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THE JOURNAL OF FINANCEVOL. LX, NO. AUGUST 2005 Predatory Trading MARKUS K. BRUNNERMEIER and LASSE HEJE PEDERSEN ∗ ABSTRACT This paper studies predatory trading, trading that induces and/or exploits the need of otherinvestorstoreducetheirpositions.Weshowthatifonetraderneedstosell,others also sell and subsequently buy back the asset. This leads to price overshooting and a reduced liquidation value for the distressed trader. Hence, the market is illiquid when liquidity is most needed. Further, a trader profits from triggering another trader’s crisis, and the crisis can spill over across traders and across markets. L ARGE TRADERS FEAR A FORCED LIQUIDATION , especially if their need to liquidate is known by other traders. For example, hedge funds with (nearing) margin calls may need to liquidate, and this could be known to certain counterparties such as the bank financing the trade. Similarly, traders who use portfolio insurance, stop loss orders, or other risk management strategies can be known to liquidate in response to price drops; a short-seller may need to cover his position if the price increases significantly or if his share is recalled (i.e., a “short squeeze”); certain institutions have an incentive to liquidate bonds that are downgraded or in default; and, intermediaries who take on large derivative positions must hedge them by trading the underlying security. A forced liquidation is often very costly since it is associated with large price impact and low liquidity. We provide a new framework for studying the strategic interaction among large traders who have market impact. Traders trade continuously and limit their trading intensity to minimize temporary price impact costs. Some of the traders may end up in financial difficulty, and the resulting need to liquidate is known by the other strategic traders. Our analysis shows that if a distressed large investor is forced to unwind his position (i.e., when he needs liquidity the most), other strategic traders initially trade in the same direction. That is, to profit from price swings, other traders ∗ Brunnermeier is affiliated with Princeton University and CEPR; Pedersen is at New York Uni- versity and NBER. We are grateful for helpful comments from Dilip Abreu, William Allen, Ed Altman, Yakov Amihud, Patrick Bolton, Menachem Brenner, Robert Engle, Stephen Figlewski, Gary Gorton, Rick Green, Joel Hasbrouck, Burt Malkiel, David Modest, Michael Rashes, Jose Scheinkman, Bill Silber, Ken Singleton, Jeremy Stein, Marti Subrahmanyam, Peter Sørensen, Nikola Tarashev, Jeff Wurgler, an anonymous referee, and seminar participants at NYU, McGill, Duke University, Carnegie Mellon University, Washington University, Ohio State University, Uni- versity of Copenhagen, London School of Economics, University of Rochester, University of Chicago, UCLA, Bank of England, University of Amsterdam, Tilburg University, Wharton, Harvard Uni- versity, and New York Federal Reserve Bank as well as conference participants at Stanford’s SITE conference and the annual meeting of the European Finance Association. Brunnermeier acknowl- edges research support from the National Science Foundation. 1825 1826 TheJ ournal of Finance conduct predatory trading and withdraw liquidity instead of providing it. This predatory activity makes liquidation costly and leads to price overshooting. Moreover, predatory trading can even induce the distressed trader’s need to liquidate; hence, predatory trading can enhance the risk of financial crisis. We show that predation is profitable if the market is illiquid and if the distressed trader’s position is large relative to the buying capacity of other traders. Fur- ther, predation is most fierce if there are few predators. These findings are in line with anecdotal evidence as summarized in Table I. A well-known example is the alleged trading against Long Term Capital Man- agement’s (LTCM’s) positions in the fall of 1998. Business Week wrote: ... if lenders know that a hedge fund needs to sell something quickly, theywillsellthesameasset—drivingthepricedownevenfaster.Goldman, Sachs & Co. and other counterparties to LTCM did exactly that in 1998. 1 Cramer (2002, p. 182) describes hedge funds’ predatory intentions in colorful terms: When you smell blood in the water, you become a shark.... when you know that one of your number is in trouble... you try to figure out what he owns and you start shorting those stocks... Also, Cai (2002) finds that “locals” on the Chicago Board of Exchange (CBOE) pits exploited knowledge of LTCM’s short positions in the treasury bond fu- tures market. Another indication of the fear of predatory trading is evident in the opposition to UBS Warburg’s proposal to take over Enron’s traders without taking over its trading positions. This proposal was opposed on the grounds that “it would present a ‘predatory trading risk’ because Enron’s traders would ef- 2 fectively know the contents of the trading book.” Similarly, many institutional investors are forced by law or their own charter to sell bonds of companies which undergo debt restructuring procedures. Hradsky and Long (1989), for example, documents price overshooting in the bond market after default announcements. Furthermore, our model shows that an adverse wealth shock to one large trader, coupled with predatory trading, can lead to a price drop that brings other traders into financial difficulty, leading in turn to further predation, and so on. This ripple effect can cause a widespread crisis in the financial sector. Ac- cordingly,thetestimonyofAlanGreenspanintheU.S.HouseofRepresentatives on October 1, 1998 indicates that the Federal Reserve Bank was worried that LTCM’s financial difficulties might destabilize the financial system as a whole: ... the act of unwinding LTCM’s portfolio would not only have a sig- nificant distorting impact on market prices but also in the process could produce large losses, or worse, for a number of creditors and counterpar- ties, and for other market participants who were not directly involved with 3 LTCM. 1 2“The Wrong Way to Regulate Hedge Funds,” Business Week,Februy26,2001, p.90. AFX News Limited, AFX—Asia, January 18, 2002. 