Research Article | Open Access

Christos E. Kountzakis, "The Completion of Real-Asset Markets by Options", *International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 139690, 20 pages, 2010. https://doi.org/10.1155/2010/139690

# The Completion of Real-Asset Markets by Options

**Academic Editor:**S. M. Gusein-Zade

#### Abstract

We combine the theory of finite-dimensional lattice subspaces and the theory of regular values for maps between smooth manifolds in order to study the completion of real asset markets by options. The strike asset of the options is supposed to be a nominal asset. The main result of the paper is like in the case of the completion of a nominal asset market by options that if the strike asset of the options is the riskless asset, then the completion of a real asset market is generically equal to .

#### 1. Introduction

The investigation of the contingent claims' hedging possibilities arising from a certain market is an old question of study in mathematical finance. It is well known that in his seminal work, Ross in [1] proved that a primitive market of nominal assets (assets whose payoff is expressed in the unique-numeraire good) can become complete by implementing call and put options on the elements of the primitive asset span if there is a portfolio whose payoff is a contingent claim which separates the states of the world (called *efficient fund*). Later on, Arditti and John in [2] proved that if an efficient fund exists, then almost every portfolio payoff in the sense of the Lebesgue measure is also an efficient fund. John in [3] introduced the notion of the *maximally efficient fund* which is a portfolio payoff which actually separates the subsets of states which can be separated by the specific asset span. John in [3] also indicated that the span of all the call and put options written on the elements of a specific asset span is the span of the characteristic functions of these subsets of states. The span of the call and put options on a maximally efficient fund is the span of the options written on the asset span and almost every portfolio payoff is a maximally efficient fund. After that in [4, 5] and recently in [6] the problem of completing a span of primitives with options when the state space is infinite was studied. The completion of a market especially by options (and not by assets in general) is related to the Pareto Optimality properties of the competitive equilibria in incomplete markets. A detailed study of this topic in the multiperiod model is contained in [7]. Kountzakis and Polyrakis in [8] proved that if is an asset span of nominal assets and is a span of strike vectors—being also a span of nominal assets—for the call and put options written on the elements on , then the span of the call and put options written on with strike vectors taken from is the sublattice of generated by the span of (Theorem 3), called *completion of ** by options with respect to * and denoted by . By examining the case where is one-dimensional, namely, and has an expansion with positive coefficients with respect to the *positive basis* of the completion (Proposition 17), the authors generalize the notion of the efficient fund, by defining the -*efficient fund* (Definition 18) and proving that almost every payoff in is an -efficient fund (Theorem 21). The results of [8] rely on the theory of finite-dimensional lattice-subspaces in function spaces, initially developed in [9, 10].

In this paper, we consider the numeraire payoff vectors of the real asset structure as primitive securities of the asset span. We actually wonder whether such generic results about taking a complete market through implementing call and put options written on the elements on an existing real-asset span with respect to a nominal asset. We actually wonder whether the span of the options written on the numeraire-good (and thus spot price-affected) payoffs of real assets with respect to a strike asset whose payoff is also expressed in terms of the numeraire good is the complete market.

#### 2. Nominal Asset-Markets and Their Completion by Options

We give some essential notions about markets of assets whose payoffs are expressed in a single (numeraire) good. This is because the asset spans we are going to study are formulated via real assets (namely, by assets whose payoffs are expressed in several goods) but they are actually single-good asset spans, being the value asset spans of them.

