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Journal of Financial Economics

Journal of Financial Economics 111 (2014) 589–608

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Trading in derivatives when the underlying is scarce$

Snehal Banerjee a,n, Jeremy J. Graveline b

a Northwestern University, Kellogg School of Management, United Statesb University of Minnesota, Carlson School of Management, United States

a r t i c l e i n f o

Article history:Received 4 March 2013Received in revised form14 June 2013Accepted 15 July 2013Available online 4 December 2013

JEL classification:G12G13

Keywords:ScarcityShort-sellingPrice distortionsDerivativesRegulation

5X/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.jfineco.2013.11.008

are grateful for comments from Darrell Duffrdan, Arvind Krishnamurthy, Xuewen Liu, Mat the Minnesota Junior Finance Conferencediation Research Society Conference (2012in Palo Alto (2012).esponding author at: Kellogg School of MUniversity, 2001 Sheridan Road, Evansto

el.: +1 847 491 8335.ail addresses: snehal-banerjee@kellogg.northrjee), jeremy@umn.edu (J.J. Graveline).mothy Ryan, chief executive officer of Secl Markets Association (SIFMA), in responsemodity derivatives proposed by the Commosion (CFTC) under the mandate of the Dl Times, December 5, 2011). In December 2ional Swaps and Derivatives Association (ISDC over the proposed position limits. In Septe

a b s t r a c t

Regulatory restrictions and market frictions can constrain the aggregate quantity of longand short positions in a security. When these constraints bind, we refer to the security asscarce, and its price becomes distorted relative to its value in a frictionless market. Weshow that an otherwise redundant derivative can reduce the price distortion of theunderlying security by relaxing its scarcity. We also show that it is especially important toanalyze the underlying and derivative markets jointly when evaluating the impact ofregulation, such as short-sales bans and position limits in derivatives, that restricts trade.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

It is critical that regulatory rulemaking ... is done right,with the proper analysis to ensure that any new rulesdo not impede the function of the markets they aremeant to protect.1

All rights reserved.

ie, Nicolae Gârleanu,oto Yogo, and parti-(2011), the Financial), and the NBER AP

anagement, North-n, IL 60208, United

western.edu

urities Industry andto the position limitsdity Futures Tradingodd-Frank Act (see011, SIFMA and theA) filed suit against

mber 2012, less than

Many recent policy changes have focused on restrictingtrade in derivatives and the securities that underlie them. Forexample, the Securities and Exchange Commission (SEC)imposed short-selling bans on financial stocks in the fall of2008, and many European countries imposed similar restric-tions for bank stocks during the recent Eurozone crisis.On the derivatives side, the European Union has bannednaked credit default swap (CDS) positions on sovereign debt,and the CFTC has proposed position limits on certaincommodity derivatives to “diminish, eliminate or prevent”excessive speculation.

By restricting trade in a security, these regulations maydistort prices. For instance, a short-sales ban preventspessimistic short-sellers from trading, but it also preventslong positions from being larger, in aggregate, than the

(footnote continued)two weeks before the limits were set to take effect, US district judgeRobert Wilkins ruled against the proposed limits, arguing that moreanalysis was required by the CFTC to assess if the proposals werenecessary and appropriate under the law.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608590

security's outstanding supply. More generally, regulatoryrestrictions and market frictions, such as transactionscosts, search costs, short-sales constraints, and marginrequirements, can constrain the aggregate quantity of longand short positions in a security. When these constraintsbind, we refer to the security as scarce, and the price of thesecurity must adjust to clear the market.

In frictionless markets, a simple derivative security isredundant because it provides exposure to the samesource of risk as the underlying asset. However, we showthat this redundancy no longer applies when the under-lying can be scarce. In fact, the presence of a derivativemay affect the price of the underlying itself. Intuitively,derivatives can reduce the scarcity of the underlying byproviding a substitute for long and short positions. Wecharacterize how equilibrium prices and trading volume inthe underlying asset and its derivative are jointly deter-mined, and provide sufficient conditions under which thepresence of a derivative reduces both the scarcity of theunderlying and the associated price distortion.

Our model provides a framework to analyze the effectof proposed regulations that restrict trade in the under-lying or derivative securities. It is important to note thatprices and quantities that are empirically observed in theabsence of these proposed trading restrictions cannot beused directly to test the impact of the restrictions. Instead,to evaluate the effects of such policy, one must jointlycharacterize the equilibrium in the underlying and deri-vative markets under the counter-factual assumption thatthe restrictions are in effect. To provide a role for regula-tory trading restrictions, we assume that some investorsmay trade for non-informational motives, and therefore,distort prices relative to the underlying asset's fundamen-tal value. In this framework, we show that even if trade inthe derivative is always accompanied by distortionarytrading by speculators, restricting trade in the derivativemay increase the overall price distortion for an underlyingasset that is scarce. Moreover, if the underlying asset isscarce and the derivative is not a perfect substitute, thenwe show that a short-sales ban can actually lower the priceof the security, instead of raising it. As such, our analysishighlights the importance of accounting for scarcity in theunderlying, and jointly modeling the underlying andderivative markets, when evaluating policy decisions.

Our analysis is applicable to any security that may bescarce, including both liquid and illiquid assets. Generallyspeaking, liquidity captures the ease with which a parti-cular security can be traded, and the literature has focusedon the role of various frictions in generating illiquidity.2

In contrast, the notion of scarcity reflects the aggregatedemand for both long and short positions in an assetrelative to the capacity for such positions that it can

2 These frictions include transactions costs (e.g., Amihud andMendelson, 1986; Duffie, 1996; Vayanos, 1998; Krishnamurthy, 2002;Acharya and Pedersen, 2005; Bongaerts, De Jong, and Driessen, 2011),search frictions (e.g., Duffie, Gârleanu, and Pedersen, 2002; Vayanos andWeill, 2008), and asymmetric information (e.g., Kyle, 1985; Wang, 1993;Gârleanu and Pedersen, 2004). See Amihud, Mendelson, and Pedersen(2005) and Vayanos and Wang (2012) for excellent surveys of theliterature on liquidity and asset prices.

support. For example, on-the-run Treasuries are extremelyliquid, but the demand for long and short positions inthese securities often exceeds the supply that are availableto be borrowed in the financing, or repo, market. Whenthis situation occurs, the Treasury is scarce — it is costly toborrow and trades “on special” in the financing market.Off-the-run Treasuries are also extremely liquid, but theyare not typically scarce, since the demand for positions ismost often concentrated in their on-the-run counterparts.Conversely, while the corporate bond market is much lessliquid than the Treasury market, corporate bonds can alsobe scarce when the demand for positions to hedge orspeculate on default risk exceeds the supply of corporatebonds that are available to be traded. Other illiquid assets,such as real-estate and commodities, can also be scarce ifthere is sufficient demand for positions but they areinherently difficult to borrow and sell short.

The rest of the paper is organized as follows. In the nextsection, we briefly discuss the related literature. In Section 3,we develop a benchmark model and characterize equili-brium prices and quantities in both markets. We also derivesufficient conditions on preferences and payoff distributionsunder which the presence of a derivative security reducesthe price distortion in the underlying. In Section 4, wedevelop a general framework that allows us to analyze theimplications of scarcity on standard policy changes such aslimits on derivative trading and short-selling bans. Section 5concludes. All proofs are in Appendix A.

2. Related literature

In general, the equilibrium prices of existing securitieschange when a derivative security, exposed to risks thatare not spanned by those securities, is introduced into aneconomy (e.g., see Detemple and Selden, 1991; Zapatero,1998; Boyle and Wang, 2001; Bhamra and Uppal, 2009).3

In contrast to these earlier papers, where derivativescomplete the market by allowing investors to trade newsources of risk, we show that the presence of a derivativesecurity can affect the price of the underlying asset, evenwhen both securities span the same risks. In our setup, thederivative makes the market more “complete” by relaxingthe constraint on the aggregate capacity for positions inthe underlying asset.4 As such, the mechanism throughwhich derivatives affect the price of the underlying is alsodistinct from, but complementary to, other channels thathave been suggested in the literature, such as reducingtransactions costs and search frictions (e.g., Merton, 1989;Gârleanu, 2009).

Our paper is closely related to the large literature thatexplores the effect of trading frictions on asset prices and,in particular, to earlier models that generate costly

3 However, much of the existing literature on derivatives assumesthat the presence of such a derivative does not affect the price of theunderlying security — this is described by Hakansson (1979) as “TheCatch 22 of Option Pricing.”

4 Jordan and Kuipers (1997) provide direct empirical evidence of theeffect of trading in a derivative on the price of the underlying security inU.S. Treasury markets.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 591

borrowing through frictions such as transaction costs (e.g.,Duffie, 1996; Krishnamurthy, 2002), and search costs (e.g.,Duffie, Gârleanu, and Pedersen, 2002; Vayanos and Weill,2008).5 While our model is more stylized than some of themodels in these papers, it offers greater tractability and allowsus to characterize how trade in derivative securities affects thecost of borrowing an underlying that may be scarce. Ourmodel also provides an intuitive and flexible framework inwhich to analyze the effects of regulatory policy that restrictstrade in the derivative or the underlying asset.

More generally, our paper relates to the literature onfinancial innovation and security design, including early workby Allen and Gale (1988), Duffie and Jackson (1989), Cuny(1993), and Rahi (1995), and more recent work by Simsek(2011), Kubler and Schmedders (2012), Shen, Yan, and Zhang(2012), and others.6 In contrast to the general approach inthese papers, our model focuses on a particular form ofmarket incompleteness (i.e., the constrained capacity forpositions in the underlying), and we specifically derive howthe price is affected by the presence of a derivative. Moreover,while some of these papers focus on how the introduction ofderivatives can increase the distortion in equilibrium alloca-tions (e.g., Simsek, 2011) and prices (e.g., Kubler andSchmedders, 2012), we highlight a channel through whicheven simple derivatives can help reduce the distortion in theequilibrium price of the underlying. As such, we view ouranalysis as complementary to these other papers.

Finally, our paper relates to the literature that studiesthe effect of short-sale constraints on asset prices. Standardintuition suggests that short-sales restrictions constrainpessimists from expressing their views and thereforeincrease the price of a security (e.g., Miller, 1977). A numberof subsequent papers characterize conditions under whichthis overpricing result fails to hold (e.g., Diamond andVerrecchia, 1987; Bai, Chang, and Wang, 2006; Gallmeyerand Hollifield, 2008). In our model, the mechanism throughwhich short-sale constraints can lower the security's priceis analogous to the one in Duffie, Gârleanu, and Pedersen(2002), but we characterize conditions under which theresult obtains in the presence of derivatives.

3. Benchmark model

We begin by developing a benchmark model to high-light the effect of derivatives on the scarcity of the under-lying asset. Section 3.1 presents the setup of the model andSection 3.2 provides a discussion of our assumptions.In Section 3.3, we present the main analysis, including acharacterization of the equilibrium in the underlying andderivative markets. In Section 3.4, we characterize suffi-cient conditions, for general utility functions and payoff

5 A number of papers document this distortion in the price of scarcesecurities, in the form of a price premium paid for securities that arecostly to borrow (i.e., trade “on special”). These papers include Jordan andJordan (1997), Krishnamurthy (2002), Goldreich, Hanke, and Nath (2005),and Banerjee and Graveline (2013), which show a positive relationbetween prices and borrowing fees in bond markets, and D'Avolio(2002), Geczy, Musto, and Reed (2002), and Ofek and Richardson(2003), which document it in equity markets.

6 See Allen and Gale (1994) and Duffie and Rahi (1995) for compre-hensive surveys and discussions of the earlier literature in this area.

distributions, under which the scarcity of the underlyingasset is reduced when investors can trade a derivative.

3.1. Model setup

Securities and payoffs. There are two dates and threesecurities in the market. The risk-free security is normal-ized to pay a net return of zero. The risky asset, which werefer to as the underlying, trades at a price P and paysoff a normally distributed fundamental value F �Nðm; νÞ inthe next period. The derivative security has a price D,and a payoff of Fþɛ in the next period, where ɛ�Nð0; δÞ isnormally distributed and independent of the fundamentalvalue F. The net supply of the underlying security is givenby Q40, and the derivative is in zero net supply. Short-sellers in the underlying security must borrow it from longinvestors and pay a borrowing fee of RZ0.

