# The Completion of Real-Asset Markets by Options

Authors: Christos E. Kountzakis
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## 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 toℝS.


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## Body

## 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 X is an asset span of nominal assets and U is a span of strike vectors—being also a span of nominal assets—for the call and put options written on the elements on X, then the span of the call and put options written on X with strike vectors taken from U is the sublattice of ℝS generated by the span Y of X∪U (Theorem 3), called completion ofXby options with respect toU and denoted by FU(X). By examining the case where U is one-dimensional, namely, U=[u] and u 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 Fu(X)-efficient fund (Definition 18) and proving that almost every payoff in Y is an Fu(X)-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 ofn 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 andS states of the world. At time-period t=0, there is uncertainty about the true state of the world, while at time-period t=1 this state is revealed. Suppose that there are nprimitive assets in the market which are nonredundant, namely, their payoff vectors x1,x2,…,xn∈ℝS at time period t=1 are linearly independent. A portfolio in this market is a vector θ=(θ1,θ2,…,θn) of ℝn in which θi, i=1,2,…,n, denotes the number of units invested to the asset i. If θi≥0, then the investment to θi units of the asset i denotes a long position to these units. If θi<0, then the investment to θi units of the asset i denotes a short position to -θi units of the asset i. The payoff of a portfolio θ, if the payoff vectors x1,x2,…,xn∈ℝS are expressed in terms of the numeraire as well, is the vector T(θ)=∑i=1nθixi. The range of the operator T:ℝn→ℝS is called asset span of the market consisted by x1,x2,…,xn. A contingent claim is any liability c∈ℝS, 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 c there is some portfolio θ such that T(θ)=c, then the contingent claim c is called replicated or hedged. Any portfolio θ∈ℝn such that T(θ)=c is called a replicating portfolio or hedging portfolio of c. The most classical examples of derivatives are options, which include call options and put options. If c∈ℝS is a replicated contingent claim, then the call option written on c with exercise pricea is understood as a derivative claim written on it and the same holds also for the put option written on c with exercise price a. The corresponding call option is the claim (c-a1)+, while the put option is (a1-c)+. The positive part x+ of a vector x∈ℝS is defined by x∨0, where if x,y∈ℝS, the lattice operations on ℝS are defined in the following way: (2.1)x∨y=sup⁡{x,y}=(x1∨y1,x2∨y2,…,xS∨yS),x∧y=inf⁡{x,y}=(x1∧y1,x2∧y2,…,xS∧yS),
while xi∨yi=max{xi,yi}, xi∧yi=min{xi,yi} for any i=1,2,…,n. The negative partx- of x∈ℝS is defined as follows: x-=(-x)∨0. The lattice identity x∨y+x∧y=x+y implies the well-known put-call parity: (2.2)(c-a1)++(a1-c)+=c-a1.
As we see, these lattice operations are established via the usual (component-wise) partial ordering on ℝS 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 c at time-period 1, if she buys the claim c at the price a independently from the state of the world and if we suppose that the state-payoffs c(s), s=1,2,…,S of c are the possible-tradable prices of c 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 c at time-period 1 if she sells c (short) at the price a independently from the state of the world and if we suppose that the state-payoffs c(s), s=1,2,…,S, of c are the possible-tradable prices of c at the states of the world. The notion of the call and put option written on some asset c can be generalized and the strike vector can be risky and different from 1. If we denote such a vector by u, the corresponding call option written on c with exercise price a with respect to u is the contingent claim whose payoff vector is (c-au)+. In the same way, the corresponding put option is (au-c)+. The last call option is denoted by cu(c,a), while the put option is denoted by pu(c,a). The call option cu(c,a) and the put option pu(c,a) are called nontrivial if cu(c,a)>0, pu(c,a)>0, 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 a is called nontrivial exercise price for c. Finally, we say that two states s1,s2∈{1,2,…,S} with s1≠s2 are separated by the contingent claim (asset)c if c(s1)≠c(s2). Two states s1,s2∈{1,2,…,S} with s1≠s2 are separated by the asset spanX if they are separated by some x∈X, where X is the range of the payoff operator T.


### 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 x1,x2,…,xn and the strike vector is u. The completion by options of the asset span X=[x1,x2,…,xn] with respect to u is defined as follows: O1={cu(x,a)∣x∈X,a∈ℝ} and X1 is the subspace of ℝS generated by O1. For n≥1, On={cu(x,a)∣x∈Xn-1,a∈ℝ} and Xn is the subspace of ℝS generated by On. The completion by options of X with respect to u is the subspace Fu(X)=⋃n=1∞Xn of ℝS. Consider the set 𝒜={x1+,x1-,x2+,x2-,…,u+,u-}. Any maximal set consisted by linearly independent vectors of 𝒜 is called a basic set of the asset span X=[x1,x2,…,xn] with respect to u. A basic set is not necessarily unique, but the cardinality of all the basic sets of X with respect to u is the same and it is denoted by r. In Theorem 3 of [8], it is proved that Fu(X)=S(Y), where S(Y) is the sublattice of ℝS generated by the subspace Y=[X∪{u}]. The sublattice S(B) generated by a nonempty set B of vectors of ℝS is the set of finite suprema of elements of the subspace [B] generated by it, but S(B) cannot be determined by using this method. By Definition 10 and Theorem 11 in [8], Fu(X)=S(𝒜). Hence the problem of the determination of the completion by options of X with respect to u is equivalent to the determination of the sublattice S(𝒜) of ℝS. If {y1,y2,…,yr} is a basic set of the asset span X with respect to the strike vector u, then the basic functionβ of y1,y2,…,yr is very important in the determination of Fu(X). β was defined in [9] and in the case where the set of states of the world is finite is defined as follows: (2.3)β(i)=(y1(i)y(i),y2(i)y(i),…,yr(i)y(i)),
for each i=1,2,…,S with y(i)>0, where y=y1+y2+⋯+yr. This function takes values on the simplex Δr-1={d∈ℝ+r∣∑j=1rdj=1} of ℝ+r. As it is proved in [8], by using Theorem 3.6 in [10], the cardinality of the range of β for a basic set of the asset span X with respect to u is the dimension of Fu(X) ([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 S(Y). The last Theorem relies on the specification of a positive basis for S(Y). The usual (component-wise) partial ordering on the space of the continuous functions C(Ω)={x∣x:Ω→ℝ,x is continuous}, where Ω is a compact, Hausdorff topological space is defined through the positive cone C+(Ω)={x∈C(Ω)∣x(t)≥0 for any t∈Ω} as follows: x≥y⇔x(t)≥y(t) for any t∈Ω, or else x≥y⇔x-y∈C+(Ω). C(Ω) endowed with this partial ordering is a vector lattice, because for any x,y∈C(Ω) the pointwise supremum x∨y and the pointwise infimum x∧y exist in C(Ω). If Z is a subspace of C(Ω), the induced partial ordering on the elements of Z is implied by the cone Z+=Z∩C+(Ω). Z endowed with this partial ordering is an ordered subspace of C(Ω). If for any x,y∈Z, supZ{x,y}, infZ{x,y} exist in Z, then Z is called a lattice-subspace of C(Ω). It is true that in this case, for any x,y∈Z, (2.4)sup⁡Z⁡{x,y}≥x∨y≥x∧y≥inf⁡Z⁡{x,y}.
