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INTRODUCTION

The pricing of an asset is one of the main problems of financial markets. When any asset is traded on market dealers need to know at what price the trade should take place. The problem of correct pricing can exist not only at the first time asset comes to the financial market. If financial asset is popular or it is a paper of big promising company it can be easily overpriced. The efficiency market hypothesis allows some deviations from a fundamental price in the short run period but if this deviation is already big and long then the financial speculative bubble exists. Black (1986) According to Black's definition of market efficiency if bubble does not grow `too big' or too long' the efficient market hypothesis accepts it.

Speculative bubble is a phenomenon in the financial market when the market price exceeds the fundamental value of an asset. If dealers and market players do not pay attention to this deviation then it leads to huge losses after the crash of the bubble. The fundamental part price always supported by an asset but added price does not carry in itself anything except expectations of market players.

History knows a lot of speculative bubbles. Examples of bubbles can be: South Sea Company 1711 (bubble in the wake of the growth of colonial trade and strong developing of industry), Railway Mania in 1836 (a lot of capital and dividends on the railroads market create a big interest of dealers to invest in railroads), Florida real estate bubble 1920 (huge amounts of money were invested in the real estate in Florida and prices on houses was many times more than its fundamental price). It was a speculative bubble on the stocks of IT companies. NASDAQ index had a 500% rise in this period (from 1000 to 5132 points) but the problem was in pricing of IT companies assets. The feasible capital was very small and twenty years ago no one could truly price an internet activity. In the 2007-2008 was a global financial crisis. It was formed because of the bubble in the mortgage market.

Bubble is a systematic deviation of price index from the fundamental value of an asset. Agents will play different strategies depend on their confidence in the fundamental value of an asset. Their actions may be different. I will focus on the behavior of arbitrageurs. To derive the model of bubble formation it is important to find two positions:

1. The equilibrium strategies of agents;

2. The point of time when people see the bubble on the market and the panic begins.

Investors will pay attention to their private signals and public information. I assume that price already absorbing the private signals. The market price only partially reflects the information contained in the private signals and only to the extent that investor's trade on their private info. The expectation of investors depends on purchased public information and private signals accumulated in price.

The model would be based on consideration of the behavior of two types of arbitrageurs. All of them have the ability to purchase for extra information from informational agencies. Their choice of buying this kind of inside or not will depend on their preferences. As actions of informed players are reflected in price some of arbitrageurs will not pay a fee to informational agencies for inside. The difference will be in exact knowledge of exogenous bubble bursting time. For agents who paid an inside fee the expected closing time will be a common knowledge. For agents who reject this opportunity there is a probability of leaving the market at post-crash level of price and losing an abnormal return and take losses from crash and transaction costs. If expected profit from buying extra information would be higher than from not buying then all players will purchase for inside information and bubble will not exist. For this reason we need to find an equilibrium quantity of inside information in the economy. This would be derived through equality of expected profits of two groups of arbitrageurs.

The time has continuous structure and functions of price and fundamental value will be continuous and constantly increasing. Players will have a risk neutral linear utility function. The amount of agents tends to infinity and none of them has a direct influence on the price (all agents are price takers and actions of one player can not affect the economic state). Each agent views the state of the market from the moment at which he receives the signal of bubble existence. Noise traders in this model will play the role of liquidity shock and their choice of position has an equal chance of being short or long.

At the random point the deviation of price from fundamental value will be constant. Rational arbitrageurs sequentially become aware of the bubble. They do not know who takes the signal early or later than them. There is no straight coordination between actions of players. Coordination can be represented as attention to the same source of information. All agents who take an inside will close before the crash. Players who try to forecast without inside information will close positions at pre-crash price only partially. Players will maximize their expected profit in the case of asymmetric information based on their preferences. Their strategies will be the same: close position and take extra profit at the time when the temptation to `ride the bubble' equal to the fear of burst and fear continues to growth. When they start to close positions the bubble burst would formed immediately.