Investor beliefs and preferences. There are two groups ofinvestors, indexed by iAfL; Sg, with constant absolute riskaversion (CARA) utility over next period's wealth, risk-tolerance τ, and common beliefs. An investor from group iis endowed with an exposure ρi to the fundamental shock F,where ρS ¼ �ρL ¼ ρ. We denote investor i's position in theunderlying asset by xi, and her position in the derivative by yi.

We focus on the range of parameters such that Linvestors are long in the underlying asset and S investorsare short. Each short position must be borrowed from along. However, since the underlying asset is in positive netsupply, not every long position can be loaned in equili-brium to a short-seller (i.e., 0r�xS=xLo1). To convey ourbasic intuition more clearly, we exogenously fix the max-imum fraction 0rγo1 of long positions that are bor-rowed by (or equivalently, loaned to) shorts. While wetake a reduced-form approach in our benchmark model, inAppendices B and C we characterize the conditions underwhich our results are robust to endogenizing this fractionγ, as discussed below.

Investor i's wealth in the next period is given by

Wi ¼Wi;0þρiFþxiðF�PþγiRÞþyiðFþɛ�DÞ; ð1Þwhere γirγo1 if i is long in the underlying (i.e., xi40)and γi ¼ 1 if i is short (i.e., xio0). The γi term captures thefact that short-sellers must borrow each unit they sell at aborrowing fee R, while only a fraction γ of long positionscan be borrowed by (loaned to) short-sellers. Investor imaximizes expected utility UiðWiÞ, given by

U Wið Þ ¼ �E exp � 1τWi

� �� �: ð2Þ

3.2. Discussion of assumptions

In any equilibrium, since the underlying is in positivenet supply, not every long position can be borrowed by (orequivalently, loaned to) a short-seller.7 There are instanceswhen the upper bound, γ, on borrowing and lending is

7 However, there may be significant heterogeneity in the fraction ofeach long investor's position that they lend to short-sellers. We abstractaway from this heterogeneity in order to maintain tractability and moreclearly highlight the intuition for our results.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608592

exogenous (e.g., due to short-sales bans or finite marketcapacity to clear trades). However, one might expect that,in general, γ is determined endogenously. We consider twosuch scenarios in the appendices.

In Appendix B, we consider a model in which longinvestors must pay a cost to lend out a fraction γ of theirposition to short-sellers. In this case, the optimal fraction γlent out in equilibrium trades off the marginal cost oflending out an additional unit with the marginal benefit ofgetting the lending fee R from doing so. In Appendix C, weconsider an alternate setup in which short-sellers mustpay a cost to search for long investors in order to borrowshares and establish a short position. In this case, theoptimal search intensity (which, in turn, determines theequilibrium fraction of long positions that are borrowed)sets the marginal cost of searching equal to the increase ina short-seller's expected utility from being able to trade inthe underlying security (i.e., from locating a long investor).The analysis in these appendices characterizes conditionsunder which the results from our benchmark modelextend to these settings.

Note that the constraint on borrowing and lending (i.e.,γ) can also be interpreted as a collateral constraint faced byinvestors (e.g., Geanakoplos, 2003; Simsek, 2013). Forinstance, suppose the initial wealth of L investors reflectstheir endowment of the risky asset (i.e., W0 ¼ QP for theaggregate long investor), and the collateral constraintimplies that their position in the risky asset can be atmost a multiple κ of their initial wealth (i.e., xLPrκW0).In this case, the link between the collateral constraint κand the borrowing constraint γ is given by γ ¼ ðκ�1Þ=κ.8

We focus exclusively on a simple derivative that isotherwise redundant if the underlying asset is not scarce.As such, we do not consider more complex derivatives (suchas options with nonlinear payoffs) that would “complete”the market in a more traditional sense by providing expo-sure to a risk that investors wish to trade but cannot do sousing only the underlying security.9 This assumption allowsus to focus exclusively on the role of derivatives in relaxingthe scarcity of the underlying.

We assume that the derivative offers a potentially noisyexposure, Fþɛ, to the fundamental risk, F, so that it maynot be a perfect substitute for the underlying asset. Forexample, the cheapest-to-deliver option in many exchangetraded derivatives introduces additional noise in theirpayoff, as does the fact that the secondary market forcustomized over-the-counter derivatives is often relativelyilliquid.10 In the extreme, the derivative markets for someassets may not even exist (e.g., futures on individual

8 This follows from comparing the constraint on collateral, given byxLPrκQP, to the constraint on long positions implied by γ, given byxLrQ=ð1�γÞ.

9 For example, if the volatility of an asset is stochastic, then an optionon that asset can complete the market (in the traditional sense) byallowing investors to explicitly trade this risk.

10 For instance, the primary market for over-the-counter interest rateswaps is generally considered to be extremely liquid, but the secondarymarket is not. As a result, rather than use the secondary market to exit anexisting swap position, participants in this market typically take anoffsetting position in a new swap (in the primary market) and are,therefore, left with some residual basis risk.

stocks and bonds), and so investors may be forced to usederivatives on related assets. In the next subsection, weillustrate that when noise in the derivative payoff is present(i.e., δ40), it effectively constrains investors' equilibriumpositions in the derivative security. In Appendix D, wedevelop a model that eliminates the noise in the derivativepayoff, but instead imposes position limits in the derivativemarket. Our main result is qualitatively similar in thismodel — increasing ease of trade in the derivative (i.e.,relaxing the limits on derivative positions) reduces the pricedistortion in the underlying asset by relaxing the effectivescarcity that investors face.11

For simplicity, we assume that ɛ is uncorrelated with F.The model can be easily adjusted for the noise ɛ to becorrelated with F by redefining Fþɛ¼ αFþη, where η is thecomponent of ɛ that is uncorrelated with F. Moreover, aswe show in Section 3.4, if we maintain the assumptionthat investors exhibit constant absolute risk aversion, ourmain result is robust to relaxing the assumption that F andɛ are normally distributed.

3.3. Market clearing and equilibrium

An equilibrium in this market is defined as the set ofprices P, D, and R, and the positions xi (in the underlying)and yi (in the derivative) for each group i of investors, suchthat (i) the positions xi and yi maximize the utility of agenti given by Eq. (2) subject to the budget constraint in Eq. (1),and (ii) the cash and financing markets for the underlyingand the market for the derivative are cleared. The deriva-tive is in zero net supply, so the market clearing conditionfor it is given by

yLþyS ¼ 0: ð3ÞThe cash market clearing condition for the underlyingasset is given by

xLþxS ¼ Q ; ð4Þand the financing market clearing condition is given by

�xSrγxL: ð5ÞThe financing market clearing condition binds with equal-ity when there is a strictly positive fee R40 to borrow thesecurity, since long investors would like to lend out asmuch of their position as possible. Moreover, since short-sellers can borrow at most a fraction γ from longs, Eqs. (4)and (5) imply that the maximum aggregate long positionin the underlying is Q=ð1�γÞ, and the maximum aggregateshort position is �γQ=ð1�γÞ.

If borrowing and lending are unconstrained, then thefrictionless price of the underlying reflects the (risk-adjusted)

11 As we discuss in Appendix D, one notable difference with thissetup is that when the constraint in the underlying and the position limitin the derivative are both binding, the price of the derivative is bounded,but not determinate without additional assumptions (e.g., bargainingbetween L and S investors over the gains from trade). This result isbecause the derivative is in zero net supply and so L and S investors mustalways pay the same price. In contrast, as we discuss below, a positiveborrowing cost R in the underlying asset allows the market to clear evenwhen aggregate positions in it are constrained, since L and S investorspay different net prices for the underlying (P�γR and P�R, respectively)when R40.

Fig. 1. Scarcity and the distortion in price of the underlying. The toppanel plots the frictionless price (horizontal, dotted line), P0, the (inverse)demand functions for long and short positions in the underlying security(downward sloping and upward sloping dashed lines, respectively), andcharacterizes the range of γ for which the underlying security is scarce.In addition, the bottom panel plots the equilibrium price (solid, nonlinearcurve), P.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 593

expected value of its payoff, and is given by

P0 ¼m� ν

2τQ : ð6Þ

Also, there is no cost to borrow the security and theequilibrium quantities are given by

�yS;0 ¼ yL;0 ¼ 0 and ð7Þ

Q�xS;0 ¼ xL;0 ¼ 12Qþρ: ð8Þ

If the constraint on borrowing and lending does bind,then the price of the underlying is distorted relative to thefrictionless price P0. The following proposition charac-terizes this distortion, as well as the rest of the equilibriumin this case.

Proposition 1. Given the economy above with L and Sinvestors, the equilibrium prices are given by

D¼m� ν

2τQ ; P ¼ P0þ

1þγ

2R; and ð9aÞ

R¼max 0;δ

δþν

11�γ

ν

τ2ρ� 1þγ

1�γQ

� �� �; ð9bÞ

and the equilibrium quantities are given by

�yS ¼ yL ¼1�γ

δ

τ

2R and Q�xS ¼ xL ¼

11�γ

Q if R40;

12Qþρ if R¼ 0:

8>><>>:

ð10Þ

The proof is a special case of Proposition 4 in Section 4that follows. To gain some intuition for the relation betweenequilibrium prices and quantities, recall from Eq. (8) thatxL;0 ¼ 1

2Qþρ and xS;0 ¼ 12Q�ρ are the equilibrium quanti-

ties in the underlying asset when its price is P0 as given byEq. (6) and the constraint on borrowing and lending doesnot bind. Market clearing implies that the maximumaggregate long position in equilibrium is Q=ð1�γÞ and themaximum aggregate short position is �γQ=ð1�γÞ. There-fore, if

xL;0 ¼12Qþρ4

11�γ

Q and xS;0 ¼12Q�ρo� γ

1�γQ ;

ð11Þthen the constraint on borrowing and lending binds, sincethe aggregate demand for long and short positions exceedsthe capacity that the underlying can support. From Eq. (11),the constraint binds if and only if,

0o2ρ� 1þγ

1�γQ : ð12Þ

The borrowing cost, R (see Eq. (9)), is positive when theconstraint binds and Eq. (12) holds. It allows the cash marketto clear because longs and shorts pay different net prices forthe underlying (P�γR and R�P per unit, respectively).

Fig. 1 illustrates the notion of scarcity and its effect onthe price of the underlying security. The top panel plotsthe inverse aggregate demand curves for long and shortpositions in the underlying. Since L investors are willing tohold larger long positions when the net price they pay for

the security is lower, the aggregate demand curve for longpositions is downward sloping. Similarly, the aggregatedemand curve for short positions is upward sloping since Sinvestors are willing to hold larger short positions only ifthe net price of the underlying is higher. The aggregatedemand curves intersect at the price P0, which is thefrictionless price for the underlying security. If the capacityfor aggregate positions beyond the outstanding is to theright of this intersection point (which in the plot is atγ=ð1�γÞ ¼ 2, so that γ ¼ 2

3), then the security is not scarce,and its price is given by P0. However, if the capacity foraggregate positions is constrained to the left of the inter-section, then the demand for long and short positionscannot clear at the same net price for long investors andshort-sellers, and the security is scarce. In this case, as thebottom panel of Fig. 1 illustrates, the cash and borrowingmarkets for the underlying can only clear if the cost toborrow the underlying security becomes positive (i.e.,R40), so that long investors and short-sellers pay differ-ent net costs for the underlying security (i.e., P�γR andP�R, respectively). As a result, the equilibrium price P ofthe underlying security (plotted as the solid curve in thebottom panel) is distorted relative to the frictionless priceP0. In particular, even though investors have the samebeliefs about fundamentals and their hedging demandsperfectly offset each other, the equilibrium price for theunderlying is higher than P0 when it is scarce.

Positions in the derivative security provide a substitutefor positions in the underlying asset, and therefore can

Fig. 2. The effect of the derivative on the price of the underlying. The plotdepicts the effect of introducing a derivative on the demand for long andshort positions in the underlying, and the resulting effect on the price ofthe underlying security. The dashed lines reflect the demand for long andshort positions when there is no derivative, while the dotted lines reflectthe demand for long and short positions in the presence of the derivative.Similarly, the solid line reflects the equilibrium price of the underlyingsecurity when there is no derivative, while the dot-dashed line reflectsthe underlying price in the presence of a derivative.

