The Completion of Real-Asset Markets by Options

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 ℝ^𝑆.

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 ℝ^𝑆 generated by the span 𝑌 of 𝑋∪𝑈 (Theorem 3), called completion of 𝑋 by options with respect to 𝑈 and denoted by 𝐹_𝑈(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 𝐹_𝑢(𝑋)-efficient fund (Definition 18) and proving that almost every payoff in 𝑌 is an 𝐹_𝑢(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 of 𝑛 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 𝑆 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,…,𝑥_𝑛∈ℝ^𝑆 at time period 𝑡=1 are linearly independent. A portfolio in this market is a vector 𝜃=(𝜃₁,𝜃₂,…,𝜃_𝑛) of ℝ^𝑛 in which 𝜃_𝑖, 𝑖=1,2,…,𝑛, denotes the number of units invested to the asset 𝑖. If 𝜃_𝑖≥0, then the investment to 𝜃_𝑖 units of the asset 𝑖 denotes a long position to these units. If 𝜃_𝑖<0, then the investment to 𝜃_𝑖 units of the asset 𝑖 denotes a short position to −𝜃_𝑖 units of the asset 𝑖. The payoff of a portfolio 𝜃, if the payoff vectors 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ^𝑆 are expressed in terms of the numeraire as well, is the vector 𝑇(𝜃)=∑^𝑛_𝑖=1𝜃_𝑖𝑥_𝑖. The range of the operator 𝑇∶ℝ^𝑛→ℝ^𝑆 is called asset span of the market consisted by 𝑥₁,𝑥₂,…,𝑥_𝑛. A contingent claim is any liability 𝑐∈ℝ^𝑆, 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∈ℝ^𝑛 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∈ℝ^𝑆 is a replicated contingent claim, then the call option written on 𝑐 with exercise price 𝑎 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∈ℝ^𝑆 is defined by 𝑥∨0, where if 𝑥,𝑦∈ℝ^𝑆, the lattice operations on ℝ^𝑆 are defined in the following way: 𝑥∨𝑦=sup𝑥,𝑦=𝑥₁∨𝑦₁,𝑥₂∨𝑦₂,…,𝑥_𝑆∨𝑦_𝑆,𝑥∧𝑦=inf𝑥,𝑦=𝑥₁∧𝑦₁,𝑥₂∧𝑦₂,…,𝑥_𝑆∧𝑦_𝑆, (2.1) while 𝑥_𝑖∨𝑦_𝑖=max{𝑥_𝑖,𝑦_𝑖}, 𝑥_𝑖∧𝑦_𝑖=min{𝑥_𝑖,𝑦_𝑖} for any 𝑖=1,2,…,𝑛. The negative part 𝑥⁻ of 𝑥∈ℝ^𝑆 is defined as follows: 𝑥⁻=(−𝑥)∨0. The lattice identity 𝑥∨𝑦+𝑥∧𝑦=𝑥+𝑦 implies the well-known put-call parity: (𝑐−𝑎𝟏)⁺+(𝑎𝟏−𝑐)⁺=𝑐−𝑎𝟏. (2.2) As we see, these lattice operations are established via the usual (component-wise) partial ordering on ℝ^𝑆 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 𝑐_𝑢(𝑐,𝑎), while the put option is denoted by 𝑝_𝑢(𝑐,𝑎). The call option 𝑐_𝑢(𝑐,𝑎) and the put option 𝑝_𝑢(𝑐,𝑎) are called nontrivial if 𝑐_𝑢(𝑐,𝑎)>0, 𝑝_𝑢(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 span 𝑋 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,…,𝑥_𝑛 and the strike vector is 𝑢. The completion by options of the asset span 𝑋=[𝑥₁,𝑥₂,…,𝑥_𝑛] with respect to 𝑢 is defined as follows: 𝑂₁={𝑐_𝑢(𝑥,𝑎)∣𝑥∈𝑋,𝑎∈ℝ} and 𝑋₁ is the subspace of ℝ^𝑆 generated by 𝑂₁. For 𝑛≥1, 𝑂_𝑛={𝑐_𝑢(𝑥,𝑎)∣𝑥∈𝑋_𝑛−1,𝑎∈ℝ} and 𝑋_𝑛 is the subspace of ℝ^𝑆 generated by 𝑂_𝑛. The completion by options of 𝑋 with respect to 𝑢 is the subspace 𝐹_𝑢(𝑋)=⋃^∞_𝑛=1𝑋_𝑛 of ℝ^𝑆. Consider the set 𝒜={𝑥⁺₁,𝑥⁻₁,𝑥⁺₂,𝑥⁻₂,…,𝑢⁺,𝑢⁻}. Any maximal set consisted by linearly independent vectors of 𝒜 is called a basic set of the asset span 𝑋=[𝑥₁,𝑥₂,…,𝑥_𝑛] 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 𝐹_𝑢(𝑋)=𝑆(𝑌), where 𝑆(𝑌) is the sublattice of ℝ^𝑆 generated by the subspace 𝑌=[𝑋∪{𝑢}]. The sublattice 𝑆(𝐵) generated by a nonempty set 𝐵 of vectors of ℝ^𝑆 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], 𝐹_𝑢(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 ℝ^𝑆. If {𝑦₁,𝑦₂,…,𝑦_𝑟} is a basic set of the asset span 𝑋 with respect to the strike vector 𝑢, then the basic function 𝛽 of 𝑦₁,𝑦₂,…,𝑦_𝑟 is very important in the determination of 𝐹_𝑢(𝑋). 𝛽 was defined in [9] and in the case where the set of states of the world is finite is defined as follows: 𝛽(𝑖)=𝑦₁(𝑖)𝑦(𝑖),𝑦₂(𝑖)𝑦(𝑖)𝑦,…,^𝑟(𝑖)𝑦(𝑖), (2.3) for each 𝑖=1,2,…,𝑆 with 𝑦(𝑖)>0, where 𝑦=𝑦₁+𝑦₂+⋯+𝑦_𝑟. This function takes values on the simplex Δ_𝑟−1={𝑑∈ℝ^𝑟₊∣∑^𝑟_𝑗=1𝑑_𝑗=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 𝐹_𝑢(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, sup_𝑍{𝑥,𝑦}, inf_𝑍{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, sup_𝑍𝑥,𝑦≥𝑥∨𝑦≥𝑥∧𝑦≥inf_𝑍𝑥,𝑦. (2.4) 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,…,𝑏_𝑟} is a basis of 𝑍 such that 𝑍₊={𝑥=∑^𝑟_𝑖=1𝜆_𝑖𝑏_𝑖∣𝜆_𝑖≥0 for any 𝑖=1,2,…,𝑟}. If the ordered subspace 𝑍 has a positive basis {𝑏₁,𝑏₂,…,𝑏_𝑟}, then for any 𝑥∈𝑍₊ the coefficients {𝜆₁,𝜆₂,…,𝜆_𝑟} of its expansion in terms of this basis are all positive real numbers, namely, 𝜆_𝑖∈ℝ₊ holds for any 𝑖=1,2,…,𝑟. Also, if 𝑍 has a positive basis, then if 𝑥,𝑦∈𝑍 and 𝑥=∑^𝑟_𝑖=1𝜆_𝑖𝑏_𝑖, 𝑦=∑^𝑟_𝑖=1𝜇_𝑖𝑏_𝑖, 𝑥≥𝑦⇔𝜆_𝑖≥𝜇_𝑖 for any 𝑖=1,2,…,𝑟. This implies inf_𝑍𝑥,𝑦=^𝑟_𝑖=1𝜆_𝑖∧𝜇_𝑖𝑏_𝑖,sup_𝑍𝑥,𝑦=^𝑟_𝑖=1𝜆_𝑖∨𝜇_𝑖𝑏_𝑖; (2.5) 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,…,𝑏_𝑟} is a positive basis of 𝑍, then {𝑘₁𝑏₁,𝑘₂𝑏₂,…,𝑘_𝑟𝑏_𝑟} is a positive basis of 𝑍 too, where 𝑘_𝑖>0,𝑖=1,2,…,𝑟. If Ω={1,2,…,𝑆}, then 𝐶(Ω)=ℝ^𝑆. The statement of Theorem 3.7 in [10] is the following (supposing that 𝑥₁,𝑥₂,…,𝑥_𝑛 are positive and linearly independent vectors of 𝐶(Ω)).