3Testimony of Alan Greenspan, U.S. House of Representatives, October 1, 1998, http://www. federalreserve.gov/boarddocs/testimony/19981001.htm. Predatory Trading 1827 Also, the Brady Report (Brady et al. (1988), p. 15) suggests that the 1987 stock market crash was partly due to predatory trading in the spirit of our model: ... This precipitous decline began with several “triggers,” which ignited mechanical, price-insensitive selling by a number of institutions following portfolio insurance strategies and a small number of mutual fund groups. The selling by these investors, and the prospect of further selling by them, encouraged a number of aggressive trading-oriented institutions to sell in anticipation of further declines. These aggressive trading-oriented insti- tutions included, in addition to hedge funds, a small number of pension andendowmentfunds,moneymanagementfirmsandinvestmentbanking houses. This selling in turn stimulated further reactive selling by portfolio insurers and mutual funds. Predation risk affects the optimal risk management strategy for large institu- tional investors who hold illiquid assets. The optimal risk management strategy should depend on the liquidity of the assets and on the positions and financial standing of other large investors. Indeed, JP Morgan Chase and Deutsche Bank recently developed a “dealer exit stress-test” to assess the risk that a rival is forced to withdraw from the market (Jeffery (2003)). Further, risk managers should consider the risk that fund outflows can lead to predatory trading, re- sulting in losses that could fuel further outflows, and so on. Hence, the more likely fund outflows are, the more liquid the fund’s asset holdings should be. The danger of predatory trading might make it impossible for a fund to raise money in order to temporarily bridge some financial short-falls, since doing so requires that it reveals its financial need. More generally, the possibility of predatory trading is an argument against very strict disclosure policy. In the same spirit, the disclosure guidelines of the IAFE Investor Risk Commit- tee (IRC) (2001) maintain that “large hedge funds need to limit granularity of reporting to protect themselves against predatory trading against the fund’s position.” Likewise, market makers at the London Stock Exchange prefer to delay the reporting of large transactions since it gives them “a chance to reduce a large exposure, rather than alerting the rest of the market and exposing them to predatory trading tactics from others.” 4 Our model also provides guidance for the valuation of large security posi- tions. We distinguish between three forms of value, with increasing emphasis on the position’s liquidity. Specifically, the “paper value” is the current mark-to- market value of a position, the “orderly liquidation value” reflects the revenue one could achieve by secretly liquidating the position, and the “distressed liqui- dation value” equals the amount which can be raised if one faces predation by other strategic traders, that is, with endogenous market liquidity. We show that under certain conditions, the paper value exceeds the orderly liquidation value, which in turn exceeds the distressed liquidation value. Hence, if a large trader estimates “impact costs” based on normal (orderly) market behavior, then he 4Financial Times,J une 5, 1990, section I, p. 12. 1828 TheJ ournal of Finance may underestimate his actual cost in case of an acute need to sell because pre- dation makes liquidity time-varying. In particular, predation reduces liquidity when large traders need it the most. Along these lines, Pastor and Stambaugh (2003) and Acharya and Pedersen (2005) find measures of liquidity risk to be priced. Our work is related to several strands of literature. First, our model provides a natural example of “destabilizing speculation” by showing that although strategic traders stabilize prices most of the time, their predatory behavior can destabilize prices in times of financial crises. Our model thus con- tributes to an old debate; see Friedman (1953), Hart and Kreps (1986), DeLong et al. (1990), and Abreu and Brunnermeier (2003). Trading based on private information about security fundamentals is studied by Kyle (1985), whereas, in our model, agents trade to profit from their information about the future order flow coming from the distressed traders. Order flow information is also studied by Madrigal (1996), Vayanos (2001), and Cao, Evans, and Lyons (2006), but these papers do not consider the strategic effects of forced liquidation. The notion of predatory trading partially overlaps with that of stock price manip- ulation, which is investigated by Allen and Gale (1992) among others. One distinctive feature of predatory trading is that the predator derives profit from the price impact of the prey and not from his own price impact. Attari, Mello, and Ruckes (2002) and Pritsker (2003) are close in spirit to our paper. Pritsker (2003) also finds price overshooting in an example with heterogeneous risk- averse traders. Attari et al. (2002) focus, in a two-period model, on a distressed trader’s incentive to buy in order to temporarily push up the price when fac- ing a margin constraint, and a competitor’s incentive to trade in the opposite direction and to lend to this trader. The systemic risk component of our paper is related to the literature on financial crisis. Bernardo and Welch (2004) pro- vide a simple model of “financial market runs” in which traders join a run out of fear of having to liquidate before the price recovers, and Morris and Shin (2004) study how sales can reinforce sales. The remainder of the paper is organized as follows. Section I introduces the model. Section II provides a preliminary result which simplifies the analysis. Section III derives the equilibrium and discusses the nature of predatory trad- ing, with both a single and multiple predators. Further, Section III shows how predation can drive an otherwise solvent trader into financial distress and dis- cusses implications for risk management. Section IV studies the valuation of largepositionsinlightofilliquiditycausedbypredation.SectionVconsidersthe buildup of the traders’ positions and implications of disclosure requirements. Front-running, circuit breakers, up-tick rule, and contagion are discussed in Section VI. Proofs are relegated to Appendix A. Appendix B provides a gener- alized model with noisy asset supply. I. Model We consider a continuous-time economy with two assets, a riskless bond and a risky asset. For simplicity we normalize the risk-free rate to 0. The risky asset Predatory Trading 1829 has an aggregate supply of S > 0 and a final payoff v at time T, where v is a random variable with an expected value of E(v) = µ. One can view the risky asset as the payoff associated