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Journal of Financial Economics

Journal of Financial Economics 115 (2015) 411–428

http://d0304-40

☆ A p“Arbitrathank HPierre CZhiguoseminarings atCowlesUniversSociety,North AUniversUniversLugano.Finance

n CorrE-m

rprieto@

journal homepage: www.elsevier.com/locate/jfec

Asset pricing with arbitrage activity$

Julien Hugonnier a,n, Rodolfo Prieto b

a École Polytechnique Fédérale de Lausanne and Swiss Finance Institute, 1015 Lausanne, Switzerlandb Department of Finance, School of Management, Boston University, United States

a r t i c l e i n f o

Article history:Received 6 November 2013Received in revised form10 April 2014Accepted 6 May 2014Available online 8 October 2014

JEL classification:D51D52G11G12

Keywords:Limits of arbitrageRational bubblesWealth constraintsExcess volatilityLeverage effect

x.doi.org/10.1016/j.jfineco.2014.10.0015X/& 2014 Elsevier B.V. All rights reserved.

revious version of this paper was circulageurs, bubbles and credit conditions”. For hengjie Ai, Rui Albuquerque, Harjoat Bhamollin Dufresne, Jérôme Detemple, BernardHe, Leonid Kogan, Mark Loewenstein, Marcparticipants at the 2013 American Mathem

Boston College, the Bank of Canada, Boston UConference on “General Equilibrium and itsity, the 2012 conference of the Financial Intethe 2014 European Finance Association Annumerican Meeting of the Econometric So

ity of Finance and Economics, the fourth Finidad Adolfo Ibáñez, University of GenevaFinancial support by the NCCR FinRisk PInstitute, and Boston University is gratefullyesponding author. Tel.: þ41 21 693 0114.ail addresses: Julien.Hugonnier@epfl.ch (J. Hubu.edu (R. Prieto).

a b s t r a c t

We study an economy populated by three groups of myopic agents: constrained agents subjectto a portfolio constraint that limits their risk taking, unconstrained agents subject to a standardnonnegative wealth constraint, and arbitrageurs with access to a credit facility. Such credit isvaluable as it allows arbitrageurs to exploit the limited arbitrage opportunities that emergeendogenously in reaction to the demand imbalance generated by the portfolio constraint. Themodel is solved in closed-form, and we show that, in contrast to existing models with frictionsand logarithmic agents, arbitrage activity has an impact on the price level and generates bothexcess volatility and the leverage effect. We show that these results are due to the fact thatarbitrageurs amplify fundamental shocks by levering up in good times and deleveraging inbad times.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Textbook asset pricing theory asserts that arbitrageopportunities cannot exist in a competitive market because

ted under the titleelpful comments wera, Pablo Castañeda,Dumas, Phil Dybvig,el Rindisbacher, andatical Society meet-niversity, the eighthApplications” at Yalermediation Researchal Meeting, the 2014ciety, SouthwesternanceUC Conference,, and University ofroject A5, the Swissacknowledged.

gonnier),

they would be instantly exploited, and thereby eliminated,by arbitrageurs. This basic principle is certainly valid forriskless arbitrage opportunities defined as trades thatrequire no initial investment and whose value can onlygrow over time.1 However, there is no reason to believe thatit should hold for risky arbitrage opportunities, such asconvergence trades, that guarantee a sure profit at somefuture date but require capital to fund potential losses atinterim dates.2 Indeed, the fact that arbitrageurs havelimited capital and are subject to solvency requirements

1 Such an opportunity arises for example when two assets that carrythe same exposure to risk offer different returns. See Gromb and Vayanos(2002) for a model in which such arbitrage opportunities arise due tomarket segmentation, and Basak and Croitoru (2000) for a model inwhich they arise due to the fact that securities are subject to differentmargin constraints.

2 Examples of such arbitrages include mispricing in equity carve-outs(Lamont and Thaler, 2003a,b), dual class shares (Lamont and Thaler, 2003a)and the simultaneous trading of shares from Siamese twin conglomeratessuch as Royal Dutch and Shell. See Rosenthal and Young (1990), Lamont and

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428412

limits their ability to benefit from such risky arbitrageopportunities and implies that they could subsist in equili-brium. In such cases, the trading activity of arbitrageurswould not suffice to close the arbitrage opportunities butwill nonetheless have an impact on the equilibrium, and thegoal of this paper is to investigate the effect of this riskyarbitrage activity on asset prices, volatilities, risk sharing andwelfare.

To address these issues a general equilibrium modelmust be constructed in which risky arbitrage opportunitiesexist in the first place. We achieve this by considering amodel of an exchange economy similar to those of Basakand Cuoco (1998) and Hugonnier (2012). We start from acontinuous-time model that includes a riskless asset inzero net supply, a dividend-paying risky asset in positivesupply and two groups of agents with logarithmic prefer-ences. Agents in both groups are subject to a standardnonnegativity constraint on wealth.3 But, while agents inthe first group are unconstrained in their portfolio choice,we assume that agents in the second group are subject to aportfolio constraint that limits their risk taking and,thereby, tilts their demand toward the riskless asset. Thisportfolio constraint generates excess demand for theriskless asset and captures in a simple way the globalimbalance phenomenon pointed out by Caballero (2006),Caballero, Farhi, and Gourinchas (2008) and Caballero andKrishnamurthy (2009), among others. This excess demandnaturally implies that the interest rate decreases and themarket price of risk increases compared with a frictionlesseconomy. But it also implies that the stock and the risklessasset are overvalued in that their equilibrium prices eachinclude a strictly positive bubble.4 The intuition is thateven though agents of both groups are price takers, thepresence of constrained agents places an implicit liquidityprovision constraint on unconstrained agents through themarket clearing conditions: At times when the portfolioconstraint binds, unconstrained agents have to hold thesecurities that constrained agents cannot, and this iswhere the mispricing finds its origin. Bubbles arise toincite unconstrained agents to provide a sufficient amountof liquidity, and they persist in equilibrium because thenonnegative wealth constraint prevents them from indefi-nitely scaling their positions.

To study the impact of arbitrage activity on equilibriumoutcomes, we then introduce a third group of agents thatwe refer to as arbitrageurs. These agents have logarithmic

(footnote continued)Thaler (2003a,b), Ashcraft, Gârleanu, and Pedersen (2010), and Gârleanu andPedersen (2011).

3 Nonnegativity constraints on wealth were originally proposed byDybvig and Huang (1988) as a realistic mechanism to preclude doublingstrategies. They are widely used, and usually considered innocuous, incontinuous-time models but are also introduced in discrete-time,infinite-horizon models. See, for example, Kocherlakota (1992) andMagill and Quinzii (1994), among others.

4 The bubble on the price of a security is the difference between themarket price of the security and its fundamental value defined as theminimal amount of capital that an unconstrained agent needs to hold toreplicate the cash flows of the security while maintaining nonnegativewealth. See Santos and Woodford (1997), Loewenstein and Willard(2000), and Hugonnier (2012), and Section 2.5 for a precise definitionand a discussion of the basic properties of asset pricing bubbles.

utility and are unconstrained in their portfolio choice, butthey differ from unconstrained agents along two impor-tant dimensions. First, these agents initially hold no capitaland thus would be able to consume only if they can exploitthe risky arbitrage opportunities that arise due to thepresence of constrained agents. Second, these agents haveaccess to a credit facility that enhances their tradingopportunities by allowing them to weather transitoryperiods of negative wealth. This facility should be thoughtof as a reduced-form for various types of uncollateralizedcredit such as unsecured financial commercial paper (see,e.g., Kacperczyk and Schnabl, 2009; Adrian, Kimbrough,and Marchioni, 2010), implicit lines of credit (see, e.g., Sufi,2009), or loan guarantees.5 To capture the fact that theavailability of arbitrage capital tends to be procyclical (see,e.g., Ang, Gorovyy, and Van Inwegen, 2011; Ben-David,Franzoni, and Moussawi, 2012) we assume that this creditfacility is proportional to the market portfolio.

We derive the unique equilibrium in closed form interms of aggregate consumption and an endogenous statevariable that measures the consumption share of con-strained agents. Importantly, because the portfolio con-straint acts as a partial hedge against bad fundamentalshocks, this state variable is negatively correlated withaggregate consumption. The analysis of the equilibriumsheds light on the disruptive role of arbitrageurs in theeconomy, and reveals that risky arbitrage activity results inan amplification of fundamental shocks that could helpexplain empirical regularities such as excess volatility andthe leverage effect. The main implications can be summar-ized as follows. First, we show that arbitrage activitybrings the equilibrium prices of both securities closer totheir fundamental values and simultaneously has a nega-tive impact on the equilibrium stock price level. The latterfinding is unique to our setting and stands in stark contrastto what happens in exchange economies with exogenousdividends and logarithmic preferences in which theimpact of frictions is entirely captured by the interest rateand market price of risk. See, for example, Detemple andMurthy (1997), Basak and Cuoco (1998) and Basak andCroitoru (2000).

Second, and related, we show that the trading ofarbitrageurs pushes the stock volatility above that of theunderlying dividend process. This excess volatility is self-generated within the system and comes from the fact thatarbitrageurs amplify fundamental shocks by optimallylevering up their positions in good times and deleveragingin bad times. The excess volatility component implied byour model is quantitatively significant and increases withboth the size of the arbitrageurs' credit facility and theconsumption share of constrained agents. Because thelatter is negatively correlated with aggregate consumptionour model implies that volatility tends to increase whenthe stock price falls. It follows that risky arbitrage activity

5 Commercial paper is among the largest source of short termfunding for both financial and non-financial institutions. For example,over the period 2010–2013 the average amount of commercial paperoutstanding was 1.04 trillion dollars, and about half of that amount isaccounted for by unsecured paper issued by financial institutions, seeFederal Reserve Economic Data (2014).

6 In the same spirit, Jurek and Yang (2007) and Liu and Timmermann(2013) study portfolio choice problems in which the value of thearbitrage opportunity follows a mean-reverting process so that theamount of time necessary to generate a profit is random.

7 The literature on speculative bubbles (see, e.g., Miller, 1977;Harrison and Kreps, 1978; Scheinkman and Xiong, 2003) uses a differentdefinition of the fundamental value that is not based on any cash flowreplication considerations and, therefore, cannot connect bubbles to theexistence of arbitrage opportunities. Furthermore, these models are ingeneral set in partial equilibrium as they assume the existence of ariskless technology in infinitely elastic supply.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 413

is consistent with the leverage effect documented by Black(1976), Schwert (1989), Mele (2007) and Aït Sahalia, Fan,and Li (2013) among others. Furthermore, because con-strained agents are partially shielded from bad fundamen-tal shocks, the market price of risk is negatively correlatedwith aggregate consumption. It follows that, in line withthe evidence in Mehra and Prescott (1985, 2003) and Famaand French (1989), our model also produces a counter-cyclical equity premium.

