Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue11-12.Oct2000:

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24 (2000) 1703}1719

Minimum-cost portfolio insurance

C.D. Aliprantis

!,

*

, D.J. Brown

"

, J. Werner

#

!Departments of Economics and Mathematics, School of Management, Purdue University,

West Lafayette, IN 47907-1310, USA

"Department of Economics, Yale University, New Haven, CT 06520, USA #Department of Economics, University of Minnesota, Minneapolis, MN 55455, USA

Accepted 30 November 1999

Abstract

Minimum-cost portfolio insurance is an investment strategy that enables an investor

to avoid losses while still capturing gains of a payo!of a portfolio at minimum cost. If

derivative markets are complete, then holding a put option in conjunction with the reference portfolio provides minimum-cost insurance at arbitrary arbitrage-free security prices. We derive a characterization of incomplete derivative markets in which the minimum-cost portfolio insurance is independent of arbitrage-free security prices. Our characterization relies on the theory of lattice-subspaces. We establish that a necessary and su$cient condition for price-independent minimum-cost portfolio insurance is that the asset span is a lattice-subspace of the space of contingent claims. If the asset span is a lattice-subspace, then the minimum-cost portfolio insurance can be easily calculated as

a portfolio that replicates the targeted payo!in a subset of states which is the same for

The authors are pleased to acknowledge the suggestions and comments of Peter Bossaerts, Phillip Henrotte and Yiannis Polyrakis. The research of C.D. Aliprantis was partially supported by the 1995 PENED Program of the Ministry of Industry, Energy and Technology of Greece and by the NATO Collaborative Research Grant d941059. Roko Aliprantis also expresses his deep appreciation for the hospitality provided by the Department of Economics and the Center for Analytic Economics at Cornell University and the Division of Humanities and Social Sciences of the California Institute of Technology where parts of this paper were written during his sabbatical leave (January}June, 1996). Jan Werner acknowledges the"nancial support of the Deutsche Forschun-gsgemainschaft, SFB 303.

*Corresponding author. Tel.: 001-765-494-4404; fax: 001-765-494-9658. E-mail address:[email protected] (C.D. Aliprantis).

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1. Introduction

Portfolio insurance enables an investor to avoid losses while still capturing gains of a portfolio payo!. It can be obtained by holding in conjunction with the reference portfolio, a put option with strike price equal to the insurance#oor. Whenever the portfolio payo! falls below the insurance#oor, the put option pays the di!erence between the #oor and the portfolio payo!. Leland (1980) provides a characterization of investors who are most likely to demand portfolio insurance.

It follows that portfolio insurance is marketed if and only if options on portfolios of securities are marketed (either as genuine option contracts or as replicating portfolios). Ross (1976) demonstrated that all options on portfolios are marketed if and only if markets for derivative securities are complete. The payo!of a derivative security depends only on the payo!s of primitive securities. Portfolio insurance is an example of a derivative security.

Options markets exist for relatively few portfolios of securities. In general, options on portfolios cannot be replicated by portfolios of options on individual securities if such are marketed. Consequently, an investor may not be able to achieve the insured payo!on a reference portfolio at a desired#oor. In such case the investor may wish to purchase a payo! that is at least as large as the infeasible insured payo!, at the minimum cost. This investment strategy enables the investor to avoid losses and capture the gains at the minimum cost, and is referred to as theminimum-cost portfolio insurance.

Cost minimization is often used as a criterion for optimal hedging of a desired contingent claim. Edirisinghe et al. (1993) and Naik and Uppal (1994) study minimum-cost hedging with transaction costs and/or portfolio constraints (see also Broadie et al. (1998) in a continuous time setting). The main advantage of the cost-minimization criterion is that it is independent of investors'preferences and of probability beliefs.

In general, the solution to the problem of minimizing the cost of a portfolio subject to the constraint that the payo!exceeds the insured payo!, depends on security prices. In this paper we provide a characterization of market structures in which the cost minimizing portfolio is independent of security prices. One case in which the cost minimizing portfolio does not depend on prices is when derivative markets are complete. Then, the insured payo! is marketed, and a portfolio that generates that payo!provides the minimum-cost insurance at any prices, as long as there are no arbitrage opportunities. We show that completeness of derivative markets is not a necessary condition for the min-imum-cost portfolio insurance to be price independent. A necessary and su$-cient condition admits a large and interesting class of incomplete derivative market structures.

Our primary characterization of market structures in which minimum-cost portfolio insurance is price independent relies on the mathematical theory of

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Riesz spaces (vector lattices) and lattice-subspaces. Riesz spaces have been used in the context of options markets by Brown and Ross (1991) and by Green and Jarrow (1987). The space of contingent claims on a"nite set of states of nature is a Riesz space with a supremum of any two contingent claimsy

1andy2de"ned

as the statewise supremum and denoted byy

1sy2. A put option on a payo!

ywith strike price kcan be represented using the operation of supremum as (k!y)s0, wherek_{is the risk-free payo!}_of_{k. The insured claim on payo!}_y_at #oor kisysk. The subspaceM_{of payo!s of all portfolios of securities (asset}

span) is a Riesz subspace if the supremum of arbitrary two payo!s inM_belongs

toM_{. If the asset span is a Riesz subspace, then every option is marketed. The}

result of Ross (1976) can be restated as saying that derivative markets are complete if and only if the asset span is a Riesz subspace of the space of contingent claims.

The asset spanMis alattice-subspaceif there exists a supremum relative to

Mof any two payo!s inM. The supremum relative toMof two payo!sy

1and

2inMis their least upper bound inM, and is denoted byy1sMy2. IfMis

a lattice-subspace, then the payo!ys

Mkexists and has the least cost among all

marketed payo!s that dominate the insured claimysk_{, for arbitrary} arbitrage-free prices of securities. Our main result is that the minimum-cost portfolio insurance is price independent if and only if the asset span M _{is a}

lattice-subspace.