12 To highlight the sufficient conditions as generally as possible, weabstract away from the underlying parameter of the payoff distribution orpreferences that decreases the equilibrium holdings of the derivative.However, what we have in mind is a change in a fundamental parameter(e.g., the variance of the noise in the derivative payoff) that affectsinvestors' positions in the derivative, and our goal in this section is tocharacterize the effect of this change in derivative positions on the priceof the underlying security.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608594

relax the scarcity in the underlying. Investors trade offtheir demand for positions in the underlying risk againstthe additional noise ɛ in the payoff of the derivative.As Fig. 2 illustrates, when investors can trade in a deriva-tive security, the aggregate demand for long and shortpositions in the underlying are smaller at each net pricefor the underlying. Therefore, in the presence of thederivative, the underlying security is scarce for a smallerrange of γ (the intersection of the demand curves for longand short positions shifts left), and the price distortion inthe underlying security is smaller. In contrast to a friction-less market where the derivative is a redundant security,the presence of the derivative in this case can affect theprice of the underlying asset. The following propositionsummarizes this result.

Proposition 2. The distortion in the price of the underlyingasset, ΔP � P�P0, decreases with the noise δ in the derivativesecurity. Therefore: (a) as the noise in the derivative (i.e., δ)becomes arbitrarily small, the price distortion in the under-lying disappears, i.e., limδ-0ΔP ¼ 0, and (b) the price distor-tion in the underlying is largest if investors do not have accessto a derivative (or equivalently, the noise in the derivative isinfinite).

Finally, one might expect that the tradeoff between thenoise δ and the constraint γ on borrowing and lendingwould be reflected in both the price and the volume oftrade in the derivative. However, in our benchmark model,the price D of the derivative is a “cleaner” measure of the(risk-adjusted) expected value of the underlying security,in that it does not depend on δ or γ. Specifically, thederivative price D is the market's risk-tolerance weightedexpectation of F, adjusted for aggregate risk due to thefundamental F. Instead, the tradeoff between γ and δmanifests itself in the equilibrium positions in the deriva-tive. All else equal, derivative positions are smaller whenthe lending constraint is less binding and when the noisein the derivative is higher (i.e., equilibrium derivativepositions are decreasing in γ and δ, respectively).

The scarcity of the underlying is closely related to thelending fee R, which is typically small in the data. However,the prices and quantities that we observe empirically reflectthe equilibrium scarcity in the presence of derivatives andin the absence of the proposed regulatory restrictions.Therefore, a small lending fee does not imply that the roleof scarcity is unimportant for evaluating regulatory policy.Instead, as we highlighted in the introduction, one mustanalyze equilibrium prices and quantities under the counter-factual assumptions of imposing the proposed restrictions.We take up this task in Section 4.

3.4. General utility functions and payoff distributions

In this subsection, we characterize general sufficientconditions under which the presence of the derivativeincreases or decreases the price distortion in the under-lying. We begin with some general notation. Let uiðWiÞ beagent i's increasing and concave utility function over wealthWi. As in our benchmark model, we assume that the payoffof the underlying, F, and the noise in the derivative payoff,ɛ, are independent and that E½ɛ� ¼ 0. However, we donot impose any additional distributional assumptions. LetxiðΠi;DÞ and yiðΠi;DÞ be agent i's optimal demand for theunderlying and derivative, respectively, where D is the pricefor the derivative and Πi ¼ P�γiR is the net price that agenti pays for the underlying. That is,

fxiðΠi;DÞ; yiðΠi;DÞg ¼ arg maxx;y

Ei½uiðWiÞ� where ð13Þ

Wi ¼W0;iþρiFþxðF�ΠiÞþyðFþɛ�DÞ; ð14Þand ρLo0oρS. Let P, R, and D denote the equilibrium pricesthat clear the cash, financing, and derivative markets,respectively. That is,

xLðP�γR;DÞþxSðP�R;DÞ ¼Q ; ð15aÞ

γxLðP�γR;DÞþxSðP�R;DÞZ0; ð15bÞand

yLðP�γR;DÞþySðP�R;DÞ ¼ 0: ð16ÞWhen the asset is scarce, Eq. (15b) binds with equality andcan be combined with Eq. (15a) to produce

xL P�γR;Dð Þ�Q ¼ γQ1�γ

¼ �xS P�R;Dð Þ: ð17Þ

Our main result in Section 3.3 is that the price of theunderlying security is higher (more distorted) when thereis no derivative. In the more general current setting, it issufficient to show that the equilibrium price of the under-lying is decreasing in the equilibrium derivative positionsof the investors, as the following result characterizes.12

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 595

Proposition 3. Suppose the underlying security is scarce (i.e.,condition (17) holds).

(i)

If for all iAfL; Sg, we have

∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂Do0

everywhere, then the price of the underlying is higher whenthere is no derivative available to trade.

(ii)

If for all iAfL; Sg, we have

∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂D40

everywhere, then the price of the underlying is lowerwhen there is no derivative available to trade.

In the proof of Proposition 3, we show that the sign of an

investor's position in the derivative must be the same as thesign of his or her position in the underlying asset. Therefore,to move towards an equilibriumwith no derivative positions,L investors must decrease their position in the derivative andS investors must increase their position (i.e., they must take asmaller short position in the derivative). Since the price ofthe underlying security can be expressed as

P ¼ 11�γ

ðP�γRÞ|fflfflfflffl{zfflfflfflffl}ΠL

� γ

1�γðP�RÞ|fflfflffl{zfflfflffl}

ΠS

; ð18Þ

a sufficient condition for the price of the underlying to increase(decrease) with smaller derivative positions is that dΠi=dyio0(dΠi=dyi40, respectively) for both i¼L and i¼S investors.

To see why the conditions in Proposition 3 are suffi-cient for the above conditions, suppose that the net pricethat agent i pays for the underlying changes by dΠi. Sincethe underlying asset is scarce, the equilibrium positions init are xL ¼Q=ð1�γÞ and xS ¼ �γQ=ð1�γÞ, and must remainunchanged for the market to continue to clear. That is,

dxi ¼∂xi∂Πi

dΠiþ∂xi∂D

dD¼ 0; ð19Þ

which, in turn, implies that the equilibrium price of thederivative must change by

dD¼ � ∂xi=∂Πi

∂xi=∂DdΠi: ð20Þ

Along this equilibrium path, the change in agent i'soptimal position in the derivative is

dyi ¼∂yi∂Πi

dΠiþ∂yi∂D

dD¼ ∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂D

� �dΠi; ð21Þ

which implies the sign of dΠi=dyi is characterized by theconditions in Proposition 3. The following result providesintuition for the conditions in Proposition 3.

Corollary 1. Assume that the underlying security is scarceand the demand curves for the underlying and the derivativeare downward sloping (i.e., ∂xi=∂Πio0 and ∂yi=∂Do0).

1.

13 In contrast, condition (23) seems less intuitive since it requiresthat the optimal demand for a security is always less sensitive to itsown price.

Suppose that, for all iAfL; Sg, either of the two followingconditions always holds:

∂yi∂Πi

o ∂xi

∂Πi

and

∂xi∂D

o ∂yi

∂D

or ð22aÞ

∂xi∂D

o ∂xi

∂Πi

and

∂yi∂Πi

o ∂yi

∂D

: ð22bÞ

Then

(a) The price of the underlying is higher in the absenceof a derivative if the substitution effect alwaysdominates the wealth effect for both securities (i.e.,∂xi=∂D40 and ∂yi=∂Πi40).

(b) The price of the underlying is lower in the absence ofa derivative if the wealth effect always dominatesthe substitution effect for both securities (i.e.,∂xi=∂Do0 and ∂yi=∂Πio0).

2.

Suppose that, for all iAfL; Sg, either of the two followingconditions always holds:

∂xi∂Πi

o ∂yi

∂Πi

and

∂yi∂D

o ∂xi

∂D

or ð23aÞ

∂xi∂Πi

o ∂xi

∂D

and

∂yi∂D

o ∂yi

∂Πi

: ð23bÞ

Then

(a) The price of the underlying is lower in the absenceof a derivative if the substitution effect alwaysdominates the wealth effect for both securities(i.e., ∂xi=∂D40 and ∂yi=∂Πi40).

(b) The price of the underlying is higher in the absenceof a derivative if the wealth effect always domi-nates the substitution effect for both securities (i.e.,∂xi=∂Do0 and ∂yi=∂Πio0).

The sufficient conditions in Corollary 1 depend on(i) the relative sensitivity of the optimal demand for eachsecurity to the price of the securities (as described inEqs. (22) and (23)) and (ii) whether the substitution effectdominates the wealth (or income) effect for both securities.The first part of condition (22) requires that, for a givenchange in the price of a security, the optimal demand for thatsecurity changes by more than the optimal demand for theother security. Alternatively, the second part of condition (22)requires that, for the same change in price of both theunderlying and the derivative, the optimal demand for asecurity is more sensitive to its own price. Under either ofthese intuitive conditions,13 the price of the underlying ishigher in the absence of a derivative if the substitution effectalways dominates the wealth effect, but lower if the wealtheffect always dominates the substitution effect. It seemscounterintuitive to assume that the wealth effect alwaysdominates the substitution effect, therefore, the sufficientconditions for 1(a) above seem most natural.

It is difficult to derive more primitive sufficient condi-tions because the partial derivatives depend on both theassumptions of preferences and the payoff distributions.However, the conditions in part (i) of Proposition 3 arealways satisfied when investors exhibit constant absoluterisk aversion. Therefore, the results from our main model

14 From Proposition 4 (below), we have

P� γτLþτSτLþτS

R¼ τLmLþτSmS

τLþτS� ν

τLþτSQþρLþρS�yB �

:

Therefore, if investors can observe both R and P, then they can infer alinear combination of expectations and aggregate supply as in standardnoisy rational expectations models (e.g., Grossman and Stiglitz, 1980).It may be possible to study a very stylized model with asymmetricinformation within our framework, but we leave this for future work.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608596

survive even when we the relax the assumption thatpayoffs are normally distributed.

Corollary 2. If both agents have preferences with constantabsolute risk aversion (CARA), then for iAfL; Sg,we always have

∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂Do0:

Therefore, if both groups of agents have CARA preferences, thenthe price of the underlying is higher when there is no derivativeavailable to trade. This result holds for any distributions of thefundamental payoff F and noise ɛ in the derivative payoff (withfinite first and second moments).

4. The general model

In this section, we generalize our benchmark model byallowing for agents with heterogeneous beliefs and tradeby speculators. The first subsection presents the setup ofthe model and Section 4.2 characterizes the equilibriumprices and quantities. In the following subsections, wecharacterize the equilibrium in special cases of the generalmodels. In Section 4.3, we assume that L and S investorscan only trade in the underlying, but speculators can tradeboth the underlying and derivative securities. In Section4.4, we analyze the equilibrium when speculators can onlytrade in the derivative security. This case is useful toconsider the effects of restricting trade in derivatives whenthere are speculative investors who can distort the under-lying price through their trades in the derivatives market.In Section 4.5, we consider the effects of introducing long-only investors (e.g., mutual funds) in the underlying onequilibrium prices for the underlying and derivative secu-rities. Finally, in Section 4.6, we analyze the special case inwhich short-sales of the underlying security are comple-tely banned and characterize the conditions under whichimposing such a ban can lead to a decrease in the price ofthe underlying, even in the presence of a derivative.

4.1. Model setup

The assumptions about security payoffs and prices areas in our benchmark model of Section 3. We refer to theL and S investors as hedgers, since they use the underlyingand derivative securities to hedge shocks to their endow-ments (i.e., ρi). In addition, we assume there exist spec-ulators, denoted by B, who have no endowment shocks(i.e., ρB ¼ 0) but instead bet on the realization of F.An investor of type i has CARA utility with risk-toleranceτi. While investors are assumed to agree on the distribu-tion of ɛ, they may have different beliefs about F.In particular, while the objective (or “true”) distributionof F is given by F �Nðm; νÞ as before, we assume thatinvestor i's beliefs about F are given by

F �Nðmi; νÞ; ð24Þwhere mLZmS. Again, we focus on the parameter spacewhere L investors have long positions and S investors haveshort positions in equilibrium.

Whether or not investors have a distortionary effect onprices depends on whether or not they are assumed to be

trading on fundamental information. For instance, if differ-ences in expectations arise due to differences in information,then equilibrium prices aggregate information and the pri-mary source of price distortion in our setup is the scarcity ofthe underlying security.14 However, more generally, thesedifferences in beliefs may represent other, non-informationalmotives for trade that are often interpreted as distortionary,and therefore may generate a role for regulation.