Third, we show that, because arbitrageurs are alwayslevered in equilibrium, arbitrage activity mitigates theportfolio imbalance induced by constrained agents andresults in an increase of the interest rate and a decrease ofthe market price of risk compared with the model inwhich arbitrageurs are absent. This liquidity provision,however, comes at a cost as arbitrage activity has anegative impact on the consumption share of constrainedagents and their welfare. This occurs through two chan-nels: Arbitrage activity reduces the stock price and hencethe initial wealth of constrained agents, but it alsoincreases the stock volatility and, therefore, tightens theportfolio constraint that limits the risk taking of con-strained agents.

Our work is related to several recent contributions inthe asset pricing literature. Basak and Croitoru (2000)show that mispricing can arise between two securitiesthat carry the same risk if all agents are subject to aportfolio constraint that prevents them from exploitingthe corresponding riskless arbitrage opportunity. If theconstraint is removed even for a small fraction of thepopulation then the unconstrained are able to close thearbitrage opportunity and mispricing becomes inconsis-tent with equilibrium. By contrast, we build a model inwhich risky arbitrage opportunities persist in equilibriumdespite the presence of unconstrained agents because theyrequire agents to hold enough capital to sustain interimlosses. Basak and Croitoru (2006) consider a productioneconomy version of Basak and Croitoru (2000) in whichthey introduce risk-neutral arbitrageurs who hold nowealth and are subject to a portfolio constraint that setsan exogenous limit on the size of their position. As in thispaper, the activity of these arbitrageurs brings prices closerto fundamentals but the linear production technology thatthe authors use determines the stock price and its volati-lity exogenously. On the contrary, all prices in our modelare endogenously determined and we show that arbitrageactivity generates excess volatility. Other key differencesare that while the arbitrageurs in Basak and Croitoru(2006) saturate their constraint and instantaneously con-sume all profits, the arbitrageurs in this paper are risk-averse, accumulate wealth over time, and never exhausttheir credit limit.

Our findings are also related to those of Gromb andVayanos (2002), who investigate the welfare implicationsof financially constrained arbitrage in a segmented marketsetting with zero net supply securities and an exogenousinterest rate. In their model arbitrageurs exploit the risk-less arbitrage opportunities that arise across markets and,thereby, allow Pareto-improving trade to occur. By con-trast, the arbitrageurs in our paper do not alleviate con-straints and their trading activity could hinder the welfare

of both constrained and unconstrained agents. In a relatedcontribution, Gârleanu, Panageas, and Yu (2013) study amodel with endogenous entry in which a continuum ofinvestors, and financial intermediaries are located on acircle. In their model, participation costs and collateralconstraints prevent agents from trading in all assets. Thisimplies that diversifiable risk is priced and exposes risklessarbitrage opportunities that cannot be eliminated due toprohibitive participation costs. By contrast, we show thatthe presence of constrained agents generates risky arbit-rage opportunities that persist in equilibrium becauseunconstrained agents and arbitrageurs are subject towealth constraints and study the feedback effects ofarbitrage activity on the equilibrium outcomes.

Most of the existing models with risky arbitrage oppor-tunities are cast in partial equilibrium. For example, Liuand Longstaff (2004) study the portfolio choice problem ofa risk-averse arbitrageur subject to margin constraints andwho can trade in a riskless asset with a constant interestrate and an exogenous arbitrage opportunity modeled as aBrownian bridge.6 Kondor (2009) considers a model inwhich a risk neutral arbitrageur exploits an exogenousarbitrage opportunity whose duration is distributed expo-nentially. As in this paper, the price gap between themispriced securities could diverge before it converges andthereby inflict interim losses on arbitrageurs. By contrast,we study the role of credit in enlarging the set of strategiesavailable to arbitrageurs and its impact on equilibriumquantities such as prices, volatilities and interest rateswithin a model in which mispricing is microfounded andall markets clear.

Haddad (2013) also considers a model in which someagents are levered and bear more aggregate risk, but themechanism is different. In his setting, agents choosedynamically whether to be levered in the stock, in whichcase they actively participate in the firm's management.The collective activism of levered agents improves thegrowth rate of dividends, and these agents are remuner-ated for this service in a competitive way that makes thefirm indifferent to the level of active capital. The analysis ofHaddad (2013) also highlights the impact of deleveragingrisk on equilibrium outcomes but, in contrast to this paper,the equilibrium of his economy features neither riskyarbitrage opportunities nor excess volatility.

Finally, we highlight the connections with the literatureon rational asset pricing bubbles.7 Santos and Woodford(1997) and Loewenstein and Willard (2000) show that, infrictionless economies with complete markets bubbles canexist on zero net supply securities, such as options andfutures, but not on positive net supply securities such as

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428414

stocks. Hugonnier (2012) shows that the presence ofportfolio constraints can generate bubbles also on positivenet supply securities even if some agents are uncon-strained, and Prieto (2012) extends these conclusions toeconomies in which agents have heterogenous risk aver-sion and beliefs. In addition to these contributions, somepapers analyze the properties of bubbles in partial equili-brium settings. In particular, Cox and Hobson (2005) andHeston, Loewenstein, and Willard (2007) study bubbles onthe price of derivatives, and Jarrow, Protter, and Shimbo(2010) analyze bubbles in models with incomplete markets.An important difference between these partial equilibriumstudies and our paper lies in the fact that they assume theexistence of a risk-neutral measure and, therefore, rule outthe existence of a bubble on the riskless asset. By contrast, weshow that such a pricing measure cannot exist in our modelbecause the existence of a bubble on the riskless asset is anecessary condition for equilibrium.

The remainder of the paper is organized as follows.In Section 2, we present our assumptions and provide abasic discussion of bubbles. In Section 3, we solve theindividual optimization problems, derive the equilibriumand discuss its main properties. In Section 4, we analyzethe equilibrium trading strategies and present the impli-cations of the model for excess volatility and the leverageeffect. In Appendix A, we show that our results remainqualitatively unchanged under an alternative, less cyclicalspecification of the credit facility that puts a constantbound on the discounted losses that the arbitrageurs isallowed incur. All proofs are gathered in Appendix B.

2. The model

In this section we present our assumptions regardingthe economic environment, characterize the feasible setsof the three groups of agents present in the economy anddiscuss basic properties of rational asset pricing bubbles.

2.1. Securities markets

We consider a continuous-time economy on an infinitehorizon.8 Uncertainty in the economy is represented by aprobability space carrying a Brownian motion Zt. In whatfollows, we assume that all random processes are adaptedwith respect to the usual augmentation of the filtrationgenerated by this Brownian motion.

Agents trade in two securities: a riskless asset in zeronet supply and a stock in positive supply of one unit. Theprice of the riskless asset evolves according to

S0t ¼ 1þZ t

0S0uru du ð1Þ

for some short rate process rt that is to be determined inequilibrium. The stock is a claim to a dividend process δt

8 We assume an infinite horizon to avoid having to keep track of timeas a state variable, but this assumption is not needed for the validity ofour conclusions. Bubbles also arise in a finite horizon version of ourmodel, and the arbitrage activity that they attract results in a pricedecrease and a countercyclical excess volatility component in equilibriumstock returns.

that evolves according to a geometric Brownian motionwith constant drift μδ and volatility σδ40. The stock priceis denoted by St and evolves according to

StþZ t

0δu du¼ S0þ

Z t

0Suðμu duþσu dZuÞ ð2Þ

for some initial value S040, some drift μt, and somevolatility σt that are to be determined endogenously inequilibrium.

2.2. Trading strategies

A trading strategy is a pair ðπt ;ϕtÞ, where πt representsthe amount invested in the stock and ϕt represents theamount invested in the riskless asset. A trading strategy issaid to be self-financing given initial wealth w and con-sumption rate ctZ0 if the corresponding wealth processsatisfies

Wt � πtþϕt ¼wþZ t

0ðϕuruþπuμu�cuÞ duþ

Z t

0πuσu dZu:

ð3ÞImplicit in the definition is the requirement that thetrading strategy and consumption plan be such that theabove stochastic integrals are well defined.

2.3. Agents

The economy is populated by three agents indexed bykAf1;2;3g. Agent k is endowed with nkA ½0;1� units of thestock and his preferences are represented by

UkðcÞ � EZ 1

0e�ρt logðctÞ dt

� �ð4Þ

for some subjective discount rate ρ40. In what follows welet wk � nkS0 denote the initial wealth of Agent k com-puted at equilibrium prices.

The three agents have homogenous preferences andbeliefs but differ in their trading opportunities. Agent 1 isfree to choose any strategy whose wealth remains non-negative at all times, and we refer to him as the uncon-strained agent. Agent 2, to whom we refer as theconstrained agent, is subject to the same nonnegativewealth requirement as Agent 1 but is also required tochoose a strategy that satisfies

πtACt � fπAR: jσtπjrð1�εÞσδWtg; ð5Þfor some given εA ½0;1�. This portfolio constraint can bethought of as limiting the amount of risk the agent isallowed to take. If σtZσδ (which we show is the case inequilibrium), then this constraint forces Agent 2 to keep astrictly positive fraction of his wealth in the riskless assetat all times, and introduces an imbalance that ultimatelygenerates bubbles. This constraint is also a special case ofthe general risk constraints in Cuoco, He, and Isaenko(2008) and is studied in both Gârleanu and Pedersen(2007) and Prieto (2012) as a constraint on conditionalvalue-at-risk.

Agent 3 is free to choose any self-financing strategy, butin contrast to the two other agents, he is not required to

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 415

maintain nonnegative wealth. Instead, we assume that thisagent has access to a credit facility that allows him towithstand short-term deficits provided that his wealthsatisfies the lower bound

W3tþψStZ0; tZ0; ð6Þand the transversality condition

lim infT-1

E½ξTWT �Z0 ð7Þ

for some exogenously fixed ψZ0, where ξt is the stateprice density process defined in Eq. (8). This agent shouldbe thought of as an arbitrageur whose funding liquidityconditions are determined by the magnitude of ψ. The factthat the amount of credit available to this arbitrageurincreases with the size of the market is meant to capturein a simple way the observation that capital availabilityimproves in times when the stock market is high. See, forexample, Ben-David, Franzoni, and Moussawi (2012) andAng, Gorovyy, and Van Inwegen (2011) for evidence onhedge fund trading.

