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The valuation problem in arbitrage price theory
Journal
of Mathematical
Economics
The valuation theory Stephen
problem
North-Holland
in arbitrage
price
A. Clark*
University of Kentucky, Submitted
22 (1993) 463478.
November
Lexington
KY, USA
1990, accepted
September
1992
Suppose a continuous, strictly positive, linear price functional p is given on a subspace M of marketed claims. The valuation problem consists of verifying whether or not there exists a continuous, strictly positive, linear extension of p from M to the entire contingent claims space X. We solve this problem when X belongs to a large class of Banach lattices including the classical Banach spaces, and also simplify some analogous results found in the literature for other types of financial models.
1. Introduction
This essay aims toward resolving a valuation problem in the foundations of arbitrage price theory. An arbitrage opportunity is essentially a feasible contingent claim with positive net return across all states of nature. In other words, an arbitrage is a ‘free lunch’. Financial economists have long recognized that a competitive market equilibrium eliminates all arbitrage opportunities and have exploited this condition to evaluate contingent claims [e.g. see Modigliani and Miller (1958), Merton (1973) or Black and Scholes (1973)]. In contrast, we follow an axiomatic approach that abstracts away from general equilibrium theory. In the next section, we present some basic conditions which describe the absence of arbitrage opportunities. Suppose the contingent claims space X is a Hausdorff, locally convex, topological vector lattice and the marketed claims space M is a subspace of X. Ross (1978) has suggested a key axiom A.1 of no arbitrage opportunities, formally demanding that there does not exist a feasible claim in the strictly positive cone. Under a mild auxiliary condition typically satisfied by a riskless marketed claim, it is straightforward to verify that A.1 characterizes a strictly positive, linear price functional p: M-R. This result is the most fundamental and elementary type of Correspondence to: Stephen A. Clark, Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA. *The author expresses gratitude to an anonymous referee who detected a logical error in the original version of this paper. 03044068/93/‘SO6.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
464
S.A. Clark, Valuation problem in arbitrage price theory
arbitrage argument. Then we present another axiom A.2, which is a topological version of A.l. Its purpose is to assure that the price functional p is also continuous. In the third section, we discuss the historical connection between arbitrage price theory and subjective probability theory [Ramsey (1926) or De Finetti (1937)]. We only need to slightly weaken our axioms, A.1 and A.2, to obtain coherency axioms, A.3 and A.4, for constructing probability measures in terms of a positive linear functional p: M +R which serves as an expectations operator. This exercise illustrates how subjective probability theory has anticipated the modern development of arbitrage price theory. In the fourth section, we introduce the valuation problem. Following Ross (1978) a valuation operator P : X -+[w is a continuous, strictly positive, linear extension of p to the entire contingent claims space X. A fundamental problem of arbitrage price theory is to characterize the existence of a valuation operator, providing a computational method for evaluating contingent claims. Recent developments in general equilibrium theory suggest that it would be useful to solve this problem for an arbitrary topological vector lattice [e.g., see Mas-Cole11 (1986) or Zame (1987)], but Hilbert lattices are especially common in financial modelling [e.g., see Harrison and Kreps (1979), Chamberlain and Rothschild (1983) or Hansen and Richard (1987)]. Here we suggest a new axiom AS, which is a weak version of Krep’s (1981) assumption A.6 of ‘no free lunches’. The latter condition has been utilized by Kreps (1981) and by Duffie and Huang (1986) to solve the valuation problem for a class of models that evoke separability assumptions. Nevertheless, we show that A.5 and A.6 are equivalent assumptions for practically all the circumstances where these authors have characterized the existence of a valuation operator. In the fifth and final section, we completely resolve the valuation problem for the classical Banach spaces Lq(S, 5,~) when 154 < a. These methods do not require separability assumptions. Instead we merely require that p is a CJfinite countably additive measure on (St), an assumption that is vital for the implementation of the Radon-Nikodym theorem. Here we show that A.5 completely characterizes the existence of a valuation operator. This result has applications to the theory of continuous trading [e.g. see Karatzas (1989)]. 2. Arbitrage equilibrium The formal model consists of a Hausdorff, locally convex, topological vector lattice X, which is interpreted as the collection of all contingent claims, and a subspace M, which is interpreted as the collection of marketed claims. By definition, a topological vector lattice X is a topological vector continuous lattice operation with a (uniformly) space equipped v :X xX+X. This lattice operation v creates a natural vector order 2 on
S.A. Clark, Valuation problem in arbitrage price theory
465
X defined by the condition x1 2x2 if and only if (x1 -x2) v 0=(x, -x2). The positive cone is given by X+ = {XE X: x20). The vector order 2 is reflexive, transitive, and antisymmetric. Let > denote its asymmetric component, so that x1 )x2 if and only if x1 2x, and x1 #x2. The strictly positive cone is Thus, X+=X++ u (0). A key property of a given by X + + ={xEX:X>O}. topological vector lattice is the closedness of X+. Indeed, many of our results generalize to an ordered topological vector space with closed positive cone. The following notation is useful. Suppose A,B are subsets of X. Let -A={x~x:-XEA}, A+B={x~X:x=a+b for some UEA,~EB}, and A -B = A +(-B). We also let A denote the topological closure of A. Presumably, there exist markets for only some of the contingent claims in and the collection of marketed claims M is the linear span of X3 say {miliol, {mi}iE,. Thus, a marketed claim me M is just a portfolio of long and short positions in the available markets. We employ the notation m =xie, Mimi for a marketed claim with the understanding that the number of shares %i of the claim mi is non-zero in only a finite number of markets. We do not necessarily assume, however, that the numbers of markets is finite. In summary, we have the defining formula
M=
x~X:x=~~imiforsome{~i}i~, icl
Of course, the requirement that M is a subspace assumes frictionless trading, e.g. unlimited short selling and no transactions costs. Suppose the market assigns a price pi to the marketed claim m, for every in I. We do not need to assume that prices are a priori positive, but rather we intend to derive this property as a consequence of the absence of arbitrage opportunities. A price correspondence p: M+R is naturally defined by the formula
p(m) =
C &pi : m = C limi for some (2i}i,, i
iel
isl
I
for every rnE M. The economic interpretation of this price correspondence is easily explained. Indeed, p(m) represents all possible prices for which the marketed claim m E M can be purchased by constructing it as a portfolio of contingent claims offered directly in the markets. We say that a marketed claim me M is feasible provided that there exists some 4 up such that 420. Let F denote the collection of all feasible claims. The principle of no arbitrage, as originally formulated by Ross (1976, 1978), asserts that there does not exist a feasible claim in the strictly positive cone X++. For any XE M, let p(x) >O denote the condition 4 >O for every q E p(x). In formal terms, we now have the following axiom:
S.A. Clark, Valuation problem in arbitrage price theory
466
A.I.
For every XEM, x>O implies
p(x)>O.
If we momentarily interpret > as a strict preference relation, then A.1 represents Harrison and Krep’s (1979) condition of ‘no simple free lunches’. More generally, their condition implies A.l, because they assume preferences are strictly montone with respect to 2. It is straightforward to determine the implications of A.1 for the price correspondence p : M -+R. Suppose there exists a marketed claim m, such that m, >O, e.g. a marketed riskless security. Then the absence of arbitrage opportunities implies p is a linear functional. In general, suppose Y is a subspace of X and p : Y + R is a linear functional on Y. Let Y + = Y n X+ and Y++=YnX++. We say that p is positive whenever p(y) 20 for every YE y+, and we say that p is strictly positive whenever p(y) >0 for every 4’E y++. We record the following result in order to provide the flavor of an elementary arbitrage argument. Theorem 1. Suppose Mi + #I$. strictly positive linear jiinctionul.
Then
A.1 holds
if and only
if p: M-+R
is a
Proqf. If p is a strictly positive linear functional, then A.1 holds trivially. Conversely, assume A.1 holds. Let m, E M+ +.Then p0 >O for any p0 up. We first show that p(m) is uniquely determined for every rn~ M. On the contrary, assume p(m) is not uniquely determined for some me M. Then there We may write m=xiE,2.imi with q= exist q, q* Ep(m) such that q >q*. *m with q* =xiEII$pi. Define a contingent claim :z;$; and m=&,,A i
.x = [I(4 - 4*)lPolmo + rT, (AT- &)m,. x E M and x = [(q - q*)/p,,]m, > 0, because (4 - q*)/pO >O. Thus, Furthermore, O=C(q-_*)lP,lP,+Ci,r(;lt-~i)PiEP(x) implies x E F. This contradicts A.l, showing that p(m) is uniquely determined for every me M. Next, we show that p: M +R is linear. On the contrary, assume m = i,m, +A,m, and p(m)#i.,p(m,)+3L,p(m,). Then p(m) is not uniquely determined, contradicting the conclusion reached in the previous paragraph. Thus, p is a linear functional. Finally, it is now obvious that A.1 implies p is strictly positive. 0 Then
XEM++.
We say that p: M+R is an arbitrage equilibrium whenever p is a stricly positive, linear functional. This terminology is motivated by the well-known connection between a market equilibrium and the absence of arbitrage
467
S.A. Clark, Valuation problem in arbitrage price theory
opportunities [e.g. see Harrison and Kreps (1979)]. For example, suppose an agent in the economy has strictly monotone preferences. Then A.1 is a necessary condition for the existence of a compensating equilibrium, because otherwise the markets cannot clear. We next study the continuity of the price system in the relative topology. Accordingly, we define another price correspondence p: A?--+!& so that the graph of p is the closure of the graph of p. A local definition of j is somewhat complicated by the fact that X does not necessarily have a countable local base, forcing us to use topological nets instead of topological sequences. In general, let {x, : r~ r} denote a net in X with directed index set r. We write x,+x to denote the convergence of the net {x,: YE r} to the point x E X, Then, p: M+[W is locally defined by the condition qER:3{m,:yEr}
P(m) =
in M such that mY-+m, and 3 { py : y E r}
in [wsuch that py Ep(m,) Vy or and pY+q
for every rnE i@. For any rnE M, let p(m) ~0 denote the condition q>O every qEp(m). We introduce the following axiom in order to eliminate possibility of approximate arbitrage opportunities. A.2.
