On the Order Form of the Fundamental Theorems of Asset Pricing

In this article, we provide an order-form of the First and the Second Fundamental Theorem of Asset Pricing both in the one-period market model for a finite and infinite state-space and in the case of multi-period model for a finite state-space and a finite time-horizon. The space of the financial positions is supposed to be a Banach lattice. We also prove relevant results in the case where the space of the financial positions is not ordered by a lattice cone.


Theorems
The First Fundamental Theorem of Asset Pricing states that the absence of arbitrage for a stochastic process X is equivalent to the existence of an equivalent martingale measure for X .It was shown in [1] that for a locally bounded d  -valued semi-martingale X the condition of No Free Lunch with Vanishing Risk is equivalent to the existence of an equivalent local martingale measure for the process X .It was proved in [2] that the local boundedness assumption on X may be dropped under the notion of equivalent σ -martingale measure.The work [3], also discussed in [4], is still essential in this topic and actually this work's results rely on what Kreps established as the viable market model consisted by an incomplete market and a linear price system on it.In the present work we are going to resolve the so-called Strictly Positive Extension Property from the financial aspect.The presence of heavy-tails in continuous time models and the possible change of frame from p L spaces to Orlicz spaces in order to fit the modelling requirements, oblige us to search for more general versions of the two FTAPs, mostly relied on the geometry of these spaces.Recently, in [5], a Fundamental Theorem of Asset Pric-ing and a Super-Replication Theorem in a model-independent framework are both proposed.But these theorems are proved in the setting of finite, discrete time and a market consisting of a risky asset S , as well as options written on this risky asset, too.Notions like the one of the strictly positive projection or that of the filtration are alike the ones met in [6].A difference between our notion of strictly positive projection and the equivalent notion in [6] is that ours is weaker.That's because if x > , it would be 0 Px > .An important difference between the article of Troitsky and ours is that we extend the framework of Definitions so as to include cases of non-discrete time spaces.Another one is that we apply these notions in order to provide a new version of the two FTAP, while in [6] an important ordered -space theory of martingales in Banach lattices is developed.Finally, markets subspaces are taken to be sublattices because of the fact that we may include layers of call and put options written on an initial market space, as we remarked in [7].The present paper is organized as follows: First, we provide some useful notions and definitions and examples for them, as well.Next, we prove the Order Form of the FTAP in the Banach-lattice case and in the next sections we provide the analog of these results in the finite-models case.We also explain the application of our results on the Black-Scholes-Merton model.We also compare them to the Example developed in [4].The case of non-lattice cones is examined in the last section of the paper, in relation with the classes of reflexive and strongly reflexive cones, mentioned in [8].The role of the existence of an unconditional basic sequence in a Banach space is also quoted in this section independently from the results provided in [8], as an important condition for the extraction of results concerning FTAP.This condition is not irrelevant to ( [9], Th. 1.1), about Lindelöf Properties of weak topology, but here it mainly concerns the construction of a Strictly Positive Projection Operator.On the other side, in the paper [10] ideals of ( ) 0 L µ are used in order to deduce an FTAP-like result ( [10], Lem.1), while our results refer to sublattices.

Useful Notions and Preliminaries
We consider two periods of time (0 and 1) and a non-empty set of states of the world Ω which is supposed to be an infinite set.The true state ω ∈ Ω that the investors face is contained in some A ∈  , where  is some σ-algebra of subsets of Ω which gives the information about the states that may occur at time-period 1.A financial position is a  -measurable random variable : x Ω →  .This random variable is the profile of this position at time-period 1.We suppose that the probability of any state of the world to occur is given by a probability measure . The financial positions are supposed to lie in some subspace E of ( ) , being a Banach lattice.
Definition 1 An incomplete market in E is some sublattice M of E .A complete market in E is some sublattice M of E , such that M E = .It is well-known that we define the positive cone F + of a subspace F of an ordered vector space to be the set F F E + + = ∩ , where + E denotes the positive cone of E .Definition 2 A positive projection : P E F → is a projection, which maps each element of E to some ele- ment of its subspace F , such that ( ) We also recall the notion of random field.
where E is a Bananch lattice, ,  is a topological space and ω ∈ , for any t ∈  .Such a random field X is called associated to the pair ( ) . We also may provide the notion of the filtration in the frame of random fields: Definition 4 A filtration associated to the pair ( ) We also give the definition of the adapted random field under this frame., it has the Martingale Property, while the filtration ( ) a a A P ∈ is consisted by strictly positive projections.
We give some examples for the previously mentioned notions.
Example 11 If  is a sub-algebra of the σ -algebra  of Ω, then since ( )