Because Agent 3 can continue trading in states ofnegative wealth, the wealth constraint of the arbitrageurin Eq. (6) allows for excess borrowing. Trades in thesestates can be considered uncollateralized as Agent 3 doesnot have enough assets to cover his liabilities in case ofinstantaneous liquidation. Note, however, that, due to hispreferences, Agent 3 never willingly stops servicing debt.In other words, Agent 3 is balance sheet insolvent butnever cash-flow insolvent in states of negative wealth,and, as a result, his debt is riskless at all times.

To emphasize the interpretation of Agent 3 as anarbitrageur, we assume from now on that n3 ¼ 0 so thathis initial wealth is zero.9 This in turn implies that theendowments of the other agents can be summarized bythe number n¼ n2Að0;1� of shares of the stock initiallyheld by the constrained agent.

2.4. Definition of equilibrium

The concept of equilibrium that we use is similar to thatof equilibrium of plans, prices and expectations introducedby Radner (1972).

Definition 1. An equilibrium is a pair of security priceprocesses ðS0t ; StÞ and an array fckt ; ðπkt ;ϕktÞg3k ¼ 1 of con-sumption plans and trading strategies such that

(a)

9

arbitendoof thsettifacilof thonesecti

Given ðS0t ; StÞ the plan ckt maximizes Uk over thefeasible set of Agent k and is financed by the tradingstrategy ðπkt ;ϕktÞ.

(b)

Markets clear: ∑3k ¼ 1ϕkt ¼ 0, ∑3

k ¼ 1πkt ¼ St and∑3

k ¼ 1ckt ¼ δt .

For simplicity we assume that the mass and initial wealth ofrageurs are exogenously fixed, but these quantities can be easilygenized by assuming that arbitrageurs are heterogenous in the sizeeir credit facility and have to pay to enter the market. In such ang an arbitrageur would enter only if the profits that his creditity allows him to generate exceed the entry cost, and the aggregationese decisions gives rise to a representative arbitrageur similar to thewe use but whose credit facility reflects the entry costs and the cross-onal distribution of individual credit facilities.

An equilibrium is said to have arbitrage activity if theconsumption plan of the arbitrageur is not identically zero.

Because the arbitrageur starts from zero wealth, itcould be that the set of consumption plans he can financeis empty. In such cases, his consumption and optimalportfolio are set to zero and the equilibrium involves onlythe two other agents. To determine conditions underwhich the arbitrageur participates it is necessary tocharacterize his feasible set. This is the issue to whichwe now turn.

2.5. Feasible sets, bubbles, and limited arbitrages

Let ðS0t ; StÞ denote the prices in a given equilibrium andassume that there are no riskless arbitrages for otherwisethe market could not be in equilibrium. As is well known(see, e.g., Duffie, 2001), this implies that μt ¼ rtþσtθt forsome process θt. This process is referred to as the marketprice of risk and is uniquely defined on the set wherevolatility is nonzero. Now consider the state price densitydefined by

ξt ¼1S0t

exp �Z t

0θu dZu�

12

Z t

0jθuj2 du

� �: ð8Þ

The following proposition shows that the ratio ξt;u ¼ ξu=ξtcan be used as a pricing kernel to characterize the feasiblesets of Agents 1 and 3 and provides conditions underwhich the arbitrageur participates in the market.

Proposition 1. Assume that limT-1E½ξTST � ¼ 0.10 A con-sumption plan ct is feasible for Agent kAf1;3g if and only if

EZ 1

0ξtct dt

� �rwkþ1fk ¼ 3gψ ðS0�F0Þ; ð9Þ

where the nonnegative process

Ft � EtZ 1

tξt;uδu du

� �ð10Þ

gives the minimal amount that Agent 1 needs to hold at timetZ0 to replicate the dividends of the stock while maintainingnonnegative wealth. In particular, the feasible set of Agent 3is nonempty if and only if ψ ðS0�F0Þ40.

Following the literature on rational bubbles (see, forexample, Santos and Woodford, 1997; Loewenstein andWillard, 2000; Hugonnier, 2012) we refer to Ft in Eq. (10)as the fundamental value of the stock and to

Bt � St�Ft ¼ St�Et

Z 1

tξt;uδu du

� �Z0 ð11Þ

as the bubble on its price. Using this terminology,Proposition 1 shows that the feasible set of the arbitrageuris empty unless two conditions are satisfied: there needsto be a strictly positive bubble on the stock, and the agent

10 This transversality condition guarantees that the deflated price ofthe stock converges to zero at infinity and allows for a simple character-ization of the feasible set of Agent 3. This condition can be relaxed at thecost of a more involved characterization (see Lemma B.1 in Appendix B)but is without loss of generality as we show that it necessarily holds inequilibrium.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428416

must have access to the credit facility. The intuition behindthis result is clear. Because the arbitrageur does not holdany initial wealth he can consume in the future only ifthere are arbitrage opportunities in the market that he isable to exploit, at least to some extent.

At first glance, it could seem that a stock bubble shouldbe inconsistent with optimal choice and, therefore, alsowith the existence of an equilibrium, because it impliesthat two assets with the same cash flows (the stock andthe portfolio that replicates its dividends) are traded atdifferent prices. The reason that this is not so is that, due towealth constraints, bubbles only constitute limited arbit-rage opportunities. To see this, consider the textbookstrategy that sells short x40 shares, buys the portfoliothat replicates the corresponding cash flows over a givenfinite time interval ½0; T�, and invests the remainder in theriskless asset. The value process of this trading strategy is

Atðx; TÞ ¼ xðS0tB0ðTÞ�BtðTÞÞ; ð12Þwhere

BtðTÞ � St�FtðTÞ ¼ St�EtZ T

tξt;uδu duþξt;TST

� �Z0 ð13Þ

gives the bubble on the price of the stock over the interval½t; T �. This trade requires no initial investment and, if thestock price includes a strictly positive bubble, then itsterminal value AT ðx; TÞ ¼ xS0TB0ðTÞ is strictly positive so itdoes constitute an arbitrage opportunity in the usualsense. But this opportunity is risky because it entails thepossibility of interim losses and, therefore, cannot beimplemented to an arbitrary scale by the agents in theeconomy. In particular, the arbitrageur can implement thistrade only up to size ψ because otherwise the solvencyconstraint (6) would not be satisfied. Similarly, the uncon-strained agent can implement this arbitrage only if heholds sufficient collateral in the form of cash or securities.

The discussion so far focuses on the stock, but bubblescan be defined on any security, including the riskless asset.Over an interval ½0; T �, the riskless asset can be viewed as aderivative security that pays a single terminal dividendequal to S0T . The fundamental value of such a security isF0tðTÞ ¼ Et ½ξt;TS0T �, whereas its market value is simplygiven by S0t. This naturally leads to defining the finitehorizon bubble on the riskless asset as

B0t Tð Þ � S0t�F0t Tð Þ ¼ S0t 1�Et ξt;TS0TS0t

� �� �: ð14Þ

As in the case of the stock, a bubble on the riskless asset isconsistent with both optimal choice and the existence ofan equilibrium in our economy. In fact, we show belowthat, due to the presence of constrained agents, bubbles onboth the stock and the riskless asset are necessary formarkets to clear.

Remark 1. Eq. (14) shows that the riskless asset has abubble over ½0; T� if and only if the process Mt � S0tξtsatisfies E½MT �oM0 ¼ 1. Because the economy is driven bya single source of risk, this process is the unique candidate forthe density of the risk-neutral measure and it follows that thepresence of a bubble on the riskless asset is equivalent to thenonexistence of the risk-neutral measure. See Loewenstein

and Willard (2000), and Heston, Loewenstein, and Willard(2007) for derivatives pricing implications.

Remark 2. Combining Eqs. (2) and (3) reveals that, underthe assumption of Proposition 1, a given consumption planis feasible for the arbitrageur if and only if there exists atrading strategy ðπn

t ;ϕn

t Þ such that the process

Wn

t ¼WtþψSt ¼ψS0þZ t

0ðruϕn

uþπn

uμu�cu�ψδuÞ du

þZ t

0πn

uσu dZu ð15Þ

is nonnegative and satisfies the transversality condition(7). This shows that the feasible set of the arbitrageurcoincides with that of an auxiliary agent who has initialcapital ψS0, receives income at rate �ψδt , and is subject toa liquidity constraint that requires him to maintain non-negative wealth at all times as in He and Pagès (1993),El Karoui and Jeanblanc-Picqué (1998), and Detemple andSerrat (2003).Given this observation, it might seem surprising that the

static characterization of the arbitrageur's feasible setinvolves only the unconstrained state price density ξtinstead of a family of shadow state price densities. Thisis due to the fact that, because the implicit income rate�ψδt in Eq. (15) is negative, the liquidity constraint of theauxiliary agent is non binding. The intuition is clear: toconsume at a nonnegative rate while simultaneouslyreceiving negative income over time, this agent mustmaintain nonnegative wealth at all times. Mathematically,it follows from El Karoui and Jeanblanc-Picqué (1998) andthe above observation that a consumption plan is feasiblefor the arbitrageur if and only if it satisfies

supΛAL

EZ 1

0ΛuξuðcuþψδuÞ du

� �rψS0; ð16Þ

where L denotes the set of nonnegative, decreasingprocesses with initial value smaller than one and, becauseψδtZ0, we have that the supremum is attained by Λn

u � 1.This important simplification implies that the marginalutility of the arbitrageur is a function of the unconstrainedstate price density and allows us to characterize theequilibrium in terms of an endogenous state variable thatfollows a standard diffusion process instead of a reflecteddiffusion process (see Proposition 3).

3. Equilibrium

In this section we solve the optimization problem of thethree agents and aggregate their decisions to construct theunique equilibrium.

3.1. Individual optimality

Combining Proposition 1 with well-known results onlogarithmic utility maximization leads to the followingcharacterization of optimal policies.

Proposition 2. Assume that limT-1E½ξTST � ¼ 0. Then theoptimal consumption and trading strategies of the three

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 417

agents are given by

ckt ¼ ρðWktþ1fk ¼ 3gψBtÞ; ð17Þand

π1t ¼ ðθt=σtÞW1t ; ð18Þ

π2t ¼ κtðθt=σtÞW2t ; ð19Þ

π3t ¼ ðθt=σtÞðW3tþψBtÞ�ψ ðΣBt =σtÞ; ð20Þ

where

κt ¼min 1;ð1�εÞσδ

jθt j

� �A 0;1½ �; ð21Þ

and the process ΣBt denotes the diffusion coefficient of the

process Bt.

The solution for the unconstrained Agent 1 is standardgiven logarithmic preferences: this agent invests in aninstantaneously mean–variance efficient portfolio and hasa constant marginal propensity to consume equal to hisdiscount rate. The solution for the constrained Agent 2follows from Cvitanić and Karatzas (1992) and shows thatthe constraint binds in states where the market price ofrisk is high. This is intuitive: because Agent 2 has loga-rithmic preferences, absent the portfolio constraint hewould invest in the stock in proportion to the marketprice of risk and the result follows by noting that theconstraint limits the amount of risk he is allowed to take.

