BUBBLES AND CRASHES. By Dilip Abreu and Markus K. Brunnermeier 1

Transcription

1 Econometrica, Vol. 71, No. 1 (January, 23), BUBBLES AND CRASHES By Dili Abreu and Markus K. Brunnermeier 1 We resent a model in which an asset bubble can ersist desite the resence of rational arbitrageurs. The resilience of the bubble stems from the inability of arbitrageurs to temorarily coordinate their selling strategies. This synchronization roblem together with the individual incentive to time the market results in the ersistence of bubbles over a substantial eriod. Since the derived trading equilibrium is unique, our model rationalizes the existence of bubbles in a strong sense. The model also rovides a natural setting in which news events, by enabling synchronization, can have a disroortionate imact relative to their intrinsic informational content. Keywords: Bubbles, crashes, temoral coordination, synchronization, market timing, overreaction, limits to arbitrage, behavioral finance. 1 introduction Notorious examles of bubbles followed by crashes include the Dutch tuli mania of the 163 s, the South Sea bubble of and more recently the Internet bubble which eaked in early 2. Standard neoclassical theory recludes the existence of bubbles, by backwards induction arguments in finite horizon models and a transversality condition in infinite horizon models (cf. Santos and Woodford (1997)). Similar conclusions emerge from no trade theorems in settings with asymmetric information (cf. Milgrom and Stokey (1982), Tirole (1982)). 2 All agents in these models are assumed to be rational. The efficient markets hyothesis also imlies the absence of bubbles. Many roonents of this view (see, for instance, Fama (1965)) are quite willing to admit that behavioral/boundedly rational traders are active in the market lace. However, they argue that the existence of sufficiently many well-informed arbitrageurs guarantees that any otential misricing induced by behavioral traders will be corrected. We develo a model that challenges the efficient markets ersective. In articular, we argue that bubbles can survive desite the resence of rational arbitrageurs who are collectively both well-informed and well-financed. The backdro of our analysis is a world in which there are some behavioral agents variously subject to animal sirits, fads and fashions, overconfidence and related sychological biases that might lead to momentum trading, trend chasing, and the like. 1 We thank seminar articiants at various universities and conferences, anonymous referees, and esecially Haluk Ergin, Ming Huang, Sebastian Ludmer, Stehen Morris, José Scheinkman, Andrei Shleifer, and an editor for helful comments. The authors acknowledge financial suort from the National Science Foundation and Princeton s CEPS rogram. 2 A more extensive review of the literature on bubbles can be found in Brunnermeier (21). 173

2 174 d. abreu and m. brunnermeier There is by now a large literature that documents and models such behavior. 3 We do not investigate why behavioral biases needing rational corrections arise in the first lace; for this we rely on the body of work very artially documented in footnote 3. We study the imact of rational arbitrage in this setting. We suose that rational arbitrageurs understand that the market will eventually collase but meanwhile would like to ride the bubble as it continues to grow and generate high returns. Ideally, they would like to exit the market just rior to the crash, to beat the gun in Keynes colorful hrase. However, market timing is a difficult task. Our arbitrageurs realize that they will, for a variety of reasons, come u with different solutions to this otimal timing roblem. This disersion of exit strategies and the consequent lack of synchronization are recisely what ermit the bubble to grow, desite the fact that the bubble bursts as soon as a sufficient mass of traders sells out. This scenario is the starting oint of our work. We resent a model that formalizes the synchronization roblem described above. Our aroach emhasizes two elements: disersion of oinion among rational arbitrageurs and the need for coordination. We assume that the rice surasses the fundamental value at a random oint in time t. Thereafter, rational arbitrageurs become sequentially aware that the rice has dearted from fundamentals. Arbitrageurs do not know whether they have learnt this information early or late relative to other rational arbitrageurs. In addition to its literal interretation, the assumtion of sequential awareness can be viewed as a metahor for a variety of factors such as asymmetric information, differences in viewoints, etc. that, in our model, find exression in the feature we seek to exlore and emhasize, that is, temoral miscoordination. Arbitrageurs have common riors about the underlying structure of the model. The coordination element in our model is that selling ressure only bursts the bubble when a sufficient mass of arbitrageurs have sold out. Arbitrageurs face financial constraints that limit their stock holding as well as their maximum shortosition. This limits the rice imact of each arbitrageur. Large rice movements can only occur if the accumulated selling ressure exceeds some threshold. In other words, a ermanent shift in rice levels requires a coordinated attack. In this resect, our model shares some features with the static second generation models of currency attacks in the international finance literature (Obstfeld (1996)). Thus, our model has both elements of cooeration and of cometition. On the one hand, at least a fraction of arbitrageurs need to sell out in order for the bubble to burst; this is the coordination/cooerative asect. On the other hand, arbitrageurs are cometitive since at most a fraction <1 of them can leave the market rior to the crash. In the equilibrium of our model, arbitrageurs stay in the market until the subjective robability that the bubble will burst in the next trading round is sufficiently high. Arbitrageurs who get out of the market just rior to the crash 3 See, for instance, Daniel, Hirshleifer, and Subrahmanyam (1998), Hirshleifer (21), Odean (1998), Thaler (1991), Shiller (2), and Shleifer (2).

3 bubbles and crashes 175 make the highest rofit. Arbitrageurs who leave the market very early make some rofit, but forgo much of the higher rate of areciation of the bubble. Arbitrageurs who stay in the market too long lose all caital gains that result from the bubble s areciation. Our results regarding market timing seem to fit well with oular accounts of the behavior of hedge fund managers during the recent internet bubble. 45 For examle, when Stanley Druckenmiller, who managed George Soros $8.2 billion Quantum Fund, was asked why he didn t get out of internet stocks earlier even though he knew that technology stocks were overvalued, he relied that he thought the arty wasn t going to end so quickly. In his words We thought it was the eighth inning, and it was the ninth. Faced with mounting losses, Druckenmiller resigned as Quantum s fund manager in Aril 2. 6 Rational arbitrageurs ride the bubble even though they know that the bubble will burst for exogenous reasons by some time t + if it has not succumbed to endogenous selling ressure rior to that time. Here t is the unknown time at which the rice ath surasses the fundamental value and arbitrageurs start getting aware of the misricing. In equilibrium each arbitrageur sells out after waiting for some interval after she becomes aware of the misricing. If arbitrageurs oinions are sufficiently disersed, there exists an equilibrium in which the bubble never bursts rior to t +. Even long after the bubble begins and after all agents are aware of the bubble, it is nevertheless the case that endogenous selling ressure is never high enough to burst the bubble. For more moderate levels of disersion of oinion (or equivalently for smaller ) endogenous selling ressure advances the date at which the bubble eventually collases. Nevertheless, the bubble grows for a substantial eriod of time. Moreover, these equilibria are unique (modulo a natural refinement). Thus, there is a striking failure of the backwards induction argument that would yield immediate collase in a standard model. The synchronization roblem arbitrageurs face is from their oint of view a blessing rather than a curse. After becoming aware of the bubble, they always have the otion to sell out, but rather otimally choose to ride the bubble over some interval. Our model rovides a natural setting in which news events can have a disroortionate imact relative to their intrinsic informational content. This is because 4 Brunnermeier and Nagel (22) rovide a detailed documentation of hedge funds stock holdings. 5 Kindleberger (1978) notes that even Isaac Newton tried to ride the South Sea Bubble in 172. He got out of the market at 7, after making a 3,5 rofit, but he decided to re-enter it thereby losing 2, at the end. Frustrated with his exerience, he concluded: I can calculate the motions of the heavenly bodies, but not the madness of eole. 6 However, fund managers can also not afford to simly stay away from a raidly growing bubble. Julian Roberts, manager of the legendary Tiger Hedge Fund, refused to invest in technology stocks since he thought they were overvalued. The Tiger Fund was dissolved in 1999 because its returns could not kee u with the returns generated by dotcom stocks. A Wall Street analyst who has dealt with both hedge fund managers vividly summarized the situation: Julian said, This is irrational and I won t lay, and they carried him out feet first. Druckenmiller said, This is irrational and I will lay, and they carried him out feet first. New York Times, Aril 29, 2, Another Technology Victim; To Soros Fund Manager Says He Overlayed Hand.