Implicit in the discussion of regulatory policy thatrestricts trade in a security is the notion that some subsetof this trade leads to undesirable distortions in prices. Forinstance, a short-sales ban is often considered whenregulators believe that short-sellers are overly pessimisticand drive asset prices below their fundamental value.Similarly, regulators may propose restrictions on deriva-tive positions in commodities if they feel that speculatorswith beliefs that are disconnected from fundamentals maydistort prices. Given a stand on investors' information andtrading motives, our framework allows us to characterizethe effect of regulatory restrictions when the underlyingsecurity is scarce. For instance, in Section 4.4, we assumethat speculators have distorted beliefs, so that, in theabsence of scarcity in the underlying (e.g., if γ ¼ 1),restricting trade in the derivative security would reducethe price distortion in the underlying. Similarly, in Section 4.6,we assume that S investors have overly pessimistic beliefsand evaluate the impact of a short-sales ban. To emphasize, itis not our objective to argue that short-sellers or speculatorsare distortionary. Instead, we analyze the effects of commonlyproposed regulatory restrictions under these assumptionswhen the underlying security may be scarce.

It is also worth noting that, rather than analyze thewelfare implications of regulatory policy, we insteadrestrict attention to price distortions relative to a friction-less benchmark. Since investors in our model have hetero-geneous beliefs about the distribution of fundamentals,it is unclear how one defines a welfare criterion — seeBrunnermeier, Simsek, and Xiong (2012) and Gilboa,Samuelson, and Schmeidler (2012) for recent approaches.In the context of our model, a frictionless benchmark priceis a less ambiguous objective.

4.2. Equilibrium characterization

The market clearing conditions for the underlying assetare given by

xLþxSþxB ¼ Q and γxLþγBxBþxSZ0; ð25Þwhere γB ¼ 1 if xBo0 and γB ¼ γ if xB40. The marketclearing condition for the derivative becomes

yLþySþyB ¼ 0: ð26Þ

15 A casual argument proposed for why speculators in the derivativesecurity should not affect the price of the underlying is based on theobservation that derivatives are in zero net supply. For instance, ifspeculators are optimistic and so have an aggregate long exposure, otherinvestors must be willing to take the other side, and so the buyingpressure should not affect the price of the underlying security. However,what this casual argument fails to recognize is that the aggregate shortexposure of these other traders (the hedgers in our model) in thederivative market will affect their positions in the underlying security,which in turn, affects the price of the underlying.

16 We are grateful to the referee for highlighting this situation.17 When there is only one group of speculators, they do not trade in

the derivative security. However, as the proof of Proposition 4 establishes,even when there are multiple groups of speculators, the net derivativeposition across them all must be zero (i.e., ∑BiyBi ¼ yB ¼ 0), and theconclusions of Corollary 3 follow.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 597

The following proposition characterizes the equilibriumprices and positions in the underlying security and thederivative.

Proposition 4. Given the economy above with L, S, and Binvestors, the equilibrium prices are given by

D¼ P0þν

τLþτSxBþyB �þ δ

τLþτSyB; ð27Þ

R¼max 0;δ

δþνR0þ

11�γ

ν

τLxþB � ν

τSx�B

� �� �� �; ð28Þ

P ¼ P0þν

τLþτSxBþyB �þ γτLþτS

τLþτSR; ð29Þ

where x�B ¼ xB1fxB o0g, xþ

B ¼ xB1fxB 40g,

P0 �τLmLþτSmS

τLþτS� ν

τLþτSQþρLþρS �

and ð30Þ

R0 �1

1�γ

ν

τS� γ

1�γQþρS

� ��mSþmL�

ν

τL

11�γ

QþρL

� �� �:

ð31ÞThe equilibrium quantities fxi; yigiA fL;S;Bg are characterized inthe proof.

The characterization of the equilibrium in the aboveresult highlights the incremental effect of the presence ofB investors. In particular, note that P0 is the frictionlessprice of the underlying asset if it is traded by L and Sinvestors only, once we allow for heterogeneity in beliefs,risk-tolerances, and endowment shocks. Similarly, in theabsence of B investors, the borrowing rate for the under-lying is given by maxf0;R0g if L and S investors cannottrade the derivative, and by maxf0;R0δ=ðδþνÞg if they cantrade the derivative.

Note that when B investors participate in the underlying(i.e., xBa0) and the underlying is scarce, a larger demandfrom speculators increases the borrowing rate R, irrespectiveof whether they are long or short. This result is because, allelse equal, their demand for positions in the underlyingincreases its scarcity. Moreover, due to a risk-sharing effect,a larger long (short) position from B investors increases(decreases, respectively) the price P.

It is also interesting to note that B investors do not alwaysacquire a position in the underlying security. Specifically, aswe characterize in the proof, when the underlying is scarce(i.e., R40), and the investors' beliefs mB are close to thehedgers' risk-adjusted valuation of the asset (i.e., P0),B investors optimally choose not to participate in the under-lying market. However, even in this case, the speculator canaffect the price of the underlying. Since B investors trade inthe derivative security, the net exposure of L and S investorsin the derivative need not be zero. This net aggregateexposure to the fundamental shock F leads to a risk adjust-ment in the price of the underlying security. For instance, ifB investors are net long in the derivative (i.e., yB40), then Land S investors must be net short, which implies that thetotal exposure of the hedgers to the fundamental shock islower — as a result, the risk-premium component of P issmaller, and therefore P is higher. Similarly, if B investors arenet short in the derivative (i.e., yBo0), hedgers must be netlong, which leads to a bigger risk-premium adjustment and

hence a lower price for the underlying.15 Moreover, eventhough speculators do not affect the borrowing rate R if theydo not trade in the underlying security (i.e., when xB ¼ 0),they do affect the net cost of being long and short in theunderlying security (i.e., P�γR and R�P, respectively)through their effect on P. In particular, if speculators havean aggregate long position in the derivative, then, all elseequal, the price P of the underlying is higher, whichincreases the net cost of a long position (i.e., P�γR), butdecreases the net cost of a short position (i.e., R�P).

While the general model is tractable and offers a greatdeal of flexibility, the equilibrium comparative statics resultsare not transparent given the large number of parameters.In the following subsections, we consider special cases of theabove model which are more parsimonious, but highlightthe important consequences of our analysis for regulatorypolicy, and the intuition for these results.

4.3. When L and S investors only trade in the underlying

There may be investors who are prohibited from acces-sing derivative markets, while speculators (or arbitrageurs)are able to able to trade in both the underlying andderivative markets.16 This setting can be analyzed as a specialcase of our model. Specifically, suppose that L and S investorshave correct beliefs (i.e., m¼mL ¼mS), while speculatorshave incorrect beliefs (i.e., mBam), but only speculators cantrade in the derivative market. The following corollarycharacterizes the equilibrium prices in this case, and theeffect of speculators on the price of the underlying.

Corollary 3. Suppose that L and S investors have correctbeliefs (i.e., mL ¼mS ¼m) but can only trade in the under-lying market (i.e., yL ¼ yS ¼ 0), while speculators have incor-rect beliefs (i.e., mBam) and can trade both the underlyingand derivative securities. Then, if the underlying security isscarce, the cost of borrowing R and the price of the under-lying P are always higher in the presence of speculators thanin their absence, irrespective of whether they have long orshort positions in equilibrium.

Since L and S investors do not trade the derivative, marketclearing implies that the net derivative position across allspeculators must be zero (i.e., yB ¼ 0), and that the borrow-ing rate R is given by the expression in (28) with δ-1, orequivalently, δ=ðδþνÞ ¼ 1.17 In this case, any distortionaryeffects of speculation are directly through the price and

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608598

borrowing rate for the underlying security. If the underlyingis scarce, an increase in speculative trading in either direction(i.e., long or short) increases the borrowing rate R. In additionto this effect, speculators increase the price of the underlyingif they are long, and decrease it if they are short. However,the first effect dominates and the overall effect of speculatorson the price of the underlying is given by

P� P0þγτLþτSτLþτS

R0

� �¼ 1

1�γ

ν

τLxþB � γ

1�γ

ν

τSx�B ; ð32Þ

which implies that both optimistic and pessimistic specula-tors (i.e., xB40 and xBo0, respectively) always increase theprice of the underlying security. Intuitively, since L and Sinvestors do not access the derivative markets, the presenceof the derivative does not reduce the scarcity of the under-lying. Therefore, any trading by speculators leads to a pricedistortion in the underlying. As we discuss in the nextsubsection, this result is in contrast to the effect of spec-ulators when they can only access the derivatives market buthedgers can trade both securities.

4.4. The effect of speculation only in the derivative market

In this subsection, we consider the case where spec-ulators can only trade in the derivative security, andhedgers are exposed to their distortionary trades throughthe derivatives market. This scenario is often cited byregulators as justification for limiting speculative tradein derivative securities. For instance, on November 15,2011, the European Parliament adopted regulation thatrestricts investors from entering into uncovered, or“naked,” CDS on sovereign debt after November 2012,in an effort to curb speculation against a country'sdefault.18 Similarly, under the mandate of the Dodd-Frank Act, the CFTC proposed position limits on commod-ity derivatives so as: “(i) To diminish, eliminate, or preventexcessive speculation as described under this section;(ii) To deter and prevent market manipulation, squeezes,and corners; (iii) To ensure sufficient market liquidity forbona fide hedgers; and (iv) To ensure that the pricediscovery function of the underlying market is notdisrupted.”19

To analyze this situation, we assume that the L and Sinvestors in our model have correct beliefs (i.e.,m¼mL ¼mS), while speculators have incorrect beliefs(i.e., mBam) and can only trade the derivative security(i.e., xB ¼ 0). To simplify the analysis, we also assume thatall investors have the same risk-tolerance (i.e., τi ¼ τ), andthat the endowment shocks of L and S investors perfectlyoffset each other (i.e., ρS ¼ �ρL ¼ ρ). The frictionless bench-mark in this setup is when there are no constraints onborrowing and lending (i.e., γ ¼ 1), and no speculators (i.e.,τB ¼ 0). In this case, the price of the underlying security

18 See http://ec.europa.eu/internal_market/securities/short_selling_en.htm.

19 See http://www.cftc.gov/LawRegulation/DoddFrankAct/Rulemakings/DF_26_PosLimits/index.htm.

simplifies to20

P0 ¼m� ν

2τQ : ð33Þ

Relative to this frictionless benchmark, if the underlying isscarce, the price distortion when investors have access tothe derivative is given by

ΔP ¼ 1þγ

2Rþ ν

2τyB ¼

1þγ

2δ

νþδR0þ

ν

2τyB; ð34Þ

and the distortion when investors do not have access tothe derivative is given by

ΔP ¼ 1þγ

2R0: ð35Þ

Therefore, despite the distortionary effects of speculativetraders, the presence of a derivative may reduce theoverall price distortion in the underlying security, throughits effect on scarcity. The following result characterizessufficient conditions for this result.

Proposition 5. Suppose that L and S investors have correctbeliefs (i.e., mL ¼mS ¼m), while speculators have incorrectbeliefs (i.e., mBam) and can only trade in the derivativesmarket (i.e., xB ¼ 0). Moreover, suppose that τi ¼ τ for all iand ρS ¼ �ρL ¼ ρ.

(i)

2

whicno no(i.e.,P0 ¼unch

If

23

mB�mð Þþ ν

3τQ

o 1þγð ÞR0;

then the price distortion in the underlying (relative to thefrictionless price) is lower in the presence of the deriva-tive with speculators than in the absence of derivatives.

(ii)

If

23

mB�mð Þþ ν

3τQ

4 νþ2δ

ν1þγð ÞR0;

then the price distortion in the underlying (relative to thefrictionless price) is higher in the presence of the deriva-tive with speculators than in the absence of derivatives.

(iii)

If speculators are long (i.e., yB40), or sufficiently short (i.e.,

yBo� 1þγð Þ δ

νþδ

τ

νR0;

then the price distortion in the underlying (relative to thefrictionless price) is increasing in the aggregate position ofspeculators (i.e., jyBj). Otherwise, if

� 1þγð Þ δ

νþδ

τ

νR0oyBo0;

then the price distortion is increasing in the aggregateposition of speculators.