We present a number of su$cient conditions for the asset span to be a subspace. The asset span of any two limited liability securities is a lattice-subspace. Using a result of Abramovich et al. (1994) we show that an asset span is a lattice-subspace if and only if there exists a set of as many states as there are securities with the property that for any state not in that set the vector of payo!s of all securities is a linear positive combination of payo! vectors in states belonging to the set. We call such set a fundamental set of states. A di!erent necessary and su$cient condition forMto be a lattice-subspace is the existence of a Yudin basis forM, i.e., a basis of limited liability payo!s such that every marketed limited liability payo!has a unique representation as a nonnegative linear combination of basis payo!s. The notion of Yudin basis is a generaliz-ation to incomplete markets of a basis of Arrow securities for complete markets (Arrow, 1953).

We show that the minimum-cost portfolio insurance has a simple form whenever it is independent of security prices. The cost minimizing portfolio replicates the insured payo!in fundamental states. Since the securities markets restricted to the fundamental states are complete, the minimum-cost portfolio insurance can be reduced to portfolio insurance in complete markets.

The paper is organized as follows: In Section 2, we de"ne the minimum-cost portfolio insurance. In Section 3, we introduce the lattice-subspace property of the asset span and show that it is necessary and su$cient for the minimum-cost portfolio insurance to be price independent in the set of arbitrage-free prices.

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1Bold1denotes theS-vector (1,2,1); similarly,kdenotes (k,2,k) for any scalark.

We also present a characterization of the lattice-subspace property in terms of fundamental states. A simple method for "nding the minimum-cost portfolio insurance is given in Section 4. Section 5 extends the results to in"nitely many states. The use of Yudin basis for the minimum-cost portfolio insurance is discussed in Section 6. The appendix contains some basic concepts and results about lattice-subspaces.

In a companion paper, Aliprantis et al. (1998b), we explore implications of the lattice-subspace property of the asset span for the existence of optimal allocations and equilibria in securities markets with in"nitely many securities.

2. Minimum-cost portfolio insurance

Suppose that there areN securities traded in a market at the beginning of a time period. End-of-period payo!s of the securities are uncertain. We shall initially assume that there are"nitely many possible end-of-period payo!s of the securities. The more general case will be treated in Section 5. We shall use a"nite set of statesS"_M1,2,SNto describe the uncertainty.

The payo!of securityninSstates is a vectorx

n3RS`. The payo!sx1,2,xN are assumed linearly independent so that there are no redundant securities (and

so N4S). We use x(s) to denote the N-dimensional vector of payo!s of all

securities in states(state-payowvector). For a portfolioh"(h₁,2,h

N)3RN, its payo!isX(h)"+Nn/1hnxn. The set of payo!s of all portfolios is the linear span of payo!sx

1,2,xNin the spaceRSof all state contingent claims and is the asset span M. A contingent claim is amarketed payowif it lies in the asset span. It is assumed that the risk-free payo! is marketed, so that 13M.1 If the asset span equals the whole space of contingent claims (i.e., ifN"S), then markets are complete.

Letp"(p

1,2,p_N)3RNbe a vector of security prices. A non-zero portfolio

h with positive payo! X(h)50 and zero or negative value p)h40 is an arbitrage portfolio. A security price vectorp3_RNisarbitrage-freeif there is no arbitrage portfolio, that is, ifp)h'0 for all non-zero portfolioshwithX(h)50.

The following simple duality result is well known.

Lemma 2.1. Ifp)h50for every arbitrage-free pricevectorp,thenX(h)50.

Proof. Suppose by way of contradiction that there is a portfoliohwithp)h50

for every arbitrage-free p whileX(h)(s)(0 for some states. Then there exist strictly positive numbersj_s for eachssuch that+_Ss_/1j_sX(h)(s)(0. Using the

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2We use the symbolsto denote the statewise supremum of two contingent claims. That is (ysz)(s)"maxMy(s),z(s)N, fors"1,2,S.

de"nition of X(h) we have +_Ss_/1j

sx(s)h(0. De"ne a price system q by q"+Ss

/1jsx(s). Clearly, qis arbitrage-free. This is, however, a contradiction,

sinceq)h(0. h

The insured payo! on a portfolio hat a `#oorak is the contingent claim X(h)sk_.2_{This contingent claim may or may not be marketed. The} minimum-cost insurance provides a payo! that dominates the insured payo! at the minimum cost. It is an investment strategy that captures the gains of holding the portfolio and limits the downside risk. Formally, the minimum-cost portfolio insuranceis de"ned by the following cost minimization problem:

min

g_|RN

p)g

subject to

X(g)5X(h)sk_.

This linear programming problem has a unique solution as long aspis arbitr-age-free. We denote the solution byhkand refer to it as theminimum-cost insured portfolio.

In general, the minimum-cost insured portfolio depends on security prices. There are, however, cases in which it is independent of arbitrage-free prices. These cases are very important. Not only that the insured portfolio can be selected without the knowledge of current security prices, but also, as we shall see in Section 4, it has a simple form. The minimum-cost insured portfolio is said to be price independentif it does not depend on arbitrage-free security prices. Needless to say, it is only the composition}not the cost}of the portfolio that does not depend on security prices.

3. The lattice property and fundamental states

In this section we provide characterizations of market structures that admit price-independent minimum-cost portfolio insurance. These characterizations rely on the notions of Riesz subspaces and lattice-subspaces, see Section 7. The asset spanM_{is a}_{Riesz subspace}_{if for every marketed payo!s}_y

1,y23Mtheir

(statewise) supremumy

1sy2belongs toM. It is alattice-subspace, if for every

marketed payo!s y

1,y23M there exists a least upper bound of y1 and

2relative toM. That is, there existsz3Msuch that (1)z5y1andz5y2(i.e,

z5y

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supremum byy

1sMy2. It is always true thaty1sMy25y1sy2. Note that if Mis a Riesz subspace, then it is also a lattice-subspace, andy

1sy2"y1sMy2.

Aderivative contingent claimis a contingent claim that depends only on the payo!s of the securities. More precisely, it is any contingent claim that has the same payo! in states in which the payo!s of all securities are the same. The insured payo!on a portfolio is an example of a derivative contingent claim. If every derivative contingent claim is marketed (complete derivative markets), then the insured payo!on every portfolio is marketed, and a portfolio that generates the insured payo!is the minimum-cost insured portfolio for every arbitrage-free price. Thus, the portfoliohksatis"esX(hk)"X(h)sk.

