ypoly/ARTICLES (POLYRAKIS)/the ccompletion i… · DOI: 10.1007/s10203-006-0059-z DEF 29, 1–21 (2006) Decisionsin Economicsand Finance c Springer-Verlag 2006 The completion of security

DOI: 10.1007/s10203-006-0059-zDEF 29, 1 – 21 (2006) Decisions in

Economics andFinancec© Springer-Verlag 2006

The completion of security markets*

Christos Kountzakis, Ioannis A. Polyrakis

Department of Mathematics, National Technical University of Athense-mail: [email protected]

Received: 24 January 2005 / Accepted: 17 November 2005 – c© Springer-Verlag 2006

Abstract. In this article we study the completion by options of a two-periodsecurity market in which the space of marketed securities is a subspace X ofR

m. Although there are important results about the completion (by options)Z of X, the problem of the determination of Z in its general form is stillopen. In this paper we solve this problem by determining a positive basis ofZ. This method of positive bases simplifies the theory of security marketsand also answers other open problems of this theory. In the classical papersof this subject, call and put options are taken with respect to the risklessbond 1 of R

m. In this article we generalize this theory by taking call and putoptions with respect to different risky vectors u from a fixed vector subspaceU of R

m. This generalization was inspired by certain types of exotic optionin finance.

Mathematics Subject Classification (2000): 46B40, 46A35, 91B28, 91B30

Journal of Economic Literature Classification: G190, D520

1. Introduction

In this article we study a two-period security market with a finite set{1, 2, . . . , m} of states. We assume that the space of marketed securities is asubspaceX of R

m which is generated by the primitive securities x1, x2, . . . , xn.

* We would like to thank two anonymous referees for their useful remarks which helpedus to improve this article. One of them suggested to us that such exotic options as lookbackoptions have risky strike vectors. This was a crucial remark for the structure of this article.

This research is supported by the HERAKLEITOS project which is co-funded by theEuropean Social Fund (75%) and National Resources (25%).

2 C. Kountzakis, I.A. Polyrakis

In the theory of security markets call and put options are taken with respectto the riskless bond u = 1. Nevertheless certain types of exotic option in fi-nance, such as the forward-start and lookback options, are taken with respectto risky strike vectors which depend on the payoffs of the securities in thedates before their expiration dates. These strike vectors are not necessarilythe same for the different securities and an important remark is that thesestrike vectors belong to a minimal subspace U of R

m. Motivated by suchoptions, we generalize the theory of security markets by taking call and putoptions with respect to risky strike vectors u from a fixed subspace U ofR

m which we call the strike subspace. If U is a one-dimensional subspacegenerated by a vector u, then the call and put options are taken with respectto the fixed risky vector u of R

m and, if u is the riskless bond 1, call and putoptions are taken with respect to 1 (classical case). Also in this article we donot assume that the primitive securities xi and the strike vectors u are neces-sarily positive. It is known that call and put options do not belong in generalto the space of marketed securities. So, in the theory of security markets,the market is completed by considering as space of marketed securities thesubspace of R

m generated by all the possible call and put options.In Section 2 we suppose that the payoff space E is a general vector lattice

and we start the study of security markets by a mathematical definition of thesubspace generated by the call and put options which we call “the completionby options of X” and denote by FU(X). If U is a one-dimensional subspacegenerated by u we denote the completion of X by Fu(X). This definitionexpresses the way this subspace is generated inductively by taking call andput options of marketed securities and incorporating them in the market. IfY is the subspace of E generated by the set X ∪ U we show, in Theorem 3,that FU(X) is the sublattice of E generated by Y . In the sequel we return tothe finite-dimensional case and we study the problem of the determinationof FU(X) in the case in which the payoff space is the space R

m.For the history of this problem recall that Ross (1976), in his seminal

paper, proved that the completion by options F1(X) of X is the whole spaceR

m if and only if X has an efficient fund. Arditti and John (1980) provedthat, if X has an efficient fund, then almost any portfolio (in the sense of theLebesgue measure of X) is an efficient fund. John (1981) studied the case inwhich the Ross assumption is not satisfied and the completion by options ofX is a proper subspace of R

m. He defined the notion of maximally efficientfund and he proved that F1(X) is generated by the call and put options writtenon a maximally efficient fund of X. As shown in John (1981), almost anyportfolio is a maximally efficient fund but there is no way to determine amaximally efficient fund of X. Also there is no way to determine whetheran efficient fund of X exists. Therefore the call and put options written onthe elements of X generate the completion by options of X but, in general,

The completion of security markets 3

we cannot determine F1(X) by taking call and put options of marketedsecurities. The most serious difficulty is that we do not know the dimensionof F1(X); therefore if we try to determine the completion of X by findinglinearly independent call and put options, at any step we do not know ifthe set of call and put options we have found generates F1(X) or a propersubspace. So, although there are very important results about the completionby options of X, the problem of the determination of F1(X) in its generalform remains open. Ross (1976), Green and Jarrow (1987) and Brown andRoss (1991) observed that, in security markets, any call and put option canbe replicated (i.e., F1(X) = X) if and only if X is a sublattice of R

m. Sincethere are no criteria to check whether or not X is a sublattice, we can saythat this problem also remains open for applications.

As shown in Theorem 2, FU(X) is the sublattice S(Y ) of Rm generated

by Y but there is no way to determine S(Y ). The only known result fromthe theory of ordered spaces says that S(Y ) is the set of finite suprema ofthe elements of Y . However we cannot use it for the determination of S(Y )

because we do not know the dimension of S(Y ); therefore we do not knowhow many linearly independent finite suprema are needed to achieve S(Y ).

In this article we use the results of Polyrakis (1999) in which a newmathematical method for the determination of the sublattice of R

m generatedby a set of linearly independent positive vectors of R

m is provided and wedetermine the completion FU(X) of X by determining a positive basis ofFU(X).A vector-valued function β which we call basic function is importantfor our study. The function β is defined on the set {1, 2, . . . , m} of statesand it is easy to determine the values of β. We prove, in Theorem 14, that thedimension of FU(X) is equal to the number of different values of β; thereforewe provide an easy criterion for the dimension of FU(X). By this result, wecharacterize the markets in which any contingent claim is replicated and themarkets where the call and put options fill the payoff space. Also based onthe values of β we determine a positive basis {bi} of FU(X). Brown andRoss (1991) posed the following question: Do basic derivative assets existso that any derivative asset is a portfolio of these derivative assets? As weshow, the vectors bi of the positive basis of FU(X) are the basic derivativeassets in the sense of Brown and Ross. Also the vectors bi are derivativeswith minimal support in FU(X), i.e., for each i, the support of bi defines aminimal block of states in the sense that there is no element (derivative) x

of Fu(X) with nonzero payoffs in a set of states “smaller” than the supportof (bi). Also we show that the vectors bi have disjoint supports; thereforethe vectors of the positive basis are “minimal disjoint derivatives” whichgenerate any other derivative. In Section 5 we assume that the call and putoptions are taken with respect to an arbitrary but fixed risky strike vector u. Inthe spirit of the articles of Ross (1976) and John (1981), we define the notion


of Fu(X)-efficient fund as a vector e of FU(X) so that the set of nontrivialcall and put options written on e generate Fu(X) and we give a formula, inTheorem 19, which determines completely the set of Fu(X)-efficient funds.Moreover we note that our results have important applications in the casewhere u = 1. In Theorem 23 we prove that the notion of F1(X)-efficientfund and that of maximally efficient fund are equivalent. Lastly we remarkthat, in the works of Green and Jarrow (1987) and Nachman (1988), theproblem of the completion by options is studied in the case where the spaceX of marketed securities is an infinite-dimensional subspace of the spaceR

� of the real-valued functions defined on the set � of states and the risklessbond is the constant function 1. For a study of two-period security marketswe refer to the book of LeRoy and Werner (2001).

