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Minimum-cost portfolio insurance
Journal of Economic Dynamics & Control 24 (2000) 1703}1719
Minimum-cost portfolio insurance夽 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$cient condition for price-independent minimum-cost portfolio insurance is that the asset span is a lattice-subspace of the space of contingent claims. If the asset span is a lattice-subspace, then the minimum-cost portfolio insurance can be easily calculated as a portfolio that replicates the targeted payo! in a subset of states which is the same for every reference portfolio. 2000 Elsevier Science B.V. All rights reserved.
夽 The authors are pleased to acknowledge the suggestions and comments of Peter Bossaerts, Phillip Henrotte and Yiannis Polyrakis. The research of C.D. Aliprantis was partially supported by the 1995 PENED Program of the Ministry of Industry, Energy and Technology of Greece and by the NATO Collaborative Research Grant C941059. Roko Aliprantis also expresses his deep appreciation for the hospitality provided by the Department of Economics and the Center for Analytic Economics at Cornell University and the Division of Humanities and Social Sciences of the California Institute of Technology where parts of this paper were written during his sabbatical leave (January}June, 1996). Jan Werner acknowledges the "nancial support of the Deutsche Forschungsgemainschaft, SFB 303.
* Corresponding author. Tel.: 001-765-494-4404; fax: 001-765-494-9658. E-mail address: [email protected] (C.D. Aliprantis). 0165-1889/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 9 1 - 3
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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 the minimum-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 minimum-cost portfolio insurance to be price independent. A necessary and su$cient condition admits a large and interesting class of incomplete derivative market structures. Our primary characterization of market structures in which minimum-cost portfolio insurance is price independent relies on the mathematical theory of
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Riesz spaces (vector lattices) and lattice-subspaces. Riesz spaces have been used in the context of options markets by Brown and Ross (1991) and by Green and Jarrow (1987). The space of contingent claims on a "nite set of states of nature is a Riesz space with a supremum of any two contingent claims y and y de"ned as the statewise supremum and denoted by y Ry . A put option on a payo! y with strike price k can be represented using the operation of supremum as (k!y)R0, where k is the risk-free payo! of k. The insured claim on payo! y at #oor k is yRk. The subspace M of payo!s of all portfolios of securities (asset span) is a Riesz subspace if the supremum of arbitrary two payo!s in M belongs to M. If the asset span is a Riesz subspace, then every option is marketed. The result of Ross (1976) can be restated as saying that derivative markets are complete if and only if the asset span is a Riesz subspace of the space of contingent claims. The asset span M is a lattice-subspace if there exists a supremum relative to M of any two payo!s in M. The supremum relative to M of two payo!s y and y in M is their least upper bound in M, and is denoted by y R y . If M is + a lattice-subspace, then the payo! yR k exists and has the least cost among all + marketed payo!s that dominate the insured claim yRk, for arbitrary arbitragefree prices of securities. Our main result is that the minimum-cost portfolio insurance is price independent if and only if the asset span M is a latticesubspace. We present a number of su$cient conditions for the asset span to be a latticesubspace. The asset span of any two limited liability securities is a latticesubspace. 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$cient condition for M to be a lattice-subspace is the existence of a Yudin basis for M, i.e., a basis of limited liability payo!s such that every marketed limited liability payo! has a unique representation as a nonnegative linear combination of basis payo!s. The notion of Yudin basis is a generalization to incomplete markets of a basis of Arrow securities for complete markets (Arrow, 1953). We show that the minimum-cost portfolio insurance has a simple form whenever it is independent of security prices. The cost minimizing portfolio replicates the insured payo! in fundamental states. Since the securities markets restricted to the fundamental states are complete, the minimum-cost portfolio insurance can be reduced to portfolio insurance in complete markets. The paper is organized as follows: In Section 2, we de"ne the minimum-cost portfolio insurance. In Section 3, we introduce the lattice-subspace property of the asset span and show that it is necessary and su$cient for the minimum-cost portfolio insurance to be price independent in the set of arbitrage-free prices.
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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 are N 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 states S"+1,2, S, to describe the uncertainty. The payo! of security n in S states is a vector x 311 . The payo!s x ,2, x L > , 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 state s (state-payow vector). For a portfolio h"(h ,2, h )31,, its payo! is X(h)" , h x . The set of , L L L payo!s of all portfolios is the linear span of payo!s x ,2, x in the space 11 of , all state contingent claims and is the asset span M. A contingent claim is a marketed payow if it lies in the asset span. It is assumed that the risk-free payo! is marketed, so that 13M. If the asset span equals the whole space of contingent claims (i.e., if N"S), then markets are complete. Let p"(p ,2, p )31, be 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 vector p31, is arbitrage-free if there is no arbitrage portfolio, that is, if p ) h'0 for all non-zero portfolios h with X(h)50. The following simple duality result is well known. Lemma 2.1. If p ) h50 for every arbitrage-free price vector p, then X(h)50. Proof. Suppose by way of contradiction that there is a portfolio h with p ) h50 for every arbitrage-free p while X(h)(s)(0 for some state s. Then there exist strictly positive numbers j for each s such that 1 j X(h)(s)(0. Using the Q Q Q Bold 1 denotes the S-vector (1,2,1); similarly, k denotes (k,2, k) for any scalar k.