Suppose that there are two periods of economic activity and states of the world. At time-period , there is uncertainty about the true state of the world, while at time-period this state is revealed. Suppose that there are *primitive assets* in the market which are *nonredundant*, namely, their payoff vectors at time period are linearly independent. A *portfolio* in this market is a vector of in which , , denotes the number of units invested to the asset . If , then the investment to units of the asset denotes a *long position* to these units. If , then the investment to units of the asset denotes a *short position* to units of the asset . The payoff of a portfolio , if the payoff vectors are expressed in terms of the numeraire as well, is the vector . The range of the operator is called *asset span* of the market consisted by . A *contingent claim* is any liability , while a *derivative* is a contingent claim whose payoff is connected through a functional form to some portfolio payoff for a market of primitive assets. If for some contingent claim there is some portfolio such that , then the contingent claim is called *replicated* or *hedged*. Any portfolio such that is called a *replicating portfolio* or *hedging portfolio* of . The most classical examples of derivatives are *options*, which include *call options* and *put options*. If is a replicated contingent claim, then the call option written on with *exercise price * is understood as a derivative claim written on it and the same holds also for the put option written on with exercise price . The corresponding call option is the claim , while the put option is . The positive part of a vector is defined by , where if , the *lattice operations* on are defined in the following way:
while , for any . The *negative part * of is defined as follows: . The lattice identity implies the well-known *put-call parity*:
As we see, these lattice operations are established via the usual (component-wise) partial ordering on and they make it a *vector lattice*. The meaning of the call option as a derivative asset may be the following: it expresses the payoff of the buyer of the claim at time-period 1, if she buys the claim at the price independently from the state of the world and if we suppose that the state-payoffs , of are the possible-tradable prices of at the states of the world. In the same way, the meaning of the put option as a derivative asset may be the following: it expresses the payoff of the seller of the claim at time-period 1 if she sells (short) at the price independently from the state of the world and if we suppose that the state-payoffs , , of are the possible-tradable prices of at the states of the world. The notion of the call and put option written on some asset can be generalized and the *strike vector* can be risky and different from . If we denote such a vector by , the corresponding call option written on with exercise price with respect to is the contingent claim whose payoff vector is . In the same way, the corresponding put option is . The last call option is denoted by , while the put option is denoted by . The call option and the put option are called *nontrivial* if , , which means that for both of these vectors all of their components are positive and at least one of them is nonzero. In such a case, the exercise price is called *nontrivial exercise price* for . Finally, we say that two states with are *separated by the contingent claim (asset) * if . Two states with are *separated by the asset span * if they are separated by some , where is the range of the payoff operator .

##### 2.1. The Completion of a Nominal-Asset Market by Options

For the sake of completeness of the present paper, we are going to present in brevity the main results from [8]. We suppose that the payoff vectors of the primitive assets are and the strike vector is . The *completion by options* of the asset span with respect to is defined as follows: and is the subspace of generated by . For , and is the subspace of generated by . The *completion by options* of with respect to is the subspace of . Consider the set . Any maximal set consisted by linearly independent vectors of is called a *basic set* of the asset span with respect to . A basic set is not necessarily unique, but the cardinality of all the basic sets of with respect to is the same and it is denoted by . In Theorem 3 of [8], it is proved that , where is the *sublattice* of generated by the subspace . The sublattice generated by a nonempty set of vectors of is the set of finite suprema of elements of the subspace generated by it, but cannot be determined by using this method. By Definition 10 and Theorem 11 in [8], . Hence the problem of the determination of the completion by options of with respect to is equivalent to the determination of the sublattice of . If is a basic set of the asset span with respect to the strike vector , then the *basic function * of is very important in the determination of . was defined in [9] and in the case where the set of states of the world is finite is defined as follows:
for each with , where . This function takes values on the simplex of . As it is proved in [8], by using Theorem 3.6 in [10], the cardinality of the range of for a basic set of the asset span with respect to is the dimension of ([8], Theorems 8 and 9). Theorem 9 in [8] is an application in Euclidean spaces of the Theorem 3.7 in [10] about the determination of the sublattice . The last Theorem relies on the specification of a *positive basis* for . The *usual (component-wise) partial ordering* on the space of the continuous functions is continuous}, where is a compact, Hausdorff topological space is defined through the positive cone for any as follows: for any , or else . endowed with this partial ordering is a *vector lattice*, because for any the pointwise supremum and the pointwise infimum exist in . If is a subspace of , the *induced partial ordering* on the elements of is implied by the cone . endowed with this partial ordering is an *ordered subspace* of . If for any , , exist in , then is called a *lattice-subspace* of . It is true that in this case, for any ,
If for any , then , is a *sublattice* of and this is also the general definition of a sublattice for an ordered subspace of a vector lattice. A sublattice is a lattice-subspace but the converse is not always true. If is finite-dimensional and its dimension is equal to , then a *positive basis * is a basis of such that for any . If the ordered subspace has a positive basis , then for any the coefficients of its expansion in terms of this basis are all positive real numbers, namely, holds for any . Also, if has a positive basis, then if and , , for any . This implies
hence is a lattice-subspace of . About the relation between positive bases and finite-dimensional lattice-subspaces, the following theorems hold.