(3) for the
For every x E Ji?f, x >O implies p(x) > 0.
The logical form of A.2 is analogous economic interpretation. Its significance
to A.l, lending itself to a similar is explained by the following result.
Theorem 2. Suppose M+ + #$. Then A.2 holds continuous, strictly positive linear functional.
if and only
if p:?a+iw
is a
Proof. It is straightforward to verify that if p is a continuous, strictly positive linear functional; then A.2 holds. Conversely, assume A.2 holds. Then Fn h?l++ =c#J,because for every rnEF we have q50 for some qEp(m). But F-M+GF implies -SF. Thus, Let mOE M++. Then m,#F. m,$F-M+. So there exists an open, convex neighborhood of m,, say N(m,), such that N(m,) n (F - M ‘) = 4. In particular, N(m,) n (F - M ‘) = 4. Let C(m,) denote the convex cone generated by N(m,), i.e. C(m,)= {y~X:y=Lx for some A>0 and xEN(m,)}. Then C(mo)n(F--M+)=4. The geometric Hahn-Banach theorem now implies there exists a continuous linear functional P : X--f [w such that P(x) 2 0 for every x E C(m,) and P(x) 5 0 for every x E F - M + . In particular, P(x) 5 0 for every x E F. Notice that A.2 implies A. 1. So we obtain that p : M-R is a strictly positive linear functional from Theorem 1. Furthermore, for every rnE M, p(m) 5 0 implies P(m) 50. Replacing m with -m in this implication, we see that for every rnc M, p(m) 20 implies P(m) 2 0. Thus, -p(m) = 0 implies P(m) =0 for every m E M. Consequently, there exists some 3,~ [w such that
J.Math
C
468
S.A. Clark, Valuation problem in arbitrage price theory
p(m)=U’(m) for every me M [see Kelly and Namioka (1963)]. An evaluation at rn,~M++ shows that 2 #O, because A.1 implies p(m,) > 0. Therefore, P is continuous on M implies p is continuous. It immediately follows that p: A+R is a continuous, positive linear functional [see Kelly and Namioka (1963)]. Finally, we must show that p is strictly positive. On the contrary, assume p is not strictly positive. Then there exists some XEM+ + such that @(x)=0. Let {x,: YE T} be a net in M such that x, --tx. Then p(x,)-+p(x)=O. Now if {x, :YE f} has a feasible subnet, then we immediately obtain a violation of A.2. So p(x,)>O for all y sufficiently large in the directed index set r. As before, let m, E M ‘+. Then A.1 implies p(m,) ~0. Define &,=p(x,)/p(m,,) for every y E r. Then E,,>O for all y sufficiently large, and ,4,-O. Define XT= x,-i.,m, for every y E r. Then X~*EM. Furthermore, p(x,*) =p(x,) &p(m,) = 0 implies XTE F. But x:--+x implies x E F. Thus, x E F n M + +, which is another violation of A.2. This contradiction shows that p is strictly positive. 0 We say that p: i%+rW is a continuous arbitrage equilibrium whenever @ is a continuous, strictly positive linear functional. Since general equilibrium theory usually relies upon a fixed point theorem or the geometric HahnBanach theorem to guarantee the existence of a continuous price functional compatible with a market equilibrium, the requirement of a continuous arbitrage equilibrium is not an extraordinary theoretical demand [e.g. see Harrison and Kreps (1979)].
3. Subjective probability theory Much of the recent work in arbitrage price theory was anticipated by Ramsey (1926) and De Finetti (1937) in their classic studies of subjective probability theory, where the probability of an event is interpreted as an individual’s degree of belief in its occurrence. We wish to especially focus upon De Finetti’s formulation of the principle of coherence, because his approach is utility free and closely related to the Ross principle of no arbitrage - A.l. Indeed, the latter principle is practically the same as the concept of strict coherence [Shimony (1955) and Kemeny (1955)], which was later rejected by De Finetti (1972) as a proper foundation for probability theory. The connection between arbitrage prices and coherent beliefs is clearly seen when we regard the contingent claims space X as a vector lattice of real-valued functions upon a state space S. Suppose E is an event in S, i.e. E is a subset of S. Let I, denote the indicator function on E, so that I,(s) = 1 for every s E E and I,(s) = 0 for every s $ E. Of course, I, can be interpreted as a contingent claim which pays off $1 if event E realizes and SO otherwise. According to De Finetti (1937), the subjective probability of event E, say
S.A. Clark,
Valuation
problem
in arbitrage
price theory
469
rc(E), is just the price an individual would be willing to pay for the contingent claim I,. Hence, Z(E) =p(l,). Therefore, we may view subjective probability theory as the following special case of our general model of financial transactions. Suppose we have an algebra r of events in the state space S. Then we construct the space of marketed claims M as the span of {I,: E E 0. Thus, M is the collection of (measurable) simple functions on S, and M is a sublattice of X. Of course, probability theory merely requires a positive linear functional instead of a strictly positive one, allowing the possibility that an event EE 5 might be assigned probability zero. This requires a suitable modification of our no arbitrage axioms. For any XE M, let p(x) 20 denote the condition that 4 20 for every q Ep(x). We introduce the following axiom in order to eliminate the relevant arbitrage opportunities. A.3.
For every x E M, x 2 0 implies p(x) 2 0.
This axiom has been discussed by Varian (1987) in the context of a finite state-space model. Its significance lies in the following result, which does not directly evoke the state-space approach. Theorem p0 up.
3.
Suppose there exists some mOE M+ + such that p,, >O for some Then A.3 holds if and only if p : M+R is a positive linear
functional. Proof.
The demonstration of this theorem is a straightforward modification of the proof to Theorem 1, and we only provide a summary sketch. If p is a positive linear functional, then it is trivial that A.3 holds. Conversely, assume A.3 holds. We first show p(m) is uniquely determined for every rnE M. Contrary to hypothesis, assume p(m) is not uniquely determined for some rnE M. Then we may purchase m at a low price, sell m at a high price, and use a positive fraction of the proceeds to purchase shares of m,. We have now constructed a portfolio that violates A.3. This contradiction shows that p(m) is uniquely determined for every me M. It also follows that p is a linear functional, because otherwise p is not uniquely determined. The positivity of p now follows trivially from A.3. 0 We remark
that the auxiliary
condition
of Theorem 3, i.e. there exists some is almost always satisfied by For example, I, is interpreted as a riskless asset in the typical securities market model. On the other hand, the subjective probability model usually requires SE< with n(S) =p(l,)= 1 as a normalization constant. The above theorem is a modest generalization of De Finetti’s well-known m,,EM+’ such that p0 > 0 for some p0 up, m, = Is as an axiom for the state-space model.
470
S.A. Clark,
Valualion
problem
in arbitrage
price theory
characterization of subjective probability [e.g. see De Finetti (1972)]. Indeed, A.3 is essentially equivalent to his coherency axiom, which asserts that p(x) >O whenever x is both marketable and uniformly positive. Thus, A.3 is the defining characteristic of a coherent probability measure rr: 5-R for (S,<). We remark that subjective probability theorists traditionally take finite additivity as the key property of a probability measure with countable additivity as an important special case [e.g. see Kyburg and Smokler (1964)]. This viewpoint is highlighted by the fact that the proof to Theorem 3 works in an arbitrary vector lattice. For any x E M, let p(x) 20 denote the condition qz0 for every REP. We next introduce the following axiom as a topological version of A.3. A.4.
For every x E M, x 2 0 implies
The following result requires some delicate
is analogous modifications.
p(x) 2 0. to Theorem
Theorem 4. Suppose there exists some rnOE M+ p0 cp(m,). Then A.4 holds and F # X if and only positive linear functionul.
2, but
its demonstration
’ such that p0 >0 fbr some if p: h?i--+R is u continuous,
Proc$ Assume p: M-+R is a continuous, positive linear functional. Then A.4 holds trivially. Furthermore, x E F implies p(x) 5 0. Thus, p(m,) =pO > 0 implies m, $ F. This verifies that F # X. Conversely, assume A.4 holds and F#X. Notice that A.4 implies A.3. Hence, p: M +R is a positive linear functional, by virtue of Theorem 3. Define F,={x~M:not @(x)20}. Since F={x~M:p(x)~0}, it is clear that F, G F. Thus, F, GF. Now A.4 directly implies F,-- Ji?f+ c F,, so that (F, - M+)G F,cF. By hypothesis, there exists some xo$F. So there also exists some open, convex neighborhood of x0, say N(x,), such that N(x,) n (F,M+) = 4. In particular, N(x,) n (F,M+) = 4. Let C(x,) denote the convex cone generated by N(x,), i.e. C(x,) = {y E X : y = Ax for some 3,> 0 and XE C(x,)}. Since F-&i+ IS a convex cone, it follows that C(x,) n (F-M’) = 4. Define C = C(x,) + M+. Then C is an open, convex cone such that CnF-q5. Let F,={x~M:p(x)<0). Notice that F,#+, because -m,EFl. We claim that F, c F,. Indeed, for any XE M, x $ F, implies p(x) 20. Since p(x) E&X), we obtain x $ F,, which verities the claim. Next, observe that C n F, =c$ implies C n F, = 4. An application of the geometric Hahn-Banach theorem yields the existence of a continuous, nonzero linear functional P : X--+ R such that P(x) 5 0 for every x E F, and P(x) 2 0 for every x E C. Assume p(m) = 0 for some m E F. Then for any 6 > 0, we obtain p(m-_m,)=p(m)-_p(m,)= -6p(m,)
S.A. Clark, Valuation problem in arbitrage price theory
471
Thus P(m- 6m,) 50 for every 6 > 0. Letting d-+0, it follows that P(m) 50 by virtue of the continuity of P. In summary, P has the property that for every x E M, p(x) 50 implies P(x) SO. The argument in the proof of Theorem 2 now applies; showing that P, p are proportional linear functionals on M. Thus, P is continuous on M implies p is continuous. Since p is a continuous, positive linear functional; p is also a continuous, positive linear functional [see Kelly 0 and Namioka (1963)]. Corollary. A.4 holds
Suppose
there
exists
if and only if fp: ii&E4
some m, E M ’ + such that p(m,) >O. Then is a continuous, positive linear functional.