A characterization of complete derivative markets has been given in Ross (1976). Ross observed that any derivative contingent claim can be generated by a portfolio of options on marketed payo!s. For instance, the insured payo! X(h)sk_satis"es _X(_h_)sk"X(h)#[(k!X(h))s0], hence it can be generated by holding the portfoliohand a put option on X(h) with strike price k. Thus, a necessary and su$cient condition for complete derivative markets is that all put and call options on marketed payo!s be marketed. Since each put and call option is a supremum of two marketed payo!s, all options are marketed if the asset span is a Riesz subspace. The following is essentially a restatement of the result due to Ross (1976).

Theorem 3.1. Derivative markets are complete if and only if the asset span is a Riesz subspace ofRS.

Proof. The theorem of Ross (1976) implies that derivative markets are complete if and only if (k!y)s0 belongs toMfor everyy3Mand everyk. This latter property is necessary and su$cient forMto be a Riesz subspace. This follows from the simple lattice identityy

1sy2"y1#[(y2!y1)s0]. h

Completeness of derivative markets is a su$cient but not a necessary condi-tion for the minimum-cost portfolio insurance to be price independent. Suppose that there exists the supremum of the payo!sX(h) and k_{relative to the asset} spanM_{, i.e.,}_X(_h₎s

Mk. A portfolio that generates the payo! X(h)sMkis the

minimum-cost insured portfolio for every arbitrage-free prices. We have the following:

Theorem 3.2. The minimum-cost insured portfolio exists and is price independent for every portfolio and at every yoor if and only if the asset span is a lattice-subspace of RS. In this case, the minimum-cost insured portfolio hk satisxes

X(hk)"X(h)s

Mk.

Proof. Assume"rst that the minimum-cost insured portfolio exists and is price

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insured portfolio. For any portfoliogsuch thatX(g)5X(h)sk_{, we have that} p)g5p)hk for every arbitrage-free price vector p. Lemma 2.1 implies that

X(g)5X(hk). Hence, the payo!X(hk) is the supremum ofX(h) andk_inM, i.e.,

X(h)s

Mk.

We have thus demonstrated that the supremum ys

Mk exists for every

marketed payo! y and every k. The simple lattice identity y

1sMy2"

[(y

1!y2#k)sMk]#y2!kproves that the supremumy1sMy2exists for two

arbitrary payo!sy

1 andy2 inM. Hence,Mis a lattice-subspace.

Conversely, if the asset span M _{is a lattice-subspace, then the supremum}

X(h)s

Mk exists for every portfolio h and every k. Now let the portfolio hk

be such that X(hk)"X(h)s

Mk. Then, for every portfolio g satisfying

X(g)5X(h)sk, we have thatX(g)5X(hk). This inequality impliesp)g5p)hk

for every arbitrage-free price vector p. Consequently, the portfolio hk is the minimum-cost insured portfolio for every arbitrage-free price. h

Theorem 3.2 characterizes the price-independent minimum-cost portfolio insurance in terms of the lattice-subspace property of the asset span. We now proceed to characterize the market structures with the lattice-subspace prop-erty. A necessary and su$cient condition for the asset spanMto be a lattice-subspace is the existence of a fundamental set of states. For any subsetF-S

of states, we shall denote by X

F the matrix of security payo!s restricted to the subset of states F. A subset F"_Ms

1,2,sNN-S of N states is called fundamental, if

1. theN]NmatrixX

F is nonsingular,and

2. for eachsNFthere exist non-negative scalarsa₁s,as₂,2,asN such that x(s)"+N

i/1

a_six(s i).

Expressing it in another way, condition (2) asserts that every state-payo! vector of a non-fundamental state lies in the cone generated by the state-payo!s of the fundamental states.

We have the following:

Theorem 3.3 (Abramovich}Aliprantis}Polyrakis, 1994). The asset span M _is a lattice-subspace ofRSif and only if there is a fundamental set of states.

It is well known that any two-dimensional subspace ofRScontaining the unit vector1is a lattice-subspace (see Theorem 3.5, Aliprantis et al., 1998a). When N"2, the state-payo!sx(s) fors3Sgenerate a closed cone inR2. If there are "nitely many states, the two extreme rays of this cone identify the two funda-mental states. Thus, the minimum-cost portfolio insurance is always price independent in the case of two securities.

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Example 3.1. We consider two securities with payo!s in three states given by

1"(1, 1, 1) and x2"(0, 1, 2).

Their asset span is a two-dimensional subspace of R3 containing the unit vector 1, and hence it is a lattice-subspace. Clearly, the state-payo! vector (1, 1) of state 2 lies in the cone generated by state-payo!s (1, 0) of state 1 and (1, 2) of state 3. In fact (1, 1)"1

2(1, 0)#12(1, 2). Thus, states 1 and 3 are

fundamental.

The insured payo!on portfolioh"(0, 1) of one share of security 2 at#oor 1 is the contingent claimx

2s1"(1, 1, 2) and is not marketed. A direct veri"cation

shows that the marketed payo!(1,3₂, 2) is the supremumx

2sM1. This payo!is

generated by the portfolio (1,1₂) which is the minimum-cost insured portfolio at every arbitrage-free price. Note that derivative markets are incomplete in this example.

However, already with three securities it is not the case that arbitrary payo!s generate an asset span which is a lattice-subspace. When the asset span is not a lattice-subspace, the minimum-cost portfolio insurance depends on prices. The following example is based on Abramovich et al. (1994).

Example 3.2. Here we consider three securities with payo!s in four states given by

1"(1, 1, 1, 1), x2"(1, 1, 2, 2) and x3"(0, 1, 1, 2).

One can see that none of the state-payo!vectors (1, 1, 0), (1, 1, 1), (1, 2, 1), and (1, 2, 2) lies in the cone generated by the remaining three state-payo! vectors. Consequently, a fundamental set of states does not exist and the asset span is not a lattice-subspace.