2. The completion by options in vector lattices

In this article we study a two-period security market with a finite numberof primitive securities labelled by the natural numbers 1, 2, . . . , n, withpayoffs the linearly independent vectors x1, x2, . . . , xn of a vector lattice(Riesz space) E which we call the payoff space. A portfolio is a vectorθ = (θ1, θ2, . . . θn) of R

n where θi is the number of units of the ith security.Then T (θ) = ∑n

i=1 θixi ∈ Rm is the payoff of θ . Since the operator T

is one-to-one, it identifies portfolios with their payoffs. Thus the vectorsx1, x2, . . . , xn are called primitive securities, the subspace

X = [x1, x2, . . . , xn]of E, generated by the vectors xi , is referred to as the space of marketedsecurities and the vectors of X as portfolios. A vector x ∈ E is marketedwhile x is replicated if x ∈ X.

Recall that, for any x, y ∈ E, we denote by x ∨ y the supremum and byx ∧ y the infimum of {x, y} in E. Also x+ = x ∨ 0, x− = (−x) ∨ 0 and|x| = x ∨ (−x) are the positive part, negative part and absolute value of x.A linear subspace Z of E is a sublattice or a Riesz subspace of E if, forany x, y ∈ X, x ∨ y and x ∧ y belong to Z. Suppose that B is a subset of E.The intersection of all sublattices of E which contain B is again a sublatticeof E and it is the smallest sublattice of E which contains B. This sublatticeis denoted by S(B) and it is called the sublattice of E generated by B. If[B] is the linear subspace generated by B, it is easy to show that S(B) is thesublattice of E generated by [B], i.e., S(B) = S([B]). It is known that S(B)

is the set of finite suprema of elements of [B], but this is only a theoreticalresult because we cannot determine S(B) by using finite suprema of [B]even in the case where E = R

m. For more details on vector lattices we referto the book of Aliprantis and Burkinshaw (1985).


For any x ∈ E, u ∈ E and any real number a the vector cu(x, a) =(x − au)+ is the call option and pu(x, a) = (au − x)+ is the put option ofx with respect to the strike vector u and exercise price a. Hence pu(x, a) =(au − x)+ = (−x − (−a)u)+ = cu(−x, −a).

In the classical case E = Rm and call and put options are taken with

respect to the riskless bond 1. In this paper we define call and put optionswith respect to the elements u of a subspace U of E. Thus here we useU to denote an arbitrary but fixed subspace of E which we call the strikesubspace. The elements of U are the strike vectors. This generalization isinspired by such options in finance as the forward-start and lookback optionswhich we present below. Note also that options with risky strike vectors canbe found in foreign exchange markets.

2.1. The forward-start and lookback options

Consider the stochastic multiperiod model of security markets with a finiteset of states � = {1, 2, . . . , m}, a finite time horizon T = {0, 1, 2, . . . , T },an increasing family δ = {�0, �1, . . . , �T } of partitions of � which rep-resents the available information during the different dates with �0 = {�}and �T = {{1}, {2}, . . . , {m}}; see Magill and Quinzii (1996), Chapter 4or LeRoy and Werner (2001), Chapter 21. A security is a financial contractissued at the period t = 0 whose payoff is described by x = (x1, . . . , xT )

where xt ∈ Rm is the payoff of x at the date t . (For any t ∈ T, the payoff

xt may be the dividends paid at date t , or the prices at which x is tradedat t). Since x is adapted with δ, for any t the payoff xt of x is constant onthe elements of �t . Thus, if we suppose that �t = {σ t

1, .., σtkt}, we have

xt (j) = ati ∈ R for each j ∈ σ t

i . We consider European call and put optionswritten on the financial contract x with expiration date T . These optionsdepend only on the payoff of x at the terminal date; therefore we considerthe payoff of x to be its payoff xT at the date T . Thus our economy can beconsidered as a two-period (0 and T ) economy where the other (intermedi-ate) dates do not affect the options. In the next types of exotic option (seeMusiela and Rutkowski (1998)), the strike vectors depend on the payoffs ofsecurities in the intermediate dates.

The forward-start call option of x with expiration time T and time ofdependence t is the call option on xT with strike vector u = xt and exerciseprice a = 1. Its payoff is cu(x

T , 1) = (xT − xt )+. Since xt is constanton the elements of �t , the strike vector xt belongs to the subspace Ut ofR

m generated by the characteristic vectors uti, i = 1, 2, . . . , kt of σ t

i , i.e.,ut

i(j) = 1 for any j ∈ σ ti and ut

i(j) = 0 for any j �∈ σ ti ; therefore Ut is

the strike space for this type of option. The lookback call option written onthe contract x is the call option on xT with strike vector u the minimum of


the vectors xt for t = 0, 1, 2, . . . , T and exercise price a = 1. Its payoff iscu(x

T , 1) = (xT −u)+. The lookback put option written on the contract x isthe put option on xT with strike vector u the maximum of the vectors xt fort = 0, 1, 2, . . . , T and exercise price a = 1, i.e., pu(x

T , 1) = (u − xT )+.

In these two cases UT is the strike subspace.

Example 1. Suppose that � = {1, 2, 3, . . . , 10}, T = {0, 1, 2, 3}, �0 =�, �1 = {{1, 2, 4, 6, 8}, {3, 5, 7, 9, 10}}, �2 = {{1, 2, 4}, {6, 8},{3, 5}, {7, 9, 10}}, and �3 = {{1}, . . . , {10}}. Consider a financial marketwith three securities (financial contracts) x1, x2, x3 so that

x01 = 13

10(1, 1, 1, 1, 1, 1, 1, 1, 1, 1), x1

1 = 1

5(7, 7, 6, 7, 6, 7, 6, 7, 6, 6),

x21 = 1

3(7, 7, 6, 7, 6, 0, 2, 0, 2, 2) and x3

1 = (2, 2, 4, 3, 0, 0, 0, 0, 1, 1)

are the payoffs of x1 at the different dates,

x02 = 19

10(1, 1, 1, 1, 1, 1, 1, 1, 1, 1), x1

2 = 1

5(7, 7, 12, 7, 12, 7, 12, 7, 12, 12),

x22 = 1

6(2, 2, 9, 2, 9, 18, 18, 18, 18, 18), x3

2 = (0, 0, 1, 1, 2, 3, 1, 3, 4, 4)

are the payoffs of x2 and the payoffs of x3 are the vectors

x03 = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1), x1

3 = 1

5(6, 6, 4, 6, 4, 6, 4, 6, 4, 4),

x23 = (2, 2, 0, 2, 0, 0,

4

3, 0,

4

3,

4

3), x3

3 = (3, 3, 0, 0, 0, 0, 4, 0, 0, 0).