4 176 d. abreu and m. brunnermeier news events make it ossible for agents to synchronize their exit strategies. Of course, large rice dros are themselves significant synchronizing events, and we investigate how an initial rice dro may lead to a full-fledged collase. Thus the model yields a rudimentary theory of overreaction and rice cascades and suggests a rationale for sychological benchmarks such as resistance lines. In addition, our model rovides a framework for understanding fads in information such as the (over-)emhasis on trade figures in the eighties and on interest rates in the nineties. Overall, the idea that the bursting of a bubble requires synchronized action by rational arbitrageurs, who might lack both the incentive and the ability to act in a coordinated way, has imortant imlications. It rovides a theoretical argument for the existence and ersistence of bubbles. It undermines the central resumtion of the efficient markets ersective that not all agents need to be rational for rices to be efficient, and hence rovides further suort for behavioral finance models that do not exlicitly model rational arbitrageurs. More generally, our model suggests a new mechanism that limits the effectiveness of arbitrage. The latter might be relevant in other contexts; see Abreu and Brunnermeier (22) for an alication of the framework develoed in this aer. The remainder of the aer is organized as follows. Section 2 illustrates how the analysis relates to the literature. In Section 3 we introduce the rimitives of the model and define the equilibrium. Section 4 shows that all arbitrageurs emloy trigger strategies in any trading equilibrium. Section 5 demonstrates that if the disersion of oinion among arbitrageurs is sufficiently large, they never burst the finite horizon bubble and it only crashes for exogenous reasons at its maximum size. For smaller disersion of oinion the bubble also ersists but arbitrageurs burst it before the end of the horizon. Section 6 investigates the role of synchronizing events including the secial case of rice events. Section 7 concludes. 2 related literature The no-trade theorems rovide sufficient conditions for the absence of seculative bubbles. Allen, Morris, and Postlewaite (1993) develo contra-ositives of the no-trade theorems highlighting necessary conditions for the existence of bubbles. They call a misricing a bubble if it is mutual knowledge that the rice is too high. Of course, mutual knowledge (that is, all traders know) does not imly common knowledge (that is, all traders know, that all traders know, and so on ad infinitum). They rovide illustrative examles that satisfy their conditions and suort bubbles. In Allen and Gorton (1993) fund managers churn bubbles at the exense of their less informed client investors (who do not know the skill of the fund manager). They take on an overvalued asset even though they know that they might be last in line and hence unable to unload the asset. The order of trades is random and exogenous in their model. In equilibrium, our sequential awareness assumtion also leads to sequential trading and uncertainty about the timing of one s trades relative to other arbitrageurs trades.

5 bubbles and crashes 177 The aers described above assume that all agents are fully rational. In contrast, our work falls within the class of models in which rational arbitrageurs interact with boundedly rational behavioral traders. In DeLong, Shleifer, Summers, and Waldmann (199b) rational arbitrageurs ush u the rice after some initial good news in order to induce behavioral feedback traders to aggressively buy stocks in the next eriod. This delayed reaction by feedback traders allows the arbitrageurs to unload their osition at a rofit. In DeLong, Shleifer, Summers, and Waldmann (199a) arbitrageurs risk aversion and short horizons limit their ability to correct the misricing. In contrast, in our model arbitrageurs initially do not even attemt to lean against the misricing even though they are riskneutral and infinitely lived. In Shleifer and Vishny (1997), rofessional fund managers forgo rofitable long-run arbitrage oortunities because the rice might deart even further from the fundamental value in the intermediate term. In that case, the fund manager would have to reort intermediate losses causing client investors to withdraw art of their money which forces her to liquidate at a loss. Knowing this might haen in advance, the fund manager only artially exloits the arbitrage oortunity. In these aers rational arbitrageurs do not have the collective ability to correct misricing either because of their risk aversion or because of exogenously assumed caital constraints. In contrast in our aer, the aggregate resources of all arbitrageurs are sufficient to bring the rice back to its fundamental value. Thus, the weakness of arbitrage in our model is articularly striking because arbitrageurs can jointly correct the misricing, but it nevertheless ersists. Our assumtion that a critical mass of seculators is needed to burst a bubble has its roots in the currency attack literature. Obstfeld (1996) highlights the necessity of coordination among seculators to break a currency eg and oints out the resulting multilicity of equilibria in a setting with symmetric information. Morris and Shin (1998) introduce asymmetric information and derive a unique equilibrium by alying the global games aroach of Carlsson and van Damme (1993). Both currency attack models are static in the sense that seculators only decide whether to attack now or never. In contrast, our model is dynamic. The disersion of oinion among arbitrageurs, which we model as sequential awareness, can be related to the idea of asynchronous clocks. The latter were first introduced in the comuter science literature (see Halern and Moses (199)). Morris (1995) alies the concet to a dynamic coordination roblem in labor economics. There are several differences between our aers. His model satisfies strategic comlementarity and the global games aroach alies. Our model has elements of both coordination and cometition. The latter leads to a reemtion motive that lays a central role in our analysis. It is imortant for Morris result that layers can only condition on their individual clocks and not on calendar time or on their ayoffs, whereas in our model they can. In articular, traders learn from the existence of the bubble in our setting. In the richer strategy set of our model strategic comlementarity is not satisfied and the global games aroach does not aly. See footnote 15 for a discussion of this oint. In the tyical global game alication the symmetric information game has multile

6 178 d. abreu and m. brunnermeier equilibria, and any degree of asymmetric information leads to a unique equilibrium. Thus, there is a discontinuity at the oint where asymmetry vanishes. In contrast, in our dynamic model the reemtion motive leads to a unique equilibrium even under symmetric information. Furthermore, as the extent of asymmetric information goes to zero, the equilibrium outcome converges to the symmetric information (no bubble) outcome. 3 the model Historically, bubbles have often emerged in eriods of roductivity enhancing structural change. Examles include the railway boom, the electricity boom, and the recent internet and telecommunication boom. In the latter case, and in many of the historical examles, sohisticated market articiants gradually understood that the immediate economic imact of these structural changes was limited and that their full imlementation would take a long time. They also realized that only a few of the comanies engaged in the new technology would survive in the long run. On the other hand, less sohisticated traders over-otimistically believed that a aradigm shift or a new economy would lead to ermanently higher growth rates. We assume the rice rocess deicted in Figure 1. This rice rocess reflects the scenario outlined above, and may be interreted as follows. Prior to t = Figure 1. Illustration of rice aths.

7 bubbles and crashes 179 the stock rice index coincides with its fundamental value, which grows at the risk-free interest rate r, and rational arbitrageurs are fully invested in the stock market. Without loss of generality, we normalize the starting oint to t = and the stock market rice at t = to = 1. From t = onwards the stock rice t grows at a rate of g>r, that is t = e gt. 7 This higher growth rate may be viewed as emerging from a series of unusual ositive shocks that gradually make investors more and more otimistic about future rosects. Until some random time t, the higher rice increase is justified by the fundamental develoment. 8 We assume that t is exonentially distributed on with the cumulative distribution function t = 1 e t. Nevertheless, the rice continues to increase at the faster rate g even after t. Hence, from t onwards, only some fraction 1 of the rice is justified by fundamentals, while the fraction reflects the bubble comonent. We assume that is a strictly increasing and continuous function of t t, the time elased since the rice dearted from fundamentals. For the secial case where the fundamental value e gt +rt t coincides with the rice ath at t and always grows at a rate r thereafter, t t = 1 e g rt t. The rice t = e gt is ket above its fundamental value by irrationally exuberant behavioral traders. They believe in a new economy aradigm and think that the rice will grow at a rate of g in eretuity. 9 As soon as the cumulative selling ressure by rational arbitrageurs exceeds, the absortion caacity of behavioral traders, the rice dros by a fraction to its ost-crash rice. From this oint onwards, the rice grows at a rate of r. 1 Even if the selling ressure never exceeds we assume that the bubble bursts for exogenous reasons as soon as it reaches its maximum size. This translates to a final date, since is strictly increasing. Let us denote this date by t +. Note that this assumtion of a final date is arguably the least conducive to the ersistence of bubbles. In a classical model it would lead to an immediate collase for the usual backwards rogramming reasons. Another imortant element of our analysis is that rational arbitrageurs become sequentially informed that the fundamental value has not ket u with the growth of the stock rice index. More secifically, a new cohort of rational arbitrageurs of mass 1/ becomes aware of the misricing in each instant t from t until t +. That is, t t + forms the awareness window. Since t is random, an 7 The analysis can be directly extended to a setting that incororates a stochastic rice rocess with a finite number of ossible rice aths and where the rice grows in exectations at a rate of g. 8 Kindleberger (1978) refers to the hase of fundamentally good news as dislacement. 9 The rice rocess we assume is a modeling simlification that facilitates a clean and simle analysis. One, admittedly simlistic, setting that rationalizes this rocess is a world with behavioral traders who are risk-neutral, but wealth constrained, and who require a rate of return of g in order to invest in the stock market. If they are overconfident (in the sense that they are mistakenly convinced that they will receive early warning of a change in fundamentals), they will urchase any quantity of shares u to their aggregate absortion caacity whenever the rice falls below e gt. See also the remark at the end of this section. 1 We refer to fundamental value as the rice that will emerge after the bubble bursts. Notice that for =, there is no dro in fundamentals at t.