If for any x,y∈Z, x∨y,x∧y∈Z then Z, is a sublattice of C(Ω) 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 Z is finite-dimensional and its dimension is equal to r, then a positive basis{b1,b2,…,br} is a basis of Z such that Z+={x=∑i=1rλibi∣λi≥0 for any i=1,2,…,r}. If the ordered subspace Z has a positive basis {b1,b2,…,br}, then for any x∈Z+ the coefficients {λ1,λ2,…,λr} of its expansion in terms of this basis are all positive real numbers, namely, λi∈ℝ+ holds for any i=1,2,…,r. Also, if Z has a positive basis, then if x,y∈Z and x=∑i=1rλibi, y=∑i=1rμibi, x≥y⇔λi≥μi for any i=1,2,…,r. This implies (2.5)inf⁡Z⁡{x,y}=∑i=1r(λi∧μi)bi,sup⁡Z⁡{x,y}=∑i=1r(λi∨μi)bi;
hence Z is a lattice-subspace of C(Ω). About the relation between positive bases and finite-dimensional lattice-subspaces, the following theorems hold.(Choquet-Kendall)
A finite-dimensional ordered vector spaceEwhose positive coneE+is closed and generating(E=E+-E+)is a vector lattice if and only ifEhas a positive basis.(I. A. Polyrakis)
A finite-dimensional ordered vector spaceEis a vector lattice if and only ifEhas 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{b1,b2,…,br} is a positive basis of Z, then {k1b1,k2b2,…,krbr} is a positive basis of Z too, where ki>0,i=1,2,…,r. If Ω={1,2,…,S}, then C(Ω)=ℝS. The statement of Theorem 3.7 in [10] is the following (supposing that x1,x2,…,xn are positive and linearly independent vectors of C(Ω)).Theorem 3.7 (I. A. Polyrakis)
LetZbe the sublattice ofC(Ω)generated byx1,x2,…,xnand letm∈ℕ.Then the statements (i) and (ii)are equivalent.(i)
dim(Z)=m.(ii)
R(β)={P1,P2,…,Pm}.If the statement (ii) is true, then Z is constructed as follows. (a)
EnumerateR(β)so that its first n vectors can be linear independent. Denote again by Pi, i=1,2,…,m,the new enumeration and let Ii=β-1(Pi), i=1,2,…,m.(b)
Define the functionsxn+k(t)=ak(t)z(t), t∈Ω, k=1,2,…,m-n,where akis the characteristic function of In+k.   (z=x1+x2+⋯+xn).(c)
Z=[x1,…,xn,xn+1,…,xm].As it is indicated in Propositions 6 and 7 in [8], F1(X) 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 thatZis a sublattice ofℝm. If the constant vector 1=(1,1,…,1)is an element of Z, thenZhas a positive basis{b1,b2,…,br}which is a partition of the unit,that is, 1=∑i=1rbi,and for each vector biwe have bi(j)=1for each j∈supp(bi).(Proposition 7, [8])
Suppose thatZis a sublattice of ℝmwith a positive basis
(2.6){b1,b2,…,br}.
Then for each ithe vector bihas minimal support in Z, that is, there is no x∈Z,x≠0, such that supp(x)⊂≠supp(bi).We show thatsupp(y)={i∈{1,2,…S}∣y(i)≠0}.According to [8], Definition 18, a vector e∈Fu(X) is an Fu(X)-efficient fund if Fu(X) is the linear subspace of ℝS which is generated by the set of nontrivial call options and the set of nontrivial put options of e. Also, the statements of [8], Theorem 19, Proposition 20, Theorem 21 are the following.(Theorem 19, [8])
Suppose that{b1,b2,…,bμ}is a positive basis ofFu(X),u=∑i=1μλibi,andλi>0for eachi.Then the vectore=∑i=1μκibi  of  Fu(X)is anFu(X)-efficient fund if and only ifκi/λi≠κj/λjfor eachi≠j.(Proposition 20, [8])
Each nonefficient subspace ofFu(X)is a proper sublattice ofFu(X).(Theorem 21, [8])
Suppose that{b1,b2,…,bμ}is a positive basis ofFu(X)and thatu=∑i=1μλibiwithλi>0for eachi.Then(i)
the nonempty setD=Y∖⋃i∈I(Y∩Zi), where{Zi∣i∈I}is the set of nonefficient subspaces of Fu(X), is the set of Fu(X)-efficient funds of Yand the Lebesgue measure of Y is supported on D;(ii)
Fu(X)is the subspace of ℝSgenerated by the set of the call options {cu(x,a)∣x∈Y,a∈ℝ}written on the elements ofY. If u∈X,Fu(X)is the subspace X1 of ℝSgenerated by the set of call options O1={cu(x,a)∣x∈X,a∈ℝ} written on the elements of X.Theorem 21 indicates that the completion is attained at the first step of the inductive procedure described above.