Consider European forward-start options with expiration date t = 3 andtime of dependence t = 1. Then the vectors x3

1 , x32 , x

33 are the payoffs of the

securities (with respect these options) and the subspace X = [x31 , x

32 , x

33 ]

of R10 generated by the vectors x3

i is the space of marketed securities. Theforward-start call option of x1, or more accurately of x3

1 , with time of depen-dence 1 is taken with respect to the risky strike vector u = x1

1 = 75u1 + 6

5u2,where u1

1 = (1, 1, 0, 1, 0, 1, 0, 1, 0, 0) and u12 = (0, 0, 1, 0, 1, 0, 1, 0, 1, 1)

are the characteristic vectors of the elements of �1 and we have cu(x31 , 1) =

(x3 − u)+ = 15(3, 3, 14, 8, 0, 0, 0, 0, 0, 0). Of course the strike vector x1

1belongs to the subspace U = U 1 of R

10 generated by the characteristicvectors u1, u2. The corresponding strike vectors for the forward-start calloption of x2, x3 are the vectors x1

2 , x13 which also belong to U . Any portfolio

x = θ1x1+θ2x2+θ3x3 ∈ X has payoff at the date 1, x1 = θ1x11 +θ2x

12 +θ3x

13 ;

therefore the strike vector x1 for the forward-start call option of x with time


of dependence 1 is again a vector of U . Finally we remark that U is a sublat-tice of R

10 because u1, u2 have disjoint supports. This example is continuedin Example 15 where a positive basis of the completion of X is determined.

2.2. The completion by options

The completion of the market is the subspace of E which arises inductivelyby adding to the market the call and put options of the marketed securitiesand by taking call and put options again. We define this subspace as follows.Denote by O1 = {cu(x, a)|x ∈ X, u ∈ U, a ∈ R} the set of call optionswritten on the elements of X and by X1 the subspace of E generated bythe set O1. For any natural number n ≥ 1 denote by On = {cu(x, a)|x ∈Xn−1, u ∈ U, a ∈ R} the set of call options written on the elements of Xn−1,and by Xn the subspace of E generated by the set On. Then Xn ⊆ Xn+1

for each n because x = x+ − x− = cu(x, 0) − cu(−x, 0) ∈ Xn+1 for anyx ∈ Xn.

Definition 2. The space FU(X) = ∪∞n=1Xn is the completion by options of

X with respect to U .

If U is a one-dimensional subspace generated by a vector u of E thencall and put options are taken with respect to the fixed strike vector u. Theninstead of FU(X) we write Fu(X) and we say that Fu(X) is the completionby options of X with respect to u.

In this article we denote by Y the subspace of E generated by X ∪ U ,i.e., Y = {λx +au

∣∣x ∈ X, u ∈ U, λ, a ∈ R}. We put S2 = {x ∨y|x, y ∈ Y }

and, for any natural number n ≥ 3, we denote by Sn the set Sn = {x ∨y|x ∈Sn−1, y ∈ Y }. Since the sublattice S(Y ) of E generated by Y is the set offinite suprema of elements of Y , it follows that S(Y ) = ∪∞

n=2Sn.

Theorem 3. In the above notation, we have

(i) Y ⊆ X1,(ii) FU(X) is the sublattice S(Y ) of E generated by Y , and(iii) if U ⊆ X, then FU(X) is the sublattice of E generated by X.

Proof. For any y = x + au ∈ Y , we have y = (x + au)+ − (−x − au)+ =cu(x, −a) − cu(−x, a) ∈ X1, and so (i) holds.

To show that FU(X) ⊆ S(Y ) it is enough to show that Xn ⊆ S(Y ) forany n. For any y ∈ O1 we have y = cu(x, a) = (x − au)+ = (x − au) ∨ 0for some x ∈ X, u ∈ U and a ∈ R; therefore y ∈ S2 ⊆ S(Y ) becausex−au ∈ Y . Hence O1 and therefore also X1 are contained in S(Y ). Supposenow that Xn ⊆ S(Y ). To show that Xn+1 ⊆ S(Y ) it is enough to show thatOn+1 ⊆ S(Y ). For any z ∈ On+1 we have z = (x−au)∨0 for some x ∈ Xn,u ∈ U and a ∈ R. Since Xn ⊆ S(Y ) we have that x ∈ Sk for some k.Also we


remark that z = −au+(x∨au); hence z ∈ Y+Sk+1 ⊆ S(Y )+S(Y ) = S(Y ).Therefore On+1 ⊆ S(Y ), hence Xn+1 ⊆ S(Y ) and FU(X) ⊆ S(Y ).

We show now that S(Y ) ⊆ FU(X). Suppose that y ∈ S2. Then y =z ∨ x, where z, x ∈ Y . Therefore y = (z1 + a1u1) ∨ (z2 + a2u2), wherez1, z2 ∈ X, u1, u2 ∈ U and a1, a2 ∈ R. Hence y = z2 + a2u + (z1 −z2 − (a2u2 − a1u1)) ∨ 0 ∈ Y + X1. As Y ⊆ X1, it follows that y ∈X1 + X1 = X1, and so S2 ⊆ FU(X). Suppose that Sn ⊆ FU(X) for somen. For any z ∈ Sn+1 we have that z = x ∨ y with x ∈ Sn and y ∈ Y .By our assumption that Sn ⊆ FU(X) it follows that x ∈ Xk for some k.Since Y ⊆ Xk we have that x − y ∈ Xk. But z = y + (x − y) ∨ 0;hence z = y + cu(x − y, 0) ∈ Y + Xk+1 ⊆ Xk+1 + Xk+1 = Xk+1, and soz ∈ FU(X) and Sn+1 ⊆ FU(X). Thus Sm ⊆ FU(X) for any m, and henceS(Y ) ⊆ FU(X). Hence FU(X) = S(Y ) and statement (ii) holds. If U ⊆ X,then X = Y , and so FU(X) = S(X).

Suppose moreover that E is a topological vector space. If the limits ofcall and put options are also considered as marketed, it is natural to definethe notion of the closed completion of X as follows: the closure of FU(X) inE is the closed completion by options of X with respect to U . If E = R

m,the completion by options of X and the closed completion by options of X

coincide because any subspace of Rm is closed.

3. Sublattices and positive bases in Rm

In this section we give the basic mathematical notions and results in Rm =

{x = (x(1), x(2), . . . , x(m))|x(i) ∈ R for any i} which are needed forthis article. We view R

m as ordered by the pointwise ordering, i.e., forany x, y ∈ R

m we have x ≥ y if and only if x(i) ≥ y(i) for each i.Then R

m+ = {x ∈ Rm∣∣x(i) ≥ 0 for each i} is the positive cone of R

m.Suppose that L is an ordered subspace of R

m, i.e., L is a subspace ofR

m again ordered by the pointwise ordering. Then L+ = Rm+ ∩ L is the

positive cone of L. A basis {b1, b2, . . . , br} of L is a positive basis of L

if L+ = {x = ∑ri=1 λibi | λi ∈ R+ for each i}. In other words, a basis of

L is positive if, for any x ∈ L, we have that x is positive if and only itscoefficients in the basis are positive. Although L has infinitely many basesthe existence of a positive basis of L is not always ensured. Moreover, L hasa positive basis if and only if L is a lattice-subspace of R

m (i.e., if L in thepointwise ordering is a vector lattice). As noted in the previous section, L isa sublattice of R

m if for any x, y ∈ L, x∨y, x∧y ∈ L.Any sublattice of Rm

has a positive basis. For more details on positive bases and lattice-subspaceswe refer to Polyrakis (1996,1999).

