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Arbitrage Theory with State-Price DeflatorsSalvador Cruz Rambaud aa Departamento de Economía y Empresa , University of Almería , SpainPublished online: 01 Aug 2013.

To cite this article: Salvador Cruz Rambaud (2013) Arbitrage Theory with State-Price Deflators, Stochastic Models, 29:3,306-327, DOI: 10.1080/15326349.2013.808902

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Stochastic Models, 29:306–327, 2013Copyright © Taylor & Francis Group, LLCISSN: 1532-6349 print/1532-4214 onlineDOI: 10.1080/15326349.2013.808902

ARBITRAGE THEORY WITH STATE-PRICE DEFLATORS

Salvador Cruz Rambaud

Departamento de Economía y Empresa, University of Almería, Spain

� The aim of this article is to develop some mathematical results for the existence of asset pricebubbles, which is consistent with the martingale analysis of financial markets. More specifically,this article studies the relation between the divergence of the sum of dividend-price ratios and theabsence of bubbles. The framework used describes a contemporary model of a securities marketbuilt in an environment with both uncertainty and a countable number of trading dates. Themarket allows trading of a finite number of corporate securities. In this context, no arbitragecondition is characterized by the existence of a state-price deflator across an infinite time horizon.

Keywords Arbitrage; Bubble; Convergence; Martingale; State-price deflator.

Mathematics Subject Classification Primary 60G42; Secondary 91G30.

1. INTRODUCTION

The existence of bubbles in an economy is an interesting issue inmathematical finance because, in times of financial crisis, price bubblesappear. There are two classes of bubbles: rational and speculative. From arational point of view, a price bubble arises when the price of a securityor an asset is higher than its fundamental value. So, the first task is tosatisfactorily establish the concept of the fundamental value of an asset.Usually, it is the present value of the future dividend payments. Accordingto Werner,[37] “price bubbles cannot exist in equilibrium in the standarddynamic asset pricing model as long as assets are in strictly positive supplyand the present value of total future resources over the infinite time isfinite.” Inside this class of bubbles, there are rational agents who believethat the money supply is exogenously controlled by a central bank, whileothers believe that the money supply is endogenously given by the banking

Received September 2012; Accepted May 2013Address correspondence to Salvador Cruz Rambaud, Departamento de Economía y Empresa,

University of Almería, Spain; E-mail: scruz@ual.es

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Arbitrage Theory 307

sector, where bubbles arise due to the existence of borrowing constraints.Here investors have to be prevented from taking part in Ponzi schemes.

On the other hand, Harrison and Kreps[18] were the first in exploringthe possibility of speculative bubbles in equilibrium under assumptions ofsales constraints and heterogeneous beliefs. Thus, bubbles can arise dueto the agent’s marginal valuation of future dividends or be based on abehavioral foundation like some model with adaptive learning.

This article introduces a martingale theory of asset bubbles forfinancial markets. Following Gilles and LeRoy,[17] we can interpret a bubbleas a payoff at infinity, since the countably additive component of marketvalue can be regarded as the present value of all payoffs attained at finitedates. But we model a bubble as a random variable instead of a charge.

The results are presented in the abstract framework of a discrete-timeprice-dividend model with uncertainty over an infinite time horizon. Onejustification for this approach is the prevalence of martingale methodsin mathematical finance. Thus, this model is capable of describing arich variety of financial instruments, including long-lived and short-livedsecurities, dividend and coupon payments, and spot and forward markets.The essential requirement is a frictionless market for which a risklessasset is always available. The Martingale Convergence Theorem providesa virtually irrefutable argument for the existence of bubbles. In this way,a bubble is identified as the gap between the fundamental value and themarket value of capital stock.

With respect to the source of an asset bubble, Malinvaud[24] regardedthe source of economic inefficiency as excess capital accumulation atinfinity. Thus, the stipulation that the bubble is zero can be identified asa financial version of Malinvaud’s Transversality Condition. Closely relatedto this condition, Clark[7] showed that the bubble at the current date onan optimal production program is the limit of the conditional expectationof future capital value as time tends toward infinity.

In a recent article, Montrucchio[26] studies this issue not from the pointof view of infinite-horizon stochastic economies efficiency, but with respectto the existence of bubbles for long-lived assets. He showed that suchan issue is closely related to the approach proposed by Cass. State-pricedeflators are endogenous in equilibrium models of dynamic asset markets.There usually are some restrictions (e.g., transversality conditions) onwhat state-price deflators may arise in equilibrium. On the other hand,dynamic models of asset markets cannot be fully frictionless. They haveto involve some constraint on portfolio holdings, for otherwise investorswould engage in Ponzi schemes. The recent literature on price bubblesshows that the form of portfolio constraints plays a crucial role in theexistence of price bubbles (see Kocherlakota[22] and Araujo, Páscoa, andTorres-Martínez[2]).

