s (cid:19) ⇔ s = s ˜ s ˜ s , from where f needs to satisfy(17) f ( λ s ) = µ λ , s f ( s ) , λ , µ λ , s > , for any s ∈ R d + . We will require (17) in our main result Theorem 4 (as well as in Theorem 5).Lemma 2 shows that a map F preserving no-arbitrage is necessarily strict inverselyconvexity preserving. Lemma 4, item 1, below provides sufficient conditions on f toestablish the said property of F ; on the other hand, Example 3 shows that the assump-tions on f , while being sufficient, are not necessary. Lemma 4.
Let f : dom X → dom X ′ be a function satisfying (17). Then, there exists aunique map F : Im X → Im X ′ which makes commutative the diagram (16). Moreover (1) If f is (strict) inversely convexity preserving then F is (strict) inversely convex-ity preserving. (2)
F is continuous iff f is continuous.Proof.
For all x ∈ Im X there exist s ∈ dom X such that X ( s ) = x . The only way todefine F is then F ( x ) = X ′ ( f ( s )) for all x ∈ R d and it is well defined by condition (17).Let’s see that F is a strict inversely convexity preserving map if f is assumed tosatisfy that property. Fix ˆ x , ˜ x ∈ Im X and let x ∈ Im X such that x = α ˆ x + ( − α ) ˜ x with 0 < α < . Then, there exists ˆ s , ˜ s ∈ dom X such that X ( ˆ s ) = ˆ x and X ( ˜ s ) = ˜ x . Moreover, since X isa strict segment preserving map (Theorem 6), there exists β ∈ ( , ) such that x = X ( β ˆ s + ( − β ) ˜ s ) . Then since X ′ is strict segment preserving, F ( x ) = X ′ ( f ( β ˆ s + ( − β ) ˜ s )) ∈ (cid:0) X ′ ( f ( ˆ s )) , X ′ ( f ( ˜ s )) (cid:1) = ( F ( ˆ x ) , F ( ˜ x )) . The proof for the case inversely convexity preserving, is similar. This gives item . For item , observe that the perspective functions X , X ′ are continuous and open.The last assertion follows because if Q is an open cube in R d , and a < b positive realnumbers, then X (( a , b ) × Q ) = [ r ∈ ( a , b ) r Q , is open in R d . Thus, by composition F is continuous iff f is continuous. (cid:3) Example 3.
The converse in item 1 of Lemma 4 is not valid. Consider f : { ( x , y , z ) ∈ R : x > } → { ( x , y , z ) ∈ R : x > } given byf ( x , y , z ) = ( xx + y , yx + y , zx + y ) . The induced function F is the identity function on R but f is not inversely convexitypreserving because ( , , ) ∈ (( , − , ) , ( , , )) , butf ( , , ) = ( , , ) / ∈ ( f ( , − , ) , f ( , , )) = (( / , − / , ) , ( / , / , )) . Under the additional hypothesis that Im F contains a nondegenerate triangle, thenext Theorem characterizes those f : dom X → dom X ′ inducing a strict inverselyconvexity preserving map F . That hypothesis on Im F is equivalent to Im F not beingcontained in a straight line. As a complement, Lemma 5 shows that this last conditionon Im F holds iff Im f is not contained in a 2 dimensional subspace.The appearance of “0” in f ( s ) L ( s ) below merely reflects our arbitrary choice of S asnumeraire, choosing S k as numeraire will result in the appearance of f k ( s ) L k ( s ) in the nextresult (see Section 6 for an example). Theorem 3.
Assume f : dom X → dom X ′ satisfying (17), induces a strict inverselyconvexity preserving map F, such that Im F is not contained in a straight line. Then (18) f ( s ) = f ( s ) L ( s ) L ( s ) , with f (its first coordinate function) satisfying (17), L : R d + → R d ′ + , a linear map,with and L , f > on s > , (L the first coordinate of L).Conversely, if f has the form (18) and satisfies the properties listed after that formuladisplay, then it induces a strict inversely convexity preserving map F.Proof. Let us consider first that dom X = { s ∈ R d + : s i > ∀ i } , so dom F = { x ∈ R d : x i > ∀ i } is convex, then by Theorem 7(19) X ′ ( f ( s )) = F ( X ( s )) = ( A ( X ( s )) + b , ..., A d ′ ( X ( s )) + b d ′ ) B ( X ( s )) + c . It then follows that for 1 ≤ i ≤ d ′ , f i ( s ) f ( s ) = F i ( X ( s )) = A i ( X ( s )) + b i B ( X ( s )) + c = a i , s s + · · · + a i , d s d s + b i B s s + · · · + B d s d s + c . O-ARBITRAGE SYMMETRIES 19
Which can be written as f i ( s ) = f ( s ) L i ( s ) L ( s ) , with L i ( s ) = b i s + a i , s + · · · + a i , d s d , and L ( s ) = cs + B s + · · · + B d s d . From where, defining L ( s ) = ( L ( s ) , L ( s ) , · · · , L d ′ ( s )) , (18) holds with the expectedconditions, since f satisfies (17) because f do, and both f , L > s > X = { s ∈ R d + : s > } , which implies that dom F = R d ,then by [P´ales (2012), Cor 1], (19) can be written with B ( x ) + c ≡
1. Consequently(18) holds with L ( s ) = s .Conversely, if f has the form (18) with the required conditions, then satisfies (17),because f , L , L satisfy (17), by hypothesis and linearity respectively, consequentlyby Lemma 4 there exists F such that F ( X ( s )) = X ′ ( f ( s )) . Let’s show that F is strictinversely convexity preserving. X ′ ( f ( s )) = L ( s ) L ( s ) , · · · , L d ′ ( s ) L ( s ) ! , where for 1 ≤ i ≤ d ′ L i ( s ) L ( s ) = a i , s + · · · + a i , d s d a , s + · · · + a , d s d = a i , + a i , s s + · · · a i , d s d s a , + a , s s · · · a , d s d s = a i , + A i ( X ( s )) a , + B ( X ( s )) . With A i ( x ) = a i , x + · · · + a i , d x d , and B ( x ) = a , x + · · · + a , d x d in the last expression.Defining A = ( A , · · · , A d ′ ) and b = ( a , , · · · , a d ′ , ) , it follows that(20) F ( x ) = b + A ( x ) a , + B ( x ) , Which is strict inversely convexity preserving by Theorem 6. (cid:3)
Lemma 5.