Suppose that {b1, b2, . . . , br} is a positive basis of L. Then it is easyto show, for any x = ∑r

i=1 λibi, y = ∑ri=1 μibi , that x ≥ y if and only


if λi ≥ μi for each i. Also each bi is an extremal point of L+. ( A vectorx0 ∈ L+, x0 �= 0 is an extremal point of L+ if, for any x ∈ L, 0 ≤x ≤ x0 implies x = λx0 for a real number λ). This property implies thata positive basis of L is unique in the sense of positive multiples. We givethe next four straightforward results without proof. Recall that the supportof a vector x = (x(1), x(2), . . . , x(m)) of R

m is the set supp(x) = {i =1, 2, . . . , m|x(i) �= 0}.Proposition 4. An ordered subspace Z of R

m with a positive basis{b1, b2, . . . , br} is a sublattice of R

m if and only if supp(bi)∩ supp(bj ) = ∅for any i �= j .

Proposition 5. IfZ is a sublattice of Rm with a positive basis {b1, b2, . . . , br},then, for any x = ∑r

i=1 ηibi, y = ∑ri=1 θibi ∈ Z, we have

(i) ηi = x(k)

bi (k), where k ∈ supp(bi),

(ii) x ∨ y = ∑ri=1(ηi ∨ θi)bi and x ∧ y = ∑r

i=1(ηi ∧ θi)bi .

Proposition 6. Suppose that Z is a sublattice of Rm. If the constant vec-

tor 1 = (1, 1, . . . , 1) is an element of Z, then Z has a positive basis{b1, b2, . . . , br} which is a partition of the unit, i.e., 1 = ∑r

i=1 bi andfor each vector bi we have bi(j) = 1 for each j ∈ supp(bi).

Proposition 7. Suppose that Z is a sublattice of Rm with a positive basis

{b1, b2, . . . , br}. Then for each i the vector bi has minimal support in Z,i.e., there is no x ∈ Z, x �= 0, such that supp(x) � supp(bi).

Suppose now that z1, z2, . . . , zr are fixed, linearly independent, positivevectors of R

m and that Z is the subspace of Rm generated by the vectors zi .

We present the results of Polyrakis (1999) for the determination of a positivebasis of the sublattice W of R

m generated by the set {z1, z2, . . . , zr}. Thefunction

β(i) =(

z1(i)

z(i),z2(i)

z(i), . . . ,

zr(i)

z(i)

)

,

for each i ∈ {1, 2, . . . , m}, with z(i) > 0, where z = z1 + z2 + . . . + zr ,is the basic function of z1, z2, . . . , zr . This definition, which is given inPolyrakis (1996), is important for the study of positive bases. The set R(β) ={β(i)

∣∣i = 1, 2, . . . , m, with z(i) > 0}, is the range of β and the cardinal

number cardR(β) of R(β) is the number of (different) elements of R(β). Inthe above notation we have the following result.

Theorem 8 (Polyrakis (1999), Theorem 3.6). The subspace Z is a sublat-tice of R

m if and only if cardR(β) = r .


If R(β) = {P1, P2, . . . , Pr}, a positive basis {b1, b2, . . . , br} of Z isgiven by the formula

(b1, b2, . . . , br)T = A−1(z1, z2, . . . , zr)

T ,

where A is the r × r matrix whose ith column is the vector Pi , for eachi = 1, 2, . . . , r , and (b1, b2, . . . , br)

T , (z1, z2, . . . , zr)T are the matrices

with rows the vectors b1, b2, . . . , br and z1, z2, . . . , zr .

Theorem 9 (Polyrakis (1999), Theorem 3.7 ). The dimension of the sub-lattice W of R

m generated by the vectors zi is equal to the cardinal numberof R(β).

In the same result of Polyrakis (1999) it is shown that, if R(β) ={P1, P2, . . . , Pμ}, the sublattice W of R

m generated by {z1, z2, . . . , zr} isconstructed by following the steps of the following algorithm.

(a) Enumerate R(β) so that its first r vectors are linearly independent (asshown in Polyrakis (1999), such an enumeration always exists). Denotethe new enumeration again by Pi, i = 1, 2, . . . , μ, and put Ir+k = {i ∈{1, 2, . . . , m}∣∣β(i) = Pr+k} for each k = 1, 2, . . . , μ − r .

(b) Define the vectors zr+k, k = 1, 2, . . . , μ − r , as follows:

zr+k(i) = z(i) if i ∈ Ir+k and zr+k(i) = 0 if i �∈ Ir+k,

where z = z1 + z2 + . . . + zr is the sum of the vectors zi .(c) W = [z1, z2, . . . , zr , zr+1, . . . , zμ].(d) A positive basis {b1, b2, . . . , bμ} of W is constructed as follows.

Consider the basic function γ of z1, z2, . . . , zr , zr+1, . . . , zμ and sup-pose that {P ′

1, P′2, . . . , P ′

μ} is the range of γ ( the range of γ has exactlyμ points). Then

(b1, b2, . . . , bμ)T = D−1(z1, z2, . . . , zμ)T ,

where D is the μ × μ matrix with columns the vectors P ′1, P

′2, . . . , P ′

μ.

4. The completion by options in Rm

In this section we continue the study of security markets of Section 2 underthe assumption that E = R

m. As shown, the completion by options FU(X)

of X with respect to U is the sublattice of Rm generated by Y , where Y is

the subspace of Rm generated by X ∪ U . In the previous section a method

for the construction of the sublattice of Rm generated by a set of linearly

independent positive vectors of Rm is given. In this section we use this

method for the determination of FU(X).


Definition 10. Any set {y1, y2, . . . , yr} of linearly independent positive vec-tors of R

m such that FU(X) is the sublattice of Rm generated by {y1, y2, . . . ,

yr}, is a basic set of the market.

We denote by A the subset of Rm which is defined as follows:

A = {x+1 , x−

1 , x+2 , x−

2 , . . . , x+n , x−

n } if U ⊆ X

and

A = {x+1 , x−

1 , x+2 , x−

2 , . . . , x+n , x−

n , u+1 , u−

1 , u+2 , u−

2 , . . . , u+d , u−

d }if U �⊆ X,

where {u1, u2, . . . , ud} is a basis of U .

Theorem 11. Any maximal subset {y1, y2, . . . , yr} of linearly independentvectors of A is a basic set of the market.

Proof. Since {y1, y2, . . . , yr} is a maximal subset of linearly independentvectors of A, these two sets generate the same linear subspace and thereforealso the same sublattice in R

m. Thus it is enough to show that FU(X) is thesublattice S(A) generated by A. For each x ∈ X ∪ U we have x+, x− ∈FU(X), and so A ⊆ FU(X) and S(A) ⊆ FU(X). For the converse wenote that xi = x+

i − x−i ∈ S(A) and also that ui = u+

i − u−i ∈ S(A)

for each i. Since FU(X) is the minimum sublattice which contains the set{x1, x2, . . . , xn, u1, u2, . . . , ud} and S(A) is a sublattice which contains thisset it follows that FU(X) ⊆ S(A) and FU(X) = S(A).

If {y1, y2, . . . , yr} is a basic set of the market, the function

β(i) =(

y1(i)

y(i),y2(i)

y(i), . . . ,

yr(i)

y(i)

)

,

for each i = 1, 2, . . . , m, with y(i) > 0, where y = y1 + y2 + . . . + yr , isthe basic function of y1, y2, . . . , yr . Recall that R(β) is the range of β andcardR(β) the cardinal number of R(β). The function β takes values in thesimplex �r = {ξ ∈ R

r+| ∑ri=1 ξi = 1} of R

r+.