The solution for the arbitrageur is novel and illustrateshow this agent is able to reap arbitrage profits, and therebyconsume, despite the fact that he holds no initial capital.Eq. (20) shows that the optimal strategy for this agent is toshort ψ shares of the stock, buy ψ units of the portfolio thatreplicates the stock dividends, and invest the strictlypositive net proceeds of these transactions into the samemean–variance efficient portfolio as Agent 1. This strategy isadmissible only because of the credit facility, and allows thearbitrageur to increase his consumption basis from W3t

to W3tþψBt ¼ e�ρtψB0=ξt . The optimal consumption inEq. (17) then follows by noting that, because the arbitrageurhas logarithmic preferences, his marginal propensity to con-sume is constant and equal to his subjective discount rate.

Remark 3. The optimal policy of the arbitrageur bears aclose resemblance to that of a hypothetical agent withlogarithmic utility and no initial wealth who receives laborincome at rate et in a complete market with state pricedensity ξt. The optimal consumption of such an agent is

ct ¼ ρðWtþHtÞ ¼ ρe�ρtðH0=ξtÞ ð22Þwhere Ht gives the fundamental value of the agent's futureincome, and an application of Itô's lemma shows that hisoptimal trading strategy is

πt ¼ ðθt=σtÞðWtþHtÞ�ðΣHt =σtÞ ð23Þ

where ΣHt denotes the diffusion coefficient of the process Ht.

This solution is isomorphic to that given in Proposition 2with one important caveat: Instead of arising exogenouslyfrom the agent's income, the process Ht ¼ψBt in this paperis endogenously generated by the profits that arbitrageursare able to reap from the market.

Remark 4. Because W3tþψSt ¼ψ ðFtþe�ρtB0=ξtÞ40 thearbitrageur never exhausts his credit limit. Although of adifferent nature, this result is reminiscent of Liu andLongstaff (2004) who study the portfolio decisions of anarbitrageur facing an exogenous arbitrage opportunitymodelled as a Brownian bridge and a margin constraint.

3.2. Equilibrium allocations and risk sharing

To characterize the equilibrium we use a representativeagent with stochastic weights that allows us to easily clearmarkets despite the imperfect risk sharing induced by thepresence of the constrained agent (see Cuoco and He,1994). The utility function of this representative agent isdefined by

u c; γ; λt� �� max

c1 þ c2 þ c3 ¼ cðlogðc1Þþλt logðc2Þþγ logðc3ÞÞ; ð24Þ

where λt40 is an endogenously determined weightingprocess that encapsulates the differences across the agentsand γZ0 is an endogenous constant that determines therelative weight of arbitrageurs in the economy.

By Proposition 2, we have that the first order condi-tion for Agents 1 and 3 can be written as ξt ¼ e�ρtðck0=cktÞfor k¼1,3. Comparing these with the first order conditionof the representative agent's problem shows that theequilibrium state price density and the equilibrium con-sumption are given by

ξt ¼ e�ρt ucðδt ; γ; λtÞucðδ0; γ; λ0Þ

¼ e�ρtδ0ð1�s0Þδtð1�stÞ

; ð25Þ

and

c2t ¼ ρW2t ¼ stδt ; ð26Þ

c1t ¼ ρW1t ¼1

1þγ1�stð Þδt ; ð27Þ

and

c3t ¼ ρ W3tþψBt� �¼ δt�c1t�c2t ¼

γ1þγ

1�stð Þδt ; ð28Þ

where the endogenous state variable

st �c2tδt

¼ λt1þγþλt

A 0;1ð Þ ð29Þ

tracks the consumption share of the constrained agent.Combining Eqs. (8), (19), (25) and (27) then allows us todetermine the equilibrium drift and volatility of this statevariable, and delivers the following explicit characteriza-tion of equilibrium.

Proposition 3. In equilibrium, the riskless rate of interest andthe market price of risk are explicitly given by

θt ¼ σδ 1þ εst1�st

� �; ð30Þ

rt ¼ ρþμδ�σδθt ¼ ρþμδ�σ2δ 1þ εst

1�st

� �; ð31Þ

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428418

and the consumption share of the constrained agent evolvesaccording to the stochastic differential equation

dst ¼ �stεσδ dZtþ st1�st

εσδ dt� �

ð32Þ

with initial condition s0 ¼ ρw2=δ0.

The above characterization of equilibrium is notable fortwo reasons. First, it follows from Eqs. (19), (21) and (30)that the portfolio of Agent 2 satisfies

σtπ2t ¼W2tð1�εÞσδoW2tθt : ð33ÞThis shows that the portfolio constraint binds at all timesin equilibrium, and it follows that Agent 2 constantly has apositive demand for the riskless asset. This, in turn, impliesthat prices should adjust to entice Agents 1 and 3 toborrow and explains why, as shown by Eqs. (30) and (31),the market price of risk increases and the interest ratedecreases compared with an unconstrained economyðε¼ 0Þ. Second, Eq. (32) shows that the consumption shareof the constrained agent is negatively correlated withdividends and, therefore, tends to decrease (increase)following sequences of positive (negative) cash flowshocks. The intuition for this result is clear. By limitingthe amount of risk that Agent 2 can take, the portfolioconstraint implies that his consumption is less sensitive tobad shocks but also limits the extent to which it benefitsfrom sequences of good shocks.

3.3. Equilibrium prices

To compute the equilibrium stock price, we rely on themarket clearing conditions which require that St ¼∑3

k ¼ 1Wkt .Combining this identity with Eqs. (30), (38) and the clearing ofthe consumption good market gives

St ¼ Pt�ψBt ¼ Pt�ψ ðSt�FtÞ; ð34Þwhere Pt ¼ δt=ρ is the stock price that would prevail inequilibrium if arbitrageurs were absent from the economy,and

Ft ¼ EtZ 1

tξt;uδu du

� �¼ δt 1�stð ÞEt

Z 1

te�ρðu� tÞ du

1�su

� �ð35Þ

gives the fundamental value of the stock. Settingα¼ψ=ð1þψ ÞA ½0;1� and solving Eq. (47) gives

St ¼ αFtþð1�αÞPt : ð36ÞTo complete the characterization of the equilibrium, it remainsto determine whether the price includes a bubble. Using theabove expression together with Eq. (42) shows that

Bt ¼ 1�αð Þ Pt�Ftð Þ ¼ ð1�αÞδt1þγþλt

EtZ 1

te�ρðu� tÞðλt�λuÞ du

� �ð37Þ

and it follows that the stock price is bubble free if and only ifthe weighting process is a martingale. An application of Itô'slemma gives

dλt ¼ 1þγ� �

dst

1�st

� �¼ �λt 1þγþλt

� � εσδ1þγ

dZt ; ð38Þ

so that the weighting process is a local martingale. However,Proposition 4 shows that this local martingale fails to be a truemartingale and, thereby, proves that any equilibrium includesa bubble on the stock and arbitrage activity.

Proposition 4. The weighting process is a strict local martin-gale. In particular, the stock price includes a strictly positivebubble in any equilibrium.

Combining the above proposition with Eqs. (23), (50)and Itô's product rule reveals that in equilibrium thediscounted gains process

ξtStþZ t

0ξuδu du¼ Et

Z 1

0ξuδu du

� �þξtBt ð39Þ

is a local martingale, as required to rule out risklessarbitrage opportunities, but not a true martingale. Thisresult shows that the distinction between local and truemartingales, which is usually perceived as a technicality,captures an important economic phenomenon, namely,the presence of an asset pricing bubble. It also clearlyindicates that continuous-time bubbles are of a differentnature than those that could arise in discrete-time models.In particular, because a discrete-time local martingale is atrue martingale over any finite horizon (see Meyer, 1972),the same arguments as in Santos and Woodford (1997)imply that a stock bubble cannot arise in a discrete-timeversion of our model. By contrast, Proposition 4 shows thatin continuous-time a bubble arises as soon as there areconstrained agents in the economy and cannot be eradi-cated by arbitrage activity unless arbitrageurs have accessto unbounded credit.

Our next result establishes the existence and unique-ness of the equilibrium and derives closed-form expres-sions for the stock price and its bubble.

Theorem 1. There exists a unique equilibrium, in which thestock price and its bubble are explicitly given by

St ¼ ð1�αsηt ÞPt ð40Þand

Bt ¼ St�Ft ¼ ð1�αÞsηt Pt ð41Þwith the constant

η� 12þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14þ 2ρðεσδÞ2

s41: ð42Þ

The consumption share of the constrained agent evolvesaccording to Eq. (45) with initial condition given by theunique s0Að0;1Þ such that s0 ¼ nð1�αsη0Þ.

Theorem 1 offers several important conclusions. First, itshows that the unique equilibrium includes a strictlypositive bubble on the stock and, therefore, generatesarbitrage activity as soon as Agent 3 has access to a creditfacility in that α40.

Second, it shows that in stark contrast to existingequilibrium models with frictions and logarithmic agents(see, e.g., Detemple and Murthy, 1997; Basak and Cuoco,1998; Basak and Croitoru, 2000) the combination of portfolioconstraints and arbitrage activity generates a price/dividendratio that is both time-varying and lower than that whichwould have prevailed in the absence of arbitrageurs. Impor-tantly, Eq. (53) shows that the price/dividend ratio is adecreasing function of the consumption share of constrainedagents and, because the latter is negatively correlated withfundamental shocks, arbitrage activity generates both excess

Fig. 1. Bubble on the stock and credit conditions. This figure plots therelative bubble on the stock at horizon T¼100 as function of the initialstock holdings of the constrained agent for different values of theparameter governing the size of the credit facility. To construct thisfigure we set, σδ ¼ 0:0357, ρ¼0.001 and ε¼0.5.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 419

volatility and the leverage effect despite the fact that allagents have logarithmic preferences. In Section 4.2 weinterpret this important property in light of the equilibriumportfolio strategies that we derive in Section 4.1.

Third, the expression for the stock price shows thatarbitrage activity has a negative impact on the equilibriumprice level. One way to understand this result is to observethat arbitrage activity makes the stock more volatile andthereby reduces the value of the collateral services itprovides. Another way to understand this result is to viewthe price decrease as rents to the arbitrage technology.Comparing the consumption of Agent 1 with that of Agent 3and using Eq. (54) shows that these rents accrue to thearbitrageur in the form of an infinitely lived stream ofconsumption at rate αδts

ηt . This additional consumption

reduces the amount of dividend available to stockholdersto δtð1�αsηt Þ, and the equilibrium stock price is simply givenby the usual logarithmic valuation formula applied to thisreduced dividend process.