8 18 d. abreu and m. brunnermeier Figure 2. Sequential awareness. individual arbitrageur does not know how many other arbitrageurs have received the signal before or after her. An agent who becomes aware of the bubble at t i has a osterior distribution for t with suort t i t i. Each agent views the market from the relative ersective of her own t i. We will refer to the arbitrageur who learns of the misricing at time t i as arbitrageur t i. From the ersective of arbitrageur t i the truncated distribution of t is t t i = e e t i t e 1 Figure 2 deicts the distributions of t for arbitrageurs t i t j, and t k, where t k = t i +, so that arbitrageur t k is the last ossible tye to become aware of the bubble given arbitrageur t i s information. Viewed more abstractly, the set of ossible tyes of arbitrageurs is. Nature s choice of t determines the active tyes t i t t + in the economy. As noted earlier, we view this secification as a modeling device that catures temoral miscoordination arising from differences of oinion and information. We assume that 1 e < g r This guarantees that arbitrageurs do not wish to sell out before they become aware of the misricing. When, for instance, is very high or g r very small, the model is uninteresting in that arbitrageurs will sell out right away. The rice t = e gt exceeds its ost-crash (fundamental) value from t onwards. However, only a few arbitrageurs are aware of the misricing at this oint. From t + onwards, the misricing is known to a large enough mass of arbitrageurs who are collectively able to correct it. We label any ersistent misricing beyond t + a bubble. Given such an environment, we want to identify the best strategy of a rational arbitrageur t i. Each arbitrageur can sell all or art of her stock holding or even go short until she reaches a certain limit where her financial constraint is binding. Each trader can also buy back shares. A trader may exit from and return to the market multile times. However, each arbitrageur is limited in the number of shares she can go short or long. Without loss of generality, we can normalize the action sace to be the continuum between 1, where indicates the maximum long osition and 1 the maximum short osition each arbitrageur can take on. This sign convention is convenient given the focus of our analysis on selling ressure.

9 bubbles and crashes 181 Let tt i denote the selling ressure of arbitrageur t i at t and the function 1 a strategy rofile. The strategy of a trader who became aware of the bubble at time t i is given by the maing t i t i + 1. Note that 1 tt i is trader t i s stock holding at time t. For arbitrary, the function t need not be measurable in t i. We confine attention to strategy rofiles that guarantee a measurable function t. Notice that no individual deviation from such a rofile has any imact on the measurability roerty. The aggregate selling ressure of all arbitrageurs at time t t is given by stt = mintt + tt i dt i.let t T t = inftstt or t = t + denote the bursting time of the bubble for a given realization of t. Recall t i denotes arbitrageur t i s beliefs about t given that t t i t i. Hence, her beliefs about the bursting date are given by tt i = dt t i T t <t Given the structure of the game, the actions of the other traders affect trader t i s ayoff only if these actions cause the bubble to burst. Each arbitrageur s ayoff deends on the rices at which she sells and buys her shares minus the transactions costs for executing the order. The execution rices of arbitrageurs orders are either the re-crash rice t or the ost-crash rice 1 t t t. In the secial case where an arbitrageur submits her sell order exactly at the instant when the bubble bursts, her order is executed at the re-crash rice t = e gt as long as the accumulated selling ressure is smaller than or equal to. If the accumulated selling ressure exceeds, then only the first randomly chosen orders will be executed at the re-crash rice while the remaining orders are only executed at the ost-crash rice. In other words, the exected execution rice 1 t + 1 t t t is a convex combination of both rices, where > if the selling ressure is strictly larger than, and = if the selling ressure is less than or equal to at the time of the bursting of the bubble. We assume that arbitrageurs incur transactions costs whenever they alter their ositions. This imlies that each arbitrageur will only change her stock holdings at most a finite number of times; strategies entailing an infinite number of changes will be strictly dominated. Consequently, we confine attention to strategies that involve at most an (arbitrary) finite number of changes of osition. We rule out rohibitively high transactions costs that reclude arbitrageurs from ever selling out desite the fear of a bursting bubble. It is algebraically convenient to assume that the resent value of transactions cost is a constant c. That is, transactions cost at t equals ce rt. This formulation together with the former assumtion guarantees that equilibrium behavior is indeendent of c, the size of the transactions cost.

10 182 d. abreu and m. brunnermeier A general secification of the ayoff function is notationally cumbersome and involves a recursive formulation; it is given in the Aendix. For the secial case that arbitrageur t i remains fully invested in the market until she comletely sells out at t and remains out of the market thereafter (i.e. does not re-enter the market), arbitrageur t i s exected ayoff for selling out at t is t e rs 1 s T 1 ssdst i + e rt t1 tt i c t i rovided that the selling ressure at t does not strictly exceed and that T is strictly increasing. It will turn out that in the unique equilibrium trader t i adots recisely such a trigger-strategy, that the selling ressure never strictly exceeds, that T is strictly increasing, and consequently T 1 is well defined. The latter imlies that if the bubble bursts at s, the bubble comonent of the asset rice is recisely s T 1 ss. 11 Remark: Our assumtion that the bubble only bursts when the selling ressure exceeds imlies that rational agents do not become aware of selling ressure by other rational agents until it crosses this threshold. We relax this simlifying assumtion somewhat in Section 6 by allowing for intermediate rice dros rior to the final crash. Nevertheless, we do not fully endogenize the rice rocess. Note that in our model, the central uncertainty is about the one-dimensional random variable t. A nonstochastic rice rocess that is strictly monotonic in selling ressure would immediately reveal t. A more comlete model would entail multi-dimensional uncertainty and noisy rices that reclude all relevant asymmetric information in the model from being inferred with certainty from the current rice level. We believe that our rincial results would be qualitatively reserved in such a setting, though their recise exression would be substantially more comlicated. 4 reliminary analysis This section shows that without loss of generality we can restrict the analysis to trigger strategies, that is, to trading strategies in which an agent who sells out at t continues to attack the bubble at all times thereafter. We also derive the sell out condition according to which each arbitrageur sells her shares exactly at the moment when the temtation to ride the bubble balances the fear of its imminent collase (Lemma 7). We use the following notion of equilibrium. Definition 1: A trading equilibrium is defined as a Perfect Bayesian Nash Equilibrium in which whenever a trader s stock holding is less than her maximum, 11 Much of the analysis in Section 4 is used to establish these roerties and in articular that, without loss of generality, attention may be restricted to trigger-strategies. Readers who wish to omit these details may, after absorbing the sell-out condition of Lemma 7, move directly to Section 5.

11 bubbles and crashes 183 then the trader (correctly) believes that the stock holding of all traders who became aware of the bubble rior to her are also at less than their resective maximum long ositions. This definition entails a restriction on beliefs that is a natural one in our setting since the earlier an arbitrageur becomes aware of the misricing, the lower is her estimate of the fundamental value, and the more inclined she is, ceteris aribus, to sell out. Indeed, an immediate conjecture is that this roerty of beliefs is an imlication of equilibrium, but we have not been able to rove this. Note that we are not a riori restricting attention to trigger-strategies: rather this restriction emerges as a result of our analysis. Lemma 1 states that, in equilibrium, an arbitrageur is either fully invested in the market, tt i =, or at her maximum short-osition, tt i = 1. Lemma 1 (No Partial Purchases or Sell Outs): tt i 1 tt i. This Lemma, in effect, reduces the er eriod action sace to or 1. It facilitates the analysis since aggregate selling ressure is now simly given by the mass of traders who are out of the market. It is a consequence of risk-neutrality and the fixed comonent of transactions costs. However, it is not essential for the main results of our aer to hold. Risk-averse arbitrageurs would gradually leave the market. This would make it necessary to kee track of each arbitrageur s osition to calculate the aggregate selling ressure. All this would add comlexity without a corresonding increase in insight. Lemma 1, together with the definition of a trading equilibrium, immediately imlies Corollary 1. It states that when arbitrageur t i sells out her shares, all arbitrageurs t j where t j t i also have already sold, or will at that moment sell, all their shares. We refer to this feature as the cut-off roerty. Corollary 1 (Cut-off Proerty): tt i = 1 tt j = 1 t j t i tt i = tt j = t j t i. and Definition 2: The function Tt i = infttt i > denotes the first instant at which arbitrageur t i sells any of her shares. Arbitrageur t + reduces her holdings for the first time at Tt +. Since, by Corollary 1, all arbitrageurs who became aware of the misricing rior to t + are also comletely out of the market at Tt +, the bubble bursts when trader t + sells out her shares, rovided that it did not already burst earlier for exogenous reasons. Corollary 2: The bubble bursts at T t = mint t + t +. We will show that in any equilibrium, each arbitrageur t i can rule out certain realizations of t at the time when she first sells out her shares. For this urose, we define t su t i.