Definition 12. The space of marketed securities X is complete by optionswith respect to U if X = FU(X).

By Theorems 8 and 9 of Section 3 we have the following criteria for thecompletion of X.

Theorem 13. The space X of marketed securities is complete by optionswith respect to U if and only if U ⊆ X and cardR(β) = n.

Theorem 14. The dimension of FU(X) is equal to the cardinal number ofR(β). Therefore FU(X) = R

m if and only if cardR(β) = m.


For the determination of a positive basis of FU(X), according to Sec-tion 3, we determine a basic set {y1, y2, . . . , yr} of the market, find the rangeR(β) of the basic function β of the vectors yi and follow the steps of thealgorithm after Theorem 9, as the following example shows.

Example 15. We continue the study of the stochastic economy of Example1. Consider European forward-start options with expiration date t = 3 andtime of dependence t = 1. We study the completion of the market withrespect to this type of option. As we have noted in Example 1, the marketedspace X is the subspace of R

10 generated by the vectors

x31 = (2, 2, 4, 3, 0, 0, 0, 0, 1, 1), x3

2 = (0, 0, 1, 1, 2, 3, 1, 3, 4, 4),

x33 = (3, 3, 0, 0, 0, 0, 4, 0, 0, 0)

and the strike subspace U is generated by the vectors u1 = (1, 1, 0, 1, 0,

1, 0, 1, 0, 0), u2 = (0, 0, 1, 0, 1, 0, 1, 0, 1, 1) which define a basis of U .According to our methodology, A = {x3

1 , x32 , x

33 , u1, u2} and {y1 = x3

1 , y2 =x3

2 , y3 = x33 , y4 = u1, y5 = u2} is a basic set of the market. The basic

function is β = 1y(y1, y2, y3, y4, y5), where y is the sum of the vectors yi

and we find that

β(1) = β(2) = (1

3, 0,

1

2,

1

6, 0) = P1, β(3) = (

2

3,

1

6, 0, 0,

1

6) = P2,

β(4) = (3

5,

1

5, 0,

1

5, 0) = P3, β(5) = (0,

2

3, 0, 0,

1

3) = P4,

β(6) = β(8) = (0,3

4, 0,

1

4, 0) = P5, β(7) = (0,

1

6,

2

3, 0,

1

6) = P6,

β(9) = β(10) = (1

6,

2

3, 0, 0,

1

6) = P7.

Therefore cardR(β) = 7 and FU(X) is a seven-dimensional subspaceof R

10. The five first vectors P1, P2, P3, P4, P5 of R(β) are linearly inde-pendent, so we preserve the same enumeration of R(β). According to ourmethodology, I6 = β−1(P6) = {7}, I7 = β−1(P7) = {9, 10} and we definethe new vectors y6 = (0, 0, 0, 0, 0, 0, 6, 0, 0, 0), y7 = (0, 0, 0, 0, 0, 0, 0, 0,

6, 6). We determine the basic function γ = 1y′ (y1, y2, . . . , y7) of yi, i =

1, 2, . . . , 7, where y ′ is the sum of these vectors. We find that

γ (1) = γ (2) = (1

3, 0,

1

2,

1

6, 0, 0, 0) = P ′

1,

γ (3) = (2

3,

1

6, 0, 0,

1

6, 0, 0) = P ′

2,

γ (4) = (3

5,

1

5, 0,

1

5, 0, 0, 0) = P ′

3, γ (5) = (0,2

3, 0, 0,

1

3, 0, 0) = P ′

4,


γ (6) = γ (8) = (0,3

4, 0,

1

4, 0, 0, 0) = P ′

5,

γ (7) = (0,1

12,

1

3, 0,

1

12,

1

2, 0) = P ′

6,

γ (9) = γ (10) = (1

12,

1

3, 0, 0,

1

12, 0,

1

2) = P ′

7.

A positive basis of FU(X) is given by the formula (b1, b2, . . . , b7)T =

A−1(y1, y2, . . . , y7)T where A is the matrix whose columns are the vectors

P ′i , i = 1, 2, . . . , 7 and we find that the vectors

b1 = (1, 1, 0, 0, 0, 0, 0, 0, 0, 0), b2 = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0),

b3 = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0), b4 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 0),

b5 = (0, 0, 0, 0, 0, 1, 0, 1, 0, 0), b6 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0),

b7 = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1)

define a positive basis of FU(X).

Example 16. Suppose that in a security market R12 is the payoff space, that

the primitive securities are

x1 = (1, 2, 2, −1, 1, −2, −1, −3, 0, 0, 0, 0),

x2 = (0, 2, 0, 0, 1, 2, 0, 3, −1, −1, −1, −2),

x3 = (1, 2, 2, 0, 1, 0, 0, 0, −1, −1, −1, −2)

and that call and put options are taken with respect to the strike vectoru = (1, 2, 2, 1, 1, 2, 1, 3, −1, −1, −1, −2), i.e., U = [u] is the strike sub-space. A maximal subset of linearly independent vectors of {x+

1 , x−1 , x+

2 ,

x−2 , x+

3 , x+3 , u+, u−} is a basic set of the market. We find that x−

2 = x−3 ,

x+3 = x+

1 , u+ = x+1 + x−

1 , u− = x−2 , and that {y1 = x+

1 , y2 = x−1 ,

y3 = x+2 , y4 = x−

2 } is a basic set of the market. The basic function ofthe vectors yi is β = 1

y(y1, y2, y3, y4), where

y = y1 + y2 + y3 + y4 = (1, 4, 2, 1, 2, 4, 1, 6, 1, 1, 1, 2).

β(1) = β(3) = (1, 0, 0, 0) = P1, β(2) = β(5) = (1

2, 0,

1

2, 0) = P2,

β(4) = β(7) = (0, 1, 0, 0) = P3, β(6) = β(8) = (0,1

2,

1

2, 0) = P4,

β(9) = β(10) = β(11) = β(12) = (0, 0, 0, 1) = P5.

Hence Fu(X) is a five-dimensional subspace. The vectors P1, P2, P3, P5

are linearly independent, so we enumerate R(β) again so that its first fourelements are linearly independent. We put


R(β) = {P1 = (1, 0, 0, 0),

P2 = (1

2, 0,

1

2, 0), P3 = (0, 1, 0, 0), P4 = (0, 0, 0, 1),

P5 = (0,1

2,

1

2, 0)}.

According to our algorithm, I5 = β−1(P5) = {6, 8} and we considerthe vector y5 of R

12 with y5(i) = y(i) if i = 6, 8 and y5(i) = 0 otherwiseso that y5 = (0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 0). The completion by optionsof X is the subspace of R

12 generated by the set {y1, y2, y3, y4, y5}. Inorder to determine a positive basis of Fu(X) we consider the basic functionγ = 1

y′ (y1, y2, y3, y4, y5), where y ′ = (1, 4, 2, 1, 2, 8, 1, 12, 1, 1, 1, 2) isthe sum of the vectors yi . We have

γ (1) = γ (3) = (1, 0, 0, 0, 0) = P ′1, γ (2) = γ (5) = 1

2, 0,

1

2, 0, 0) = P ′

2,

γ (4) = γ (7) = (0, 1, 0, 0, 0) = P ′3, γ (6) = γ (8) = (0,

1

4,

1

4, 0,

1

2) = P ′

4,

γ (9) = γ (10) = γ (11) = γ (12) = (0, 0, 0, 1, 0) = P ′5.