with an arbitrage strategy consisting of multiple assets. The price of the risky asset at time t is denoted by p(t). The economy has two kinds of agents: large strategic traders (arbitrageurs) and long-term investors. We can think of the strategic traders as hedge funds and proprietary trading desks, and the long-term investors as pension funds and individual investors. Strategic traders, i ∈ {1, 2,... ,I }, are risk neutral and seek to maximize their expected profit. Each strategic trader is large, and hence, his trading impacts the equilibrium price. He therefore acts strategically and takes his price impact into account when trading. Each strategic trader i has a given initial endowment, x (0), of the risky asset and he can continuously trade the i i asset by choosing his trading intensity, a (t). Hence, at time t his position, x (t), in the risky asset is ▯ t i i i x (t) = x (0) + a (τ)dτ. (1) 0 We assume that each large strategic trader is restricted to hold i x (t) ∈ [−¯,¯x]. (2) This position limit can be interpreted more broadly as a risk limit or a capi- tal constraint. The specific constraint on asset holdings is not crucial for our results. What is crucial is that strategic traders cannot take unlimited posi- tions, because if they could, they would drive the price to the expected value p = µ,a trivial outcome. To consider the case of limited capital, we assume that ¯I < S. Strategic traders are subject to a risk of financial distress at time t .W e 0 consider both the case in which an exogenous set of agents is in distress (Sec- tion III.A) and the case of endogenous distress (Section III.B). In any case, we d denote the set of distressed strategic traders by I and the set of unaffected strategic traders, the “predators,” by I . Similarly, the number of distressed d p traders is I and the number of predators is I .A strategic trader in financial distress must liquidate his position in the risky asset, that is, i A a (t) ≤− I if x(t) > 0 and t > t0 d i i ∈ I ⇒ a (t) =f 0 i x(t) = 0 and t > t0 (3) i A a (t) ≥ I if x(t) < 0 and t > t0 where A ∈ R is related to the market structure described below. This statement says that a distressed trader must liquidate his position at least as fast as A/I until he reaches his final position x (T) = 0. Below we show that this is the 5All random variables are defined on a probability space (▯, F, P). 1830 TheJ ournal of Finance fastest rate at which an agent can liquidate without risking temporary price impact costs. 6 The assumption of forced liquidation can be explained by (external or in- ternal) agency problems. Bolton and Scharfstein (1990) show that an optimal financial contract may leave an agent cash constrained even if the agent is 7 subject to predation risk. Also, the need to liquidate can be the result of a com- pany’s own risk management policy. We note that our results do not depend qualitatively on the nature of the troubled agents’ liquidation strategy, nor do they depend on the assumption that such agents must liquidate their entire position. It suffices that a troubled large trader must reduce his position before time T. In addition to the strategic traders, the market is populated by long-term investors. The long-term traders are price-takers and have, at each point in time, an aggregate demand of Y (p) = 1 (µ − p), (4) λ depending on the current price p. This demand schedule by long-term traders is based on two assumptions. First, it is downward sloping since in order to get long-term traders to hold more of the risky asset, they must be compensated in terms of lower prices. This could be because of risk aversion or because of insti- tutional frictions that make the risky asset less attractive for long-term traders. For instance, long-term traders may be reluctant to buy complicated derivatives such as asset-backed securities. (This institutional friction, of course, is what makes it profitable for strategic traders to enter the market.) A downward sloping demand curve also arises in a price pressure model al a Grossman and Miller (1988) since the competitive but risk-averse market-making sector is only willing to absorb the selling pressure at a lower price. Price pressure im- plies a temporary price decline, and, similarly, in our model the price decline vanishes at time T. Alternatively, if strategic traders have private information about the fundamental value v, then the long-term traders face an adverse selection problem that naturally leads to a downward sloping demand curve (Kyle (1985)). As in Kyle (1985), λ measures the market liquidity of the risky asset.8 The second assumption underlying (4) is that long-term traders’ demand depends only on the current price p. That is, they do not attempt to profit from price swings. This behavior by the long-term investors is motivated by 6We will see later that, in equilibrium, a troubled trader who must liquidate maximizes his profit by liquidating at this speed. Liquidating fast minimizes the costs of front-running by other traders. 7 8Bolton and Scharfstein (1990) consider predation in product markets, not in financial markets. While the long-term traders have a downward sloping demand curve, we shall see that the strategictraders’actionstendtoflattenthecurve,exceptduringcrisisperiods.Empirically,Shleifer (1986), Chan and Lakonishok (1995), Wurgler and Zhuravskaya (2002), and others document down- ward sloping demand curves, disputing Scholes (1972) who concludes that the demand curve is almost flat. Predatory Trading 1831 an assumption that they do not have sufficient information, skills, or time to predict future price changes. Thetradingmechanismworksinthefollowingway.Themarketclearingprice p(t) solves Y(p(t)) + X(t) = S, where X is the aggregate holding of the risky asset by strategic traders, I ▯ i X (t) = x (t). (5) i=1 Market clearing and (4) imply that the price is p(t) = µ − λ(S − X (t)). (6) Hence, while in the “long term” at time T, the price is expected to be µ, in the “medium term” the demand curve is downward sloping as described in (6). Further, in a given instant, that is, “in the very short term,” the strategic investors do not have immediate access to the entire demand curve (6). As Longstaff (2001) documents, in the real world one cannot trade infinitely fast in illiquid markets. To capture this phenomenon, we assume that strategic traders can as a whole trade at most A ∈ R shares per time unit at the current price p(t). Rather than simply assuming that orders beyond A cannot be executed, we assume that