Now consider the portfolioh"(0, 0, 1). The insured payo! on the portfolio

hat the#oork"1 is the contingent claimx

3s1"(1, 1, 1, 2) and is not in the

asset span. Next, consider the two arbitrage-free price vectors:p"(1,11₈, 1) and q"(1,13₈, 1). The minimum-cost insured portfolio at pricesp is the portfolio (0, 1, 0) with payo!(1, 1, 2, 2) and the cost of11₈. At pricesq, the minimum-cost insured portfolio is the portfolio (2,!1, 1) with payo!(1, 2, 1, 2) and the cost of

11₈. Note that the portfolio (2,!1, 1) is more expensive than (0, 1, 0) at pricesp. The reverse is true at pricesq.

4. Computing the minimum-cost portfolio insurance

In this section we present a simple method of "nding the minimum-cost insured portfolio whenever it is price independent.

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For a fundamental set of statesFand two contingent claimsy

1,y23RS, we

writey

15Fy2 ify1dominatesy2in the fundamental states. That is,

15Fy2 if y1(s)5y2(s) for every states3F.

Similarly, we writey₁"

Fy2, ify1andy2are the same in the fundamental states. Theorem 4.1. IfF"Ms

1,s2,2,s_NNis a fundamental set of states and y₁,y₂ are payows in the asset spanM,then

15y2 if and only if y15Fy2.

Proof. The su$ciency part is obvious. To prove the necessity, leth1andh2be

two portfolios such that X(h1)"y

1 and X(h2)"y2. Then X(h1)5FX(h2) means

N + n/1

n(si)(h1n!h2n)50 for every fundamental state s

i3F. Since for every sNF we can write

n(s)"+Ni/1asixn(si) withasi50, we see that X(h1)(s)!X(h2)(s)"+N

n/1

n(s)(h1n!h2n) "+N

i/1

a_si

C

+N n/1

n(si)(h1n!h2n)

D

50 for each states. Thus,X(h1)5X(h2), or y

15y2. h

Our next result provides a simple characterization of the minimum-cost insured portfolios when there exists a fundamental set of states.

Theorem 4.2. Suppose that there exists a fundamental set of statesFfor the asset spanM.Then for every arbitrage-free price systempand for every portfoliohand

yoork,the minimum-cost insured portfoliohkis the unique portfolio that replicates the insured payowX(h)sk_{in the fundamental states. That is}_,

X(hk)"

FX(h)sk.

The portfolio hk is the solution to the equation X

Fhk"FX(h)sk, that is,

hk"X~1

F [X(h)sk]F.

Proof. If there exists a fundamental set of statesF, then, by Theorem 3.3, the

asset span is a lattice-subspace. By Theorem 3.2, the minimum-cost insured portfolio hk satis"es X(hk)"X(h)s

Mk. We have to show that

X(h)s

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Letz3Mbe the (unique) marketed payo!such thatz"

FX(h)sk. (Note that since the matrixX

Fis non-singular such a marketed payo!zalways exists and is unique.) We must show thatzis the supremum ofX(h) andk_{relative to the asset} spanM. First, let us verify thatzis an upper bound ofX(h) andk_{. Indeed, since} z"

FX(h)sk, it follows thatz5FX(h) andz5Fk, and so from Theorem 4.1 we infer thatz5X(h) andz5k_.

To verify thatzis the least upper bound ofX(h) andk_in_{M, let}_y3M_satisfy

y5X(h) andy5k_{. Then,}_y5X(h)sk, and hencey5

FX(h)sk. Consequently,

Fz. Now Theorem 4.1 implies y5z. Therefore, z is the supremum

X(h)s

Mk. h

Loosely put, Theorem 4.2 says that, if there exists a fundamental set of states, then only the fundamental states are relevant for the minimum-cost portfolio insurance. The minimum-cost portfolio insurance in incomplete markets co-incides with the portfolio insurance in complete security markets restricted to the fundamental states.

Example 4.1. In Example 3.1 of two securities with payo!s x

1"1 and

2"(0, 1, 2), the insured payo! on security 2 at #oor k"1 is the contingent

claim x

2s1"(1, 1, 2) and is not in the asset span. Since states 1 and 3 are

fundamental, the minimum-cost insurance on security 2 replicates the claim (1, 1, 2) in states 1 and 3. The portfolio (1,1₂) has payo!(1,3₂, 2) and provides the minimum-cost insurance at arbitrary arbitrage-free prices.

5. Portfolio insurance with in5nitely many states

Our results on minimum-cost portfolio insurance extend without any essen-tial di!erence to the case of in"nitely many states. We assume that the set of states is a compact topological space _D and we take the space C(_D) of all continuous functions on _D as the space of contingent claims (see Brown and Ross, 1991). Note that the space C(D) is a Riesz space under the pointwise ordering and contains the constant random variable1representing the risk-free payo!.

A derivative contingent claim is now formally de"ned as a contingent claim that can be represented by a continuous function whose domain is the range of the security payo!s. That is, a contingent claimz3C(D) is a derivative claim if z(s)"f(x(s)) for every s3_D, for some continuous function f:RNP_R. Ross's characterization of complete derivative markets (Theorem 3.1) has been ex-tended to in"nitely many states by Brown and Ross (1991) and by Green and Jarrow (1987).

A necessary and su$cient condition for markets admitting price-independent minimum-cost portfolio insurance is that the asset span is a lattice-subspace of

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the payo!space}a condition that allows derivative markets to be incomplete. This is so because Theorem 3.2 (and also Lemma 2.1) can be easily extended to the present setting. A fundamental set of states is de"ned in exactly the same way as in Section 3: It is a subset of N states in which the security payo!s are non-redundant, and such that the state-payo!vector of each non-fundamental state lies in the cone generated by state-payo!vectors of the fundamental states. A more subtle argument is required to extend Theorem 3.3 to the setting of in"nitely many states. In general, it is not true that a"nite dimensional subspace of an arbitrary Riesz space is a lattice-subspace if and only if there is a funda-mental set of states. However, if the Riesz space is the space of continuous functions and the subspace contains the order unit (risk-free payo!), then Theorem 3.3 extends to the following:

Theorem 5.1. The asset span M-C(_D) containing the risk-free payow 1 is a lattice-subspace if and only if there is a fundamental set of states.