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The literature on rational bubbles arising in sequential economiesis rather large (see, for example, Diba and Grossman[12]; Santos andWoodford[31]; or Montrucchio and Privileggi[27]). But, in this article,like Montrucchio, our aim will be to investigate the behavior of aconcrete series in connection with the existence of price bubbles.Thus, inspired in Santos and Woodford,[31] we will use the distinctionbetween ambiguous and non-ambiguous bubbles. The consideration of thestochastic fundamental value of an asset (see Back and Pliska[4]) will giverise to the existence of a bubble component for some state prices and notfor some others. Montrucchio’s article introduces the following short-runpricing equilibria:

atpt = E t [at+1(pt+1 + dt+1)],where pt is the spot price vector of an asset, dt is its dividend vector, and atdenotes a state-price process. He also studies the relation existing betweenarbitrage-free prices, verifying this equation, and the behavior of the series:

∞∑t=0

|dt ||pt | ,

where | · | is the vector l1 norm. Several results showed the closeconnection between the divergence of this series and the absence ofbubbles. The main one is Theorem 2, which establishes the non-existence,unambiguously, of pricing bubbles, provided the series goes to infinityuniformly. Moreover, he presents other results that deliver weaker versionsof this basic result. On the other hand, Theorem 8 is the converse ofTheorem 2, and it establishes that the series diverges uniformly under anassumption of impatience of some long-lived agent.

In a different way, we will use the usual state-price deflators of finance,and so the present article aims to develop some mathematical results forthe existence of asset price bubbles by introducing a state-price deflatorand its characteristics. It is well known that the existence of a state-price deflator is equivalent to No Arbitrage (NA) condition. Additionally,Kabanov and Kramkov[21] demonstrated that NA at date t is equivalent tothe existence of an equivalent martingale measure Q (t), i.e., a probabilitymeasure Q (t) on (�,�t+1) such that Q (t) is equivalent to P and that

EQ (t)(zit+1 + dit+1 |�t) = zt �

If dQ (t)dP is the associated Radon-Nikodym derivative, then

�t+1 = dQ (t)dP

�t ,

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Arbitrage Theory 309

where �0 = 1, defines a state-price deflator. In this way, Dalang, Morton,and Willinger[10] demonstrate that NA is equivalent to the existenceof an equivalent martingale measure Q across a finite time horizon.Nevertheless, this result does not always hold over an infinite time horizon.

The existence of an equivalent measure Q across an infinite timehorizon

EQ (zit+1 + dit+1 |�t) = zt

has been characterized by Schachermayer[32] with the condition called NoFree Lunches with Bounded Risk (NFLBR). Nevertheless, the conversetheorem does not hold, since Back and Pliska[4] provided the firstknown example of an infinite time horizon market for which NA holds,but for which there does not exist an equivalent martingale measure.Jarrow, Protter, and Shimbo[20] imposed a condition weaker than NFLBR,the No Free Lunch with Vanishing Risk (NFLVR) (see Delbaen andSchachermayer[11]) demonstrating that this hypothesis is equivalent tothe existence of an Equivalent Local Martingale Measure (ELMM) (FirstFundamental Theorem of Asset Pricing) in a continuous securities market.Moreover, the complete market condition guarantees that the ELMM isunique (Second Fundamental Theorem of Asset Pricing).

Nevertheless, in the present article, although the notion of a riskneutral measure is not central, we will not renounce to characterizethe existence of an equivalent martingale measure Q by imposing someconditions to the state-price deflator, in a discrete securities market withinfinite time horizon, which seems more natural from an economicsperspective. These conditions refer to uniform integrability and theclosure property of the state-price deflator.

Moreover, another argument to continue with the characterization ofan equivalent martingale measure is that, according to Jarrow and Protter,[19] “it is commonly believed that in economic theory both discrete andcontinuous time models are equivalent in the sense that one can always beused to approximate the other, or equivalently, any economic phenomenapresent in one is also present in the other. Unfortunately, [� � � ] (strict) localmartingales [� � � ] exist in continuous time models, but not in discrete timemodels.”

The organization of this article is the following: Section 2 presents thestochastic model in which No Arbitrage (NA) condition will be embodiedand introduces the concept of a state-price deflator whose existence (notuniqueness) is a necessary and sufficient condition for NA. In Section 3,we introduce the stochastic version of the Fundamental Equation of AssetPricing showing the possible existence of a price bubble. Moreover, thissection helps in understanding whether and how price bubbles can arise.Here we demonstrate some necessary and sufficient conditions for the

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non-existence of a price bubble with respect to any state-price deflator.In Section 4, we present other necessary and sufficient conditions basedon the concept of a closed martingale and the convergence in mean ofrandom variables. Finally, Section 5 summarizes and concludes.

2. SECURITIES MARKET MODEL AND STATE-PRICE DEFLATORS

Let us consider a complete probability space (�,� ,P ) with state space�, �-algebra � , and probability measure P . There are a countable numberof trading dates t indexed by the non-negative integers. The availableinformation is represented by a filtration ��t�

∞t=0 such that �0 = �∅,��

and that �t is a refinement of �t−1 for every date t ≥ 0, i.e., �t is acollection of partitions of the events in �t−1. Moreover, � = �

( ⋃∞t=0 �t

).

The information �t is accessible to all market agents at precisely date t(see, for example, Clark and Srinivasan[8]).

Let Xt denote the vector space of all �t -measurable random variablesfor every date t ≥ 0. A sequence of random variables �xt�∞

t=0 is adaptedprovided that xt ∈ Xt for every t ≥ 0. The market consists of a finitenumber n of corporate securities paying dividends. Let the finite set Iindex the corporate securities. The ith corporate security is represented bythe (ex dividend) adapted price process �zit �

∞t=0 and the adapted dividend

process �dit �

∞t=0 with di

0 = 0. Both the price and the dividend processes for acorporate security are positive, by virtue of limited liability, and adapted tothe filtration. Let zt := [z1t , z2t , � � � , znt ] and dt := [d1

t , d2t , � � � , d

nt ], being zit and

dit the price and the dividend corresponding to the ith corporate security

at time t , respectively. All equalities and inequalities are presumed to holdalmost surely in the following development.

