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Asset Prices and Trading Volume under FixedTransactions CostsAndrew W. LoMassachusetts Institute of Technology and National Bureau of Economic ResearchHarry MamayskyMorgan Stanley and Yale UniversityJiang WangMassachusetts Institute of Technology, China Center for Financial Research, and National Bureauof Economic ResearchWe propose a dynamic equilibrium model of asset prices and tradingvolume when agents face fixed transactions costs. We show that evensmall fixed costs can give rise to large “no-trade” regions for eachagent’s optimal trading policy. The inability to trade more frequentlyreduces the agents’ asset demand and in equilibrium gives rise to asignificant illiquidity discount in asset prices.I.IntroductionIt is now well established that transactions costs in asset markets are animportant factor in determining the trading behavior of market particWe thank John Heaton, Leonid Kogan, Mark Lowenstein, Svetlana Sussman, DimitriVayanos, Greg Willard, and seminar participants at the University of Chicago, Columbia,Cornell, the International Monetary Fund, Massachusetts Institute of Technology, University of California at Los Angeles, Stanford, Yale, the NBER 1999 Summer Institute, andthe Eighth World Congress of the Econometric Society for helpful comments and discussion. Research support from the MIT Laboratory for Financial Engineering and theNational Science Foundation (grant SBR-9709976) is gratefully acknowledged.[Journal of Political Economy, 2004, vol. 112, no. 5]䉷 2004 by The University of Chicago. All rights reserved. 0022-3808/2004/11205-0007 10.001054

asset prices and trading volume10551ipants. Consequently, transactions costs should also affect market liquidity and asset prices in equilibrium.2 However, the direction andmagnitude of their effects on asset prices, trading volume, and othermarket variables are still subject to considerable controversy and debate.Early studies of transactions costs in asset markets relied primarily onpartial equilibrium analysis. For example, by comparing exogenouslyspecified returns of two assets—one with transactions costs and anotherwithout—that yield the same utility, Constantinides (1986) argued thatproportional transactions costs have only a small impact on asset prices.However, using the present value of transactions costs under a set ofcandidate trading policies as a measure of the liquidity discount in assetprices, Amihud and Mendelson (1986b) concluded that the liquiditydiscount can be substantial, despite relatively small transactions costs.More recently, several authors have developed equilibrium models toaddress this issue. For example, Heaton and Lucas (1996) numericallysolve a model in which agents trade to share their labor income riskand conclude that symmetric transactions costs alone do not affect assetprices significantly. Vayanos (1998) develops a model in which agentstrade to smooth lifetime consumption and shows that the price impactof proportional transactions costs is linear in the costs and that forempirically plausible magnitudes their impact is small. Huang (2003)considers agents who are exposed to surprise liquidity shocks and whoare able to trade in a liquid and an illiquid financial asset. He also findsthat in the absence of additional constraints, the liquidity premium issmall.A common feature of these equilibrium models is that agents haveonly infrequent trading needs. Such models may understate the effectof transactions costs on asset prices, given the much higher levels oftrading activity that we observe empirically. This suggests the need fora more plausible model of trading behavior to fully capture the economic implications of transactions costs in financial markets.In this paper, we provide such a model by investigating the impactof fixed transactions costs on asset prices and trading behavior in acontinuous-time equilibrium model with heterogeneous agents. Investors are endowed with a nontradable risky asset, and in a frictionless1The literature on optimal trading policies in the presence of transactions costs is vast(see, e.g., Constantinides 1976, 1986; Eastham and Hastings 1988; Davis and Norman1990; Dumas and Luciano 1991; Morton and Pliska 1995; Schroeder 1998). The impactof transactions costs on agents’ economic behavior has been studied in many other contextsas well (see, e.g., Baumol 1952; Tobin 1956; Arrow 1968; Rothschild 1971; Bernanke 1985;Pindyck 1988; Dixit 1989).2See, e.g., Demsetz (1968), Garman and Ohlson (1981), Amihud and Mendelson(1986a, 1986b), Grossman and Laroque (1990), Aiyagari and Gertler (1991), Dumas(1992), Tuckman and Vila (1992), Heaton and Lucas (1996), Vayanos (1998), Vayanosand Vila (1999), and Huang (2003).