Assume f : dom X → dom X ′ is a function satisfying (17) and F the inducedfunction as in Lemma 4. Then, Im F is contained in a straight line iff Im f is containedin a 2-dimensional subspace.Proof. Assume that f ( s ) = ( y , . . . , y d ′ ) then F ( X ( s )) = X ′ ( f ( s )) = y ( y , . . . , y d ′ ) . It follows that
Im F = { z ∈ R d ′ : ( , z ) ∈ λ ( Im f ) , for some λ > } . If Im f ⊂ π , a 2-dimensional subspace, then Im F ⊂ { z ∈ R d ′ : ( , z ) ∈ π } , and this set is contained in the straight line π ∩ { y = } ⊂ R d ′ + .Conversely, assume there exist s , s , s ∈ dom X such that f ( s ) , f ( s ) , f ( s ) are l.i.Since Im F is contained in a straight line, it follows that there exists α ∈ R such that F ( X ( s )) = α F ( X ( s )) + ( − α ) F ( X ( s )) = α X ′ ( f ( s )) + ( − α ) X ′ ( f ( s )) , which leads to the contradiction f ( s ) = f ( s ) f ( s ) α f ( s ) + f ( s ) f ( s ) ( − α ) f ( s ) . (cid:3)
5. N O A RBITRAGE I NVARIANCE
This section studies a class of transformations that do not change a given node’slocal properties of being arbitrage-free (this latter notion as per Definition 8). Wealso provide an explicit characterization for such symmetry transformations, this isachieved under a general, and weak, condition restricting their ranges.As a special case, we will prove that the no-arbitrage property is unchanged undera change of num´eraire. We also describe the similar results that apply for the prop-erty of 0-neutral and, therefore, need also pursue some developments that apply to thisconcept as well. In general, the class of transformations studied should represent sym-metries obeyed by any type of functional relationship among asset’s prices resultingfrom no arbitrage considerations. In particular, if prices S satisfy a h ( S ) = h ( S ′ ) = S and S ′ are related by a no-arbitrage symmetry asper Definition 11 below. This fact is illustrated with an example in Section 6.Let M = S × H and M ′ = S ′ × H ′ be trajectory based markets, with d + d ′ + M onto M ′ , will be givenby a function f : R d + → R d ′ + which will be called a market transformation . Thatis, to a trajectory S = ( S , W , m ) ∈ M corresponds a trajectory S ′ = ( S ′ , W ′ , m ′ ) ∈ M ′ ,where S ′ k = f ( S k ) , k ≥
0, and W ′ , m ′ are transformed in consequence. For instance, if W represents the quadratic variation of the logarithm of the assets prices, then W ′ k = k − ∑ i = ( log f ( S i + ) − log f ( S i )) . This example illustrates a case when W ′ can be obtained from S ′ . In other cases, whenthis is not possible, W ′ and m ′ should be prescribed but, how this is actually done doesnot affect the developments in the present section. Definition 11.
A market transformation f , as above, which leaves invariant the arbitrage-free property ( -neutral property), as per Definition 8, of a given market’s node willbe called a no-arbitrage symmetry ( -neutral symmetry). Therefore, if the node ( S , k ) is arbitrage-free so will be ( S ′ , k ) if f is a no-arbitragesymmetry (similarly for a 0-neutral symmetry). This remark also shows that the com-position of no-arbitrage symmetries (0-neutral symmetries) is a no-arbitrage symmetry(0-neutral symmetry). We may refer to either type of symmetry as NAS (No-ArbitrageSymmetries) when there is no need to be specific. Remark 6.
The above notions depend on a choice of numeraire through Definition 8but we will prove in Corollary 2 that a symmetry transformation remains as such undera numeraire change. Of course the interest is in general symmetry transformationsf : R d + → R d ′ + that behave so for any possible node in any possible trajectory O-ARBITRAGE SYMMETRIES 21 market (with corresponding dimension d) and that is the type of characterization wepursue.
Recall from Definition 7 that local conditions are based on properties of the incre-ment set ∆ X ( S ( S , k ) ) , where ( S , k ) is a node of the market model. This set is totallydetermined by the values taken by the trajectories in the stage k + S k . To make this fact explicit, for each node ( S , k ) we introduce a notation for the setof reachable prices: Σ k ( S ) ≡ { ˆ S k + : ˆ S = ( ˆ S , ˆ W , ˆ m ) ∈ S ( S , k ) } ⊆ R d + . (21)The next proposition (which follows from Lemma 1 in Section 4.2) shows that localconditions can be rewritten in terms of the set Σ k ( S ) . Proposition 8.