Theorem 3.7 (I. A. Polyrakis)
Let𝑍be the sublattice of𝐶(Ω)generated by𝑥₁,𝑥₂,…,𝑥_𝑛and let 𝑚∈ℕ. Then the statements (i) and (ii)are equivalent.(i)dim(𝑍)=𝑚. (ii)𝑅(𝛽)={𝑃₁,𝑃₂,…,𝑃_𝑚}. If the statement (ii) is true, then Z is constructed as follows. (a)Enumerate 𝑅(𝛽)so that its first 𝑛 vectors can be linear independent. Denote again by 𝑃_𝑖, 𝑖=1,2,…,𝑚,the new enumeration and let 𝐼_𝑖=𝛽⁻¹(𝑃_𝑖), 𝑖=1,2,…,𝑚. (b)Define the functions 𝑥_𝑛+𝑘(𝑡)=𝑎_𝑘(𝑡)𝑧(𝑡), 𝑡∈Ω, 𝑘=1,2,…,𝑚−𝑛,where 𝑎_𝑘is the characteristic function of 𝐼_𝑛+𝑘. (𝑧=𝑥₁+𝑥₂+⋯+𝑥_𝑛). (c)𝑍=[𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1,…,𝑥_𝑚].

As it is indicated in Propositions 6 and 7 in [8], 𝐹_𝟏(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 that 𝑍is a sublattice ofℝ^𝑚. If the constant vector 𝟏=(1,1,…,1)is an element of 𝑍, then𝑍has a positive basis{𝑏₁,𝑏₂,…,𝑏_𝑟}which is a partition of the unit, that is, 𝟏=∑^𝑟_𝑖=1𝑏_𝑖,and for each vector 𝑏_𝑖we have 𝑏_𝑖(𝑗)=1for each 𝑗∈sup𝑝(𝑏_𝑖).

(Proposition 7, [8])
Suppose that 𝑍is a sublattice of ℝ^𝑚with a positive basis 𝑏₁,𝑏₂,…,𝑏_𝑟. (2.6) Then for each 𝑖the vector 𝑏_𝑖has minimal support in 𝑍, that is, there is no 𝑥∈𝑍,𝑥≠0, such that sup𝑝(𝑥)⊂_≠sup𝑝(𝑏_𝑖).