*( Choquet-Kendall)*

A finite-dimensional ordered vector spacewhose positive coneis closed and generatingis a vector lattice if and only ifhas a positive basis.

*( I. A. Polyrakis)*

A finite-dimensional ordered vector spaceis a vector lattice if and only ifhas a positive basis.

While a subspace may have several bases, if it has a positive basis this basis is unique under the multiplication of its elements by positive numbers. If is a positive basis of , then is a positive basis of too, where . If , then . The statement of Theorem 3.7 in [10] is the following (supposing that are positive and linearly independent vectors of ).

*Theorem 3.7 ( I. A. Polyrakis)*

Letbe the sublattice ofgenerated by

*and let*

*.*Then the statements (i) and (ii)are equivalent.(i). (ii). If the statement (ii) is true, then is constructed as follows. (a)Enumerate so that its first vectors can be linear independent. Denote again by , the new enumeration and let , . (b)Define the functions , , where is the characteristic function of . (). (c).

As it is indicated in Propositions 6 and 7 in [8], has a positive basis which is a *partition of the unit* and the vectors of it have disjoint supports. Let us remember their statements.

*(Proposition 6, [8])*

Suppose that is a sublattice of. If the constant vector is an element of , thenhas a positive basiswhich is a partition of the unit*, *that is, and for each vector we have for each .

*(Proposition 7, [8])*

Suppose that is a sublattice of with a positive basis
Then for each the vector has minimal support in , that is, there is no , such that .

We show that .

According to [8], Definition 18, a vector is an -*efficient fund* if is the linear subspace of which is generated by the set of nontrivial call options and the set of nontrivial put options of . Also, the statements of [8], Theorem 19, Proposition 20, Theorem 21 are the following.

*(Theorem 19, [8])*

Suppose thatis a positive basis of*, **, *andfor each*. *Then the vector of is an*-*efficient fund if and only iffor each*. *

*(Proposition 20, [8])*

Each nonefficient subspace ofis a proper sublattice of*. *

*(Theorem 21, [8])*

Suppose thatis a positive basis ofand thatwithfor each*. *Then(i)the nonempty set , whereis the set of nonefficient subspaces of , is the set of *-*efficient funds of and the Lebesgue measure of is supported on ; (ii)is the subspace of generated by the set of the call options written on the elements of. If is the subspace of generated by the set of call options written on the elements of .

Theorem 21 indicates that the completion is attained at the first step of the inductive procedure described above.

#### 3. Real Assets and Their Importance

Suppose that there are two periods of economic activity and states of the world. At time-period , there is uncertainty about the true state of the world, while at time-period this state is revealed. We also consider goods being consumed at time-period 1, independently from the state of the world faced by the individuals. We also consider a numeraire good in terms of which all the values are expressed. The spot prices of these goods consumed at time-period 1 and if the state is faced by the individuals are represented by a vector , each component of which denotes the price of one unit of every such good in terms of the numeraire. We also suppose that there exist assets, whose payoffs are expressed initially in terms of the goods consumed in the economy. We suppose that , a condition which is directly connected to the incompleteness of the spot markets for the numeraire good at the time-period 1. The payoff of the -asset at time-period 1 if the state is faced by the individuals is a “consumption” goods’ bundle for all the assets and for all the states of the world , whether the state occurs.