A state price is the price of one dollar in each state of the world(for an introduction on state-price deflators, see Fisher[15] or Focardi andFabozzi[16]). On the other hand, the state-price deflator equals the stateprice divided by the probability of the state; in other words, it is the valueof a unit payout in a given state conditional on the occurrence of the state.Thus, the value of an asset is:

asset value =∑s∈S

payouts × state prices

=∑s∈S

payouts × (state-price deflators × probabilitys)

=∑s∈S

(payouts × state-price deflators) × probabilitys , (1)

where S is the set of states of the world at all future times.

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Arbitrage Theory 311

In this article, the state-price deflator will be considered as a stochasticprocess that evolves through time. The dynamics of the state-price deflatorare intimately related to the interest rate and the price of risk, the twocomponents of the price system (for asset pricing). The interest ratecharacterizes the expected change in the state prices, while the price ofrisk characterizes the volatility of states prices.

Equation (1) reveals the relation between the absence of arbitrageopportunities1 and the existence of the state-price deflator.

It can be shown that if all state prices are positive, then there areno arbitrage opportunities. Since the probabilities of the states are allpositive (by definition), the absence of arbitrage opportunities impliesthe existence of a (strictly positive) state-price deflator. The equivalencebetween NA and the existence of a state-price deflator can be found inSantos and Woodford.[31]

Definition 1. A state-price deflator (SPD) is a real-valued, strictly positivemartingale ��t�

∞t=0 such that �0 = 1 and

EP [�t+1(zit+1 + dit+1) |�t ] = �t zit , (2)

for every corporate security i ∈ I and date 0 ≤ t < ∞. In vectorial notation

EP [�t+1(zt+1 + dt+1) |�t ] = �t zt , (3)

for every date 0 ≤ t < ∞.

The process ��t�∞t=0 has a variety of other different names: deflator,

pricing kernel, and stochastic discounting function. State-price deflators providea useful theoretical framework when we are working in a multicurrencysetting. In such a setting, there is a different risk-neutral measure Qi foreach currency i . In contrast, the measure used in pricing with state-pricedeflators is unaffected by the base currency.

In order to justify some of the computations of conditionalexpectations, we will assume that the martingales composing the SPDare bounded away from zero everywhere. Thus, the reciprocal of thesecurity prices is in L-infinity. In what follows, we will consider both thefinite-state case and the case of infinite states. But in order to guaranteethat the various conditional expectations operations are well defined, wewill assume that all prices and dividends are integrable.

1In Section 5, a stronger concept of absence of arbitrage will be used: the NFLBR (No FreeLunches with Bounded Risk) condition.

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3. THE STOCHASTIC FUNDAMENTAL EQUATION OFASSET PRICING

Suppose one share of the ith corporate security is bought at date T andsold at a later date t , generating the revenue stream (di

T+1, diT+2, � � � , z

it +

dit ). An application of the Law of Iterated Expectations (e.g., Elliott[14];

Clark[6]) shows that

EP

(�t zit +

t∑s=T+1

�sd is |�T

)= �T ziT (4)

for every t > T . In particular,

EP

(�t zit +

t∑s=1

�sd is

)= zi0� (5)

If we regard the left-hand side of Equation (5) as a measure of thefundamental value of the corporate security, then we have constructed astochastic version of the Fundamental Equation of Asset Pricing. Lettingthe time horizon t converge to infinity, we further obtain

limt→∞

EP

(�t zit

) +∞∑s=1

EP

(�sd i

s

) = zi0� (6)

Equation (4) shows that ��t�∞t=0 allows us to obtain the prices at instant

T of a corporate security at any instant t > T using the factors �t�T

from tto T :

EP

(�t

�Tzit +

t∑s=T+1

�s

�Td is |�T

)= ziT � (7)

In particular, Equation (5) proves that ��t�∞t=0 gives us the prices at time

0 of a corporate security at any time t . In Munk,[28] a sequence of factors(�t

�t−1

)from t to t − 1 is deduced from ��t�

∞t=0. Nevertheless, Equation (7)

generalizes this construction.The following construction is inspired in Montrucchio,[26] Montrucchio

and Privileggi,[27] Sethi et al.[33–35] The aim of the rest of this section is topresent some sufficient conditions whereby the price bubble vanishes withrespect to a specific state-price deflator for an infinite time horizon.

In effect, by definition of a SPD,

EP [�t(zit + dit ) |�t−1] = �t−1zit−1,

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Arbitrage Theory 313

for every date 0 < t < ∞, from where:

EP

[�t zit

(1 + di

t

zit

)|�t−1

]= �t−1zit−1�

Multiplying both sides of the former equality by∏t−1

k=0

(1 + dik

zik

):

EP

[�t zit

t∏k=0

(1 + di

k

zik

)|�t−1

]= �t−1zit−1

t−1∏k=0

(1 + di

k

zik

)� (8)

Thus,{�t zit

∏tk=0

(1 + dik

zik

)}∞t=0

is a martingale. So, by recursively applyingthe Law of Iterated Expectations (see Tucker,[36] p. 212 or Malliaris andBrock,[25] p. 15):

EP

[�t zit

t∏k=0

(1 + di

k

zik

)]= zi0� (9)

In this section, we will see that the former product is intimately relatedto the sum of dividends divided by prices of an asset. Thus, this sum, beinginfinite, and another condition are sufficient to derive the absence of pricebubble for that asset.