Proof. This is a special case of a general result due to Polyrakis (1996); see Theorem 7.5 in Section 7. h

The characterization of minimum-cost insurance in terms of fundamental states extends to this general case of in"nitely many states. The following example illustrates these results.

Example 5.1. The contingent claim space is C[0, 1]. There are two securities: a riskless bond with payo! x

1"1, and a risky stock with payo!x2(s)"sfor

eachs3[0, 1]. The asset span is

M"_Mz3C[0, 1]:z(s)"_h

1#h2sfor eachs3[0, 1] and some (h1,h2)3R2N

and consists of all linear functions. It is a two-dimensional subspace ofC[0,1], and hence a lattice-subspace.

States 0 and 1 form a fundamental set of states for the asset spanM. Indeed, we have

x(s)"(1,s)"(1!s)(1, 0)#s(1, 1)

for every 0(s(1. The state-payo!vectors of states 0 and 1, (1, 0) and (1, 1), are the extreme rays of the cone generated by the state-payo!vectors of all states.

The insured payo!on the stock at #oorkisz"x

2sk. Thus,

z(s)"

G

k if 0

4s4k,

s ifk(s41.

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The minimum-cost insured portfolio replicates z in states 0 and 1. Since z(0)"k and z(1)"1, that portfolio is hk"(k, 1!k). Its payo! is X(hk)(s)"(1!k)s#k. Thus, the minimum-cost insured portfolio on the stock at #oor k consists of k bonds and 1!k stocks, regardless of the prevailing arbitrage-free security price.

When security payo!s are their resale prices next period, then the compact set

Dis taken to be the normalized cone of arbitrage-free price vectors in the next period market (see Henrotte, 1996). Securities become price-contingent con-tracts, a notion due to Kurz (1974). For a portfolio h3_RN, its payo! is X(h)(q)"q)hforq3D-RN. The asset span is

M"Mz3C(_D): z(q)"q)hfor each q3D, for someh3RNN.

Assuming that the risk-free payo!1is marketed, Theorem 7.5 implies that the asset span M _{is a lattice-subspace if and only if there exist} _N _{price vectors}

q1,2,qNinDsuch that every price vector inDis a linear positive combination ofq1,2,qN. Expressing it in another way,Mis a lattice-subspace if and only if the cone of arbitrage-free price vectorsDis generated by someNprice vectors.

6. Portfolio insurance with Yudin basis

A Yudin basis of the asset spanMis a set ofNpositive payo!se

1,2,e_N3M that spanMwith the extra property that a payo!y"+N

n/1jnen3Mis positive if and only ifj

n50 for eachn(for details, see appendix). By Theorem A.2, the asset spanMis a lattice-subspace of the contingent claim spaceXif and only if there exists a Yudin basis forM_.

Example 6.1. A Yudin basis for the asset span of all linear functions on [0, 1] in Example 5.1 consists of two functions e

1 and e2 given by e1(s)"s and

e₂(s)"1!s, for eachs3[0, 1]. IfMe

1,2,e_NNis a Yudin basis ofM, then for any two payo!sy₁"+Nn_/1j1_ne_n andy

2"+Nn/1j2nen inM, we have y

15y2 if and only ifj1n5j2n for eachn.

The following result explains the use of Yudin basis for portfolio insurance.

Theorem 6.1. Suppose that there exists a Yudin basise

1,2,eNfor the asset span M_._{For an arbitrary portfolio}_h_{and an arbitrary}_y_oor_k,_let_X(_h₎"+N

n/1jnen and letk"+N

n/1cnen.Then the minimum-cost insured portfoliohksatisxes X(hk)"+N

n/1

maxMj

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Proof. This is straightforward from the de"nition of the minimum-cost insured portfolio and the property of the dominance order of payo!s we noted above (see also Theorem A.3). h

The portfoliosg1,2,gNthat generate the payo!se₁,2,e_Nof a Yudin basis of M _{can be thought of as mutual funds. They provide the same spanning}

opportunities as the securities. A portfolio of mutual funds has a positive payo! if and only if none of the funds is held short.

The portfoliohkof Theorem 6.1 is given by

hk"+N n/1

maxMj

n,cnNgn.

A characterization of a Yudin basis of an asset span when the set of statesSis "nite can be derived from the Abramovich}Aliprantis}Polyrakis Theorem A.4. If the asset span M is a lattice-subspace of RS, then there exists a set F"Ms

1,2,s_NNof fundamental states and the Yudin basis for Mconsists of payo!se

n forn"1,2,Ngiven by e

n(s)"

G

1 if s"s n, 0 if sOs

n, S3F,

a_sn if sNF,

wherea_snis thenth (non-negative) coe$cient in the expansionx(s)"+Ni

/1asix(si) in the de"nition of the fundamental states.

The Yudin basis of an asset span in the"nite-dimensional contingent claim spaceRSresembles the Arrow securities for the fundamental states. The payo! e

nequals one in the fundamental statesn, zero in all other fundamental states, and is non-negative in non-fundamental states.

Example 6.2. In Example 3.1 of two securities with payo!s x

1"1 and

2"(0, 1, 2), a Yudin basis consists of payo!se1"(1,21, 0) ande2"(0,12, 1). The

minimum-cost insured portfolio on security 2 at #oor 1 can be found using Theorem 6.1 as follows: we"rst write the payo!of security 2 asx

2"2e2 and

the risk-free payo! as 1"e

1#e2; the payo! of the minimum-cost insured

portfolio is maxM1, 0Ne

1#maxM1, 2Ne2"e1#2e2"(1,32, 1)

as already seen.

Appendix. Lattice-subspaces,Yudin bases and projections

We shall discuss here brie#y the basic properties of lattice-subspaces and Yudin bases. For details and proofs we refer the reader to Abramovich et al.

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(1994) and Aliprantis et al. (1998a). For details about the theory of Riesz spaces the reader can consult the books by Aliprantis and Burkinshaw (1978, 1985), and Luxemburg and Zaanen (1971).

An order relation5on a vector spaceXis said to bea partial orderif, in addition to being re#exive, antisymmetric and transitive, it is also compatible with the algebraic structure ofX in the sense thatx5yimplies:

(a)x#z5y#z for each z and (b)ax5_ay for alla50.