The value of the payoff of the -asset in terms of the numeraire is The value payoff vector under prices of the -asset at time-period 1 is The value payoff matrix under prices of the assets is denoted by and it is actually the matrix whose columns are the vectors , . The prices for the goods are expressed in terms of the numeraire, or else the vectors are normalized and for this reason taken to vary on the interior of the simplex in terms of the induced topology of the Euclidean space , which is an -dimensional manifold (the interior of the simplex is consisted by those vectors which have nonzero components).

Though the real assets’ payoffs are initially expressed in goods of consumption, we may give a financial interpretation to them. The -real asset can be viewed as an asset whose payoff is expressed in different currencies in every state of the world. can be viewed as the payoff vector of this asset at time-period 1 if the state is the true state of the world, where in this case denotes the units of the -currency which delivers to the owner of one unit of this asset, where . The prices are related to the exchange rates of another (“numeraire”) currency with respect to these payoff currencies. For example, denotes the amount of the numeraire received by selling one unit of the -currency at time-period 1 and if the state becomes true, where . These “spot” prices allow for the expression which is the payoff vector of every such “multicurrency” asset in terms of the numeraire currency across the states of the world.

For a simple definition of the real asset structures, see in [11]. The examples of real assets mentioned in [11] are the *futures contracts* and the *equity contracts*.

A future contract for the good is the financial contract which promises its owner one unit of the good independently of which is the state of the world that is going to be true at the time-period 1. The numeraire payoff of this contract at the time-period 1 is equal to , . The -payoff matrix of this contract with respect to the consumption goods is a matrix which has all its columns equal to zero except the column which corresponds to the -good, whose entries are all equal to 1. The numeraire payoff vector in this case has the form we described.

An equity contract is connected to a stochastic production plan of some firm, or else it is connected to the decision of production made under the state which is true. If we suppose that the goods enter the production of a firm, then a stochastic production plan is a vector where every is a production plan in a technology set . The payoff vector of an equity contract relies on the vector , whose components denote the profits earned by the firm at any state of the world under the relevant spot prices and the selection of the production plan .

The *corporate bonds* and the earnings of the shareholders of a firm are straightly connected to the way we define a call option on the numeraire payoff of a real asset with respect to a nominal asset. For example, the shareholders’ payoff of a firm whose total nominal value of its bond is and its equity payoff vector written on the production value vector , as we mentioned before, is the call option . This is the payoff vector of the *equity contract* of the firm.

Magill and Shafer in [12] study the completeness properties of a real asset market structure under the light of the rational expectations equilibrium studied in [13]. In their paper, the authors indicate that (Proposition 2) if the number of real assets is at least as great as the number of states of the world, then generically the asset span is equivalent to the complete numeraire spot market. However, in this paper we refer to the case where a real asset market structure is not equivalent to a complete spot market for the numeraire. This is assured for example, by the condition that we pose on the number of the real assets. Hence we wonder whether the span can be completed by implementing call and put options written on elements of this asset span with respect to some nominal asset whose payoff is expressed in terms of the numeraire good across the states of the world. This nominal asset is denoted by and it is supposed to have positive payoffs in any state of the world. The nominal asset may be the riskless bond, but we may select to be some risky asset. We define the completion of the asset span by options with respect to the strike asset in a way which is the same to the one described to the Definition 2 in [8]. We first prove that generically both in the prices and in the payoffs , , of the assets, the numeraire payoffs , , are nonredundant. For this goal, we use the Preimage Theorem for smooth maps between manifolds. We also prove that the set of prices and consumption-good payoffs , , of the real assets, such that the numeraire payoff vectors separate the states of the world is generic in . This implies that the completion of the market by options is the whole , because in this case the dimension of the sublattice of generated by , is the number of the different values of the basic function of the set of vectors , according to what is proved in [10], Theorem 3.7. We actually prove this result for the case where . But in this case, generically we have that this number of different values is exactly equal to , hence the completion of the asset span by options with respect to the nominal asset is the whole space . This is the main result of the present paper, which is equivalent to the one which holds for the markets of nominal assets.