## 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 x1,x2,…,xn and the strike vector is u. The completion by options of the asset span X=[x1,x2,…,xn] with respect to u is defined as follows: O1={cu(x,a)∣x∈X,a∈ℝ} and X1 is the subspace of ℝS generated by O1. For n≥1, On={cu(x,a)∣x∈Xn-1,a∈ℝ} and Xn is the subspace of ℝS generated by On. The completion by options of X with respect to u is the subspace Fu(X)=⋃n=1∞Xn of ℝS. Consider the set 𝒜={x1+,x1-,x2+,x2-,…,u+,u-}. Any maximal set consisted by linearly independent vectors of 𝒜 is called a basic set of the asset span X=[x1,x2,…,xn] with respect to u. A basic set is not necessarily unique, but the cardinality of all the basic sets of X with respect to u is the same and it is denoted by r. In Theorem 3 of [8], it is proved that Fu(X)=S(Y), where S(Y) is the sublattice of ℝS generated by the subspace Y=[X∪{u}]. The sublattice S(B) generated by a nonempty set B of vectors of ℝS is the set of finite suprema of elements of the subspace [B] generated by it, but S(B) cannot be determined by using this method. By Definition 10 and Theorem 11 in [8], Fu(X)=S(𝒜). Hence the problem of the determination of the completion by options of X with respect to u is equivalent to the determination of the sublattice S(𝒜) of ℝS. If {y1,y2,…,yr} is a basic set of the asset span X with respect to the strike vector u, then the basic functionβ of y1,y2,…,yr is very important in the determination of Fu(X). β was defined in [9] and in the case where the set of states of the world is finite is defined as follows: (2.3)β(i)=(y1(i)y(i),y2(i)y(i),…,yr(i)y(i)),
for each i=1,2,…,S with y(i)>0, where y=y1+y2+⋯+yr. This function takes values on the simplex Δr-1={d∈ℝ+r∣∑j=1rdj=1} of ℝ+r. As it is proved in [8], by using Theorem 3.6 in [10], the cardinality of the range of β for a basic set of the asset span X with respect to u is the dimension of Fu(X) ([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 S(Y). The last Theorem relies on the specification of a positive basis for S(Y). The usual (component-wise) partial ordering on the space of the continuous functions C(Ω)={x∣x:Ω→ℝ,x is continuous}, where Ω is a compact, Hausdorff topological space is defined through the positive cone C+(Ω)={x∈C(Ω)∣x(t)≥0 for any t∈Ω} as follows: x≥y⇔x(t)≥y(t) for any t∈Ω, or else x≥y⇔x-y∈C+(Ω). C(Ω) endowed with this partial ordering is a vector lattice, because for any x,y∈C(Ω) the pointwise supremum x∨y and the pointwise infimum x∧y exist in C(Ω). If Z is a subspace of C(Ω), the induced partial ordering on the elements of Z is implied by the cone Z+=Z∩C+(Ω). Z endowed with this partial ordering is an ordered subspace of C(Ω). If for any x,y∈Z, supZ{x,y}, infZ{x,y} exist in Z, then Z is called a lattice-subspace of C(Ω). It is true that in this case, for any x,y∈Z, (2.4)sup⁡Z⁡{x,y}≥x∨y≥x∧y≥inf⁡Z⁡{x,y}.
If for any x,y∈Z, x∨y,x∧y∈Z then Z, is a sublattice of C(Ω) 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 Z is finite-dimensional and its dimension is equal to r, then a positive basis{b1,b2,…,br} is a basis of Z such that Z+={x=∑i=1rλibi∣λi≥0 for any i=1,2,…,r}. If the ordered subspace Z has a positive basis {b1,b2,…,br}, then for any x∈Z+ the coefficients {λ1,λ2,…,λr} of its expansion in terms of this basis are all positive real numbers, namely, λi∈ℝ+ holds for any i=1,2,…,r. Also, if Z has a positive basis, then if x,y∈Z and x=∑i=1rλibi, y=∑i=1rμibi, x≥y⇔λi≥μi for any i=1,2,…,r. This implies (2.5)inf⁡Z⁡{x,y}=∑i=1r(λi∧μi)bi,sup⁡Z⁡{x,y}=∑i=1r(λi∨μi)bi;
hence Z is a lattice-subspace of C(Ω). About the relation between positive bases and finite-dimensional lattice-subspaces, the following theorems hold.(Choquet-Kendall)
A finite-dimensional ordered vector spaceEwhose positive coneE+is closed and generating(E=E+-E+)is a vector lattice if and only ifEhas a positive basis.(I. A. Polyrakis)
A finite-dimensional ordered vector spaceEis a vector lattice if and only ifEhas 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{b1,b2,…,br} is a positive basis of Z, then {k1b1,k2b2,…,krbr} is a positive basis of Z too, where ki>0,i=1,2,…,r. If Ω={1,2,…,S}, then C(Ω)=ℝS. The statement of Theorem 3.7 in [10] is the following (supposing that x1,x2,…,xn are positive and linearly independent vectors of C(Ω)).Theorem 3.7 (I. A. Polyrakis)
LetZbe the sublattice ofC(Ω)generated byx1,x2,…,xnand letm∈ℕ.Then the statements (i) and (ii)are equivalent.(i)
dim(Z)=m.(ii)
R(β)={P1,P2,…,Pm}.If the statement (ii) is true, then Z is constructed as follows. (a)
EnumerateR(β)so that its first n vectors can be linear independent. Denote again by Pi, i=1,2,…,m,the new enumeration and let Ii=β-1(Pi), i=1,2,…,m.(b)
Define the functionsxn+k(t)=ak(t)z(t), t∈Ω, k=1,2,…,m-n,where akis the characteristic function of In+k.   (z=x1+x2+⋯+xn).(c)
Z=[x1,…,xn,xn+1,…,xm].As it is indicated in Propositions 6 and 7 in [8], F1(X) 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 thatZis a sublattice ofℝm. If the constant vector 1=(1,1,…,1)is an element of Z, thenZhas a positive basis{b1,b2,…,br}which is a partition of the unit,that is, 1=∑i=1rbi,and for each vector biwe have bi(j)=1for each j∈supp(bi).(Proposition 7, [8])
Suppose thatZis a sublattice of ℝmwith a positive basis
(2.6){b1,b2,…,br}.
Then for each ithe vector bihas minimal support in Z, that is, there is no x∈Z,x≠0, such that supp(x)⊂≠supp(bi).We show thatsupp(y)={i∈{1,2,…S}∣y(i)≠0}.According to [8], Definition 18, a vector e∈Fu(X) is an Fu(X)-efficient fund if Fu(X) is the linear subspace of ℝS which is generated by the set of nontrivial call options and the set of nontrivial put options of e. Also, the statements of [8], Theorem 19, Proposition 20, Theorem 21 are the following.(Theorem 19, [8])
Suppose that{b1,b2,…,bμ}is a positive basis ofFu(X),u=∑i=1μλibi,andλi>0for eachi.Then the vectore=∑i=1μκibi  of  Fu(X)is anFu(X)-efficient fund if and only ifκi/λi≠κj/λjfor eachi≠j.(Proposition 20, [8])
Each nonefficient subspace ofFu(X)is a proper sublattice ofFu(X).(Theorem 21, [8])
Suppose that{b1,b2,…,bμ}is a positive basis ofFu(X)and thatu=∑i=1μλibiwithλi>0for eachi.Then(i)
the nonempty setD=Y∖⋃i∈I(Y∩Zi), where{Zi∣i∈I}is the set of nonefficient subspaces of Fu(X), is the set of Fu(X)-efficient funds of Yand the Lebesgue measure of Y is supported on D;(ii)
Fu(X)is the subspace of ℝSgenerated by the set of the call options {cu(x,a)∣x∈Y,a∈ℝ}written on the elements ofY. If u∈X,Fu(X)is the subspace X1 of ℝSgenerated by the set of call options O1={cu(x,a)∣x∈X,a∈ℝ} written on the elements of X.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 andS states of the world. At time-period t=0, there is uncertainty about the true state of the world, while at time-period t=1 this state is revealed. We also consider L goods being consumed at time-period 1, independently from the state of the world s faced by the individuals. We also consider a numeraire good in terms of which all the values are expressed. The spot prices of these L goods consumed at time-period 1 and if the state s is faced by the individuals are represented by a vector p(s)∈ℝ++L, 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 n assets, whose payoffs are expressed initially in terms of the goods consumed in the economy. We suppose that n<S, 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 i-asset at time-period 1 if the state s is faced by the individuals is a “consumption” goods’ bundle Ai(s)∈ℝ+L for all the assets i=1,2,…,n and for all the states of the world s=1,2,…,S, whether the state s occurs.The value of the payoff of thei-asset in terms of the numeraire is (3.1)p(s)⋅Ai(s),s=1,2,…,S.