Our next result shows that, in addition to a stockbubble, the equilibrium price system includes a bubbleon the riskless asset over any investment horizon. Impor-tantly, combining this result with Remark 1 shows that inthe unique equilibrium of our economy a risk-neutralprobability measure does not exist.

Proposition 5. In the unique equilibrium, the price of theriskless asset is

S0t ¼ eqtPt

P0

sts0

� �1=ε

ð43Þ

with the constant q¼ ρ�12 1�εð Þσ2

δ . Over the time interval½t; T�, the stock and the riskless asset include bubble compo-nents that satisfy

BtðTÞSt

¼H T�t; st ;2η�1� �Bt

StrH T�t; st ;2=ε�1

� �¼ B0tðTÞS0t

;

ð44Þwhere we have set

d7 τ; s; að Þ � logðsÞεσδ

ffiffiffiτ

p 7a2εσδ

ffiffiffiτ

p; ð45Þ

Hðτ; s; aÞ �Nðdþ ðτ; s; aÞÞþs�a Nðd� ðτ; s; aÞÞ; ð46Þand the function N(x) denotes the cumulative distributionfunction of a standard normal random variable.

Proposition 5 shows that in relative terms, i.e., for eachdollar of investment, the bubble on the riskless asset islarger than that on the stock over any investment horizon.Because an agent subject to a nonnegative wealth con-straint cannot short both assets, we naturally expect thatAgent 1 chooses a strategy that exploits the bubble on theriskless asset because it requires less collateral per unit ofinitial profit. This intuition is confirmed in Section 4.1where we show that the equilibrium strategy of theunconstrained agent can be seen as the combination ofan all equity portfolio and a short position in the risklessasset bubble.

Comparing the results of Theorem 1 and Proposition 5reveals that arbitrage activity has a different effect on thestock and riskless asset bubbles. Indeed, Eq. (54) shows

that arbitrage activity impacts the stock bubble bothdirectly through the constant 1�α and indirectly throughthe initial value of the constrained agent's consumptionshare process, while Eq. (57) shows that only the latterchannel is at work for the riskless asset bubble. Animmediate consequence of this observation is that whilethe stock bubble disappears when arbitrageurs have accessto unlimited credit ðα-13ψ-1Þ, the riskless assetbubble is always strictly positive. The reason for thisdifference is that our formulation of the wealth constraintgives the arbitrageur a comparative advantage over agent 1in exploiting the stock bubble, but not the riskless assetbubble.

Because the right hand side of Eq. (42) is increasing andconcave in the weighting process, it follows from Jensen'sinequality and Proposition 4 that st is a supermartingale.This implies that the consumption share of constrainedagents is expected to decline over time and a directcalculation provided in Appendix B shows that s1 ¼ 0 sothat constrained agents, and the bubbles that their pre-sence generates, disappear in the long run. A natural wayto correct this behavior and thereby ensure that bubblessubsist even in the long run is to allow for birth and deathof agents of the various kinds in such a way as tocontinuously repopulate the group of constrained agents.See Gârleanu and Panageas (2014) for a recent contribu-tion along these lines.

To illustrate the magnitude of the bubbles and theimpact of arbitrage activity we plot in Figs. 1 and 2 therelative bubbles on the stock and the riskless asset asfunctions of the initial stock holdings of the constrainedagent in an economy in which agents have discount rateρ¼0.001, the constraint is set at ε¼0.5, and the volatilityof the aggregate consumption is taken to be σδ ¼ 0:0357 asproposed by Basak and Cuoco (1998) to match the esti-mates of Mehra and Prescott (1985). As shown by thefigures, the bubbles can account for a significant fraction ofthe equilibrium prices and this fraction decreases both asthe fraction of the market held by constrained agentsdecreases and as credit conditions improves. To gain more

Fig. 2. Bubble on the riskless asset and credit conditions. This figure plotsthe relative bubble on the riskless asset at horizon T¼100 as function ofthe initial stock holdings of the constrained agent for different values ofthe parameter governing the size of the credit facility. To construct thisfigure we set, σδ ¼ 0:0357, ρ¼0.001 and ε¼0.5.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428420

insight into these properties it is necessary to determinehow arbitrage activity affects the path of the constrainedagent's consumption share, and this is the issue we turnto next.

3.4. Comparative statics

In equilibrium, the consumption share of the con-strained agent plays the role of an endogenous statevariable. Therefore, to understand the impact of arbitrageactivity we need to understand how the credit facilityparameter α influences the path of this state variable. Tostate the result, let stðαÞ denote the constrained agent'sconsumption share seen as a function of time and thecredit facility parameter.

Proposition 6. We have

� ns0ðαÞ1þη

s0ðαÞþnηs0ðαÞηstðαÞs0ðαÞ

� �r∂stðαÞ

∂αr0: ð47Þ

The consumption share of the constrained agent, the stockprice, and the stock bubble are decreasing in arbitrageactivity at all dates.

The above result shows that arbitrage activity reducesthe consumption share of the constrained agent at everypoint in time and, hence, also his welfare. Furthermore, adirect calculation based on Eq. (47) and the definition ofthe initial value s0ðαÞ shows that the consumption share ofthe arbitrageur

s3t αð Þ ¼ γðαÞ1þγðαÞ 1�st αð Þð Þ ¼ αs0ðαÞη

1�stðαÞ1�s0ðαÞ

ð48Þ

is increasing in the credit facility parameter α at all dates,while the consumption share of the unconstrained agent

s1t αð Þ ¼ 11þγðαÞ 1�st αð Þð Þ ¼ 1�αs0ðαÞη�s0ðαÞ

� �1�stðαÞ1�s0ðαÞ

ð49Þis decreasing in α at the initial time but non-monotone atsubsequent dates. The intuition for these findings is the

following. As α increases the consumption share of theconstrained agent decreases so that more consumptionbecomes available for Agents 1 and 3, but the repartition ofthat consumption simultaneously tilts toward the arbitra-geur as a higher α improves his comparative advantageover the unconstrained agent. This generates an initialdecrease in the consumption share of the unconstrainedagent but the effect could reverse itself over time depend-ing on the path of the economy.

4. Analysis

In this section, we study the equilibrium tradingstrategies and discuss the implications of arbitrage activityfor the equilibrium volatility of the risky asset.

4.1. Equilibrium portfolio strategies

Proposition 2 and Theorem 1 allow us to derive inclosed-form the trading strategies employed in equili-brium by each of the three groups of agents.

Proposition 7. Define a strictly positive function by setting

v sð Þ ¼ v s;αð Þ � εηαsη

1�αsη

� �: ð50Þ

Then, the equilibrium trading strategies of the three agents,and their respective signs, are explicitly given by

π1t ;ϕ1t

� �¼ 1�ð1�εÞstð1þγÞð1þvðstÞÞ

;vðstÞ�ðεþvðstÞÞstð1þγÞð1þvðstÞÞ

� �PtARþ � R� ;

ð51Þ

π2t ;ϕ2t

� �¼ 1�ε1þvðstÞ

;εþvðstÞ1þvðstÞ

� �stPtAR2

þ ; ð52Þ

and

π3t ;ϕ3t

� �¼ γ π1t ;ϕ1t

� �þ εη�11þvðstÞ

; �εηþvðstÞ1þvðstÞ

� �αsηt PtAR7 � R�

ð53Þwith the strictly positive constant γ ¼ c30=c10.

Because the portfolio constraint binds at all times inequilibrium [see Eq. (33)], Agent 2 constantly keeps astrictly positive fraction of his wealth in the riskless asset,and Proposition 7 shows that this long position in theriskless asset is offset by the borrowing positions held bythe two other agents.

As can be seen from Eq. (51), the unconstrained Agent 1uses this borrowing to invest larger amounts in the stockdespite the fact that its price includes a strictly positivebubble. To understand this finding recall that Agent 1 isrequired to maintain nonnegative wealth and, therefore,cannot simultaneously exploit both bubbles. Because theriskless asset bubble requires less collateral per dollar ofinitial profit, we expect this agent to short the risklessasset bubble and use the stock as collateral. Our next resultconfirms this intuition by showing that the wealth ofAgent 1 is the outcome of a dynamic strategy that buysthe stock and shorts the riskless asset bubble.

Proposition 8. The wealth of Agent 1 expressed as a self-financing strategy in the stock and the riskless asset bubble

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 421

over the interval ðt; T� is given by

W1t ¼ϕS1tðTÞþϕB0

1t ðTÞ; ð54Þwhere

ϕS1t Tð Þ ¼ εstþð1�Σ0tðTÞÞð1�stÞ

ð1þγÞð1þvðstÞ�Σ0tðTÞÞPtZ0; ð55Þ

ϕB01t Tð Þ ¼W1t�ϕS

1t Tð Þ ¼ vðstÞ�ðεþvðstÞÞstð1þγÞð1þvðstÞ�Σ0tðTÞÞ

Ptr0;

ð56Þand the process Σ0tðTÞ is the diffusion coefficient of

ð1=σδÞlogB0tðTÞ.Turning to the last group of agents, Proposition 7 shows

that the equilibrium trading strategy of the arbitrageur canbe decomposed into two parts: a short position of size ψ inthe stock bubble and a long position in the same strategyas the unconstrained agent. The short position in the stockbubble is worth Wat �W3t�γW1t ¼ �αsηt Pt , and an appli-cation of Itô's lemma shows that this part of the arbitra-geur's equilibrium portfolio corresponds to a self-financingtrading strategy that holds

nat �ðηε�1Þαsηt

1�αsηt ð1�ηεÞ ð57Þ

units of the stock and invests

ϕat �Wat�natSt ¼ � ηεαsηt Pt

1�αsηt ð1�ηεÞr0 ð58Þ

in the riskless asset. While the stock position can be eitherpositive or negative, we have that the position in theriskless asset is negative and decreasing in δt ¼ ρPt . Thisimplies that, compared with the unconstrained agent, thearbitrageur levers up in good times and delevers in badtimes. This feature of the equilibrium is in line with theevidence in Ang, Gorovyy, and Van Inwegen (2011) andleads to an amplification of fundamental shocks thatgenerates excess volatility and the leverage effect, as weexplain next.

4.2. Equilibrium volatility

Combining our explicit formula for the equilibriumstock price with the comparative statics result ofProposition 6 directly leads to the following characteriza-tion of the equilibrium volatility of the stock.

Proposition 9. The equilibrium volatility of the stock is givenby

σt ¼ ð1þvðst ;αÞÞσδ; ð59Þwhere the nonnegative function vðs;αÞ is defined as inEq. (50). In particular, the equilibrium volatility of the stockincreases with arbitrage activity.