Given a trajectory based set S , S = { ( S i , W i , m ) } i ≥ ∈ S and aninteger k ≥ . (1) The node ( S , k ) is arbitrage-free if, and only if,X ( S k ) ∈ ri ( co ( X ( Σ k ( S )))) . (2) The node ( S , k ) is -neutral if, and only if,X ( S k ) ∈ cl ( co ( X ( Σ k ( S )))) . Theorem 4 (Arbitrage-free invariance) . Assume f : dom X → dom X ′ to be a map sat-isfying (17) and that the function F, induced by Lemma 4, is strict inversely convexitypreserving. Given a trajectory based market M = S × H , let S ∈ S and k ≥ . If ( S , k ) is an arbitrage-free node, then ( S ′ , k ) , where S ′ i = f ( S i ) i ≥ , is an arbitrage-freenode in a transformed market M ′ = S ′ × H ′ , i.e. ∈ ri ( co ( { X ′ ( f ( ˆ S k + )) − X ′ ( f ( S k )) : ˆ S ∈ S ( S , k ) } )) ⊆ R d ′ and so f is a no-arbitrage symmetry.Proof. We know from Lemma 4 that there exists F : dom X → dom X ′ given by F ( x ) = X ′ ( f ( s )) , where s ∈ dom X such that X ( s ) = x . Thus, since by hypothesis it is a strictinversely convexity preserving map, from Lemma 2 and Lemma 1, it follows (by taking x = X ( S k ) ) that 0 ∈ ri ( co ( { F ( x ) − F ( x ) : x ∈ X ( Σ k ( S )) } )) , or, equivalently, 0 ∈ ri ( co ( { X ′ ( f ( ˆ S k + )) − X ′ ( f ( S k )) : ˆ S ∈ S ( S , k ) } )) . (cid:3) Remark 7.
By Lemma 4 item 1, if f is strict inversely convexity preserving, then theinduced F satisfies the hypothesis of Theorem 4. Also notice that if Im F contains anondegenerate triangle, by Theorem 3, f is of the form given by (18).
Corollary 1 (Explicit Characterization) . Assume f : dom X → dom X ′ satisfies (17). (1) If f is a no-arbitrage symmetry (as per Definition 11) for any market and Im fis not contained in a -dimensional subspace then f is characterized by ex-pression (18). (2) Conversely if f has the form (18) then it is a no-arbitrage symmetry for anypossible market.Proof.
We recall that (17) assures the existence of the induced function F as in Lemma4. Assume f is a no-arbitrage symmetry from a market M onto a market M ′ . Then, byTheorem 4 and Lemma 2, the induced F must be strict inversely convexity preserving.Moreover, if Im f is not contained in a 2-dimensional subspace Lemma 5 implies thatIm F is not contained in a straight line. Finally by Theorem 3 f takes the form (18).This proves 1.For the converse, if f has the form (18), the converse of Theorem 3 implies thatthe induced function F is strict inversely convexity preserving. Thus by Lemma 2 andTheorem 4 f is a no-arbitrage symmetry for any possible market. (cid:3) Observe that the composition of no-arbitrage symmetries of the form (18) is againof this form.A transformation of interest in financial terms is the one that changes the marketmodel’s num´eraire. Let’s assume that the first asset S is strictly positive for everytrajectory in S , so the first coordinate can take the place of an alternative num´erairefor the model. For each S ∈ S , we will denote the sequence of prices relative to S by Y ( S ) = { ( Y ( S i ) , W i , m ) } i ≥ where Y : D ′ ⊂ R d + → R d is the perspective functionover the second coordinate:(22) Y ( s ) ≡ (cid:18) s s , s s , . . . , s d s (cid:19) D ′ ≡ { s = ( s , . . . , s d ) ∈ R d + : s > } . Y j ( S i ) represents the value of the j -th asset in units of the new num´eraire. We willprove next the following proposition that will be useful for the coming results. Proposition 9.
Let σ be the permutation on R d + that interchanges the first coordinatewith the second and X the perspective function on R d + (defined in (3) ). Then, Y = X ◦ σ over D ” ≡ { s ∈ dom X : s > } . Furthermore, σ is a strict segment preservingmap.Proof. Fix s ∈ D ”, then σ ( s ) ∈ dom X and ( X ◦ σ )( s ) = X (cid:16) s , s , . . . , s d (cid:17) = (cid:18) s s , . . . , s d s (cid:19) = Y ( s ) . Since σ is a linear map, it follows from Theorem 6 in Appendix B, that it is a strictsegment preserving map. (cid:3) We are now in a position to show that the arbitrage-free condition on a trajectorybased market M is independent of the choice of num´eraire. For this, we will state thefollowing Corollary. O-ARBITRAGE SYMMETRIES 23
Corollary 2.