Proof. Notice that p(mO) > 0 implies implies p0 > 0 for any F(m,) >O immediately. 0
m, $ F.
Thus, F#X. In p0 Ep(m,). The result
addition, follows
We remark that the auxiliary condition in this corollary, i.e. there exists some m,E M++ such that p(m,) > 0, is just a convenient combination of the auxiliary condition in Theorem 4 and the condition F#X. In view of A.4, we can just as well replace it with the condition that there exists some m, E M+ + such that O$p(m,,). If the markets are incomplete in the sense that G #X, then the extra condition F#X is automatically satisfied. Some additional insight is provided when we study our auxiliary conditions in the context of a state-space model. Then mO= I, is the natural candidate to fulfill both roles, so that we might assume p(Z,) >O or just 0 $ p(1,). In fact, assume some m, E M satisfies p(mO) > 0. If m, is (essentially) bounded, i.e. there exists 1 >O such that /m,(s)/ 51 for (almost) every s E S; then m, is dominated by a positive scalar multiple of I,, i.e. ,lIs2m,. Thus, A.4 implies @(Al,) 2 p(m,), so that p(1,) >O. The following state-space model is useful in the foundations of probability theory. Suppose 5 is a a-algebra of events in S and p is a Lebesgue measure on (S, 5). Let X =L”(S, 4,~) denote the Banach lattice of all essentially bounded, measurable, real-valued functions on S equipped with the essential supremum norm. Then the topological dual of X is isomorphic to the lattice of all tinitely additive, signed measures on (S, {) [Yosida and Hewitt (1952)]. Notice that Theorem 4 readily applies to this model. On the other hand, suppose we endow X=L”(S, 5,~) with the topology of pointwise convergence. Since X is the dual of L’(S, 5, cl), it is straightforward to verify that this topology is the same as the weak-star topology. Let (Ej}jm_ 1 denote a sequence of events in 5 that partitions S. Let D,= us= 1 Ej whenever 15 k< co, so that the sequence {I&T= 1 converges (monotonely) to I, is the weak-star topology. We now examine the Corollary to Theorem 4 under the presumption that m,=I,. Notice that O$p(Z,) implies that it is impossible to assign subjective probabilities to the events in t such that
472
S.A. Clark, Valuation problem in arbitrage price theory
rc(Ej) = 0 whenever 15 j < co and rr(S) = 1. This property was discussed by De Finetti (1972) to distinguish between finitely additive and countably additive probability measures. More importantly, notice that the continuity of p: M-+LQ implies that n is countably additive, when we work in the framework of the weak-star topology. In general, continuity does not pose a special problem when X is a Banach lattice for the following reason. Suppose p: M +[w is a (strictly) positive linear functional. Then p has a positive linear extension to X provided that M contains a radial point of X+ [see Kelly and Namioka (1963)]. Yet a positive linear functional P on a Banach lattice X is necessarily continuous [see Schaeffer (1974)]. A fortiori, p is continuous on M, and fi is continuous on ii%. Therefore, we may deduce A.4 from A.3 whenever X is a Banach lattice and the appropriate auxiliary condition holds, e.g. I, E M and 1 E ~(1,).
4. The valuation operator Suppose p : Ii?l+R is a continuous arbitrage equilbrium. In forma1 terms, a valuation operator is a continuous, strictly positive, linear extension of p to the entire contingent claims space X. In other words, a valuation operator is a continuous, strictly positive, linear functional P:X-+R such that the restriction of P to the subspace M is identical to p. This concept was introduced by Ross (1976, 1978) in order to provide a computational approach to arbitrage price theory. Furthermore, it plays a key role in the construction of an equivalent martingale measure for a dynamic model of the securities markets [e.g. see Harrison and Kreps (1979) or Duffie and Huang (1986)]. Thus, the provision of conditions for the existence of a valuation operator is a vital problem in financial economics. It is straightforward to verify that the following condition is necessary for the existence of a valuation operator.
{x,:y~r} in
A.5. If there exists a net and there exists a net (m,: y E r} not x>O.
X such that x7-+x for some XEX, in F such that my 2 x, for every y E r; then
In more concise terms, this axiom asserts that X+ + n (F-X’) = 4. We regard A.5 as a weak version of Krep’s (1981) condition of ‘no free lunches’, which is expressed by the following assumption. A.6. If there exists a net {x,: YES} in X such that xy+x for some XEX, and there exists a net {m,:y~r} in A such that liminf{p(m,): YES} 50 and my 2 x, for every y E r; then not x > 0. Kreps
demonstrates
that A.6 is both
a necessary
and sufficient
condition
for
S.A. Clark, Valuation problem in arbitrage price theory
413
the existence of a valuation operator whenever the contingent claims space X has an appropriate structure, e.g. X is normable and its dual space has weakstar separable subsets. Duffie and Huang (1986) derive the same type of result whenever X is a separable Banach space. Nevertheless, we claim that A.5 and A.6 are equivalent conditions under all the circumstances for which these authors have characterized the existence of a valuation operator. It is more convenient here to focus upon general methods, rather than attempting a tedious replication of their arguments, to show that A.6 can be replaced with A.5. Hence, we offer the following strict separation theorem. Theorem 5. Suppose J, K are non-empty convex cones (with vertices at the origin) in a separable Banach space X. Then there exists a continuous linear functional P : X-+ R such that P(x) > 0 for every x E J and P(x) 5 0 for every XEK ifand only ifJn(K-J)=$. to verify that if there exists a continuous linear Proof. It is straightforward functional P: X+R such that P(x) > 0 for every x E J and P(x) 5 0 for every XEK, then J n (K- J)=#. Conversely, assume J n (K - J)=4. Then for every x E J, there exists an open, convex neighborhood of x, say N(x), such that N(x) n (K-J) = 4. Let C(x) be the convex cone generated by J + N(x), i.e. C(x) = {z~X:z=Ay for some 2~0 and some YE J+N(x)}. It is straightforward to verify that C(x) is an open, convex cone such that C(x) n K =c$. So an application of the geometric Hahn-Banach theorem yields the existence of a continuous, non-zero linear functional P,: X-R such that P,(y) 2 0 for every YE C(x) and P,(y) 5 0 for every y E K. Now P,(y) 20 for every y E C(x), which implies P,(y) 2 0 for every y E J. Furthermore, P,(x) > 0, because C(x) is open and x E C(x). In summary, for every x E J, there exists a continuous linear functional P, : X-t R such that (i) P,(y) 2 0 for every y E J, (ii) P,(y) 50 for every YE K, and (iii) P,(x)>O. Notice that IIPxl[>O. Without loss of generality, we may also assume that (iv) IIPxll= 1 for every XE J, by renormalization if necessary. Let X* denote the (topological) dual of X, and let B* denote the unit ball in X*, i.e. B*={PEX*:IIPII~~}. S’mce X is a separable Banach space, B* is metrizable with respect to the relative weak-star topology [e.g. see Dunford and Schwartz (1971)]. It follows that B* satisfies the second axiom of countability in the relative weak-star topology. Let Y= {Px: x E J}. Since 9~ B*, 9 also satisfies the second axiom of countability, and hence Y is weak-star separable. Let {Pxi>,“_, denote a countable, weak-star dense subset of 9. We now define a linear functional P:X+R by the formula P(x)= XI’= 1 2_‘Px,(x) for every XEX. Then P is a well-defined continuous linear functional because {x1= I 2 -‘P,,},“= 1 is a Cauchy sequence in the Banach space X*. Furthermore, it is clear that P(x) 50 for every XE K. Since {Px,}~l
414
S.A. Clark,
Valuation
problem
in arbitrage
price theory
is weak-star dense in 9, it follows that for every XEJ, there exists positive integer i such that II’,(x)--P,,(x)/0 P(x)>0 for every xEJ. 0
some and
The significance of the above theorem to arbitrage price theory can be explained as follows. If we select J =X+ + and K = F, then the condition J n (K -J) =4 is equivalent to axiom A.5. Furthermore, the condition P(x) 5 0 for every x E F implies P and p are proportional linear functionals on J@ (recall the argument in the proof to Theorem 2). So we may assume P extends p without loss of generality. Therefore, A.5 characterizes the existence of a valuation operator whenever X is a separable Banach space. The analogous result by Duffte and Huang (1986) is an immediate corollary. They analyze a dynamic securities market model whose contingent claims space X is a L’ probability space. Suppose a tinite number of securities can be traded at only a finite number of dates. If we identify a state of nature as a realized stream of net return vectors, then the state space S is finite dimensional and, hence, separable. Let 5 denote a a-algrebra on S, and let rr denote a countably additive probability measure on (S, 5). As pointed out by Duffte and Huang (1986) the contingent claims space X=L’(S, 5,~) is a separable Banach space, so that Theorem 5 readily applies. It is straightforward to verify that the proof to Theorem 5 remains valid when we replace the hypothesis that X is separable with the hypothesis that X* has weak-star separable subsets. This result generalizes an analogous proposition due to Kreps (1981, Theorem 3). Indeed, we may replace A.6 with A.5 under all the circumstances for which he has verified the existence of a valuation operator. Nevertheless, we are heavily indebted to Kreps, because the latter half of the proof to Theorem 5 is a simple adaptation of his methods. A similar technique was employed by Duffie and Huang (1986). Another important issue is the relation between A.5 and A.l. Although it may appear that A.5 is stronger than A.l, these axioms are actually equivalent under various circumstances. We present the following lemma in order to establish their equivalence in the case when M is finite-dimensional. Recall that a polyhedral cone J by definition has a finite base (j,}~=r of non-zero vectors such that x EJ if and only if x=~~~= 1Ai,ji some sequence {Ii}:= 1 of non-negative real numbers. Suppose J is u polyhedral cone and K is a closed convex cone in a Lemmu. Hausdorjji locally convex, topological vector space X. lf J n K = {0}, then K-J is a closed convex cone. Proof
We demonstrate base {ji}l= 1 for J. First topological net in K-J
this result by induction on the dimension n of the assume n= 1 and J n K = (0). Let {X~:YE T} be a such that x,-+x. We want to show that XEK-J.