In effect, denoting �it := �t zit∏t

k=0

(1 + dik

zik

), it is verified that

supt

EP (|�it |) = supt

EP (�it) = zi0 < +∞,

because, by Equation (9), all �it have the same mathematical expectation.Then, by the Martingale Convergence Theorem (Malliaris and Brock,[25]

p. 32 or Loève,[23] p. 393), �it → �i < +∞ almost surely, where �i is arandom variable such that

EP (|�i |) = EP (�i) < +∞� (10)

Let us denote by A� the maximal set verifying that �i > 0 on A�. Observethat eventually A� can be the empty set.

Montrucchio[26] demonstrates that if the limit limt→∞∑t

k=0ditzit

= +∞holds uniformly, then the price process �zit �

∞t=0 and the dividend process

�dit �

∞t=0 involve no bubble, that is, limt→∞ EP (�t zit ) = 0, for every state-price

deflator ��t�∞t=0, i.e., unambiguously. Later, when trying to find a weaker

condition, he assumes that the limit limt→∞∑t

k=0ditzit

= +∞ holds almostsurely. But he needs another condition to guarantee the implication and,in order to do this, he supposes that the random sequence ��t zit �

∞t=0

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is equicontinuous: for all � > 0, there is some = (�) > 0 such thatEP (1A�t zit ) ≤ � for all t and all A ∈ � with P (A) ≤ .

Montrucchio recognizes that the so-defined equicontinuity propertyis not so simple to check. So we are going to propose the definitionof a locally bounded random sequence, but first we need the followingconstruction of a semimetric in � .

This idea (construction) is inspired by Billingsley[5] (p. 31). For aprobability measure P on a �-field � , define:

d(A,B) = P (A � B), (11)

for A and B in � , being A � B the symmetric difference of A and B. Thisis a semimetric on � , because A = B may not follow from d(A,B) = 0.Nevertheless, identifying sets at distance 0 makes � into a metric space. Ineffect, if d is a semimetric, then the binary relation defined by

A ∼ B if d(A,B) = 0

is an equivalence relation, and d defines a metric d̂ on the set ofequivalence classes by

d̂([A], [B]) = d(A,B) (12)

(see Aliprantis and Border,[1] p. 71).Given a positive random sequence �Xt�

∞t=0, consider the following

family of real-valued functions:

ft : �/ ∼−→ �

[A] �→ ft([A])

defined by

ft([A]) = ‖1AXt‖∞ = inf�M > 0 : |1A()Xt()| ≤ M for P − almost all �,

where ‖ · ‖∞ is the essential sup norm and ‖1AXt‖∞ is the ‖ · ‖∞-norm ofthe P -measurable function 1AXt : � −→ �. Here the convention inf∅ = ∞applies (see Aliprantis and Border,[1] p. 462). This function is well defined,because [A] = [B] implies A ∼ B and P (A � B) = 0. Therefore, P [(A\B) ∪(B\A)] = 0 and so P (A\B) = P (B\A) = 0. Moreover, A = (A\B) ∪ (A ∩ B),and B = (B\A) ∪ (A ∩ B). As P (A\B) = 0 and P (B\A) = 0, A ∩ B applies.

Thus, we have defined a family of real-valued functions

ft : �/ ∼−→ ��

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Arbitrage Theory 315

�/ ∼ has the metric � and � has its usual metric. Now, we areinterested in the local boundedness of this family of functions whenconsidering the corresponding metric spaces:

ft : (�/ ∼,�) −→ (�, | · |)�Definition 2. A family of functions �ft�∞

t=0 is locally bounded provided thatfor every x0 ∈ X , there exists a neighborhood U (x0) and a positive numberM = M (x0) such that |ft(x)| ≤ M , for every x ∈ U (x0) and for every t .

More specifically, we are interested in the local boundedness of ourfamily at A = ∅. In effect, if �ft�∞

t=0 is locally bounded at A = ∅, applyingdefinition 2, there exists a neighborhood of ∅, U (∅), and a positivenumber M = M (∅) such that |ft(A)| ≤ M , for every A ∈ U (∅) and forevery t . But there exists a � > 0 such that �A ∈ �/d(∅,A) < �� ⊆ U (∅). Asd(∅,A) = P (A), this means that this condition is valid for all events with ameasure less than �. In summary,

Definition 3. A random sequence �Xt�∞t=0 is locally bounded if there exists

�0 > 0 and a positive number M such that |Xt | ≤ M , on every A withP (A) < �0 and for every t .

Observe that this definition is similar to that of locally boundedrandom sequence in Elliott[14] (p. 58).

Notice that, in general, when dealing with positive and integrablerandom variables, local boundedness implies equicontinuity. In effect, forall � > 0, we can take � = min��0, �

M �. Thus, for all A ∈ � with P (A) ≤ �,P (A) < �0 (and so Xt ≤ M ) and P (A) < �

M , whereby

EP (1AXt) =∫AXtdP ≤ M · P (A) < �� (13)

Therefore, in order to improve Theorem 2 in Montrucchio,[26] it isnecessary to introduce a weaker definition of locally bounded randomsequence.