Let M = S × H a semi-bounded trajectory based market such thatS > for all S ∈ S and H contains the class of restricted portfolios (11). If M isarbitrage-free with S as num´eraire, then M is arbitrage-free with S as num´eraire.Proof. From Proposition 9 above, it follows that Y = X ◦ σ on the set D ” ≡ { s ∈ dom X : s > } , where σ is the permutation of the first coordinate by the second. Also,since M is arbitrage-free, it follows from Proposition 4 that M is locally arbitrage-free. Then, any node ( S , k ) in the market is arbitrage-free (all notions with S asnum´eraire). As σ verifies the hypothesis of Theorem 4, we can ensure that0 ∈ ri (cid:0) co (cid:0) ∆ Y ( S ( S , k ) ) (cid:1)(cid:1) ≡ ri (cid:0) co (cid:0) { Y ( ˆ S k + ) − Y ( S k ) : ˆ S ∈ S ( S , k ) } (cid:1)(cid:1) , for all ( S , k ) , or, in other words, every node is arbitrage-free with respect to the num´eraire S . Then, it follows from Theorem 1, that M is arbitrage-free with respect to thenum´eraire S . (cid:3) Our goal is now to find market transformations f that preserve 0-neutral nodes (i.e.0-neutral symmetries). From Lemma 3 we know that the induced transformation F needs to be continuous and inversely convexity preserving in order to preserve the clo-sure of convex sets. The following Theorem shows that these conditions are, somehow,sufficient to obtain a 0-neutral symmetry as per Definition 11. Theorem 5 (0-neutral invariance) . Let f : dom X → dom X ′ be a continuous mapsatisfying (17) and the function F, induced by Lemma 4, is inversely convexity pre-serving. Given a trajectory based market M = S × H , let S ∈ S and k ≥ . If ( S , k ) is a -neutral node, then ( S ′ , k ) is a -neutral node in the transformed market M ′ = S ′ × H ′ , i.e. X ′ ( f ( S k )) ∈ cl (cid:0) co (cid:0) X ′ ( f ( S ( S , k ) )) (cid:1)(cid:1) and so f is a -neutral symmetry.Proof. We know from Lemma 4, in Section 4.2, that there exists a continuous map F : R d → R d ′ given by F ( x ) = X ′ ( f ( s )) , where s ∈ dom X such that X ( s ) = x . Moreover,by hypothesis it is inversely convexity preserving.Thus, since by hypothesis X ( S k ) ∈ cl (cid:0) co (cid:0) X ( S ( S , k ) ) (cid:1)(cid:1) , from Lemma 3 in Section4.2, it follows that, X ′ ( f ( S k )) ∈ cl (cid:0) co (cid:0) X ′ ( f ( S ( S , k ) )) (cid:1)(cid:1) . (cid:3) By the converse of Theorem 3, if f is given as in the expression (18), with theprescribed conditions, then the induced funtion F has the expression (20). Therefore,it is also inversely convexity preserving and continuous by Theorem 6; then f preserves0-neutral nodes and so it is 0-neutral symmetry.In the 0-neutral market definition, the selection of an explicit num´eraire is required.Consider, as in Corollary 2, a trajectory based markets such that S > Corollary 3.
Given a semi bounded trajectory based market M = S × H such thatS > for all S ∈ S . If ( S , k ) is a -neutral node with respect to the num´eraire S ,then it is also with respect to the num´eraire S . In particular, if M is locally -neutral,it will also be -neutral for any choice of num´eraire.Proof. From Proposition 9 it follows Y = X ◦ σ over the set D ” = { s ∈ dom X : s > } ,where σ is the permutation of the first coordinate by the second. Since σ verifies thehypothesis of Theorem 5, then0 ∈ cl (cid:0) co (cid:0)(cid:8) Y ( ˆ S k + ) − Y ( S k ) : ˆ S ∈ S ( S , k ) (cid:9)(cid:1)(cid:1) and ( S , k ) is a 0-neutral node with respect to the num´eraire S . We can conclude that if M is locally 0-neutral with respect to the num´eraire S , then it will also be 0-neutralwith respect to the num´eraire S . Therefore, it follows from Theorem 2, that if M islocally 0-neutral for S , M will be 0-neutral for any other choice of num´eraire. (cid:3)
6. E
XAMPLE
We will provide a slightly non-traditional development on the call-put parity rela-tionship. This is a simple relation among prices of certain assets; it is derived in manytextbooks and can be obtained through a no-arbitrage based proof. We will derive itunder the weaker hypothesis of 0-neutrality and relate the relationship to NAS (No-Arbitrage Symmetries). Our main point of revisiting the call-put parity is that it willallow us to provide an explicit example of NAS (besides a change of numeraire) aswell as to illustrate their meaning in this context.6.1.
Call-Put Parity Under -Neutrality. Consider an arbitrary time evolution offour assets S t ≡ ( C t , P t , Y t , B t ) , ≤ t ≤ T . We require, C T = ( Y T − B T ) + , P T = ( B T − Y T ) + , and B T = K where K is a constant . That is: C is a European call written on asset Y , with strike K and expiration T . Simi-larly for the European put P . B is a bond. Clearly ( C T − P T − Y T + B T ) =
0, which canbe thought as a boundary condition. Under an appropriate no-arbitrage assumption thecall-put parity is the following result [Musiela & Rutkowski (2005), Cor 1.4.2]:(23) ( C t − P t − Y t + B t ) = , ∀ ≤ t ≤ T . That is, no-arbitrage, under the said conditions, constraints the evolution of the fourassets accordingly to (23).We will add details on dimensions that are neglected in the above formulation, dis-pensing with units/dimensions is standard in the literature but making them explicit isrelevant to our philosophy as a change of units should be a NAS (but we do not explorethis view in the paper). We will insert appropriate dimensions/units whenever relevantbut switch (or alternate) to suppressing units (as usual) whenever the relevant dimen-sions have been made clear. We write Z = ( Z )[ Z ] where ( Z ) is the (dimensionless)numerical value and [ Z ] the dimensional units of the variable Z respectively. O-ARBITRAGE SYMMETRIES 25