The proof of the above result follows from the expres-sions for the price distortion in the presence and absenceof derivatives with speculators. For parts (i) and (ii), note

0 Alternatively, one can set the frictionless benchmark as the case inh there are no constraints on borrowing and lending (i.e., for γ ¼ 1 ),ise in the derivative (i.e., δ¼ 0), and speculators have correct beliefsmB ¼m). In this case, the price of the underlying security ism�Qν=ð3τÞ. As is apparent, the results in Proposition 5 remainanged for this benchmark.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 599

that the price distortion with the derivative is lower when

� 1þγ

2νþ2δνþδ

R0oν

2τyBo

1þγ

2ν

νþδR0; ð36Þ

and higher when either

ν

2τyB4

1þγ

2ν

νþδR0 or

ν

2τyBo� 1þγ

2νþ2δνþδ

R0; ð37Þ

and that the optimal derivative position for the speculatorsis given by

yB ¼τ

δþνmB�Dð Þ ¼ 2

3τ

δþνmB�mð Þþ 1

3ν

δþνQ : ð38Þ

Intuitively, the presence of the derivative reduces the pricedistortion in the underlying when the security is scarce(i.e., R0 is large), the outstanding supply Q is small, and thedisagreement between speculators and hedgers is notlarge (i.e., jmB�mj is small). Notably, whether the presenceof the derivative reduces price distortion is unaffected bythe noise in the payoff (i.e., δ), although the magnitude ofthe change in the price distortion is obviously affected.Finally, part (iii) emphasizes that even in the presence of aderivative, completely restricting trade by speculators maylead to a larger price distortion in the underlying thanallowing for some trade by (short) speculators.

This result highlights the tradeoff that regulators facewhen restricting trade in derivatives. While allowing forunrestricted trade in derivatives in the presence of aggres-sive speculators may increase the price distortion in theunderlying, excessive trading restrictions can make theunderlying more scarce and, thereby, distort prices.

4.5. The effect of speculation only in the underlying

As documented by Kaplan, Moskowitz, and Sensoy(2013), Rizova (2011), Evans, Ferreira, and Porras Prado(2012), and others, a large part of the equity lendingmarket has recently been dominated by long-only inves-tors who lend stocks (e.g., mutual funds). An open ques-tion in this literature is whether the lending activity ofsuch buy-and-hold investors distorts the price of theunderlying security.21 We can analyze this situationas a special case of our model in which B investors arerestricted to have only long positions in the underlyingsecurity and are unable to trade in the derivative.22

In particular, suppose that L and S investors have correctbeliefs (i.e., m¼mL ¼mS), and B investors can only trade inthe underlying market (i.e., yB ¼ 0). The following corollarycharacterizes the equilibrium prices in this case, and theeffect of long-only (or short-only) investors on the price ofthe underlying and the derivative.

Corollary 4. Suppose that L and S investors have correctbeliefs (i.e., mL ¼mS ¼m), and speculators can only trade inthe underlying security (i.e., yB ¼ 0). Then:

21 We thank the referee for pointing out this case.22 While we interpret B investors as speculators in the earlier

subsections, we do not think of mutual funds as speculators. However,it is notationally convenient to denote the mutual funds as B investors, sothat we can appeal to the general results in Section 4.2.

(i)

In the presence of long-only speculators (i.e., xB40), theprice of the derivative D, the borrowing cost R, and the priceof the underlying P are all higher than in their absence.

(ii)

In the presence of short-only speculators (i.e., xBo0), theprice of the derivative D is always lower and theborrowing cost R is always higher than in their absence.The price of the underlying is higher in their presence ifτSð1�γÞ=ðτSþγτLÞoδ=ðδþνÞ, and lower if τSð1�γÞ=ðτSþγτLÞ4δ=ðδþνÞ.

As discussed before, when the underlying security isscarce, the effect of speculative trading in the underlying istwo-fold. First, higher demand from either side leads to anincrease in the borrowing rate since it increases the scarcityin the underlying. Second, due to risk-sharing in the under-lying, a larger long position from speculators increases theprice of the underlying, while a larger short positiondecreases the price. The overall effect on the underlyingprice then depends on the relative magnitudes of the twoeffects, and their signs, as is characterized by the expression:

P� P0þγτLþτSτLþτS

δ

νþδR0

� �

¼ ν

τLþτSxBþ

γτLþτSτLþτS

δ

δþν� 11�γ

ν

τLxþB � ν

τSx�B

� �: ð39Þ

In the case of long-only speculators, the two effects reinforceeach other, and as such, unambiguously increase the price ofthe underlying. However, for short-only speculators in theunderlying, the overall effect depends on

P� P0þγτLþτSτLþτS

δ

νþδR0

� �¼ 1� γτLþτS

τSð1�γÞδ

δþν

� �ν

τLþτSxB:

ð40Þ

In particular, note that when the derivative is a perfectsubstitute (i.e., δ¼ 0), the price of the underlying decreaseswith shorting demand from B investors, while in the absenceof the derivative (i.e., δ¼1), the price of the underlyingincreases (as in Section 4.3).

Finally, it is important to note that even though Binvestors do not trade the derivative security, they affectthe price of the derivative through their trading in theunderlying since the two securities are substitutes. If Binvestors have a net long position in the underlying, theyincrease the price of the underlying. All else equal, thismakes the derivative a more attractive substitute for longinvestors, which drives up its price. Similarly, when Binvestors have a net short position in the underlying,this drives down the price of the derivative. Therefore,our model generates an interesting prediction: increasedshort-selling by investors in the underlying alwaysdecreases the derivative price, but increases the price ofthe underlying when the derivative payoff is sufficientlynoisy.

4.6. The effect of short-sales bans

Regulations that restrict short-selling are often justifiedas a means to limit the effect of pessimistic investors whowould otherwise distort prices below their fundamental

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608600

value. For instance, in September 2008, the SEC tempora-rily banned short-selling in financial stocks “to combatmarket manipulation that threatens investors and capitalmarkets.”23 More recently, a number of EU membercountries imposed short-sales bans for bank stocks to“restrict the benefits that can be achieved by spreadingfalse rumors.”24

To capture this scenario, we assume that there are nospeculators in the market (i.e., τB ¼ 0), that L investorshave correct beliefs about fundamentals (i.e., mL ¼m),and that S investors are pessimistic (i.e., mSom). In thiscase, while short-sellers are pessimistic, the followingproposition shows that for some parameter values, banningshort-sales (i.e., setting γ ¼ 0) actually reduces the price ofthe underlying asset, even in the presence of derivatives.

Corollary 5. Suppose there are no speculators (i.e., τB ¼ 0),L investors have correct beliefs (i.e., mL ¼m), and S investorsare pessimistic (i.e., mSom).

(i)

2

2

feabd

If short-selling is allowed, then the underlying price isgiven by

P ¼ τLmþτSmS

τLþτS� ν

τLþτSQþρSþρL �þ γτLþτS

τLþτSR; where

ð41Þ

R¼max 0;

ν

τSρS�

γ

1�γQ

� �� ν

τL

11�γ

QþρL

� �þm�mS

1�γð Þ 1þν

δ

� 8>><>>:

9>>=>>;:

ð42Þ

(ii)

If short-selling is not allowed, then the underlying priceis given by

Pns ¼τLmþτSmS

τLþτS� ν

τLþτSQþρSþρL �þ τS

τLþτSRns

ð43Þwhere

Rns ¼max 0;

ν

τSρS�

ν

τLQþρL �þm�mS

1þν

δ

� 8><>:

9>=>;: ð44Þ

The price of the derivative in either case is given by

D¼ τLmþτSmS

τLþτS� ν

τLþτSQþρSþρL �

: ð45Þ

Even though short-selling is not allowed, it is instruc-tive to decompose the underlying price into a frictionless

component and a shadow cost of borrowing the security(i.e., Rns). For instance, in the special case when trading inderivatives is not allowed (i.e., δ-1 which impliesδ= δþνð Þ ¼ 1) and short-selling is not allowed (i.e., γ ¼ 0),the shadow cost of borrowing the security is given by

Rns ¼ν

τSρS�

ν

τLQþρL �þm�mS; ð46Þ

3 See http://www.sec.gov/news/press/2008/2008-211.htm.4 See http://www.ft.com/intl/cms/s/0/9a55839a-c42d-11e0-ad9a-00144c0.html.

and the price of the underlying reduces to the familiarexpression

Pns ¼τLmþτSmS

τLþτS� ν

τLþτSQþρSþρL �þ τS

τLþτSRns ð47Þ

Pns ¼m� ν

τLQþρL �

: ð48Þ

That is, if L investors must hold the underlying security andcannot trade the derivative with the S investors, then the priceof the underlying security is completely determined by thebeliefs and preferences of L investors. If short-selling is notallowed, but S investors are allowed to trade the derivativesecurity, then their beliefs affect the value of the underlyingsecurity. In fact, lower noise in the derivative payoff (i.e., lowerδ), decreases the price Pns when short-sales are not allowed,since S investors are able to express their beliefs morestrongly. In the limit, if there is no noise in the derivative(i.e., δ¼ 0), then imposing the short-sales constraint has noeffect on the price of the underlying. In contrast, regardlessof the noise in the derivative payoff, imposing a short-sellingban has no effect on the price of the derivative securityitself. However, positions in the derivative security do changein response to the ban — as in our benchmark model inSection 3, the optimal demand for the derivative securitydepends on the (shadow) cost of borrowing R, which dependson whether the short-selling ban is in effect.

Finally, note that the derivative of P with respect to γ atγ ¼ 0 is given by

∂P∂γ

γ ¼ 0

¼ δ

δþνm�msþν

ρSτS

� 1τL

ρLþ2Q �� �� �

; ð49Þ

which is positive when Q is small. Thus, if the underlyingsecurity is scarce relative to its outstanding supply, its pricemay be lower when short-sales are banned (i.e., γ ¼ 0).Intuitively, when the underlying is scarce but short-sellingis not allowed, the shadow cost of borrowing the underlyingis not zero (i.e., Rnsa0), but none of this borrowing cost isreflected in the price Pns. In this case, relaxing the short-selling ban increases the price of the underlying since itallows L investors to lend some of their positions at Rns,which increases the price they are willing to pay for theunderlying asset. The above result generalizes a similarresult in Duffie, Gârleanu, and Pedersen (2002) to a settingin which investors can trade derivatives.

To summarize, we have the following results concerningthe effect of a short-sales ban on an underlying asset thatmay be scarce, when investors have access to a derivative.

Proposition 6. Suppose that there are no speculators (i.e.,τB ¼ 0), L investors have correct beliefs (i.e., mL ¼m), Sinvestors are pessimistic (i.e., mSom). Then:

(i)

if the aggregate supply of the underlying is low enough,imposing a short-sales ban can lower the price of the under-lying security, even when L and S can trade the derivative,

(ii)

the price of the underlying security decreases as thenoise in the derivative decreases (i.e., δ decreases), evenin the presence of a short-sales ban,

(iii)

imposing a short-sales ban has no effect on the price ofthe underlying if investors can trade a derivative with nonoise (i.e., δ¼0),

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 601

(iv)

2

markscarcderiv

imposing a short-sales ban has the largest effect on theprice of the underlying if investors cannot trade aderivative (i.e., δ-1, or equivalently, δ=ðδþνÞ ¼ 1), and

(v)

imposing a short-sales ban has no effect on the deriva-tive price but increases the size of equilibrium derivativepositions.

26 For examples, see Detemple and Selden (1991), Zapatero (1998),Boyle and Wang (2001), and Bhamra and Uppal (2009).

27 For example, in October 2011, the average daily trading volume inStandard and Poor's 500 (S&P 500) futures on the Chicago Mercantile

As these results highlight, regulation that bans short-selling to mitigate the distortionary effects of trading bypessimistic investors may be ineffective if investors cantrade derivatives, and can actually decrease the underlyingprice if the security is sufficiently scarce. These results arein contrast to the over-pricing effect of short-sale con-straints in many standard models (e.g., Miller, 1977) that isoften used to motivate such regulation.

While beyond the scope of our model, in settings wherethe noise in the derivative payoff is endogenously affected bythe constraint on borrowing and lending in the underlying(i.e., if δ is affected by γ), parts (i) and (ii) of the aboveproposition may be particularly relevant. For instance, con-sider a setting in which market makers in the derivativemarket must hedge their positions by trading in the under-lying security. In this case, one might expect that restrictingshort-sales (i.e., decreasing γ) makes it more difficult formarket makers to hedge their derivative exposures andtherefore increases the noise δ in the derivative payoff.25

As a result, in this setting, (i) and (ii) predict opposite effectson the underlying price. In particular, the first part of theresult implies that a decrease in γmight lead to lower P, whilethe second part implies that an increase in δ leads to a higherP. In an alternative setting, suppose that investors face searchfrictions in trading both the underlying and derivative secu-rities. In this case, increasing scarcity in the underlying byrestricting short-sales might lead to increased liquidity andlower search costs in the derivative security, as more investorscoordinate their trading in the latter security. As a result, thetwo effects could reinforce each other a decrease in γ and adecrease in δ might both lead to a decrease in P.