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de"nition of X(h) we have 1 j x(s)h(0. De"ne a price system q by Q Q q" 1 j x(s). Clearly, q is arbitrage-free. This is, however, a contradiction, Q Q since q ) h(0. 䊐 The insured payo! on a portfolio h at a `#oora k is the contingent claim X(h)Rk. This contingent claim may or may not be marketed. The minimumcost 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 insurance is de"ned by the following cost minimization problem: min p ) g EZ1, subject to X(g)5X(h)Rk. This linear programming problem has a unique solution as long as p is arbitrage-free. We denote the solution by hI and refer to it as the minimum-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 independent if 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 span M is a Riesz subspace if for every marketed payo!s y , y 3M their (statewise) supremum y Ry belongs to M. It is a lattice-subspace, if for every marketed payo!s y , y 3M there exists a least upper bound of y and y relative to M. That is, there exists z3M such that (1) z5y and z5y (i.e, z5y Ry ), and (2) z 5z for every z 3M with property (1). We denote that We use the symbol R to denote the statewise supremum of two contingent claims. That is (yRz)(s)"max+y(s), z(s),, for s"1,2, S.
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supremum by y R y . It is always true that y R y 5y Ry . Note that if + + M is a Riesz subspace, then it is also a lattice-subspace, and y Ry "y R y . + A derivative contingent claim is 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 portfolio hI satis"es X(hI)"X(h)Rk. 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)Rk satis"es X(h)Rk"X(h)#[(k!X(h))R0], hence it can be generated by holding the portfolio h and a put option on X(h) with strike price k. Thus, a necessary and su$cient condition for complete derivative markets is that all put and call options on marketed payo!s be marketed. Since each put and call option is a supremum of two marketed payo!s, all options are marketed if the asset span is a Riesz subspace. The following is essentially a restatement of the result due to Ross (1976). Theorem 3.1. Derivative markets are complete if and only if the asset span is a Riesz subspace of 11. Proof. The theorem of Ross (1976) implies that derivative markets are complete if and only if (k!y)R0 belongs to M for every y3M and every k. This latter property is necessary and su$cient for M to be a Riesz subspace. This follows from the simple lattice identity y Ry "y #[(y !y )R0]. 䊐 Completeness of derivative markets is a su$cient but not a necessary condition for the minimum-cost portfolio insurance to be price independent. Suppose that there exists the supremum of the payo!s X(h) and k relative to the asset span M, i.e., X(h)R k. A portfolio that generates the payo! X(h)R k is 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 latticesubspace of 11. In this case, the minimum-cost insured portfolio hI satisxes X(hI)"X(h)R k. + Proof. Assume "rst that the minimum-cost insured portfolio exists and is price independent. Consider an arbitrary portfolio h and a #oor k. Let hI denote the
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insured portfolio. For any portfolio g such that X(g)5X(h)Rk, we have that p ) g5p ) hI for every arbitrage-free price vector p. Lemma 2.1 implies that X(g)5X(hI). Hence, the payo! X(hI) is the supremum of X(h) and k in M, i.e., X(h)R k. + We have thus demonstrated that the supremum yR k exists for every + marketed payo! y and every k. The simple lattice identity y R y " + [(y !y #k)R k]#y !k proves that the supremum y R y exists for two + + arbitrary payo!s y and y in M. Hence, M is a lattice-subspace. Conversely, if the asset span M is a lattice-subspace, then the supremum X(h)R k exists for every portfolio h and every k. Now let the portfolio hI + be such that X(hI)"X(h)R k. Then, for every portfolio g satisfying + X(g)5X(h)Rk, we have that X(g)5X(hI). This inequality implies p ) g5p ) hI for every arbitrage-free price vector p. Consequently, the portfolio hI is the minimum-cost insured portfolio for every arbitrage-free price. 䊐 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 property. A necessary and su$cient condition for the asset span M to be a latticesubspace is the existence of a fundamental set of states. For any subset F-S of states, we shall denote by X the matrix of security payo!s restricted to $ the subset of states F. A subset F"+s ,2, s ,-S of N states is called , fundamental, if 1. the N;N matrix X is nonsingular, and $ 2. for each s , F there exist non-negative scalars aQ , aQ ,2, aQ such that , , x(s)" aQ x(s ). G G G 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 of 11 if and only if there is a fundamental set of states. It is well known that any two-dimensional subspace of 11 containing the unit vector 1 is a lattice-subspace (see Theorem 3.5, Aliprantis et al., 1998a). When N"2, the state-payo!s x(s) for s3S generate a closed cone in 1. If there are "nitely many states, the two extreme rays of this cone identify the two fundamental 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 x "(1, 1, 1) and x "(0, 1, 2). Their asset span is a two-dimensional subspace of 1 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, 0)#(1, 2). Thus, states 1 and 3 are fundamental. The insured payo! on portfolio h"(0, 1) of one share of security 2 at #oor 1 is the contingent claim x R1"(1, 1, 2) and is not marketed. A direct veri"cation shows that the marketed payo! (1, , 2) is the supremum x R 1. This payo! is + generated by the portfolio (1, ) 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 x "(1, 1, 1, 1), x "(1, 1, 2, 2) and x "(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 portfolio h"(0, 0, 1). The insured payo! on the portfolio h at the #oor k"1 is the contingent claim x R1"(1, 1, 1, 2) and is not in the asset span. Next, consider the two arbitrage-free price vectors: p"(1, , 1) and q"(1, , 1). The minimum-cost insured portfolio at prices p is the portfolio (0, 1, 0) with payo! (1, 1, 2, 2) and the cost of . At prices q, the minimum-cost insured portfolio is the portfolio (2,!1, 1) with payo! (1, 2, 1, 2) and the cost of . Note that the portfolio (2,!1, 1) is more expensive than (0, 1, 0) at prices p. The reverse is true at prices q.