A positive basis of Fu(X) is given by the formula (b1, b2, b3, b4, b5)T =

A−1(y1, y2, y3, y4, y5)T , where A is the matrix with columns the vectors

P ′i . Thus we find that the vectors

b1 = (1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0),

b2 = (0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0),

b3 = (0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0),

b4 = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2),

andb5 = (0, 0, 0, 0, 0, 8, 0, 12, 0, 0, 0, 0)

define a positive basis of Fu(X).

4.1. Basic derivative assets

Brown and Ross (1991) posed the following question: Do basic derivativeassets exist so that any derivative asset is a portfolio of these derivativeassets? Note that call and put options are positive vectors of FU(X) andalso that, by the term “portfolio”, Brown and Ross mean a positive linearcombination of the basic derivative assets. If we assume that {bi} is a positivebasis of FU(X), then any positive vector (derivative) of FU(X) is a positivelinear combination of the vectors bi , and so the basis {bi} is actually a


set of basic derivative assets. Thus, according to Brown and Ross, we callthe elements of a positive basis of FU(X) basic derivative assets or basicderivatives. We give below interesting extra properties of a positive basisof FU(X) which arise from Propositions 4, 5 and 6 of Section 3.

Proposition 17. Any positive basis {b1, b2, . . . , bμ} of FU(X) satisfies thefollowing properties:

(i) the vectors bi have disjoint supports,(ii) each bi has minimal support in FU(X), i.e., there is no a vector x ∈

FU(X), x �= 0, with supp(x) � supp(bi).

If 1 ∈ FU(X), then FU(X) has a positive basis which is a partition of theunit.

Since the vectors bi have minimal support, for each i the support ofbi defines a minimal block of states in the sense that there is no element(derivative) x of FU(X) with nonzero payoffs in a smaller (than supp(bi))set of states. Therefore the elements of a positive basis can be defined inde-pendently as the derivatives with nonzero payoff in a minimal set of statesso that any other derivative is a linear combination of these derivatives.

5. Efficient funds

In this section we assume that call and put options are taken with respectto a fixed strike vector u �= 0 of R

m. Thus we assume that U is the one-dimensional subspace of R

m generated by u and Y the linear subspace ofR

m generated by X∪{u}. Instead of FU(X) we write Fu(X) and we say thatFu(X) is the completion by options of X with respect to u. As noted in theintroduction, efficient funds have been studied in many economic articles.Although they are important for the theory of security markets, there is noway to determine the efficient funds of the market. In this section we solvethis problem by using the theory of positive bases.

Suppose that x is an element of Fu(X). If (x−au)+ > 0 and (x−au)− >

0, then we say that cu(x, a) is a nontrivial call option and also that pu(x, a) isa nontrivial put option of x. The following definition generalizes the notionof efficient fund and the more general notion of maximally efficient fund.

Definition 18. A vector e ∈ Fu(X) is an Fu(X)-efficient fund if Fu(X) isthe linear subspace of R

m which is generated by the set of nontrivial calloptions and the set of nontrivial put options of e.

Theorem 19. Suppose that {b1, b2, . . . , bμ} is a positive basis of Fu(X),u = ∑μ

i=1 λibi, and λi > 0 for each i. Then the vector e = ∑μ

i=1 κibi ofFu(X) is an Fu(X)-efficient fund if and only if κi

λi�= κj

λjfor each i �= j .


Proof. Suppose that k1, k2, . . . , kμ are real numbers such that κi

λi�= κj

λjfor

each i �= j . Then there exists an enumeration {i1, i2, . . . , iμ} of the set{1, 2, . . . , μ} with

κi1λi1

>κi2λi2

> . . . >κiμ

λiμ. Then e = ∑μ

ν=1 κiν biν and

it is easy to show that cu(e, a) is a nontrivial call option if and only ifκiμ

λiμ< a <

κi1λi1

. Also it is easy to show that

cu

(

e,κi2

λi2

)

=(

e − κi2

λi2

u

)+=

(

κi1 − λi1

κi2

λi2

)

bi1,

cu

(

e,κi3

λi3

)

=(

e − κi3

λi3

u

)+=

(

κi1 − λi1

κi3

λi3

)

bi1 +(

κi2 − λi2

κi3

λi3

)

bi2,

......

cu

(

e,κiμ

λiμ

)

=(

e − κiμ

λiμ

u

)+= (κi1 − λi1

κiμ

λiμ

)bi1 +(

κi2 − λi2

κiμ

λiμ

)

bi2

+ . . . +(

κiμ−1 − λiμ−1

κiμ

λiμ

)

biμ−1,

and

pu

(

e,κiμ−1

λiμ−1

)

=(

λiμ

κiμ−1

λiμ−1

− κiμ

)

biμ.

From the above triangular form and the last equation we see that the linearsubspace generated by the nontrivial call and put options of e is the wholespace Fu(X). Therefore e is an Fu(X)-efficient fund.

For the converse suppose that the vector e = ∑μ

i=1 κibi is an Fu(X)-efficient fund. If we suppose that κi

λi= κj

λj= θ for some i �= j , then, for any

nontrivial call option cu(e, a), we have cu(e, a) = (e − au)+ = ∑μν=1(kν −

aλν)+bν . Since (ki − aλi)

+ = (θ − a)+λi , (kj − aλj )+ = (θ − a)+λj , we

have that the expansion of cu(e, a) in the positive basis of Fu(X) has theterm (θ − a)+[λibi + λjbj ]. In a similar way, any nontrivial put option hasthe term (a − θ)+[λibi + λjbj ]. Therefore the space W generated by thenon-trivial call and put options of e is a proper subspace of Fu(X) becauseW is contained in the (μ− 1)-dimensional subspace of Fu(X) generated bythe vector λibi + λjbj and the vectors bf with f �= i, j . This is a contra-diction, whence κi

λi�= κj

λjfor each i, j with i �= j , and hence the converse is

true.

By the above theorem we conclude that an element x of Fu(X) is not anFu(X)-efficient fund if and only if x belongs to one of the (μ−1)μ

2 subspacesof Fu(X) which are defined by the equations

κ1

λ1= κ2

λ2,κ1

λ1= κ3

λ3, . . . ,

κμ−1

λμ−1= κμ

λμ

,


provided of course that all the coefficients λi of u in the positive basis {bi} ofFU(X) are greater than zero. We refer to these subspaces as the non-efficientsubspaces of Fu(X). For example the non-efficient subspace κ1

λ1= κ2

λ2of

Fu(X) is {∑μ

i=1 κibi ∈ Fu(X)∣∣ κ1λ1

= κ2λ2

}. By using the above theorem, it iseasy to determine the efficient funds and the non-efficient subspaces of themarket. Note that the notion of the non-efficient subspace appears, althoughnot in a formal way, in the papers of Ross (1976) and John (1981).

Proposition 20. Each non-efficient subspace of Fu(X) is a proper sublatticeof Fu(X).