Proposition 9 shows that, contrary to existing modelswith frictions and logarithmic agents, our model withconstrained agents and risky arbitrage activity generatesexcess volatility. Furthermore, this excess volatility is self-generated within the system and therefore provides a newexample of the endogenous risks advocated by Danielsson

and Shin (2003), He and Krishnamurthy (2012), andBrunnermeier and Sannikov (2014) among others. In ourmodel, excess volatility is explicitly given by

σt�σδ ¼ v st ;αð Þσδ ¼ ηεαsηt

1�αsηt

� �σδ ð60Þ

and increases with both the constrained agent's consump-tion share and the amount α of arbitrage activity. Becauseshocks to the consumption share process are negativelycorrelated with fundamental shocks this implies that thestock volatility is negatively correlated with fundamentalshocks and it follows that our model is consistent with theleverage effect (see Black, 1976; Schwert, 1989; Mele, 2007;Aït Sahalia, Fan, and Li, 2013) according to which stockvolatility tends to increase when prices fall. In addition,because the equilibrium market price of risk in Eq. (30) isalso negatively correlated with fundamental shocks we havethat the equity premium

σtθt ¼ σ2δ 1þv st ;αð Þð Þ 1þ εst

1�st

� �Zσ2

δ ð61Þ

is countercyclical which is consistent with the evidencepresented by Mehra and Prescott (1985, 2003) and Famaand French (1989), among others.

To illustrate the amount of excess volatility that ourmodel generates, we plot in Fig. 3 the volatility of the stockas a function of the consumption share of constrainedagents for different values of the parameter ψ ¼ α=ð1�αÞthat governs the size of the credit facility. As shown by thefigure the amplification of fundamental shocks induced byarbitrage activity is sizable and increases with both theconsumption share of constrained agents and the amountof arbitrage activity. For example, with 80% of constrainedagents the stock volatility varies between 1.5 and 2.03times that of the underlying dividend depending on thesize of the credit facility.

As can be seen from Eq. (60), the model generatesexcess volatility if and only if there is arbitrage activity inthat α40. Therefore, intuition suggests that the source ofthe excess volatility lies in what arbitrageurs do inresponse to fundamental shocks and, more precisely, inthe short stock bubble position that their preferentialaccess to credit allows them to implement. To confirmthis intuition, recall from Section 4.1 that this positionrequires the arbitrageur to borrow

jϕat j ¼ jWat�natSt j ¼ηεαsηt δt

ρð1�αsηt ð1�ηεÞÞ ð62Þ

from the constrained agent, and assume that the economysuffers a negative fundamental shock so that the dividenddecreases. As in the model without arbitrageurs (see, forexample, Hugonnier, 2012) this triggers a relative decreaseof the same magnitude in the price of the stock. However,because ϕat is negative and decreasing in δt ¼ ρPt we havethat this decrease in δt prompts the arbitrageur to delever.This puts additional downward pressure on the stockcompared with the case without arbitrage activity andleads to a larger total decrease of the stock price. Symme-trically, the arbitrageur tends to lever up his position inreaction to positive fundamental shocks. This puts addi-tional upward pressure on the price and, therefore,

Fig. 3. Volatility and credit conditions. This figure plots the equilibriumstock volatility as a function of the consumption share of the constrainedagent for various values of the parameter that governs the size of the creditfacility. To construct the figure, we set σδ ¼ 0:0357, ρ¼0.001, and ε¼0.5.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428422

amplifies the effect of the shock compared with the casewithout arbitrage activity. Because

∂2jϕat j∂δt∂st

¼ εαη2s1þηt

ρð1�αsηt ð1�ηεÞÞ2Z0; ð63Þ

the magnitude of the amplification increases with theconsumption share of the constrained agent. This propertyexplains the convexity of the stock volatility function thatis apparent from Eq. (59) and is intuitive as we know fromSection 3.3 that the size of the stock bubble, and hence theinfluence of the arbitrageur, increases with the consump-tion share of the constrained agent.

5. Conclusion

In this paper we derive a novel and analytically tractableequilibrium model of dynamic arbitrage. We consider aneconomy populated by three groups of agents: constrainedagents who are subject to a portfolio constraint that tiltstheir demand toward the riskless asset, unconstrained agentswho are subject only to a nonnegativity constraint onwealth,and arbitrageurs who have no initial wealth but have accessto an credit facility that allows them to weather transitorylosses.

We show that the presence of constrained agents in theeconomy gives rise to risky arbitrage opportunities in theform of asset pricing bubbles and that these bubbles makethe credit facility valuable by allowing arbitrageurs toconsume despite the fact that they initially hold no wealth.We solve for the equilibrium in closed-form and show thatit is characterized by bubbles on both traded assets, a timevarying price–dividend ratio, and a sizable countercyclicalexcess volatility component.

Appendix A. Alternative credit facility

In this appendix, we show that our results remainqualitatively unchanged if we replace the stock by theriskless asset in the arbitrageur's wealth constraint. We fix

a constant ℓZ0 and assume that the arbitrageur is subject to

W3tþℓS0tZ0; tZ0: ð64ÞProceeding as in the baseline model, we obtain the followingcharacterization of the feasible consumption set for thearbitrageur.

Proposition A.1. A consumption plan is feasible for Agent 3 ifand only if

EZ 1

0ξtc3t dt

� �rℓ 1� lim

T-1E½ξTS0T �

� �: ð65Þ

The feasible set of Agent 3 is non-empty if and only if theprice of the riskless asset includes a bubble over somehorizon.

Going through the same steps as in Section 3 showsthat the state price density and the optimal consumptionsof Agents 1 and 2 are given by Eqs. (25)–(27) where theconsumption share follows Eq. (32). Itô's lemma thenshows that the market price of risk and the interest rateare given by Eqs. (30) and (31), and it follows that over theinterval ½t; T � the riskless asset includes a bubble that isgiven by

B0t Tð Þ ¼ S0t 1�EtξTS0TξtS0t

� �� �¼ S0tH T�t; st ;2=ε�1

� �40:

ð66ÞIn particular, we have that limT-1E½ξTS0T � ¼ 0, and it thusfollows from the same arguments as in the proof ofProposition 2 that the optimal consumption of the arbi-trageur is given by c3t ¼ ρðW3tþℓS0tÞ. Combining thisidentity with Proposition 5 and the market clearing con-ditions then shows that

St ¼δtρ�ℓS0t ¼ Pt 1�eqt

ℓP0

sts0

� �1=ε" #

; ð67Þ

and it now remains to prove that, under appropriate para-metric assumptions, the initial value of the consumptionshare process can be chosen in such a way that the agents'consumption and the stock price are positive at all dates.

Theorem A.2. Denote by x̂Að0;1Þ the unique solution tox̂ε ¼ nð1� x̂Þ, and let q be defined as in Proposition 5.

(a)

If q40 and ℓZ x̂P0, then no equilibrium exists. (b) If qr0 and 0rℓo x̂P0 then there exists a unique

equilibrium in which the price of the traded securitiesare given by Eqs. (43) and (67) where the consumptionshare process evolves according to Eq. (32) with initialvalue s0 ¼ nð1�ℓ=P0Þ.

The first part shows that the equilibrium fails to existwhenever agents are sufficiently impatient and the creditlimit exceeds a threshold that depends positively on theinitial endowment of the constrained agent and the initialsize of the economy as measured by P0. The intuition isclear: If ℓ is too large relative to P0, then the consumptionc3t�ρW3t generated by the arbitrageur's profits exceedsthe available stock of the consumption good, which leadsthe stock price in Eq. (67) to take negative values that areincompatible with limited liability.

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 423

Our next result shows that in equilibrium the stockprice includes a bubble that is dominated in relative termsby the riskless asset bubble over any horizon.

Corollary A.3. Under the conditions of Theorem A.2(b) theequilibrium price of the stock includes a bubble componentthat satisfies

0rBt Tð Þ ¼ sηt PtH T�t; st ;2η�1� ��ℓB0t Tð Þr St

S0tB0t Tð Þ;

ð68Þwith η41 and the function Hðτ; s; aÞ as in Proposition 5.

Comparing Theorem A.2 and Corollary A.3 with theresults of Sections 3 and 4 shows that the asset pricingimplications of Eq. (64) are qualitatively similar to those ofour main model. First, the equilibrium includes bubbles thatallow arbitrageurs to consume despite the fact that they holdno initial capital. Second, arbitrage activity has a negativeimpact on the stock price and the bubbles on both assets. Itfollows that arbitrage activity reduces mispricing, but we canno longer show that arbitrageurs eradicate the stock bubblein the limit of infinite credit because equilibrium fails to existfor large values of ℓ. Third, the price/dividend ratio is timevarying and the equilibrium stock volatility

σ st ℓð Þð Þ ¼ σδ 1�eqtℓP0

stðℓÞs0ðℓÞ

� �1=ε" #�1

Zσδ ð69Þ

is increasing in the size of the credit limit, decreasing in theconsumption share of constrained agents and negativelycorrelated with fundamental shocks so that arbitrage activitygenerates both excess volatility and the leverage effect.

Appendix B. Proofs

Proof of Proposition 1. The static budget constraint forAgent 1 is well known and follows from Lemma B.1 byletting bt ¼Πt ¼ 0. The static budget constraint for thearbitrageur is novel and follows from Lemma B.1 by lettingbt ¼ψδt , Πt ¼ψSt and using the assumption thatlimT-1E½ξTΠT � ¼ 0. □

Consider an asset with dividend rate btZ0 and priceprocess

Πt ¼Π0þZ t

0ðrsΠsþΣΠ

s θs�bsÞ dsþZ t

0ΣΠ

s dZsZ0 ð70Þ

for some diffusion coefficient ΣΠt such that the above

integrals are well defined. To simultaneously cover themodels of Section 2 and Appendix A, consider an agentwho is subject to the transversality condition (7) and thewealth constraint

W0 ¼ 0rWtþΠt ; tZ0: ð71ÞLemma B.1 provides a static characterization of the feasibleset associated with these constraints and implies theresults of Propositions 1 and A.1.

Lemma B.1. A consumption plan ct is feasible if and only if

EZ 1

0ξscs ds

� �rΠ0� lim

T-1E ξTΠT þ

Z T

0ξsbs ds

� �: ð72Þ

The feasible set of the arbitrageur is non empty if and only ifΠt includes a strictly positive bubble component.