Definition 3 (bis). A random sequence �Xt�∞t=0 is locally bounded if there

exists �0 > 0 such that for every A with P (A) < �0, �Xt�∞t=1 is almost surely

bounded on A, i.e., there is a scalar M (A) with |Xt | ≤ M (A) a.s. on A, forevery t .

Some remarks:

• First of all, the term local boundedness in the general theory of stochasticprocesses is reserved to describe processes for which there is a localizing

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sequence of stopping times such that each stopped process is almostsurely bounded.

• Under the assumption that the space is non-atomic, the concept oflocal boundedness coincides with boundedness. Indeed, pick m ∈ �such that 1/m < �0 and write � = ⋃m

i=1 �i , where P (�i) = 1/m for alli = 1, 2, � � � ,m. For each i = 1, 2, � � � ,m, pick Mi > 0 such that |Xt | ≤ Mi

holds almost surely on �i for all t ∈ �. Then, with M = maxi=1,2,���,m Mi ,we have |Xt | ≤ M almost surely for all t ∈ �, i.e., the process is bounded.

• Local boundedness does not imply equicontinuity and vice versa.

In these conditions, we can enunciate the following:

Theorem 1. Let ��t�∞t=0 be a state-price deflator. For each fixed asset i, if∑∞

t=1ditzit

= +∞ almost surely and ��t zit �∞t=0 is locally bounded, then the price process

�zit �∞t=0 and the dividend process �di

t �∞t=0 involve no bubble with respect to ��t�

∞t=0

(ambiguously).

Proof. As{�t zit

∏tk=0

(1 + dik

zik

)}∞t=0

is a martingale, the Martingale

Convergence Theorem (e.g., Malliaris and Brock,[25] p. 32) implies thatthis process almost surely converges to the random variable �i < +∞. Onthe other hand, in general it is verified that

1 +t∑

k=0

dik

zik≤

t∏k=0

(1 + di

k

zik

)� (14)

Taking into account that∑t

k=0dikzik

diverges almost surely, then∏t

k=0

(1 + dik

zik

)also diverges almost surely, whereby necessarily

�t zit → 0

almost surely. By Proposition 5 in Cruz Rambaud,[9] ��t zit �∞t=0 converges in

mean and so EP (�t zit ) → 0. Hence there is not price bubble. �

Observe that, in the proof of Theorem 1, the convergence in mean isjust a direct application of Fatou’s lemma for lim inf and reverse-Fatou’slemma for lim sup where we need ��t zit �

∞t=0 to be locally bounded. The

following definition is inspired in Aliprantis and Border[1] (p. 322).

Definition 4. A random sequence �Xt�∞t=0 converges in order (or is order

convergent) to X , denoted by Xto→ X , if there is a sequence �Yt�

∞t=0 such

that Yt ↓ 0 (pointwise convergence) and |Xt − X | ≤ Yt , a.s., for every t .

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Corollary 1. Let ��t�∞t=0 be a state-price deflator. For each fixed asset i, if∑∞

t=1ditzit

= +∞ almost surely and ��t zit �∞t=0 is order convergent, then the price

process �zit �∞t=0 and the dividend process �di

t �∞t=0 involve no bubble with respect to

��t�∞t=0 (ambiguously).

Proof. It is a direct consequence of the Lebesgue’s DominatedConvergence Theorem (Billingsley,[5] p. 213), because |�t zit | ≤ Yt < Y1, a.s.,for every t (recall that Y1 is integrable). �

Corollary 2. Let ��t�∞t=0 be a state-price deflator. For each fixed asset i, if∑∞

t=1ditzit

= +∞ almost surely and ��t zit �∞t=0 is decreasing, then the price process

�zit �∞t=0 and the dividend process �di

t �∞t=0 involve no bubble with respect to ��t�

∞t=0

(ambiguously).

The main difference of our approach with respect to Montrucchio’s isthat this author allows both the price and the dividend to be equal to 0on a non-null set. Nevertheless, our approach assumes that both randomvariables are positive (almost surely).

Next, we are going to look for other sufficient conditions to guaranteethe non-existence of a price bubble, but first we will introduce thefollowing:

Definition 5. Suppose �Xt�∞t=0 is a positive supermartingale. Then �Xt�

∞t=0

is said to be a potential if limt→∞ EP (Xt) = 0.The following Lemma is inspired in the Theorem of Riesz

Decomposition and is a version of the Krickeberg Decomposition Theorem(Resnick,[29] pp. 386–387).

Lemma 1. Let �Xt�∞t=0 be a positive submartingale convergent in L1(�,� ,P )

to X∞ a.s. positive. Then �Xt�∞t=0 can be decomposed as the difference of a positive

martingale �Yt�∞t=0 with a positive limit, and a potential �Zt�

∞t=0.

Proof. It suffices to construct Yt := EP (X∞ |�t) and Zt := Yt − Xt . Thus,the result immediately follows. �

The following Theorem is inspired in Ash.[3]

Theorem 2. Let ��t�∞t=0 be a state-price deflator such that, for every p > 1,

EP (�t+pzit+p |�t) ≤ �t+p−1zit+p−1� (15)

In these conditions, for each fixed asset i, if∑∞

t=1ditzit

= +∞ almost surely, thenthe price process �zit �

∞t=0 and the dividend process �di

t �∞t=0 involve no bubble with

respect to ��t�∞t=0 (ambiguously).

