QED. Queen s Economics Department Working Paper No. 1148

Transcription

1 QED Queen s Economics Department Workin Paper No The Role of Lare Players in a Dynamic Currency Attack Game Mei Li Queen s University Frank Milne Queen s University Department of Economics Queen s University 94 University Avenue Kinston, Ontario, Canada K7L 3N

2 The Role of Lare Players in a Dynamic Currency Attack Game Mei Li Frank Milne October 23, 2007 Abstract We establish a dynamic currency attack model in the presence of a lare player (LP) based on Abreu and Brunnermeier (2003), which differs from most existin oneperiod static currency attack models. In an attack on a fixed exchane rate reime with a radually overvaluin currency, both the inability of speculators to synchronize their attack and their incentive to time the collapse of the reime lead to the persistent overvaluation of the currency. We find that the presence of an LP, who is defined as a speculator with more wealth and superior information, can accelerate or delay the collapse of the reime, dependin on his incentives to preempt other speculators or to ride the overvaluation. When an LP s incentive to preempt other speculators is dominant, the presence of an LP will accelerate the collapse of the reime. However, when an LP s incentive to ride the overvaluation is dominant, the presence of an LP will delay the collapse of the reime. The latter case provides valuable insihts into the role that LP s play in currency attacks: it differs from the usual perception that the presence of LPs will facilitate arbitrae in an asset market and alleviate asset mispricin due to their capability and willinness to arbitrae. Keywords: Lare Player, Currency Attack JEL Classification: D80, F31 Department of Economics, Queen s University, Kinston, Ontario, Canada K7L 3N6. s: lim@qed.econ.queensu.ca, milnef@qed.econ.queensu.ca 1

3 1 Introduction We often observe that lare players such as hede funds play an active role in currency attacks aainst fixed exchane rate reimes. They launch currency attacks by employin a lare amount of wealth to build lare short positions. Afterward they try to influence market sentiment by publicly announcin their short positions and beliefs that devaluation is inevitable. This causes herdin amon small traders, and/or deters contrarians from takin opposite positions. It seems that the presence of lare players facilitates coordination amon speculators and increases financial instability in the attacked currencies, specially in small economy currencies. This is sometimes called the bi elephants in small ponds effect. We establish a formal model to study the role that lare players play in a currency attack, based on the model developed by Abreu and Brunnermeier (2003). In their model, rational arbitraeurs in an asset market become aware of an asset bubble sequentially. Due to the lack of common knowlede about the bubble, and need for coordination to burst the bubble, the bubble will be persistent and its burstin time depends on the incentives of the arbitraeurs to ride the bubble, as opposed to incentives to preempt other arbitraeurs in sellin the asset. Similar to their model, we assume that a currency beins to be overvalued in a fixed exchane rate reime after a certain time. Speculators have dispersed opinions in the sense that they only become aware of the overvaluation sequentially. In addition, we assume that the fixed exchane rate reime will collapse only when attackin pressure reaches a threshold level. This assumption captures the main feature of currency attacks: there is a necessity for coordination amon speculators to break a currency pe. This coordination feature is emphasized in Obstfeld (1996) and other currency attack models (see especially Morris and Shin (1998)). In our setup, the speculators try to choose the optimal time to launch their attack, driven by two competin incentives: first, the incentive to ride the overvaluation; and second, the incentive to preempt other speculators. The speculators incentive to ride the overvaluation stems from two sources in our model: first, they can reap hiher benefits from the devaluation if the overvaluation lasts loner. Second, if they time their attack more precisely, they will save on attackin costs. The speculator attempts to preempt other speculators, 2

4 because only the speculator attackin early will ain from the collapse of the reime. The late speculators will ain nothin. Abreu and Brunnermeier (2003) consider a symmetric ame with a continuum of atomistic small arbitraeurs. We are more interested in a richer market structure where both a lare player and a continuum of small players are present. More specifically, we are interested in studyin how the presence of a lare player will chane equilibrium outcomes. In our model, a lare player is defined by two characteristics: first, he has more precise information about the fundamental value of a currency. Here we assume the extreme case where a lare player has perfect information about the time when the overvaluation beins. Second, a lare player can employ substantially larer amounts of wealth to launch a currency attack. The wealth that a lare player employs can come from his own capital, or more importantly, from his accessibility to credit due to his reputation. This is how hihly leveraed financial institutions finance their speculation. Our model differs from most currency attack literature in several aspects. First, it is one of the few papers studyin currency attacks in a dynamic setup. Most existin literature uses static models, which miss the complicated market dynamics in currency attacks. Second, while most existin currency attack literature simply assumes exoenously the existence of the overvaluation under a fixed exchane rate reime, we model endoenously the oriin of currency overvaluation in the presence of rational arbitraeurs. A key contribution of Abreu and Brunnermeier (2003) lies in that they offer a eneral explanation of how asset mispricin arises. Even in the presence of rational arbitraeurs, who are capable of correctin the mispricin, they choose not to do it due to their incentive to ride the bubble. This mechanism can be comfortably applied to explainin how currency overvaluation arises in a fixed exchane rate reime. Our approach is consistent with the microstructure method for modelin exchane rates. We believe that forein exchane market participants hold hihly dispersed opinions about exchane rates. As arued by Lyons (2001), even if all market participants have the same information about exchane rates, the ways in which they interpret or model the implications of that information can be different. Thus, they 3