Proof. Suppose that {b1, b2, . . . , bμ} is a positive basis of Fu(X) and thatW is a non-efficient subspace of Fu(X). Then W is defined by an equation ofthe form κi

λi= κj

λj. For simplicity suppose that W is defined by the equation

κ1λ1

= κ2λ2

. Then, for each x = ∑μ

i=1 κibi ∈ W , we see that κ1 = θλ1, κ2 =θλ2, and so x = θ(λ1b1 + λ2b2) + ∑μ

i=3 κibi . Therefore W is generated bythe vectors λ1b1 +λ2b2, b3, b4, . . . , bμ. Since the elements of the basis {bi}have disjoint supports we deduce that the vectors λ1b1 + λ2b2, b3, . . . , bμ

have disjoint supports and by Proposition 4 of Section 3, W is a sublatticeof R

m. It is clear that W is a proper subspace of Fu(X).

Theorem 21. Suppose that {b1, b2, . . . , bμ} is a positive basis of Fu(X) andthat u = ∑μ

i=1 λibi with λi > 0 for each i. Then:

(i) the nonempty set D = Y \ ∪i∈I (Y ∩ Zi), where {Zi |i ∈ I } is the set ofnon-efficient subspaces of Fu(X), is the set of Fu(X)-efficient funds ofY and the Lebesgue measure of Y is supported on D;

(ii) Fu(X) is the subspace of Rm generated by the set of the call options

{cu(x, a)|x ∈ Y, a ∈ R} written on the elements of Y .If u ∈ X, Fu(X) is the subspace X1 of R

m generated by the set of calloptions O1 = {cu(x, a)|x ∈ X, a ∈ R} written on the elements of X.

Proof. Suppose that Y is contained in a non-efficient subspace Zi of Fu(X).As shown before, each non-efficient subspace of Fu(X) is a proper sublatticeof R

m, and so the sublattice Fu(X) generated by Y is contained in Zi , acontradiction. Hence Y ∩Zi is a proper subspace of Y . Now Y \∪i∈I Y ∩Zi �=∅ because Y cannot be the union of a finite number of its proper subspaces.Since any proper subspace of Y is of measure zero, the Lebesgue measureof Y is supported on D. Hence Y has Fu(X)-efficient funds, and thereforethe set of call options written on the elements of Y generate Fu(X).

6. The classical case

In this section we assume that the primitive securities x1, x2, . . . , xn arelinearly independent, positive vectors of R

m and that call and put options are


taken with respect to the riskless bond 1 = (1, 1, . . . , 1) ∈ Rm. According

to our notation, F1(X) is the completion by options of X with respect to 1.As noted in the introduction, our previous results are also important for thiscase. First of all they provide a method for the determination of a positivebasis {b1, b2, . . . , bμ} of F1(X). Also it is easy to check whether any optionis replicated or whether F1(X) = R

m by checking the cardinality of the rangeof the basic function β. As shown before, the vectors of the positive basis ofF1(X) define minimal blocks of states and also the vectors bi are the basicderivative assets in the sense of Brown and Ross (1991). Moreover F1(X)

has a positive basis which is a partition of the unit, as seen in Proposition 17,and therefore the vectors of any positive basis {bi} of F1(X) are constant ontheir support. The next result follows directly from Theorem 19.

Theorem 22. If {b1, b2, . . . , bμ} is a positive basis of F1(X) which is apartition of the unit, then the vector e = ∑μ

i=1 κibi is an F1(X)-efficientfund if and only if κi �= κj for each i �= j .

It is clear that each efficient fund is an F1(X)-efficient fund (a positivevector e of F1(X) is an efficient fund if e(i) �= e(j) for any i �= j ) but theconverse is not always true. John (1981) defined the notion of maximallyefficient fund (portfolio) as follows. If A is the payoff matrix of the primitivepayoffs x1, x2, . . . , xn, i.e., A is the matrix with columns the vectors xi , thena positive element e ∈ F1(X) is a maximally efficient fund if e(i) �= e(j)

for any i, j with A(i) �= A(j), where A(k) is the vector defined by the kthrow of A. John (1981) remarks that the space generated by the call optionswritten on a maximally efficient fund is the completion by options of X.Thus, according to our definition, if an element e of F1(X) is a maximallyefficient fund then e is an F1(X)-efficient fund. We show below that theconverse is also true. So, by using Theorem 21, we can determine exactlythe set of efficient funds and the set of maximally efficient funds.

Theorem 23. An element e of F1(X) is a maximally efficient fund if and onlyif e is an F1(X)-efficient fund.

Proof. We may suppose that {y1, y2, . . . , yr} is a basic set of the market.Thus, if 1 ∈ X, then r = n and yi = xi for each i = 1, 2, . . . , n and in thecase where 1 �∈ X we have r = n + 1, yi = xi for each i = 1, 2, . . . , n

and yr = 1. Suppose that β is the basic function of the vectors yi . In thealgorithm for the determination of the completion by options, suppose thatR(β) = {P1, P2, . . . , Pμ} is the range of β after the new enumeration, sothat its first r vectors are linearly independent and suppose that Ij = {i ∈{1, 2, . . . , m}∣∣β(i) = Pj }, for each j = 1, 2, . . . , μ, is the inverse imageof Pj . In the sequel, according to the algorithm, we define the vectors yi ,i = r + 1, . . . , μ, so that yr+k(i) = y(i) if i ∈ Ir+k and yr+k(i) = 0 ifi �∈ Ir+k, where y = y1+y2+. . .+yr . The basic function of y1, y2, . . . , yμ is


γ (i) =(

y1(i)

y ′(i),y2(i)

y ′(i), . . . ,

yμ(i)

y ′(i)

)

,

for each i = 1, 2, . . . , m, with y ′(i) > 0, where y ′ = y1 + y2 + . . . + yμ.For each i ∈ Ir+k we have yr+k(i) = y(i) and yr+j (i) = 0 for each j �= k,whence y ′(i) = 2y(i). We determine the range R(γ ) = {P ′

1, P′2, . . . , P ′

μ}of γ . By the definition of the new vectors yi and the above remarks it followsthat, for each i ∈ Ij , we have γ (i) = (Pj , 0) = P ′

j for each j ≤ r and γ (i) =12 (Pj , ej−r ) = P ′

j if j = r +1, . . . , μ, where {e1, e2, . . . , eμ−r} is the usualbasis of R

μ−r . Based on the above remarks it is easy to show that Ij = {i ∈{1, 2, . . . , m}∣∣γ (i) = P ′

j } for each j = 1, 2, . . . , μ. We show now thatIj = supp(bj ) for each j = 1, 2, . . . , μ. By statement (c) of Theorem 3.6 ofPolyrakis (1996) we see that γ (i) = P ′

j implies that bj (i) > 0 and bk(i) = 0for each k �= j ; therefore γ −1(P ′

j ) ⊆ supp(bj ). Suppose that i ∈ supp(bj ).Theny ′(i) > 0 because, if we suppose thaty ′(i) = 0, thenyk(i) = 0 for eachk and by the formula giving the positive basis we conclude that bj (i) = 0, acontradiction. Therefore for any i ∈ supp(bj ), the function γ is defined on i

and we assert that γ (i) = P ′j . Indeed if we assume that γ (i) = P ′

k we see thati ∈ γ −1(Pk) ⊆ supp(bk), and so k = j because the vectors of the positivebasis have disjoint supports. Therefore γ −1(P ′

j ) = supp(bj ) for each j .Suppose that e is an F1(X)-efficient fund. To show that e is a maxi-

mally efficient fund it is enough to show that A(i) �= A(j) implies thate(i) �= e(j). Suppose that A(i) �= A(j). Then we can show that β(i) �= β(j)

as follows. As remarked in the beginning of this proof in the case where1 ∈ X, we have r = n and yi = xi for each i = 1, 2, . . . , n; there-fore β(k) = A(k)

‖A(k)‖1for each k, where ‖A(k)‖1 is the �1-norm of A(k).