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 states F and two contingent claims y , y 311, we write y 5 y if y dominates y in the fundamental states. That is, $ y 5 y if y (s)5y (s) for every state s3F. $ Similarly, we write y " y , if y and y are the same in the fundamental states. $ Theorem 4.1. If F"+s , s ,2, s , is a fundamental set of states and y , y are , payows in the asset span M, then y 5y if and only if y 5 y . $ Proof. The su$ciency part is obvious. To prove the necessity, let h and h be two portfolios such that X(h)"y and X(h)"y . Then X(h)5 X(h) $ means , x (s )(h!h)50 L G L L L for every fundamental state s 3F. Since for every s , F we can write G x (s)" , aQ x (s ) with aQ 50, we see that L G G L G G , X(h)(s)!X(h)(s)" x (s)(h!h) L L L L , , " aQ x (s )(h!h) 50 G L G L L G L for each state s. Thus, X(h)5X(h), or y 5y . 䊐
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 states F for the asset span M. Then for every arbitrage-free price system p and for every portfolio h and yoor k, the minimum-cost insured portfolio hI is the unique portfolio that replicates the insured payow X(h)Rk in the fundamental states. That is, X(hI)" X(h)Rk. $ The portfolio hI is the solution to the equation X hI" X(h)Rk, that is, $ $ hI"X\[X(h)Rk] . $ $ Proof. If there exists a fundamental set of states F, then, by Theorem 3.3, the asset span is a lattice-subspace. By Theorem 3.2, the minimum-cost insured portfolio hI satis"es X(hI)"X(h)R k. We have to show that + X(h)R k" X(h)Rk. + $
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Let z3M be the (unique) marketed payo! such that z" X(h)Rk. (Note that $ since the matrix X is non-singular such a marketed payo! z always exists and is $ unique.) We must show that z is the supremum of X(h) and k relative to the asset span M. First, let us verify that z is an upper bound of X(h) and k. Indeed, since z" X(h)Rk, it follows that z5 X(h) and z5 k, and so from Theorem 4.1 we $ $ $ infer that z5X(h) and z5k. To verify that z is the least upper bound of X(h) and k in M, let y3M satisfy y5X(h) and y5k. Then, y5X(h)Rk, and hence y5 X(h)Rk. Consequently, $ y5 z. Now Theorem 4.1 implies y5z. Therefore, z is the supremum $ X(h)R k. 䊐 + 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 coincides 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 and x "(0, 1, 2), the insured payo! on security 2 at #oor k"1 is the contingent claim x R1"(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, ) has payo! (1, , 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 essential 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 variable 1 representing 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 claim z3C(D) is a derivative claim if z(s)"f (x(s)) for every s3D, for some continuous function f : 1,P1. Ross's characterization of complete derivative markets (Theorem 3.1) has been extended 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 fundamental 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. 䊐 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, and a risky stock with payo! x (s)"s for each s3[0, 1]. The asset span is M"+z3C[0, 1]: z(s)"h #h s for each s3[0, 1] and some (h , h )31, and consists of all linear functions. It is a two-dimensional subspace of C[0,1], and hence a lattice-subspace. States 0 and 1 form a fundamental set of states for the asset span M. 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 #oor k is z"x Rk. Thus, k if 04s4k, z(s)" s if k(s41.
Clearly, for 0(k(1 the contingent claim z is not in the asset span.
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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 hI"(k, 1!k). Its payo! is X(hI)(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 D is taken to be the normalized cone of arbitrage-free price vectors in the next period market (see Henrotte, 1996). Securities become price-contingent contracts, a notion due to Kurz (1974). For a portfolio h31,, its payo! is X(h)(q)"q ) h for q3D-1,. The asset span is M"+z3C(D): z(q)"q ) h for each q3D, for some h31,,. Assuming that the risk-free payo! 1 is marketed, Theorem 7.5 implies that the asset span M is a lattice-subspace if and only if there exist N price vectors q,2, q, in D such that every price vector in D is a linear positive combination of q,2, q,. Expressing it in another way, M is a lattice-subspace if and only if the cone of arbitrage-free price vectors D is generated by some N price vectors. 6. Portfolio insurance with Yudin basis A Yudin basis of the asset span M is a set of N positive payo!s e ,2, e 3M , that span M with the extra property that a payo! y" , j e 3M is positive L L L if and only if j 50 for each n (for details, see appendix). By Theorem A.2, the L asset span M is a lattice-subspace of the contingent claim space X if and only if there exists a Yudin basis for M. 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 and e given by e (s)"s and e (s)"1!s, for each s3[0, 1]. If +e ,2, e , is a Yudin basis of M, then for any two payo!s y " , je , L L L and y " , je in M, we have L L L y 5y if and only if j5j for each n. L L The following result explains the use of Yudin basis for portfolio insurance. Theorem 6.1. Suppose that there exists a Yudin basis e ,2, e for the asset span , M. For an arbitrary portfolio h and an arbitrary yoor k, let X(h)" , j e and L L L let k" , c e . Then the minimum-cost insured portfolio hI satisxes L L L , X(hI)" max+j , c ,e . L L L L
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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). 䊐 The portfolios g,2, g, that generate the payo!s e ,2, e of 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 portfolio hI of Theorem 6.1 is given by , hI" max+j , c ,gL. L L L A characterization of a Yudin basis of an asset span when the set of states S is "nite can be derived from the Abramovich}Aliprantis}Polyrakis Theorem A.4. If the asset span M is a lattice-subspace of 11, then there exists a set F"+s ,2, s , of fundamental states and the Yudin basis for M consists of , payo!s e for n"1,2, N given by L 1 if s"s , L e (s)" 0 if sOs , S3F, L L aQ if s , F, L where aQ is the nth (non-negative) coe$cient in the expansion x(s)" , aQx(s ) L G G G in the de"nition of the fundamental states. The Yudin basis of an asset span in the "nite-dimensional contingent claim space 11 resembles the Arrow securities for the fundamental states. The payo! e equals one in the fundamental state s , zero in all other fundamental states, L L and is non-negative in non-fundamental states.