We will have [ C t ] = $ C which would require (see below) the insertion of a dimen-sional constant a with units [ a ] = S C with ( a ) representing the number of shares asso-ciated to a call option. We will take ( a ) = ( a ) will have theeffect of multiplying the call-put parity by ( a ) (usually, in practice ( a ) = a represents the number of shares per call contract, this is not an artificial insertion as itis a feature of traded options. Similarly P t will contain a dimensional constant b with [ b ] = S P with ( b ) representing the number of shares associated to a put option, we willtake ( b ) = ( a ) = ( a ) ).In order to provide a derivation of (23) under 0-neutrality, we first express the abovesetting in our trajectorial framework. The above formulation is in continuous timebut we consider this to be a nonessential point (as we argue below). We will workwith trajectories of the form S i = ( S i , t i , m ) = ( S i , S i , S i , S i , t i , m ) = ( C i , P i , Y i , B i , t i , m ) where 0 = t < t < . . . < t m = T . Clearly, the times t i are trajectory dependent; as aparticular case we could take t i = i TM , 0 ≤ i ≤ M for a given constant M . Given that theargument will apply to any trajectory set with these coordinates we can approximateany arbitrary time t by taking M larger.Let S denote any 0-neutral trajectory set with the above introduced coordinates andthat obeys S M ( S ) = C M ( S ) = a ( S M ( S ) − [ K ] S M ( S )) ) + = a ( Y M ( S ) − K $ B ) + , S M ( S ) = P M ( S ) = b ([ K ] S M ( S ) − S M ( S )) ) + = b ( K $ B − Y M ( S ) ) + , and S M ( S ) = B M ( S ) = ( K ) $ B for all S . K is a dimensional constant with [ K ] = B S , K represents the number of bond units per share and so K $ B is the strike price. So wehave [ C i ] = [ a ] $ Y , [ P i ] = [ b ] $ Y , [ Y i ] = $ Y and [ B i ] = $ B . Moreover, assume M ( S ) = m tobe a stopping time in the sense that if S ′ k = S k for all 0 ≤ k ≤ M ( S ) then M ( S ′ ) = M ( S ) .Finally, we also assume t M ( S ) = T . Such S will be called admissible .The previous call-put parity is now written with units and taking ( a ) = ( b ) :(24) β ( S i ) ≡ ( C C i − P P i − Y Y i + B B i ) = , ∀ ≤ i ≤ M . That is, no-arbitrage, under the said conditions, constraints the evolution of the four as-sets accordingly to (24). (24) holds if and only if B π ( X ( S i )) ≡ B [( C i B i ) − ( P i B i ) − ( Y i B i ) + )] = X ( S M ( S ) ) = ( S M ( S ) S M ( S ) , S M ( S ) S M ( S ) , S M ( S ) S M ( S ) ) = ( C M ( S ) B M ( S ) , P M ( S ) B M ( S ) , Y M ( S ) B M ( S ) ) (notice that we are abusing the notation by using S as numeraire instead of the usual S ).To establish (24), we will return now to the usual practice of suppressing the units,in particular, in the proof below when we write X ( S i ) it will be interpreted as thecoordinates without the dimensions i.e. (( C i B i ) , ( P i B i ) , ( Y i B i )) . Proof of Call-Put Parity.
Let Π ≡ { x ∈ R : π ( x , x , x ) = x − x − x + = } .Consider an admissible trajectory set as described above; according to Proposition 5.2item 2: X ( S M ( S ) − ) ∈ cl ( co ( X ( Σ M ( S ) − ( S )))) . Clearly cl ( co ( X ( Σ M ( S ) − ( S )))) ⊆ Π and therefore π ( X ( S M ( S ) − )) =
0. Continuing the argument by induction we obtain π ( X ( S i )) = [( C i B i ) − ( P i B i ) − ( Y i B i ) + ] = ≤ i ≤ M ( S ) which is our version of thecall-put parity. The result is here established solely under the hypothesis of 0-neutralitythat is weaker than the no-arbitrage assumption.6.3. An Example of a NAS.
Let us introduce the following transformation: C i → C ′ i = P i Y i B i , P i → P ′ i = C i Y i B i , Y i → Y ′ i = Y i , B i → B ′ i = B i . So ( C i , P i , Y i , B i ) → ( C ′ i , P ′ i , Y ′ i , B ′ i ) = Y i B i ( P i , C i , B i , Y i , ) . We then have (we are disregarding dimensional constants with numerical value 1): C ′ M ( S ) = ( Y ′ M ( S ) − B ′ M ( S ) ) + = ( Y M ( S ) − K ) + P ′ M ( S ) = ( B ′ M ( S ) − Y ′ M ( S ) ) + = ( K − Y M ( S ) ) + . In financial terms, the transformed variables C ′ i , P ′ i are prices of a call and a putoptions, respectively, but now depending on the price of the same asset Y i but expressedin terms of shares per currency unit. This is not equivalent to using Y as the numeraire.Notice that C ′ i − P ′ i − Y ′ i + B ′ i = Y i B i ( P i − C i − B i + Y i ) = . In fact, we will argue that → is indeed a NAS. We change notation to touch basiswith the formal notation in the paper, let: s → s ′ being given by s ′ = f ( s ) . where,with the notation s ≡ ( s , s , s , s ) , f ( s , s , s , s ) = ( s , s , s , s ) s s and notice that if L ( s , s , s , s ) ≡ ( s , s , s , s ) , a linear function, we obtain: f ( s ) = f ( s ) L ( s ) L ( s ) . So f has the form (18) and by Corollary 1 item 2, f is a no-arbitrage symmetry. In fact, f preserves 0-neutrality as well and this follows from the converse of Theorem 3 andTheorem 5.6.4. Call-Put Parity Under a No-Arbitrage Symmetry.