S.A. Clark, Valuation problem in arbitrage price theory
475
Since x,EK-J, we may write x,=k,-I,,j, for some k,EK and n,ZO. We claim that {x?:y~Z’} is bounded. On the contrary, assume {x?:YE~} is unbounded. Then we assume I,+ cc and 1, > 0 for every y E r without loss of generality, by passing to a subnet if necessary. Consider the identity x,/E.,= k,/l,j,. Since xy+x and ,++co, it follows that x,/2,+0. Thus, k,/,$,+ j,. But k,ll, E K and K is closed implies j, E K. Since J n K = (O}, we obtain the contradiction j, =O. This contradiction shows that (1,: y E r} is bonded. We may now assume A,,--+;1for some AZ0 without loss of generality, by passing to a subnet if necessary. Then k,=x,+;l,j, -+x+ nj,. It follows that x + nj, = k for some k E K, because K is closed. Therefore, x = k -lj, implies x E K -J. This verities the lemma in the case n = 1. The inductive step proceeds as follows. Assume the result is true for a polyhedral cone J, with base of dimension n. Assume J is a polyhedral cone with base {ji}lZf of dimension n+ 1 such that J n K= (0). Let J, be the polyhedral cone with base {j “+i}, and let J, be the polyhedral cone with base {j,}~=,. Then J= J, + J, implies J, n (K- J,)= (0). Notice that J, n K = {0}, so that the inductive hypothesis implies (K-J,) is a closed convex cone. Thus, J, n (K-J,) = (0) implies (K-J,) - J, = K - J is a closed convex cone. cl Theorem
6.
Suppose
M is a finite-dimensional
subspace
of X.
Then
A.5
is
equivalent to A.I. Proof. Recall that A.1 asserts F n X+ + =c$ and that A.5 asserts X+ + n (F-X+)=4. It is obvious that X++n(F-X+)=4 implies FnX++=4, because FE(-). Conversely, assume F n X+ + = 4. Then F n X+ = (0).
Furthermore, M is finite-dimensional and F is a half-space in M implies F= F is a polyhedral cone. Thus, the previous lemma implies (F-X+) = F-X+. Finally, it is straightforward to verify that F n X+ + =$ implies X ++ n(F-X+)=4, 0 so that X++ n(F-X+)=4. This theorem is readily applied to a finite state-space model consisting of only a finite number of markets. In view of Theorem 5, notice that the existence of a valuation operator is completely characterized by the simplest arbitrage axiom A.1 in this situation. In the context of dynamic securities models, Back and Pliska (1991) have constructed a counterexample which shows that A.1 does not generally imply the existence of a valuation operator. A potential gap is suggested by comparing Theorems 1 and 2, which implies that a strictly positive price functional p: M+lR is not necessarily continuous. On the other hand, if M is a closed sublattice of a Banach lattice X, then p is continuous [e.g., see Schaeffer (1974), so that A.1 is equivalent to A.2. Another potential gap exists between A.2 and AS, but we have nothing else to offer besides Theorem 6 for closing it in this paper.
J.Math
D
476
S.A. Clark,
5. Classical
Valuation
problem
in arbitrage
price them)
Banach spaces
Consider a classical Banach space Lq(S, t,n); where (S, i) is a Bore1 measurable state space, 7c is a countably additive probability measure on (S, t), and 15 q < co. These spaces have a number of well-known features which make them attractive for financial modelling [e.g. see Dunford and Schwartz (1971)]. Suppose X=Lq(S, 5, Z) is the contingent claims space. According to the Riesz representation theorem, a continuous linear functional P: X+Iw has the representation P(x) =jx~ld~, where y E L,P(S, <, n) is fixed and l/p+ l/q= 1. If P is a valuation operator, then the strict positivitl of P further implies y>O a.s. (7~). Define another measure p on (S. <) by the formula p(E)= P(I,)/P(Z,) for every event EEL. Obviously, /I is a finitely additive probability measure, and the countable additivity of 11 is easil! derived from the representation
and the Lebesgue monotone convergence theorem. Notice that J:/I’(I,~) is the Radon-Nikodym derivative dp/drc. Since dp/dn >O a.s. (n), it follows that /1 and 71 are equivalent, i.e. they are mutually absolutely continuous. Furthermore, we obtain
for every m E M, so that p is normalized into a p-expectations operator. In the framework of a dynamic securities market, these properties imply /L is an equivalent martingale measure [e.g. see Harrison and Kreps (1979) or Duffie and Huang (1986)]. The state space S is often taken to be all realizable streams of security prices over a finite time interval [0, r] endowed with a suitable metric topology. Then r can be taken as the predictable a-algebra generated by the natural filtration on S. For example, a simple model for one-dimensional Brownian motion specifies S as the collection of all continuous functions on [O. T] endowed with the uniform metric topology, and imposes the Wiener measure 7-r on (S,r). It is noteworthy that this space is separable [e.g. see Billingsley (1968)], so that Lq(S, 4, n) is also separable whenever 1 zq < Y. Hence, Theorem 5 applies and the existence of a valuation operator is characterized by A.5. On the other hand if we take S as the collection of all right-continuous, left-limit functions on [O, T] endowed with the uniform metric topology; then we encounter an inseparable state space [Billingsley (1968)]. Although this difficulty is sometimes avoided by working with the Skohorod topology, it points toward the importance of solving the valuation problem in an inseparable topological vector lattice. Thrown
7.
Suppose
X = Lq(S, 5, p): where
(S, <) is a Bowl measurable
space.
S.A. Clark,
Valuation
problem
in arbitrage
price theory
471
p is a a-finite countable additive measure on (S, 0, and 1 sq < co. Further suppose p : M-+R is a continuous, strictly positive, linear price functional on the subspace M of marketed claims in X. Then p admits a valuation operator if and only
tf condition
A.5 holds.
Proof. The necessity of A.5 is obvious, so we turn to its sufficiency. Assume A.5 holds. The method explain in the first paragraph of Theorem 5 shows there exists a positive linear functional P,:X+[w that for every XEX++ that P, induces a extending p such that 11 Pxll = 1 and P,(x) >O. Notice countably additive, finite measure A, on (S, 4) defined by E.,(E) =Px(I,) for by p, the Halmosevery E E l. Since the family {i_: x E X + + } is dominated Savage theorem [see Halmos and Savage (1949, Lemma 7)] implies there exists a countable equivalent subfamily. say {i.x, : 15 i < cc ). Thus, A,(E) = 0 for every x E X + + if and only if i,,,(E) = 0 for every 15 i < CC. We now define a linear functional P:X-+R by the formula P(x)= positive linear xi”= 1 2_‘P,,(x) for every XEX. Clearly, P is a well-defined, functional extending p (see the last paragraph of the proof of Theorem 5). We claim that P is strictly positive and, hence, a valuation operator. Indeed, the Riesz representation theorem implies that for every x E X+ + the linear functional P,: X--+R has the representation P,(y) = l yzx d/l = 1 y d2,
for some z,~L~(S,s,p), where l/p+l/q=l. Thus. P,(x)=jxd&>O. The definition of the Lebesgue integral now implies there exists E,E < and a positive number fi, such that &(E,) >O and xz6,1Ex. It follows that &(I?,) > 0 for some 15 i < a, and that P,,(x) = s x d&, 2 j fi,lEx di.,, = 6,i.,r(E,) >O. Therefore, P(x) ~2-‘PxI(x) ~0. 0 The proof to the above theorem has a measure-theoretic flavor in contrast to the purely geometric flavor of the proof to Theorem 5. It utilizes the Halmos-Savage theorem, which asserts that a family of countably additive measures is dominated by a a-finite countably additive measure if and only if there exists a countable equivalent subfamily [Halmos and Savage (1949)]. The fact that a a-finite countably additive measure is equivalent to a countably additive probability measure is well-known and easy to verify. Although n is usually assumed to be a probability measure in financial modelling, we have elected to express Theorem 7 in terms of a a-finite measure in order to make the method of proof more transparent. Neither Theorem 5 nor Theorem 7 effectively handle the inseparable Banach space X=L”(S, 5.7~) endowed with the essential supremum norm topology. Nevertheless, we might impose the weak-star topology on L”(S, 5, rc), exploiting the fact that L”(S, 5, n) is the dual of L’(S, 5, rr). Then a weak-star continuous linear functional P : X+ R has the representation P(x) =j xy dn for some fixed y E L’(S, r, rr) [e.g. see Kelly and Namioka (1963,
478
S.A. Clark, Valuation problem in arbitrage price theory
ch. 5)]. The method of proof for Theorem 7 is easily adopted situation, again showing that A.5 characterizes the existence operator.