The goal of my work is to define the equilibrium strategy of players in environment of asymmetric information. The price index, volume of investments and size of the bubble depend on the behavior of players. If the panic will be different than the moment of exogenous (depends on non-modeled variables). Another word the model of bubble formation can be derived through the strategies of players. If bubble started to rise then all agents would try to `ride the bubble' for abnormal return. This is a starting point of model's logic. Using the backward induction from optimal strategy of each agent we may find the point of time when deviation of price index was started and also may find the beginning of panic which leads to exogenous burst. This paper will show how the presence of inside information and its absorption in the price of the asset will affect the strategy of the players and the collapse of the bubble.

Firstly, we will look at the simple model of the bubble. This part is important to show the logic of model and a visual demonstration of the equilibrium search method. All methodology in the model and its extension will be same. The difference is only distributions of variables and effects taken from literature I used in my work. Section 2 of paper illustrates how the analysis in this in this work relates to the literature. Section 3 stresses the main assumptions, setup and analysis of the first model with derivation of equilibrium. In Section 4 I will try to analyze more complex and general model. Section 5 concludes.

1. LITERATURE REVIEW

speculative financial market bubble

There are a lot of models which explain the formation of bubble. To my analysis I will rely on the papers with information asymmetry and signaling role of information. The paper written by Dilip Abreu and Markus K. Brunnermeier (2003) “BUBBLES AND CRASHES describe the model of bubbles because of structural changes. Their model describes the existence of bubble on efficient market with rational arbitrageurs. They show endogenous and exogenous reasons of burst of bubble even if arbitrageurs want to stay on the market to get maximum profit. This model challenges the efficient market hypothesis. They show that bubbles can exist despite the presence of rational arbitrageurs even if they are well-informed. In their model arbitrageurs want to stay on the market until the crash and leave it with maximum possible arbitrage profit. The equilibrium of this model is staying of arbitrageurs on the market until the burst in the next period is higher than `cost-benefit' ratio. The equilibrium in this model is unique. After becoming aware of the bubble traders can short their assets, but for them optimally choose to “ride the bubble” before their fear of crash is not high. In this model the asymmetry of information is also represented by the difference in time when next player sees the bubble. Every player thinks that the time of formation is the same as a time that they understand that bubble exists.

The assumptions for construction the equilibrium on the market and the logic behind my paper were inspired by the paper of Sanford Grossman & Joseph Stiglitz (1980) `ON THE IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKET'. This paper explains the concept of informative price systems. The price reflects the information of informed individual players. The idea is that then more and more players purchase for the inside information than the actual price index absorb the existing information. In my work arbitrageurs have the opportunity to buy some inside information but also can rely only on the price. This paper shows that there is no equilibrium on the market if costs of information are positive if price reflects inside information fully. My paper will show slightly another case. The equilibrium strategy of players will be unique. I refer to this article because I think that approach of the signaling price and the absorption of information in the work of Grossman and Stiglitz can help me to define the unique bubble equilibrium.

Another paper I used in my work is Antonio Doblas-Madrid (2012) “A ROBUST MODEL OF BUBBLES WITH MULTIDIMENSIONAL UNCERTAINTY”. The model of Abreu and Brunneimeier made a big influence on work of Doblas-Madrid. He made a discrete-time version of Abreu and Brunneimeier. The bubble formed because of the overreaction to events that are initially fundamental in nature. In this model the assumption of unobservability of bubble starting time is also exists. One part of my study is to find the correct starting point of formation of bubble. Also Doblas-Madrid writes that private signals are “overvalued” for agents. This can be explained by price of public information. If player wants to purchase for private info he need to find financial analytic and pay him for the job. In this paper the private information is overvalued. One of the main differences in this work from Abreu and Brunneimeier is absence of behavioral agents.

The third main resource for my study is the book “Rational Herds” by Christophe P. Chamley. In this book only part 16.3 explain the model of the bubble. This model also based on information and informed agents. The main point of this study is very close to two previous papers. The evolution of price and information here is almost the same as it was before. In this book a very good explanation of fractional gap between the bubble price and fundamental. Also there is a good explanation of difference between exogenous crash and endogenous (like in the work of Abreu and Brunneimeier). The first can occur than the bubble component reach its maximum value and the second case occur if the critical mass of agents who short their assets achieved. This book explains the strategy of traders in equilibrium. The agents hold the asset in a long position before the probability of crash is high because their gain from the arbitrage is higher than their losses. Here the costs of transactions and information takes place. The distribution of time I will take from this book. There are the same propositions as in Abreu and Brunneimeier work.