Example 6.2. In Example 3.1 of two securities with payo!s x "1 and x "(0, 1, 2), a Yudin basis consists of payo!s e "(1, , 0) and e "(0, , 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 as x "2e and the risk-free payo! as 1"e #e ; the payo! of the minimum-cost insured portfolio is max+1, 0,e #max+1, 2,e "e #2e "(1, , 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 relation 5 on a vector space X is said to be a partial order if, in addition to being re#exive, antisymmetric and transitive, it is also compatible with the algebraic structure of X in the sense that x5y implies: (a) x#z5y#z for each z and (b) ax5ay for all a50. A vector space equipped with a vector space order is called a partially ordered vector space or simply an ordered vector space. The set X>"+x3X: x50, is referred to as the positive cone of X. An arbitrary cone C of a vector space X de"nes a partial order on X by letting x5y if 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 space X is said to be a vector lattice or a Riesz space if it is also a lattice. That is, a partially ordered vector space X is a vector lattice if for every pair of vectors x, y3X their supremum (least upper bound) and inxmum (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 vectors x, y in a vector lattice are denoted by xRy and xy, respectively. In a vector lattice, the elements x>"xR0, x\"(!x)R0 and "x""xR(!x) are called the positive part, the negative part, and the absolute value of x. We always have the identities x"x>!x\ and "x""x>#x\. Theorem A.1. A partially ordered space X is a Riesz space if and only if there exists a vector a3X such that xRa exists for every x3X. Proof. See Luxemburg and Zaanen (1971, Theorem 1.5, p. 55).
䊐
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 +x, y, exists in > when ordered by the cone >>">5X>. If > is a lattice-subspace of X, then we shall denote the supremum and in"mum of the set +x, y,-> in > by xR y and x y, respectively. That is, 7 7 xR y"min+z3>: z5x and z5y, 7 and x y"max+z3>: z4x and z4y,. 7
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A vector subspace > of a vector lattice X is said to be a vector sublattice or a Riesz subspace if for each x, y3> we have xRy and xy 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 a partially ordered normed space. A norm "" ) "" on a vector lattice is said to be a lattice norm if "x"4"y" implies ""x""4""y"". A normed vector lattice is a vector lattice equipped with a lattice norm. A complete normed vector lattice is called a Banach lattice. A classical example of a Banach lattice is C(D), the vector space of all continuous real-functions on a compact topological space D, equipped with the pointwise order and the sup norm. Dexnition A.2. If > is a "nite-dimensional vector subspace of a partially ordered space X, then a basis +e ,( of > consisting of positive vectors is called a Yudin G G basis whenever the vectors x" ( j e of > satisfy x3>>">5X> if and G G G only if j 50 for each i. G 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, Proposition 1.5, p. 9). Theorem A.2 (Yudin). A xnite dimensional vector subspace > of a partially ordered vector 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 +e ,( and + fl ,(l are two Yudin bases for a subspace > of G G a partially ordered vector space, then each e is a scalar multiple of some fl and G each fl is a scalar multiple of some e . G Moreover, if > is a lattice-subspace and +e ,2, e , is a Yudin basis for >, then ( for any two vectors x" ( a e and y" ( b e in >, we have G G G G G G ( ( xR y" (a Rb )e and x y" (a b )e . 7 G G G 7 G G G G G 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 xnite-dimensional space 11, the following statements are equivalent.
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1. > is an N-dimensional lattice-subspace of 11. 2. There is a fundamental set F"+s ,2, s , of N states. , 3. There exist a Yudin basis of non-negative vectors +e ,2, e , in > and a subset , of N states F"+s ,2, s , such that e (s )"d for all i, j"1,2, N. , G H GH A positive vector u in a vector lattice X is said to be an order unit, if for each x3X there exists some j'0 such that ju5x. Theorem A.5. A xnite-dimensional subspace >-C(D) containing the order unit 1 is 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). 䊐 There is another characterization of lattice-subspaces in terms of positive projections. Recall that a positive linear operator P: XPX on a Riesz space is said to be a positive projection if P"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 projection P: C(D)PC(D) having range >, i.e., P(C(D))">. Moreover, if this unique positive projection P exists, then for each pair x, y3> we have xR y"P(xRy) and x y"P(xy). 7 7 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. Brown, D.J., Ross, S.A., 1991. Spanning, valuation and options. Economic Theory 1, 3}12.