A vector space equipped with a vector space order is called apartially ordered

vector spaceor simply anorderedvector space. The setX`"_Mx3X:x50Nis referred to as thepositive coneofX.

An arbitrary coneCof a vector spaceXde"nes a partial order onXby letting

x5yif x!y3C, in which case X_`"C. On the other hand, if (X,5) is an

ordered vector space, then X` is a cone. Partial order relations and cones correspond in one-to-one fashion.

A partially ordered vector spaceXis said to be avector latticeor aRiesz space

if it is also a lattice. That is, a partially ordered vector spaceXis a vector lattice if for every pair of vectorsx,y3Xtheirsupremum(least upper bound) andinxmum

(greatest lower bound) exist in X. Any cone of a vector space that makes it a Riesz space is referred to as a lattice cone. As usual, the supremum and in"mum of a pair of vectorsx,yin a vector lattice are denoted byxsyandx'y,

respectively. In a vector lattice, the elements x`"xs0,x~"(!x)s0 and

DxD"xs(!x) are called the positive part, thenegative part, and theabsolute

valueof x. We always have the identities

x"x`!x~ and DxD"x`#x~.

Theorem A.1. A partially ordered spaceXis a Riesz space if and only if there exists avectora3Xsuch thatxsaexists for everyx3X.

Proof. See Luxemburg and Zaanen (1971, Theorem 1.5, p. 55). h

De,nition A.1. A vector subspace>_{of a partially ordered vector space}_X_{is said}

to be a lattice-subspace if > _{under the induced ordering from} _X _{is a vector}

lattice in its own right. That is,>_{is a lattice-subspace if for every}_x,_y3>_the

least upper bound of the set Mx,yN exists in > _{when ordered by the cone} >_`">_W_X_`_.

If > _{is a lattice-subspace of} _{X, then we shall denote the supremum and}

in"mum of the setMx,yN->_in>_by_xs

Yyandx'Yy, respectively. That is,

Yy"minMz3>:z5x andz5yN and

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A vector subspace>_{of a vector lattice}_X _{is said to be a}_v_{ector sublattice}_or

aRiesz subspaceif for each x,y3>_{we have}_xsy_and _x_'_y_in >_{. Every Riesz}

subspace is automatically a lattice-subspace but a lattice-subspace need not be a Riesz subspace.

A normed space which is also a partially ordered vector space is apartially ordered normed space. A normDD)DDon a vector lattice is said to be alattice normif

DxD4_DyD impliesDDxDD4_DDyDD. Anormedvector latticeis a vector lattice equipped with a lattice norm. A complete normed vector lattice is called aBanach lattice. A classical example of a Banach lattice is C(D), the vector space of all continuous real-functions on a compact topological spaceD, equipped with the pointwise order and the sup norm.

Dexnition A.2. If>_{is a}_{"nite-dimensional vector subspace of a partially ordered}

spaceX, then a basisMe

iNJi/1of>consisting of positive vectors is called aYudin basis whenever the vectorsx"+Ji

/1jiei of > satisfyx3>`">WX` if and only ifj

i50 for eachi.

The fundamental connection between lattice-subspaces and Yudin bases is given next. This result is due to Yudin (1939) (see also Luxemburg and Zaanen, 1971, Theorem 26.11, p. 152) and justi"es the name`Yudin basisa. It can be also proven using the classical Choquet}Kendall theorem; see Peressini (1967, Prop-osition 1.5, p. 9).

Theorem A.2 (Yudin). A xnite dimensional vector subspace > _{of a partially}

orderedvector space is a lattice-subspace if and only if it has a Yudin basis. When Yudin bases exist, they are essentially unique. For a proof of the next result see Aliprantis et al. (1998a, Lemma 8, p. 6).

Theorem A.3. If Me

iNJi/1 andMflNJl

/1 are two Yudin bases for a subspace >of a partially orderedvector space, then eache

i is a scalar multiple of somefl and

eachfl is a scalar multiple of somee

i. Moreover,if>_{is a lattice-subspace and}_M_e

1,2,e_JNis a Yudin basis for>,then for any twovectorsx"+J

i/1aiei andy"+iJ/1biei in>,we have

Yy"+J i/1

(a_is_b

i)ei and x'Yy"+J i/1

(a_i'b_i)e_i.

For "nite-dimensional spaces, we have the following result of Abramovich et al. (1994).

Theorem A.4 (Abramovich}Aliprantis}Polyrakis). For a subspace > _{of some}

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1. >_{is an N-dimensional lattice-subspace of}_RS_.

2. There is a fundamental setF"Ms

1,2,s_NNofNstates. 3. There exist a Yudin basis of non-negativevectorsMe

1,2,e_NNin>and a subset ofNstatesF"Ms

1,2,s_NNsuch thate_i(s_j)"d_ij for all i,j"1,2,N. A positive vectoruin a vector latticeXis said to be anorder unit, if for each x3Xthere exists somej'0 such thatju5x.

Theorem A.5. A xnite-dimensional subspace>-C(D)containing the order unit

1is a lattice-subspace if and only if there is a fundamental set of states.

Proof. See the proof of Proposition 3.5 in Polyrakis (1996, p. 2801). h

There is another characterization of lattice-subspaces in terms of positive projections. Recall that a positive linear operatorP:XPXon a Riesz space is said to be apositive projectionifP2"P.

Theorem A.6. Assume that a xnite-dimensional subspace>-C(D) contains the order unit 1. Then > _{is a lattice-subspace if and only if there exists a} ₍_unique₎

positive projectionP:C(D)PC(D)having range>_,_i.e._,_P(C(_D₎₎">_._Moreo_v_er_,_if

this unique positive projection Pexists,then for each pairx,y3>_{we ha}_v_e

Yy"P(xsy) and x'Yy"P(x'y).

For a proof of this theorem and more details see Abramovich et al. (1994). For more about lattice-subspaces, the reader can consult the work of Polyrakis (1994) and the references therein.

References

Abramovich, Y.A., Aliprantis, C.D., Polyrakis, I.A., 1994. Lattice-subspaces and positive projections. Proceedings of the Royal Irish Academy 94a, 237}253.