1056journal of political economyeconomy they wish to trade continuously in the market, in amounts thatare cumulatively unbounded, to hedge their nontraded risk exposure.But in the presence of a fixed transactions cost, they choose to tradeonly infrequently. Indeed, we find that even small fixed costs can giverise to large “no-trade” regions for each agent’s optimal trading policy.Moreover, the uncertainty regarding the optimality of the agents’ assetpositions between trades reduces their asset demand, leading to a decrease in the asset price in equilibrium. We show that this price decrease—an “illiquidity discount”—satisfies a power law with respect tothe fixed cost; that is, it is approximately proportional to the squareroot of the fixed cost, implying that small fixed costs can have a significant impact on asset prices. Moreover, the size of the illiquidity discount increases with the agents’ trading needs at high frequencies andis very sensitive to their risk aversion.Our model also allows us to examine how transactions costs can influence the level of trading volume. The apparently high level of volumein financial markets has often been considered puzzling from a rationalasset-pricing perspective (see, e.g., Ross 1989), and some have evenargued that additional trading frictions or “sand in the gears,” in theform of a transactions tax, ought to be introduced to actively discouragehigher-frequency trading (see, e.g., Tobin 1984; Stiglitz 1989; Summersand Summers 1990). Yet in the absence of transactions costs, most dynamic equilibrium models will show that it is quite rational and efficientfor trading volume to be infinite when the information flow to the marketis continuous, for example, a diffusion. An equilibrium model with fixedtransactions costs can reconcile these two disparate views of tradingvolume. In particular, our analysis shows that while fixed costs do implyless than continuous trading and finite trading volume, an increase insuch costs has only a slight effect on volume at the margin.We develop the basic structure of our model in Section II and discussthe nature of market equilibrium under fixed transactions costs in Section III. We derive an explicit solution for the dynamic equilibrium inSection IV and analyze the solution in Section V. Section VI reports theresults of a calibration exercise of our model, and we present conclusionsin Section VII. Proofs appear in the Appendix in the online edition ofthe article.II.The ModelOur model consists of a continuous-time dynamic equilibrium in whichheterogeneous agents trade with each other over time to hedge theirexposure to nontraded risk. Our interest in the trading process requiresthat we consider more than one agent, and because we seek to captureboth the time of trade and the quantity of trade in an equilibrium setting,

asset prices and trading volume1057we develop our model in continuous time. However, for tractability andeconomic clarity, we maintain parsimony in modeling the heterogeneityamong agents, their trading motives, and the economic environment.A.The EconomyOur economy is defined over a continuous-time horizon [0, ) andcontains a single commodity that is used as the numeraire. The underlying uncertainty of the economy is characterized by a two-dimensionalstandard Brownian motion B p {(B 1t, B 2t) : t 0} defined on its filteredprobability space (Q, F, F, P), where the filtration F p { F t : t 0} represents the information revealed by B over time.There are two traded securities: a risk-free bond and a risky stock.The bond pays a positive, constant interest rate r. Each stock share paysa cumulative dividend Dt, where冕tDt p a Dt jDdB 1s p a Dt jD B 1t(1)0and āD and jD are positive constants. The securities are traded competitively in a securities market. Let P p {Pt : t 0} denote the stock priceprocess.Transactions in the bond market are costless, but transactions in thestock market are costly. For each stock transaction, the buyer and sellerhave to pay a combined fixed cost of k that is exogenous and independent of the amount transacted. The allocation of this fixed cost betweenbuyer and seller, denoted by k and k , respectively, is determined endogenously in equilibrium. More formally, the transactions cost for atrade d is given byk k(d) p 0k for d 1 0for d p 0for d ! 0,{(2)where d is the signed volume (positive for purchases and negative forsales), k is the cost to the buyer, k is the cost to the seller, and thesum k k p k is fixed.There are two agents in the economy, indexed by i p 1, 2, and eachagent is initially endowed with no bonds and v̄ shares of the stock. Inaddition, agent i is endowed with a stream of nontraded risky incomewith cumulative cash flow Nti, where冕tN p it0i( 1)XsdB 1s,(3a)