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Proof. Obviously, Xt := ∑tk=1 �kd

ik is a submartingale, because

EP (�t+1dit+1 |�t) ≥ 0 implies EP (

∑t+1k=1 �kd

ik |�t) ≥ ∑t

k=1 �kdik . So, we can apply

Lemma 1 and construct the martingale Yt = �t zit + ∑tk=1 �kd

ik and the

potential Zt = �t zit . Now, it suffices to take t = 0 in Equation (15) and todeduce that EP (�pzip) ≤ �p−1zip−1 → 0 a.s. as p → ∞. �

In general, the converse of Theorem 1 is not true. More specifically,the absence of bubbles does not imply the almost surely divergence of theseries

∑∞t=1

ditzit. But we can enunciate the following:

Theorem 3. Let ��t�∞t=0 be a state-price deflator verifying that P (A�) = 12. For

each fixed asset i, if the price process �zit �∞t=0 and the dividend process �di

t �∞t=0 involve

no bubble with respect to ��t�∞t=0, then

∑∞t=1

ditzit

= +∞.

Proof. If ��t�∞t=0 is a state-price deflator, the deflated price plus cumulative

deflated dividend {�t zit +

t∑h=1

�hd ih

}∞

t=0

is a martingale. So, the Martingale Convergence Theorem (Elliott,[14]

pp. 20–22) implies that this process almost surely converges to the randomvariable

limt→∞

�t zit +∞∑h=1

�hd ih �

Thus, the limiting deflated share price limt→∞ �t zit definitely exists. So,we have a sequence ��t zit �

∞t=1 of non-negative random variables converging

to limt→∞ �t zit finite almost surely. Thus, by Fatou’s Lemma (Doob,[13]

p. 629) and the fact that limt→∞ �t zit ≥ 0,

EP

(limt→∞

�t zit)

≤ limt→∞

EP

(�t zit

)�

If there is not a price bubble, then limt→∞ EP (�t zit ) = 0 and thenEP (limt→∞ �t zit ) = 0 and limt→∞ �t zit = 0, almost surely. As

limt→∞

�t zit ·∞∏k=0

(1 + di

k

zik

)> 0

holds almost surely, we can state that∏∞

k=0

(1 + dik

zik

) = +∞, almost surely. �

2Recall that A� was defined at the beginning of Section 3.

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On the other hand, for every ∈ �, such that P (��) = 0, considerthe set ��c and let M be the intersection of all of them:

M =⋂∈�

P (��)=0

��c � (16)

Corollary 3. Let ��t�∞t=0 be a state-price deflator verifying that A� ⊇ M �= ∅. For

each fixed asset i, if the price process �zit �∞t=0 and the dividend process �di

t �∞t=0 involve

no bubble with respect to ��t�∞t=0, then

∑∞t=1

ditzit

= +∞.

Corollary 4. Let ��t�∞t=0 be a state-price deflator. For each fixed asset i, if the

price process �zit �∞t=0 and the dividend process �di

t �∞t=0 involve no bubble with respect

to ��t�∞t=0, then we can find a new complete probability space (�,�,P ) where∑∞

t=1ditzit

= +∞.

Proof. It suffices to take � = �F ∩ M , F ∈ � �. �

Theorem 4. Let ��t�∞t=0 be a state-price deflator. Assume that, for each fixed asset

i, the price process �zit �∞t=0 and the dividend process �di

t �∞t=0 verify the following

condition

z∞ + d∞ > 0�

Then there is not a price bubble with respect to ��t�∞t=0 if and only if �∞ = 0

almost surely.

Proof. If there is not a price bubble with respect to ��t�∞t=0, then, by

definition, limt→∞ EP (�t zt) = 0. By Fatou’s Lemma,

EP (�∞z∞) ≤ limt→∞

EP (�t zt),

whereby �∞z∞ = 0 almost surely.If �∞ > 0 on a non-null set A, then z∞ = 0 on A, which implies that

d∞ > 0 on A and that �∞d∞ > 0 on A. This is a contradiction because, byEquation (6),

∑∞t=0 �t dt converges almost surely and so necessarily �∞d∞ =

0 almost surely.The proof of the converse implication is analogous. �

The following result provides a case in which P (A�) = 1. Denote theratio dik

zikby r ik :

r ik := dik

zik(17)

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and represent by r it the arithmetic mean of r i1, ri2, � � � , r

it . In these

conditions, we can enunciate the following:

Proposition 1. If r it ≤ r it and limt→∞ t�t d it > 0, then

limt→∞

[�t zit

t∏k=1

(1 + di

k

zik

)]> 0, a.s.

Proof.

limt→∞

[�t zit

t∏k=1

(1 + di

k

zik

)]≥ lim

t→∞

[�t zit

(1 +

t∑k=1

dik

zik

)]

= limt→∞

�t zit + limt→∞

�t zit

t∑k=1

dik

zik

= limt→∞

�t zit + limt→∞

�t zit t rit

≥ limt→∞

�t zit + limt→∞

�t zit td it

zit= lim

t→∞�t zit + lim

t→∞t�t d i

t > 0�

So the proposition is demonstrated. �

We can summarize all the results in Figure 1.

FIGURE 1 Scheme with all theorems. LB means Local boundedness condition; F-L, Fatou’sLemma; T1, Theorem 1; and T3, Theorem 3.