The value payoff vector under prices p=(p(1),…,p(S)) of the i-asset at time-period 1 is (3.2)p•Ai=(p(1)⋅Ai(1),…,p(S)⋅Ai(S)).
The value payoff matrix under prices p of the n assets is denoted by (3.3)p•A
and it is actually the matrix whose columns are the vectors p•Ai, i=1,2,…,n. The prices for the goods are expressed in terms of the numeraire, or else the vectors p=(p(1),p(2),…,p(S)) are normalized and for this reason taken to vary on the interior of the simplex (3.4)Δ={p∈R+SL∣∑s=1S‖p(s)‖1=1}
in terms of the induced topology of the Euclidean space ℝSL, which is an SL-1-dimensional manifold M (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. Thei-real asset can be viewed as an asset whose payoff is expressed in L different currencies in every state of the world. Ai(s) can be viewed as the payoff vector of this asset at time-period 1 if the state s is the true state of the world, where in this case Aij(s)≥0 denotes the units of the j-currency which delivers to the owner of one unit of this asset, where j=1,2,…,L. The prices p(s)∈ℝ++L are related to the exchange rates of another (“numeraire”) currency with respect to these L payoff currencies. For example, pj(s) denotes the amount of the numeraire received by selling one unit of the j-currency at time-period 1 and if the state s becomes true, where j=1,2,…,L. These “spot” prices allow for the expression p•Ai 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 goodl=1,2,…,L is the financial contract which promises its owner one unit of the good l 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 pl(s), s=1,2,…,S. The S×L-payoff matrix A 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 l-good, whose entries are all equal to 1. The numeraire payoff vector p•A 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 theL goods enter the production of a firm, then a stochastic production plan is a vector y=(y(s),s=1,2,…,S) where every y(s) is a production plan in a technology set Y∈ℝL. The payoff vector of an equity contract relies on the vector p•y, whose components denote the profits p(s)·y(s) earned by the firm at any state of the world s=1,2,…,S under the relevant spot prices p(s) and the selection of the production plan y(s).Thecorporate 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 b and its equity payoff vector written on the production value vector p•y, as we mentioned before, is the call option (p•y-b1)+. 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 n is at least as great as the number of states of the world, then generically the asset span [p•A1,p•A2,…,p•An] 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 n<S we pose on the number of the real assets. Hence we wonder whether the span [p•A1,p•A2,…,p•An] 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 u 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 u to be some risky asset. We define the completion of the asset span [p•A1,p•A2,…,p•An] by options with respect to the strike asset u 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 p and in the payoffs Ai, i=1,2,…,n, of the assets, the numeraire payoffs p•Ai, i=1,2,…,n, are nonredundant. For this goal, we use the Preimage Theorem for smooth maps between manifolds. We also prove that the set of prices p and consumption-good payoffs Ai, i=1,2,…,n, of the real assets, such that the numeraire payoff vectors p•A1,p•A2,…,p•An separate the states of the world is generic in M×ℝ++SLn. This implies that the completion of the market by options is the whole ℝS, because in this case the dimension of the sublattice of ℝS generated by p•A1,p•A2,…,p•An, u is the number of the different values of the basic function of the set of vectors {p•A1,p•A2,…,p•An,u}, according to what is proved in [10], Theorem 3.7. We actually prove this result for the case where u=1. But in this case, generically we have that this number of different values is exactly equal to S, hence the completion of the asset span [p•A1,p•A2,…,p•An] by options with respect to the nominal asset u=1 is the whole space ℝS. 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 assetAi, i=1,2,…,n, with respect to the nominal asset u∈ℝ++S under spot prices p and exercise price k∈ℝ is the derivative on the numeraire payoff p•Ai of Ai whose payoff vector is (4.1)(p•Ai-ku)+.
This indicates that the payoff of this option at the state s is (4.2)max⁡{p(s)⋅Ai(s)-ku(s),0},
expressed in units of the numeraire. We denote this option by cu(p•Ai,k). The equivalent put option is defined in a similar way, namely, it is the claim (ku-p•Ai)+ and it is denoted by pu(p•Ai,k).The completion by options of the asset span[p•A1,p•A2,…,p•An] with respect to the nominal asset u is determined as follows: (4.3)O1={cu(x,k),k∈R,x∈[p•A1,p•A2,…,p•An]},X1=[O1],
and for any natural number m≥1(4.4)Om={cu(x,k),k∈R,x∈Xm-1},Xm=[Om].
In the above definition X0=X=A(p).We denote byA(p) the numeraire-asset span [p•A1,p•A2,…,p•An] of the real asset structure consisted by the assets Ai, i=1,2,…,n, if spot prices are equal to p. We also use A(p) in order to denote the S×n matrix whose columns are the vectors p•Ai, i=1,2,…,n.Definition 4.1.
The completion by options ofA(p)with respect to u is the following subspace Fu(A(p)) of ℝS:
(4.5)Fu(A(p))=⋃m=1∞Xm
of ℝS.By Theorem 3 in [8] what is proved is that Fu(A(p)) is the sublattice S(Y(p)) of ℝS generated by the elements of the span Y(p) generated by the vectors p•A1,p•A2,…,p•An,u of ℝS.