1058journal of political economyX t p jXB 2t,(3b)iand jX is a positive constant. For future reference, we let X ti { ( 1)Xt.iThe term B 2t specifies the nontraded risk, and X t gives agent i’s exposureto the nontraded risk at time t. Since X t1 X t2 p 0 for all t, there is noaggregate nontraded risk. In addition, the nontraded risk is assumedto be perfectly correlated with stock dividends, allowing the agents touse the stock to hedge their nontraded risk. Since each agent’s exposureto nontraded risk, X ti, is stochastic, he desires to trade in the stock marketcontinuously to hedge his nontraded risk as it changes over time. Thepresence of this high-frequency trading need is essential in analyzinghow transactions costs—which prevent the agents from trading continuously—affect their asset demands and equilibrium prices.Each agent chooses his consumption and trading policy to maximizethe expected utility from his lifetime consumption. Let C denote theagents’ consumption space, which consists of F-adapted, integrable consumption processes c p {ct : t 0}. The agents’ stock trading policy spaceconsists of only “impulse” trading policies, defined as follows.Definition 1. Let { {1, 2, }. An impulse trading policy {(tk ,dk) : k 苸 } is a sequence of trading times tk and trade amounts dk suchthat (1) 0 tk tk 1 almost surely for all k 苸 , (2) tk is a stoppingtime with respect to F, (3) dk is measurable with respect to Ftk, (4) dk d̄ ! , and (5) E 0[e gkn(s)] ! , wheren(s) {冘{k : 0 tk s}1gives the number of trades in time [0, s].Conditions 1–3 are standard for impulse policies. Conditions 4 and5 are imposed here for technical reasons. Condition 4 requires thattrade sizes be finite.3 Condition 5 requires that trading not be frequentenough to generate infinite trading costs. These are fairly weak conditions that any optimal policy should satisfy.Agent i’s stock holding at time t is vti, given byvti p v0i 冘{k : tki t}dki ,(4)where v0i is his initial endowment of stock shares, which is assumed tobe v̄.3Limiting trade sizes to be finite rules out potential doubling strategies. Effectively, werequire the trading policy to be in the L space, which is a standard condition in continuous-time settings.

asset prices and trading volume1059itLet M denote agent i’s bond position at t (in value). The term M tirepresents agent i’s liquid financial wealth. Then冕titM p冕t(rM cs)ds is0(vsidDs dNsi) 0冘{k : 0 tik t}(Ptki dik kki),(5)where kki p k(dki ) is given in (2). Equation (5) defines agent i’s budgetconstraint. Agent i’s consumption/trading policy (c, d) is budget feasibleif the associated M t process satisfies (5).Both agents are assumed to maximize expected utility of the form[冕 u(c) p E 0 ]e rt gc tdt ,0(6)where r and g (both positive) are the time discount coefficient and therisk aversion coefficient, respectively. To prevent agents from implementing a “Ponzi scheme,” we impose the following constraint on theirpolicies for all g 1 0: E 0[ e rt r g(Mt vtPt) r gp X t] ! Gt 0,iii(7) where p is an arbitrary positive number, representing the shadow pricefor future nontraded income.4 The set of budget feasible policies thatalso satisfies the constraint (7) gives the set of admissible policies, whichis denoted by V.For the economy to be properly defined, we require the followingcondition:4g 2jX2 1,(8)which limits the volatility in each agent’s exposure to the nontradedrisk.B.Definition of EquilibriumDefinition 2. An equilibrium in the stock market is defined bya price process P p {Pt : t 0} that is progressively measurable withrespect to F,b. an allocation of the transactions cost (k , k ) between buyer andseller, anda.4A more conventional constraint is imposed only on the terminal data. In the absenceof nontraded income, the usual terminal condition is limtr E0[ e rt rgWt] p 0, whereWt p Mt vPt is the terminal financial wealth. In this paper, for convenience, we imposea stronger condition (7), which limits agents from running an unbounded financial deficitat any point in time, not just in the limit.

1060c.journal of political economyagents’ trading policies {(t , d ) : k 苸 }, i p 1, 2, given the priceprocess and the allocation of transactions costs,ikiksuch that1.each agent’s consumption/trading policy maximizes his lifetime expected utility:[冕 J { sup E i(c,d)苸V2.]e rt gc tdt ,0i(9)the stock market clears: for all k 苸 ,tk1 p tk2(10a)d1k p dk2.(10b)andC.DiscussionBy assu