to handle this of a valuation
References Back, K. and S.R. Pliska, 1991, On the fundamental theorem of asset pricing with an inlinite state space, Journal of Mathematical Economics 20, no. 1, 1-18. Billingsley, P., 1968, Convergence of probability measures (Wiley, New York). Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 3, 637-654. Chamberlain, G. and M. Rothschild, 1983, Arbitrage, factor structure, and mean-variance analysis on large asset markets, Econometrica 51, 1281-1304. De Finetti, B., 1937, Foresight: Its logical laws, its subjective sources, Annales de ‘l’lnstitute Henri Poincare 7 [Reprinted in: H.E. Kyburg akid H.E. Smokler, eds., Studies in subjective probability (Krieger, Huntington, NY) 19641. De Finetti, B., 1972, Probability, induction and statistics (Wiley, New York). DufIie, D. and C. Huang, 1986, Multiperiod security markets with differential information, Journal of Mathematical Economics 15, 283-303. Dunford, N. and J.T. Schwartz, 1971, Linear operators, Vol. 1 (Interscience, New York). Halmos, P.R. and L.J. Savage, 1949, Application of the Radon-Nikodym theorem to the theory of sufficient statistics, Annals of Mathematical Statistics 20, 225-241. Hansen, L.P. and SF. Richard, 1987, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica 55, no. 3, 5877613. Harrison, M.J. and D.M. Kreps, 1979, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory 20, 381408. Karatzas, I., 1989, Optimization problems in the theory of continuous trading, SIAM Journal of Control and Optimization 27, no. 6, 1221-1259. Kelly, J.L. and I. Namioka, 1963, Linear topological spaces (Van Nostrand, Princeton, NJ). Kemeny, J., 1955, Fair bets and inductive probabilities, Journal of Symbolic Logic 20, 263-273. Kreps, D.M., 1981, Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics, 15-35. Kyburg, H.E. and H.E. Smokier, 1964, Studies in subjective probability (Krieger, Huntington, NY). Mas-Colell, A., 1986, The price equilibrium existence problem in topological vector lattices, Econometrica 54, no. 5, 1039-1053. Merton, R.C., 1973, The theory of rational option pricing, Bell Journal of Economics and Management Science 4, 141-183. Modigliani, F. and M. Miller, 1958, The cost of capital, corpoi_,e finance and the theory of investment, American Economic Review 48, 261-297. Ramsey, F.P., 1926, Truth and probability [Reprinted in: H.E. Kyburg and H.E. Smokler, eds., Studies in subjective probability (Krieger, Huntington, NY) 19641. Ross, S., 1976, Risk, return, and arbitrage, in: I. Friend and J. Bicksler, eds., Risk and return in finance (Ballinger, Cambridge, MA). Ross, S., 1978, A simple approach to the valuation of risky streams, Journal of Business 51, no. 3, 453485. Schaeffer, H., 1974, Banach lattices and positive operators (Springer-Verlag, New York). Shimony, A., 1955, Coherence and the axioms of conlirmation, Journal of Symbolic Logic 20, l-28. Varian, H.R., 1987, The arbitrage principle in Iinancial economics, Journal of Economic Perspectives 1, no. 2, 55-72. Yosida, K. and E. Hewitt, 1952, Finitely additive measures, Transactions of the American Mathematical Society 72, 4666. Zame, W.R., 1987, Competitive equilibria in production economies with an intinite dimensional commodity space, Econometrica 55, no. 5, 1075-l 108.
of Mathematical
Economics
The valuation theory Stephen
problem
North-Holland
in arbitrage
price
A. Clark*
University of Kentucky, Submitted
22 (1993) 463478.
November
Lexington
KY, USA
1990, accepted
September
1992
Suppose a continuous, strictly positive, linear price functional p is given on a subspace M of marketed claims. The valuation problem consists of verifying whether or not there exists a continuous, strictly positive, linear extension of p from M to the entire contingent claims space X. We solve this problem when X belongs to a large class of Banach lattices including the classical Banach spaces, and also simplify some analogous results found in the literature for other types of financial models.
1. Introduction
This essay aims toward resolving a valuation problem in the foundations of arbitrage price theory. An arbitrage opportunity is essentially a feasible contingent claim with positive net return across all states of nature. In other words, an arbitrage is a ‘free lunch’. Financial economists have long recognized that a competitive market equilibrium eliminates all arbitrage opportunities and have exploited this condition to evaluate contingent claims [e.g. see Modigliani and Miller (1958), Merton (1973) or Black and Scholes (1973)]. In contrast, we follow an axiomatic approach that abstracts away from general equilibrium theory. In the next section, we present some basic conditions which describe the absence of arbitrage opportunities. Suppose the contingent claims space X is a Hausdorff, locally convex, topological vector lattice and the marketed claims space M is a subspace of X. Ross (1978) has suggested a key axiom A.1 of no arbitrage opportunities, formally demanding that there does not exist a feasible claim in the strictly positive cone. Under a mild auxiliary condition typically satisfied by a riskless marketed claim, it is straightforward to verify that A.1 characterizes a strictly positive, linear price functional p: M-R. This result is the most fundamental and elementary type of Correspondence to: Stephen A. Clark, Department of Statistics, University of Kentucky, Lexington, KY 40506-0027, USA. *The author expresses gratitude to an anonymous referee who detected a logical error in the original version of this paper. 03044068/93/‘SO6.00
0
1993-Elsevier
Science Publishers
B.V. All rights reserved
464
S.A. Clark, Valuation problem in arbitrage price theory
arbitrage argument. Then we present another axiom A.2, which is a topological version of A.l. Its purpose is to assure that the price functional p is also continuous. In the third section, we discuss the historical connection between arbitrage price theory and subjective probability theory [Ramsey (1926) or De Finetti (1937)]. We only need to slightly weaken our axioms, A.1 and A.2, to obtain coherency axioms, A.3 and A.4, for constructing probability measures in terms of a positive linear functional p: M +R which serves as an expectations operator. This exercise illustrates how subjective probability theory has anticipated the modern development of arbitrage price theory. In the fourth section, we introduce the valuation problem. Following Ross (1978) a valuation operator P : X -+[w is a continuous, strictly positive, linear extension of p to the entire contingent claims space X. A fundamental problem of arbitrage price theory is to characterize the existence of a valuation operator, providing a computational method for evaluating contingent claims. Recent developments in general equilibrium theory suggest that it would be useful to solve this problem for an arbitrary topological vector lattice [e.g., see Mas-Cole11 (1986) or Zame (1987)], but Hilbert lattices are especially common in financial modelling [e.g., see Harrison and Kreps (1979), Chamberlain and Rothschild (1983) or Hansen and Richard (1987)]. Here we suggest a new axiom AS, which is a weak version of Krep’s (1981) assumption A.6 of ‘no free lunches’. The latter condition has been utilized by Kreps (1981) and by Duffie and Huang (1986) to solve the valuation problem for a class of models that evoke separability assumptions. Nevertheless, we show that A.5 and A.6 are equivalent assumptions for practically all the circumstances where these authors have characterized the existence of a valuation operator. In the fifth and final section, we completely resolve the valuation problem for the classical Banach spaces Lq(S, 5,~) when 154 < a. These methods do not require separability assumptions. Instead we merely require that p is a CJfinite countably additive measure on (St), an assumption that is vital for the implementation of the Radon-Nikodym theorem. Here we show that A.5 completely characterizes the existence of a valuation operator. This result has applications to the theory of continuous trading [e.g. see Karatzas (1989)]. 2. Arbitrage equilibrium The formal model consists of a Hausdorff, locally convex, topological vector lattice X, which is interpreted as the collection of all contingent claims, and a subspace M, which is interpreted as the collection of marketed claims. By definition, a topological vector lattice X is a topological vector continuous lattice operation with a (uniformly) space equipped v :X xX+X. This lattice operation v creates a natural vector order 2 on
S.A. Clark, Valuation problem in arbitrage price theory
465
X defined by the condition x1 2x2 if and only if (x1 -x2) v 0=(x, -x2). The positive cone is given by X+ = {XE X: x20). The vector order 2 is reflexive, transitive, and antisymmetric. Let > denote its asymmetric component, so that x1 )x2 if and only if x1 2x, and x1 #x2. The strictly positive cone is Thus, X+=X++ u (0). A key property of a given by X + + ={xEX:X>O}. topological vector lattice is the closedness of X+. Indeed, many of our results generalize to an ordered topological vector space with closed positive cone. The following notation is useful. Suppose A,B are subsets of X. Let -A={x~x:-XEA}, A+B={x~X:x=a+b for some UEA,~EB}, and A -B = A +(-B). We also let A denote the topological closure of A. Presumably, there exist markets for only some of the contingent claims in and the collection of marketed claims M is the linear span of X3 say {miliol, {mi}iE,. Thus, a marketed claim me M is just a portfolio of long and short positions in the available markets. We employ the notation m =xie, Mimi for a marketed claim with the understanding that the number of shares %i of the claim mi is non-zero in only a finite number of markets. We do not necessarily assume, however, that the numbers of markets is finite. In summary, we have the defining formula
M=
x~X:x=~~imiforsome{~i}i~, icl
Of course, the requirement that M is a subspace assumes frictionless trading, e.g. unlimited short selling and no transactions costs. Suppose the market assigns a price pi to the marketed claim m, for every in I. We do not need to assume that prices are a priori positive, but rather we intend to derive this property as a consequence of the absence of arbitrage opportunities. A price correspondence p: M+R is naturally defined by the formula
p(m) =
C &pi : m = C limi for some (2i}i,, i
iel
isl
I
for every rnE M. The economic interpretation of this price correspondence is easily explained. Indeed, p(m) represents all possible prices for which the marketed claim m E M can be purchased by constructing it as a portfolio of contingent claims offered directly in the markets. We say that a marketed claim me M is feasible provided that there exists some 4 up such that 420. Let F denote the collection of all feasible claims. The principle of no arbitrage, as originally formulated by Ross (1976, 1978), asserts that there does not exist a feasible claim in the strictly positive cone X++. For any XE M, let p(x) >O denote the condition 4 >O for every q E p(x). In formal terms, we now have the following axiom:
S.A. Clark, Valuation problem in arbitrage price theory
466
A.I.
For every XEM, x>O implies
p(x)>O.
If we momentarily interpret > as a strict preference relation, then A.1 represents Harrison and Krep’s (1979) condition of ‘no simple free lunches’. More generally, their condition implies A.l, because they assume preferences are strictly montone with respect to 2. It is straightforward to determine the implications of A.1 for the price correspondence p : M -+R. Suppose there exists a marketed claim m, such that m, >O, e.g. a marketed riskless security. Then the absence of arbitrage opportunities implies p is a linear functional. In general, suppose Y is a subspace of X and p : Y + R is a linear functional on Y. Let Y + = Y n X+ and Y++=YnX++. We say that p is positive whenever p(y) 20 for every YE y+, and we say that p is strictly positive whenever p(y) >0 for every 4’E y++. We record the following result in order to provide the flavor of an elementary arbitrage argument. Theorem 1. Suppose Mi + #I$. strictly positive linear jiinctionul.
Then
A.1 holds
if and only
if p: M-+R
is a
Proqf. If p is a strictly positive linear functional, then A.1 holds trivially. Conversely, assume A.1 holds. Let m, E M+ +.Then p0 >O for any p0 up. We first show that p(m) is uniquely determined for every rn~ M. On the contrary, assume p(m) is not uniquely determined for some me M. Then there We may write m=xiE,2.imi with q= exist q, q* Ep(m) such that q >q*. *m with q* =xiEII$pi. Define a contingent claim :z;$; and m=&,,A i
.x = [I(4 - 4*)lPolmo + rT, (AT- &)m,. x E M and x = [(q - q*)/p,,]m, > 0, because (4 - q*)/pO >O. Thus, Furthermore, O=C(q-_*)lP,lP,+Ci,r(;lt-~i)PiEP(x) implies x E F. This contradicts A.l, showing that p(m) is uniquely determined for every me M. Next, we show that p: M +R is linear. On the contrary, assume m = i,m, +A,m, and p(m)#i.,p(m,)+3L,p(m,). Then p(m) is not uniquely determined, contradicting the conclusion reached in the previous paragraph. Thus, p is a linear functional. Finally, it is now obvious that A.1 implies p is strictly positive. 0 Then
XEM++.