5 can come to different conclusions about exchane rates based on the same information. So it is well justified to assume that speculators do not have perfect information about the time when the overvaluation arises, and only become radually aware of the overvaluation. In such an opinion-dispersed market where coordination is required for a successful currency attack, the incentive for ridin the overvaluation naturally arises and leads to the persistent overvaluation of the currency. This explanation is consistent with empirical observations that currency attacks often lead to substantial and sudden devaluations, causin extreme volatility in an economy. Due to the features of our model, our study of lare players in currency attacks focuses on a different aspect compared to the standard, more static models. Most existin literature on lare players in currency attacks focuses on the possibility of the collapse of a fixed exchane rate reime, and on whether the presence of lare players will increase this possibility or not. In our model, currency devaluation is inevitable, and the issue that we focus on is when it will happen. Thus our study focuses on whether the presence of lare players will accelerate or delay a currency attack. Here we do not ive a formal welfare analysis to examine whether the presence of a lare player is beneficial or harmful to an economy. However, in eneral, we believe that a currency overvaluation is harmful to an economy, and early correction is always better than a late one if the correction is inevitable. In this sense, a late collapse of the reime will do more harm to an economy than an early one. Usin our model, we find some interestin results. First, we find that the presence of a lare player will not necessarily accelerate the collapse of an overvalued fixed exchane rate reime. This result is important because lare players are usually believed to facilitate arbitrae in an asset market and reduce asset mispricin. In our model, the presence of a lare player will accelerate or delay the collapse of a fixed exchane rate reime, dependin on whether his incentive to ride the overvaluation is dominated, or not, by his incentive to preempt the mass of small speculators. If his incentive to preempt is dominant, his presence will accelerate the collapse of the reime. He can do so not only because his wealth facilitates the attack, but also because his presence makes other small speculators attack earlier. Conversely, if a lare player s incentive to preempt other speculators is dominated by the incentive to 4

6 ride the overvaluation, his presence will delay the collapse of the reime. He can do so not only because he will wait loner, but also because his presence makes other speculators wait loner too. The lare player s incentive to ride the overvaluation makes the existence of a lare player a mechanism for delayin any asset mispricin. Instead, he will use his market power to make reater profits from larer, later asset mispricin. The rest of the paper consists of six sections. Section 2 provides a literature survey. Section 3 discusses the basic model and characterization of a dynamic currency attack. The model is a variation of that established by Abreu and Brunnermeier (2003): the model has a continuum of small arbitraeurs tradin in a currency with a fixed exchane rate, that is open to a currency attack. We provide a characterization of the equilibrium and comparative statics. Section 4 introduces a lare player, proves that there is a unique equilibrium and characterizes that equilibrium. Section 5 conducts comparative statics for the model. Section 6 observes that our results can be applied to the oriinal Abreu and Brunnermeier (2003) set-up with a stock market. Section 7 concludes with observations on further possible extensions. 2 Literature Survey Abreu and Brunnermeier (2003) construct a dynamic coordination ame to explain the existence of asset bubbles, even in the presence of rational arbitraeurs who are capable of burstin the bubble. We have already discussed their model in detail. (Our paper is an application of their model to currency attacks.) In their model, only a continuum of atomistic speculators exists. Since we focus on the study of the role that a lare player plays in a currency attack, our model exhibits a richer market structure where both a lare player and a continuum of atomistic speculators co-exist. Both Rochon (2006) and Gara Minuez-Afonso (2007) apply Abreu and Brunnermeier (2003) to currency attacks and try to explain the devaluation that we observe when a fixed exchane rate reime collapses. The most important difference between our model and theirs is that our model focuses on the role of lare players in a currency attack with imperfect common knowlede, while they study currency attacks 5