#### 4. The Completion of a Real Assets' Span with Respect to a Nominal Asset

The call option written on the real asset , , with respect to the nominal asset under spot prices and exercise price is the derivative on the numeraire payoff of whose payoff vector is This indicates that the payoff of this option at the state is expressed in units of the numeraire. We denote this option by . The equivalent put option is defined in a similar way, namely, it is the claim and it is denoted by .

The completion by options of the asset span with respect to the nominal asset is determined as follows: and for any natural number In the above definition .

We denote by the numeraire-asset span of the real asset structure consisted by the assets , , if spot prices are equal to . We also use in order to denote the matrix whose columns are the vectors , .

*Definition 4.1. *The completion by options ofwith respect to is the following subspace of :
of .

By Theorem 3 in [8] what is proved is that is the sublattice of generated by the elements of the span generated by the vectors of .

#### 5. On the Generic Determination of the Completion

In the rest of the paper, we use two assumptions. (A)There is not any zero element in the matrix whose columns are the vectors , or else for any . (B)No free-goods are available, that is, .

We also suppose in the rest of the paper that . These conditions about both the numbers of the assets and the number of the states are crucial for the validity of the results of the present paper, since they are related to the applicability of the theorems of Differential Topology mentioned in the next paragraphs. The matrix can be also identified with a vector of . We will use by both ways.

Supposing that , our aim is to determine the *basic set of the asset span * in the sense of Definition 10 and Theorem 11 in [8], generically in the prices and the payoffs of the real assets. We remind that a basic set of the asset span whose completion is taken with respect to is a maximal subset of linearly independent vectors among the set of positive and negative parts of the vectors . But these vectors are all positive, hence what we have to check is whether the variation of and the payoffs , , allow for these vectors to be linearly independent except negligible sets.

We show that the set is a manifold itself. We are going to provide some notions about manifolds in Euclidean spaces contained in many papers and books, such as in the books [14–16] and in the paper [17, pages 52-53].

The definition of a smooth manifold contained in [15], for example, is the following: * A subset ** is called a smooth manifold (of class **) of dimension ** if each ** has a neighborhood ** that is diffeomorphic to an open subset of the Euclidean space **. *

Suppose that is an -dimensional smooth manifold. A subset of is a *null set* in if for any there is a chart of around such that has Lebesgue measure zero in (). We show that a *chart* of around is a triple in which is an open subset of , is a subset of containing , and is a -diffeomorphism. Every such subset of is called an *open neighborhood* of in .

A subset of is a *set of full measure* in if is a null set in .

The above definitions are taken from [16, page 149].

A subset of full measure in is called *generic*.

If is a generic subset of and is a generic subset of , then is also a generic subset of . The implication holds for a finite number of such inclusions.

If a particular property depends on the elements of the manifold and this property is true for any element in a generic set in , we say that this property holds *generically*, or *almost everywhere* or *almost always*.

A smooth function between the smooth manifolds , is *regular* at a point if the derivative of has full rank at the point , or else if the differential is a surjection. In this case, is called a *regular point* of .

A smooth function between the smooth manifolds , is *critical* at a point if is not regular at . In this case, is called a *critical point* of .

A smooth function between the smooth manifolds is *transversal* at a point if any is a regular point of . An alternative name for such a is *regular value* of .

A which belongs to the range of values of a smooth function between the smooth manifolds is called a *critical value* of if it is not a regular one.

The above definitions are taken form [16, pages 79-80], and [17, pages 52-53].