5. Conclusions

When the demand for long and short positions exceeds anasset's capacity to support such positions, it becomes scarceand its price is distorted relative to its value in a frictionlessmarket. In this case, we show that even simple derivatives areno longer redundant since the presence of a derivative canalleviate the scarcity in the underlying asset, and, therefore,reduce the distortion in its price. Finally, we show thataccounting for scarcity is important in analyzing the effectsof regulatory policy, such as short-selling bans and derivativeposition limits, that restrict trade in an underlying asset or itsderivative.

To highlight the role of derivatives in relaxing the scarcityof the underlying asset, we focus exclusively on a simple(linear) derivative that is otherwise redundant. Our modelpredicts that, all else equal, assets with simple derivatives

5 In the extreme, investors may be unwilling or unable to makeets in the derivative security when the underlying is sufficientlye (e.g., single name futures contracts), and instead only offeratives on correlated securities (e.g., broader index futures).

that are close substitutes should have lower borrowing feesand therefore smaller price distortions. A number of empiri-cal papers that study how the introduction of optionsimpacts the underlying asset find evidence consistent withour model's predictions for simple derivatives. For example,Sorescu (2000) finds that the price of the underlying stocktends to fall when (nonlinear) options on the stock areintroduced. Danielsen and Sorescu (2001) provide empiricalevidence that is consistent with the notion that the intro-duction of options mitigates short-sale constraints. However,one must be cautious in interpreting these results asconclusive evidence for our model. While options may relaxthe scarcity of the underlying asset, they can also affect theequilibrium by allowing investors to trade new sources ofrisk (such as the volatility of the asset).26

In testing the empirical predictions of our model, it isalso important to account for a number of features of themarket that we have abstracted from in order to moreclearly highlight the role of derivatives in relaxing thescarcity of the underlying. For instance, we do not considerthe effect of financial intermediaries who make markets inthe derivative securities. If market makers need to hedgetheir derivative positions with positions in the underlying,then increased trade in derivatives can lead to an increasein the scarcity of the underlying. One must also keep inmind that the introduction of derivatives is endogenous —it may be more likely that derivatives are introduced onunderlying securities that are scarce. In particular, it isimportant to control for this endogeneity in any cross-sectional tests of our model's predictions.

Finally, the notion that simple derivatives can “complete”the market by allowing long investors and short-sellers totake larger aggregate positions in the same source of risk asthe underlying, may also be important for understandingthe relative size of derivative markets across various assets.For instance, although U.S. equity indexes are extremelyliquid, they are often accompanied by very large futuresmarkets, which may be driven by the fact that it can bedifficult to simultaneously short-sell all of the componentsof an index.27 Similarly, the fact that many corporate bondsare often difficult to borrow (and short-sell) may be animportant driver of the recent surge in the size of thecorporate CDS markets (e.g., Gupta and Sundaram, 2011).Our model may also be useful for understanding why evenextremely liquid securities like U.S. Treasuries may beaccompanied by extremely large futures markets.28 As ourmodel highlights, the size of the derivative market may bedriven, not by the constraint on borrowing and lending inthe underlying per se, but by the demand for both long andshort positions relative to the trading capacity of the under-lying that results from this constraint.

Exchange (CME) was about $270B notional, while trade in all stocks onthe NYSE, Nasdaq, and SPDRs (an exchange traded fund that mimics theS&P 500) was about half that.

28 In October 2011, the average daily trading volume in all Treasurysecurities was around $500B, while trade in four Treasury futures on theChicago Mercantile Exchange was about $220B notional.

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608602

Appendix A. Proofs of main results

Proof of Proposition 3. We begin by making the followingobservations:

1.

In any equilibrium, unless ɛ¼ 0, P�RoDoP�γR.If D4P�γR, then long investors would short thederivative and so would shorts, so markets cannotclear. If DoP�R, then short-sellers would be long thederivative and so would longs, so markets cannot clear.

2.

Given the above, yL40 and ySo0. If yLo0, then thelong could do better by selling some of the underlyinginstead since DoP�γRoP. Similarly, if yS40, thenshort-sellers would be better off reducing their shortposition a little, since P�RoD.

3.

29 More specifically, since u′ðWÞ ¼ e�kW and therefore,

u′

E½u′� ¼ g Fð Þh ɛð Þ where E g Fð Þ½ � ¼ 1¼ E h ɛð Þ� �;

so that

E F�E Fu′

E½u′�

� �� �ɛ�E ɛ

u′

E½u′�

� �� �u′

E½u′�

� �¼ E ðF�E½FgðFÞhðɛÞ�Þðɛ�E½ɛgðFÞhðɛÞ�ÞgðFÞhðɛÞ� �

¼ E½ðF�E½FgðFÞ�ÞgðFÞ�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0

E½ðɛ�E½ɛhðɛÞ�ÞhðɛÞ�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0

:

.

If the constraints are binding before the derivative, theywill be binding after the derivative, unless ɛ¼ 0. If not,R¼0, which implies P¼D. But since ɛa0, no one tradesthe derivative, and so we have a contradiction.

These observations imply that generically, L investors havepositive positions in the derivative, S investors havenegative positions in the derivative, and the rest of theargument follows the text of the paper. Specifically, theequilibrium position in the underlying does not change foreither investor i.e.,

dxi ¼∂xi∂Πi

dΠiþ∂xi∂D

dD¼ 0; ð50Þ

which implies the price of the derivative must change by

dD¼ � ∂xi=∂Πi

∂xi=∂DdΠi: ð51Þ

Along this path, the change in investor i's optimal positionin the derivative is

dyi ¼∂yi∂Πi

dΠiþ∂yi∂D

dD¼ ∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂D

� �dΠi: ð52Þ

Hence,

∂yi∂Πi

� ∂yi∂D

∂xi=∂Πi

∂xi=∂Do0 ð53Þ

is a sufficient condition for dyi=dΠio0, and vice versa,which, combined with the observation that yL ¼ �yS andP ¼ΠL=ð1�γÞ�ΠSγ=ð1�γÞ, gives us the result. □

Proof of Corollary 2. Dropping the subscript i, the firstorder conditions of the optimal portfolio problem inEq. (13) are

0¼ E½fF�Πgu′ðW0þρFþx½F�Π�þy½Fþɛ�D�Þ�; ð54aÞ

0¼ E½fFþɛ�Dgu′ðW0þρFþx½F�Π�þy½Fþɛ�D�Þ�: ð54bÞDifferentiating each of these equations with respect toΠ and D yields

∂x∂Π

¼ ðAþ2BþCÞE½u′þxðF�ΠÞu″��ðAþBÞxE½ðFþɛ�DÞu″�AC�B2 ;

ð55aÞ

∂y∂Π

¼ AxE½ðFþɛ�DÞu″��ðAþBÞE½u′þxðF�ΠÞu″�AC�B2 ; ð55bÞ

∂x∂D

¼ ðAþ2BþCÞyE½ðF�ΠÞu″��ðAþBÞE½u′þyðFþɛ�DÞu″�AC�B2 ;

ð55cÞ

∂y∂D

¼ AE½u′þyðFþɛ�DÞu″��ðAþBÞyE½ðF�ΠÞu″�AC�B2 ; ð55dÞ

where

A¼ E½ðF�ΠÞ2u″�; ð56aÞ

B¼ E½ðF�ΠÞðɛ�DþΠÞu″�; ð56bÞ

C ¼ E½ðɛ�DþΠÞ2u″�: ð56cÞFor an investor with CARA utility, we have u″ ¼ �ku′ forsome constant k40, so that

E½ðF�ΠÞu″� ¼ �kE½ðF�ΠÞu′�|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}0 by Eq: ð54aÞ

¼ 0; ð57Þ

E½ðFþɛ�DÞu″� ¼ �kE½ðFþɛ�DÞu′�|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}0 by Eq: ð54bÞ

¼ 0: ð58Þ

Also,

E½ðF�ΠÞðɛ�DþΠÞu″� ð59aÞ

¼ E F�E Fu′

E½u′�

� �� �ɛ�E ɛ

u′E½u′�

� �� �u″

� �by Eq: 54ð Þ

ð59bÞ

¼ �kE u′� �E F�E F

u′E½u′�

� �� �ɛ�E ɛ

u′E½u′�

� �� �u′

E½u′�

� �ð59cÞ

¼ �kE u′� �E F�E F

u′E½u′�

� �� �u′

E½u′�

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0

E ɛ�E ɛu′

E½u′�

� �� �u′

E½u′�

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0

ð59dÞ

¼ 0: ð59eÞTo move from Eq. (59c) to Eq. (59d), note that u′=E½u′�defines a change of probability measure under which F andɛ are independent (since they are independent under theoriginal measure).29 Combining these results, we have

∂x∂Π

¼ ðAþCÞE½u′�AC

¼ E½ðF�ΠÞ2u″�E½u′�þE½ðɛ�DþΠÞ2u″�E½u′�E½ðF�ΠÞ2u″�E½ðɛ�DþΠÞ2u″�

;

ð60aÞ

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 603

∂y∂Π

¼ ∂x∂D

¼ � ∂y∂D

¼ � AE½u′�AC

¼ � E½ðF�ΠÞ2u″�E½u′�E½ðF�ΠÞ2u″�E½ðɛ�DþΠÞ2u″�

;

ð60bÞwhich establishes the result. □

Proof of Proposition 4. We consider a more general speci-fication than in the proposition, in which there are twotypes of speculators, indexed by iAfBO;BPg (for optimistsand pessimists, respectively). The first-order conditions forinvestor iAfL; S;BO;BPg imply that

τiðmi�PþγiRÞ ¼ νðxiþyiþρiÞ and ð61Þ

τiðmi�DÞ ¼ νðxiþyiþρiÞþδyi: ð62ÞLet P0 ¼ ðτLmLþτSmSÞ=ðτLþτSÞ�ðν=ðτLþτSÞÞðQþρLþρSÞ, andlet xB ¼ xBOþxBP , and yB ¼ yBOþyBP . The market clearingcondition for the derivative implies that

τLmLþτSmS�ðτLþτSÞD¼ νðQþρLþρS�ðxBþyBÞÞ�δyBð63Þ

) D¼ P0þν

τLþτSxBþyB �þ δ

τLþτSyB ð64Þ

and market clearing in the cash-market implies

P ¼ P0þν

τLþτSxBþyB �þ γτLþτS

τLþτSR: ð65Þ

Finally, comparing the risk-return tradeoffs for L and Simplies that

R 1�γð Þ ¼ ν

τSxSþρSþyS ��mS�

ν

τLxLþρLþyL �þmL ð66Þ

and since ðν=τSÞyS�ðν=τLÞyL ¼ ðν=δÞðγ�1ÞR, we have

R¼max 0;

ν

τSxSþρS ��mSþmL�

ν

τLxLþρL �

1�γð Þ 1þν

δ

� 8><>:

9>=>;: ð67Þ

When R40, the lending constraint binds since longs wantto lend as much as they can,

xS ¼ � γ

1�γQ�x�

B and xL ¼1

1�γQ�xþ

B ; ð68Þ

where

x�B ¼ ðxBO1fxBO o0g þxBP1fxBP o0gÞ and ð69Þ

xþB ¼ ðxBO1fxBO 40gþxBP1fxBP 40gÞ: ð70Þ

xBi ¼τBiν

mBi�P0ð Þ�τBiν

τSτSþτL

1�γð ÞR0þν

τLxþBj �

ν

τSx�Bj

� ��τL

1þ τBiτLþτS

þτBiτL

τSτLþτS

xBi ¼τBiν

mBi�P0ð ÞþτBiν

τLτLþτS

1�γð ÞR0þν

τLxþBj �

ν

τSx�Bj

� ��τL

1þ τBiτLþτS

þτBiτS

τLτLþτS

This implies

R¼max 0;δ

δþνR0þ

11�γ

ν

τLxþB � ν

τSx�B

� �� �� �; ð71Þ

where

R0 �1

1�γ

ν

τS� γ

1�γQþρS

� ��mSþmL�

ν

τL

11�γ

QþρL

� �� �:

The equilibrium quantities can be computed by plugging inthe expressions for the price into the first-order conditionsfor each type of investor. Since yBi ¼ ðτBi=δÞðP�γBiR�DÞ andxBiþyBi ¼ ðτBi=νÞðmBi�PþγBiRÞ, we have for ja i:

yBi ¼τBiδ

γτLþτSτLþτS

R� δ

τLþτSyBOþyBP ��γBiR

� �ð72Þ

) 1þ τBiτLþτS

� �yBi ¼

τBiδ

γτLþτSτLþτS

�γBi

� �R� τBi

τLþτSyBj: ð73Þ

But this implies

1þ τBiτLþτS

� �xBi

¼ τBiν

mBi�P0ð Þ� τBiτLþτS

xBjþτBiν

þ τBiδ

� γBi�

γτLþτSτLþτS

� �R: ð74Þ

When R¼0, the above reduces to

1þ τBiτLþτS

� �yBi ¼ � τBi

τLþτSyBj ) yBi ¼ � τBi

τLþτSyB ) yBi ¼ 0

ð75Þand this implies

xBi ¼τBiν

τLþτSþτBOþτBPð ÞmBi�ðτBOmBOþτBPmBPÞ�ðτLþτSÞP0

τLþτSþτBOþτBP

� �:

ð76ÞWhen R40, then

xBi 1þ τBiτLþτS

� �� τBi

νγBi�

γτLþτSτLþτS

� �1

1�γ

ν

τLxþBi �

ν

τSx�Bi

� �¼ τBi

νmBi�P0ð Þ� τBi

τLþτSxBj

þ τBiν

γBi�γτLþτSτLþτS

� �R0þ

11�γ

ν

τLxþBj �

11�γ

ν

τSx�Bj

� �:

ð77ÞThis implies that

(i)

τBiþτS

x

τBiþτS

x

If xB40, then γB ¼ γ, and so

Bj

40; ð78Þ

and so mBi�P04ðτS=ðτSþτLÞÞð1�γÞR0þðν=τLÞxþBj is

sufficient.