The propositions about information market I take from the paper “INFORMATION MARKETS AND THE COMOVEMENT OF ASSET PRICES” by Laura Veldkamp. As I will analyze the different markets and indexes, the analysis of covariance between prices on different assets take place. If the private information is costly investors prefer to purchase for a subset of the assets. Also suppliers of information make prices on information on glamour subsets smaller and agents buy it more often. This creates an information cascade, because all players pay more attention to the same information and with the time it becomes a common knowledge. Therefore, several ideas for my research were taken from this book.

The paper of Graham “HERDING AMONG INVESTMENT NEWSLETTER” has idea of `mimic' actions. If the reputation of information supplier is high then the private information is highly correlated across analysts. The aggregate information is overwhelming and informational cascade occurs. Because of authoritative source of information too many investors will make the same choice.

2. THE MODEL

2.1 SETUP

An economy populated by a continuum of traders. All players are risk-neutral. As they want to create an abnormal return I assume that all their free money they will put into market with the bubble. Another word, portfolio consist only one asset.

At trader i gets a signal that there is a bubble. The financial bubble is a long and big deviation of price from the fundamental value of an asset. Upon getting the signal, trader i decides to acquire information or not.Ex-ante all players are identical. After purchasing for inside they are divided into informed and uninformed. In equilibrium, a fraction of traders buy info. The fraction build their strategies based on available data. They can see only price index ex-ante and ex-post. Informed traders (insiders) can see the time of the bubble begins. The motivation of buying the extra information is ability to derive the and the time of exogenous burst. Uninformed traders can not see this but they see the actual price index. As insiders play on the market than price partially reflects the insider information.

Insiders purchase informational fee to the agencies (information suppliers). They want to maximize their profit from inside information sales and decide to choose .

The formation of bubble started at . All arbitrageurs become aware of the bubble sequentially. The distribution of agents is uniform. They may receive a signal on the time interval . of agents becomes aware of bubble at each moment of this interval. Assume that price index growing at constant rate and the function of price will be . . Also, conditionally to is improper uniform distribution . At time bubble arises and for all time period the fundamental value of asset explained by function .

Figure 1

All informed traders will have the same strategies. Also, with the time the insider's actions will affect the price index which is influence the return of uninformed agents Sanford Grossman & Joseph Stiglitz (1980) `ON THE IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKET'. Then we can say that return of uninformed traders is positively correlated with informed agent's arbitrage profit. An increase of informational qualityincreases the informativeness of the price. Also, less the costs of information makes this factor (informativeness of the price) smaller. For risk-neutral players positions on the market can be very long which creates higher absorption of insider's information in the price index. Also there is a negative dependence between the share of informed arbitrageurs and noise in price. If noise increases the proportion of insiders also increases. Similarly, if the noise in purchased information rises than more people forced to buy extra information because of less accurate private forecasts. Sanford Grossman & Joseph Stiglitz (1980) `ON THE IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKET'. These are the comparative statics implications from the Grossman &Stiglitz setup. In my paper this information is included to show the logic and motivation of my work. The relationship of insider's actions and price index can be interpreted in this passage. They tell about quantity and quality of inside information and how it relates to the strategies and payoffs of arbitrageurs.From this we can conclude that noise has no influence on price informativeness, because the changes in noise create more accurate forecast of uninformed (if noise goes down) or create a bigger share of informed agents (if noise goes up).

2.2 ANALYSIS

The strategy of players is to maximize their arbitrage profit by closing the positions before the bursting date. It will be a Perfect Bayesian equilibrium in which a trader's asset holding maximum long-position or maximum short position , so the restrictions for selling pressure from each investor (and cumulative selling pressure) can be only at two points: {0,1}. A trading equilibrium is defined as PBNE (Perfect Bayesian Nash Equilibrium) in which traders hold their stock at the same quantity as all other players.