12 184 d. abreu and m. brunnermeier Definition 3: The function t su t i denotes the lower bound of the suort of trader t i s osterior beliefs about t,attt i. Lemma 2 derives a lower bound for t su t i. Lemma 2 (Preemtion): In equilibrium, arbitrageur t i believes at time Tt i that at most a mass of arbitrageurs became aware of the bubble rior to her. That is, t su t i t i. To see this, suose to the contrary that arbitrageur t i believes that t can also be strictly smaller than t i. For these t s, the selling ressure at Tt i strictly exceeds. Consequently, if the bubble bursts at Tt i, arbitrageur t i does not receive the re-crash rice for her holding of the asset. Furthermore, arbitrageur t i has to believe that the bubble bursts with strictly ositive robability at Tt i. Hence, she has a strict incentive to reemt and sell out slightly rior to Tt i.in contrast to the roof in the Aendix, this intuitive argument imlicitly assumes Tt i >t i. The Preemtion Lemma allows us to derive further roerties of the bursting time T. Lemmas 3 and 4 in the Aendix show that T is strictly increasing and continuous. It follows that T 1 T, the inverse of T, is well defined. Lemma 5 in the Aendix establishes that T is also continuous. Let B c t denote the event that the bubble has not burst until time t. Lemma 6 (Zero Probability): For all t i >, arbitrageur t i believes that the bubble bursts with robability zero at the instant Tt i. That is, PrT 1 T t i t i, B c T t i = for all t i >. The following roosition establishes that in any equilibrium arbitrageurs necessarily use trigger-strategies. That is, each arbitrageur t i sells out at Tt i and never re-enters the market. Proosition 1 (Trigger-strategy): In equilibrium, arbitrageur t i maintains the maximum short osition for all t Tt i, until the bubble bursts. Proof: Suose not, and that there exists an equilibrium in which arbitrageur t i sells out at Tt i and re-enters the market later. Given transaction costs c> and the receding lemma, in equilibrium arbitrageur t i must stay out of the market for a strictly ositive time interval. She will stay out of the market at least until tye t i + exits the market, for some > (indeendent of t i ). 12 By the cut-off roerty arbitrageur t i cannot re-enter the market until after arbitrageur 12 Exiting the market and aying the transaction cost c can only be justified if the bubble bursts with a certain robability. Since the bubble bursts with zero robability exactly when trader t i enters, she has to stay out of the market until at least a certain mass of (younger) traders also exit the market. Note is indeendent of t i.

13 bubbles and crashes 185 t i + first re-enters. The same reasoning alies to arbitrageur t i + with resect to arbitrageur t i + 2. Proceeding this way we conclude that arbitrageur t i stays out of the market until the bubble bursts at t + or arbitrageur t i + exits and then re-enters the market. Of course, the latest ossible date at which the bubble bursts from arbitrageur t i s viewoint is when t i + exits the market. Q.E.D. We have roved that T 1 exists and is strictly increasing and continuous. We confine attention to equilibria for which the latter function is absolutely continuous such that t = T 1 t is also absolutely continuous. Let t denote its associated density. Recall that t = 1 e t and tti is arbitrageur t i s conditional cumulative distribution function of the bursting date at time t i. Similarly, tt i denotes the associated conditional density. The trigger-strategy characterization greatly simlifies the analysis of equilibrium and the secification of ayoffs. As noted earlier the ayoff to selling out at time t is t e rs 1 s T 1 ssst i ds+ e rt t1 tt i c t i Differentiating the ayoff function with resect to t yields the sell-out condition stated in Lemma 7. Note that htt i = tt i 1 tt i is the hazard rate that the bubble will burst at t. Lemma 7 (Sell-out Condition): If arbitrageur t i s subjective hazard rate is smaller than the cost-benefit ratio, i.e. g r htt i < t T 1 t trader t i will choose to hold the maximum long osition at t. Conversely, if htt i > g r t T 1 t she will trade to the maximum short osition One could easily extend the analysis to cature the reutational enalty, due to relative erformance evaluation, that institutional investors face for staying out of the market while the bubble grows at the exected rate g. In a setting with a reutational enalty equal to a fraction k of the rice level, the term t i + t condition generalizes to s t e ru ku dust i ds needs to be added to the ayoff and the sell-out g r k htt i <> t T 1 t The reutational term does not modify the structure of the results, but would be relevant in obtaining realistic calibrations of the model.

14 186 d. abreu and m. brunnermeier Recall that if the bubble bursts at t its size is recisely t T 1 tt. To understand the sell-out condition intuitively, consider the first-order benefits and costs of attack at t versus t +, resectively. The benefits are given by htt i tt T 1 t, the size of the bubble times the robability that the bubble will burst over the small interval. In the case that the bubble does not burst, the costs of being out of the market for a short interval are ( ) t + t 1 htt i rt Note that t + t rt > since the bubble areciates faster than the riskfree rate. Dividing by t and letting yields the attack condition. Lemma 7 also conforms with our earlier result that trader t i either wholeheartedly attacks or holds the maximum long osition. 5 ersistence of bubbles The imossibility of bubbles emerging in standard asset ricing models is most transarent in a finite horizon setting in which they are ruled out by a straightforward backward induction argument. Interestingly, a similar argument involving the iterative removal of non-best-resonse symmetric trigger-strategies rovides some intuition for the main results of this section. Even if no arbitrageur sells her shares, the bubble ultimately bursts at t + by assumtion. Since each arbitrageur becomes aware of the misricing only after t, she knows for sure that the bubble will never last beyond t i +. But it might burst even before this time since from arbitrageur t i s oint of view, the ultimate bursting date t + is distributed between t i + and t i +. If the bubble bursts at t +, arbitrageur t i s best resonse is to exit the market before t i +. More secifically, let t i + 1 be arbitrageur t i s best resonse exit time if she conjectures that other arbitrageurs never attack, and consequently that the bubble bursts only at t +. The conjecture that all arbitrageurs ride the bubble for 1 eriods leads to a new bursting date and a new best resonse exit time t i + 2. Proceeding in this way, we inductively obtain 3 4 and so on. In the standard model this yields lim n n =, which recludes the emergence of bubbles. We show in Section 5.1 that this backward induction rocedure has no bite if 1 e g r In articular, lim n n = 1 and furthermore, the bubble only bursts at t +. Conversely, for 1 e > g r

15 bubbles and crashes 187 we show in Section 5.2 that lim n n < 1 but lim n n >. Hence, this backward rocedure does bite, but not as much as in the classical case. Note that this induction argument is restricted to symmetric strategies. The formal analysis below emloys a different reasoning that also addresses nonsymmetric strategies. Sequential awareness blunts the cometition amongst arbitrageurs uon which the efficient markets view is remised and leads to sufficient ambiguity amongst arbitrageurs for a strategy of market timing to be rofitably ursued. Hence, from the arbitrageurs oint of view, the lack of synchronization is not a roblem but rather a blessing. 51 Exogenous Crashes Recall that arbitrageurs become aware of the bubble sequentially in a random order and furthermore have a nondegenerate osterior distribution over t. All arbitrageurs become aware of the bubble during the interval t t +, where we have interreted to be a measure of differences in oinion and other heterogeneities across layers. From t + onwards, more than arbitrageurs are aware of the bubble and have collectively the ability to burst it. We show that if 1 e g r then in the unique trading equilibrium the bubble only bursts for exogenous reasons when it reaches its maximum size relative to the rice. In this case the endogenous selling ressure of the rational arbitrageurs has absolutely no influence on the time at which the bubble bursts. Definition 4: The function t i = Tt i t i denotes the length of time arbitrageur t i chooses to ride the bubble subsequent to becoming aware of the misricing. It is worth noting that our result holds desite the fact that it is ossible within our model for traders to coordinate selling out on articular dates, say Friday, 13th of Aril, 21 by adoting (asymmetric) strategies that entail nonconstant t i. Proosition 2: Suose 1 e g r Then there exists a unique trading equilibrium. In this equilibrium all traders sell out 1 = 1 ( ) g r ln < g r eriods after they become aware of the bubble and stay out of the market thereafter. Nevertheless, for all t, the bubble bursts for exogenous reasons recisely when it reaches its maximum ossible size.

16 188 d. abreu and m. brunnermeier Proof: Ste 1: 1 defines a symmetric equilibrium. Suose arbitrageur t i believes that the bubble bursts at t +. Conditioning on t t i t i her osterior distribution over t is t t i = e e t i t e 1 and her osterior distribution over bursting dates t = t i + is t i + t i = e e e 1 at t i. The corresonding hazard rate is ht i + t i = t i + t i 1 t i + t i = 1 e Suose that the bubble bursts for exogenous reasons. Then = in the exression for the hazard rate. The latter is deicted in Figure 3 as an increasing function of. Since the exogenous bursting date t + is indeendent of, soisthe cost-benefit ratio of the sell-out condition (Lemma 7). This ratio equals g r/ and is reresented by the horizontal line in Figure 3. The oint of intersection 1 solves ht i + 1 t i = g r/. Clearly, 1 = 1 ( ) g r ln g r By the sell-out lemma arbitrageur t i is otimally out of the market for > 1 and conversely for < 1 she holds on to her shares. Under our assumed condition Figure 3. Exogenous crash.