We show that sup𝑝(𝑦)={𝑖∈{1,2,…𝑆}∣𝑦(𝑖)≠0}.

According to [8], Definition 18, a vector 𝑒∈𝐹_𝑢(𝑋) is an 𝐹_𝑢(𝑋)-efficient fund if 𝐹_𝑢(𝑋) is the linear subspace of ℝ^𝑆 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,…,𝑏_𝜇}is a positive basis of𝐹_𝑢(𝑋), 𝑢=∑^𝜇_𝑖=1𝜆_𝑖𝑏_𝑖, and𝜆_𝑖>0for each𝑖. Then the vector𝑒=∑^𝜇_𝑖=1𝜅_𝑖𝑏_𝑖  of  𝐹_𝑢(𝑋)is an𝐹_𝑢(𝑋)-efficient fund if and only if𝜅_𝑖/𝜆𝑖≠𝜅_𝑗/𝜆𝑗for each𝑖≠𝑗.

(Proposition 20, [8])
Each nonefficient subspace of𝐹_𝑢(𝑋)is a proper sublattice of𝐹_𝑢(𝑋).

(Theorem 21, [8])
Suppose that{𝑏₁,𝑏₂,…,𝑏_𝜇}is a positive basis of𝐹_𝑢(𝑋)and that𝑢=∑^𝜇_𝑖=1𝜆_𝑖𝑏_𝑖with𝜆_𝑖>0for each𝑖. Then(i)the nonempty set 𝐷=𝑌⧵⋃_𝑖∈𝐼(𝑌∩𝑍_𝑖), where{𝑍_𝑖∣𝑖∈𝐼}is the set of nonefficient subspaces of 𝐹_𝑢(𝑋), is the set of 𝐹_𝑢(𝑋)-efficient funds of 𝑌and the Lebesgue measure of 𝑌 is supported on 𝐷; (ii)𝐹_𝑢(𝑋)is the subspace of ℝ^𝑆generated by the set of the call options {𝑐_𝑢(𝑥,𝑎)∣𝑥∈𝑌,𝑎∈ℝ}written on the elements of𝑌. If 𝑢∈𝑋,𝐹_𝑢(𝑋)is the subspace 𝑋₁ of ℝ^𝑆generated by the set of call options 𝑂₁={𝑐_𝑢(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 and 𝑆 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 𝐴_𝑖(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 the 𝑖-asset in terms of the numeraire is 𝑝(𝑠)⋅𝐴_𝑖(𝑠),𝑠=1,2,…,𝑆. (3.1) The value payoff vector under prices 𝑝=(𝑝(1),…,𝑝(𝑆)) of the 𝑖-asset at time-period 1 is 𝑝•𝐴_𝑖=𝑝(1)⋅𝐴_𝑖(1),…,𝑝(𝑆)⋅𝐴_𝑖(𝑆). (3.2) The value payoff matrix under prices 𝑝 of the 𝑛 assets is denoted by 𝑝•𝐴 (3.3) and it is actually the matrix whose columns are the vectors 𝑝•𝐴_𝑖, 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 Δ=𝑝∈ℝ₊^𝑆𝐿∣^𝑆_𝑠=1𝑝(𝑠)₁=1 (3.4) in terms of the induced topology of the Euclidean space ℝ^𝑆𝐿, 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. The 𝑖-real asset can be viewed as an asset whose payoff is expressed in 𝐿 different currencies in every state of the world. 𝐴_𝑖(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 𝐴_𝑖𝑗(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, 𝑝_𝑗(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•𝐴_𝑖 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 𝑙=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 𝑝_𝑙(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 the 𝐿 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∈ℝ^𝐿. 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).

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 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•𝐴_𝑛] 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•𝐴_𝑛] 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•𝐴_𝑛] 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 𝐴_𝑖, 𝑖=1,2,…,𝑛, of the assets, the numeraire payoffs 𝑝•𝐴_𝑖, 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 𝐴_𝑖, 𝑖=1,2,…,𝑛, of the real assets, such that the numeraire payoff vectors 𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛 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 ℝ^𝑆, because in this case the dimension of the sublattice of ℝ^𝑆 generated by 𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛, 𝑢 is the number of the different values of the basic function of the set of vectors {𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛,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•𝐴_𝑛] by options with respect to the nominal asset 𝑢=𝟏 is the whole space ℝ^𝑆. 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 𝐴_𝑖, 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•𝐴_𝑖 of 𝐴_𝑖 whose payoff vector is 𝑝•𝐴_𝑖−𝑘𝑢⁺. (4.1) This indicates that the payoff of this option at the state 𝑠 is max𝑝(𝑠)⋅𝐴_𝑖(𝑠)−𝑘𝑢(𝑠),0, (4.2) expressed in units of the numeraire. We denote this option by 𝑐_𝑢(𝑝•𝐴_𝑖,𝑘). The equivalent put option is defined in a similar way, namely, it is the claim (𝑘𝑢−𝑝•𝐴_𝑖)⁺ and it is denoted by 𝑝_𝑢(𝑝•𝐴_𝑖,𝑘).

The completion by options of the asset span [𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛] with respect to the nominal asset 𝑢 is determined as follows: 𝑂₁=𝑐_𝑢(𝑥,𝑘),𝑘∈ℝ,𝑥∈𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛,𝑋₁=𝑂₁, (4.3) and for any natural number 𝑚≥1𝑂_𝑚=𝑐_𝑢(𝑥,𝑘),𝑘∈ℝ,𝑥∈𝑋_𝑚−1,𝑋_𝑚=𝑂_𝑚. (4.4) In the above definition 𝑋₀=𝑋=𝐴(𝑝).