Proof. Assume that ct is feasible and let Wt satisfyingEqs. (7) and (71) denote the corresponding wealth process.An application of Itô's lemma shows that

ξtðWtþΠtÞþZ t

0ξsðcsþbsÞ ds ð73Þ

is a nonnegative local martingale and therefore a super-martingale. Using this property together with Eq. (71) andthe monotone convergence theorem then gives

EZ 1

0ξscs ds

� �¼ lim

T-1EZ T

0ξscs ds

� �ð74Þ

r limT-1

Π0 � E ξT WT þΠTð ÞþZ T

0ξsbs ds

� �� �ð75Þ

rΠ0�lim infT-1

E ξTWT

� lim infT-1

E ξTΠT þZ T

0ξsbs ds

� �ð76Þ

rΠ0 � limT-1

E ξTΠT þZ T

0ξsbs ds

� �; ð77Þ

where the last inequality follows from the fact thatbecause

Yt ¼ ξtΠtþZ t

0ξsbs ds¼Π0þ

Z t

0ξsðΣΠ

s �ΠsθsÞ dZs ð78Þ

is a nonnegative local martingale the function T↦E½YT �rY0 isdecreasing and hence admits a well-defined limit. Conversely,assume that the consumption plan ct satisfies Eq. (72) with anequality. Because YtZ0 is a local martingale, it follows fromFatou's lemma and Doob's supermartingale convergencetheorem that the limit

L¼ limT-1

ξTΠT ¼ limT-1

YT �Z 1

0ξsbs ds ð79Þ

is well defined and nonnegative. Now consider the process

Wt ¼ �Πtþ1ξtEtZ 1

tξsðcsþbsÞ dsþaL

� �; ð80Þ

where the nonnegative constant a is defined by a¼0 if therandom variable L is almost surely equal to zero and by

a¼ limT-1

E½ξTΠT �E½L� A 1;1ð Þ ð81Þ

otherwise. The martingale representation theorem, Itô'slemma, and the definition of a imply that Wt is the wealthprocess of a self-financing strategy that starts from no initialcapital and consumes at rate ctþbt . It is clear from thedefinition that Eq. (71) holds and using the definition of aand the monotone convergence theorem we deduce that

lim infT-1

E½ξTWT � ¼ � limT-1

E½ξTΠT �þaE½L� ¼ 0 ð82Þ

and the proof is complete. □

Proof of Proposition 2. The optimal strategy of the uncon-strained agent follows from well known results. See, forexample, Duffie (2001, Chapter 9.E) and Karatzas and

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428424

Shreve (1998, Chapter 3). Letting pt ¼ πt=W2t denote theproportion of wealth that Agent 2 invests in the stock wehave that the portfolio constraint can be written as

ptAfpAR: jσtpjr ð1�εÞσδg; ð83Þand the optimal strategy of the constrained agent nowfollows from Cvitanić and Karatzas (1992, Section 11). Letus now turn to the optimal strategy of the arbitrageur.Using Proposition 1, his optimization problem can beformulated as

maxcZ0

EZ 1

0e�ρs log cs ds

� �subject to E

Z 1

0ξscs ds

� �rψB0:

ð84ÞUsing the concavity of the utility function together withthe same arguments as in the second part of the proof ofLemma B.1, shows that the solution to this problem andthe corresponding wealth process are explicitly given byc3t ¼ ðy3eρtξtÞ�1 and

W3tþψBt ¼1ξtEtZ 1

tξsc3s ds

� �¼ c3t

ρ¼ e�ρtψB0

ξtð85Þ

for some Lagrange multiplier y340 that is determined insuch a way that W30 ¼ 0. This shows that Eq. (17) holds forthe arbitrageur. Applying Itô's lemma to the right handside of the above expression and using the fact that

Bt ¼ B0þZ t

0ðrsBs dsþΣB

s ðdZsþθs dsÞÞ; ð86Þ

we deduce that the optimal wealth process evolvesaccording to

W3t ¼Z t

0ððrsW3s�c3sÞ dsþððW3sþψBsÞθs

�ψΣBs ÞðdZsþθs dsÞÞ; ð87Þ

and the desired result now follows by comparing thisexpression with Eq. (3). □

Proof of Proposition 3. To determine the dynamics of theconsumption share, assume that dst ¼mt dtþkt dZt forsome drift mt and volatility kt. Applying Itô's lemma toEqs. (8) and (25) and comparing the results shows that

θt ¼ σδ�kt

1�st; ð88Þ

and

rt ¼ ρþμδ�σ2δþ

σδkt�mt

1�st� kt

1�st

� �2

: ð89Þ

Proposition 2 shows that, along the optimal path,W2t ¼ stδt=ρ. Applying Itô's lemma to this expression,and matching terms with the dynamic budget constraint(3), shows that the drift and volatility of the consumptionshare process are related by

mtþk2t

1�st¼ 0; ð90Þ

and

σtπ2t ¼W2t σδþktst

� �: ð91Þ

Substituting Eq. (88) into Eq. (19) and comparing the resultwith Eq. (91) then shows that

σδ ¼kt

1�stþ σδþ

ktst

� �max 1;

jσδð1�stÞ�kt jð1�εÞð1�stÞσδ

� �: ð92Þ

Solving Eqs. (90) and (92) gives the drift and volatility ofthe consumption share in Eq. (32) and the remainingclaims follow by substituting into Eqs. (88) and (89). □

Lemma B.2. We have

qt Tð Þ ¼Z T

tρe�ρðu� tÞλt�Et ½λu�

1þγþλtdu¼ sηt H τ; st ;2η�1

� ��e�ρτstHðτ; st ;1Þ; ð93Þ

where τ¼ T�t and the function Hðτ; s; aÞ is defined as inEq. (46).

Proof. See Hugonnier (2012, Lemma A.3). □

Lemma B.3. Let aAR be a constant. Then the stochasticdifferential equation

Yt að Þ ¼ 1�Z t

0Yu að Þ aþ su

1�su

� �εσδ dZu ð94Þ

admits a unique strictly positive solution that satisfies

Et ½YtþT ðaÞ� ¼ YtðaÞð1�HðT ; st ;2a�1ÞÞ ð95Þwhere the function Hðτ; s; aÞ is defined as in Eq. (46). Theprocess Yt(a) is a strictly positive local martingale but not amartingale.

Proof. Itô's lemma and Eq. (32) show that the processΛt ¼ st=ð1�stÞ evolves according to the driftless stochasticdifferential equation

dΛt ¼ �Λtð1þΛtÞεσδ dZt ; ð96Þand the results follow fromHugonnier (2012, Lemma A.5). □

Proof of Proposition 4. By application of Itô's lemma wehave that the weighting process evolves according to

dλt ¼ �λt 1þγþλt� � εσδ

1þγdZt ¼ �λt 1þΛt

� �εσδ dZt : ð97Þ

Therefore, the uniqueness of the solution to Eq. (94)implies that λt ¼ Ytð1Þ and the desired result now followsfrom Lemma B.3. □

Proof of Theorem 1. Combining Eq. (37) with the monotoneconvergence theorem shows that the bubble on the stocksatisfies

Bt

ð1�αÞPt¼ ρ1þγþλt

EtZ 1

te�ρðu� tÞðλt�λuÞ du

� �ð98Þ

¼ limT-1

ρ1þγþλt

EtZ T

te�ρ u�tð Þ λt � λu

� �du

� �¼ lim

T-1qt Tð Þ: ð99Þ

Taking the limit in Eq. (93) then gives Eq. (41) andsubstituting the result in Eq. (34) produces the formulafor the stock price. Furthermore, the fact that αsηT r1 andthe supermartingale property of the weighting process

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 425

implies that

E ξTST ¼ e�ρTP0

1þγþλ0E 1þγþλT� �

1�αsηT� �

re�ρTP0 ð100Þ

and because ρ40, it follows that the assumption ofPropositions 1 and 2 hold. To establish the existence anduniqueness of the equilibrium, it suffices to show thatwhenever nAð0;1� and α40 there are unique constantss0A ð0;1Þ and γ40 such that

w2 ¼ nð1�αsη0ÞP0 ¼ s0P0; ð101Þand

w1 ¼ 1�nð Þ 1�αsη0� �

P0 ¼1

1þγ1�s0ð ÞP0: ð102Þ

A direct calculation shows that the pair ðs0; γÞAð0;1Þ �ð0;1Þ is a solution to this system of equations if and only if

g s0ð Þ � n 1�αsη0� ��s0 ¼ 0 and γ ¼ γ s0ð Þ � n�s0

s0ð1�nÞ: ð103Þ

Because g(s) is continuous and strictly decreasing withgð0Þ ¼ n and gðnÞo0, we have that the nonlinear equationgðs0Þ ¼ 0 admits a unique solution s0 that lies in the openinterval ð0;nÞ, and it follows that the constant γðs0Þ isstrictly positive. □

Proof of Proposition 5. Applying Itô's lemma to the righthand side of Eq. (43) and using Eq. (32) we obtain that

eqtPt

P0

sts0

� �1=ε

¼ 1þZ t

0equ

Pu

P0

sus0

� �1=ε

� ρþμδ�σ2δ 1þ εsu

1�su

� �� �du ð104Þ

and the formula for the riskless asset price now followsfrom Eqs. (1) and (31). Let us now turn to the finite horizonbubbles. Using Eq. (13) and the law of iterated expecta-tions, we obtain

BtðTÞ ¼ St�EtZ T

tξt;uδu duþξt;TST

� �ð105Þ

¼ St�EtZ 1

tξt;uδu du�

Z 1

Tξt;uδu duþξt;TST

� �ð106Þ

¼ Bt�Et ξt;T ST � ET

Z 1

TξT ;uδu du

� �� �¼ Bt�Et ξt;TBT

:

ð107ÞTo compute the term on the right-hand side, we start byobserving that due to Eq. (93) and the monotone conver-gence theorem we have

sηu ¼ limΘ-1

qu Θ� �¼ Eu

Z 1

uρe�ρðk�uÞ λu�λk

1þγþλudk

� �; uZ0:

ð108ÞUsing this identity in conjunction with Lemmas B.2 andB.3, the definition of the equilibrium state price densityand Eq. (29) then gives

Bt�Et ½ξt;TBT �ð1�αÞPt

¼ sηt �e�ρðT� tÞEt1þγþλT1þγþλt

sηT

� �ð109Þ

¼ sηt � ρEtZ 1

Te�ρ u�tð Þ λT � λu

1þγþλtdu

� �ð110Þ

¼ sηt þqt Tð Þ� limΘ-1

qt Θ� ��e�ρ T�tð ÞEt

λT � λt1þγþλt

� �ð111Þ

¼ sηt H T � t; st ;2η� 1� � ð112Þ

and the formula of the statement now follows fromEq. (41). Using Eq. (14) we have that the finite horizonbubble on the riskless asset is given by

B0t Tð Þ ¼ S0t 1�EtξTS0TξtS0t

� �� �: ð113Þ

Because the process Mt ¼ ξtS0t evolves according to

Mt ¼ 1�Z t

0Msθs dZs ¼ 1�

Z t

0Msð1=εþXsÞϵσδ dZs; ð114Þ

we have that Mt ¼ Ytð1=εÞ by the uniqueness of thesolution to Eq. (94) and the formula of the statementnow follows from Lemma B.3. To complete the proof, itremains to show that the relative bubble on the stock isdominated by the relative bubble on the riskless asset overany horizon. Consider the function defined by