Let us now see the effecton the call-put parity relation after applying a no-arbitrage symmetry. Towards thisgoal, consider f to be a no-arbitrage symmetry satisfying (17) and such that Im f is notcontained in a 2-dimensional subspace. From Corollary 1, we have f ( S i , . . . , S i ) = f ( S i ) = f ( S i ) L ( S i ) L ( S i ) where f , L , L are as in Theorem 3. Then F ( X ( S i )) = F ( S i S i , S i S i , S i S i ) = ( C ′ i B ′ i , P ′ i B ′ i , Y ′ i B ′ i ) = ( L ( S i ) L ( S i ) , L ( S i ) L ( S i ) , L ( S i ) L ( S i ) ) . O-ARBITRAGE SYMMETRIES 27
All in all, we will then take (with some abuse of notation): F ( x ) = A ( x ) + bB ( x ) + c where ( B ( x ) + c ) > , with A : R → R and B : R → R both linear transformations (notice that we havereproduced computations from Theorem 3).Before proceeding to a computation we need to impose that the boundary conditionbehaves as follows:(25) C ′ T = ( Y ′ T − B ′ T ) + , P ′ T = ( B ′ T − Y ′ T ) + , that is, the corresponding transformed price coordinates are prices of a call and a puton the transformed asset. Such an imposition is necessary for the derivation to followand prescribes that the boundary condition is invariant under F .We briefly sketch an argument establishing(26) π ( F ( X ( S i ))) = a F B ( x ) + c π ( X ( S i )) , where a F ≡ ( a , − a , − a , − a , ) and a j , k are the matrix coordinates of a matrixrepresentation of A . The relationship (26) makes it immediate that π ( X ( S i )) = π ( F ( X ( S i ))) = F .The implication π ( X ( S i )) = = ⇒ π ( F ( X ( S i ))) = f is a no-arbitrage symmetry and so a 0-neutral symmetryand given that S is assumed to be 0-neutral so will then be S ′ (this trajectory setobtained from S by acting with f on the trajectories S ∈ S ).Given that π is linear it is enough to consider the case F ( x ) = A ( x ) and establish theexistence of a F such that π ( F ( X ( S i ))) = a F π ( X ( S i )) .To start, substracting the two equations in (25) we obtain:(27) ( a , − a , ) C T + ( a , − a , ) P T + ( a , − a , ) Y T + ( a , − a , ) B T =( a , − a , ) C T + ( a , − a , ) P T + ( a , − a , ) Y T + ( a , − a , ) B T . It turns out, that in order to establish π ( F ( X ( S i ))) = a F π ( X ( S i )) , we will only need toobtain some relationships among the matrix entries a i , j . For reasons of space we onlysketch the derivations which follow from (27). First, let Y T > B T and equating coeffi-cients of Y T (equating coefficients of variables does require some minimal assumptionson Y T and B T which we do not make explicit) we obtain(28) ( a , − a , ) + ( a , − a , ) = ( a , − a , ) + ( a , − a , ) , a similar relation is obtained for the coefficients of B T . Two more analogous relation-ships among coefficients are obtained from the case Y T < B T . The said relationships allow to evaluate as follows π ( F ( X ( S i ))) = π ( A ( S i )) = a , C i + a , P i + a , S i + a , B i − a , C i − a , P i − a , S i − a , B i − a , C i − a , P i − a , S i − a , B i + a , C i + a , P i + a , S i + a , B i = ( a , − a , − a , + a , ) C i + ( a , − a , − a , + a , ) P i + ( a , − a , − a , + a , ) S i + ( a , − a , − a , + a , ) B i = a F π ( X ( S i )) .
7. C
ONCLUSION
The paper poses and solves the following basic question: what transformations, act-ing on financial events, leave the no-arbitrage property invariant? Such transformationsare called no-arbitrage symmetries (NAS) and are interpreted as mapping financialevents to financial events. We make use of results from convex analysis and a generalnon-probabilistic framework to characterize and provide explicit expressions for theNAS. We take advantage of a formulation of arbitrage free markets (as per Section 4)in terms of geometric assumptions of the trajectories in discrete time. The problemformulation naturally provides the characterization, in a local sense, of no-arbitragepreserving transformations.The transformed variables, i.e. the output values of NAS, do require an interpre-tation as the original setting is abstract and general. For example, in the example ofSection 6 we have to impose that boundary conditions should also be invariant underNAS and in so doing we required that two of the transformed variables acted as calland put options on the two remaining transformed variables. From such a general pointof view we think that the result of applying a NAS to financial events are admissibleprices for financial events but the latter will require an interpretation that will dependon the context and the specific NAS under consideration.A
PPENDIX
A. R
ESULTS AND P ROOFS FROM S ECTION
Proposition 10.
A trajectory based market M = S × H is -neutral if and only if,for each Φ ∈ H and ε > there exist S ε ∈ S such that (29) N Φ ( S ε ) − ∑ i = H i ( S ε ) · ∆ i X ( S ε ) < ε . Proof.
Suppose first M is 0-neutral. From the definition follows that for any ε > S ′ ∈ S " N Φ ( S ′ ) − ∑ i = H i ( S ′ ) · ∆ i X ( S ′ ) ≤ < ε O-ARBITRAGE SYMMETRIES 29 for all Φ ∈ H . Then, for each Φ there exist S Φ ∈ S such that N Φ ( S Φ ) − ∑ i = k H i ( S Φ ) · ∆ i X ( S Φ ) < ε for any ε >
0. Thus we proved the necessary condition.For the sufficient condition, fix ε >
0, then, by hypothesis, for each Φ ∈ H there is S ε ∈ S such that N Φ ( S ε ) − ∑ i = H i ( S ε ) · ∆ i X ( S ε ) < ε . Then, for each Φ ∈ H inf S ′ ∈ S " N Φ ( S ′ ) − ∑ i = H i ( S ′ ) · ∆ i X ( S ′ ) < ε . Since ε > S ′ ∈ S " N Φ ( S ′ ) − ∑ i = H i ( S ′ ) · ∆ i X ( S ′ ) ≤ Φ ∈ H . Therefore, since 0 ∈ H , we conclude that M is 0-neutral. (cid:3) Proof of Theorem 1
Proof.