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4. CLOSED SPD AND CONVERGENCE IN MEAN

In the following paragraph, we are going to study the conditionsof existence of the limiting share price (zi∞). In effect, as indicated inthe proof of Theorem 3, if ��t�

∞t=0 is a state-price deflator, the limiting

deflated share price limt→∞ �t zit definitely exists. Moreover, if ��t�∞t=0 is itself

a martingale, the Martingale Convergence Theorem also implies that

�∞ := limt→∞

�t (18)

is a non-negative random variable. Taking into account that

zi∞ := limt→∞

zit = limt→∞

�t zit�t

= limt→∞ �t zit�∞

, (19)

we can deduce that, if �∞ > 0, then zi∞ exists.

Definition 6. A martingale ��t�∞t=0 is closed provided that EP (�∞ |�t) = �t

at every date 0 ≤ t < ∞.

Let �St�∞t=0 be a stochastic process of �t -measurable �n -valued random

variables. Following Schachermayer,[32] define K0 to be the vector space ofeasy stochastic integrals

K0 = span�(h(), St() − Ss())�,

where (·, ·) denotes the inner product in �n , s and t run the pairs in �0

with s < t and h an �n -valued �s -measurable. Note that K0 is a subspaceof L0(�,� ,P ), which is the space of � -measurable, real-valued functions.Denote by C0 the convex cone (K0 − L0

+(�,� ,P )), i.e., those elements ofL0 that are dominated by some f ∈ K0. Denote by C the convex cone C0 ∩L∞. Note that C consists of those elements of L∞ that are dominated bysome f ∈ K0. We shall denote by C̃ the set of all limits of �∗-convergentsequences in C . Clearly, C̃ is a convex cone in L∞.

Definition 7. A stochastic process �St�∞t=0 satisfies NFLBR (no free lunches

with bounded risk) condition if

C̃ ∩ L∞+ = �0�� (20)

Theorem 5. For a state-price deflator ��t�∞t=0, the following four conditions are

equivalent:

(i) ��t�∞t=0 is uniformly integrable.

(ii) ��t�∞t=0 is closed and �∞ > 0.

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(iii) ��t�∞t=0 verifies that �∞ is the Radon-Nikodym derivative of an equivalent

martingale measure Q with respect to P such that{zit +

t∑h=1

dit

}∞

t=0

is a martingale with respect to (�,� ,Q ).(iv)

{�t zit + ∑t

h=1 �hdih

}∞t=0

verifies NFLBR condition.

Proof. (i) ⇒ (ii). It is a consequence of the uniform integrability of��t�

∞t=0 as a martingale and the closure property (see Elliott,[14] pp. 20–22)

EP (�∞ |�t) = �t

at every date 0 ≤ t < ∞. Moreover, under the hypothesis, �∞ > 0 if andonly if each �t > 0. Thus, condition (ii) follows.

(ii) ⇒ (iii). In effect, if ��t�∞t=0 is closed, the state-price deflator

condition and the conditional expectation calculation properties imply thefollowing chain of equalities:

EP

[�t(zit + di

t ) |�t−1

] = �t−1zit−1,

EP

[EP (�∞ |�t)(zit + di

t ) |�t−1

] = EP (�∞ |�t−1)zit−1,

EP

[�∞(zit + di

t ) |�t−1

] = EP (�∞zit−1 |�t−1),

EP

[�∞(�zit + di

t ) |�t−1

] = 0,

being �zit := zit − zit−1.From the closure condition of ��t�

∞t=0 again and the Law of Iterated

Expectations (Elliott,[14] p. 4–5), it can be deduced that EP (�∞) = 1. As,moreover, �∞ > 0, it can be deduced that �∞ is the Radon-Nikodymderivative dQ /dP for a probability measure Q equivalent to P associatedwith ��t�

∞t=0. Therefore, Q is an equivalent martingale measure, i.e., an

equivalent probability measure such that

EQ

[�zit + di

t |�t−1

] = 0, (21)

for every i ∈ I and for every 1 ≤ t < ∞.

(iii) ⇒ (i). Assume that there is an equivalent martingale measure.Let Q be the martingale measure equivalent to P . Denote

�∞ := dQ /dP , (22)

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Arbitrage Theory 323

that is to say, �∞ is the Radon-Nikodym derivative of Q . Define thefollowing state-price deflator ��t�∞

t=0 associated with Q

�t := EP (�∞ |�t)�

This construction has the following features (see Clark[7]):

1. Since �∞ is �t -measurable, the process ��t�∞t=0 is adapted to the filtration.

2. As �∞ is a strictly positive random variable, the process ��t�∞t=0 is strictly

positive.3. A conditional expectations calculation shows that

EP (�t+1 |�t) = EP �E(�∞ |�t+1) |�t = EP (�∞ |�t) = �t

for every 0 ≤ t < ∞, so that the process ��t�∞t=0 is a martingale.

4. Finally, equation EQ [�zit + dit |�t−1] = 0 directly translates into equation

EP [�t(zit + dit ) |�t−1] = �t−1zit−1.

By the Theorem of Radon-Nikodym (see Malliaris and Brock,[25] p. 12),�∞ is integrable and so (see Lemma, Billingsley,[5] p. 491) �t = EP (�∞ |�t)is uniformly integrable.