7 only in a model without lare players. In addition, even in our basic model without lare players, the way in which we model a currency attack is also slihtly different from theirs. We model the payoff structure of speculators who try to ain from the devaluation, while they model the payoff structure of the attackers who try to avoid a capital loss associated with devaluation. Morris and Shin (1998) study currency attacks in a one-period lobal ame setup. They demonstrate that, althouh a self-fulfillin currency attack ame has multiple equilibria when economic fundamentals are common knowlede, it has a unique equilibrium when speculators can only observe the fundamentals with small noise. Successfully overcomin the problem of indeterminacy of multiple equilibria models, their model allows the analysis of policy implications. Corsetti, Dasupta, Morris and Shin (2004) extend the model established by Morris and Shin (1998) to one with a lare player. They analyze two cases where the lare player has, and has not, a sinallin function. They find that in both cases the presence of a lare player does increase the possibility of the collapse of a fixed exchane rate reime, and make small speculators more aressive. Correstti, Pesenti, and Roubini (2001) ive a comprehensive survey on the role that lare players play in currency attacks. In the theoretical section of their survey, they apply a traditional coordination ame with perfect information, and then a lobal ame established by Corsetti, Dasupta, Morris and Shin (2004) to the study of the role of a lare player in a currency market. In the empirical section, they combine both econometric analysis and case studies to explore examples of currency attacks. Their conclusion is that both theoretical and empirical studies reveal that lare players do have a sinificant role in currency attacks, and more academic research is required to address a number of issues, includin the dynamics of currency attacks or crises. Bannier (2005) modifies the model established by Corsetti, Dasupta, Morris and Shin (2004) by chanin the assumption about a central bank s stratey. Due to that modification, both the lare player and small speculators strateies are symmetric and analytical results are available. She finds that this modification chanes the results iven by Corsetti, Dasupta, Morris and Shin (2004). Now a lare player can increase the possibility of a reime collapse only when market sentiment is pessimistic. 6

8 However, the presence of a lare player will decrease the possibility of a reime collapse when the market sentiment is optimistic. 3 The Benchmark Model without a Lare Player 3.1 Environment This model is a simple modification of the Abreu and Brunnermeier (2003) model. We capture the essence of their idea that the difficulty in coordination amon arbitraeurs, toether with their incentive to time the market, can cause asset mispricin. We modify the model to apply it to forein exchane markets. Assume that there is a country with a fixed exchane rate reime where a central bank commits to maintainin the exchane rate at a fixed level until it exhausts all of its forein reserves, whose level is denoted by k > 0. From time t 0 > η, the exchane rate becomes overvalued relative to its fundamental value, at a rate of. Denote the initial exchane rate as E 0. The fundamental exchane rate at t is E 0 when t < t 0 and E 0 (1 + (t t 0 )) when t t 0. Here the exchane rate is denominated in the domestic currency, say wons. So E 0 means that 1 dollar can exchane for E 0 wons. Without any currency attacks, the fixed exchane rate reime will collapse at some exoenously iven time t 0 +τ. This assumption captures the idea that any asset mispricin is not sustainable in the lon run. We follow Abreu and Brunnermeier (2003) in makin this simplified assumption to avoid ever reater currency overvaluations. Fiure 1 shows how the fundamental exchane rate chanes with time. There is a continuum of atomistic speculators of mass 1. Each speculator is financially constrained and can only access the credit whose worth is normalized to 1 dollar. Each speculator has to choose from two strateies: attackin or refrainin. When t < t 0 + τ, the exchane rate will devalue to the fundamental value if and only if attackin pressure exceeds k. This assumption follows that of Obstfeld (1996) and Morris and Shin (1998) and captures the idea of market liquidity. We specify the payoff structure of speculators as follows: if they choose refrainin, 7

9 E E = E 0 (1+(t t 0 )) E 0 0 t 0 τ t Fiure 1: How the fundamental exchane rate E chanes with time t which means that they will do nothin, they will ain zero. If they choose to attack, they will borrow wons from the banks of the attacked country, then exchane them into dollars from the central bank. The costs of attackin consist of two parts. One part is the fixed transaction costs associated with the currency exchanes, which is denoted by c F. We assume that the fixed transaction costs are not so hih that they prevent the speculators from ever attackin, despite the awareness of the overvaluation. The other part is the interest differential between wons and dollars, since we assume that the interest rate of wons is hiher than that of dollars. Let c denote the interest differential. Thus, if a speculator keeps attackin durin a time interval t, he will incur the cost of c. t. The payoffs of speculators from attackin is as follows. If the reime collapses at instant t, the payoffs of a speculator attackin at instant t with the wealth of 1 dollar will depend on how many other speculators are attackin. If the attackin mass is less than or equal to k, his payoffs are E 0.(t t 0 ). If the attackin mass is reater than k, only the first randomly chosen mass k of attackin speculators will ain the payoffs of E 0.(t t 0 ). So iven the attackin pressure α > k, the expected payoffs of a speculator are iven by k E α 0.(t t 0 ). For simplicity of the analysis, we assume that no partial attackin is allowed. 8