Aliprantis, C.D., Brown, D.J., Polyrakis, I.A., Werner, J., 1998a. Yudin cones and inductive limit topologies. Atti del Seminario Matematico e Fisico del' Universita di Modena 46, 389}412.

Aliprantis, C.D., Brown, D.J., Polyrakis, I.A., Werner, J., 1998b. Portfolio dominance and optimality in in"nite securities markets. Journal of Mathematical Economics 30, 347}366.

Aliprantis, C.D., Burkinshaw, O., 1978. Locally Solid Riesz Spaces. Academic Press, New York. Aliprantis, C.D., Burkinshaw, O., 1985. Positive Operators. Academic Press, New York. Arrow, K.J., 1953. Le role des valeurs boursieres pour la repartition la meillure des risques.

Econometrie 40, 41}47. English translation: The role of securities in the optimal allocation of risk-bearing. 1964. Review of Economic Studies. 31, 91}96.

Broadie, M., Cvitanic, J., Soner, H.M., 1998. Optimal replication of contingent claims under portfolio constraints. Review of Financial Studies 11, 59}81.

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Edirisinghe, C., Naik, V., Uppal, R., 1993. Optimal replication of options with transaction costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117}139.

Green, R., Jarrow, R.A., 1987. Spanning and completeness in markets with contingent claims. Journal of Economic Theory 41, 202}210.

Henrotte, P., 1996. Construction of a state space for interrelated securities with an application to temporary equilibrium theory. Economic Theory 8 (3), 423}459.

Kurz, M., 1974. The Kesten}Stigum model and the treatment of uncertainty in equilibrium theory. In: Balch, M.S., McFadden, D.L., Wu, S.Y. (Eds.), Essays on Economic Behavior Under Uncertainty. North-Holland, New York, pp. 389}399.

Leland, H., 1980. Who should buy portfolio insurance. Journal of Finance 35, 581}594. Luxemburg, W.A.J., Zaanen, A.C., 1971. Riesz Spaces I. North-Holland, Amsterdam.

Naik, V., Uppal, R., 1994. Leverage constraints and the optimal hedging of stock and bond options. Journal of Financial and Quantitative Analysis 29, 199}223.

Peressini, A., 1967. Ordered Topological Vector Spaces. Harper & Row, New York.

Polyrakis, I.A., 1994. Lattice-subspaces of C[0,1] and positive bases. Journal of Mathematical Analysis and Applications 184, 1}18.

Polyrakis, I.A., 1996. Finite-dimensional lattice-subspaces ofC(X) and curves ofRn. Transactions of the American Mathematical Society 348, 2793}2810.

Ross, S.A., 1976. Options and e$ciency. Quarterly Journal of Economics 90, 75}89.

Yudin, A.I., 1939. A solution of two problems in the theory of partially ordered spaces. Doklady Akademy Nauk SSSR 23, 418}422.

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The minimum-cost insured portfolio replicates z in states 0 and 1. Since

z(0)"k and z(1)"1, that portfolio is hk"(k, 1!k). Its payo! is

X(hk)(s)"(1!k)s#k. Thus, the minimum-cost insured portfolio on the stock at #oor k consists of k bonds and 1!k stocks, regardless of the prevailing arbitrage-free security price.

When security payo!s are their resale prices next period, then the compact set Dis taken to be the normalized cone of arbitrage-free price vectors in the next period market (see Henrotte, 1996). Securities become price-contingent con-tracts, a notion due to Kurz (1974). For a portfolio h3RN, its payo! is

X(h)(q)"q)hforq3D-RN. The asset span is

M"_Mz3C(_D): z(q)"q)hfor each q3D, for someh3RNN.

Assuming that the risk-free payo!1is marketed, Theorem 7.5 implies that the asset span M _{is a lattice-subspace if and only if there exist} _N _{price vectors}

q1,2,qNinDsuch that every price vector inDis a linear positive combination ofq1,2,qN. Expressing it in another way,M_{is a lattice-subspace if and only if} the cone of arbitrage-free price vectorsDis generated by someNprice vectors.

6. Portfolio insurance with Yudin basis

A Yudin basis of the asset spanMis a set ofNpositive payo!se

1,2,e_N3M that spanMwith the extra property that a payo!y"+N

n/1jnen3Mis positive if and only ifj

n50 for eachn(for details, see appendix). By Theorem A.2, the asset spanMis a lattice-subspace of the contingent claim spaceXif and only if there exists a Yudin basis forM_.

Example 6.1. A Yudin basis for the asset span of all linear functions on [0, 1] in Example 5.1 consists of two functions e

1 and e2 given by e1(s)"s and

e₂(s)"1!s, for eachs3[0, 1].

IfMe₁,2,e_NNis a Yudin basis ofM, then for any two payo!sy₁"+Nn_/1j1_ne_n andy

2"+Nn/1j2nen inM, we have

15y2 if and only ifj1n5j2n for eachn.

The following result explains the use of Yudin basis for portfolio insurance. Theorem 6.1. Suppose that there exists a Yudin basise

1,2,eNfor the asset span M_._{For an arbitrary portfolio}_h_{and an arbitrary}_yoor_k_,_let_X₍_h₎"+N

n/1jnen and letk"+N

n/1cnen.Then the minimum-cost insured portfoliohksatisxes

X(hk)"+N

n/1 maxMj

(2)

Proof. This is straightforward from the de"nition of the minimum-cost insured portfolio and the property of the dominance order of payo!s we noted above (see also Theorem A.3). h

The portfoliosg1,2,gNthat generate the payo!se₁,2,e_Nof a Yudin basis of M _{can be thought of as mutual funds. They provide the same spanning} opportunities as the securities. A portfolio of mutual funds has a positive payo! if and only if none of the funds is held short.

The portfoliohkof Theorem 6.1 is given by

hk"+N n/1

maxMj

n,cnNgn.

F"Ms

1,2,s_NNof fundamental states and the Yudin basis for Mconsists of payo!se

n forn"1,2,Ngiven by

n(s)"

G

1 if s"s

n, 0 if sOs

n, S3F,

a_sn if sNF,

wherea_snis thenth (non-negative) coe$cient in the expansionx(s)"+Ni

/1asix(si) in the de"nition of the fundamental states.