We say that p: M+R is an arbitrage equilibrium whenever p is a stricly positive, linear functional. This terminology is motivated by the well-known connection between a market equilibrium and the absence of arbitrage
467
S.A. Clark, Valuation problem in arbitrage price theory
opportunities [e.g. see Harrison and Kreps (1979)]. For example, suppose an agent in the economy has strictly monotone preferences. Then A.1 is a necessary condition for the existence of a compensating equilibrium, because otherwise the markets cannot clear. We next study the continuity of the price system in the relative topology. Accordingly, we define another price correspondence p: A?--+!& so that the graph of p is the closure of the graph of p. A local definition of j is somewhat complicated by the fact that X does not necessarily have a countable local base, forcing us to use topological nets instead of topological sequences. In general, let {x, : r~ r} denote a net in X with directed index set r. We write x,+x to denote the convergence of the net {x,: YE r} to the point x E X, Then, p: M+[W is locally defined by the condition qER:3{m,:yEr}
P(m) =
in M such that mY-+m, and 3 { py : y E r}
in [wsuch that py Ep(m,) Vy or and pY+q
for every rnE i@. For any rnE M, let p(m) ~0 denote the condition q>O every qEp(m). We introduce the following axiom in order to eliminate possibility of approximate arbitrage opportunities. A.2.
(3) for the
For every x E Ji?f, x >O implies p(x) > 0.
The logical form of A.2 is analogous economic interpretation. Its significance
to A.l, lending itself to a similar is explained by the following result.
Theorem 2. Suppose M+ + #$. Then A.2 holds continuous, strictly positive linear functional.
if and only
if p:?a+iw
is a
Proof. It is straightforward to verify that if p is a continuous, strictly positive linear functional; then A.2 holds. Conversely, assume A.2 holds. Then Fn h?l++ =c#J,because for every rnEF we have q50 for some qEp(m). But F-M+GF implies -SF. Thus, Let mOE M++. Then m,#F. m,$F-M+. So there exists an open, convex neighborhood of m,, say N(m,), such that N(m,) n (F - M ‘) = 4. In particular, N(m,) n (F - M ‘) = 4. Let C(m,) denote the convex cone generated by N(m,), i.e. C(m,)= {y~X:y=Lx for some A>0 and xEN(m,)}. Then C(mo)n(F--M+)=4. The geometric Hahn-Banach theorem now implies there exists a continuous linear functional P : X--f [w such that P(x) 2 0 for every x E C(m,) and P(x) 5 0 for every x E F - M + . In particular, P(x) 5 0 for every x E F. Notice that A.2 implies A. 1. So we obtain that p : M-R is a strictly positive linear functional from Theorem 1. Furthermore, for every rnE M, p(m) 5 0 implies P(m) 50. Replacing m with -m in this implication, we see that for every rnc M, p(m) 20 implies P(m) 2 0. Thus, -p(m) = 0 implies P(m) =0 for every m E M. Consequently, there exists some 3,~ [w such that
J.Math
C
468
S.A. Clark, Valuation problem in arbitrage price theory
p(m)=U’(m) for every me M [see Kelly and Namioka (1963)]. An evaluation at rn,~M++ shows that 2 #O, because A.1 implies p(m,) > 0. Therefore, P is continuous on M implies p is continuous. It immediately follows that p: A+R is a continuous, positive linear functional [see Kelly and Namioka (1963)]. Finally, we must show that p is strictly positive. On the contrary, assume p is not strictly positive. Then there exists some XEM+ + such that @(x)=0. Let {x,: YE T} be a net in M such that x, --tx. Then p(x,)-+p(x)=O. Now if {x, :YE f} has a feasible subnet, then we immediately obtain a violation of A.2. So p(x,)>O for all y sufficiently large in the directed index set r. As before, let m, E M ‘+. Then A.1 implies p(m,) ~0. Define &,=p(x,)/p(m,,) for every y E r. Then E,,>O for all y sufficiently large, and ,4,-O. Define XT= x,-i.,m, for every y E r. Then X~*EM. Furthermore, p(x,*) =p(x,) &p(m,) = 0 implies XTE F. But x:--+x implies x E F. Thus, x E F n M + +, which is another violation of A.2. This contradiction shows that p is strictly positive. 0 We say that p: i%+rW is a continuous arbitrage equilibrium whenever @ is a continuous, strictly positive linear functional. Since general equilibrium theory usually relies upon a fixed point theorem or the geometric HahnBanach theorem to guarantee the existence of a continuous price functional compatible with a market equilibrium, the requirement of a continuous arbitrage equilibrium is not an extraordinary theoretical demand [e.g. see Harrison and Kreps (1979)].
3. Subjective probability theory Much of the recent work in arbitrage price theory was anticipated by Ramsey (1926) and De Finetti (1937) in their classic studies of subjective probability theory, where the probability of an event is interpreted as an individual’s degree of belief in its occurrence. We wish to especially focus upon De Finetti’s formulation of the principle of coherence, because his approach is utility free and closely related to the Ross principle of no arbitrage - A.l. Indeed, the latter principle is practically the same as the concept of strict coherence [Shimony (1955) and Kemeny (1955)], which was later rejected by De Finetti (1972) as a proper foundation for probability theory. The connection between arbitrage prices and coherent beliefs is clearly seen when we regard the contingent claims space X as a vector lattice of real-valued functions upon a state space S. Suppose E is an event in S, i.e. E is a subset of S. Let I, denote the indicator function on E, so that I,(s) = 1 for every s E E and I,(s) = 0 for every s $ E. Of course, I, can be interpreted as a contingent claim which pays off $1 if event E realizes and SO otherwise. According to De Finetti (1937), the subjective probability of event E, say
S.A. Clark,
Valuation
problem
in arbitrage
price theory
469
rc(E), is just the price an individual would be willing to pay for the contingent claim I,. Hence, Z(E) =p(l,). Therefore, we may view subjective probability theory as the following special case of our general model of financial transactions. Suppose we have an algebra r of events in the state space S. Then we construct the space of marketed claims M as the span of {I,: E E 0. Thus, M is the collection of (measurable) simple functions on S, and M is a sublattice of X. Of course, probability theory merely requires a positive linear functional instead of a strictly positive one, allowing the possibility that an event EE 5 might be assigned probability zero. This requires a suitable modification of our no arbitrage axioms. For any XE M, let p(x) 20 denote the condition that 4 20 for every q Ep(x). We introduce the following axiom in order to eliminate the relevant arbitrage opportunities. A.3.
For every x E M, x 2 0 implies p(x) 2 0.
This axiom has been discussed by Varian (1987) in the context of a finite state-space model. Its significance lies in the following result, which does not directly evoke the state-space approach. Theorem p0 up.
3.
Suppose there exists some mOE M+ + such that p,, >O for some Then A.3 holds if and only if p : M+R is a positive linear
functional. Proof.
The demonstration of this theorem is a straightforward modification of the proof to Theorem 1, and we only provide a summary sketch. If p is a positive linear functional, then it is trivial that A.3 holds. Conversely, assume A.3 holds. We first show p(m) is uniquely determined for every rnE M. Contrary to hypothesis, assume p(m) is not uniquely determined for some rnE M. Then we may purchase m at a low price, sell m at a high price, and use a positive fraction of the proceeds to purchase shares of m,. We have now constructed a portfolio that violates A.3. This contradiction shows that p(m) is uniquely determined for every me M. It also follows that p is a linear functional, because otherwise p is not uniquely determined. The positivity of p now follows trivially from A.3. 0 We remark
that the auxiliary
condition
of Theorem 3, i.e. there exists some is almost always satisfied by For example, I, is interpreted as a riskless asset in the typical securities market model. On the other hand, the subjective probability model usually requires SE< with n(S) =p(l,)= 1 as a normalization constant. The above theorem is a modest generalization of De Finetti’s well-known m,,EM+’ such that p0 > 0 for some p0 up, m, = Is as an axiom for the state-space model.
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price theory
characterization of subjective probability [e.g. see De Finetti (1972)]. Indeed, A.3 is essentially equivalent to his coherency axiom, which asserts that p(x) >O whenever x is both marketable and uniformly positive. Thus, A.3 is the defining characteristic of a coherent probability measure rr: 5-R for (S,<). We remark that subjective probability theorists traditionally take finite additivity as the key property of a probability measure with countable additivity as an important special case [e.g. see Kyburg and Smokler (1964)]. This viewpoint is highlighted by the fact that the proof to Theorem 3 works in an arbitrary vector lattice. For any x E M, let p(x) 20 denote the condition qz0 for every REP. We next introduce the following axiom as a topological version of A.3. A.4.
For every x E M, x 2 0 implies
The following result requires some delicate
is analogous modifications.
p(x) 2 0. to Theorem
Theorem 4. Suppose there exists some rnOE M+ p0 cp(m,). Then A.4 holds and F # X if and only positive linear functionul.