10 The speculators only have imperfect information about t 0, the time at which the overvaluation beins. More specifically, all the speculators have a prior belief about t 0, which is denoted by Φ(t 0 ). We assume that the speculators have an improper uniform belief about t 0 over [0, ). From t 0, a new cohort of small speculators with mass 1 η overvaluation in each instant from t 0 until t 0 + η. Conditional on t i, speculator t i s belief about t 0 is iven by the CDF where t [t i η, t i ]. t i. becomes aware of the Φ(t 0 t i ) = t t i + η, (1) η Given such a setup, we try to find the equilibrium stratey of a rational speculator Let σ(t, t i ) denote the stratey of speculator t i and the function σ : [0, ) [0, ) {0, 1} a stratey profile. Speculator t i s stratey is iven by σ(., t i ) : [0, t i + τ ] {0, 1}, where 0 means refrainin and 1 means attackin. The areate attackin pressure of all the speculators at time t t 0 is iven by s(t, t 0 ) = min{t,t0 +η} t 0 σ(t, t i )dt i. (2) Let T (t 0 ) = inf{t s(t, t 0 ) k or t = t 0 + τ } (3) denote the collapse time of the fixed exchane rate reime for a iven realization of t 0. Recall that Φ(. t i ) denotes speculator i s belief about t 0 iven that t 0 [t i η, t i ]. Hence, his belief about the collapse time is iven by Π(t t i ) = dφ(t 0 t i ). T (t 0 )<t The time t i expected payoffs of speculator t i, who remains refrainin until he beins to attack at time t and keeps attackin afterward until the reime collapses, are iven by ti +τ t E 0 (s T 1 (s)) c(s t)dπ(s t i ) c F, 9

11 provided that the attackin pressure at t does not strictly exceed k and that T (.) is strictly increasin. Later we will show that in equilibrium all the conditions will hold. If we normalize the initial exchane rate to 1, we et: ti +τ t (s T 1 (s)) c(s t)dπ(s t i ) c F. (4) 3.2 Equilibrium Characterization We confine our attention to symmetric trier strateies. We can prove that there is a unique symmetric trier stratey equilibrium. In this equilibrium, each speculator t i will attack at the instant t i + τ and keep attackin until the reime collapses. Dependin on parameter values of η, k, and c, the reime can collapse exoenously or endoenously. Here we will focus on the endoenous collapse case. Rochon (2006) proves in a similar setup that this symmetric trier stratey equilibrium is a stronly rational expectation equilibrium in the set of strateies with the only restriction bein that speculators act after bein informed. Proposition 1. Given τ > c kη and c, there is a unique symmetric trier stratey equilibrium where the reime collapses endoenously. In this equilibrium, each speculator t i beins to attack at the instant t i + τ and keeps attackin until the reime collapses, where τ = c kη. In equilibrium the reime collapses exactly at the instant t 0 + kη + τ. Given τ > kη and c <, there is a unique symmetric trier stratey equilibrium where the reime collapses endoenously. In this equilibrium, each speculator t i beins to attack at the instant t i and keeps attackin until the reime collapses. In equilibrium the reime collapses exactly at the instant t 0 + kη. Proof: Let τ define a symmetric trier equilibrium. That is, all the speculators bein to attack at t i + τ. Given such a stratey, the reime will collapse when speculator t 0 + kη attacks, and the collapsin time will be t 0 + kη + τ. 10

12 Now consider the optimal stratey of speculator t i iven that all the other speculators take the stratey τ. Thus the reime will collapse at t 0 +ζ, where ζ = kη +τ. Speculator t i believes that t 0 [t i η, t i ], the CDF of his posterior belief about t 0 is iven by Φ(t t i ) = t t i + η. (5) η Since the collapsin time is t 0 + ζ, he believes that t 0 + ζ [t i η + ζ, t i + ζ]. The CDF of his posterior belief about the collapsin date t 0 + ζ at time t i + τ is iven by Π(t i + τ t i ) = t i + τ (t i η + ζ) η = τ + η ζ. (6) η Speculator t i s expected payoff from attackin at t and keepin attackin until the reime collapses is iven by: ti +ζ t ((s T 1 (s)) c(s t))dπ(s t i ) c F. (7) The first order condition ives the optimal τ for him to attack: π(t i + τ t i ) 1 Π(t i + τ t i ) = c (t i + τ T 1 (t i + τ)). (8) We also check the second order condition, which turns out that the second order derivative is neative and the second order condition is satisfied. Takin Equation (6) into the left hand side of the first order condition ives us: π(t i + τ t i ) 1 Π(t i + τ t i ) = 1 ζ τ. (9) In addition, in this symmetric equilibrium, the duration between the time when the reime collapses and the time when the overvaluation happens is iven by: t i + τ T 1 (t i + τ) = τ + kη = ζ. This is because each speculator will delay a period of τ and the reime will collapse exactly at the moment t 0 +kη+τ when the speculator t 0 + kη launches his attack. So we find: 1 τ + kη τ = c (τ + kη). (10) 11