C.D. Aliprantis et al. / Journal of Economic Dynamics & Control 24 (2000) 1703}1719 1719 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 of C(X) and curves of 1L. Transactions of the American Mathematical Society 348, 2793}2810. Ross, S.A., 1976. Options and e$ciency. Quarterly Journal of Economics 90, 75}89. Yudin, A.I., 1939. A solution of two problems in the theory of partially ordered spaces. Doklady Akademy Nauk SSSR 23, 418}422.
Minimum-cost portfolio insurance夽 C.D. Aliprantis *, D.J. Brown, J. Werner Departments of Economics and Mathematics, School of Management, Purdue University, West Lafayette, IN 47907-1310, USA Department of Economics, Yale University, New Haven, CT 06520, USA Department of Economics, University of Minnesota, Minneapolis, MN 55455, USA Accepted 30 November 1999
Abstract Minimum-cost portfolio insurance is an investment strategy that enables an investor to avoid losses while still capturing gains of a payo! of a portfolio at minimum cost. If derivative markets are complete, then holding a put option in conjunction with the reference portfolio provides minimum-cost insurance at arbitrary arbitrage-free security prices. We derive a characterization of incomplete derivative markets in which the minimum-cost portfolio insurance is independent of arbitrage-free security prices. Our characterization relies on the theory of lattice-subspaces. We establish that a necessary and su$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 every reference portfolio. 2000 Elsevier Science B.V. All rights reserved.
夽 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 C941059. 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 Forschungsgemainschaft, SFB 303.
* Corresponding author. Tel.: 001-765-494-4404; fax: 001-765-494-9658. E-mail address: [email protected] (C.D. Aliprantis). 0165-1889/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 9 1 - 3
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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 the minimum-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 minimum-cost portfolio insurance to be price independent. A necessary and su$cient condition admits a large and interesting class of incomplete derivative market structures. Our primary characterization of market structures in which minimum-cost portfolio insurance is price independent relies on the mathematical theory of
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Riesz spaces (vector lattices) and lattice-subspaces. Riesz spaces have been used in the context of options markets by Brown and Ross (1991) and by Green and Jarrow (1987). The space of contingent claims on a "nite set of states of nature is a Riesz space with a supremum of any two contingent claims y and y de"ned as the statewise supremum and denoted by y Ry . A put option on a payo! y with strike price k can be represented using the operation of supremum as (k!y)R0, where k is the risk-free payo! of k. The insured claim on payo! y at #oor k is yRk. The subspace M of payo!s of all portfolios of securities (asset span) is a Riesz subspace if the supremum of arbitrary two payo!s in M belongs to M. If the asset span is a Riesz subspace, then every option is marketed. The result of Ross (1976) can be restated as saying that derivative markets are complete if and only if the asset span is a Riesz subspace of the space of contingent claims. The asset span M is a lattice-subspace if there exists a supremum relative to M of any two payo!s in M. The supremum relative to M of two payo!s y and y in M is their least upper bound in M, and is denoted by y R y . If M is + a lattice-subspace, then the payo! yR k exists and has the least cost among all + marketed payo!s that dominate the insured claim yRk, for arbitrary arbitragefree prices of securities. Our main result is that the minimum-cost portfolio insurance is price independent if and only if the asset span M is a latticesubspace. We present a number of su$cient conditions for the asset span to be a latticesubspace. The asset span of any two limited liability securities is a latticesubspace. 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$cient condition for M to be a lattice-subspace is the existence of a Yudin basis for M, i.e., a basis of limited liability payo!s such that every marketed limited liability payo! has a unique representation as a nonnegative linear combination of basis payo!s. The notion of Yudin basis is a generalization to incomplete markets of a basis of Arrow securities for complete markets (Arrow, 1953). We show that the minimum-cost portfolio insurance has a simple form whenever it is independent of security prices. The cost minimizing portfolio replicates the insured payo! in fundamental states. Since the securities markets restricted to the fundamental states are complete, the minimum-cost portfolio insurance can be reduced to portfolio insurance in complete markets. The paper is organized as follows: In Section 2, we de"ne the minimum-cost portfolio insurance. In Section 3, we introduce the lattice-subspace property of the asset span and show that it is necessary and su$cient for the minimum-cost portfolio insurance to be price independent in the set of arbitrage-free prices.
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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 are N 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 states S"+1,2, S, to describe the uncertainty. The payo! of security n in S states is a vector x 311 . The payo!s x ,2, x L > , 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 state s (state-payow vector). For a portfolio h"(h ,2, h )31,, its payo! is X(h)" , h x . The set of , L L L payo!s of all portfolios is the linear span of payo!s x ,2, x in the space 11 of , all state contingent claims and is the asset span M. A contingent claim is a marketed payow if it lies in the asset span. It is assumed that the risk-free payo! is marketed, so that 13M. If the asset span equals the whole space of contingent claims (i.e., if N"S), then markets are complete. Let p"(p ,2, p )31, be 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 vector p31, is arbitrage-free if there is no arbitrage portfolio, that is, if p ) h'0 for all non-zero portfolios h with X(h)50. The following simple duality result is well known. Lemma 2.1. If p ) h50 for every arbitrage-free price vector p, then X(h)50. Proof. Suppose by way of contradiction that there is a portfolio h with p ) h50 for every arbitrage-free p while X(h)(s)(0 for some state s. Then there exist strictly positive numbers j for each s such that 1 j X(h)(s)(0. Using the Q Q Q Bold 1 denotes the S-vector (1,2,1); similarly, k denotes (k,2, k) for any scalar k.