(iii) ⇔ (iv). This equivalence is due to Theorem A inSchachermayer.[32]

�

Corollary 5. Let ��t�∞t=0 be a state-price deflator verifying any of the four

equivalent conditions of Theorem 5. If �zit �∞t=0 is uniformly integrable, then

limt→∞

EP (�t zit ) = EP ( limt→∞

�t zit )�

Proof. It is a direct consequence of the fact that ��t�∞t=0 and �zit �

∞t=0

are both uniformly integrable, and so ��t zit �∞t=0. Hence Theorem 16.13 in

Billingsley[5] (p. 220) applies. �

Corollary 6. Let ��t�∞t=0 be a state-price deflator verifying any of the four

equivalent conditions of Theorem 5. If �zit �∞t=0 is uniformly bounded, then

limt→∞

EP (�t zit ) = EP ( limt→∞

�t zit )�

Proof. It is a direct consequence of the fact that uniform boundednessimplies uniform integrability (see Theorem 5.3 in Billingsley,[5] p. 72).Then Corollary 5 applies. �

Observe that Corollary 5 and 6 are another version of VitaliConvergence Theorem (Rudin,[30] p. 134).

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Corollary 7. Let ��t�∞t=0 be a state-price deflator verifying any of the four

equivalent conditions of Theorem 5. If �zit �∞t=0 is uniformly integrable, for each fixed

asset i, then∑∞

t=1ditzit

= +∞ almost surely implies that the price process �zit �∞t=0 and

the dividend process �dit �

∞t=0 involve no bubble with respect to ��t�∞

t=0 (ambiguously).

Proof. By Corollary 5,

limt→∞

EP (�t zit ) = EP ( limt→∞

�t zit )�

Thus,∑∞

t=1ditzit

= +∞ almost surely implies ��t zit �∞t=0 → 0 and so, due to

the uniform integrability of the process ��t zit �∞t=0,

limt→∞

EP (�t zit ) = 0�

Therefore, there is not a price bubble. �

Theorem 6. With previous notations, if ��it�∞t=0 converges to �i ∈ L1, then

�i > 0. In this case, the price process �zit �∞t=0 and the dividend process �di

t �∞t=0

involve no bubble with respect to ��t�∞t=0 if and only if

∑∞t=1

ditzit

= +∞ a.s.

Proof. Let us see that the condition is necessary. As a result oflimt→∞

∑tk=0

dikzik

= +∞, it is verified that limt→∞ �t zit = 0. Moreover, from

Billingsley[5] (p. 220), ��it�∞t=0 is uniformly integrable. As �t zit < �it , this

implies the uniform integrability of ��it zit �

∞t=0. So ��it z

it �

∞t=0 converges in

mean:

EP

(limt→∞

�t zit)

= limt→∞

EP (�t zit )�

As limt→∞ �t zit = 0, due to the uniform integrability of ��t zit �∞t=0, it can

be deduced that limt→∞ EP (�t zit ) = 0 and so the price process �zit �∞t=0 and

the dividend process �dit �

∞t=0 involve no bubble with respect to ��t�

∞t=0.

Let us see that the condition is sufficient. In effect, assume that �i > 0in A�. As ��it�

∞t=0 converges in mean:

EP

(1A� · �i) = lim

t→∞EP (1A� · �it)�

Moreover, as E(1A� · Xt) ≤ E(1A�) · E(Xt), taking limits:

zi0 ≤ P (A�) · zi0,

then P (A�) = 1. So, �i > 0 (a.s.). Therefore, by Theorem 3,∑∞

t=1ditzit

= +∞a.s. �

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Arbitrage Theory 325

Corollary 8. With previous notations, if ��it�∞t=0 converges almost surely

uniformly to �i ∈ L1, then �i > 0. In this case, the price process �zit �∞t=0 and the

dividend process �dit �

∞t=0 involve no bubble with respect to ��t�

∞t=0 if and only if∑∞

t=1ditzit

= +∞ a.s.

Proof. It is a direct application of Theorem 6 and Proposition 5 in CruzRambaud.[9]

�

5. CONCLUSION

This article extends the results in Montrucchio.[26] It considers asequential asset trading model with a general probabilistic structureand a countable infinite time horizon. In this way, we have introducedthe concept of a state-price deflator whose existence (not uniqueness)is equivalent to require the No Arbitrage (NA) condition. Take intoaccount that this concept facilitates the study of NA condition becausedeflated capital values have various martingale properties that make thecomputation of present values easier. As a consequence, this articleprovides a tool to handle NA in a multicurrency scenario.

On the other hand, we have introduced the stochastic version of theFundamental Equation of Asset Pricing showing the possible existence of aprice bubble. This article studies the relation between arbitrage free prices(see definition 1) and the behavior of the series

∞∑t=0

dit

zit,

where ditzit

is the dividend-price ratio. Several results illustrate theconnection between the divergence of this series and the absence ofbubbles (e.g., see Theorems 1, 3, and 6). Moreover, we have demonstratedsome necessary and sufficient conditions for the non-existence of a pricebubble by requiring some conditions to the state-price deflator. Thus,we have to highlight the nice characterization of uniformly integrabilitypresented in Theorem 5 and the result obtained in Theorem 6.

An open question is the importance and consequences derived fromthe uniqueness of the SPD.

ACKNOWLEDGMENTS

This article has been partially supported by the project “Valoraciónde proyectos gubernamentales a largo plazo: obtención de la tasa socialde descuento,” reference: P09-SEJ-05404, Proyectos de Excelencia de la

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Junta de Andalucía and Fondos FEDER. I am very grateful for the valuablecomments and suggestions of an anonymous referee.

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