The exogenous burst of bubble will be at the moment . If all players have inside then bubble bursts at and its satisfied the efficient market hypothesis. The equivalency of utilities condition is

Sustainability of equilibrium also can be represented as profits ratio. The maximization of profit for risk-neutral agents is the main goal, so the ratio of profits also should be equal to one in equilibrium. From this equation we can find the optimal share of insiders in the economy. Another word, to find the equilibrium share of insiders in the economy utilities should be equal. If one of them will be greater than other, all traders chose the strategy which gives them higher return.

Also, to find the correct informational fee what agents ready to pay for inside information then individual rationality condition (IR) and incentive compatibility condition (IC) should be satisfied Jean Tirole THE THEORY OF INDUSTRIAL ORGANISATION:

Grossman and Stiglitz in their paper in the paper `ON THE IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKET'1980 wrote that utility from buying information and from not buying should be equal. In the state of the market with bubble it is an important point. If utility from inside information will be higher than all rational individuals will purchase it. All available information will be absorbed by the price and market is efficient which is meaning that there is no bubble on the market.Sanford Grossman & Joseph Stiglitz `ON THE IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKET'1980 From this paradox we can impose more strict restrictions for our model:

If the informational fee does not contradict these two conditions then investors would be ready to purchase for inside to info suppliers. This can help us to derive the optimal fee which agencies will set as a price for inside information.

2.3 EQUILIBRIUM

As the strategy of risk-neutral players is maximization of expected profit, we can change the equivalency condition:



The strategy of uninformed players will be:

Derive an optimal time of closing positions:

The expected profit of uninformed traders with the optimal time of closing positions is

As they do not have an inside information, their expected profit based on the expectations about the bursting date.

Informed traders can observe the life-cycle of bubble and can make more accurate expectations. Their expected profit will be based on exact dates. We do not need to solve a maximization problem for this group of traders. If agent chooses to purchase for information his expected profit will have a form:

Now, equilibrium can be found:

Derive the information fee function:

Derive the optimal share of investors through the maximization of information fee function with respect to :

Fraction of informed traders depends positively on the life span of the bubble and negatively on the awareness window . The reason to this may be the increasing the opportunity of players to receive the signal. The increasing of the life span eliminates the share of informed traders because of the confidence of arbitrageurs. They think that in the long period of bubble formation they can ride the bubble without any inside information. Also, the reason may be a very long and `invisible' formation of systematic deviation of price index from the fundamental value. With the equilibrium share of insiders in the economy we can derive an optimal informational fee that traders should pay to be informed:

It is an equilibrium informational fee which creates an optimal share of inside traders. It is positively depended on quantity of players received the signal (awareness window) and growth rate. Increasing in time of exogenous burst of bubble decreases the informational fee. From this we can say that with higher life-cycle of bubble the demand for inside information would be smaller. The reason of this can be the confidence of arbitrageurs on the long time periods. This result is not intuitive in terms of entering the market. The information fee does not depend on time of entry because of two reasons:

1. There is no discount factor in the model

2. As the function is linear, result will be the same at each moment of time

Figure 2

For the model with not constant growth the informational fee will be depended on time of entering the market.From this section we can make several propositions.

PROPOSITION 1: Fraction of informed traders positively depends on the life-time of the bubble. With increasing of life-cycle of bubble, the share of insiders increasing .

PROPOSITION 2: Informational fee positively depends on awareness window .

In the next section we will look at the more difficult model, based on the setup of Abreu &Brunnermeier. This is important part of the work because it may shows the optimal point of leaving the market by uninformed traders, dependence of share of informed traders and time and also add more realistic assumptions about market.

3. EXTENSION OFTHE MODEL

The main setup of this model was taken from D. Abreu and M. Brunnermeier (2003) `BUBBLES AND CRASHES'. Distributions of price index, fundamental value and I took from this work. I changed this model by introducing the separation of players into two groups: informed and uninformed. Also, I changed the bubble component. Now it consists of two parts. Also, I tried to add a dependence of price index from actions of informed arbitrageurs.