If we assume that β(i) = β(j), then A(i) = A(j)‖A(i)‖1‖A(j)‖1

, and therefore

xk(i) = xk(j)‖A(i)‖1‖A(j)‖1

for each k. Since 1 ∈ X we have 1 = ∑nk=1 μkxk,

and therefore 1 = 1(i) = ∑nk=1 μkxk(i) and 1 = 1(j) = ∑n

k=1 μkxk(j).Thus ‖A(i)‖1

‖A(j)‖1

∑nk=1 μkxk(j) = ∑n

k=1 μkxk(j), whence ‖A(i)‖1‖A(j)‖1

= 1 whichimplies that A(i) = A(j), a contradiction. Therefore β(i) �= β(j). If weassume that 1 �∈ X, then r = n + 1, yi = xi for each i = 1, 2, . . . , n andyr = 1. Thus β(k) = ( A(k)

1+‖A(k)‖1, 1

1+‖A(k)‖1) for each k. If we assume that

β(i) = β(j), then 11+‖A(i)‖1

= 11+‖A(j)‖1

, and so ‖A(i)‖1 = ‖A(j)‖1. AlsoA(i)

1+‖A(i)‖1= A(j)

1+‖A(j)‖1whence A(i) = A(j), a contradiction, and again

β(i) �= β(j). Hence A(i) �= A(j) implies β(i) �= β(j). Thus, if wesuppose that β(i) = Pi1, β(j) = Pj1 , then Pi1 �= Pj1 which implies thatP ′

i1�= P ′

j1, where γ (i) = P ′

i1, γ (j) = P ′

j1. Since supp(bi1) = γ −1(P ′

i1) and

supp(bj1) = γ −1(P ′j1), we conclude that bi1 �= bj1 .

Since 1 ∈ F1(X), the vectors {bi} are constant in their support. In particu-lar, if we suppose that 1 = ∑μ

k=1 λkbk then bk(i) = 1λk

for each i ∈ supp(bk)


therefore the basis {b′k = λkbk} is a positive basis of F1(X) which is a par-

tition of the unit.Since i ∈ supp(bi1), j ∈ supp(bj1) we have that i ∈ supp(b′

i1), j ∈

supp(b′j1). If e = ∑μ

k=1 θkb′k is the expansion of e in the basis {b′

k}, thene(i) = θi1 and e(j) = θj1 and θi1 �= θj1because e is an F1(X)-efficient fund,and so e(i) �= e(j). Hence e is a maximally efficient fund.

6.1. Separation of states

Two different states i, j are separated (or discriminated) with respect toF1(X) if there exists x ∈ F1(X) so that x(i) �= x(j). If there exists x ∈ X

with x(i) �= x(j), i, j are separated with respect to X.

Proposition 24. The states i, j are separated with respect to X if and onlythey are separated with respect to F1(X).

Proof. Suppose that i, j are separated with respect to F1(X). If we assumethat x(i) = x(j) for any x ∈ X, then z(i) = z(j) for any z ∈ Y , where Y isthe subspace of R

m generated by X ∪ {1}. Thus, for any call or put optiony of an element z ∈ Y with respect to 1 we have y(i) = y(j). Since F1(X)

is generated by the set of call options written on the elements of Y , see inTheorem 21, it follows that i, j are not separated with respect to F1(X), acontradiction, and the proposition follows.

Theorem 25. Suppose that {b1, b2, . . . , bμ} is a positive basis of F1(X) andthat i, j are two different states. Then i, j are not separated with respect toF1(X) if and only if there exists an element bk of the positive basis for whichi, j ∈ supp(bk).

Proof. Suppose that i, j are not separated. If we suppose that i ∈ supp(bk1),j ∈ supp(bk2) with bk1 �= bk2 , then bk1(i) > 0 and bk1(j) = 0 becausethe elements of the positive basis have disjoint supports, a contradiction.Therefore i, j belong to the support of the same vector bk. For the conversesuppose that i, j ∈ supp(bk). Since 1 ∈ F1(X), the vectors of {bi} are con-stant in their support. Suppose that bk(l) = θk for any l ∈ supp(bk). Then forany x = ∑μ

i=1 λibi we have that x(l) = θkλk for any l ∈ supp(bk) becausethe elements of the positive basis have disjoint supports, and therefore thestates i, j are not separated.

Recall that our security market is resolving if 1 ∈ X and for any twodifferent states i, j there exists a primitive security xk such that xk(i) �=xk(j). It is known that the completion by options of a resolving securitymarket is the whole space R

m. By using the theory of positive bases we canprove it as follows. Suppose that the support of a vector bk of a positive


basis of F1(X) is not a singleton. Then i, j ∈ supp(bk) for at least two statesi, j , and so these states are not separated, a contradiction. Hence the supportof any element of the basis is a singleton. This implies that the basis hasm different vectors, and hence F1(X) = R

m. The security market is calledstrongly resolving if 1 ∈ X and for any choice of n states (n is the number ofprimitive securities) and any contingent claim y ∈ F1(X) there is a uniqueelement x ∈ X such that y and x coincide in these n states. In Aliprantis andTourky (2002), it is proved that, in a strongly resolving security market withn ≤ m+1

2 , any nontrivial option is not replicated. The proof in Aliprantisand Tourky (2002) is short but clever. By using the theory of positive baseswe can give a proof, similar to theirs.

References

Aliprantis, C.D., Burkinshaw, O. (1985): Positive operators. Academic Press, Orlando, FL

Aliprantis, C.D., Tourky, R. (2002): Markets that don’t replicate any option. EconomicsLetters 76, 443–447

Arditti, F.D., John, K. (1980): Spanning the state space with options. Journal of Financialand Quantitative Analysis 15, 1–9

Brown, D.J., Ross, S.A. (1991): Spanning, valuation and options. Economic Theory 1, 3–12

Green, R.C., Jarrow, R.A. (1987): Spanning and completeness in markets with contingentclaims. Journal of Economic Theory 41, 202–210

John, K. (1981): Efficient funds in a financial market with options: a new irrelevance propo-sition. The Journal of Finance 36, 685–695

LeRoy, S.F., Werner, J. (2001): Principles of financial economics. Cambridge UniversityPress, Cambridge, MA

Magill, M., Quinzii, M. (1996): Theory of incomplete markets. I. MIT Press, Cambridge,MA

Musiela, M., Rutkowski, M. (1997): Martingale methods in financial modelling. Springer,Berlin

Nachman, D.C. (1988): Spanning and completeness with options. The Review of FinancialStudies 1, 311–328

Polyrakis, I.A. (1996): Finite-dimensional lattice-subspaces of C(�) and curves of Rn.

Transactions of the American Mathematical Society 348, 2793–2810

Polyrakis, I.A. (1999): Minimal lattice-subspaces. Transactions of the American Mathemat-ical Society 351, 4183–4203

Ross, S.A. (1976): Options and efficiency. The Quarterly Journal of Economics 90, 75–89

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ypoly/ARTICLES (POLYRAKIS)/the ccompletion i… · DOI: 10.1007/s10203-006-0059-z DEF 29, 1–21 (2006) Decisionsin Economicsand Finance c Springer-Verlag 2006 The completion of security