We denote by 𝐴(𝑝) the numeraire-asset span [𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛] of the real asset structure consisted by the assets 𝐴_𝑖, 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•𝐴_𝑖, 𝑖=1,2,…,𝑛.

Definition 4.1. The completion by options of𝐴(𝑝)with respect to 𝑢 is the following subspace 𝐹_𝑢(𝐴(𝑝)) of ℝ^𝑆: 𝐹_𝑢𝐴𝑝=^∞_𝑚=1𝑋_𝑚 (4.5) of ℝ^𝑆.

By Theorem 3 in [8] what is proved is that 𝐹_𝑢(𝐴(𝑝)) is the sublattice 𝑆(𝑌(𝑝)) of ℝ^𝑆 generated by the elements of the span 𝑌(𝑝) generated by the vectors 𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛,𝑢 of ℝ^𝑆.

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 𝐴 whose columns are the vectors 𝐴_𝑖, or else 𝐴_𝑖∈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 that 𝑛≥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 ℝ^S𝐿𝑛. We will use 𝐴 by both ways.

Supposing that 𝐴_𝑖∈ℝ^𝑆𝐿₊₊, our aim is to determine the basic set of the asset span 𝐴(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•𝐴_𝑛,𝑢. But these vectors are all positive, hence what we have to check is whether the variation of 𝑝 and the payoffs 𝐴_𝑖, 𝑖=1,2,…,𝑛, allow for these vectors to be linearly independent except negligible sets.

We show that the set intΔ=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 subset 𝑀⊆ℝ^𝑘 is called a smooth manifold (of class 𝐶^𝑟,𝑟≥1) of dimension 𝑚 if each 𝑥∈𝑀 has a neighborhood 𝑊∩𝑀 that is diffeomorphic to an open subset of the Euclidean space ℝ^𝑚.

Suppose that 𝐹 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 ℝ^𝑚 (𝑉⊆𝐹,𝐷⊆ℝ^𝑚). We show that a chart of 𝐹 around 𝑥∈𝐹 is a triple (𝜓,𝐷,𝑉) in which 𝐷 is an open subset of ℝ^𝑚, 𝑉 is a subset of 𝐹 containing 𝑥, and 𝜓⁻¹∶𝑉→𝐷 is a 𝐶^𝑟-diffeomorphism. Every such subset 𝑉 of 𝐹 is called an open neighborhood of 𝑥 in 𝐹.

A subset 𝐻 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 in 𝐹 is called generic.

If 𝐹^′′ 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 manifold 𝐹 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 function 𝑓∶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 function 𝑓∶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 function 𝑓∶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.

A 𝑦∈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 an 𝑚-dimensional manifold 𝑀 in a Euclidean space ℝ^𝑘 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 a𝐶^𝑘-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 a𝐶^𝑟-map into𝑁, where the𝐶^𝑟-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 map𝐹_𝑥∶𝑀→𝑁, 𝐹_𝑥(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•𝐴_𝑛} are linearly independent.

Proof. If 𝐴(𝑝) is the matrix whose columns are the elements of the set 𝑝•𝐴₁,…,𝑝•𝐴_𝑛, (5.1) we have to show that for a set of full measure in 𝑀×ℝ^𝑆𝐿𝑛₊₊, we have rank𝐴(𝑝)=𝑛. We define 𝑇_𝜎∶𝑀×ℝ^𝑆𝐿𝑛₊₊→ℝ, where 𝑇_𝜎(𝑝,𝐴)=det𝐴_𝜎(𝑝) and 𝐴_𝜎(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). 𝑇_𝜎 is transversal to 0, namely, 0 is a regular value of 𝑇_𝜎. Since this is true by Lemma 5.2, then 𝑇_𝜎⁻¹(0)=𝑃_𝜎 is either empty or a submanifold of 𝑀×ℝ^𝑆𝐿𝑛₊₊ of dimension 𝑆𝐿𝑛+𝑆𝐿−2, namely, a null set 𝑁_𝜎 of 𝑀×ℝ^𝑆𝐿𝑛₊₊. Repeating this for all combinations 𝜎∈𝐶_𝑛, 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∈𝐶𝑛}𝑁_𝜎, being a null set of 𝑀×ℝ^𝑆𝐿𝑛₊₊, where 𝐶_𝑛 denotes the set of 𝑛-combinations of the 𝑆 objects. Namely, the generic matrix 𝐴(𝑝) is of full rank.

Lemma 5.2. The map 𝑇_𝜎∶𝑀×ℝ^𝑆𝐿𝑛₊₊→ℝ, where 𝑇_𝜎(𝑝,𝐴)=det𝐴_𝜎(𝑝), is transversal to 0 for any combination 𝜎 of 𝑛 objects out of 𝑆.