traders suffer temporary impact costs if ▯ ▯ ▯ ▯ i ▯ ▯ a (t)▯> A. (7) i Orders are executed with equal priority in the sense that trader i incurs a cost of ▯ ▯ G(a (t), a (t)) := γ max 0, a − a ¯, a − a , (8) ¯ −i −i where a ¯ = ¯(a (t)) and a = a(a (t)) are, respectively, the unique solutions to ¯ ¯ ▯ a¯ + min{a ,¯a}= A, (9) j,▯ i ▯ a + max{a , a}= A, (10) ¯ ¯ j,▯ i and where a (t): = (a (t), ... , a i−1(t), a+1(t), ... , a (t)). In words¯ (a)s the ¯ highest intensity with which trader i can buy (sell) without incurring the cost associated with a temporary price impact. Further, G is the product of the per- share cost, γ, multiplied by the number of shares exceeding a ¯ or ¯.We assume for simplicity that the temporary price impact is large, that is, γ ≥ λIx ¯. There are several possible interpretations of this market structure. First, we can think of a limit order book with a finite depth as follows: each instant, long-term traders submit A new buy-limit orders and A new sell-limit orders at the current price level, while old limit orders are canceled. This implies that 1832 TheJ ournal of Finance the depth of the limit order book is always a flow of Adt . Hence, as long as the strategic traders trade at a total speed lower than A, their orders are absorbed by the limit order book, new limit orders flow in, and the price walks up or down the demand curve (6). Orders that exceed A cannot be executed. More generally, one could assume that such excess orders would hit limit orders far away from the current price, and consequently suffer temporary impact costs 9 in line with our model. Alternatively, one could interpret the model as an over-the-counter market in which it takes time to find counterparties. In order to trade, strategic traders must make time-consuming phone calls to long-term traders. As each strategic trader goes through his “rolodex”—his list of customer phone numbers ordered by reservation value—they walk along the demand curve. 10If strategic traders share the same customer base, they face an aggregate speed constraint in line with our model. If traders’ customer bases are distinct, the speed constraint is trader-specific. 11 Importantly,ourqualitativeresultsdonotdependonthespecificassumptions of the model; for example, they also arise in a discrete-time setting. The results rely on: (i) strategic traders have limited capital, that is, x ¯ << ∞, (otherwise, the price is always µ = E(v)); and (ii) markets are illiquid in the sense that large trades move prices (λ> 0), and traders avoid trading arbitrarily fast (A < ∞). The latter assumption is relaxed in Section VI. B in which all long-term traders participate in a batch auction and orders of any size can be executed immediately. Strategic trader i’s objective is to maximize his expected wealth subject to the i constraints described above. His wealth is the final value, x (T )v,o f his stock holdings reduced by the cost, a (t)p(t) + G,o f buying the shares, where G is the temporary impact cost. That is, a strategic trader’s objective is ▯ ▯ T ▯ i i i −i iax Eix (T)v − [a (t)p(t) + G(a (t), a (t))]dt , (11) a (·)∈A 0 i i where A is the set of feasible trading processes, that is, the {F }-adapted t piecewise-continuous processes that satisfy (2) and (3). The filtration {F } rep- i t resents trader i’s information. We assume that each strategic trader learns, at time t , which traders are in distress. We consider both the case in which 0 the size of any distressed trader’s position is disclosed at t 0 and the case in 9 Our interpretation of the limit order book implicitly assumes that new orders arrive close to the current price, even if some trader hits limit orders far away from the current price. If new orders flow in at the last execution price, then hitting orders far away from the current price becomes even more costly as it permanently moves the price. 10If the strategic traders must contact the long-term traders in random order, the model needs to be slightly adjusted, but would qualitatively be the same. Duffie, Garleanu, and Pedersen (2003a, 2003b) provide a search framework for over-the-counter markets. 11Longstaff (2001) assumes that an agent must choose a limited trading intensity, that is, |a (t)|≤ constant. Making this assumption separately for each trader would not change our re- sults qualitatively. Predatory Trading 1833 which it is not. With no disclosure of positions, the filtration {F i} is generated d t by I 1 (t≥0 ) ith disclosure of positions, the filtration is additionally generated i d by x (t0)1 (t≥0 )or i ∈ I . 1 I D EFINITION 1: An equilibrium is a set of feasible processes (a , ... , a ) such that, for each i, ai ∈ A solves (11), taking a −i = (a , ... , a−1, ai+1, ... , a ) as given. If investors could learn from the price, then they could essentially infer other traders’ actions since there is no noise in our model. Assuming that the strate- gic traders can perfectly observe the actions of other strategic traders seems unrealistic and complicates the game. Therefore, in Appendix B, we consider a more general economy with supply uncertainty and show that, even though traders observe prices, they cannot infer other traders’ actions. For ease of ex- position, we analyze a setting with the same equilibrium actions but which abstracts from supply uncertainty; that is, we simply consider a filtration {F } t that does not include the price and the temporary impact costs. This means that a trader’s strategy depends on the current time, whether or not he is in 12 distress, and how many other traders are in distress. II. Preliminary Analysis In this section, we show how to solve a trader’s problem. For this, we rewrite trader i’s problem (11) as a constant (which depends on x(0)) plus ▯ ▯ T ▯ i 1 i 2 i −i i −i E λSx (T) − λ[x (T)] − [λa (t)X (t) + G(a (t), a (t))]dt , (12) 2 0 where we use E(v) = µ, expression (6) for the price, equation (1), which implies ▯ T i i 1 i 2 T that 0 a (t) x (t)dt = 2 [x (t) 0 , and where we define −i ▯ j X (t): = x (t). (13) j=1,...,I▯ i= Under our standing assumptions, p(t) < E(v)a t any time, and hence, any op- timal trading strategy satisfies x (T) = x ¯ if trader i is not in distress. That is, the trader ends up with the maximum capital in the arbitrage position. Fur- thermore, it is not optimal to incur the temporary impact cost, that is, each trader optimally keeps his trading intensity within his bounds a and a ¯. These ¯ considerations imply that the trader’s problem can be reduced to minimizing the third term in (12), which is useful in solving a trader’s optimization problem and in deriving the equilibrium. 