Also, it is easy to see that an -dimensional manifold in a Euclidean space is a set of -dimensional Lebesgue measure zero. This arises especially by Proposition 11 in Chapter 6 in [16].

We mention three well-known theorems of the Differential Topology which we are going to use in the following.

*(Morse-Sard's Theorem)*

Ifis a*-*class map between the manifolds, with, , and, then the set of critical values ofis a null set in*. *

*(Preimage Theorem)*

Ifis a smooth map between the smooth manifolds, of dimensions, , respectively, whileis a regular value of, then the setis either empty (in the case where) or a smooth manifold of dimension*. *

*(Transversality Theorem)*

Letbe a*-*map into, where the*-*smooth manifolds, , are of dimensions, , , respectively. Then ifandis transversal to, then there exists a set of full measuresuch that for anythe map, , is also transversal to*. *

Finally, the statements of the three above theorems are contained in [16, pages 150, 84, 151], respectively.

Proposition 5.1. *For the generic element , the vectors are linearly independent.*

*Proof. *If is the matrix whose columns are the elements of the set
we have to show that for a set of full measure in , we have . We define , where and is the matrix which arises by selecting lines-states among the lines of the matrix which correspond to the combination (a combination is a selection of objects out of objects, without having their order in mind). is transversal to 0, namely, 0 is a regular value of . Since this is true by Lemma 5.2, then is either empty or a submanifold of of dimension , namely, a null set of . Repeating this for all combinations , we take that the set of pairs in for which every submatrix of is singular is actually the set , being a null set of , where denotes the set of -combinations of the objects. Namely, the generic matrix is of full rank.

Lemma 5.2. *The map , where , is transversal to 0 for any combination of objects out of .*

*Proof. *Suppose that is nonempty. Suppose that is a selection of states out of . The image of of at is the determinant of the matrix whose -row is
Then we have
where is the set of permutations over symbols, is any permutation in , and is the sign of the permutation . Also, is the spot price of the -good corresponding to the state , where and . In a similar way, is the amount of the consumption good that the owner of one unit of the asset is going to receive at the time-period 1 if the state occurs.

In order to show that is transversal to 0, we have to prove that for any we have . To show that, we have to verify that for any at least one partial derivative
is nonzero. But this is true, since and all of the products
are nonzero.

By the fact that and by repeating the previous proof, we may take the following.

Corollary 5.3. *The map , where , is transversal to 0 for any combination of objects out of , for all .*

By using the Transversality theorem, we can prove almost the same thing.

Corollary 5.4. *The map , where , is transversal to 0 for any combination of objects out of , where , for every which lies in a set of full measure in .*

*Proof. *From the previously proved Lemma 5.2, is transversal to zero for any combination and for any . Note that , , and . Also note that is a -map, hence we may suppose that it is also a -map with . Hence the projection , where for a specific in a set of full measure , is transversal to zero.

By Transversality theorem, we can also prove the following.

Proposition 5.5. *For a set of full measure and for any asset structure , the set of prices for which the vectors are linearly independent is generic in .*

*Proof. *Since from Lemma 5.2, is transversal to zero for any combination and for any and by Transversality theorem, there is a set of full measure of such that for any fixed in this set, the projection is transversal to zero for each . Hence, we may apply the Preimage theorem for whenever lies in . Then we get that is a smooth submanifold of of dimension . Hence, is a null set in , for any combination . Consider the set of full measure in . For any and for any , we have that is a null set in . Given an , the set of prices in for which the vectors are linearly dependent is the intersection , being a null set in .

For any , we define the *basic function * (see in [9], page 2797) of the vectors
where
for all the states such that . Note that is the simplex of the positive cone of .

Note that under the assumptions (A) and (B), is well defined for all .