Example 12
The subspace of partially linear functions M in the space [ ] due to the Stone-Weierstrass Theorem.We notice that the partially linear functions defined on [ ] 0,1 is actually the sublattice generated by the bi-set of functions { } We notice that in this case, the span of this bi-set is a lattice-subspace of [ ] (see also [11]).
. As a latticesubspace, it actually has a positive basis with nodes ( [11], Pr. 2.2), hence the equivalent positive projection is defined as follows: , , , 0,1 are the nodes of the positive basis of M .
Example 14 A sequence of sublattices ( ) characterized by increasing non-terminal parts of the sequence , , , 0,1 .
 , this does not imply 0 x = if 0 x ≥ .For example, ( ) ( ) is a positive basis of n M itself.This is the case for 1 0 , , ,1

Order Versions for the Fundamental Theorems of Asset Pricing
In the proof of the two next Theorems we use the following: , where E is a Banach lattice and M is a positive sublattice of it, is a continuous operator.
Proof: Obvious, because every positive operator from a Banach lattice into to a locally solid Riesz space, is continuous.
Theorem 20 (Order 1st Fundamental Theorem of Asset Pricing) Let E be a Banach lattice and M be a sublattice of E .If M admits a strictly positive projection, then every strictly positive and continuous func- tional : f M →  , admits a strictly positive, continuous extension on E .Also, if E is a Banach lattice and M is a sublattice of E such that every strictly positive and continuous functional : f M →  , admits a strictly positive, continuous extension on E , then M admits a strictly positive projection.
Proof: The adjoint operator of the strictly positive projection : →  is a continuous, strictly positive functional of E .This is due to the duality: For the proof of the opposite, we have the following: We define the projection : . M P is a positive operator from a Banach lattice into a locally solid Riesz space.Hence it is continuous.By duality for some f strictly positive, continuous functional f of M , ( ) ( ) , , , .
Hence if we suppose that there is some But this leads to a contradiction.
Corollary 21 If E is a Banach lattice which has the Strictly Positive Martingale Property with respect to some filtration ( ) such that a a P E M = , then every strictly positive and continuous functional : a f M →  , admits a strictly positive, continuous extension on E .Corollary 22 Let E be a Banach lattice of financial positions and M be an incomplete market, such that ( ) , M f is a market model.If M admits a strictly positive projection, then for every price system : The existence of a strictly positive projection may be replaced by the Strictly Positive Martingale Property with respect to some filtration in the statement of the above Theorem.The term viable is the one established in the seminal work of D.M. Kreps (see [3], p. [18][19]. Theorem 23 (Order 2nd Fundamental Theorem of Asset Pricing) Let E be a Banach lattice and M be a dense sublattice of E .If M admits a strictly positive projection, then every strictly positive and continuous functional : f M →  , admits a unique strictly positive, continuous extension on E .Also, let E be a Banach lattice and M be a sublattice of E such that M admits a strictly positive projection.Moreover, every strictly positive and continuous functional : f M →  , admits a unique strictly positive, continuous extension on E .Then M is dense in E .
Proof: Since M is a dense sublattice of E, the adjoint (linear by the duality ( ) ( ) Corollary 25 Let E be a Banach lattice of financial positions and M be a complete market, such that ( ) M f is a market model.If M admits a strictly positive projection, then for every price system The term viable is the one established in the seminal work of D.M. Kreps (see [3], pp.[18][19].We may notice that our Theorem does not make any reference to the No -Free Lunch Condition, but it simply extends the No-Arbitrage Property all over the space E .Theorem 23 is the analog of the usual 2nd FTAP, which implies that the (local) Equivalent Martingale Measures' set of a complete market is a singleton, while under this class of market spaces the uniqueness of the (strictly positive) extension of a price system all over the space of financial positions is achieved under no presence of the No-Free Lunch Condition, too.
Let us see some Examples which confirm the connection of the above Theorems to well-known models of Mathematical Finance.