13 Since it is a symmetric equilibrium, τ = τ. Solvin the above equation, we et τ = (c )kη. (11) Given c kη < τ, the reime will collapse at t 0 + kη + τ < t 0 + τ endoenously. Notice that τ 0 if and only if c. When c <, we will et the corner solution of τ = 0. Q.E.D The intuition of the equilibrium is as follows. Given that all the speculators bein their attack at t i +τ, the instantaneous probability that the reime collapses at t i +τ of speculator t i is iven by: π(t i + τ t i ) 1 Π(t i + τ t i ) = 1 τ + kη τ. (12) If the reime exactly collapses at t i +τ, the ains from attackin will be (τ +kη). Thus, the expected marinal benefits of speculator t i attackin at t i + τ are iven by: (τ π(t i + τ t i ) + kη) 1 Π(t i + τ t i ) = (τ 1 + kη) τ + kη τ. Meanwhile, the marinal costs incurred by attackin at time t i +τ are c, which are constant. From the above equations we can see that the expected marinal ains from attackin are strictly increasin in τ, since the speculator t i s subjective instantaneous probability that the reime collapses at time t i +τ is strictly increasin in τ. So there is a unique level of τ, where the expected marinal ains from attackin are exactly equal to the marinal costs incurred by attackin. And it is the optimal time for speculator t i to attack. Fiures 2 and 3 explain the intuition. 3.3 Comparative Statics This section studies how the chanes in parameters of the model influence equilibrium results. We know that in equilibrium τ = (c )kη. 12

14 marinal costs / benifits c 0 τ τ * τ * +kη Fiure 2: How the marinal costs and benefits chane in τ in the case of the interior solution of τ marinalcosts / benefits c 0 τ τ * +kη Fiure 3: How the marinal costs and benefits chane in τ in the case of the corner solution of τ First, we can see that the speculators will wait loner with hiher c. The intuition is simple. Hiher c means that it will cost more if a speculator launches an attack early. Hence a speculator would like to wait loner to reduce the costs of attackin. 13

15 Second, we find that the speculators will wait loner with both hiher k and η. This result is also intuitive. Hiher η means more dispersed opinions amon the speculators and hiher k means a hiher requirement for coordination. Both will increase the difficulties in coordination and induce the speculators to wait loner. We know that c, k and η are all parameters indicatin how difficult it is to arbitrae in a forein exchane market. We find that now the frictions in the market become a blessin for the speculators, since more frictions will induce the speculators to wait loner and make hiher profits from the overvaluation. Third, we find that the speculators will wait loner with lower, the rate at which the currency is overvalued. In this case, hiher increases the speculators incentive to preempt other speculators and makes the speculators less patient. In the extreme case when > c, speculators will launch an attack immediately after they become aware of the overvaluation. Finally, there is an interestin result about the exchane rate level when the reime collapses, which determines the manitude of the devaluation. It is iven by ckη. We can see that does not play a role in determinin the manitude of the devaluation. This is because the speed at which the fundamental value of the currency decreases has two opposite effects: First, it affects the optimal delay time of speculators. Second, it affects the fundamental exchane rate at time t. The net result from these two effects is that will not influence the exchane rate when the reime collapses at all. 4 The Model with a Lare Player In this section we introduce a lare player into the basic model. We keep the model as simple as possible, by assumin that the speculators consist of one lare player with wealth λ < k and a continuum of small speculators of mass 1 with total wealth of 1. Here we assume λ < k such that the lare player cannot independently break the pe. This assumption is realistic because even a lare player like Soros in financial markets cannot sinle-handedly break a currency pe. Moreover, we assume that the lare player has perfect information about t 0 ; 14

16 that is, he always becomes aware of the overvaluation at t 0 when the overvaluation happens. In addition, we assume that the action of the lare player will not be observed by other speculators. Now we need to define the equilibrium in such a setup. Given all the assumptions unchaned for small speculators, we will prove that there is a unique trier stratey equilibrium in this ame. Proposition 2. Given τ > (c )(k λ)η and c > k, there is a unique trier stratey k λ equilibrium where the reime collapses endoenously. In this equilibrium, each small speculator t i beins to attack at the instant t i + τ SP and keeps attackin until the reime collapses. The lare player beins to attack at t 0 + (k λ)η + τ SP. Here τ SP = (c )(k λ)η. The reime collapses exactly at t 0 + c(k λ)η, when the lare player launches the attack. Given τ > c(k+λ)η and k < c < 1, there is a unique trier stratey equilibrium λ+k where the reime collapses endoenously. In this equilibrium, each small speculator beins to attack at the instant t i + τ SP and keeps attackin until the reime collapses. Here τ SP = (c )kη+cλη. The lare player beins to attack at t 0 +kη +τ SP. The reime collapses exactly at the time when the lare player launches the attack. Given τ > kη and c < k, there is a unique trier stratey equilibrium where λ+k the reime collapses endoenously. In this equilibrium, each small speculator beins to attack at the instant t i and keeps attackin until the reime collapses. The lare player beins to attack at t 0 + kη. The reime collapses exactly at the time when the lare player launches the attack. Proof Since the lare player has perfect information about t 0, he will choose the optimal time t 0 + τ LP to maximize his profits, iven the equilibrium strateies taken by small players. Since small players are identical ex ante and atomically small, they will take symmetric strateies. Suppose that each small player plays the symmetric tradin stratey t i + τ SP in equilibrium. From the moment of t 0 + (k λ)η on, the total wealth of the lare player and small players exceeds the threshold level k. Thus, the 15