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de"nition of X(h) we have 1 j x(s)h(0. De"ne a price system q by Q Q q" 1 j x(s). Clearly, q is arbitrage-free. This is, however, a contradiction, Q Q since q ) h(0. 䊐 The insured payo! on a portfolio h at a `#oora k is the contingent claim X(h)Rk. This contingent claim may or may not be marketed. The minimumcost 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 insurance is de"ned by the following cost minimization problem: min p ) g EZ1, subject to X(g)5X(h)Rk. This linear programming problem has a unique solution as long as p is arbitrage-free. We denote the solution by hI and refer to it as the minimum-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 independent if 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 span M is a Riesz subspace if for every marketed payo!s y , y 3M their (statewise) supremum y Ry belongs to M. It is a lattice-subspace, if for every marketed payo!s y , y 3M there exists a least upper bound of y and y relative to M. That is, there exists z3M such that (1) z5y and z5y (i.e, z5y Ry ), and (2) z 5z for every z 3M with property (1). We denote that We use the symbol R to denote the statewise supremum of two contingent claims. That is (yRz)(s)"max+y(s), z(s),, for s"1,2, S.
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supremum by y R y . It is always true that y R y 5y Ry . Note that if + + M is a Riesz subspace, then it is also a lattice-subspace, and y Ry "y R y . + A derivative contingent claim is 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 portfolio hI satis"es X(hI)"X(h)Rk. 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)Rk satis"es X(h)Rk"X(h)#[(k!X(h))R0], hence it can be generated by holding the portfolio h and a put option on X(h) with strike price k. Thus, a necessary and su$cient condition for complete derivative markets is that all put and call options on marketed payo!s be marketed. Since each put and call option is a supremum of two marketed payo!s, all options are marketed if the asset span is a Riesz subspace. The following is essentially a restatement of the result due to Ross (1976). Theorem 3.1. Derivative markets are complete if and only if the asset span is a Riesz subspace of 11. Proof. The theorem of Ross (1976) implies that derivative markets are complete if and only if (k!y)R0 belongs to M for every y3M and every k. This latter property is necessary and su$cient for M to be a Riesz subspace. This follows from the simple lattice identity y Ry "y #[(y !y )R0]. 䊐 Completeness of derivative markets is a su$cient but not a necessary condition for the minimum-cost portfolio insurance to be price independent. Suppose that there exists the supremum of the payo!s X(h) and k relative to the asset span M, i.e., X(h)R k. A portfolio that generates the payo! X(h)R k is 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 latticesubspace of 11. In this case, the minimum-cost insured portfolio hI satisxes X(hI)"X(h)R k. + Proof. Assume "rst that the minimum-cost insured portfolio exists and is price independent. Consider an arbitrary portfolio h and a #oor k. Let hI denote the
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insured portfolio. For any portfolio g such that X(g)5X(h)Rk, we have that p ) g5p ) hI for every arbitrage-free price vector p. Lemma 2.1 implies that X(g)5X(hI). Hence, the payo! X(hI) is the supremum of X(h) and k in M, i.e., X(h)R k. + We have thus demonstrated that the supremum yR k exists for every + marketed payo! y and every k. The simple lattice identity y R y " + [(y !y #k)R k]#y !k proves that the supremum y R y exists for two + + arbitrary payo!s y and y in M. Hence, M is a lattice-subspace. Conversely, if the asset span M is a lattice-subspace, then the supremum X(h)R k exists for every portfolio h and every k. Now let the portfolio hI + be such that X(hI)"X(h)R k. Then, for every portfolio g satisfying + X(g)5X(h)Rk, we have that X(g)5X(hI). This inequality implies p ) g5p ) hI for every arbitrage-free price vector p. Consequently, the portfolio hI is the minimum-cost insured portfolio for every arbitrage-free price. 䊐 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 property. A necessary and su$cient condition for the asset span M to be a latticesubspace is the existence of a fundamental set of states. For any subset F-S of states, we shall denote by X the matrix of security payo!s restricted to $ the subset of states F. A subset F"+s ,2, s ,-S of N states is called , fundamental, if 1. the N;N matrix X is nonsingular, and $ 2. for each s , F there exist non-negative scalars aQ , aQ ,2, aQ such that , , x(s)" aQ x(s ). G G G 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 of 11 if and only if there is a fundamental set of states. It is well known that any two-dimensional subspace of 11 containing the unit vector 1 is a lattice-subspace (see Theorem 3.5, Aliprantis et al., 1998a). When N"2, the state-payo!s x(s) for s3S generate a closed cone in 1. If there are "nitely many states, the two extreme rays of this cone identify the two fundamental 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 x "(1, 1, 1) and x "(0, 1, 2). Their asset span is a two-dimensional subspace of 1 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, 0)#(1, 2). Thus, states 1 and 3 are fundamental. The insured payo! on portfolio h"(0, 1) of one share of security 2 at #oor 1 is the contingent claim x R1"(1, 1, 2) and is not marketed. A direct veri"cation shows that the marketed payo! (1, , 2) is the supremum x R 1. This payo! is + generated by the portfolio (1, ) 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 x "(1, 1, 1, 1), x "(1, 1, 2, 2) and x "(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 portfolio h"(0, 0, 1). The insured payo! on the portfolio h at the #oor k"1 is the contingent claim x R1"(1, 1, 1, 2) and is not in the asset span. Next, consider the two arbitrage-free price vectors: p"(1, , 1) and q"(1, , 1). The minimum-cost insured portfolio at prices p is the portfolio (0, 1, 0) with payo! (1, 1, 2, 2) and the cost of . At prices q, the minimum-cost insured portfolio is the portfolio (2,!1, 1) with payo! (1, 2, 1, 2) and the cost of . Note that the portfolio (2,!1, 1) is more expensive than (0, 1, 0) at prices p. The reverse is true at prices q.