17 bubbles and crashes 189 t + 1 +>t +. Hence, the symmetric trigger 1 does define an equilibrium, which results in an exogenous crash at t +. Ste 2: Uniqueness. Let us suose that there is another equilibrium. Consider t j = arg min t i t i. 14 Since t i 1 for all t i, it must be the case that t j <1. Consider t su t j. (i) By Lemma 2 (Preemtion) t su t j t j. (ii) If t su t j >t j, then arbitrageur t j does not delay selling out at t j + t j only out of fear of an exogenous bursting of the bubble. By the argument of the first art, it follows that t j = 1. This contradicts the initial assumtion that t j <1. (iii) Finally, suose t su t j = t j. In this case the hazard rate that the bubble bursts, at the time when t j sells out, is at most /1 e since t j = arg mint i. Since this is in turn less than g r/, the sell-out condition is violated. Q.E.D. In equilibrium, each arbitrageur otimally rides the bubble sufficiently long that by the end of the horizon less than will have sold out, so that the bubble indeed bursts urely for exogenous reasons. We note here that the global games aroach does not aly in our setting, since our game does not satisfy strategic comlementarity. This is both because the assumtion that <1 introduces a cometitive element and the fact that traders infer information from the fact that the bubble still exists Endogenous Crashes The revious section demonstrates that arbitrageurs never burst the bubble if the disersion of oinion among them,, and the absortion caacity of behavioral traders,, are sufficiently large. In this section we examine the oosite case when this condition is not satisfied. We show that our roosed backward iteration rocedure does bite in this case. When no other arbitrageur ever sells out, the bubble bursts at t +, which induces arbitrageurs to sell out at t i + 1. In this scenario, the bubble will burst at t + 1 +, which is now strictly earlier than t + given the assumed smaller arameter values for and. Given that the bubble bursts by t at the latest, arbitrageurs seek to sell out even 14 Whenever arg min and arg max are not defined, the corresonding arguments can be rewritten in terms of infimums and suremums resectively. 15 This is robably best illustrated by means of an examle, wherein we restrict the strategy sace to trigger strategies. Consider a trader t i who starts attacking the bubble at t = t i + i, rovided that all other traders attack immediately when they become aware of the bubble. Given this strategy rofile, trader t i can infer a lower bound for t from the fact that the bubble still exists. Comare this with a situation where other traders do not start attacking immediately when they become aware of the bubble but only at, say, t. In this case trader t i cannot derive a lower bound for t from the existence of the bubble. Consequently, she has a greater incentive to attack the bubble at t. This is exactly the oosite of what strategic comlementarity would rescribe.

18 19 d. abreu and m. brunnermeier earlier, 2 < 1 eriods after they became aware of the bubble. Proceeding in this way leads to a decreasing sequence which converges to some that in fact defines the unique symmetric trigger-strategy Perfect Bayesian Nash equilibrium. The iteration of this argument does not eliminate bubbles. The reason is that the iterative rocedure loses bite gradually. As the bursting date advances, the size of the bubble also diminishes, which in turn increases the incentive to ride the bubble by reducing the cost of not exiting rior to the crash. Though the bubble bursts for endogenous reasons it may survive for a substantial eriod; at the time of the crash the bubble comonent is = 1 e g r By an aroriate choice of arameter values, the latter term can be made arbitrarily close to, which in turn can be chosen to be close to 1. Note that for the secial case where the fundamental value is e gt at t and grows at a rate of r thereafter, t t = 1 e g rt t and = 1 ( ) g r ln g r1 e Proosition 3 derives a symmetric equilibrium, where each arbitrageur sells her shares eriods after she becomes aware of the bubble. It also demonstrates that this trading equilibrium is unique. Proosition 3: Suose 1 e > g r Then there exists a unique trading equilibrium, in which arbitrageurs t i with t i leave the market ( ) g r = 1 1 e eriods after they become aware of the bubble. All arbitrageurs t i such that t i < sell out at +. Hence, the bubble bursts when it is a fraction = 1 e g r of the re-crash rice. Proof: Ste 1: defines a symmetric equilibrium. Suose that all arbitrageurs with t i sell out their shares at t i + and arbitrageurs with t i <at + for some. Then the bubble bursts at t + for endogenous reasons, where = +. Substituting this into the hazard rate /1 e as derived in Ste 1 of Proosition 2 yields the equilibrium hazard rate h = /1 e. It is indeendent of and reresented by the horizontal line in Figure 4.

19 bubbles and crashes 191 The cost-benefit ratio g r/ + of the sell-out condition (Lemma 7) is declining in, since + is (by assumtion) an increasing function. See Figure 4. In equilibrium 1 e = g r + Hence, ( ) g r = 1 1 e Note that the assumtion that 1 e < g r (see Section 3) imlies that >, while the condition 1 e > g r imlies that <. Hence, indeed defines a symmetric equilibrium. The next stes establish that this equilibrium is unique. Ste 2: The bubble always bursts for endogenous reasons when 1 e > g r Figure 4. Endogenous crash.

20 192 d. abreu and m. brunnermeier Let 1 solve ht i + 1 t i = 1 e = g r 1 Each trader would exit the market at t i + 1 if she believed that the bubble would burst for exogenous reasons when it reached its maximum ossible size. Under our assumtions t + 1 +<t +. Hence, the bubble does not always burst for endogenous reasons only if t i > 1 for at least some t i. Consider arbitrageur t j, where t j = arg max ti t i. We establish Ste 2 by showing that t j > 1 leads to a contradiction. Observe that: (i) By the Preemtion Lemma t su t j t j. (ii) If t su t j >t j, then arbitrageur t j s hazard rate at Tt j >t j + 1 that the bubble will burst for exogenous reasons strictly exceeds g r/. Since this is also the case at Tt j for sufficiently small >, she has an incentive to sell out strictly rior to Tt j. (iii) Finally, suose t su t j = t j. Since t j t i for all t i, the hazard rate that the bubble will burst at Tt j is at least /1 e. Since the bubble bursts at t ++t j >t ++ 1, the size of the bubble +t j exceeds + 1. It follows that the sell-out condition for arbitrageur t j is violated at Tt j. Ste 3: Minimum and maximum of t i coincide for t i. By Ste 2, Tt + <t + t. That is, the bubble only bursts for endogenous reasons. By the Preemtion Lemma, t su t i t i. Furthermore, t su t i >t i can be ruled out since arbitrageur t i would be strictly better off selling out at some > time after her ostensible equilibrium sell out date Tt i. (Recall that by Lemma 5, T is continuous.) Hence, given t su t i = t i, arbitrageur t i s conditional density of t at Tt i is t i t i B c T t i = e e 1 which is indeendent of t i.lett i arg min t i and t i arg max t i and suose that max t i >min t i. By the continuity of T shown in Lemma 5 and the definitions of t i and t i, it follows that Tt i + t ibc T t i <T t i + t i B c T t i for all > and conversely for <. Consequently, However, ht t i t ibc T t i <ht t i t i B c T t i + t i + t i Thus, the sell-out condition cannot be satisfied for both arbitrageurs t i and t i,a contradiction.

21 bubbles and crashes 193 Ste 4: Fort i < T t i = T. Clearly, no t i should sell out rior to T and by the cut-off roerty will sell out at T. Q.E.D. Notice that the maximum relative bubble size increases as the disersion of oinions among arbitrageurs increases. The comarative statics for the absortion caacity of the behavioral traders are the same as for. A larger requires more coordination among arbitrageurs and thus extends the bubble size. Finally, is also increasing in the excess growth rate of the bubble g r. The faster the bubble areciates, the more temting it is to ride it. Remarks: The usual backward induction argument ruling out bubbles begins at a terminal date that is common knowledge. In our model, rior to the actual bursting of the bubble it is never common knowledge even among a mass <1ofthe arbitrageurs that a bubble exists. To see this, note that at t +, (a mass of) traders know of the bubble. That is, at t + the existence of the bubble is mutual knowledge among traders. But they do not know that traders know of the bubble. This is the case only at t + 2. More generally, nth level mutual knowledge among arbitrageurs is achieved recisely at t + n. See Rubinstein (1989) for an early exloration of the crucial distinction between common knowledge and high levels of mutual knowledge. Note that in contrast to Tirole (1982) and Milgrom and Stokey (1982), our trading game is not a zero-sum game. The rofits of rational arbitrageurs come at the exense of the behavioral traders. 6 synchronizing events A central theme of our analysis is that disersion of oinion makes it difficult for arbitrageurs to synchronize selling out of the bubble asset. This difficulty rovides crucial cover for rofitable riding of the bubble. A natural conjecture is that the resence of synchronizing events unctures this cover. Perhas the most obvious such events are dates like Friday, Aril 13th, 21. In fact, the analysis of the receding section allows arbitrageurs to condition on absolute time. Nevertheless, equilibrium was found to be unique and symmetric: all arbitrageurs wait the same number of eriods after becoming aware of the misricing before attacking. Hence, erfectly anticiated events have no imact on the analysis. In Section 6.1 we consider unanticiated synchronizing events. The imortant imlications of this analysis are that: (i) news may have an imact disroortionate to any intrinsic (fundamental) informational content; (ii) fads and fashions may arise in the use of different kinds of information; (iii) while synchronizing events facilitate coordinated attacks, when such attacks fail, they lead to a temorary strengthening of the bubble. Finally in Section 6.2 we consider rice events. These may also serve as synchronization devices and lead to full correction (a rice cascade) or market rebounds.