17 payoffs of the lare player from attackin at t 0 + (k λ)η + τ SP + t are iven by where 0 t λη. λ[(k λ)η + τ SP k + t] k + t η = λkη (k λ)η + τ SP + t, kη + t Notice that t λη, or the reime will collapse solely due to the attackin pressure from small players, and the lare player will ain zero. The lare player will choose an optimal level of t to maximize his expected payoff. Solvin the maximization problem, we et that t = 0 iven λη τ SP < 0, and t = λη iven λη τ SP > 0. Therefore, the optimal stratey for the lare player is as follows. Given λη τ SP < 0, the lare player will launch the attack at t 0 + τ LP, where τ LP = (k λ)η + τ SP. Given λη τ SP > 0, the lare player will launch the attack at t 0 + τ LP, where τ LP = kη + τ SP. (The intuition for the above results is as follows. When the lare player delays his attackin, there are two effects on his payoffs. First, he will ain more from the larer devaluation when the reime collapses. Second, he will ain less due to the smaller share in the total attackin wealth. The shorter τ SP larer λ and η are, the more the lare player will ain from delayin.) is, and the Now let us look at the best responses of small players. Our previous proof for the unique symmetric trier stratey equilibrium still holds in this case. Only now the optimal attackin time t i + τ SP is determined by the followin conditions. Given the optimal stratey of the lare player, τ LP = (k λ)η+τ SP, in equilibrium the reime collapses at T = t 0 + ζ = t 0 + (k λ)η + τ SP. Therefore, the first order condition ives π(t i + τ SP t i ) 1 Π(t i + τ SP t i ) = 1 ζ τ SP = 1 (k λ)η = c ζ. In equilibrium, ζ = (k λ)η + τ SP. Thus we et τ SP = (c )(k λ)η. The lare player s equilibrium stratey is τ LP = (k λ)η+τ SP = c(k λ)η. Checkin the condition inducin the lare player to choose τ LP = (k λ)η + τ SP, we et: of τ LP λη τ SP < 0 c > k k λ. Now let us look at the case in which the lare player takes the equilibrium stratey = kη + τ SP. In equilibrium T = t 0 + ζ = t 0 + kη + τ SP. Given the lare 16

18 player s equilibrium stratey, the first order condition for small players is iven by 1 ζ τ = c SP k (kη + τ k+λ SP ) = 1 ηk = c(k + λ) k(kη + τ SP ). τ SP Solvin the above equation, we et τ SP. Therefore, τ LP = kη +. We need to check the condition inducin the lare player to choose = c(k+λ)η τ LP = kη + τ SP, which is = (c )kη+cλη λη τ SP > 0 c < 1. Moreover, notice that in order to ensure τ SP is positive, we have c(λ+k) k > 0, or c > k. When c < k, we et the corner solution of τ SP = 0. Since λη τ SP > 0 λ+k λ+k in this case, the condition required for the lare player to choose the stratey of τ LP = kη + τ SP still holds. Thus, the eneral condition for the lare player to take the stratey of τ LP = kη + τ SP is c < 1. Q.E.D. 5 The Role of a Lare Player In this section, we analyze the role that a lare player plays in a currency attack. Our model reveals that a lare player can both accelerate or delay the collapse of a fixed exchane rate reime, dependin on the circumstances. This result is important because it differs from the usual perception that the presence of a lare player in a forein exchane market will facilitate arbitrae, therefore helpin to reduce the mispricin of exchane rates. From Proposition 6, we can see that there are two possible equilibria. We will analyze these two cases respectively. 5.1 The Case in Which a Lare Player Accelerates the Attack Given c > k, the presence of the lare player will accelerate the collapse of the k λ reime. The followin are some results we find in this case. 17

19 1. The collapse of the reime is accelerated due to two reasons: first, the lare player has perfect information about t 0. Thus, more speculators are aware of the mispricin from t 0 on. Second, the presence of a lare player makes small players more aressive and shortens their delay time. The reime will collapse at t 0 + c(k λ)η, which is earlier than the reime collapse time without a lare player (which is t 0 + c kη). Here the collapse time is earlier, for two reasons: first, the lare player will bein to attack exactly when there is enouh wealth to correct the overvaluation. Thus, the reime collapses as lon as mass of k λ of small players attacks, instead of mass of k in the case without a lare player. Second, with the presence of the lare player, the small players equilibrium stratey, which is the waitin time between becomin aware of the overvaluation and before startin an attack, is shorter. In the case without a lare player, the small players stratey is τ = c kη. With the lare player it becomes τ SP η. In this case, the presence of a lare player makes = (c )(k λ) small players take more aressive strateies and accelerates the arbitrae to correct the overvaluation. 2. The collapsin time is strictly decreasin in λ, and the devaluation will be also smaller at the collapse time with larer λ. However, λ must be low enouh to ensure the existence of this equilibrium. Since t 0 + c(k λ)η, the more wealth a lare player has, the faster the fixed exchane rate reime will collapse. The exchane rate at the collapse time is iven by E 0 (1 + c(k λ)η), which is also decreasin in λ. However, in order for this acceleratin equilibrium to exist, we must have c > k (c )k. That is, λ <. k λ c Thus, this equilibrium will exist only when the wealth of the lare player is low enouh. 3. The lare player can make the most profits from the attack when λ = k 2. The profits of the lare player are iven by: λ c(k λ)η. 18