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 states F and two contingent claims y , y 311, we write y 5 y if y dominates y in the fundamental states. That is, $ y 5 y if y (s)5y (s) for every state s3F. $ Similarly, we write y " y , if y and y are the same in the fundamental states. $ Theorem 4.1. If F"+s , s ,2, s , is a fundamental set of states and y , y are , payows in the asset span M, then y 5y if and only if y 5 y . $ Proof. The su$ciency part is obvious. To prove the necessity, let h and h be two portfolios such that X(h)"y and X(h)"y . Then X(h)5 X(h) $ means , x (s )(h!h)50 L G L L L for every fundamental state s 3F. Since for every s , F we can write G x (s)" , aQ x (s ) with aQ 50, we see that L G G L G G , X(h)(s)!X(h)(s)" x (s)(h!h) L L L L , , " aQ x (s )(h!h) 50 G L G L L G L for each state s. Thus, X(h)5X(h), or y 5y . 䊐
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 states F for the asset span M. Then for every arbitrage-free price system p and for every portfolio h and yoor k, the minimum-cost insured portfolio hI is the unique portfolio that replicates the insured payow X(h)Rk in the fundamental states. That is, X(hI)" X(h)Rk. $ The portfolio hI is the solution to the equation X hI" X(h)Rk, that is, $ $ hI"X\[X(h)Rk] . $ $ Proof. If there exists a fundamental set of states F, then, by Theorem 3.3, the asset span is a lattice-subspace. By Theorem 3.2, the minimum-cost insured portfolio hI satis"es X(hI)"X(h)R k. We have to show that + X(h)R k" X(h)Rk. + $
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Let z3M be the (unique) marketed payo! such that z" X(h)Rk. (Note that $ since the matrix X is non-singular such a marketed payo! z always exists and is $ unique.) We must show that z is the supremum of X(h) and k relative to the asset span M. First, let us verify that z is an upper bound of X(h) and k. Indeed, since z" X(h)Rk, it follows that z5 X(h) and z5 k, and so from Theorem 4.1 we $ $ $ infer that z5X(h) and z5k. To verify that z is the least upper bound of X(h) and k in M, let y3M satisfy y5X(h) and y5k. Then, y5X(h)Rk, and hence y5 X(h)Rk. Consequently, $ y5 z. Now Theorem 4.1 implies y5z. Therefore, z is the supremum $ X(h)R k. 䊐 + 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 coincides 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 and x "(0, 1, 2), the insured payo! on security 2 at #oor k"1 is the contingent claim x R1"(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, ) has payo! (1, , 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 essential 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 variable 1 representing 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 claim z3C(D) is a derivative claim if z(s)"f (x(s)) for every s3D, for some continuous function f : 1,P1. Ross's characterization of complete derivative markets (Theorem 3.1) has been extended 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 fundamental 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. 䊐 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, and a risky stock with payo! x (s)"s for each s3[0, 1]. The asset span is M"+z3C[0, 1]: z(s)"h #h s for each s3[0, 1] and some (h , h )31, and consists of all linear functions. It is a two-dimensional subspace of C[0,1], and hence a lattice-subspace. States 0 and 1 form a fundamental set of states for the asset span M. 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 #oor k is z"x Rk. Thus, k if 04s4k, z(s)" s if k(s41.
Clearly, for 0(k(1 the contingent claim z is not in the asset span.
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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 hI"(k, 1!k). Its payo! is X(hI)(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 D is taken to be the normalized cone of arbitrage-free price vectors in the next period market (see Henrotte, 1996). Securities become price-contingent contracts, a notion due to Kurz (1974). For a portfolio h31,, its payo! is X(h)(q)"q ) h for q3D-1,. The asset span is M"+z3C(D): z(q)"q ) h for each q3D, for some h31,,. Assuming that the risk-free payo! 1 is marketed, Theorem 7.5 implies that the asset span M is a lattice-subspace if and only if there exist N price vectors q,2, q, in D such that every price vector in D is a linear positive combination of q,2, q,. Expressing it in another way, M is a lattice-subspace if and only if the cone of arbitrage-free price vectors D is generated by some N price vectors. 6. Portfolio insurance with Yudin basis A Yudin basis of the asset span M is a set of N positive payo!s e ,2, e 3M , that span M with the extra property that a payo! y" , j e 3M is positive L L L if and only if j 50 for each n (for details, see appendix). By Theorem A.2, the L asset span M is a lattice-subspace of the contingent claim space X if and only if there exists a Yudin basis for M. 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 and e given by e (s)"s and e (s)"1!s, for each s3[0, 1]. If +e ,2, e , is a Yudin basis of M, then for any two payo!s y " , je , L L L and y " , je in M, we have L L L y 5y if and only if j5j for each n. L L The following result explains the use of Yudin basis for portfolio insurance. Theorem 6.1. Suppose that there exists a Yudin basis e ,2, e for the asset span , M. For an arbitrary portfolio h and an arbitrary yoor k, let X(h)" , j e and L L L let k" , c e . Then the minimum-cost insured portfolio hI satisxes L L L , X(hI)" max+j , c ,e . L L L L
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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). 䊐 The portfolios g,2, g, that generate the payo!s e ,2, e of 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 portfolio hI of Theorem 6.1 is given by , hI" max+j , c ,gL. L L L A characterization of a Yudin basis of an asset span when the set of states S is "nite can be derived from the Abramovich}Aliprantis}Polyrakis Theorem A.4. If the asset span M is a lattice-subspace of 11, then there exists a set F"+s ,2, s , of fundamental states and the Yudin basis for M consists of , payo!s e for n"1,2, N given by L 1 if s"s , L e (s)" 0 if sOs , S3F, L L aQ if s , F, L where aQ is the nth (non-negative) coe$cient in the expansion x(s)" , aQx(s ) L G G G in the de"nition of the fundamental states. The Yudin basis of an asset span in the "nite-dimensional contingent claim space 11 resembles the Arrow securities for the fundamental states. The payo! e equals one in the fundamental state s , zero in all other fundamental states, L L and is non-negative in non-fundamental states.