(ii) If xBo0, then γB ¼ 1 and so

Bj

o0 ð79Þ

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608604

and so mBi�P0o�ðτL=ðτLþτSÞÞð1�γÞR0þðν=τSÞx�Bj is

sufficient.

(iii) Finally, if

� τLð1�γÞτLþτS

R0þν

τSx�Bj omBi�P0o

τSð1�γÞτSþτL

R0þν

τLxþBj ;

then xB ¼ 0 and yB ¼ τBðmB−DÞ=ðδþ νÞ.

The expressions in the statement of the propositionfollow from the above conditions and noting there is onlyone group of speculators (i.e., xB ¼ xBi and xBj ¼ 0 forja i). □

Appendix B. Costly lending by long investors

In our main model of Section 3, we exogenously fix themaximum fraction γ of their position that long investorscan lend out. While this assumption is made for tract-ability, in this subsection, we show that our results arequalitatively similar when the fraction γ lent by longs isdetermined endogenously in equilibrium. In particular, iflong investors face a cost cðγÞ to lend out a fraction γ oftheir position, then their wealth is given by

WL ¼WL;0þρLFþxL F�PþγR�cðγÞð ÞþyLðFþɛ�DÞ: ð80ÞWe show that in this case, the equilibrium is characterizedby the following proposition.

Proposition 7. Suppose that L investors pay a per-unit costcðγÞ in order to lend out a fraction γ of their portfolio, wherecðγÞ is non-negative, non-decreasing, convex, cð0Þ ¼ 0, andc′ð0Þ ¼ 0. Then, the equilibrium prices are given by

D¼ P0; P ¼ P0þΔP and R¼ τLþτSγnτLþτS

ΔPþ τLτLþτS

c γn �� �

;

ð81Þwhere the price distortion ΔP relative to the frictionless priceP0 in Eq. (6) is given by

ΔP ¼ γnτLþτSðτLþτSÞð1�γnÞmax 0;

δ

δþν

mL� ντL

ρLþ 11� γn Q

� �mSþ ν

τSρS� γn

1� γn Q�

264

375�c γn

�8><>:

9>=>;

� τLcðγnÞτLþτS

ð82Þ

and the optimal fraction γn lent out is characterized byR¼ cγðγnÞ. The equilibrium positions are given by

�yS ¼ yL ¼1δ

1�γnþ 1Rc γn �� �

τLτSγnτLþτS

ΔPþ τLτLþτS

c γn �� �

;

ð83Þ

Q�xS ¼ xL ¼

11�γn

Q if R40;

τLðQþρSþρLÞτLþτS

�ρL if R¼ 0:

8>>><>>>: ð84Þ

Proof. The first-order conditions for investor L imply that

cγ ¼ R; ð85Þ

yLδ¼ τLðP�γRþc�DÞ; ð86Þ

ðxLþyLþρLÞν¼ τLðmL�PþγR�cÞ; ð87Þ

while the first-order conditions for the S investors aregiven by

ySδ¼ τSðP�R�DÞ; ð88Þ

ðxSþySþρSÞν¼ τSðmS�PþRÞ: ð89ÞSuppose that an interior optimal fraction γn exists (weshall verify this below). For notational simplicity, let γdenote the optimal fraction γn that is characterized bycγðγnÞ ¼ RðγnÞ, and let c denote the cost at that optimalfraction (i.e., c¼ cðγnÞ). The market clearing conditions forthe derivative and underlying cash-market imply

D¼ P� τSþγτLτSþτL

Rþ τLτSþτL

c; ð90Þ

P ¼ τLmLþτSmS

τLþτSþ τSþγτL

τSþτLR� τL

τSþτLc� ν

τLþτSQþρLþρS �

;

ð91Þwhich in turn imply that the equilibrium positions in thederivative are given by

yS ¼ � 1�γþ cR

� τSτLδðτSþτLÞ

R¼ �yL: ð92Þ

Finally, note that

mS�PþR� mL�PþγR�cð Þ ¼ ν

τSxSþySþρS �� ν

τLxLþyLþρL �

;

ð93Þ

and

δ

τSyS�

δ

τLyL ¼ P�R�D� P�γRþc�Dð Þ; ð94Þ

which imply

R 1�γð Þþc¼mL�mSþν

τSxSþySþρS �� ν

τLxLþyLþρL �

;

ð95Þ

) R¼mL�mSþ

ν

τSxSþρS �� ν

τLxLþρL �

1�γð Þ 1þν

δ

� � c1�γ

: ð96Þ

Finally, to establish the existence of the optimal fraction γn,note that it must solve the equation

cγ γð Þ ¼mL�

ν

τL

11�γ

QþρL

� ��mSþ

ν

τSρS�

γ

1�γQ

� �1�γð Þ 1þν

δ

� � cðγÞ1�γ

;

ð97Þ

or equivalently,

c γð Þþ 1�γð Þc′ γð Þ ¼mL�

ν

τLρL�mSþ

ν

τSρS�

ν

τSγþ ν

τL

� �1

1�γQ

1þν

δ

� :

ð98Þ

Since cð�ÞZ0, c′ð�ÞZ0, and c″ð�ÞZ0, the left hand side(LHS) is non-negative and increasing in γ. On the otherhand, the right hand side (RHS) is positive for γ ¼ 0,decreasing in γ, and strictly negative for γ ¼ 1. Therefore,

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 605

if the LHS is less than the RHS at γ ¼ 0, i.e., if

c 0ð Þþc′ 0ð ÞomL�

ν

τLρL�mSþ

ν

τSρS�

ν

τLQ

1þν

δ

� ; ð99Þ

then there is a solution γnA 0;1½ � to the above equation.In particular, if cð0Þ ¼ 0 and c′ð0Þ ¼ 0, then the condition issatisfied, which completes the proof. □

The conditions on the cost function cð�Þ ensure that theequilibrium fraction lent out by long investors, γn, isbetween zero and one. Comparing the above result toProposition 1, an endogenous fraction γ leads to a numberof differences in equilibrium prices and quantities. First,note that there is always a price distortion, given by−τLcðγ⁎Þ=ðτL þ τSÞ, relative to the frictionless price P0. Thisprice distortion reflects the (risk-tolerance weighted) costthat long investors incur to lend out the equilibrium fractionγn of the underlying security. Second, in comparison tocondition (12), the underlying security is scarce only when

δ

δþνmL�mSþ

ν

τSρS�

γn

1�γnQ

� �� ν

τL

11�γn

QþρL

� �� ��c γn �

40:

ð100ÞSince the long investor pays Pþc�γR per unit position in theunderlying, while the short pays R�P, longs and shorts paydifferent prices even when the cost of borrowing (i.e., R) iszero. The security is scarce only when, for a zero lending fee(i.e., R¼0), the aggregate demand for long and short posi-tions do not clear the market.

When there is excess demand for the underlying secur-ity (i.e., condition (102) holds), both prices (i.e., R and P) andquantities (i.e., γ) adjust in equilibrium. The degree to whicheach adjusts depends on the specific cost function cðγÞ, andthe equilibrium quantity γn is pinned down by the Linvestors' indifference condition RðγÞ ¼ cγðγÞ.30 As before,there is trade in the derivative security only when theunderlying is scarce. The next result establishes that eventhough the borrowing constraint is endogenous in thissetting, the distortion in the price of the underlyingincreases in the noise δ in the derivative, and is highest inthe absence of the derivative.

Proposition 8. Suppose that L investors pay a per-unit costcðγÞ in order to lend out a fraction γ of their portfolio, wherecðγÞ is non-negative, non-decreasing, convex, cð0Þ ¼ 0, andc′ð0Þ ¼ 0. Then the price distortion ΔP in the underlyingsecurity increases with the variance δ of the noise in thederivative security. This implies that, all else equal, the pricedistortion in the underlying is largest when investors do nothave access to a derivative (or equivalently, the noise in thederivative becomes arbitrarily large).

Proof. As the proof of Proposition 7 establishes, theassumptions on the cost function ensure that there is anoptimal γn in equilibrium between zero and one. Moreover,recall that the cost function is non-decreasing and convex(i.e., cγZ0 and cγγZ0). Recall that the cost of borrowingcan be expressed as R¼ R0�cðγnÞ=ð1�γnÞ, where R0 is

30 Note that this is the indifference condition for the L investorswhen they take the lending fee R as given.

given by

R0 ¼mL�mSþ

ν

τSρS�

γn

1�γnQ

� �� ν

τL

11�γn

QþρL

� �1�γnð Þ 1þν

δ

� :

ð101ÞThis implies that

Rδ ¼∂R∂δ

¼ ν

δðνþδÞR0 and ð102Þ

Rγ ¼ ∂R∂γ

¼ �Q

1τS

þ 1τL

� �ð1�γÞ3 1

νþ1δ

� � þ 11�γ

R0� 11�γ

cγþ 1ð1�γÞ2

c

!:

ð103ÞFinally, since R¼ c′ðγnÞ in equilibrium, we have thatdR=dδ¼ RδþRγdγ=dδ must be equal to dcγ=dδ¼ cγγdγ=dδ,which implies

dγdδ

¼ Rδ

cγγ�Rγ¼

ν

δðνþδÞ R0

cγγþQ

1τS

þ1τL

� �ð1� γÞ3

1ν

þ1δ

� �40: ð104Þ

Since R¼ cγðγnÞ and cγγ40, we have that for γoγn,cγðγÞocγðγnÞ, and so

ΔP ¼ τLγnþτSτLþτS

R� τLτLþτS

c¼ τSτLþτS

Rþ τLτLþτS

γnR�c �

Z0:

ð105ÞMoreover, change in the price distortion due to a change inδ can be expressed as

dΔPdδ

¼ � τLτLþτS

cγdγdδ

þ τLτLþτS

Rdγdδ

þ τLγþτSτLþτS

dRdδ

; ð106Þ

dΔPdδ

¼ τLγþτSτLþτS

dRdδ

¼ τLγþτSτLþτS

dcγdδ

¼ τLγþτSτLþτS

cγγdγdδ

40: ð107Þ

Hence, the distortion in price is increasing in the noiseδ. □

Appendix C. Costly search by short sellers

In this appendix, we develop a search model in whichthe equilibrium fraction of the outstanding supply of theunderlying asset that is borrowed/lent is determinedendogenously. Specifically, suppose short-sellers pay autility cost cðλÞ to search, which results in successfullymeeting a long investor with probability λ. Conditional onmeeting a long investor, they submit demand fxS;1; yS;1g.On the other hand, if they do not meet a long investor,they submit demand fxS;0 � 0; yS;0g. As a result, short-sellers solve the following problem:

maxλ;xS;0 ;xS;1 ;yS;0 ;yS;1

λE½uðW0þxS;1ðF�ðP�RÞÞþyS;1ðFþɛ�DÞþρSFÞ�þð1�λÞE½uðW0þxS;0ðF�ðP�RÞÞþyS;0ðFþɛ�DÞþρSFÞ�

�cðλÞ:ð108Þ

Similarly, if a long investor meets a short-seller (whichhappens with probability λ), he submits a demand

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608606

fxL;1; yL;1g. On the other hand, if they do not meet a short-seller, they submit demand fxL;0; yL;0g. As a result, short-sellers solve the following problem:

maxxL;0 ;xL;1 ;yL;0 ;yL;1

λE½uðW0þxL;1ðF�ðP�RÞÞþyL;1ðFþɛ�DÞþρLFÞ�þð1�λÞE½uðW0þxL;0ðF�PÞþyL;0ðFþɛ�DÞþρLFÞ�:

ð109ÞThe market clearing conditions are given by the following:

λðxL;1þxS;1Þþð1�λÞðxS;0þxL;0Þ ¼Q ; ð110Þ

λxS;1þλxL;1Z0; ð111Þ

λðyL;1þyS;1Þþð1�λÞðyS;0þyL;0Þ ¼ 0; ð112Þ

xS;0 ¼ 0: ð113ÞTo simplify notation, we shall restrict attention to theparameter assumptions from Section 3 of the benchmarkmodel. Specifically, suppose ρS ¼ �ρL ¼ ρ and suppose therisk-tolerance of both groups is given by τ. In this case, thefollowing proposition characterizes the equilibrium.