1If one assumes that prices and temporary impact costs are observable, then our equilibrium remains a Nash equilibrium since it is optimal to choose a strategy that is a function of time if everyone else does so. This assumption would, however, raise additional technical issues re- lated to differential games (see, e.g., Clemhout and Wan (1994)). Also, such an assumption might lead to multiple equilibria, for instance, because deviations could be detected and followed by a punishment. 1834 TheJ ournal of Finance L EMMA 1: If T > 2¯ xI/A and X −i ≥ 0, trader i’s problem can be written as ▯ ▯ ▯ T i −i min E a (t)X (t)dt (14) a (·)∈A 0 ▯ T s.t. x (T) = x (0) + a (t)dt = x¯i ∈ I p (15) 0 i −i −i a (t) ∈ [a(a (t)), ¯(a (t))]. (16) ¯ d i Note that a distressed trader i ∈ I must have x (T ) = 0i n order to have a feasible strategy a ∈ A that satisfies (3). The lemma shows that the trader’s ▯ i −i problem is to minimize a (t)X (t)dt, that is, to minimize his trading cost, not taking into account his own price impact. This is because the model is set up such that the trader cannot make or lose money based on the way his own trades affect prices. (For example, λ is assumed to be constant.) Rather, the trader makes money by exploiting the way in which the other traders affect prices (through X −i). This distinguishes predatory trading from price manipulation. III. The Predatory Phase (t ∈ [t , T]) 0 We first consider the “predatory phase,” that is, the period [t , T]in 0 which some strategic traders face financial distress. In Section V, we analyze the full game including the “investment phase” [0, t )i n w0ich traders decide the size of their initial (arbitrage) positions. We assume that each strategic large trader has the same position, x(t ) 0 (0, x ¯], in the risky asset at time t 0 Furthermore, we assume for simplicity that there is “sufficient” time to trade, that is, t + 0 2xI/A < T. Weproceed in two stages: in Section A, certain traders are already in distress and we analyze the behavior of the undistressed predators. Section B endog- enizes agents’ distress and studies how predation and “panic” can lead to a widespread crisis. A. Exogenous Distress Here, we take as given the set of distressed traders, I , and the common initial holding, x(t0), of all strategic traders. A distressed trader j sells, in equi- librium, his shares at constant speed a =− A/I from t until t + x(t )I/A, and j 0 0 0 thereafter a = 0. This behavior is optimal, as will be clear later. This liquidation strategy is known, in equilibrium, by all the strategic traders. The predators’ strategies are more interesting. We first consider the simplest case in which there is a single predator, and we subsequently consider the case with multiple competing predators. Predatory Trading 1835 p A.1. Single Predator (I = 1) In the case with a single predator, the strategic interaction is simple: the predator, say i,i s merely choosing his optimal trading strategy given the known liquidation strategy of the distressed traders. Specifically, the distressed traders’ total position, X −i,i s decreasing to 0, and it is constant thereafter. Hence, using Lemma 1, we get the following equilibrium. P ROPOSITION 1: With I = 1, the following describes an equilibrium: 13 each dis- x (0 ) tressed trader sells with constant speed A/If or τ = A/I periods. The predator sells as fast as he can without incurring temporary impact costs for τ time peri- ods, and then buys back for x ¯/A periods. That is, −A/ rIf t ∈ [t0, 0 + τ), i∗ ▯ x¯▯ a (t) = o A f t ∈ t 0+ τ, t0+ τ + A , (17) x¯ 0 for t ≥ t0+ τ + A. The price overshoots; the price dynamics are p(t0) − λA[t − t0] for t ∈ ▯0 ,0t + τ), ▯ p (t) = p(t ) − λIx (t ) + λA[t − (t + τ)] for t ∈ t + τ, t + τ + x¯, (18) 0 0 0 0 0 A µ + λ[¯x − S] for t ≥ t + τ + x¯, 0 A where p(t 0 = µ + λ(Ix(t 0 − S). We see that although the surviving strategic trader wants to end up with all his capital invested in the arbitrage position (x (T) = x ¯), he is selling as long as the liquidating trader is selling. He is selling to profit from the price swings that occur in the wake of the liquidation. The predatory trader would like to “front- run” the distressed trader by selling before him and buying back shares after the distressed trader has pushed down the price further. Since both traders can sell at the same speed, the equilibrium is that they sell simultaneously and the predator buys back in the end. (The case in which predators can sell earlier than distressed traders is considered in Section VI.A.) The selling by the predatory trader leads to price “overshooting.” The price falls not only because the distressed trader is liquidating, but also because the predatory trader is selling as well. After the distressed trader is done selling, the predatory trader starts buying until he is at his capacity x ¯, and this pushes the price up toward its new equilibrium level. The predatory trader profits from the distressed trader’s liquidation for two reasons. First, the predator can sell his assets for an average price that is higher than the price at which he can buy them back after the distressed trader has left the market. Second, the predator can buy additional units cheaply until 13 The predator’s profit does not depend on how fast he buys back his shares as long as he does not incur temporary impact costs and he ends fully invested. Hence, there are other equilibria in which the predator buys back at a slower rate. These equilibria are, however, qualitatively the same as the one stated in the proposition, and there are no other equilibria than these. 