22 194 d. abreu and m. brunnermeier 61 Uninformative Events Since the rimary emhasis of this section is to understand the role of news in causing rice changes beyond its informational content, we restrict our formal analysis to nonfundamental events that serve as ure coordination devices. In articular, we extend the earlier model to allow for the arrival of synchronizing signals at a Poisson arrival rate. It is natural to assume that only arbitrageurs who are aware of the bubble observe synchronizing events. Indeed, we assume that such events are only observed by traders who became aware of the bubble more than e eriods ago. The stronger assumtion that e > catures in a simle way the idea that arbitrageurs become more wary the longer they have been aware of the bubble, and, hence, look out more vigilantly for signals that might reciitate a bursting of the bubble. An alternative justification for some arbitrageurs not recognizing synchronizing events is suggested by the second remark at the end of this subsection. Synchronizing events both allow arbitrageurs to synchronize sell outs and also convey valuable information in the event that attacks are unsuccessful. The bubble will survive a synchronized sell out if an insufficient number of arbitrageurs have observed the synchronizing event, that is, if t is sufficiently high. Let t e be the date of such a synchronizing event. By the cut-off roerty, there exists e e such that all arbitrageurs t i who observed the sunsot at least e eriods after becoming aware of the misricing, sell out after observing the synchronizing event at t e. If the bubble does not burst then all arbitrageurs learn that t >t e e. It follows that all arbitrageurs t i with t i t e e will re-enter the market until tye t e e first exits in equilibrium, or until the next synchronizing event occurs. Even traders who left the market rior to the arrival of the synchronizing event buy back their shares. In contrast to the revious sections, there is no hoe of finding a unique equilibrium in this generalized setting. For examle, the equilibrium behavior of the revious section would be exactly relicated if all arbitrageurs were to simly ignore all synchronizing events, and there are otentially numerous intermediate levels of resonsiveness to synchronizing signals. We focus on resonsive equilibria. Definition 6: A resonsive equilibrium is a trading equilibrium in which each arbitrageur believes that all other traders will synchronize (sell out) at each synchronizing event. This raises the bar for our analysis since bubbles are harder to sustain in a resonsive equilibrium. This is the olar oosite of the equilibrium in which all arbitrageurs ignore all synchronizing events. We establish that there exists a unique resonsive equilibrium. It is symmetric in the sense that as long as an arbitrageur does not observe a synchronizing event, each arbitrageur stays in the market for a fixed number eriods after

23 bubbles and crashes 195 becoming aware of the bubble. We assume that <g r/ to ensure that the ossible occurrence of a synchronizing event does not by itself justify selling out. As in Subsection 5.2 (endogenous crashes), we focus on the arameter values that satisfy g r < 1 e < g r The latter inequality guarantees that no arbitrageur finds it otimal to sell out even before she becomes aware of the misricing. Proosition 4: There exists a unique resonsive equilibrium. In this equilibrium, each arbitrageur t i always sells out at the instances of synchronizing events t e t i + e. Furthermore, she stays out of the market for all t t i + excet in the event that the last synchronized attack failed in which case she re-enters the market for the interval t t e t e + e, unless a new synchronizing event occurs in the interim. Absent the arrival of a synchronizing event the bubble will burst at t ++. The arrival of synchronizing events might accelerate the bursting date. After a failed sell out at the latest synchronizing event t e, even traders who started selling out rior to t e, that is, traders with t i <t e, buy back shares. Sell outs only resume at t e + e. Market sentiment bounces back after a failed synchronized attack. However, even in the event of failed attacks and re-entry by arbitrageurs into the market, the bubble bursts no later than t + +. The roof of Proosition 4 is summarized in Aendix A.2. Notice that given the structure of our setu, the equilibrium is fully characterized by : In any trading equilibrium, all arbitrageurs who sold out after the last synchronizing event will re-enter the market after a failed attack and sell out again exactly when the first trader, who did not articiate in the synchronized sell out, exits the market. Remarks: The news events above are secial in that they convey no information about the value of the asset. Nevertheless, they facilitate synchronization. An immediate consequence is that, in general, news events may have an imact quite out of roortion to their fundamental content. The issue of what events are considered by market articiants to be synchronizing events is a subtle one. There is obviously an unlimited suly of otential synchronizing events (such as the weather, the levels of various economic indicators, etc.). But which do articiants view as relevant? Their very abundance emhasizes the difficulty and suggests a higher level synchronization roblem, and the ossibility of multile equilibria. The above issue is closely related to fads and fashion in the acquisition and use of information. For examle, trade figures drove the market during the 198 s. In contrast, in the late 199 s Alan Greensan s statements moved

24 196 d. abreu and m. brunnermeier stock rices, while trade figures were ignored. The model in this section may be extended to allow for multile tyes of signals, A B C, etc. In some equilibria, arbitrageurs may condition on A and C, in others only on B, etc. More interestingly, some equilibria may initially condition on A and then after some histories switch to B and subsequently back to A, etc. Such a model would yield a rudimentary theory of fads and fashion. These themes are reminiscent of Keynes (1936) comarison of the stock market with a beauty contest: rofessional investment may be likened to those newsaer cometitions in which the cometitors have to ick out the six rettiest faces from a hundred hotograhs, the rize being awarded to the cometitor whose choice most nearly corresonds to the average references of the cometitors as a whole; so that each cometitor has to ick, not those faces which he himself finds rettiest, but those which he thinks likeliest to catch the fancy of the other cometitors, all of whom are looking at the roblem from the same oint of view. 62 Price Events Price Cascades and Market Rebounds Arguably, the most visible synchronizing events on Wall Street are large ast rice movements or breaks through sychological resistance lines. In this section, we allow random temorary rice dros to occur, which are ossibly due to mood changes by behavioral traders. This enables us to illustrate how a large rice decline might either lead to a full blown crash or to a rebound. In the latter event the bubble is strengthened in the sense that all arbitrageurs are in the market for some interval, including those who had reviously exited rior to the rice shock. In our simle and highly stylized model, exogenous rice dros occur with a Poisson density at the end of a random trading round t. We suose that behavioral traders mood changes temorarily and that they are only willing to hold on to shares if the rice is less than or equal to 1 t. If the bubble does not burst in the subsequent trading round, behavioral traders regain their confidence and are willing to sell and buy at a rice of t until their absortion caacity is reached or another rice dro occurs. Arbitrageurs who exit the market immediately after a rice dro receive 1 t er share. Should the synchronized attack after a rice dro fail, arbitrageurs can only buy back their shares at a rice of t. In other words, leaving the market even only for an instant is very costly desite the absence of transactions costs, which we set equal to zero. Hence, only traders who are sufficiently certain that the bubble will burst after the rice dro will leave the stock market. Let H n = t1 t n denote a history of ast temorary rice dros. Proosition 5 states that only traders who became aware of the misricing more than H n eriods earlier will choose to leave the market and attack the bubble after a rice dro. Notice that, although H n is derived endogenously for the subgame after a rice dro, it serves the same role as the exogenous e in subsection 6.1. The structure of the equilibrium is similar to the unique resonsive equilibrium of the receding subsection, with the main difference being that all arbitrageurs know H n, and H n affects H n. Analogous to the revious subsection, denotes how long each arbitrageur rides the bubble after t i if there has been no rice dro so far.

25 bubbles and crashes 197 Proosition 5: There exists an equilibrium in which arbitrageur t i exits the market after a rice dro at t n+1 if t n+1 t i + H n. Furthermore, she is out of the market at all t t i + excet in the event that the last attack failed, in which case she re-enters the market for the interval t t n+1 t n+1 + H n. Proosition 5 shows that a rice dro that is not followed by a crash leads to a rebound and temorarily strengthens the bubble. In this case all arbitrageurs can rule out that the bubble will burst within t n+1 t n+1 + H n for endogenous reasons. Within this time interval, the rice grows at a rate of g with certainty, excet if another exogenous rice dro occurs. Consequently, all arbitrageurs re-enter the market after a failed attack and buy back shares at a rice of t, even if they have sold them an instant earlier for 1 t. Furthermore, the failed attack at t n increases the endogenous threshold H n. More secifically, a failed attack at t n makes arbitrageurs more cautious about the rosects of mounting a successful attack after a rice dro at t n+1 >t n. In articular, if t n+1 is close to a failed synchronized sell out at t n, arbitrageurs will not find it otimal to exit again at t n conclusion This aer argues that bubbles can ersist even though all rational arbitrageurs know that the rice is too high and they jointly have the ability to correct the misricing. Though the bubble will ultimately burst, in the intermediate term, there can be a large and long-lasting dearture from fundamental values. A central, and we believe, realistic, assumtion of our model is that there is disersion of oinion among rational arbitrageurs concerning the timing of the bubble. This assumtion serves both as a general metahor for differences of oinion, information and beliefs among traders, and, more literally, as a reduced-form modeling of the temoral exression of heterogeneities amongst traders. While it is well understood that aroriate deartures from common knowledge will ermit bubbles to ersist, we believe that our articular formulation is both natural and arsimonious. The model rovides a setting in which overreaction and self-feeding rice dros leading to full-fledged crashes will naturally arise. It also rovides a framework that allows one to rationalize henomena such as resistance lines and fads in information gathering. 16 The equilibrium described in Proosition 5 is also maximally resonsive in that rice dros are resonded to maximally in the chronological order in which they aear. However, the bubble is more likely to burst at t n+1 in an equilibrium in which the earlier event at t n was not resonded to than in the equilibrium where arbitrageurs sold out at t n and the attack failed. We do not have a uniformly most resonsive equilibrium in this section, but rather one that is resonsive in chronological order to the maximum extent ossible. In short, the above equilibrium is not necessarily the one in which the bubble bursts earliest for any ossible sequence of rice dros. Note that the same issue also arises in a setting with unanticiated ublic events and strictly ositive transactions costs c>. We abstracted from these effects in Section 6.1 by assuming c =.