Proof. Suppose that 𝑇_𝜎⁻¹(0) is nonempty. Suppose that 𝜎={𝑠₁,𝑠₂,…,𝑠_𝑛} is a selection of 𝑛 states out of 𝑆. The image of 𝑇_𝜎(𝑝,𝐴) of 𝑇_𝜎 at (𝑝,𝐴) is the determinant of the 𝑛×𝑛 matrix 𝐴_𝜎(𝑝) whose 𝑏-row is 𝑝𝑠_𝑏⋅𝐴₁𝑠_𝑏,𝑝𝑠_𝑏⋅𝐴₂𝑠_𝑏,…,𝑝𝑠_𝑏⋅𝐴_𝑛𝑠_𝑏,𝑏=1,2,…,𝑛. (5.2) Then we have 𝑇_𝜎𝑝,𝐴=_{𝜃∈𝑆𝑛}𝜖(𝜃)𝑝𝑠₁⋅𝐴_𝜃(𝑠1)𝑠₁𝑝𝑠₂⋅𝐴_𝜃(𝑠2)𝑠₂⋯𝑝𝑠_𝑛⋅𝐴_{𝜃(𝑠𝑛)}𝑠_𝑛=_{𝜃∈𝑆𝑛}𝜖(𝜃)^𝐿_𝑑𝑠1=1𝑝𝑠₁𝑑_𝑠1𝐴_𝜃(𝑠1)𝑠₁𝑑_𝑠1⋯^𝐿_{𝑑𝑠𝑛=1}𝑝𝑠_𝑛𝑑_𝑠𝑛𝐴_{𝜃(𝑠𝑛)}𝑠_𝑛𝑑_𝑠𝑛=_{𝜃∈𝑆𝑛}^𝐿_𝑑𝑠1^𝐿_{=1𝑑𝑠2=1}⋯^𝐿_{𝑑𝑠𝑛=1}𝜖(𝜃)𝑝𝑠₁𝑑_𝑠1⋯𝑝𝑠_𝑛𝑑_𝑠𝑛𝐴_𝜃(𝑠1)𝑠₁𝑑_𝑠1⋯𝐴_{𝜃(𝑠𝑛)}𝑠_𝑛𝑑_𝑠𝑛, (5.3) where 𝑆_𝑛 is the set of permutations over 𝑛 symbols, 𝜃 is any permutation in 𝑆_𝑛, and 𝜖(𝜃) is the sign of the permutation 𝜃. Also, 𝑝(𝑠_𝑎)(𝑑_𝑠𝑎) is the spot price of the 𝑑_𝑠𝑎-good corresponding to the state 𝑠_𝑎, where 𝑎=1,2,…,𝑛 and 𝑑_𝑠𝑎=1,2,…,𝐿. In a similar way, 𝐴_{𝜃(𝑠𝑎)}(𝑠_𝑎)(𝑑_𝑠𝑎) is the amount of the 𝑑_𝑠𝑎 consumption good that the owner of one unit of the asset 𝜃(𝑠_𝑎) is going to receive at the time-period 1 if the state 𝑠_𝑎 occurs.
In order to show that 𝑇_𝜎 is transversal to 0, we have to prove that for any (𝑝,𝐴)∈𝑇_𝜎⁻¹(0) we have rank(𝐷𝑇_𝜎(𝑝,𝐴))=1. To show that, we have to verify that for any 𝑝∈𝑇_𝜎⁻¹(0) at least one partial derivative𝜕𝑇_𝜎𝑝,𝐴𝜕𝑝𝑠_𝑎𝑑_𝑠𝑎,𝑎=1,2,…,𝑛,𝑠_𝑎∈𝜎,𝑑_𝑠𝑎=1,2,…,𝐿, (5.4) is nonzero. But this is true, since 𝑝∈𝑀 and all of the products 𝐴_𝜃(𝑠1)𝑠₁𝑑_𝑠1𝐴_𝜃(𝑠2)𝑠₂𝑑_𝑠2⋯𝐴_{𝜃(𝑠𝑛)}𝑠_𝑛𝑑_𝑠𝑛 (5.5) are nonzero.

By the fact that 𝐴∈ℝ^𝑆𝐿𝑛₊₊ and by repeating the previous proof, we may take the following.

Corollary 5.3. The map 𝑇^𝐴_𝜎∶𝑀→ℝ, where 𝑇^𝐴_𝜎(𝑝)=det𝐴_𝜎(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 map 𝑇^𝐴_𝜎∶𝑀→ℝ, where 𝑇^𝐴_𝜎(𝑝)=det𝐴_𝜎(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 Lemma 5.2, 𝑇_𝜎 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 𝑇_𝜎 is a 𝐶^∞-map, hence we may suppose that it is also a 𝐶^𝑟-map with 𝑟>(𝑆𝐿−1)−1=𝑆𝐿−2. Hence the projection 𝑇^𝐴_𝜎∶𝑀→ℝ, where 𝑇^𝐴_𝜎(𝑝)=𝑇_𝜎(𝑝,𝐴) for a specific 𝐴 in a set of full measure 𝐺_𝜎⊆ℝ^𝑆𝐿𝑛₊₊, is transversal to zero.

By Transversality theorem, we can also prove the following.

Proposition 5.5. For a set of full measure 𝐺⊆ℝ^𝑆𝐿𝑛₊₊ and for any asset structure 𝐴∈𝐺, the set of prices for which the vectors {𝑝•𝐴₁,…,𝑝•𝐴_𝑛} are linearly independent is generic in 𝑀.