Example 26 Let ( )
, ,µ Ω  be a probability space endowed with an m -dimensional Brownian motion the filtration that this Brownian motion generates, i.e., ( ) . We assume a financial market consisting of 1 + n assets whose prices are modelled by an  -adapted, ( ) where: where ( ) sents the price of a riskless asset (where ( ) r ⋅ is the interest rate process which is supposed to have bounded values), while the i -th component X ⋅ , represents the evolution of the price of the i -th asset (stock).The price of the riskless asset may be used as numeraire.Suppose that Z ∈ is a stochastic exponential, then as it is well-known, the following relation holds: where Q is the probability measure defined on T  as follows: ( )

according to the
Girsanov-Cameron-Martin Theorem.Taking mean values over µ we have: which in terms of evaluation maps' values is interpreted as follows: The equivalent Riesz pairs are: where the strictly positive projection : , , , , and its strictly positive extension , , . This Example gives also a Hilbert space taste, due to the presence of 2  L -spaces, see also [12].Example 27 holds for the unique possible change of measure Q , if the market is complete for example in the Black-Scholes model and this arises indeprendently from the unique solution of the market-price-of-risk equation.
Finally, we may revisit the Example constructed in [4], in order to quote it.

Example 28
The actual form of the elements of the subspace M of ∞  is described by the following strictly positive projection: ( ) ( ) , while according to Theorem 3, a strictly positive extension of π all over ∞  exists, through duality relation ( ) ( ) ( ) ,

The Finite-State, One Period-Model Case
We will show how the above Theorems 3, 23 are applied in finite -state space models.
Let us consider the two-date market model in which the number of states of the world is denotes by S , while the time-periods are denoted by 0 and 1, respectively.We also consider an incomplete market of primitive assets whose time-period −1 payoffs are the positive, linearly independent vectors 1  , whose span is denoted by X .We suppose that X contains the riskless asset 1 , while J S < , which implies standard incompleteness.We also assume a time-period 0 , no-arbitrage price , , , J q q q q =  for the primitive assets.As it is well-known from ( [7], p. 4), ( ) generated by X .We also remind of the following Projection Basis Theorem for sublattices of S  , which arises from both ( [13], Th. 3.7), ( [14], Th. 9).
Theorem 29 Let X be a J -dimensional subspace of S  with J S < generated by the positive elements , , , J y y y  in which the riskless bond 1 is a marketed asset ( ) . Suppose that the range ( ) , , , . Suppose that the first J vectors of this set are linearly independent.If we suppose that the vectors , 1, 2, , (which are the vectors indicated by ([13], Th. 3.7), then, , , , 2) 2 , 1, 2, , .
then the vectors i b ~ defined by: ( ) , , , , , , where A is the J J × matrix whose columns are the vectors , 1, 2, ,  are a basis of X called pro- jection basis.This basis has the property: The J first coordinates of an element X x ∈ in the positive basis of ( ) S X coincide with the coordinates of the expansion of x in the basis { } , 1, 2, , . Also, according to what is mentioned in [7] about the completion of an incomplete market X by options and by following the notation we introduced, ( ) , , , y y y µ  is a maximal set of linearly independent, positive vectors of ( ) are portfolios of call and put options written on elements of X , especially since X ∈ 1 .The dimension equation which holds in the case of the no-arbitrage price q , is: where W denotes the subspace of generated by the columns of the payoff matrix ( ) , W q X of the primitive securities, while W ⊥ denotes the orthogonal subspace of it.Due to the characterization of the absence of arbitrage in the primitive asset market (see [15], Th. 9.2), there is at least one if by X we also denote the J S × matrix whose columns are the vectors 1 2 , , , J x x x  . The last relation arises from ( ) of it.We also have the following: implies a no-arbitrage price ( ) of the portfolio µ λ ∈  or else the price of the asset  ( )

to be equal to the price of the same asset under
The above vector satisfies the following equalities: The definition of the vector ( ) q π allows us to prove that it is a no-arbitrage price in the subspace generated by the vectors 1 2 , , , y y y µ  which is the completion by options ( ) , this means that: ∑ in this case, which is equal to the valuation of the portfolio ( ) , , , J λ λ λ  of the primitive assets under q .We remind that µ  is the space of the financial positions, since ( ) F X is actually equal to this space according to ( [7], Pr. 6).
Theorem 31 (First Order Finite Fundamental Theorem of Asset Pricing) For any subspace [ ]  are linearly independent, every strictly positive linear functional of X has a strictly positive extension on ( )

Proof: Every strictly positive functional :
f X →  defines a no -arbitrage price ( ) q f on X as follows:  .According to Theorem 30, 1 p for some 1 , supp , 1, 2, ,


. Proof: In the last part of [7], a brief proof was given about the fact that resolving markets have the property ( ) It is also well-known that resolving matrices are in general position, namely the complement of the set of them is a null-set in the vector space of the matrices S J × , whose entries are real numbers.Hence the super-set of all the S J × -matrices (markets), such that [ ] are linearly independent and they have the property that ( ) 1 S F X =  are also in general position.