2, but
its demonstration
’ such that p0 >0 fbr some if p: h?i--+R is u continuous,
Proc$ Assume p: M-+R is a continuous, positive linear functional. Then A.4 holds trivially. Furthermore, x E F implies p(x) 5 0. Thus, p(m,) =pO > 0 implies m, $ F. This verifies that F # X. Conversely, assume A.4 holds and F#X. Notice that A.4 implies A.3. Hence, p: M +R is a positive linear functional, by virtue of Theorem 3. Define F,={x~M:not @(x)20}. Since F={x~M:p(x)~0}, it is clear that F, G F. Thus, F, GF. Now A.4 directly implies F,-- Ji?f+ c F,, so that (F, - M+)G F,cF. By hypothesis, there exists some xo$F. So there also exists some open, convex neighborhood of x0, say N(x,), such that N(x,) n (F,M+) = 4. In particular, N(x,) n (F,M+) = 4. Let C(x,) denote the convex cone generated by N(x,), i.e. C(x,) = {y E X : y = Ax for some 3,> 0 and XE C(x,)}. Since F-&i+ IS a convex cone, it follows that C(x,) n (F-M’) = 4. Define C = C(x,) + M+. Then C is an open, convex cone such that CnF-q5. Let F,={x~M:p(x)<0). Notice that F,#+, because -m,EFl. We claim that F, c F,. Indeed, for any XE M, x $ F, implies p(x) 20. Since p(x) E&X), we obtain x $ F,, which verities the claim. Next, observe that C n F, =c$ implies C n F, = 4. An application of the geometric Hahn-Banach theorem yields the existence of a continuous, nonzero linear functional P : X--+ R such that P(x) 5 0 for every x E F, and P(x) 2 0 for every x E C. Assume p(m) = 0 for some m E F. Then for any 6 > 0, we obtain p(m-_m,)=p(m)-_p(m,)= -6p(m,)
S.A. Clark, Valuation problem in arbitrage price theory
471
Thus P(m- 6m,) 50 for every 6 > 0. Letting d-+0, it follows that P(m) 50 by virtue of the continuity of P. In summary, P has the property that for every x E M, p(x) 50 implies P(x) SO. The argument in the proof of Theorem 2 now applies; showing that P, p are proportional linear functionals on M. Thus, P is continuous on M implies p is continuous. Since p is a continuous, positive linear functional; p is also a continuous, positive linear functional [see Kelly 0 and Namioka (1963)]. Corollary. A.4 holds
Suppose
there
exists
if and only if fp: ii&E4
some m, E M ’ + such that p(m,) >O. Then is a continuous, positive linear functional.
Proof. Notice that p(mO) > 0 implies implies p0 > 0 for any F(m,) >O immediately. 0
m, $ F.
Thus, F#X. In p0 Ep(m,). The result
addition, follows
We remark that the auxiliary condition in this corollary, i.e. there exists some m,E M++ such that p(m,) > 0, is just a convenient combination of the auxiliary condition in Theorem 4 and the condition F#X. In view of A.4, we can just as well replace it with the condition that there exists some m, E M+ + such that O$p(m,,). If the markets are incomplete in the sense that G #X, then the extra condition F#X is automatically satisfied. Some additional insight is provided when we study our auxiliary conditions in the context of a state-space model. Then mO= I, is the natural candidate to fulfill both roles, so that we might assume p(Z,) >O or just 0 $ p(1,). In fact, assume some m, E M satisfies p(mO) > 0. If m, is (essentially) bounded, i.e. there exists 1 >O such that /m,(s)/ 51 for (almost) every s E S; then m, is dominated by a positive scalar multiple of I,, i.e. ,lIs2m,. Thus, A.4 implies @(Al,) 2 p(m,), so that p(1,) >O. The following state-space model is useful in the foundations of probability theory. Suppose 5 is a a-algebra of events in S and p is a Lebesgue measure on (S, 5). Let X =L”(S, 4,~) denote the Banach lattice of all essentially bounded, measurable, real-valued functions on S equipped with the essential supremum norm. Then the topological dual of X is isomorphic to the lattice of all tinitely additive, signed measures on (S, {) [Yosida and Hewitt (1952)]. Notice that Theorem 4 readily applies to this model. On the other hand, suppose we endow X=L”(S, 5,~) with the topology of pointwise convergence. Since X is the dual of L’(S, 5, cl), it is straightforward to verify that this topology is the same as the weak-star topology. Let (Ej}jm_ 1 denote a sequence of events in 5 that partitions S. Let D,= us= 1 Ej whenever 15 k< co, so that the sequence {I&T= 1 converges (monotonely) to I, is the weak-star topology. We now examine the Corollary to Theorem 4 under the presumption that m,=I,. Notice that O$p(Z,) implies that it is impossible to assign subjective probabilities to the events in t such that
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rc(Ej) = 0 whenever 15 j < co and rr(S) = 1. This property was discussed by De Finetti (1972) to distinguish between finitely additive and countably additive probability measures. More importantly, notice that the continuity of p: M-+LQ implies that n is countably additive, when we work in the framework of the weak-star topology. In general, continuity does not pose a special problem when X is a Banach lattice for the following reason. Suppose p: M +[w is a (strictly) positive linear functional. Then p has a positive linear extension to X provided that M contains a radial point of X+ [see Kelly and Namioka (1963)]. Yet a positive linear functional P on a Banach lattice X is necessarily continuous [see Schaeffer (1974)]. A fortiori, p is continuous on M, and fi is continuous on ii%. Therefore, we may deduce A.4 from A.3 whenever X is a Banach lattice and the appropriate auxiliary condition holds, e.g. I, E M and 1 E ~(1,).
4. The valuation operator Suppose p : Ii?l+R is a continuous arbitrage equilbrium. In forma1 terms, a valuation operator is a continuous, strictly positive, linear extension of p to the entire contingent claims space X. In other words, a valuation operator is a continuous, strictly positive, linear functional P:X-+R such that the restriction of P to the subspace M is identical to p. This concept was introduced by Ross (1976, 1978) in order to provide a computational approach to arbitrage price theory. Furthermore, it plays a key role in the construction of an equivalent martingale measure for a dynamic model of the securities markets [e.g. see Harrison and Kreps (1979) or Duffie and Huang (1986)]. Thus, the provision of conditions for the existence of a valuation operator is a vital problem in financial economics. It is straightforward to verify that the following condition is necessary for the existence of a valuation operator.
{x,:y~r} in
A.5. If there exists a net and there exists a net (m,: y E r} not x>O.
X such that x7-+x for some XEX, in F such that my 2 x, for every y E r; then
In more concise terms, this axiom asserts that X+ + n (F-X’) = 4. We regard A.5 as a weak version of Krep’s (1981) condition of ‘no free lunches’, which is expressed by the following assumption. A.6. If there exists a net {x,: YES} in X such that xy+x for some XEX, and there exists a net {m,:y~r} in A such that liminf{p(m,): YES} 50 and my 2 x, for every y E r; then not x > 0. Kreps
demonstrates
that A.6 is both
a necessary
and sufficient
condition
for
S.A. Clark, Valuation problem in arbitrage price theory
413
the existence of a valuation operator whenever the contingent claims space X has an appropriate structure, e.g. X is normable and its dual space has weakstar separable subsets. Duffie and Huang (1986) derive the same type of result whenever X is a separable Banach space. Nevertheless, we claim that A.5 and A.6 are equivalent conditions under all the circumstances for which these authors have characterized the existence of a valuation operator. It is more convenient here to focus upon general methods, rather than attempting a tedious replication of their arguments, to show that A.6 can be replaced with A.5. Hence, we offer the following strict separation theorem. Theorem 5. Suppose J, K are non-empty convex cones (with vertices at the origin) in a separable Banach space X. Then there exists a continuous linear functional P : X-+ R such that P(x) > 0 for every x E J and P(x) 5 0 for every XEK ifand only ifJn(K-J)=$. to verify that if there exists a continuous linear Proof. It is straightforward functional P: X+R such that P(x) > 0 for every x E J and P(x) 5 0 for every XEK, then J n (K- J)=#. Conversely, assume J n (K - J)=4. Then for every x E J, there exists an open, convex neighborhood of x, say N(x), such that N(x) n (K-J) = 4. Let C(x) be the convex cone generated by J + N(x), i.e. C(x) = {z~X:z=Ay for some 2~0 and some YE J+N(x)}. It is straightforward to verify that C(x) is an open, convex cone such that C(x) n K =c$. So an application of the geometric Hahn-Banach theorem yields the existence of a continuous, non-zero linear functional P,: X-R such that P,(y) 2 0 for every YE C(x) and P,(y) 5 0 for every y E K. Now P,(y) 20 for every y E C(x), which implies P,(y) 2 0 for every y E J. Furthermore, P,(x) > 0, because C(x) is open and x E C(x). In summary, for every x E J, there exists a continuous linear functional P, : X-t R such that (i) P,(y) 2 0 for every y E J, (ii) P,(y) 50 for every YE K, and (iii) P,(x)>O. Notice that IIPxl[>O. Without loss of generality, we may also assume that (iv) IIPxll= 1 for every XE J, by renormalization if necessary. Let X* denote the (topological) dual of X, and let B* denote the unit ball in X*, i.e. B*={PEX*:IIPII~~}. S’mce X is a separable Banach space, B* is metrizable with respect to the relative weak-star topology [e.g. see Dunford and Schwartz (1971)]. It follows that B* satisfies the second axiom of countability in the relative weak-star topology. Let Y= {Px: x E J}. Since 9~ B*, 9 also satisfies the second axiom of countability, and hence Y is weak-star separable. Let {Pxi>,“_, denote a countable, weak-star dense subset of 9. We now define a linear functional P:X+R by the formula P(x)= XI’= 1 2_‘Px,(x) for every XEX. Then P is a well-defined continuous linear functional because {x1= I 2 -‘P,,},“= 1 is a Cauchy sequence in the Banach space X*. Furthermore, it is clear that P(x) 50 for every XE K. Since {Px,}~l
414
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problem
in arbitrage
price theory
is weak-star dense in 9, it follows that for every XEJ, there exists positive integer i such that II’,(x)--P,,(x)/
some and
The significance of the above theorem to arbitrage price theory can be explained as follows. If we select J =X+ + and K = F, then the condition J n (K -J) =4 is equivalent to axiom A.5. Furthermore, the condition P(x) 5 0 for every x E F implies P and p are proportional linear functionals on J@ (recall the argument in the proof to Theorem 2). So we may assume P extends p without loss of generality. Therefore, A.5 characterizes the existence of a valuation operator whenever X is a separable Banach space. The analogous result by Duffte and Huang (1986) is an immediate corollary. They analyze a dynamic securities market model whose contingent claims space X is a L’ probability space. Suppose a tinite number of securities can be traded at only a finite number of dates. If we identify a state of nature as a realized stream of net return vectors, then the state space S is finite dimensional and, hence, separable. Let 5 denote a a-algrebra on S, and let rr denote a countably additive probability measure on (S, 5). As pointed out by Duffte and Huang (1986) the contingent claims space X=L’(S, 5,~) is a separable Banach space, so that Theorem 5 readily applies. It is straightforward to verify that the proof to Theorem 5 remains valid when we replace the hypothesis that X is separable with the hypothesis that X* has weak-star separable subsets. This result generalizes an analogous proposition due to Kreps (1981, Theorem 3). Indeed, we may replace A.6 with A.5 under all the circumstances for which he has verified the existence of a valuation operator. Nevertheless, we are heavily indebted to Kreps, because the latter half of the proof to Theorem 5 is a simple adaptation of his methods. A similar technique was employed by Duffie and Huang (1986). Another important issue is the relation between A.5 and A.l. Although it may appear that A.5 is stronger than A.l, these axioms are actually equivalent under various circumstances. We present the following lemma in order to establish their equivalence in the case when M is finite-dimensional. Recall that a polyhedral cone J by definition has a finite base (j,}~=r of non-zero vectors such that x EJ if and only if x=~~~= 1Ai,ji some sequence {Ii}:= 1 of non-negative real numbers. Suppose J is u polyhedral cone and K is a closed convex cone in a Lemmu. Hausdorjji locally convex, topological vector space X. lf J n K = {0}, then K-J is a closed convex cone. Proof
We demonstrate base {ji}l= 1 for J. First topological net in K-J
this result by induction on the dimension n of the assume n= 1 and J n K = (0). Let {X~:YE T} be a such that x,-+x. We want to show that XEK-J.