## 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 matrixA whose columns are the vectors Ai, or else Ai∈ℝ++SL for any i=1,2,…,n.(B)
No free-goods are available, that is,p∈intΔ=M.We also suppose in the rest of the paper thatn≥1,S≥2,L≥2. 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 A can be also identified with a vector of ℝSLn. We will use A by both ways.Supposing thatAi∈ℝ++SL, our aim is to determine the basic set of the asset spanA(p) 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 A(p) whose completion is taken with respect to u is a maximal subset of linearly independent vectors among the set of positive and negative parts of the vectors p•A1,p•A2,…,p•An,u. But these vectors are all positive, hence what we have to check is whether the variation of p and the payoffs Ai, i=1,2,…,n, allow for these vectors to be linearly independent except negligible sets.We show that the setintΔ=M 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 subsetM⊆ℝkis called a smooth manifold (of classCr,r≥1) of dimensionmif eachx∈Mhas a neighborhoodW∩Mthat is diffeomorphic to an open subset of the Euclidean spaceℝm.Suppose thatF is an m-dimensional smooth manifold. A subset N of F is a null set in F if for any x∈N there is a chart (ψ,D,V) of F around x such that ψ(D∩N) has Lebesgue measure zero in ℝm (V⊆F,D⊆ℝm). We show that a chart of F around x∈F is a triple (ψ,D,V) in which D is an open subset of ℝm, V is a subset of F containing x, and ψ-1:V→D is a Cr-diffeomorphism. Every such subset V of F is called an open neighborhood of x in F.A subsetH of F is a set of full measure in F if F∖H is a null set in F.The above definitions are taken from [16, page 149].A subset of full measure inF is called generic.IfF′′ is a generic subset of F′ and F′ is a generic subset of F, then  F′′ is also a generic subset of F. The implication holds for a finite number of such inclusions.If a particular property depends on the elements of the manifoldF and this property is true for any element in a generic set G in F, we say that this property holds generically, or almost everywhere or almost always.A smooth functionf:X→Y between the smooth manifolds X, Y is regular at a point x∈X if the derivative of f has full rank at the point x, or else if the differential (df)x is a surjection. In this case, x is called a regular point of f.A smooth functionf:X→Y between the smooth manifolds X, Y is critical at a point x∈X if f is not regular at x. In this case, x is called a critical point of f.A smooth functionf:X→Y between the smooth manifolds X,Y is transversal at a point y∈Y if any x∈f-1(y) is a regular point of f. An alternative name for such a y is regular value of f.Ay∈Y which belongs to the range of values of a smooth function f:X→Y between the smooth manifolds X,Y is called a critical value of f 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 anm-dimensional manifold M in a Euclidean space ℝk is a set of (m+1)-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)
IfF:M→Nis aCk-class map between the manifoldsM, NwithdimM=m, dimN=n, andk>max{0,m-n}, then the set of critical values ofFis a null set inN.(Preimage Theorem)
IfF:M→Nis a smooth map between the smooth manifoldsM, Nof dimensionsm, n, respectively, whiley∈Nis a regular value ofF, then the setF-1(y)is either empty (in the case wherem<n) or a smooth manifold of dimensionm-n.(Transversality Theorem)
LetF:M×Ω→Nbe aCr-map intoN, where theCr-smooth manifoldsM, Ω, Nare of dimensionsm, p, n, respectively. Then ifr>max{m-n,0}andFis transversal toy, then there exists a set of full measureΩ*⊆Ωsuch that for anyx∈Ω*the mapFx:M→N, Fx(t)=F(t,x), t∈Mis also transversal toy.Finally, the statements of the three above theorems are contained in [16, pages 150, 84, 151], respectively.Proposition 5.1.
For the generic element(p,A)∈M×ℝ++SLn, the vectors {p•A1,…,p•An} are linearly independent.Proof.
IfA(p) is the matrix whose columns are the elements of the set
(5.1){p•A1,…,p•An},
we have to show that for a set of full measure in M×ℝ++SLn, we have rankA(p)=n. We define Tσ:M×ℝ++SLn→ℝ, where Tσ(p,A)=detAσ(p) and Aσ(p) is the n×n matrix which arises by selecting n lines-states among the S lines of the matrix A(p) which correspond to the combination σ (a combination is a selection of n objects out of S objects, without having their order in mind). Tσ is transversal to 0, namely, 0 is a regular value of Tσ. Since this is true by Lemma 5.2, then Tσ-1(0)=Pσ is either empty or a submanifold of M×ℝ++SLn of dimension SLn+SL-2, namely, a null set Nσ of M×ℝ++SLn. Repeating this for all combinations σ∈Cn, we take that the set of pairs (p,A) in M×ℝ++SLn for which every n×n submatrix of A(p) is singular is actually the set ⋂σ∈CnNσ, being a null set of M×ℝ++SLn, where Cn denotes the set of n-combinations of the S objects. Namely, the generic matrix A(p) is of full rank.Lemma 5.2.
The mapTσ:M×ℝ++SLn→ℝ, where Tσ(p,A)=detAσ(p), is transversal to 0 for any combination σ of n objects out of S.Proof.
Suppose thatTσ-1(0) is nonempty. Suppose that σ={s1,s2,…,sn} is a selection of n states out of S. The image of Tσ(p,A) of Tσ at (p,A) is the determinant of the n×n matrix Aσ(p) whose b-row is
(5.2)(p(sb)⋅A1(sb),p(sb)⋅A2(sb),…,p(sb)⋅An(sb)),b=1,2,…,n.
Then we have
(5.3)Tσ(p,A)=∑θ∈Snϵ(θ)(p(s1)⋅Aθ(s1)(s1))(p(s2)⋅Aθ(s2)(s2))⋯(p(sn)⋅Aθ(sn)(sn))=∑θ∈Snϵ(θ)[∑ds1=1Lp(s1)(ds1)Aθ(s1)(s1)(ds1)]⋯[∑dsn=1Lp(sn)(dsn)Aθ(sn)(sn)(dsn)]=∑θ∈Sn∑ds1=1L∑ds2=1L⋯∑dsn=1Lϵ(θ)p(s1)(ds1)⋯p(sn)(dsn)Aθ(s1)(s1)(ds1)⋯Aθ(sn)(sn)(dsn),
where Sn is the set of permutations over n symbols, θ is any permutation in Sn, and ϵ(θ) is the sign of the permutation θ. Also, p(sa)(dsa) is the spot price of the dsa-good corresponding to the state sa, where a=1,2,…,n and dsa=1,2,…,L. In a similar way, Aθ(sa)(sa)(dsa) is the amount of the dsa consumption good that the owner of one unit of the asset θ(sa) is going to receive at the time-period 1 if the state sa occurs.
In order to show thatTσ is transversal to 0, we have to prove that for any (p,A)∈Tσ-1(0) we have rank(DTσ(p,A))=1. To show that, we have to verify that for any p∈Tσ-1(0) at least one partial derivative(5.4)∂Tσ(p,A)∂p(sa)(dsa),a=1,2,…,n,sa∈σ,dsa=1,2,…,L,
is nonzero. But this is true, since p∈M and all of the products
(5.5)Aθ(s1)(s1)(ds1)Aθ(s2)(s2)(ds2)⋯Aθ(sn)(sn)(dsn)
are nonzero.By the fact thatA∈ℝ++SLn and by repeating the previous proof, we may take the following.Corollary 5.3.
The mapTσA:M→ℝ, where TσA(p)=detAσ(p), is transversal to 0 for any combination σ of n objects out of S, for all A∈ℝ++SLn.By using the Transversality theorem, we can prove almost the same thing.Corollary 5.4.
The mapTσA:M→ℝ, where TσA(p)=detAσ(p), is transversal to 0 for any combination σ of n objects out of S, where n<S, for every A which lies in a set of full measure in ℝ++SLn.Proof.