26 198 d. abreu and m. brunnermeier Many of the assumtions of our simle model may be viewed as being conducive to arbitrage. In articular, we assume that all rofessionals are in agreement that assets are overvalued, while arguably there are substantial differences in oinion even amongst rofessionals regarding the ossibility that current valuations indeed reflect a new era of higher roductivity growth, lower wages, and inflation, et cetera. Presumably incororating these realistic comlications would reinforce our conclusions. The dynamic game develoed in this aer might have useful alications in other settings. These include models of currency attacks, bank-runs, and R&D investment games. Det. of Economics, Princeton University, Fisher Hall, Princeton, NJ , U.S.A.; [email protected]; htt:// dabreu/ and Det. of Economics and Bendheim Center for Finance, Princeton University, 26 Prosect Avenue, Princeton, NJ , U.S.A.; [email protected]; htt:// markus/. Manuscrit received December, 21; final revision received May, 22. APPENDIX A: General Payoff Secification Let Bt = t T t <tbe the set of t that lead to a bursting of the bubble strictly rior to t. Let B c t denote the comlement of Bt and bt = t T t = t be the set of t that lead to a bursting exactly at t. The value at time t 1 of a osition consisting of x 1 t 1 stocks that is maintained unchanged until the bubble bursts is (from the viewoint of arbitrageur t i ): V 1 x 1 t 1 t i = x 1 e rt t t 1 1 T t t T t dt t i B c t 1 where Bt i + t i B c t 1 = t i 1 t 1 t i is arbitrageur t i s distribution over t, conditional uon the time t i when arbitrageur t i became aware of the misricing, and conditional uon the existence of the bubble at t 1. Proceeding iteratively from the final change of osition, the value at t 2 <t 1 of x 2 stocks of which x 2 x 1 are sold at t 1 is given by V 2 x 2 t 2 x 1 t 1 t i = x 2 e rt t t 2 1 T t t T t dt t i B c t 2 Bt 1 \Bt Bt 1 t i B c t 2 { e rt2 c + e rt1 t 2 V 1 x 1 t 1 t i + x 2 x 1 t 1 At 1 t i B c t 2 } The adjustment term At 1 t i B c t 2 takes care of the secial case where the bubble bursts exactly at t 1 and the order of x 2 x 1 shares is not executed at the rice t 1 but at 1 t t t t t. The latter term deends on the uncertain value of t. More recisely, At 1 t i B c t 2 = x 2 x 1 t 1 t t t t dt t i B c t 2 bt 1

27 bubbles and crashes 199 Relacing suerscrit 2 by k + 1 and suerscrit 1 with suerscrit k gives us a general recursive definition of the ayoff function. Arbitrageur t i s exected ayoff from a strategy involving K 1 changes in her ortfolio is then given by the function V K x K t K x 1 t 1 where x K = 1t K = t i denotes her initial osition. APPENDIX B: Details of Section 4 Lemma 1 (No artial urchases or sell-outs): tt i 1 tt i. Proof: Consider an equilibrium strategy t i that involves a change in osition at t k from the receding osition adoted at t k+1 <t k. (The initial osition is reresented by K.) Suose that x k t k 1 and t i is otimal. The exected ayoff from t k+1 onwards is given by V k+1 lus the value of bond holdings. Notice that V k+1, excluding the transaction costs ce rtk+1, is linear in x k and furthermore must be strictly ositive. Hence, for x k x k+1 >< the ayoff is strictly dominated by x k = 1 x k =. This contradicts the initial resumtion that t i is otimal. Q.E.D. Lemma 2 (Preemtion): In equilibrium, arbitrageur t i believes at time Tt i that at most a mass of arbitrageurs became aware of the bubble rior to her. That is, t su t i t i. Proof: Let t su t i = t i a. For arbitrageur t i to believe that a>, it is necessarily the case that for all t j t i a t i, arbitrageurs t j do not sell out rior to Tt i, since otherwise by Lemma 1 and Corollary 1 the bubble would have burst already. Furthermore, the aggregate selling ressure at t = Tt i is s t t >with strictly ositive robability, since (by Lemma 1 and Corollary 1) all traders t k t i will also be fully out of the market at Tt i. Hence, with strictly ositive robability all traders t j who first sell out at Tt i will only receive the ost-crash rice for their sales. Consequently, all traders t j t i a t i have an incentive to reemt (attack slightly earlier) the ossible crash at Tt i, a contradiction. Q.E.D. Lemma 3 (Bursting Time): The function T is strictly increasing. Proof: Recall T t = mint t + t +. Since T is weakly increasing by the cut-off roerty, so is T. Now suose that there exists t > t such that T t = T t. Clearly, t + T t. It follows that for all t t t T t + = T t = T t.lett i = t +. By Lemma 2 t su i t. However, t su t i t <t t i, a contradiction. Q.E.D. Lemma 4 (Continuity of T ): The function T is continuous. Proof: (i) Since T is increasing, we only need to rule out uward jums. Suose that there is an uward jum at t. Then for small enough > and s t t T s = Ts+. Let T = lim s t T s and T = lim s t T s, and suose < T T. Recall that transactions cost ce rt is incurred for any change of osition at t. This recludes an arbitrageur from changing osition at t and t if the robability of a crash between t and t is small relative to c. Hence, for small enough >, tye t i = t + is strictly better off selling out at T >T than at Tt i <T, a contradiction. Q.E.D. Lemma 5 (Continuity of T): The function T is continuous. Proof: This argument is almost identical in structure to the roof of Lemma 4 and is omitted. Lemma 6 (Zero Probability): For all t i >, arbitrageur t i believes that the bubble bursts with robability zero at the instant Tt i. That is, PrT 1 T t i t i B c T t i = for all t i >.

28 2 d. abreu and m. brunnermeier Proof: Observe that the conditional c.d.f. t t i B c T t i = t t su t i t i t su t i Clearly, the latter is continuous. Q.E.D. Aendix C: Details of Section 6 C1 Restriction to Interim-Trigger-Strategies Let an interim-trigger-strategy be one for which an agent follows a trigger-strategy between two successive synchronizing events. We argue that, in equilibrium, agents use interim-trigger-strategies. Let Htt i = t 1 e t n e denote a history of ast events at times t 1 e <t 2 e < <t n e strictly rior to t and observed by arbitrageur t i.lett n e Htt i be the time of the most recent synchronizing event observed by arbitrageur t i.lett a equal t i if arbitrageur t i has not observed any synchronizing event, and equal t n e otherwise. Denote arbitrageur t i s osterior distribution over t at t = t a by t i B c t a Ht a t i. In other words, trader t i conditions uon her awareness date, the fact that the bubble did not burst rior to t a and the history of observed ast synchronizing events if t a = t n e. The event that the bubble did not burst rior to t a B c t a, deends on the realization of t and also on the sequence of synchronizing events. To simlify notation we denote t i B c t a Ht a t i by I ti t a. Definition 5: (i) The function T e t i = infttt i >Htt i = denotes the first instant at which arbitrageur t i sells out any of her shares conditional uon not having observed a synchronizing event thus far. (ii) The function T t e = mint e t + t + secifies the date at which the bubble bursts, absent the observation of a synchronizing event by trader t +. T e The analysis for T and T of Section 4 alies to T e and T e in this section. In articular, is strictly increasing and continuous. The arguments are analogous and are not reeated here. Since each arbitrageur incurs ositive transactions cost c when selling her shares, and the bubble bursts with zero robability at each instant, she has to leave the market at least for an interval. Arbitrageur t i stays out of the market at least until arbitrageur t i + exits the market. Recall that the occurrence of synchronization events alone does not justify the loss in areciation from exiting over a certain interval, since <g r. Hence, a generalized version of Proosition 1 alies between synchronization events, leading to what we have termed interim-trigger-strategies. Furthermore, the distribution function that the bubble will burst in the absence of further synchronizing events is given by ti i t a = T 1 e ti i t a Since in Section 6 arbitrageurs re-enter the market in equilibrium, transactions costs are otentially incurred reeatedly. Keeing track of all transactions costs comlicates the analysis in uninteresting ways. In what follows we exloit the strategic restrictions obtained above (i.e. interim-triggerstrategies), while setting transactions costs to zero. C2 Generalized Sell-out Condition Lemma 8 (Generalized Sell-out Condition): Suose arbitrageur t i has not observed a failed sell out. Then she sells out at t satisfying hti ti t T 1 e t + t tt t dt I ti = g r t t e