20 It is straihtforward to see that the optimal λ to maximize the lare player s payoffs is λ = k. So there is not a monotonically increasin relationship between 2 the wealth of the lare player and the payoffs it reaps from the attack. The intuition is that there is a tradeoff with the increase of the wealth of the lare player. On the one hand, more wealth ensures that the lare player can claim a hiher proportion of the attackin wealth that profits from the collapse of the reime. On the other hand, hiher wealth will accelerate the collapse of the reime, leadin to less devaluation, and therefore lower profits when the reime collapses. 5.2 The Case in Which a Lare Player Delays the Attack Given c < 1, the presence of a lare player delays or causes no acceleration of the reime collapse. The followin are some results that we et in this case. 1. The collapse of the reime is delayed or is not accelerated for two reasons: First, a lare player chooses to ride the overvaluation. Second, the presence of a lare player makes small players less aressive and wait loner before launchin the attack. Given k < c < 1, the reime will collapse at t λ+k 0 + kη + τ SP, where τ SP = (c )kη+cλη > 0. However, in the case without a lare player, we et the corner solution of τ = 0, which we can interpret as bein that the speculators will launch an attack as soon as they become aware of the overvaluation. There are two reasons to explain why small players delay their attack. First, small players are aware that the lare player will ride the overvaluation. Second, due to the presence of the lare player, the ains of small players from the attack will be less, which reduces the incentive of small players to preempt other players. In order to see this, recall that the equilibrium equation to determine τ SP is iven by: 1 ηk = c k (kη + τ k+λ SP ), 19

21 which is slihtly different from that in the case without a lare player: 1 ηk = c (kη + τ ). The only difference between the above two equations is that with the presence of a lare player, the expected payoffs of a small player will be the proportion of k λ+k of the total devaluation, instead of the whole devaluation. Given c < k, the presence of a lare player will at least cause no acceleration λ+k of the collapse of the reime. In this case we et the corner solution of τ SP = 0. Thus the collapse time of the reime will be t 0 + kη, which is the same as the case without a lare player. This result differs from our common belief that the presence of lare players facilitates the arbitrae and alleviates the mispricin in forein exchane markets. The intuition here is that the market power of the lare players, due to their superior information and more wealth, ives them the ability to time the collapse of the reime and ride the overvaluation. In certain circumstances they prefer to wait loner to reap the most profits from the currency overvaluation. 2. The collapse time of the reime is strictly increasin in λ. The devaluation will also be reater when the reime collapses with larer λ. Given k < c < 1, the collapse time is iven by t λ+k 0 + c(k+λ)η, which is strictly increasin in λ. Moveover, the exchane rate at the collapse of the reime is iven by E 0 (1 + c(k + λ)η), which is also strictly increasin in λ. Therefore, the more wealth the lare player has, the later the reime will collapse, and the larer the devaluation will be at the time of the collapse. Notice that in order for τ SP > 0, λ > ( c)k. So in this equilibrium λ has to be lare enouh to induce c small speculators to wait some time after bein aware of the overvaluation. 3. The profits of the lare player are strictly increasin in λ. This is straihtforward to see from the payoff function of the lare player: c(k + λ)η λ 20 = cλ(k + λ),

22 which is strictly increasin in λ. Our model reveals that the ratio of c is critical to determine whether the presence of a lare player will accelerate or delay a currency attack. When c > > 1, the presence of a lare player will accelerate the attack. When c < 1, the presence of a lare player will delay or at least will not accelerate the attack. The intuition is as follows. and c are key to determinin τ SP, that is, how lon small players will wait before launchin an attack. Moveover, τ SP is critical to determinin the ains a lare player will et from delay, relative to the losses from delay. The shorter τ SP is, the more are the ains relative to the losses. Thus, hiher and lower c lead to shorter τ SP, inducin the lare player to choose the delay equilibrium. Meanwhile, lower and hiher c lead to loner τ SP, inducin the lare player to choose the acceleratin equilibrium. In summary, only when is lare enouh and c is low enouh, is the incentive of a lare player to ride the overvaluation stron enouh to dominates the incentive to preempt small speculators, and therefore to make him wait loner. Therefore, his presence delays the reime collapse, and leads to severe currency overvaluation in the attacked country. Here we arue that the presence of a lare player will be harmful to a small economy in the sense that a lare player will employ his market power in a small economy to maximize his profits at the expense of the small economy. His presence prevents the small economy from correctin its overvalued currency in time and, we k k λ infer, causes more fluctuations in an economy once the devaluation happens. 6 A Note on the Application to the Stock Market The basic results in our model can be extended to the stock market. Suppose that we introduce a lare player into the stock market in the Abreu-Brunnermeier (2003) model. In their model, a continuum of small speculators have to decide the optimal time to sell their radually overvalued stocks. Similar to the aruments we have developed above, we can introduce a lare player who has perfect information about the time when the stock becomes overvalued. Given that small players will take a symmetric trier stratey t i + τ SP, the lare player will choose his optimal stratey 21