From the previously proved Lemma5.2, Tσ is transversal to zero for any combination σ and for any (p,A)∈M×ℝ++SLn. Note that dimM=SL-1, dimℝ++SLn=SLn, and dimℝ=1. Also note that Tσ is a C∞-map, hence we may suppose that it is also a Cr-map with r>(SL-1)-1=SL-2. Hence the projection TσA:M→ℝ, where TσA(p)=Tσ(p,A) for a specific A in a set of full measure Gσ⊆ℝ++SLn, is transversal to zero.By Transversality theorem, we can also prove the following.Proposition 5.5.
For a set of full measureG⊆ℝ++SLn and for any asset structure A∈G, the set of prices for which the vectors {p•A1,…,p•An} are linearly independent is generic in M.Proof.
Since from Lemma5.2, Tσ is transversal to zero for any combination σ and for any (p,A)∈M×ℝ++SLn and by Transversality theorem, there is a set of full measure Gσ of ℝ++SLn such that for any fixed A in this set, the projection TσA is transversal to zero for each σ. Hence, we may apply the Preimage theorem for TσA:M→ℝ whenever A lies in Gσ. Then we get that (TσA)-1(0) is a smooth submanifold of M of dimension SL-2=(SL-1)-1. Hence, (TσA)-1(0) is a null set in M, for any combination σ∈Cn. Consider the set of full measure ⋂σ∈CnGσ=G in ℝ++SLn. For any A∈G and for any σ, we have that (TσA)-1(0) is a null set in M. Given an A∈G, the set of prices in M for which the vectors p•A1,…,p•An are linearly dependent is the intersection ⋂σ∈Cn(TσA)-1(0), being a null set in M.For any(p,A)∈M×ℝ++SLn, we define the basic functionβ:{1,2,…,S}→Δn-1 (see in [9], page 2797) of the vectors p•A1,p•A2,…,p•An(5.6)β(p,A)(s)=1y(p,A)(s)(y1(p,A)(s),y2(p,A)(s),…,yn(p,A)(s)),
where (5.7)y(p,A)i(s)=p(s)⋅Ai(s),i=1,2,…,n,y(p,A)(s)=∑i=1ny(p,A)i(s),
for all the states s such that y(p,A)(s)>0. Note that Δn-1 is the simplex of the positive cone ℝ+n of ℝn.Note that under the assumptions (A) and (B),β(p,A) is well defined for all s=1,2,…,S.Proposition 5.6.
For the generic element(p,A)∈M×ℝ++SLn, the vectors p•A1, p•A2,…, and p•An are linearly independent and they separate the states of the world.Proof.
The set of(p,A)∈M×ℝ++SLn for which there is at least one pair (s,s′) of disjoint states of the world s≠s′ such that p(s)·Ai(s)=p(s′)·Ai(s′) for some asset i=1,2,…,n is denoted by E. This set of (p,A) is actually the set ⋃i=1n⋃(s,s′):s≠s′(Fs,s′,i-1(0))c, where Fs,s′,i-1:M×ℝ++SLn→ℝ is the map Fs,s′,i(p,A)=p(s)·Ai(s)-p(s′)·Ai(s′) and (Fs,s′,i-1(0))c is the complement of Fs,s′,i-1(0) in M×ℝ++SLn. Let us prove that Fs,s′,i is transversal to 0. We have to prove that for all (p,A)∈Fs,s′,i-1(0), rankDFs,s′,i(p,A)=1. Note that for any such (p,A), we have ∂Fs,s′,i(p,A)/∂p(s)(d)=Ai(d)>0, where d=1,2,…,L denotes any of the consumption goods. Thus, by applying the Preimage Theorem, Fs,s′,i-1(0) is a submanifold of M×ℝ++SLn of dimension SLn+SL-2, hence a null set. Namely, the union ⋃i=1n⋃(s,s′):s≠s′(Fs,s′,i-1(0))c is a set of full measure in M×ℝ++SLn. Also, the set of (p,A) in M×ℝ++SLn for which p•A1,p•A2,…,p•An are linearly independent in ℝS is a set of full measure as it is shown by Lemma 5.2, since this set is the complement of ⋂σ∈CnTσ-1(0) in M×ℝ++SLn, being a null set (where Cn denotes the set of n-combinations of the S objects). Hence the set of (p,A) such that p•A1,p•A2,…,p•An are linearly independent and every pair of different states (s,s′) is separated by some p•Ai, i=1,2,…,n, is the set
(5.8)(⋃σ∈Cn(Tσ-1(0))-1)c∩(⋃i=1n⋃(s,s′):s≠s′(Fs,s′,i-1(0))c),
being the intersection of two sets of full measure, hence a set of full measure in M×ℝ++SLn.Theorem 5.7.
For the generic element(p,A)∈M×ℝ++SLn, the completion by options of the span [p•A1,p•A2,…,p•An] with respect to the riskless bond is ℝS.Proof.
Consider the generic subsetK of M×ℝ++SLn such that the states of the world are separated by the vectors of {p•A1,p•A2,…,p•An}, while these vectors are linearly independent. This set is indicated in the end of Proposition 5.6. Consider K1⊆K to be the subset of those (p,A)∈K for which 1∈[p•A1,p•A2,…,p•An]. Then the basic set of the market defined in [8] in the case where (p,A)∈K1  consists of the elements of {p•A1,p•A2,…,p•An}; see also Definition 10 and Theorem 11 in [8]. Hence the completion of the numeraire asset span A(p)=[p•A1,p•A2,…,p•An] with respect to the riskless bond 1 is ℝS for the elements of K1. This is true because the values of the basic function β(p,A) of the elements {p•A1,p•A2,…,p•An} of the basic set are disjoint and their cardinality is equal to S, for any (p,A)∈K1. Then, due to Theorem 14 in [8] for any (p,A)∈K1, F1(A(p))=ℝS. Let us verify this fact. Suppose that there is some (p,A)∈K1 and some pair of states of the world (s,s′) with s≠s′ such that β(p,A)(s)=β(p,A)(s′). Then the cardinality of R(β) is less than S in this case and according to Theorem 9 in [8] and Theorem 3.7 in [10], this implies dimF1(A(p))<S. But if β(p,A)(s)=β(p,A)(s′), this implies
(5.9)p(s)⋅Aj(s)∑i=1np(s)⋅Ai(s)=p(s′)⋅Aj(s′)∑i=1np(s′)⋅Ai(s′)
for this (p,A) and for any j=1,2,…,n. These equations imply
(5.10)((p•A1)(s),(p•A2)(s),…,(p•An)(s))=((p•A1)(s′),(p•A2)(s′),…,(p•An)(s′))∑i=1n(p•Ai)(s)∑i=1n(p•Ai)(s′),
namely,
(5.11)(p(s)⋅A1(s),p(s)⋅A2(s),…,p(s)⋅An(s))=(p(s′)⋅A1(s′),p(s′)⋅A2(s′),…,p(s′)⋅An(s′))∑i=1np(s)⋅Ai(s)∑i=1np(s′)⋅Ai(s′).