S.A. Clark, Valuation problem in arbitrage price theory
475
Since x,EK-J, we may write x,=k,-I,,j, for some k,EK and n,ZO. We claim that {x?:y~Z’} is bounded. On the contrary, assume {x?:YE~} is unbounded. Then we assume I,+ cc and 1, > 0 for every y E r without loss of generality, by passing to a subnet if necessary. Consider the identity x,/E.,= k,/l,j,. Since xy+x and ,++co, it follows that x,/2,+0. Thus, k,/,$,+ j,. But k,ll, E K and K is closed implies j, E K. Since J n K = (O}, we obtain the contradiction j, =O. This contradiction shows that (1,: y E r} is bonded. We may now assume A,,--+;1for some AZ0 without loss of generality, by passing to a subnet if necessary. Then k,=x,+;l,j, -+x+ nj,. It follows that x + nj, = k for some k E K, because K is closed. Therefore, x = k -lj, implies x E K -J. This verities the lemma in the case n = 1. The inductive step proceeds as follows. Assume the result is true for a polyhedral cone J, with base of dimension n. Assume J is a polyhedral cone with base {ji}lZf of dimension n+ 1 such that J n K= (0). Let J, be the polyhedral cone with base {j “+i}, and let J, be the polyhedral cone with base {j,}~=,. Then J= J, + J, implies J, n (K- J,)= (0). Notice that J, n K = {0}, so that the inductive hypothesis implies (K-J,) is a closed convex cone. Thus, J, n (K-J,) = (0) implies (K-J,) - J, = K - J is a closed convex cone. cl Theorem
6.
Suppose
M is a finite-dimensional
subspace
of X.
Then
A.5
is
equivalent to A.I. Proof. Recall that A.1 asserts F n X+ + =c$ and that A.5 asserts X+ + n (F-X+)=4. It is obvious that X++n(F-X+)=4 implies FnX++=4, because FE(-). Conversely, assume F n X+ + = 4. Then F n X+ = (0).
Furthermore, M is finite-dimensional and F is a half-space in M implies F= F is a polyhedral cone. Thus, the previous lemma implies (F-X+) = F-X+. Finally, it is straightforward to verify that F n X+ + =$ implies X ++ n(F-X+)=4, 0 so that X++ n(F-X+)=4. This theorem is readily applied to a finite state-space model consisting of only a finite number of markets. In view of Theorem 5, notice that the existence of a valuation operator is completely characterized by the simplest arbitrage axiom A.1 in this situation. In the context of dynamic securities models, Back and Pliska (1991) have constructed a counterexample which shows that A.1 does not generally imply the existence of a valuation operator. A potential gap is suggested by comparing Theorems 1 and 2, which implies that a strictly positive price functional p: M+lR is not necessarily continuous. On the other hand, if M is a closed sublattice of a Banach lattice X, then p is continuous [e.g., see Schaeffer (1974), so that A.1 is equivalent to A.2. Another potential gap exists between A.2 and AS, but we have nothing else to offer besides Theorem 6 for closing it in this paper.
J.Math
D
476
S.A. Clark,
5. Classical
Valuation
problem
in arbitrage
price them)
Banach spaces
Consider a classical Banach space Lq(S, t,n); where (S, i) is a Bore1 measurable state space, 7c is a countably additive probability measure on (S, t), and 15 q < co. These spaces have a number of well-known features which make them attractive for financial modelling [e.g. see Dunford and Schwartz (1971)]. Suppose X=Lq(S, 5, Z) is the contingent claims space. According to the Riesz representation theorem, a continuous linear functional P: X+Iw has the representation P(x) =jx~ld~, where y E L,P(S, <, n) is fixed and l/p+ l/q= 1. If P is a valuation operator, then the strict positivitl of P further implies y>O a.s. (7~). Define another measure p on (S. <) by the formula p(E)= P(I,)/P(Z,) for every event EEL. Obviously, /I is a finitely additive probability measure, and the countable additivity of 11 is easil! derived from the representation
and the Lebesgue monotone convergence theorem. Notice that J:/I’(I,~) is the Radon-Nikodym derivative dp/drc. Since dp/dn >O a.s. (n), it follows that /1 and 71 are equivalent, i.e. they are mutually absolutely continuous. Furthermore, we obtain
for every m E M, so that p is normalized into a p-expectations operator. In the framework of a dynamic securities market, these properties imply /L is an equivalent martingale measure [e.g. see Harrison and Kreps (1979) or Duffie and Huang (1986)]. The state space S is often taken to be all realizable streams of security prices over a finite time interval [0, r] endowed with a suitable metric topology. Then r can be taken as the predictable a-algebra generated by the natural filtration on S. For example, a simple model for one-dimensional Brownian motion specifies S as the collection of all continuous functions on [O. T] endowed with the uniform metric topology, and imposes the Wiener measure 7-r on (S,r). It is noteworthy that this space is separable [e.g. see Billingsley (1968)], so that Lq(S, 4, n) is also separable whenever 1 zq < Y. Hence, Theorem 5 applies and the existence of a valuation operator is characterized by A.5. On the other hand if we take S as the collection of all right-continuous, left-limit functions on [O, T] endowed with the uniform metric topology; then we encounter an inseparable state space [Billingsley (1968)]. Although this difficulty is sometimes avoided by working with the Skohorod topology, it points toward the importance of solving the valuation problem in an inseparable topological vector lattice. Thrown
7.
Suppose
X = Lq(S, 5, p): where
(S, <) is a Bowl measurable
space.
S.A. Clark,
Valuation
problem
in arbitrage
price theory
471
p is a a-finite countable additive measure on (S, 0, and 1 sq < co. Further suppose p : M-+R is a continuous, strictly positive, linear price functional on the subspace M of marketed claims in X. Then p admits a valuation operator if and only
tf condition
A.5 holds.
Proof. The necessity of A.5 is obvious, so we turn to its sufficiency. Assume A.5 holds. The method explain in the first paragraph of Theorem 5 shows there exists a positive linear functional P,:X+[w that for every XEX++ that P, induces a extending p such that 11 Pxll = 1 and P,(x) >O. Notice countably additive, finite measure A, on (S, 4) defined by E.,(E) =Px(I,) for by p, the Halmosevery E E l. Since the family {i_: x E X + + } is dominated Savage theorem [see Halmos and Savage (1949, Lemma 7)] implies there exists a countable equivalent subfamily. say {i.x, : 15 i < cc ). Thus, A,(E) = 0 for every x E X + + if and only if i,,,(E) = 0 for every 15 i < CC. We now define a linear functional P:X-+R by the formula P(x)= positive linear xi”= 1 2_‘P,,(x) for every XEX. Clearly, P is a well-defined, functional extending p (see the last paragraph of the proof of Theorem 5). We claim that P is strictly positive and, hence, a valuation operator. Indeed, the Riesz representation theorem implies that for every x E X+ + the linear functional P,: X--+R has the representation P,(y) = l yzx d/l = 1 y d2,
for some z,~L~(S,s,p), where l/p+l/q=l. Thus. P,(x)=jxd&>O. The definition of the Lebesgue integral now implies there exists E,E < and a positive number fi, such that &(E,) >O and xz6,1Ex. It follows that &(I?,) > 0 for some 15 i < a, and that P,,(x) = s x d&, 2 j fi,lEx di.,, = 6,i.,r(E,) >O. Therefore, P(x) ~2-‘PxI(x) ~0. 0 The proof to the above theorem has a measure-theoretic flavor in contrast to the purely geometric flavor of the proof to Theorem 5. It utilizes the Halmos-Savage theorem, which asserts that a family of countably additive measures is dominated by a a-finite countably additive measure if and only if there exists a countable equivalent subfamily [Halmos and Savage (1949)]. The fact that a a-finite countably additive measure is equivalent to a countably additive probability measure is well-known and easy to verify. Although n is usually assumed to be a probability measure in financial modelling, we have elected to express Theorem 7 in terms of a a-finite measure in order to make the method of proof more transparent. Neither Theorem 5 nor Theorem 7 effectively handle the inseparable Banach space X=L”(S, 5.7~) endowed with the essential supremum norm topology. Nevertheless, we might impose the weak-star topology on L”(S, 5, rc), exploiting the fact that L”(S, 5, n) is the dual of L’(S, 5, rr). Then a weak-star continuous linear functional P : X+ R has the representation P(x) =j xy dn for some fixed y E L’(S, r, rr) [e.g. see Kelly and Namioka (1963,
478
S.A. Clark, Valuation problem in arbitrage price theory
ch. 5)]. The method of proof for Theorem 7 is easily adopted situation, again showing that A.5 characterizes the existence operator.
to handle this of a valuation
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