1836 TheJ ournal of Finance he reaches his capacity. Since the price of the predator’s existing position x(t 0) goes down, the predator may appear to be losing money on a mark-to-market basis as the liquidation takes place. In the real world, holding a position that is loosing money on a mark-to-market basis can be problematic and this could further entice the predator’s selling. The predatory behavior by the surviving agent makes liquidation excessively costly for the distressed agent. To see this, suppose a trader estimates the liq- uidity in “normal times,” that is, when no trader is in distress. The liquidity—as definedbythepricesensitivitytodemandchanges—isgivenbyλinequation(6). When liquidity is needed by the distressed trader, however, the liquidity is lower due to the fact that the market becomes “one-sided” since the predator is selling as well. Specifically, the price moves by Iλ for each unit the distressed trader is selling. The distressed trader’s excess liquidation cost equals the predator’s profit from preying. Note that the predator does not exploit the group of long-term investors. The price overshooting implies that long-term investors are buying and selling shares at the same price. Hence, it does not ma14er for the group of long-term investors whether the predator preys or not. Numerical example.W e illustrate this predatory behavior with a numerical example. The supply of the risky asset is S = 40 and there are I = 2 strategic traders, each of whom has a capacity of x ¯ = 10 shares. At time t = 5, each d 0 trader has a position of x(t 0 = 8 and I = 1 trader becomes distressed while the other trader acts as a predator. At time T = 7 the asset is liquidated with expected value µ = 140 (or, equivalently, the market becomes perfectly liquid). Before that time, the price liquidity factor is λ = 1 and A = 20 shares can be traded per time unit without temporary impact costs. Figure 1 (Panel A) illustrates the holdings of the distressed trader: this trader starts liquidating his position of 8 shares at time t0= 5 with a trading intensity of A/2 = 10 shares per time unit. He is done liquidating at time 5.8. At time 5, the predator knows that this liquidation will take place, and, further, he realizes that the price will drop in response. Hence, he wants to sell high and buy back low. The predator optimally sells all his 8 shares simultaneously with the distressed trader’s liquidation, and, thereafter, he buys back x ¯ = 10 shares as shown in Figure 1 (Panel B). Figure 2 shows the price dynamics. The price is falling from time 5 to time 5.8, when both strategic traders are selling. Since 16 shares are sold and λ = 1, the price drops 16 points, falling to 100. As the predator rebuilds his position from time 5.8 to time 6.3, the price recovers to 110. Hence, there is a price overshooting of 10 points. It is intriguing that the predator is selling even when the price is below its long-run level 110. This behavior is optimal because, as long as the distressed 14 Recall that even though long-term investors could profit from using a predatory strategy themselves, we assume that they do not have sufficient information or skills to do so. Predatory Trading 1837 10 10 8 8 t6 t6 x x 4 4 2 2 0 0 4.5 5 5.5 6 6.5 7 4.5 5 5.5 6 6.5 7 time time Panel A Panel B Figure 1. Holdings of distressed trader (Panel A) and of single predator (Panel B). Start- ing with an initial holding of x (t ) = 8, both traders sell at maximum intensity of A/2 = 10 from x (t ) 0 t0 = 5 until t0+ A/2 = 5.8. By then, the distressed trader has completed his liquidation and sub- sequently the predator buys back shares. 116 112 108 price 104 ” marginal” ” marginal” selling price buying 100 price 4.5 5 5.5 6 6.5 7 time Figure 2. Price dynamics with single predator. The price falls as the distressed trader and the predator sell from time 5 to time 5.8, and rebounds as the predator buys back. The predation leads to price overshooting and a low liquidation value for the distressed trader—the market is illiquid when the distressed trader needs liquidity. The dotted line represents the hypothetical price dynamics if the predator sells one share less, that is, if only the distressed trader sells from time 5.7 to time 5.8. This hypothetical behavior is not optimal since the last “marginal” share can be sold at an average price of 101.00 and then can be bought back cheaper at 100.50. 1838 TheJ ournal of Finance trader is selling, the price will drop further and the predator can profit from selling additional shares and later repurchasing them. To further explain this point, we consider the predator’s profit if he sells one share less. In this case, the predator sells 7 shares from time 5 to time 5.7, waits for the distressed trader to finish selling at time 5.8, and then buys 9 shares from time 5.8 to time 6.25. The price dynamics in this case are illustrated by the dotted line in Figure 2. We see that the 9 shares are bought back at the same prices as the last 9 shares were bought in the case in which the predator continues to sell as long as the distressed trader does. Hence, to compare the profit in the two th cases, we focus on the price at which the 10 (and last) share is sold and bought back. This share is sold at prices between 102 and 100, that is, at an average price of 101. It is bought back at prices between 100 and 101, that is, at an averagepriceof100.50.Hence,this“extra”tradeisprofitable,earningaprofitof 101 − 100.50 = 0.50. A.2. Multiple Predators (I ≥ 2) We saw in the previous example how a single predatory trader has an in- centive to “front-run” the distressed trader by selling as long as the distressed trader is selling. With multiple surviving traders this incentive remains, but another effect is introduced: these predators want to end up with all their capi- tal in the arbitrage position and they want to buy their shares sooner than the other strategic traders do. The proposition below shows that, in equilibrium, predators trade off these incentives by selling for a while and then start buying back before the distressed traders have finished their liquidation. p PpROPOSITION 2: In the unique symmetric equilibrium with I ≥ 2 and x(t ) ≥ 0 I −1 x¯, each distressed trader sells with constant speed A/Ifor x 0t periods. Each I −1 Ip− 1 A/I x 0t )I − 1 predator sells at trading intensity A/Ifor τ := A/I periods and buys back AId x 0t ) shares at a trading intensity of I(I −1) until t0+ A/I . That is, −rAo/I f t ∈ [0, 0 + τ), ▯ ▯ i∗ AId for t ∈ t + τ, t + x 0t , a (t) = I(I −1) 0 0 A/I (19) x 0t ) 0 for t ≥ t 0 A/I . The price overshoots; the price dynamics are p(t ) − λA[t − t ] for t ∈ [t , t + τ), 0 0 ▯0 0 ▯ ∗ AId x 0t ) p (t) = p(t 0 − λAτ + λ I(I −1)[t − (t0+ τ)] for t ∈ t 0 τ, t +0 A/I , (20) p x 0t ) µ + λ[¯xI − S] for t ≥ t 0 A/I , where p(t 0 = µ + λIx(t )0− λS. Predatory Trading 1839 The proposition shows that price overshooting also occurs in the case of 15 multiple predators if x(t )i s l0rge relative to x ¯. This is because the preda- tors strategically sell excessively at first, and start buying relatively late. It is instructive to consider why it cannot be an equilibrium that there is no price overshooting and predators start buying back already at time t < t + τ ▯ 0 when the price reaches its long-run level. To see that, suppose all predators ▯ start buying back at