Then
(5.12)p(s)⋅Aj(s)=p(s′)⋅Aj(s′)∑i=1np(s)⋅Ai(s)∑i=1np(s′)⋅Ai(s′)
for any j=1, 2,…,n. Since we supposed that 1∈[p•A1,p•A2,…,p•An], there are real numbers kj, j=1,2,…,n such that
(5.13)1=∑j=1nkj(p•Aj).
Then, 1=1(s)=∑j=1nkj(p•Aj)(s) and 1=1(s′)=∑j=1nkj(p•Aj)(s′). We get that
(5.14)∑j=1nkj(p•Aj)(s)=∑j=1nkj(p•Aj)(s′),
or else
(5.15)1=∑j=1nkj(p(s)⋅Aj(s))=∑j=1nkj(p(s′)⋅Aj(s′)).
Then
(5.16)1=∑j=1nkj(p(s′)⋅Aj(s′))∑i=1np(s)⋅Ai(s)∑i=1np(s′)⋅Ai(s′)=∑j=1nkj(p(s′)⋅Ai(s′)).
Hence, due to (5.9), ∑i=1n(p(s)·Ai)(s)=∑i=1n(p(s′)·Ai)(s′), which indicates that
(5.17)p(s)⋅Ai(s)=p(s′)⋅Ai(s′),
for any i=1,2,…,n. This is a contradiction, since this implies that (p,A) does not belong to the set K such that the elements of {p•A1,p•A2,…,p•An} separate the states of the world, since in this case the states of the pair (s,s′) where s≠s′ are not separated by the assets of the set {p•A1,p•A2,…,p•An}. We were led to a contradiction because we supposed that there exist some (p,A)∈K1 such that β(p,A)(s)=β(p,A)(s′) for some pair of states (s,s′), where s≠s′. Then there is not any such (p,A) in K1, which implies that for any (p,A)∈K1 the values of β(p,A) are disjoint; hence for the completion by options, the equality F1(A(p))=ℝS holds. Also, consider the complement of K1 in K, denoted by K2. In this case, the basic set of the market is consisted by the elements of {p•A1,p•A2,…,p•An,1}; see also Definition 10 and Theorem 11 in [8]. Hence the completion of the numeraire asset span A(p)=[p•A1,p•A2,…,p•An] with respect to the riskless bond 1 is ℝS for the elements of K2. This is true because the values of the basic function δ(p,A) of the elements {p•A1,p•A2,…,p•An} of the basic set are disjoint and their cardinality is equal to S, for any (p,A)∈K2. Then, due to Theorem 14 in [8] for any (p,A)∈K2, F1(A(p))=ℝS. Let us verify this fact. The basic function δ(p,A) of the basic set {p•A1,p•A2,…,p•An,1} of the market for an element (p,A)∈K2 has the form
(5.18)δ(p,A)(s)=1∑i=1np(s)⋅Ai(s)+1(p(s)⋅A1(s),p(s)⋅A2(s),…,p(s)⋅An(s),1),
for any s=1,2,…,S. If we suppose that the dimension of the completion F1(A(p)), which is equal to the cardinality of the range of δ(p,A), is less than S for some (p,A)∈K2, we will be led to a contradiction. If δ(p,A)(s)=δ(p,A)(s′) for some pair (s,s′) of states where s≠s′, then
(5.19)1∑i=1np(s)⋅Ai(s)+1=1∑i=1np(s′)⋅Ai(s′)+1,
an equation which corresponds to the equality of the last components of these vectors. This equation implies ∑i=1np(s)·Ai(s)=∑i=1np(s′)·Ai(s′). Also, for each j=1,2,…,n, we have
(5.20)p(s)⋅Aj(s)∑i=1np(s)⋅Ai(s)+1=p(s′)⋅Aj(s′)∑i=1np(s′)⋅Ai(s′)+1.
But from equation ∑i=1np(s)·Ai(s)=∑i=1np(s′)·Ai(s′), this also implies p(s)·Aj(s)=p(s′)·Aj(s′) for any j=1,2,…,n. This is a contradiction, since this means that the states s,s′ are not separated by the vectors p•A1,p•A2,…,p•An, namely, that (p,A) does not belong to K. Then there is not any such (p,A) in K2, which implies that for any (p,A)∈K2 the values of δ(p,A) are disjoint; hence the for the completion by options, the equality F1(A(p))=ℝS holds for any (p,A)∈K2. Hence in any case for any (p,A)∈K the completion is equal to ℝS.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.
ConsiderS=4, L=2, and n=2, 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 A1 and A2 in the market. Suppose that the payoff-matrices of the two assets are
(6.1)A1=[14231511],A2=[64845523],
respectively, where the rows of these matrices correspond to the states and the columns correspond to the goods, or else the sl-element of the matrix Ai, i=1,2, for s=1,2,…,S and l=1,2,…,L denotes the units of the l-good received at the time-period 1 by the owner of one unit of the i-asset if the state s occurs (l=1,2). Also, suppose that the spot prices for the numeraire are p=(p(1),p(2),p(3),p(4)) where p(1)=(1/12,2/12), p(2)=(3/12,1/12), p(3)=(1/12,1/12), and p(4)=(2/12,1/12). We show that we take p to be normalized with respect to the ∥·∥1-norm in ℝSL and each p(s), s=1,2,3,4, is consisted by the unitary prices of the two consumed goods at the time-period 1 whether the state s occurs. Namely, the numeraire payoff-vectors for the two assets i=1,2 are
(6.2)p•A1=[34341214],p•A2=[767356712],
respectively. Note that the riskless bond 1=(1,1,1,1) does not belong to the span of p•A1,p•A2. The basic function β of these three vectors y1=p•A1, y2=p•A2, andy3=1 is calculated as follows. First, we have that y1+y2+y3=(35/12,49/12,7/3,11/6). Then the values β(s)=(1/y(s))(y1(s),y2(s),y3(s)) of β calculated at the states s=1,2,3,4 are the following:
(6.3)β(1)=(935,1435,1235),β(2)=(949,2849,1249),β(3)=(314,514,614),β(4)=(18132,42132,72132).