Proposition 9. Suppose that S investors pay a per-unit costcðλÞ in order to search for L investors with probability λ,where cðλÞ is non-negative, non-decreasing, and strictlyconvex. Then, the equilibrium prices are given by

D¼m� ν

2τQ ; R¼max 0;

δ

δþν

ν

τρ� 1

1�λQ

� �� �and

ð114Þ

P ¼DþR� ν

2τδ

δþνxL;1þxS;1�Q �� �

ð115Þ

where

xL;1 ¼ρ if R401

1þλQþ2λρð Þ if R¼ 0;

8<: ð116Þ

xS;1 ¼�ρ if R40

� 11þλ

2ρ�Qð Þ if R¼ 0;

8<: ð117Þ

and λ is the unique solution to

c′ λð Þ ¼E½uðW0þxS;1ðF�ðP�RÞÞþyS;1ðFþɛ�DÞþρSFÞ�

�E½uðW0þyS;0ðFþɛ�DÞþρSFÞ� :

ð118Þ

Proof. The expressions for the prices follow from comput-ing the optimal demand for the underlying and thederivative in each scenario and applying the marketclearing conditions. In particular, one can show that:

�

If the underlying is scarce, then we have

yL;1 ¼ yS;1 ¼ν

νþδ

12Q ; ð119Þ

yL;0 ¼ν

νþδ

12

2ρ� 1þλ

1�λQ

� �; yS;0 ¼ � ν

νþδρ� 1

2Q

� �;

ð120Þ

xL;1 ¼ �xS;1 ¼ ρ; xL;0 ¼1

1�λQ : ð121Þ

�

If the underlying is not scarce, i.e., R¼0

yL;1 ¼ yS;1 ¼ yL;0 ¼12

ν

νþδ

1�λ

1þλ2ρ�Qð Þ; ð122Þ

yS;0 ¼ � 12

ν

νþδ2ρ�Qð Þ; ð123Þ

xL;1 ¼ xL;0 ¼1

1þλQþ2λρð Þ; xS;1 ¼ � 1

1þλ2ρ�Qð Þ:

ð124Þ

Finally, the optimal λ is pinned down by the first-ordercondition for λ in the S investors' optimization problem,given by

c′ðλÞ ¼DEU where DEU � EU1�EU0; ð125Þ

EU1 � E½uðW0þxS;1ðF�ðP�RÞÞþyS;1ðFþɛ�DÞþρSFÞ�;ð126Þ

EU0 � E½uðW0þyS;0ðFþɛ�DÞþρSFÞ�: ð127ÞNote that

EU0 ¼ �ef� ð1=τÞðW0 þmρþð1=8τÞðν=ðνþ δÞÞðQνðQ �4ρÞ�4δρ2ÞÞg ð128Þwhether or not the underlying is scarce. On the otherhand,

EU1 ¼ �ef� ð1=τÞðW0 þmρþð1=8τÞðν=ðνþ δÞÞQνðQ �4ρÞÞg ð129Þif the underlying is scarce, and

EU1 ¼ �ef� ð1=τÞðW0 þmρþð1=8τÞðν=ðνþ δÞÞð4δðQ �2ρÞ2=ð1þ λÞ2 þQνðQ �4ρÞ�4δρ2ÞÞg

ð130Þif the underlying is not scarce. As such, DEU only dependson λ through EU1 when the underlying is scarce, andmoreover, DEU is non-increasing in λ. Therefore, if cðλÞ isstrictly convex, i.e., c′ðλÞ is increasing, then there is aunique solution λn to Eq. (125), which characterizes theequilibrium search probability λ. □

The above result establishes that many of the resultsfrom our benchmark model in Section 3 continue to holdin this setting, in which the equilibrium fraction of theoutstanding supply of the underlying asset that is lent/borrowed is determined as a function of the endogenoussearch probability λ. For instance, note that: (i) the price ofthe derivative does not depend on the search probability λor the noise in the derivative payoff δ, and (ii) the repo rateR and the price distortion ΔP ¼ P�P0 are decreasing in λ.Moreover, as in the benchmark model of the paper, for afixed search probability, increasing the noise in the deri-vative δ makes the underlying more scarce and, therefore,increases the price distortion.

However, when short-sellers can endogenously choosetheir search probability (which determines the scarcity ofthe underlying) in response to the noise in the derivativepayoff, then increasing the noise in the derivative maydecrease the scarcity and price distortion in the underlying.Intuitively, the equilibrium search probability λ is deter-mined by the short-seller's first-order condition which

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608 607

sets the marginal cost of searching c′ðλÞ equal to theincrease in her expected utility from being able to tradein the underlying security (i.e., from locating a longinvestor). As such, more noise in the derivative increasesthe relative benefit (in expected utility) for each short-seller from meeting a long investor, which incentivizes herto search more aggressively in equilibrium. However, bysearching more aggressively, short-sellers reduce the scar-city in the underlying, and therefore reduce the pricedistortion.

The following proposition characterizes the conditionsunder which an increase in δ leads to an increase in theprice distortion, and shows that this is always the casewhen δ¼ 0, i.e., when the derivative security is initiallyfrictionless.

Proposition 10. Suppose that S investors pay a per-unit costcðλÞ in order to search for L investors with probability λ,where cðλÞ is non-negative, non-decreasing, and strictlyconvex and does not depend on δ. Then, the optimal searchprobability λn which is characterized by c′ðλÞ ¼DEU isincreasing in δ, i.e., λδ � ∂λn=∂δ40. The price distortion inthe underlying increases in the noise of the derivative payoff(i.e., δ), if λδ is small enough (as characterized in the proof), orif δ¼ 0.

Proof. Note that then the underlying is scarce,

DEU ¼ �e�ð1=τÞðW0 þmρþð1=8τÞðν=ðνþ δÞÞQνðQ �4ρÞÞð1�eðρ2=2τ2Þðνδ=ðνþ δÞÞÞ;

ð131Þ

while when the underlying is not scarce,

DEU ¼ �e�ð1=τÞðW0 þmρþð1=8τÞðν=ðνþ δÞÞðQνðQ �4ρÞ�4δρ2ÞÞ

ðe�ð1=2τ2Þðν=ðνþ δÞÞδðQ �2ρÞ2=ð1þ λÞ2 �1Þ: ð132Þ

In both cases, one can show that ð∂=∂δÞDEU40. Since cð�Þdoes not depend on δ, this implies that for all else equal, anincrease in δ (i.e., more noise in the derivative) shifts theDEU curve up, which implies λn will be higher, i.e.,ð∂=∂δÞλn δð Þ40. As a result, δ can have an ambiguous effecton the price distortion. In particular, note that when theunderlying is scarce, the price distortion is given by

ΔP ¼ δ

δþν

ν

τρ� 1

1�λQ

� �þ ν

2τδ

δþνQ

� �ð133Þ

) ∂∂δ

ΔP ¼ 1τ

ν

δþν

ν

νþδρþQ

12� 1

1�λ

� �� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

40

� δ

1�λQλδ|fflfflfflfflffl{zfflfflfflfflffl}

40

26664

37775:

ð134Þ

This implies that the effect of δ on ΔP depends on therelative magnitude of λδ, which in turn depends on thecost function c. Moreover, note that the derivative ispositive when δ¼ 0 (as long as λδ is bounded) or if

λδo

ν

νþδρþQ

12� 11�λ

� �� �δ

1�λQ

:

Similarly, when the underlying is not scarce, the pricedistortion is

ΔP ¼ ν

2τ2ρ�Qð Þ δ

νþδ

1�λ

1þλð135Þ

) ∂∂δ

ΔP ¼ ν

2τ2ρ�Qð Þ ðνð1�λÞ�2δðδþνÞλδÞ

ðνþδÞ2ð1þλÞ2: ð136Þ

Again, the derivative is positive when δ¼ 0, or ifλδoνð1−λÞ=ð2δðδþ νÞÞ. □

Appendix D. Position limits

Consider a version of the model in Section 3, in whichthe payoff of the derivative is F (i.e., there is no noise), butthere are position limits that restrict the size of derivativepositions to y, i.e., yL ¼ �ySry. In this case, the first-orderconditions for investor i are given by

νðxiþyiþρiÞ ¼ τðm�ðP�γiRÞÞ and ð137Þ

νðxiþyiþρiÞ ¼ τðm�DÞ; ð138Þwhere γLrγo1 and γS ¼ 1. As in the model, the marketclearing conditions are given by

∑ixi ¼Q ; ∑

iyi ¼ 0 and γxLþxS ¼ 0: ð139Þ

As in the benchmark model, the optimal equilibriumallocations in the economy without any frictions (and inthe absence of derivatives) is xL ¼ 1

2Qþρ and xS ¼ 12Q�ρ.

This implies there are two cases to consider.Case1: If 1

2Qþρo 1= 1�γð Þ �Qþy (or equivalently, if

12Q�ρZ� γ= 1�γð Þ �

Q�y), then the underlying securityis not scarce. In this case, R¼0 and the price of thederivative and the underlying are given by

P ¼D¼m� ν

2τQ : ð140Þ

Since the payoffs and prices of the underlying and thederivative are identical, this implies a certain degree onindeterminacy in the equilibrium quantities held by inves-tors. We can recover a complete characterization if weimpose some type of tie-breaking rule — for instance,suppose investors trade in the underlying when indifferentbetween the two securities. In this case, the equilibriumpositions are given by

xL ¼ Q�xS ¼min12Qþρ;

11�γ

Q� �

and ð141Þ

yL ¼ �yS ¼max 0;12Qþρ� 1

1�γQ

� �: ð142Þ

Case2: If 12QþρZ 1= 1�γð Þ �

Qþy, then the underlyingis scarce, and R40. Specifically, market clearing implies:

R¼ 11�γ

ν

τ2ρ� 1þγ

1�γQ�2yL

� �: ð143Þ

Given the payoff of the derivative and the underlying areidentical, the price of the derivative is bounded by the netcost to L and S investors for the underlying, i.e.,

P�γRZDZP�R: ð144ÞMoreover, for any price D that is strictly in between thesebounds, L and S investors take the largest derivative

S. Banerjee, J.J. Graveline / Journal of Financial Economics 111 (2014) 589–608608

position they can, i.e., yL ¼ �yS ¼ y. Within these bounds,the derivative price is indeterminate without additionalassumptions (e.g., bargaining between L and S investorscan help pin down D), but the equilibrium quantities andthe price of the underlying security are pinned down asfollows:

xL ¼Q�xs ¼1

1�γQ ; yL ¼ �yS ¼ y; P ¼m� ν

2τQþ 1þγ

2R;

ð145Þ

R¼max 0;1

1�γ

ν

τ2 ρ�yð Þ� 1þγ

1�γQ

� �� �: ð146Þ

As such, this version of the model also delivers the maincomparative static result of the paper. Increasing ease oftrade in the derivative (in this case, by increasing y),reduces the price distortion in the underlying by relaxingthe scarcity that investors face.

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