time t . Then, if a single predator deviated and postponed ▯ buying, the price would continue to fall after t . Hence, this deviating predator could buy back his position cheaper after other traders have completed their liquidations, and hence, increase his profit. The equilibrium has the property that, from each predator’s perspective, −i X (t) (the total asset holdings of other strategic traders) is declining until t0+ τ and is constant thereafter. Since predator i also sells until t + τ, aggre- 0 gate stock holdings X(t) and the price overshoot. The price overshooting is lower if there are more predators since more preda- tors behave more competitively: p P ROPOSITION 3: Keep constant the fraction, I /I, of predators, the total arbitrage capacity, Ix ¯, and the total initial stock holding, Ix(t ), and assume that Ix(t ) ≥ 0 0 I x ¯. Then, the price overshooting p (i) is strictly positive for all nonzero I < ∞; p (ii) is decreasing in the number of predators I ; and, p (iii) approaches zero as I approaches inﬁnity. Numerical example.W e consider cases with a total number of traders I = 3, 9, and 27. For each case, we assume that a third of the traders are in dis- d tress, that is, I /I = 1/3. As in the previous example, we let λ = 1, µ = 140, S = 40, t 0 5, T = 7, the total trading speed be A = 20, the total initial holding be x(t )I = 16, and the total trader holding capacity be x ¯I = 20. Figure 3 (Panel 0 A) illustrates the asset holdings of predators and Figure 3 (Panel B) shows the price dynamics. We see that there is a substantial price overshooting when the number of predators is small, and that the overshooting is decreasing as the number 15 i We assume that all strategic traders’ positions at t 0re the same, that is, x (t 0 = x(t0)∀i. The analysis extends to a setting in which strategic traders hold different positions at t .In0 such a setting the equilibrium strategies are described as follows: initially, all predators and distressed sellers sell at full speed . Hence, each trader’s X−i is declining. When X −(t) = X −i(T) = x¯(I − 1) I i for the strategic trader with the smallest initial position x (t 0, all predators start repurchasing shares at a speed of Id A. Note that this speed guarantees that each predator’s X −iis flat. I[I− 1] When the predator with the highest x(t )re0ch es his final holding ¯,h es tops buying shares and the remaining predators increase their trading intensity to Id A. Similarly, as predators I[(I− 1) − 1] d complete their repurchases, the remaining predators adjust their trading speed to remaining A. I[I − 1] Interestingly, for fixed aggregate holdings of all predators at t 0t he price overshooting increases with the dispersion of the initial holdings. To see this, note that the length of the initial selling i −i spree is determined by the predator with the smallest initial position x (t ), 0hose X (t )is 0 the highest. 1840 TheJ ournal of Finance 7 117 I=3 6 116 I=9 I=3 115 I=27 5 I=9 I=27 114 (4 x price 3 112 2 111 1 110 4.5 5 5.5 6 6.5 7 4.5 5 5.5 6 6.5 7 time time Panel A Panel B Figure 3. Holdings (Panel A) and price dynamiis (Panel B) with multiple predators. The solid line shows each predator’s holdings x (t) and the price dynamics for the case in which two predators prey on one distressed trader. The dashed line shows holdings and prices when six predators prey on three distressed traders. The dotted lines correspond to the case with 18 predators and 9 distressed traders. As the number of predators increases, the predators start buying back earlier and the price overshooting decreases. of predators increases. With more predators, the competitive pressure to buy shares early is larger. Hence, the liquidation cost for a distressed trader is de- creasing in the number of predators (even holding the total trading capacity fixed). Collusion. The predators can profit from collusion. In particular, they could increase their revenue from predation by selling until the troubled traders were finished liquidating and only then start rebuilding their positions. Hence, through collusion, the predators could jointly act like a single predator (with the slight modification that multiple predators have more capital). Collusive and noncollusive outcomes are qualitatively different. A collusive outcome is characterized by predators buying shares only after the troubled traders have left the market and by a large price overshooting. In contrast, a noncollusive outcome is characterized by predators buying all the shares they need by the time the troubled traders have finished liquidating and by a relatively smaller price overshooting. Collusion could potentially occur through an explicit arranged agreement or implicitly without arrangement, called “tacit” collusion. Tacit collusion means that the collusive outcome is the equilibrium in a noncooperative game. In our model, tacit collusion cannot occur. However, if strategic traders could observe (or infer) each others’ trading activity, then tacit collusion might arise because 16 predators could “punish” a predator that deviates from the collusive strategy. 16 If traders could observe each others’ trades, then we would have to change our definition of strategies and equilibrium accordingly. A rigorous analysis of such a model is beyond the scope of this paper. Predatory Trading 1841 Large amounts of sidelined capacity, x¯ − x(t 0). Proposition 2 states that predatory trading and the overshooting occur as long as traders’ initial holding p is large enough relative to their position limit, that is, x(t ) ≥ 0 I −1 x¯. Proposi- p I −1 tion 2 analyzes the complementary case in which x(t ) < 0 I −1 x¯, that is, the I −1 capacity on the sideline is large relative to the selling of the distressed traders. Since the amount of available (sidelined) capacity is large, the competitive pres- sure among undistressed traders to buy shares overwhelms the incentive to front-run, and therefore there is no predatory trading. Instead, undistressed traders start buying immediately. P ROPOSITION 2 : In the unique symmetric equilibrium with I ≥ 2 and x(t ) < 0 I −1 I −1 ¯, each distressed trader sells with constant speed A/I. Each predator buys A(I + I ) (I −1)x 0t )−(I −1)x¯ initially at the high trading intensity of IPI for τ :=− I + I peri- A(1− IpI ) ods and goes on buying at the lower trading intensity of AI d until t + x 0t.) I(I −1) 0 A/I The price is increasing. In Section V we study the equilibrium determination of x(t ) and show tha0 x(t )i ss o large that predatory trading happens with positive probability. 