Assume M is locally arbitrage-free and semi-bounded; fix Φ ∈ H once andfor all. If for all nodes ( S , k ) , H k ( S ) · ∆ k X ( S ′ ) = S ′ ∈ S ( S , k ) then G Φ N Φ ( S ) = N Φ ( S ) − ∑ i = H i ( S ) · ∆ i X ( S ) = S ∈ S and so V Φ N Φ ( S ) = V Φ + G Φ N Φ ( S ) = V Φ , ∀ S ∈ S ;therefore, Φ is not an arbitrage opportunity.We may then assume that there exists a trajectory S ( ) ∈ S and an integer k ≥ ( S ( ) , k ) , H k ( S ( ) ) · ∆ k X ( S ) = S ∈ S ( S ( ) , k ) .Then, by Definition 7, 1., it is possible to choose k , 0 ≤ k ≤ k , the smallest integersuch that, for 0 ≤ j < k , H j ( S ) · ∆ j X ( S ) = S ∈ S ( S ( ) , j ) , and there exists S ( ) ∈ ( S ( ) , k ) such that k ∑ i = H i ( S ( ) ) · ∆ i X ( S ( ) ) < . Consider the case when for all k < k ≤ N Φ ( S ( ) ) , H k ( S ( ) ) · ∆ k X ( S ( ) ) = ( ∗ )); then G Φ N Φ ( S ( ) ) = N Φ ( S ( ) ) − ∑ i = H i ( S ( ) ) · ∆ i X ( S ( ) ) < , under condition ( ∗ ) we have then established that Φ is not an arbitrage opportunity.Otherwise, i.e. when the case ( ∗ ) does not hold, we proceed by induction. Assumethat for i ≥ ( k j ) ij = and S ( j ) ∈ S ( S ( j − ) , k j ) , ≤ j ≤ i , such that for k j − < k < k j , ( k = ) , H k ( S ( j ) ) · ∆ k X ( S ( j ) ) =
0, and H k j ( S ( j ) ) · ∆ k j X ( S ( j ) ) <
0. In particular k i ∑ j = H j ( S i ) · ∆ j X ( S i ) < . The same argument that we used for the node ( S ( ) , k ) above, but now applied to ( S ( i ) , k i ) , and the inductive hypothesis gives the logical alternatives: a ) Φ is not an arbitrage opportunity by condition (*), b ) the inductive hypothesis holds for i + M is semi-bounded and that Φ is fixed, we remark thatthe alternative b ) becomes, eventually, empty and so the alternative a ) holds for i largeenough. Since Φ is arbitrary, M is arbitrage free. (cid:3) Proof of Proposition 4
Proof.
We proceed by contrapositive. Assume S is not locally arbitrage-free. There-fore, there is a node ( S , k ) which is not arbitrage-free, i.e. by Proposition 6 (in subsec-tion 4.1), ∆ X ( S ( S , k ) ) is disperse, so there exists ξ ∈ R d such that • ξ · ∆ k X ( S ′ ) ≥ S ′ ∈ S ( S , k ) , and • there exists S ∗ ∈ S ( S , k ) such that ξ · ∆ k X ( S ∗ ) > Ξ ( S , k ) belongs to H , it follows from Proposition 1 that V Ξ ( S , k ) N Ξ ( S , k ) ( S ′ ) = ξ · ∆ k X ( S ′ ) ≥ , for all S ′ ∈ S and there exists S ∗ ∈ S such that V Ξ ( S , k ) N Ξ ( S , k ) ( S ∗ ) = ξ · ∆ k X ( S ∗ ) > . Therefore, Ξ ( S , k ) is an arbitrage opportunity. (cid:3) Proof of Theorem 2
Proof.
Fix Φ ∈ H and ε >
0. We are going to show that there exists S ε ∈ S such that(29) holds.Fix S ∈ S , given that ( S , ) is a 0-neutral node w.r.t. H , it follows that there exists S ( ) ∈ S = S ( S , ) such that H ( S ) · ∆ X ( S ( ) ) < ε .Then, if N Φ ( S ( ) ) = N Φ ( S ( ) ) − ∑ i = H i ( S ( ) ) · ∆ i X ( S ( ) ) < ε < ε . O-ARBITRAGE SYMMETRIES 31 If N Φ ( S ( ) ) >
1, in the same way than before, we can choose a finite sequence ( S ( j ) ) nj = with n ≤ n Φ such that for 2 ≤ j ≤ n , S ( j ) ∈ S ( S ( j − ) , j − ) and j − ∑ i = H i ( S ( j ) ) · ∆ i X ( S ( j ) ) < j ∑ i = ε i < ε . Since M is semi-bounded, there exists 0 ≤ n ≤ n Φ such that N Φ ( S ( n ) ) − ∑ i = H i ( S ( n ) ) · ∆ i X ( S ( n ) ) < n ∑ i = ε i < ε . So ( ) holds with S ε = S ( n ) . Thus, since Φ ∈ H was chosen arbitrarily, it followsfrom Proposition 10 that M is 0-neutral. (cid:3) Proof of Proposition 5
Proof.
Suppose M is 0-neutral but some node ( S , k ) is not 0-neutral, it then followsfrom Proposition 7 (in subsection 4.1) that there exists ξ ∈ R d satisfyinginf S ′ ∈ S ( S , k ) ξ · ∆ k X ( S ′ ) > S ′ ∈ S ( S , k ) . By hypothesis, Ξ ( S , k ) ∈ H (see the definition preceding Proposition 4). Theninf S ′ ∈ S ( S , k ) N Ξ ( S , k ) ( S ′ ) − ∑ i = k Ξ ( S , k ) i ( S ′ ) · ∆ k X ( S ′ ) > , which is a contradiction. Therefore ( S , k ) is a 0-neutral node. (cid:3) A PPENDIX
B. C
ONVEX A NALYSIS
For x , y ∈ R d we define the closed segment [ x , y ] and the open segment ( x , y ) by [ x , y ] ≡ { tx + ( − t ) y : 0 ≤ t ≤ } and ( x , y ) ≡ { tx + ( − t ) y : 0 < t < } . To begin, let’s remember the notion of relative interior which will be very important inthe characterizations of local properties.
Definition 12 (Relative interior) . Let E ⊂ R d a convex set. The relative interior of E,that we will denote by ri ( E ) , is the interior of the set relative to its affine hull, that is, ri ( E ) = { x ∈ E : B ( x , r ) ∩ aff E ⊆ E for some r > } . The following property relates the notions of closure and relative interior.