Since the four values of β are distinct, from the Theorem 14 in [8], we get that
(6.4)dim⁡F1(A(p))=4,
namely, that the completion F1(A(p)) of the numeraire asset span
(6.5)A(p)=[p•A1,p•A2]
with respect to the riskless bond 1 is the whole space ℝ4. If we would like to indicate a call option on p•A1, we may consider the nontrivial call option with exercise price k=1/2 with respect to the numeraire. The payoff vector of this call option is (p•A1-(1/2)1)+=(1/4,1/4,0,0). More specifically and according to the algorithm for the determination of the positive basis of F1(A(p)) indicated in Theorem 8 and Theorem 9 of [8] and Theorem 3.7 of [10], by enumerating the values of β, we get β(1)=P1, β(2)=P2, β(3)=P3, andβ(4)=P4. The vectors P1, P2, P3 are linearly independent. We notice that β(4)=P4; hence we define y4′=(0,0,0,11/6). We also consider y1′=y1, y2′=y2, y3′=y3. The values of γ, being the basic function of y1′, y2′, y3′, y4′, are the following:
(6.6)γ(1)=(935,1435,1235,0)=P1′,γ(2)=(949,2849,1249,0)=P2′,γ(3)=(314,514,614,0)=P3′,γ(4)=(18264,42264,72264,12)=P4′.
We denote by D the 4×4 matrix whose columns are the vectors P1′,P2′,P3′,P4′. The vectors of the positive basis {b1,b2,b3,b4} of F1(A(p)) are determined through the matrix equation (b1,b2,b3,b4)T=D-1·(y1′,y2′,y3′,y4′)T, where (b1,b2,b3,b4)T is the 4×4 matrix whose rows are the vectors b1, b2, b3, b4 and (y1′,y2′,y3′,y4′)T is the 4×4 matrix whose rows are the vectors y1′, y2′, y3′, y4′. We have that
(6.7)D-1=[15-52-65127544-14372-712-744-28307-28110002]
and the vectors of the positive basis {b1,b2,b3,b4} are b1=(35/12,0,0,0), b2=(0,49/12,0,0), b3=(0,0,7/3,0), andb4=(0,0,0,11/3). Hence the basis b1′=(1,0,0,0), b2′=(0,1,0,0), b3′=(0,0,1,0), b4′=(0,0,0,1) is also a positive basis of Fu(A(p)), which is equal to ℝ4. According to Theorem 21 of [8], F1(A(p)) is the subspace of ℝ4 generated by {c1(x,a)∣x∈Y(p),a∈ℝ}, where Y(p) is the subspace of ℝ4 generated by p•A1,p•A2,1. As an example, if we take 1, c1(p•A2,7/12), c1(p•A2,5/6), c1(p•A2,7/6), we have respectively the set of vectors: (1,1,1,1), (7/12,21/12,3/12,0), (1/3,3/2,0,0), and(0,7/6,0,0) which are linearly independent and they generate the whole space ℝ4 which is the completion by options of A(p) with respect to u=1.Example 6.2.
Continuing with the previous example, we have that the general element of the manifoldM is p=(p(1),p(2),p(3),p(4)) where p(1)=(p1(1),p2(1)), p(2)=(p1(2),p2(2)), p(3)=(p1(3),p2(3)), p(4)=(p1(4),p2(4)), where the index i denotes the good, namely, pi(s) is the price—in units of the numeraire—for the ownership of one unit of the i-good at the time-period 1 if the state s occurs. The numeraire payoffs of the assets are
(6.8)p•A1=[p1(1)+4p2(1)2p1(2)+3p2(2)p1(3)+5p2(3)p1(4)+p1(4)],p•A2=[6p1(1)+4p1(1)8p1(2)+4p2(2)5p1(3)+5p2(3)2p1(4)+3p2(4)].
The prices of M for which these payoffs are redundant are those for which all the 2×2 determinants consist of coordinates of these vectors which correspond to a certain pair of states of the world are zero. The possible such pairs are
(6.9)(1,2),(1,3),(1,4),(2,3),(2,4),(3,4).
The determinants are those of the corresponding 2×2 matrices
(6.10)H(1,2)=[p1(1)+4p2(1)6p1(1)+4p2(1)2p1(2)+3p2(2)8p1(2)+4p2(2)],H(1,3)=[p1(1)+4p2(1)6p1(1)+4p1(1)p1(3)+5p2(3)5p1(3)+5p2(3)],H(1,4)=[p1(1)+4p2(1)6p1(1)+4p2(1)p1(4)+p1(4)2p1(4)+3p2(4)],H(2,3)=[2p1(2)+3p2(2)8p1(2)+4p2(2)p1(3)+5p2(3)5p1(3)+5p2(3)],H(2,4)=[2p1(2)+3p2(2)8p1(2)+4p2(2)p1(4)+p1(4)2p1(4)+3p2(4)],H(3,4)=[p1(3)+5p2(3)5p1(3)+5p2(3)p1(4)+p1(4)2p1(4)+3p2(4)].
We are going to prove that the set of prices p=(p(1),p(2),p(3),p(4)) where p(1)=(p1(1),p2(1)), p(2)=(p1(2),p2(2)),p(3)=(p1(3),p2(3)), andp(4)=(p1(4),p2(4)) in M, with these vectors being whose components are all positive, such that the determinants of the above six matrices are zero, is a negligible set (actually a null set) in the manifold M={p=(p(1),p(2),p(3),p(4))∣p(s)=(p1(s),p2(s)), where pi(s)>0 for every i=1,2 and for every s=1,2,3,4}. For two disjoint states s,s′, we define the map Ts,s′:M→ℝ with Ts,s′(p)=detH(s,s′). The determinant of a matrix is a C∞-map; hence we may take it to be a Cr-map for r large enough. Hence in order to apply the Preimage Theorem, we may take r>7-1=6, since dimM=7=SL-1,dimℝ=1. Hence, if Ts,s′-1(0) is a nonempty set, it is a submanifold of M of dimension 6, hence a null set in it. If we apply the Preimage Theorem for all the pairs (s,s′) of disjoint states, we may find that the set of prices in M for which p•A1,p•A2 are non-redundant is generic in M. This is due to the fact that either in the case that Ts,s′-1(0) is nonempty or not for some pair of states (s,s′) where s≠s′, the set of prices, for which the value payoffs p•A1,p•A2 of the real assets A1,A2 are redundant, is the subset ⋂(s,s′):s≠s′Ts,s′-1(0) of M. This set is actually a null set. Let us verify whether some Ts,s′-1(0) is a nonempty set or not in this case. If we would like to see whether T1,2-1(0) is a nonempty set in M, we calculate the determinant of the matrix H(1,2) and we may find that it is actually a nonempty set, since p=(p(1),p(2),p(3),p(4)) with p(1)=(1/16,1/16), p(2)=(5/128,5/64), and, for example, p(3)=(48/128,8/128), p(4)=(13/128,28/128) belong to T1,2-1(0). Hence, T1,2-1(0) is a six-dimensional submanifold of a seven-dimensional manifold which is M itself. Hence, T1,2-1(0) is a closed and zero-Lebesgue measure set of M, namely, a null set. The same search of nonemptiness can be repeated for the rest pairs of states (s,s′), where s≠s′.



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*Source: 139690-2011-02-22.xml*