Example 6.2. In Example 3.1 of two securities with payo!s x "1 and x "(0, 1, 2), a Yudin basis consists of payo!s e "(1, , 0) and e "(0, , 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 as x "2e and the risk-free payo! as 1"e #e ; the payo! of the minimum-cost insured portfolio is max+1, 0,e #max+1, 2,e "e #2e "(1, , 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 relation 5 on a vector space X is said to be a partial order if, in addition to being re#exive, antisymmetric and transitive, it is also compatible with the algebraic structure of X in the sense that x5y implies: (a) x#z5y#z for each z and (b) ax5ay for all a50. A vector space equipped with a vector space order is called a partially ordered vector space or simply an ordered vector space. The set X>"+x3X: x50, is referred to as the positive cone of X. An arbitrary cone C of a vector space X de"nes a partial order on X by letting x5y if 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 space X is said to be a vector lattice or a Riesz space if it is also a lattice. That is, a partially ordered vector space X is a vector lattice if for every pair of vectors x, y3X their supremum (least upper bound) and inxmum (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 vectors x, y in a vector lattice are denoted by xRy and xy, respectively. In a vector lattice, the elements x>"xR0, x\"(!x)R0 and "x""xR(!x) are called the positive part, the negative part, and the absolute value of x. We always have the identities x"x>!x\ and "x""x>#x\. Theorem A.1. A partially ordered space X is a Riesz space if and only if there exists a vector a3X such that xRa exists for every x3X. Proof. See Luxemburg and Zaanen (1971, Theorem 1.5, p. 55).
䊐
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 +x, y, exists in > when ordered by the cone >>">5X>. If > is a lattice-subspace of X, then we shall denote the supremum and in"mum of the set +x, y,-> in > by xR y and x y, respectively. That is, 7 7 xR y"min+z3>: z5x and z5y, 7 and x y"max+z3>: z4x and z4y,. 7
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A vector subspace > of a vector lattice X is said to be a vector sublattice or a Riesz subspace if for each x, y3> we have xRy and xy 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 a partially ordered normed space. A norm "" ) "" on a vector lattice is said to be a lattice norm if "x"4"y" implies ""x""4""y"". A normed vector lattice is a vector lattice equipped with a lattice norm. A complete normed vector lattice is called a Banach lattice. A classical example of a Banach lattice is C(D), the vector space of all continuous real-functions on a compact topological space D, equipped with the pointwise order and the sup norm. Dexnition A.2. If > is a "nite-dimensional vector subspace of a partially ordered space X, then a basis +e ,( of > consisting of positive vectors is called a Yudin G G basis whenever the vectors x" ( j e of > satisfy x3>>">5X> if and G G G only if j 50 for each i. G 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, Proposition 1.5, p. 9). Theorem A.2 (Yudin). A xnite dimensional vector subspace > of a partially ordered vector 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 +e ,( and + fl ,(l are two Yudin bases for a subspace > of G G a partially ordered vector space, then each e is a scalar multiple of some fl and G each fl is a scalar multiple of some e . G Moreover, if > is a lattice-subspace and +e ,2, e , is a Yudin basis for >, then ( for any two vectors x" ( a e and y" ( b e in >, we have G G G G G G ( ( xR y" (a Rb )e and x y" (a b )e . 7 G G G 7 G G G G G 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 xnite-dimensional space 11, the following statements are equivalent.
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1. > is an N-dimensional lattice-subspace of 11. 2. There is a fundamental set F"+s ,2, s , of N states. , 3. There exist a Yudin basis of non-negative vectors +e ,2, e , in > and a subset , of N states F"+s ,2, s , such that e (s )"d for all i, j"1,2, N. , G H GH A positive vector u in a vector lattice X is said to be an order unit, if for each x3X there exists some j'0 such that ju5x. Theorem A.5. A xnite-dimensional subspace >-C(D) containing the order unit 1 is 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). 䊐 There is another characterization of lattice-subspaces in terms of positive projections. Recall that a positive linear operator P: XPX on a Riesz space is said to be a positive projection if P"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 projection P: C(D)PC(D) having range >, i.e., P(C(D))">. Moreover, if this unique positive projection P exists, then for each pair x, y3> we have xR y"P(xRy) and x y"P(xy). 7 7 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.
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