Proposition 11 ([Rockafeller (1970), Teorema 6.1]) . Let E ⊂ R d a non empty convexset. Then, for each x ∈ ri ( E ) , α x + ( − α ) y ∈ ri ( E ) for all y ∈ cl ( E ) and for all α ∈ ( , ] . The Proposition that follows describes one of the most important properties of theclosure and the relative interior of convex sets.
Proposition 12.
Let E ⊂ R d a convex set. Then cl ( E ) and ri ( E ) are convex sets. The following characterizations of the relative interior for convex sets are useful.
Proposition 13 ([Rockafeller (1970), Corollary 6.4.1]) . Let E ⊂ R d a convex set. Thenthe relative interior of E is the set of all points x ∈ E such that for all y ∈ E there existsome ε > with x − ε ( y − x ) ∈ E . Corollary 4. x ∈ ri ( E ) iff for any b ∈ E there exists a ∈ E such that x ∈ ( a , b ) .Proof. From Proposition 13, if x ∈ ri ( E ) , for b ∈ E there exist some ε > a ≡ x − ε ( b − x ) ∈ E , so x = + ε a + ε + ε b ∈ ( a , b ) . Conversely if x = t a + ( − t ) b , with t ∈ ( , ) , then x − − tt ( b − x ) = a ∈ E . (cid:3) In the following results we will describe some operations that preserve convexity.These operations are helpful in determining or establishing when a set is convex. Givena map g : R d → R d ′ , we are going to present two properties of preservation of convexsets by g introduced in [P´ales (2012)]. We say g preserves convexity if g ( E ) is convexfor all convex subset E ⊆ R d . Analogously, we say that g − preserves convexity or g is inversely convexity preserving if g − ( E ′ ) is convex whenever E ′ is a convex subsetof g ( R d ′ ) . The following results are immediate. Proposition 14.
Let g : R d → R d ′ , (1) g preserves convexity if and only if [ g ( x ) , g ( y )] ⊆ g ([ x , y ]) for all x , y ∈ R d . (2) g is inversely convexity preserving if and only if g ([ x , y ]) ⊆ [ g ( x ) , g ( y )] for allx , y ∈ R d . Note that it follows from the previous Proposition that a convexity preserving func-tion which is, at the same time, inversely convexity preserving satisfy [ g ( x ) , g ( y )] = g ([ x , y ]) for all x , y ∈ R d . This motivates the following definition. Definition 13 (Segment preserving) . We say that a map g : R d → R d ′ preserves seg-ments if [ g ( x ) , g ( y )] = g ([ x , y ]) for all x , y ∈ R d . If ( g ( x ) , g ( y )) = g (( x , y )) for allx , y ∈ R d we say that g preserves segments strictly . Then, g preserves segments if and only if g preserves convexity and preserves con-vexity inversely. Clearly, if g preserves segments strictly, then preserves segments, theconverse, however, may not be valid.The obvious candidates to be functions that preserve segments strictly are the affinefunctions. Recall that a function g : R d → R d ′ is affine if it is the sum of a linearfunction plus a constant, that is, g ( x ) = Ax + b , where A ∈ R d × d ′ and b ∈ R d ′ . There isa larger class of functions which also preserve segments strictly. O-ARBITRAGE SYMMETRIES 33
Theorem 6 ([P´ales (2012), Thm 1]) . Let A : R d → R d ′ and B : R d → R linear functions,b ∈ R d ′ and c ∈ R . Let D = { x ∈ R d : B ( x ) + c > } , then: g : D → R d ′ given by (30) g ( x ) = A ( x ) + bB ( x ) + cpreserves segments strictly. Consider the function X : R d + → R d with dom X = { x ∈ R : x > } × R d definedin (3) by X ( x ) = x ( x , x , . . . , x d ) . This function called, perspective function , scales or normalizes vectors, so the firstcomponent is one, and then drops the first component. Since it has the form (30), thenpreserves segments strictly.The following result is key to our analysis.
Theorem 7 ([P´ales (2012), Thm 2]) . Let C ⊂ V be a nonempty convex subset andg : C → V ′ be a strict inversely convexity preserving function such that Im f contains anondegenerate triangle. Then, there exist A : V → V ′ , and B : V → R linear functions,b ∈ V ′ , and c ∈ R , such that B ( x ) + c > for x ∈ C , and (31) g ( x ) = A ( x ) + bB ( x ) + c . Moreover, by [P´ales (2012), Thm 1] g preserves segments strictly, this latter notionmeans that equality holds in (14).We will present below the Separation Theorem that we use to prove the Proposition6. Theorem 8 ([F¨ollmer & Schied (2011), Prop A.1]) . Suposse E ⊂ R d is a non emptyconvex set such that / ∈ E. Then, there exists a ∈ R d such that a · x ≥ for all x ∈ Eand a · x > for at least one x ∈ E. Furthermore, if inf x ∈ E k x k d > , then we can finda ∈ R d such that inf x ∈ E | a · x | > . Next we will define the convex hull of a set.
Definition 14 (Convex hull) . The convex hull of a set E ⊂ R d , that we will denote by co ( E ) , is the smallest convex set containing E. One of the most important characterizations of the convex hull is the Carath´eodoryTheorem.
Theorem 9 (Carath´eodory theorem) . Let E ⊂ R d . Then co ( E ) = ( d + ∑ i = λ i x i : x i ∈ E , λ i ≥ , d + ∑ i = λ = ) . F UNDING
The research of S.E. Ferrando is supported in part by an NSERC grant.The research of I.L. Degano and A.L. Gonz´alez is supported in part by NationalUniversity of Mar del Plata, Argentina [EXA902/18].R
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