Theorem 33 (Second Order Finite Fundamental Theorem of Asset Pricing) For almost any subspace
[ ]  , where ( )  are linearly independent, every strictly positive linear functional of X has a unique strictly positive extension on ( ) Proof: Every strictly positive functional : f X →  defines a no-arbitrage price ( ) q f on X as follows:

The Finite Multi-Period Model Case
Let us see what happens in the multi-period framework.We consider the event -tree model as it is presented in [15], according to which there is a finite time -horizon in the sense that for any the event-tree corresponding to the family of partitions  .Every event-tree  is a model of information re- vealing along the time-periods of  .We also consider J assets (financial contracts) whose payoff vectors are 1 2 , , , J V V V ∈    and if we denote by n the physical number which is equal to the cardinality of the nodes of the event-tree  , these are actually vectors of n  .We also suppose that the price vectors of the assets are 1 2 , , , J q q q  , where ( ) and the set T  denotes the set of nodes of the event-tree corresponding to the time-period T .If we suppose that these price vectors do not provide arbitrage opportunities in the market of the assets 1, 2, , j J =  , then since the market is incomplete there is at least one node-price vector , W q V is the payoff matrix of this market as it is indicated in ( [15], Ch. 4).In order to simplify things, we may suppose that ( ) We also suppose that one of the assets of the market is riskless, or else that for any ξ ∈ which corresponds to the same time-period, its payoff is the same.Also, this asset's initial price ( ) 0 q ξ is equal to 1.The submatrix ( ) ( ) ξ × -matrix whose rows are the vectors ( ) ( )

−
As a reference for these options we append to ( [16], Par.9.2).The market is (dynamically) complete if and only if it is one-period complete for any non-terminal node , and moreover there is a non-terminal node ξ − ∈ such that for the corresponding submatrix ( ) ( )  holds, we may add forward-start options of the form  to make it complete.In the same way we may talk about the completion by options of the span with respect to the asset ( ) 1 b ξ which may be denoted by ∈ .In a way similar to [7], the dimension of the completion It is obvious that we may reach a complete market if and only if ( ) ( ) A question which also arises in this case is how the new assets introduced in a submarket ( ) in order to reach ( ) µ ξ are priced.The answer is given in the next Theorem, being equivalent to Theorem 29.
Theorem 36 For any submarket ( ) , , , J q q q q =  is a no-arbitrage price vector for the assets is a price vector which assigns the price ( )( ) . Specifically, the price of the asset ( ) , , , , where ( ) ( ) . The above vector satisfies the following equalities: The definition of the vector ( )( ) q π ξ allows us to prove that it is a no-arbitrage price in the subspace gen- erated by the vectors , , , y y y µ ξ  which is the completion by options If for a portfolio ( ) . Also, from the Projection Basis Theorem, if , this means that . Hence ( )( ) in this case, which is equal to the valuation of the portfolio ( ) , , , J λ λ λ  of the primitive assets under ( ) q ξ .This concludes the proof.Theorem 37 (First Order Event-Tree Fundamental Theorem of Asset Pricing).For any submarket ( ) Proof: If ( ) ( ) , X ξ ξ − ∈ , then this implies a no-arbi- trage price ( ) 1 J f q ξ ∈  and since q is given, ( ) ( ) The extension of ( ) p ξ for some ( ) is a strictly positive extension of f on ( ) , where ( ) is the support of the vector j b of the positive basis of 7], Th. 6), since 1 π is unique.