Proof. Since from Lemma 5.2, 𝑇_𝜎 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 𝐺_𝜎 of ℝ^𝑆𝐿𝑛₊₊ such that for any fixed 𝐴 in this set, the projection 𝑇^𝐴_𝜎 is transversal to zero for each 𝜎. Hence, we may apply the Preimage theorem for 𝑇^𝐴_𝜎∶𝑀→ℝ whenever 𝐴 lies in 𝐺_𝜎. Then we get that (𝑇^𝐴_𝜎)⁻¹(0) is a smooth submanifold of 𝑀 of dimension 𝑆𝐿−2=(𝑆𝐿−1)−1. Hence, (𝑇^𝐴_𝜎)⁻¹(0) is a null set in 𝑀, for any combination 𝜎∈𝐶_𝑛. Consider the set of full measure ⋂_{𝜎∈𝐶𝑛}𝐺_𝜎=𝐺 in ℝ^𝑆𝐿𝑛₊₊. For any 𝐴∈𝐺 and for any 𝜎, we have that (𝑇^𝐴_𝜎)⁻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•𝐴_𝑛 are linearly dependent is the intersection ⋂_{𝜎∈𝐶𝑛}(𝑇^𝐴_𝜎)⁻¹(0), being a null set in 𝑀.

For any (𝑝,𝐴)∈𝑀×ℝ^𝑆𝐿𝑛₊₊, we define the basic function 𝛽∶{1,2,…,𝑆}→Δ_𝑛−1 (see in [9], page 2797) of the vectors 𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛𝛽(𝑠)=1𝑝,𝐴𝑦(𝑠)𝑦𝑝,𝐴¹𝑝,𝐴(𝑠),𝑦₂𝑝,𝐴(𝑠),…,𝑦_𝑛𝑝,𝐴(𝑠), (5.6) where 𝑦𝑝,𝐴_𝑖(𝑠)=𝑝(𝑠)⋅𝐴_𝑖(𝑠)𝑦(𝑠)=,𝑖=1,2,…,𝑛,𝑝,𝐴^𝑛_𝑖=1𝑦𝑝,𝐴_𝑖(𝑠), (5.7) for all the states 𝑠 such that 𝑦(𝑝,𝐴)(𝑠)>0. Note that Δ_𝑛−1 is the simplex of the positive cone ℝ^𝑛₊ of ℝ^𝑛.

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•𝐴_𝑛 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)⋅𝐴_𝑖(𝑠)=𝑝(𝑠^′)⋅𝐴_𝑖(𝑠^′) for some asset 𝑖=1,2,…,𝑛 is denoted by 𝐸. This set of (𝑝,𝐴) is actually the set ⋃^𝑛_𝑖=1⋃_{(𝑠,𝑠′)∶𝑠≠𝑠′}(𝐹⁻¹_{𝑠,𝑠′,𝑖}(0))^𝑐, where 𝐹⁻¹_{𝑠,𝑠′,𝑖}∶𝑀×ℝ^𝑆𝐿𝑛₊₊→ℝ is the map 𝐹_{𝑠,𝑠′,𝑖}(𝑝,𝐴)=𝑝(𝑠)⋅𝐴_𝑖(𝑠)−𝑝(𝑠^′)⋅𝐴_𝑖(𝑠^′) and (𝐹⁻¹_{𝑠,𝑠′,𝑖}(0))^𝑐 is the complement of 𝐹⁻¹_{𝑠,𝑠′,𝑖}(0) in 𝑀×ℝ^𝑆𝐿𝑛₊₊. Let us prove that 𝐹_{𝑠,𝑠′,𝑖} is transversal to 0. We have to prove that for all (𝑝,𝐴)∈𝐹⁻¹_{𝑠,𝑠′,𝑖}(0), rank𝐷𝐹_{𝑠,𝑠′,𝑖}(𝑝,𝐴)=1. Note that for any such (𝑝,𝐴), we have 𝜕𝐹_{𝑠,𝑠′,𝑖}(𝑝,𝐴)/𝜕𝑝(𝑠)(𝑑)=𝐴_𝑖(𝑑)>0, where 𝑑=1,2,…,𝐿 denotes any of the consumption goods. Thus, by applying the Preimage Theorem, 𝐹⁻¹_{𝑠,𝑠′,𝑖}(0) is a submanifold of 𝑀×ℝ^𝑆𝐿𝑛₊₊ of dimension 𝑆𝐿𝑛+𝑆𝐿−2, hence a null set. Namely, the union ⋃^𝑛_𝑖=1⋃_{(𝑠,𝑠′)∶𝑠≠𝑠′}(𝐹⁻¹_{𝑠,𝑠′,𝑖}(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•𝐴_𝑛 are linearly independent in ℝ^𝑆 is a set of full measure as it is shown by Lemma 5.2, since this set is the complement of ⋂_{𝜎∈𝐶𝑛}𝑇_𝜎⁻¹(0) in 𝑀×ℝ^𝑆𝐿𝑛₊₊, being a null set (where 𝐶_𝑛 denotes the set of 𝑛-combinations of the 𝑆 objects). Hence the set of (𝑝,𝐴) such that 𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛 are linearly independent and every pair of different states (𝑠,𝑠^′) is separated by some 𝑝•𝐴_𝑖, 𝑖=1,2,…,𝑛, is the set _{𝜎∈𝐶𝑛}𝑇_𝜎⁻¹(0)^−1𝑐∩^𝑛_{𝑖=1𝑠,𝑠′∶𝑠≠𝑠′}𝐹⁻¹_{𝑠,𝑠′,𝑖}(0)^𝑐, (5.8) 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•𝐴_𝑛] with respect to the riskless bond is ℝ^𝑆.