23 t 0 + τ LP to maximize his payoffs and vice versa. Here the lare player also has the incentive to ride the bubble to maximize his payoffs from the burst of the bubble. So in principle, the presence of the lare player can also delay the burstin of the bubble. Further analysis of the model is needed to obtain specific conditions under which the incentive of the lare player to ride the bubble will lead to the delay from burstin the bubble. 7 Conclusions and Future Research In this paper we study the role that lare players play in currency attacks in a dynamic currency attack ame where speculators have to determine when to attack, based on their incentives both to ride the overvaluation and to preempt other speculators. Our main findin is that a lare player can accelerate or delay the collapse of a fixed exchane rate reime, dependin on which incentive is dominant. More specifically, we find that when the incentive of a lare player to ride the currency overvaluation dominates the incentive to preempt other speculators, the presence of a lare player will delay the collapse of a fixed exchane rate reime. This findin is especially interestin because it differs from the common belief that the presence of lare players will facilitate arbitrae and reduce asset mispricin. One direction in which to extend the current model is to introduce multiple lare players and to examine how equilibrium outcomes will chane. In addition, in the current model with a lare player, we assume the extreme case that the lare player has perfect information about the time when the currency overvaluation beins. We can relax this assumption to a more eneral case where the lare player has imperfect information about the time when the currency overvaluation beins. 22

24 References [1] Abreu, Dilip and Markus K. Brunnermeier, 2003, Bubbles and Crushes, Econometria, Vol.71, No.1, [2] Bannier, Christina, 2005, Bi Elephants in Small Ponds: Do Lare Traders Make Financial Markets More Aressive? Journal of Monetary Economics, 52(2005), [3] Blustein, Paul, 2003, The Chastenin: Inside the Crisis That Rocked the Global Financial System and Humbled the IMF, PublicAffairs Press. [4] Carlsson, H., and van Damme, 1993, Global Games and Equilibrium Selection, Econometrica, 61, [5] Chamley, Christophe P., 2004, Rational Herdin: Economic Models of Social Learnin, Cambride University Press. [6] Chamley, Christophe P., 2003, Dynamic Speculative Attacks, American Economic Review, 93, [7] Cooper, Russell W., 1999, Coordination Games-complementarities and macroeconomics, Cambride University Press. [8] Corsetti, Giancarlo, Amil Dasupta, Stephen Morris, and Hyun Son Shin, 2004, Does One Soros Make a Difference? A Theory of Currency Crises with Lare and Small Traders, Review of Economic Studies, 71(1), [9] Corsetti, Giancarlo, Paolo Pesenti and Nouriel Roubini, 2001, The Role of Lare Players in Currency Crises, NBER Workin Paper, No [10] Diamond, Doulas W. and Philip H. Dybvi, 1983, Bank Runs, Deposit Insurance, and Liquidity, Journal of Political Economy, 91(3), [11] Fourcans, Andre, and Raphael Franck, 2003, Currency Crises: A Theoretical and Empirical Perspective, Edward Elar Publishin Limited. 23

25 [12] Goodhart, Charles and Gerhard Illin (eds.), 2002, Financial Crises, Contaion, and the Lender of Last Resort, Oxford University Press. [13] Jeanne, Olivier, 2000, Currency crises : a perspective on recent theoretical developments, Princeton University Press. [14] Kruman, P., 1979, A Model of Balance-of-payments Crises, Journal of Money, Credit and Bankin, 11(3), [15] Lyons, Richard K., 2001, The Microstructure Approach to Exchane Rates, The MIT Press. [16] Minuez-Afonso, Gara, 2007, Imperfect Common Knowlede in First- Generation Models of Currency Crises, International Journal of Central Bankin, Vol. 3 No [17] Morris, Stephen, 1995, Co-operation and Timin, CARESS Workin Paper [18] Morris, Stephen and Hyun Son Shin, 1998, Unique Equilibrium in a Model of Self-Fulfillin Currency Attacks, The American Economic Review, Vol.88, [19] Morris, Stephen and Hyun Son Shin, 2003, Global Games: Theory and Applications, Mathias Dewatripont, Lars Peter Hansen, and Stephen J. Turnovsky (eds.), Advances in Economics and Econometrics, Cambride University Press, [20] Obstfeld, M., 1986, Rational and Self-fulfillin Balance-of-payments Crieses, American Economic Review, 76(1), [21] Rochon, Celine, 2006, Devaluation without Common Knowlede, Journal of International Economics, 70(2),

26