Gðτ; s; aÞ ¼ sð1þaÞ=2Hðτ; s; aÞ: ð115ÞA direct calculation using Eq. (46) shows that

∂∂a

G τ; s; að Þð Þ ¼ sð1�aÞ=2 log sð ÞG0 sð Þ; ð116Þ

∂∂a

s�aG τ; s;2a�1ð Þ� �¼ 2s1�2a log 1=s� �

N d� τ; s;2a�1ð Þð ÞZ0

ð117Þwith the function defined by

G0ðsÞ ¼ ½saNðdþ ðτ; s; aÞÞ�Nðd� ðτ; s; aÞÞ�=2; ð118Þand because

G0ð0Þ ¼ 0oNðdþ ðτ;1; aÞÞ�1=2¼ G0ð1Þ; ð119Þ

G00ðsÞ ¼ ða=2Þxb�1Nðdþ ðτ; s; aÞÞZ0; ð120Þ

we conclude that the functions Gðτ; s; aÞ and s�aGðτ; s;2a�1Þ are respectively decreasing and increasing in a.Using these properties together with the first part of theproof and the fact that stAð0;1Þ, η41 and εA ½0;1�, wethen deduce that

BtðtþTÞ=St ¼ GðT ; st ;2η�1Þ=ð1þψ ð1�sηt ÞÞ ð121Þ

rG T ; st ;2η� 1� � ð122Þ

rG T ; st ;1ð Þ ð123Þ

rs�1t G T ; st ;1ð Þ ð124Þ

rs�1=εt G T ; st ;2=ε� 1

� �¼ B0t tþTð Þ=S0t ð125Þand the proof is complete. □

Asymptotic behavior of the consumption share process.Because the consumption share process is a nonnegativesupermartingale, it converges to a well-defined limit. Anapplication of Itô's lemma to Eq. (32) shows that

0rst ¼ s0e�R t

0ððεσδÞ2=ð1� suÞÞ du�ð1=2ÞðεσδÞ2t�εσδZt

rst ¼ s0e�ð1=2ÞðεσδÞ2t�εσδZt ; ð126Þ

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428426

and the desired result then follows from the fact that, bywell known results on geometric Brownian motion, theprocess st converges to zero. □

Proof of Proposition 6. By Protter (2004, Theorem 39,p. 305) we know that the first order derivative ∇tðαÞ ¼∂stðαÞ=∂s0ðαÞ exists and satisfies

∇t αð Þ ¼ 1þZ t

0∇u αð Þ dsuðαÞ

suðαÞ�suðαÞðεσδÞ2ð1�suðαÞÞ2

du

" #: ð127Þ

Solving this linear stochastic differential equation andusing the fact that the consumption share is nonnegativethen shows that

∇t αð Þrexp �Z t

0

suðαÞðεσδÞ2ð1�suðαÞÞ2

du

!stðαÞs0ðαÞ

r stðαÞs0ðαÞ

: ð128Þ

Using the chain rule, we deduce that ∂stðαÞ=∂α¼∇tðαÞ∂s0ðαÞ=∂α and the desired result now from the factthat

s0 αð Þ ¼ n 1�αs0ðαÞη� �

⟹�∂s0ðαÞ∂α

¼ ns0ðαÞ1þη

s0ðαÞþnηs0ðαÞηð129Þ

by application of the implicit function theorem. □

Proof of Proposition 7. Let σt denote the stock volatility. ByProposition 3 and Theorem 1 we have that, along theequilibrium path, the agents' wealth are given by

W2t ¼ stPt ; ð130Þ

W1t ¼ð1�stÞPt

1þγ; ð131Þ

and

W3t ¼ St�W1t�W2t ¼ γW1t�αsηt Pt ; ð132Þand the expression for the equilibrium trading strategiesreported in the statement now follows by noting that

πit ¼1σt

∂Wit

∂PtPtσδ�

∂Wit

∂ststεσδ

� �ð133Þ

and ϕit ¼Wit�πit as a result of Eqs. (3), (32) and Itô'slemma. The signs of π1t, π2t, and ϕ2t follow from theexpressions in the statement. To establish the sign of ϕ1t

we argue as follows. Using the fact that vðsÞZ0 is non-negative, we deduce that

sign ϕ1t

¼ sign½vðsÞ�sðεþvðsÞÞ�¼ sign½�1þαsη�1ðsþηð1�sÞÞ�: ð134Þ

Denote by h(s) the continuous function inside the bracket.Because η41, this function is increasing with hð0Þ ¼ �1and hð1Þ ¼ α�1 and the result follows by continuity.Finally, because �ϕat ¼ γϕ1t�ϕ3tZ0, we have thatϕ3tr0 and the proof is complete. □

Proof of Proposition 8. To establish the desired result, wehave to find a pair of adapted processes such that

W1t ¼1

1þγ1�stð ÞPt ¼ϕS

1t Tð ÞþϕB01t Tð Þ ð135Þ

and

dW1t ¼ϕS1t Tð Þ dSt

StþδtSt

dt� �

þϕB01t Tð ÞdB0tðTÞ

B0tðTÞ�ρW1t dt; ð136Þ

where St denotes the equilibrium stock price and B0tðTÞdenotes the bubble on the riskless asset at horizon T.Expanding the dynamics of these two processes and usingEq. (3) shows that these equations are equivalent to

ϕB01t ðTÞþϕS

1tðTÞ ¼W1t ; ð137Þ

ϕB01t ðTÞΣ0tðTÞþϕS

1tðTÞð1þvðstÞÞ ¼ π1tð1þvðstÞÞ; ð138Þand solving this system gives the desired decomposition.The sign of the riskless asset bubble position follows bynoting that sign ϕB0

1t ðTÞ ¼ sign ϕ1t . To establish the sign ofthe position in the stock and complete the proof it sufficesto show that Σ0tðTÞr0 or, equivalently, that the functionHðτ; s; aÞ is increasing in s. Differentiating in Eq. (46) showsthat

s1þa∂H∂s

τ; s; að Þ ¼ a N d� τ; s; að Þð Þþ 2εσδ

ffiffiffiτ

p N0 d� τ; s; að Þð Þ;

ð139Þand the required result now follows by noting that thefunction on the right hand side is increasing on ½0;1� andequal to zero at zero. □

Proof of Proposition 9. The expression for the stock volati-lity follows from Eq. (40) and Itô's lemma. On the otherhand, differentiating with respect to α gives

∂σtðαÞ∂α

¼ εηsηt ðαÞð1�αsηt ðαÞÞ2

1þ αηstðαÞ

∂stðαÞ∂α

� �; ð140Þ

and the desired result now follows from the derivativebound of Proposition 6. □

Proof of Proposition A.1. The result follows from that ofLemma B.1 by taking bt¼0 and Πt ¼ ℓS0t . We omit thedetails. □

Proof of Theorem A.2. Because individual optimality andmarket clearing are already taken into account, we havethat the existence of an equilibrium is equivalent to theexistence of a pair ðs0; γÞA ð0;1Þ � ð0;1Þ such that

w1 ¼ nðP0�ℓÞ ¼ s0P0; ð141Þ

w2 ¼ 1�nð Þ P0�ℓð Þ ¼ ð1�s0ÞP0

1þγ; ð142Þ

and P½ft: St40g� ¼ 1, where the candidate equilibriumprice process is defined as in Eq. (67). The unique solutionto the system formed by Eqs. (141) and (142) satisfiess0 ¼ nð1�ℓ=P0Þ and it follows that ℓoP0 is necessary forthe existence of an equilibrium. Assume that this conditionis satisfied and consider the non existence result. If q40,then

StPt

¼ 1�eqtℓP0

sts0

� �1=ε

r1� ℓP0

sts0

� �1=ε

; ð143Þ

and it follows that

P ft: Str0g½ �ZP t: stZb� s0ℓP0

� ��ε� �� �ð144Þ

If ℓZ x̂P0, then bAð0;1Þ and it only remains to show thatthe probability on the right hand side is strictly positive.A direct calculation using Eq. (32) shows that a scale

J. Hugonnier, R. Prieto / Journal of Financial Economics 115 (2015) 411–428 427

function for the consumption share process is given bypðxÞ ¼ ð2x�1Þ=ð4ð1�xÞÞ and the result now follows fromKaratzas and Shreve (1991, Proposition 5.22). Let us nowturn to the existence result. If qr0o x̂P0�ℓ, then thedefinition of x̂ implies that

StPt

¼ 1�eqtℓP0

sts0

� �1=ε

Z1� ℓP0

1s0

� �1=ε

40; ð145Þ

and the desired result now follows from the fact that theaggregate dividend process is strictly positive at alltimes. □

Proof of Corollary A.3. Because the state price density is thesame as in the basic model we have that the fundamentalvalue of the stock is Ft ¼ ð1�sηt ÞPt and it thus follows fromEq. (67) that the infinite horizon bubble on the stock isgiven by

Bt ¼ St� 1�sηt� �

Pt ¼ sηt Pt�ℓS0t ¼ Pt sηt �eqtℓP0

sts0

� �1=ε" #

:

ð146ÞUnder the conditions of Theorem A.2.b ηr1=ε and it thusfollows from the second inequality in Eq. (145) that

Bt ¼ s�ηt Pt 1�eqts1=ε�η

tℓP0

1s0

� �1=ε" #

Zs�ηt Pt 1� ℓ

P0

1s0

� �1=ε" #

40:

ð147ÞUsing the above identities in conjunction with Eqs. (66),(67), the fact that str1rηr1=ε and the monotonicity ofGðτ; s; aÞ we obtain that

BtðtþTÞ ¼ Bt�Et ½ξt;TBT � ¼ sηt Pt�Et ½ξt;T sηTPT ��ℓðS0t�Et ½ξt;TS0T �Þð148Þ

¼ sηt PtH T�t; st ;2η�1� ��ℓB0t Tð Þ ð149Þ

¼ sηt PtH T�t; st ;2η�1� ��ℓS0tH T�t; st ;2=ε�1

� �ð150Þ

¼ StH T�t; st ;2=ε�1� �

þPt sηt H T�t; st ;2η�1� ��H T�t; st ;2=ε�1

� �� �ð151Þ

rStH T�t; st ;2=ε�1� �þPt G T�t; st ;2η�1

� ���G T�t; st ;2=ε�1� �� ð152Þ

rStH T�t; st ;2=ε�1� � ð153Þ

and the desired result now follows from Eq. (66). □

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