Proof. Consider the generic subset 𝐾 of 𝑀×ℝ^𝑆𝐿𝑛₊₊ such that the states of the world are separated by the vectors of {𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛}, 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•𝐴_𝑛]. 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•𝐴_𝑛}; see also Definition 10 and Theorem 11 in [8]. Hence the completion of the numeraire asset span 𝐴(𝑝)=[𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛] with respect to the riskless bond 𝟏 is ℝ^𝑆 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•𝐴_𝑛} 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, 𝐹_𝟏(𝐴(𝑝))=ℝ^𝑆. 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 dim𝐹_𝟏(𝐴(𝑝))<𝑆. But if 𝛽(𝑝,𝐴)(𝑠)=𝛽(𝑝,𝐴)(𝑠^′), this implies 𝑝(𝑠)⋅𝐴_𝑗(𝑠)∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)=𝑝𝑠^′⋅𝐴_𝑗𝑠^′∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′ (5.9) for this (𝑝,𝐴) and for any 𝑗=1,2,…,𝑛. These equations imply 𝑝•𝐴₁(𝑠),𝑝•𝐴₂(𝑠),…,𝑝•𝐴_𝑛(𝑠)=𝑝•𝐴₁𝑠^′,𝑝•𝐴₂𝑠^′,…,𝑝•𝐴_𝑛𝑠^′∑^𝑛_𝑖=1𝑝•𝐴_𝑖(𝑠)∑^𝑛_𝑖=1𝑝•𝐴_𝑖𝑠^′, (5.10) namely, 𝑝(𝑠)⋅𝐴₁(𝑠),𝑝(𝑠)⋅𝐴₂(𝑠),…,𝑝(𝑠)⋅𝐴_𝑛(𝑠)=𝑝𝑠^′⋅𝐴₁𝑠^′,𝑝𝑠^′⋅𝐴₂𝑠^′,…,𝑝𝑠^′⋅𝐴_𝑛𝑠^′∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′. (5.11) Then 𝑝(𝑠)⋅𝐴_𝑗(𝑠)=𝑝𝑠^′⋅𝐴_𝑗𝑠^′∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′ (5.12) for any 𝑗=1, 2,…,𝑛. Since we supposed that 𝟏∈[𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛], there are real numbers 𝑘_𝑗, 𝑗=1,2,…,𝑛 such that 𝟏=^𝑛_𝑗=1𝑘_𝑗𝑝•𝐴_𝑗. (5.13) Then, 1=𝟏(𝑠)=∑^𝑛_𝑗=1𝑘_𝑗(𝑝•𝐴_𝑗)(𝑠) and 1=𝟏(𝑠^′)=∑^𝑛_𝑗=1𝑘_𝑗(𝑝•𝐴_𝑗)(𝑠^′). We get that ^𝑛_𝑗=1𝑘_𝑗𝑝•𝐴_𝑗(𝑠)=^𝑛_𝑗=1𝑘_𝑗𝑝•𝐴_𝑗𝑠^′, (5.14) or else 1=^𝑛_𝑗=1𝑘_𝑗𝑝(𝑠)⋅𝐴_𝑗(𝑠)=^𝑛_𝑗=1𝑘_𝑗𝑝𝑠^′⋅𝐴_𝑗𝑠^′. (5.15) Then 1=^𝑛_𝑗=1𝑘_𝑗𝑝𝑠^′⋅𝐴_𝑗𝑠^′∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′=^𝑛_𝑗=1𝑘_𝑗𝑝𝑠^′⋅𝐴_𝑖𝑠^′. (5.16) Hence, due to (5.9), ∑^𝑛_𝑖=1(𝑝(𝑠)⋅𝐴_𝑖)(𝑠)=∑^𝑛_𝑖=1(𝑝(𝑠^′)⋅𝐴_𝑖)(𝑠^′), which indicates that 𝑝(𝑠)⋅𝐴_𝑖(𝑠)=𝑝𝑠^′⋅𝐴_𝑖𝑠^′, (5.17) 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•𝐴_𝑛} 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•𝐴_𝑛}. 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 𝐹_𝟏(𝐴(𝑝))=ℝ^𝑆 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•𝐴_𝑛,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•𝐴_𝑛] with respect to the riskless bond 𝟏 is ℝ^𝑆 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•𝐴_𝑛} 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, 𝐹_𝟏(𝐴(𝑝))=ℝ^𝑆. Let us verify this fact. The basic function 𝛿(𝑝,𝐴) of the basic set {𝑝•𝐴₁,𝑝•𝐴₂,…,𝑝•𝐴_𝑛,𝟏} of the market for an element (𝑝,𝐴)∈𝐾₂ has the form 𝛿(𝑠)=1𝑝,𝐴∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)𝑝(𝑠)+1⋅𝐴₁(𝑠),𝑝(𝑠)⋅𝐴₂(𝑠),…,𝑝(𝑠)⋅𝐴_𝑛(𝑠),1, (5.18) for any 𝑠=1,2,…,𝑆. If we suppose that the dimension of the completion 𝐹_𝟏(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 1∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)=1+1∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′,+1 (5.19) an equation which corresponds to the equality of the last components of these vectors. This equation implies ∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)=∑^𝑛_𝑖=1𝑝(𝑠^′)⋅𝐴_𝑖(𝑠^′). Also, for each 𝑗=1,2,…,𝑛, we have 𝑝(𝑠)⋅𝐴_𝑗(𝑠)∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)=𝑝𝑠+1^′⋅𝐴_𝑗𝑠^′∑^𝑛_𝑖=1𝑝𝑠^′⋅𝐴_𝑖𝑠^′.+1 (5.20) But from equation ∑^𝑛_𝑖=1𝑝(𝑠)⋅𝐴_𝑖(𝑠)=∑^𝑛_𝑖=1𝑝(𝑠^′)⋅𝐴_𝑖(𝑠^′), this also implies 𝑝(𝑠)⋅𝐴_𝑗(𝑠)=𝑝(𝑠^′)⋅𝐴_𝑗(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•𝐴_𝑛, 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 𝐹_𝟏(𝐴(𝑝))=ℝ^𝑆 holds for any (𝑝,𝐴)∈𝐾₂. Hence in any case for any (𝑝,𝐴)∈𝐾 the completion is equal to ℝ^𝑆.