The Yudin basis of an asset span in the"nite-dimensional contingent claim spaceRSresembles the Arrow securities for the fundamental states. The payo!

nequals one in the fundamental statesn, zero in all other fundamental states, and is non-negative in non-fundamental states.

Example 6.2. In Example 3.1 of two securities with payo!s x

1"1 and

2"(0, 1, 2), a Yudin basis consists of payo!se1"(1,12, 0) ande2"(0,12, 1). The minimum-cost insured portfolio on security 2 at #oor 1 can be found using Theorem 6.1 as follows: we"rst write the payo!of security 2 asx

2"2e2 and the risk-free payo! as 1"e

1#e2; the payo! of the minimum-cost insured portfolio is

maxM1, 0Ne₁#maxM1, 2Ne₂"e

1#2e2"(1,32, 1) as already seen.

Appendix. Lattice-subspaces,Yudin bases and projections

We shall discuss here brie#y the basic properties of lattice-subspaces and Yudin bases. For details and proofs we refer the reader to Abramovich et al.

(3)

(1994) and Aliprantis et al. (1998a). For details about the theory of Riesz spaces the reader can consult the books by Aliprantis and Burkinshaw (1978, 1985), and Luxemburg and Zaanen (1971).

(a)x#z5y#z for each z and (b)ax5_ay for alla50.

A vector space equipped with a vector space order is called apartially ordered

vector spaceor simply anorderedvector space. The setX`"_Mx3X:x50Nis referred to as thepositive coneofX.

An arbitrary coneCof a vector spaceXde"nes a partial order onXby letting

x5yif x!y3C, in which case X_`"C. On the other hand, if (X,5) is an ordered vector space, then X` is a cone. Partial order relations and cones correspond in one-to-one fashion.

A partially ordered vector spaceXis said to be avector latticeor aRiesz space if it is also a lattice. That is, a partially ordered vector spaceXis a vector lattice if for every pair of vectorsx,y3Xtheirsupremum(least upper bound) andinxmum (greatest lower bound) exist in X. Any cone of a vector space that makes it a Riesz space is referred to as a lattice cone. As usual, the supremum and in"mum of a pair of vectorsx,yin a vector lattice are denoted byxsyandx'y, respectively. In a vector lattice, the elements x`"xs0,x~"(!x)s0 and

DxD"xs(!x) are called the positive part, thenegative part, and theabsolute

valueof x. We always have the identities

x"x`!x~ and DxD"x`#x~.

Theorem A.1. A partially ordered spaceXis a Riesz space if and only if there exists avectora3Xsuch thatxsaexists for everyx3X.

Proof. See Luxemburg and Zaanen (1971, Theorem 1.5, p. 55). h

De,nition A.1. A vector subspace>_{of a partially ordered vector space}_X_{is said}

to be a lattice-subspace if > _{under the induced ordering from} _X _{is a vector}

lattice in its own right. That is,>_{is a lattice-subspace if for every}_x_,_y3>_the

least upper bound of the set Mx,yN exists in > _{when ordered by the cone} >_`">_WX_`_.

If > _{is a lattice-subspace of} _X_{, then we shall denote the supremum and}

in"mum of the setMx,yN->_in>_by_xs

Yyandx'Yy, respectively. That is,

Yy"minMz3>:z5x andz5yN and

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A vector subspace>_{of a vector lattice}_X _{is said to be a}_v_{ector sublattice}_or

aRiesz subspaceif for each x,y3>_{we have}_xsyand x'yin >. Every Riesz subspace is automatically a lattice-subspace but a lattice-subspace need not be a Riesz subspace.

A normed space which is also a partially ordered vector space is apartially ordered normed space. A normDD)DDon a vector lattice is said to be alattice normif

Dexnition A.2. If>_{is a}_{"nite-dimensional vector subspace of a partially ordered}

spaceX, then a basisMe

iNJi/1of>consisting of positive vectors is called aYudin basis whenever the vectorsx"+Ji

/1jiei of > satisfyx3>`">WX` if and only ifj

i50 for eachi.

Theorem A.2 (Yudin). A xnite dimensional vector subspace > _{of a partially}

Theorem A.3. If Me_iNJ_i/1 andMflNJl

/1 are two Yudin bases for a subspace >of a partially orderedvector space, then eache

i is a scalar multiple of somefl and eachfl is a scalar multiple of somee

i. Moreover,if>_{is a lattice-subspace and}_Me

1,2,e_JNis a Yudin basis for>,then for any twovectorsx"+J

i/1aiei andy"+i/1J biei in>,we have

xs Yy"+J

i/1 (a_is_b

i)ei and x'Yy"+J i/1

(a_i'b_i)e_i.

For "nite-dimensional spaces, we have the following result of Abramovich et al. (1994).

Theorem A.4 (Abramovich}Aliprantis}Polyrakis). For a subspace > _{of some}

(5)

1. >_{is an N-dimensional lattice-subspace of}RS_.

2. There is a fundamental setF"Ms

1,2,s_NNofNstates.

3. There exist a Yudin basis of non-negativevectorsMe₁,2,e_NNin>and a subset ofNstatesF"Ms

1,2,s_NNsuch thate_i(s_j)"d_ij for all i,j"1,2,N. A positive vectoruin a vector latticeXis said to be anorder unit, if for each

x3Xthere exists somej'0 such thatju5x.

Theorem A.5. A xnite-dimensional subspace>-C(D)containing the order unit 1is a lattice-subspace if and only if there is a fundamental set of states.

Proof. See the proof of Proposition 3.5 in Polyrakis (1996, p. 2801). h There is another characterization of lattice-subspaces in terms of positive projections. Recall that a positive linear operatorP:XPXon a Riesz space is said to be apositive projectionifP2"P.

Theorem A.6. Assume that a xnite-dimensional subspace>-C(D) contains the order unit 1. Then > _{is a lattice-subspace if and only if there exists a} _(unique)

positive projectionP:C(D)PC(D)having range>_,_i.e.,_P₍_C₍_D₎₎">_._Moreo_v_er,_if

this unique positive projection Pexists,then for each pairx,y3>_{we ha}_v_e xs