29 bubbles and crashes 21 The roof is similar to that of Lemma 7. The first term reflects the ossibility that the bubble might burst in the next instant due to endogenous selling ressure. From trader t i s viewoint this occurs at the hazard rate htt i = ti t i 1 ti ti In this case, the relative size of the bubble is t T 1 e t. Arbitrageur t i also has to take into account the ossibility that an unexected synchronizing announcement might occur at t with a density of. The synchronizing event is followed by a crash if t t e. In this case the relative size of the bubble t t deends on the realization of t. Note that in the case of a synchronizing event, trader t i still has a chance to leave the market at the re-crash rice with robability 1 t t. Hence, one has to multily the advantage of attacking rior to the arrival of a synchronizing event by t t. Recall that 1 t t is the fraction of orders executed at the re-crash rice over all orders submitted immediately after a synchronizing event. The exact ayoff secification and a more formal roof of this lemma can be found in an earlier version of our aer (Abreu and Brunnermeier (21)). C3 Proof of Proosition 4 Ste 1: Derive symmetric equilibrium Recall that = T e t i t i t i denotes the number of eriods an arbitrageur waits to sell out if she does not observe any synchronizing event. Note that fully characterizes the symmetric resonsive equilibrium outlined in Proosition 4. In any trading equilibrium all arbitrageurs who sold out after the last synchronizing event will re-enter the market after a failed attack and sell out again exactly when the first trader, who did not articiate in this common sell out, exits the market. This follows directly from the cut-off roerty. Hence, we can restrict our attention to the trading activity of traders who have not observed a synchronized event as yet. By describing their trading strategy we fully characterize any trading equilibrium There exists a unique symmetric trading equilibrium. Note that each arbitrageur t i s osterior about t at T e t i exactly coincides with the osterior she had at Tt i in a setting without synchronizing events. That is, t t i B c T e t i HT e t i t i = = t t i B c T t i To see this, observe that in any symmetric equilibrium, the suort of t at T e t i is also t i t i (as in Section 5). Furthermore, any unobserved synchronizing event at t e <t i + e would not have led to a bursting of the bubble, since t e <t + + e for all ossible t t i t i. Hence, synchronizing events do not serve to distinguish between t in the latter interval. Since the osteriors are the same, so is the hazard rate hti ti at the time that trader t i sells out in either setting Clearly, if e, then synchronizing events have no imact on the bursting of the bubble, since trader t + does not observe a signal rior to t + +. Hence in this case =. Suose now that e <. Define = ht i + I ti + + t st i + t dt I ti g r t t i + e For equilibrium it is necessary that =. We argue that such that = is (i) unique, and (ii) exists. By Ste 1.2 ht i + I ti is constant across equilibrium. However, is strictly increasing, t I ti is the same, and the uer bound of the integral is increasing in. Thus, is strictly increasing in equilibrium. Uniqueness follows directly. For < e the sell out condition reduces to ht i + I ti + g r =. There does not exist a < e that solves this equation. This follows directly from the fact that we focus on e <

30 22 d. abreu and m. brunnermeier and that solves the receding equation uniquely (as established in Section 5). Hence, < for e. Moreover, lim, since ht i +I ti. Continuity of and the intermediate value theorem imly existence of. Stes 1.4 and 1.5 show that the unique indeed defines an equilibrium Immediately after a synchronizing event each arbitrageur who observes the synchronizing event is assumed to sell out in a resonsive equilibrium. Hence, from each arbitrageur s oint of view, a bubble bursts with strictly ositive robability at each t e in a resonsive equilibrium. Given these beliefs, and the fact that an instantaneous attack is costless for c =, it is indeed (strictly) otimal for an arbitrageur who observes the synchronizing event to sell out at this time To fully secify all relevant strategies, it only remains to consider continuation strategies after a failed sell out attemt. After a failed attack, arbitrageurs learn that fewer than traders have observed the synchronizing event. That is, t >t e e. Since all other arbitrageurs who did not observe the synchronizing event only sell out at t i +, all arbitrageurs who observed the synchronizing event at t e can rule out the ossibility that the bubble bursts rior to t e + e e rovided that no new synchronizing event occurs. Note that the bubble will not burst for exogenous reasons rior to t e + e, since the endogenous bursting time t + + occurs strictly before t + + <t +. The bubble might only burst rior to t e + e if a new synchronizing event occurs. After t = t e + e e, the analysis coincides with a setting without a synchronizing event at t e. At this oint all arbitrageurs who had articiated in the failed sell out and subsequently re-entered the market, exit again. The next two stes of the roof establish that equilibria are necessarily symmetric. They are analogous to Stes 2 and 3 of the roof of Proosition 3 and are not reeated here. See Abreu and Brunnermeier (21) for more details. C4 Proof of Proosition 5 Lemma 9: There exists a function such that for all n = 1 2 and all histories of rice dros H n, after a rice dro at tn+1, all traders t i who became aware of the misricing rior to t n+1 H n leave the market and all other arbitrageurs remain in the market. Proof: Recall that denotes the (ost-awareness) time interval after which each arbitrageur leaves the market in the absence of a rice shock. We roceed inductively, defining H k given H l where l = 1 2k 1. We are looking for the smallest such that in a trading equilibrium arbitrageur t i = t k is indifferent between exiting and staying in the market. At t k, arbitrageurs with t i <t are already out of the market rovided that t k > maxt k j + H k j 1 j=12. (When the latter condition is violated, traders have returned to the market because of a rior failed synchronized sell out.) Arbitrageur t i = t k is indifferent between staying in the market or not if ( ) + t t 1 tt dt I ti t where I ti t t t k + H k 1 t >t k ( H k 1 ) dt I ti t = denotes the information set t i = t k t t k 1 ( H k 1 t H k 1. That is, )t t 1 tt dt I ti t = Denote the left-hand side by f k I ti. Note that if the bubble does not burst, then all t arbitrageurs sell their shares at the rice of 1 t. If the bubble does burst, only the first 1/t k t orders ( orders) are executed at 1 t if t k < maxt k j I t i t, then H k 1 + j=12. The term 1 tt reflects this fact. If f k =. If I t i t < look for min such that f k I ti t. If such a does not exist, set = t. H k j 1 f k

31 bubbles and crashes 23 It remains to check that all arbitrageurs with t i <t k market and all traders with t i >t k H k 1 H k 1 strictly refer to leave the refer to remain in the market. By looking at trader t i s distribution t, it is easy to check that f k I ti t is decreasing in t i. Notice that each trader can rule out any t <t i. From the fact that the bubble did not burst before t k for endogenous reasons all arbitrageurs can rule out t <t k. Finally, since the bubble survived all sell out attemts after revious rice dros, t maxt k j H k j 1 j=12.for j=12, the lower bound is all traders with t i maxt k maxt k j H k j 1 the same, while the uer bound is given by t i. Hence, for arbitrageurs with lower t i the density on t for which t t H k 1 is higher. In addition, the (conditional) exonential distribution has the nice roerty that the relative likelihood of ossible states in the suort is the same across arbitrageurs. The distribution for arbitrageurs whose suort is t i t i is symmetric in t i. Hence, f k I ti t is also decreasing in t i in this case. Q.E.D. After establishing the critical value H k 1 in Lemma 9, the rest of the roof of Proosition 5 is analogous to the roof of Proosition 4. There are three differences: all arbitrageurs observe each rice dro, the first orders after the rice dro are only executed at a rice of 1 t instead of the re-crash rice t, and all H k 1 are history deendent. Note that H k 1 is known to all traders in equilibrium since all arbitrageurs can observe the ast rice rocess. REFERENCES Abreu, D., and M. K. Brunnermeier (21): Bubbles and Crashes, Working Paers in Economic Theory 1F3, Princeton University. (22): Synchronization Risk and Delayed Arbitrage, Journal of Financial Economics, forthcoming. Allen, F., and G. Gorton (1993): Churning Bubbles, Review of Economic Studies, 6, Allen, F., S. Morris, and A. Postlewaite (1993): Finite Bubbles with Short Sale Constraints and Asymmetric Information, Journal of Economic Theory, 61, Brunnermeier, M. K. (21): Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis and Herding. Oxford: Oxford University Press. Brunnermeier, M. K., and S. Nagel (22): Arbitrage at its Limits: Hedge Funds and the Technology Bubble, Mimeo, Princeton University. Carlsson, H., and E. van Damme (1993): Global Games and Equilibrium Selection, Econometrica, 61, Daniel, K., D. Hirshleifer, and A. Subrahmanyam (1998): Investor Psychology and Security Market Under- and Overreaction, Journal of Finance, 53, DeLong, J. B., A. Shleifer, L. H. Summers, and R. J. Waldmann (199a): Noise Trader Risk in Financial Markets, Journal of Political Economy, 98, (199b): Positive Feedback Investment Strategies and Destabilizing Rational Seculation, Journal of Finance, 45, Fama, E. F. (1965): The Behavior of Stock-Market Prices, Journal of Business, 38, Halern, J. Y., and Y. Moses (199): Knowledge and Common Knowledge in a Distributed Environment, Journal of the Association of Comuting Machinery, 37, Hirshleifer, D. (21): Investor Psychology, Journal of Finance, 56, Keynes, J. M. (1936): The General Theory of Emloyment, Interest and Money. London: Macmillan. Kindleberger, C. P. (1978): Manias, Panics and Crashes: A History of Financial Crises, 3rd edition, London: Macmillan. Milgrom, P. R., and N. Stokey (1982): Information, Trade and Common Knowledge, Journal of Economic Theory, 26, Morris, S. (1995): Co-oeration and Timing, CARESS Working Paer Morris, S., and H. Shin (1998): Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review, 88,

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