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 𝑆=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 𝐴₁=14231511,𝐴₂=64845523, (6.1) 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 𝐴_𝑖, 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 ℝ^𝑆𝐿 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 𝑝•𝐴₁=34341214,𝑝•𝐴₂=767356712, (6.2) 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: 𝛽(1)=9,3514,3512(2)=935,𝛽,4928,4912,𝛽(3)=349,514,614(4)=14,𝛽18,13242,13272.132 (6.3) Since the four values of 𝛽 are distinct, from the Theorem 14 in [8], we get that dim𝐹_𝟏𝐴𝑝=4, (6.4) namely, that the completion 𝐹_𝟏(𝐴(𝑝)) of the numeraire asset span 𝐴𝑝=𝑝•𝐴₁,𝑝•𝐴₂ (6.5) 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 𝐹_𝟏(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 𝑦^′₄=(0,0,0,11/6). We also consider 𝑦^′₁=𝑦₁, 𝑦^′₂=𝑦₂, 𝑦^′₃=𝑦₃. The values of 𝛾, being the basic function of 𝑦^′₁, 𝑦^′₂, 𝑦^′₃, 𝑦^′₄, are the following: 𝛾(1)=9,3514,351235,0=𝑃^′₁(2)=9,𝛾,4928,491249,0=𝑃^′₂,𝛾(3)=3,514,61414,0=𝑃^′₃,𝛾(4)=18,26442,26472,12642=𝑃^′₄. (6.6) We denote by 𝐷 the 4×4 matrix whose columns are the vectors 𝑃^′₁,𝑃^′₂,𝑃^′₃,𝑃^′₄. The vectors of the positive basis {𝑏₁,𝑏₂,𝑏₃,𝑏₄} of 𝐹_𝟏(𝐴(𝑝)) are determined through the matrix equation (𝑏₁,𝑏₂,𝑏₃,𝑏₄)^𝑇=𝐷⁻¹⋅(𝑦^′₁,𝑦^′₂,𝑦^′₃,𝑦^′₄)^𝑇, where (𝑏₁,𝑏₂,𝑏₃,𝑏₄)^𝑇 is the 4×4 matrix whose rows are the vectors 𝑏₁, 𝑏₂, 𝑏₃, 𝑏₄ and (𝑦^′₁,𝑦^′₂,𝑦^′₃,𝑦^′₄)^𝑇 is the 4×4 matrix whose rows are the vectors 𝑦^′₁, 𝑦^′₂, 𝑦^′₃, 𝑦^′₄. We have that 𝐷⁻¹=515−2−651275−4414372−7−712−4428307−28110002 (6.7) 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 𝑏^′₁=(1,0,0,0), 𝑏^′₂=(0,1,0,0), 𝑏^′₃=(0,0,1,0), 𝑏^′₄=(0,0,0,1) is also a positive basis of 𝐹_𝑢(𝐴(𝑝)), which is equal to ℝ⁴. According to Theorem 21 of [8], 𝐹_𝟏(𝐴(𝑝)) is the subspace of ℝ⁴ generated by {𝑐_𝟏(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, 𝑐_𝟏(𝑝•𝐴₂,7/12), 𝑐_𝟏(𝑝•𝐴₂,5/6), 𝑐_𝟏(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 manifold 𝑀 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, 𝑝_𝑖(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 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). (6.8) 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 (1,2),(1,3),(1,4),(2,3),(2,4),(3,4). (6.9) The determinants are those of the corresponding 2×2 matrices 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). (6.10) 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 𝑝_𝑖(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 𝑇_𝑠,𝑠′∶𝑀→ℝ with 𝑇_𝑠,𝑠′(𝑝)=det𝐻_{(𝑠,𝑠′)}. The determinant of a matrix is a 𝐶^∞-map; hence we may take it to be a 𝐶^𝑟-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 𝑇⁻¹_𝑠,𝑠′(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 𝑇⁻¹_𝑠,𝑠′(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′}𝑇⁻¹_𝑠,𝑠′(0) of 𝑀. This set is actually a null set. Let us verify whether some 𝑇⁻¹_𝑠,𝑠′(0) is a nonempty set or not in this case. If we would like to see whether 𝑇⁻¹_1,2(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 𝑇⁻¹_1,2(0). Hence, 𝑇⁻¹_1,2(0) is a six-dimensional submanifold of a seven-dimensional manifold which is 𝑀 itself. Hence, 𝑇⁻¹_1,2(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^′.