General Cones Revisited
Let us consider a Banach space E of financial positions, partially ordered by a closed cone C , which is not a lattice cone.Such a cone is for example a Bishop-Phepls cone, see ( [17], pp.126-127), which is well-based and it has also interior points, hence it is not a lattice cone, according to ( [17], Th. 4.4.4).Of course, the set of strictly positive functionals of such a cone has not to be empty.This is the reason due to which the Lindelöf Property mentioned in [9] about the weak topology ( ) defined on a dual system , X Y is important.Of course, there are cones which do not admit continuous strictly positive functionals.Such a cone is the positive cone of an ( ),1 space, where Γ is uncountable.Also, in this section, the definition of (in)completeness are altered.Definition 39 If M is a infinite-dimensional subspace of E ordered by the cone C, a market is an infinite- Then, the following versions of the Second and the First Fundamental Theorem of Asset Pricing are deduced, respectively.
Theorem 41 Let E be a Banach space with an unconditional basis.Then a non-lattice one exists, which makes E a complete market and every strictly positive functional of this cone admits a unique strictly positive extension.
Proof: As it is well-known from ( [18], Th. 4.2.22), the cone of the unconditional basis ( ) .Hence, P may be taken as a strictly positive projection, and consequently we may repeat the proof of Theorem 3.
In the proof of ([8], Th. 5.7) the strongly reflexive cone's construction relies exactly on the existence of an unconditional basis for the Banach space E. Then we may understand that the crucial point for the above Theorems is the existence of a basic sequence for the Banach space E. We may remind the seminal work by Bessaga-Pelczynski [19] essentials on this topic.

Appendix
In this Section, we give some essential notions and results from the theory of partially ordered linear spaces which are used in this paper.For these notions and definitions, see ( [17] Suppose that C is a wedge of where E is a normed linear space, is called uniformly monotonic functional of C if there is some real number 0 a > such that ( ) In case where a uniformly monotonic func-  ([20], Ch. 8), and [11], respectively.

1 , 1 F
obtain the last relation.As it is implied in[7] ( ) X is determined by the positive basis { } by the Projection Basis Theorem.Proof: Consider the vector ( )

Definition 35
indicating the payoffs and the ex-payoff price of the J primitive securities at the node ξ ξ + ′ ∈ .The cardinality of , is denoted by ( ) b ξ .The market of the securities is complete or as it is usually said the securities' markets are dynamically complete, if every contingent claim  .In order to understand the next, we remind of the following,Definition 34 The forward -start call option written on a contingent claim The forward -start put option written on a contingent claim = are the vectors indicated by the Projection Basis Theorem.Proof: Consider the vector ( )( [8]ach lattice under an equivalent norm.According to ([8], Th. n P n∈  are continuous, according to ([18], Cor.4.2.26) the operator , Ch. 1, Ch. 2, Ch. 3).linear structure of E ), is called partially ordered linear space.The binary relation ≥ in this case is a partial ordering on E .The set E′ denotes the linear space of all linear functionals of E , called algebraic dual while * E is the norm dual of E , in case where E is a normed linear space.
, then P is a cone.
is the dual wedge of C in E * .Also, .It can be easily proved that if C is a closed wedge of a reflexive space, then 00 C C = .If C is a wedge of E * , then the set , where : E E * * → denotes the natural embedding map from E to the second dual space E * * of E .Note that if for two wedges where f is a strictly positive functional of C is the base of C defined by f .fB is bounded if and only if f is uniformly monotonic.If B is a bounded base of C such that 0 B ∉ then C is called well-based.If C is well-based, then a bounded base of C defined by a If C is generating, then 0C is a cone of E * in case where E is a normed linear space.Also, f E * ∈ is a uniformly monotonic functional of C if and only if is called order-unit of E. If E is a normed linear space, then if every interior point of C is an order-unit of E. If E is moreover a Banach space and C is closed, then every order-unit of E is an interior point of C. The partially ordered vector space E is a vector lattice if for any with respect to the partial ordering defined by P exist in E. In this case , then E is called normed lattice.If a normed lattice is a Banach space, then it is called Banach lattice.A Banach lattice E whose norm has the property A set S in a vector lattice E is called solid if y x ≤ and S x ∈ implies S y ∈ .A solid vector subspace of a vector lattice is called ideal.An ideal I is a sublattice of E, i.e., a subspace of E such that ∈ .If D is also an ideal, then D is called band.A Banach lattice has order continuous norm, if for any net { } Kantorovich-Banach space.If S is a subset of a vector lattice E, then its disjoint complement is the set If for a vector lattice E a band B satisfies the property = ⊕ , then B is called projection band.Finally, if E is a partially ordered Banach space whose positive cone is + E , if E has a Schauder basis ( ) ∈ , this basis is called positive basis if and only if * * E 0 C .If E is partially ordered by C, then any set of the form e  → .A set D in E is order closed if { } a x ↓ holds.A Banach lattice E which is a band in its second dual (in the sense of norm topology) is called d E B B