0 B. Endogenous Distress, Systemic Risk, and Risk Management So far, we have assumed that certain strategic traders fall into financial distress, without specifying the underlying cause. In this section, we endogenize distress and study how predatory activity can lead to contagious default events. We assume that a trader must liquidate if his wealth drops to a threshold level W. This is because of margin constraints, risk management, or other ¯ considerations in connection with low wealth. Trader i’s wealth at t consists of i hisposition,x (t),oftheassetthatouranalysisfocuseson,aswellaswealthheld in other assets O (t). That is, his mark-to-market wealth is W (t) = x (t)p(t) + i i i O (t). The value of the other holdings, O (t), is subject to an exogenous shock at time t , which can be observed by all traders. At other times, O (t)i s constant. 0 Obviously, if the wealth shock ▯O at t is so neg0tive that W (t ) ≤ W, the i 0 ¯ trader is immediately in distress and must liquidate. Smaller negative shocks that result in W (t ) > W can, however, also lead to an endogenous distress, 0 ¯ since the potential selling behavior of predators and other distressed traders may erode the wealth of trader i even further. A trader who knows that he must liquidate in the future finds it optimal to start selling already at time t because 0 he foresees the price decline caused by the selling pressure of other strategic traders.Interestingly,whetheranagentanticipateshavingtoliquidatedepends on the number of other agents who are expected to be in distress. As in the d previous sections, we consider the set I of liquidating traders. d We let W(I )b e the maximum wealth at t such that tra0er i cannot avoid financial distress if I traders are expected to be in distress. More precisely, for d i I > 0, it is the maximum wealth W (t ) such th0t 1842 TheJ ournal of Finance max min W (t, a , a ) ≤ W,i (21) a ∈A it∈[0 ,T] ¯ where a −i has I − 1 strategies of liquidating and I − I strategies of preying d d in a time period of τ(I ). Further, for I = 0, W(0) = W.T o understand this definition, suppose trader i expects that I − 1 other traders will be in distress d with resulting selling pressure. Further, he expects that I − I other traders will act as predators, preying with a vigor that corresponds to I d defaults. That is, the predators sell in anticipation of all of the defaults including trader i’s own default. If, under these circumstances, trader i will sooner or later be in d default no matter what he does, then his wealth is less than W(I ). With this definition of W(I ), it follows directly that—in an equilibrium 17 d ¯ p d in which I traders immediately liquidate and I = I − I traders prey as in Propositions 1, 2, and 2 —every distressed trader i ∈ I d has wealth W (t ) ≤ d p i d 0 W(I ), and every predator i ∈ I has wealth W (t ) > W(I ). 0 ¯ Interestingly, the higher the expected number, I ,o f distressed traders, the d higher is the “survival hurdle” W(I ). ¯ P ROPOSITION 4: The more traders are expected to be in distress, the harder it is to survive. That is, W(I ) is increasing in I . d ¯ This insight follows from two facts: first, even without predatory trading, a higher number of distressed traders leads to more sell-offs and a larger price decline, thereby eroding each trader’s wealth. Second, a higher number of dis- tressed traders also makes predation more fierce since there are fewer compet- ing predators and more prey to exploit. This fierce predation lowers the price even further, making survival more difficult. Proposition 4 is useful in understanding systemic risk. Financial regula- tors are concerned that the financial difficulty of one or two large traders can drag down many more investors, thereby destabilizing the financial sec- tor. Our framework helps explain why this spillover effect occurs. To see this, consider the economy depicted in Figure 4 (Panel A). Trader A’s wealth is in the range of (W(1), W(2)], trader B’s wealth is in (W(2), W(3)], and trader C’s is in ¯ ¯ ¯ ¯ (W(3), W(4)]. The three remaining traders (D, E, and F) have enough reserves to fight off any crisis, that is, their wealth is above W(I). ¯ With these wealth levels, the unique equilibrium is such that no strategic trader is in distress and all of them immediately start to increase their position from x(t 0to ¯ x.T o see this, note first that it cannot be an equilibrium that one agent defaults. If one agent is expected to default, no one defaults because no one has wealth below W(1). Similarly, it is not an equilibrium that two traders default, because only trader A has wealth below W(2), and so on. ¯ 1There may be other kinds of equilibria in which a surviving trader does not prey because of fear of driving himself in distress. For ease of exposition, we do not consider these equilibria. Equilibria of the form that we consider exist under certain conditions on the initial holdings and wealth levels. Predatory Trading 1843 Figure 4. Systemic risk in setting with endogenous distress. This figure shows the wealth levels of traders A,... ,E and several survival hurdles W(Ithat is, the wealth necessary to d ¯ survive if the market believes that I traders will be in distress. In Panel A, traders’ wealth levels are high enough that all traders survive in the unique equilibrium. In Panel B, trader D is in distress because of a wealth shock. This leads to predatory trading which can drag traders A, B, and C into distress too. D On the other hand, if trader D faces a wealth shock at t such 0hat W (t ) < 0 W,h e can drag down traders A, B, and C, as shown in Figure 4 (Panel B). If ¯ it is expected that four traders will be in distress, then traders A, B, C, and D will liquidate their position since their wealth is below W(4). Intuitively, the ¯ fact that trader D is forced to liquidate his position encourages predation and the price is depressed. This, in turn, brings three other traders into financial difficulty. This situation captures the notion of systemic risk. The financial difficulty of one trader endangers the financial stability of three other traders. In the economy of Figure 4 (Panel B), there are also other equilibria in which 1, 2, or 3 traders face distress. For instance, it is an equilibrium that only trader D liquidates, since if everybody expects that only trader D will go under, traders A, B, C, E, and F prey only briefly and buy back after a short while. The predat

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