Proposition 5.6. *For the generic element , the vectors , , and are linearly independent and they separate the states of the world.*

*Proof. *The set of for which there is at least one pair of disjoint states of the world such that for some asset is denoted by . This set of is actually the set , where is the map and is the complement of in . Let us prove that is transversal to 0. We have to prove that for all , . Note that for any such , we have , where denotes any of the consumption goods. Thus, by applying the Preimage Theorem, is a submanifold of of dimension , hence a null set. Namely, the union is a set of full measure in . Also, the set of in for which are linearly independent in is a set of full measure as it is shown by Lemma 5.2, since this set is the complement of in , being a null set (where denotes the set of -combinations of the objects). Hence the set of such that are linearly independent and every pair of different states is separated by some , , is the set
being the intersection of two sets of full measure, hence a set of full measure in .

Theorem 5.7. *For the generic element , the completion by options of the span with respect to the riskless bond is .*

*Proof. *Consider the generic subset of such that the states of the world are separated by the vectors of , while these vectors are linearly independent. This set is indicated in the end of Proposition 5.6. Consider to be the subset of those for which . Then the basic set of the market defined in [8] in the case where consists of the elements of ; see also Definition 10 and Theorem 11 in [8]. Hence the completion of the numeraire asset span with respect to the riskless bond is for the elements of . This is true because the values of the basic function of the elements of the basic set are disjoint and their cardinality is equal to , for any . Then, due to Theorem 14 in [8] for any , . Let us verify this fact. Suppose that there is some and some pair of states of the world with such that . Then the cardinality of is less than in this case and according to Theorem 9 in [8] and Theorem 3.7 in [10], this implies . But if , this implies
for this and for any . These equations imply
namely,
Then
for any , . Since we supposed that , there are real numbers , such that
Then, and . We get that
or else
Then
Hence, due to (5.9), , which indicates that
for any . This is a contradiction, since this implies that does not belong to the set such that the elements of separate the states of the world, since in this case the states of the pair where are not separated by the assets of the set . We were led to a contradiction because we supposed that there exist some such that for some pair of states , where . Then there is not any such in , which implies that for any the values of are disjoint; hence for the completion by options, the equality holds. Also, consider the complement of in , denoted by . In this case, the basic set of the market is consisted by the elements of ; see also Definition 10 and Theorem 11 in [8]. Hence the completion of the numeraire asset span with respect to the riskless bond is for the elements of . This is true because the values of the basic function of the elements of the basic set are disjoint and their cardinality is equal to , for any . Then, due to Theorem 14 in [8] for any , . Let us verify this fact. The basic function of the basic set of the market for an element has the form
for any . If we suppose that the dimension of the completion , which is equal to the cardinality of the range of , is less than for some , we will be led to a contradiction. If for some pair of states where , then
an equation which corresponds to the equality of the last components of these vectors. This equation implies . Also, for each , we have
But from equation , this also implies for any . This is a contradiction, since this means that the states are not separated by the vectors , namely, that does not belong to . Then there is not any such in , which implies that for any the values of are disjoint; hence the for the completion by options, the equality holds for any . Hence in any case for any the completion is equal to .

Note that some arguments of the last proof are the same to the ones contained in Theorem 23 of [8].

#### 6. Examples

First, we give an example of calculation for the completion of a real asset structure.

*Example 6.1. *Consider , , and , or else we suppose that there are four states of the world, two goods in which the payoffs of the assets are primarily expressed and we suppose that there are two assets and in the market. Suppose that the payoff-matrices of the two assets are
respectively, where the rows of these matrices correspond to the states and the columns correspond to the goods, or else the -element of the matrix , , for and denotes the units of the -good received at the time-period 1 by the owner of one unit of the -asset if the state occurs (). Also, suppose that the spot prices for the numeraire are where , , , and . We show that we take to be normalized with respect to the -norm in and each , , is consisted by the unitary prices of the two consumed goods at the time-period 1 whether the state occurs. Namely, the numeraire payoff-vectors for the two assets are
respectively. Note that the riskless bond does not belong to the span of . The basic function of these three vectors