Price reflects an inside information but only partially. An observable part of information will be denote as . Covariance between these two parts is zero. This part can be observed by players only with fee - . Before buying an inside information all players are identical. After purchasing a fee to informational agency they are divided into informed and uninformed. Demand on asset of informed trades depends on observable part of bubble and actual price index. At the same time demand of uninformed based only on the actual price index. As price cannot reflect all inside information there exist a noise - . This is not only one available cost of traders. They also should pay some amount as a transaction costs - . Recall that all arbitrageurs become aware of the bubble sequentially. They receive a signal about bubble formation on the time interval . of agents becomes aware of bubble at each moment of this interval. The last player who receives the signal of bubble formation denoted as .After this interval all players know about the existence of bubble and do not take any new signals. Correct time of deviation beginning is unknown and distributed exponentially. Assume that after deviation starting the growth of fundamental value equal to risk-free rate which is less than growth before the deviation. After deviation price is rising with the same rate as it was before the bubble formation.

Abreu and Brunnermeier for simplicity said that selling pressure is only the share of agents who leave the market. Without loss of generality I will take this condition. They are limited in the choice because of existing the positive transaction costs . This is an important point. If agents can freely change the side the number of transactions will tend to infinity and selling pressure would be not defined. So, all players will hold only maximum long position (1) or maximum short position (0) in equilibrium.

3.1 SETUP

As in previous model, economy is populated by a continuum of traders. Each trader puts all his free money in the asset at which they can create an abnormal return. Portfolio of each trader consists of only a single asset. All agents have the same utility function and the same goal to take a maximum abnormal return. Share of insiders in economy will be denoted as , all of them purchased for extra information and can observe the part of the bubble.

The price of an asset exceeds the fundamental value after some period of . This is the time of bubble begins. From this point arbitrageurs can generate an abnormal return. In the life cycle of bubble there exists a time interval at witch market players gradually learn about the existence of a bubble and the possibility of arbitrage . This called an `awareness window'. If players distributed uniformly than the rate at which the share of them knew about the bubble is .

Figure 3

The price function is and the function of the fundamental value is . Bubble exists the continuum period of time from till its exogenous burst in . The growth of price index from the beginning of bubble is higher than the growth of its fundamental value. The problem of bubbles is unknown starting point and we assume that it is exponentially distributed with cumulative distribution function . In the period of existence of bubble only a part of the price index is supported by the fundamental value which is equal to
, where is a bubble component in some point of time t differ from . The price is increasing faster than fundamental value it leads to , where is a growth rate of the price and is a growth of fundamental value. For simplicity assume that is a risk free rate.

The return for arbitrageurs is defined as difference between price and fundamental value. This difference consists of two parts,

,

where is observed component but players can identify this component only with purchasing for inside information, is unobservable component not only for market makers but also for information suppliers. It can be the forecast mistake. Both components distributed normally and uncorrelated with each other. From this we can derive next points:

With this distribution the variance of return will be

Informed traderscan use to make their own predictions. The one thing inside information does not contains is the amount of players who already informed or will be informed after trader . All this assumptions can create the price as an endogenous variable . There is an assumption that with increasing the share of informed players, the more informative is price and more information available for the uninformed traders.

Players have a risk neutral linear utility function. The amount of agents tends to infinity and none of them has influence on the price. Each agent views the state of the market from the moment at which he receives the signal of bubble existence. If agent aware of bubble than he has a truncated exponential distribution of in the interval . The number of arbitrageurs is finite and all of them receive the signal in the `awareness window'. If all arbitrageurs know about the bubble existence than the moment of time is higher than `awareness window'. Each player has his own thoughts about expected bursting time. When they short their positions they are sure that the probability of crash is equal to zero at this moment. After the first selling of asset at time the selling pressure starts to rise because all traders will sell their asset sequentially. This sequential will depend on the time of the awareness about bubble. So, for each agent will be different and arbitrageurs who saw it first would sell asset early. There is a probability that some of players do not catch the moment before crash and close their positions only after bursting time.

The strategy for all agents will be the same: close position and take extra profit at the time when the temptation to `ride the bubble' equal to the fear of burst. Dilipp Abreu and Markus Brunnermeier in their paper introduce the `hazard rate'. D. Abreu and M. Brunnermeier (2003) BUBBLES AND CRASHES It is an indicator of fear of the burst of bubble. It is calculated through the expected bursting date of each market maker. This rate should be compared with `cost-benefit' ratio. In the case of my model this kind of ratio would be equal to



where is an expected closing price, expected fundamental value of an asset and is a bubble component at some period of time greater than represents the deviation of price index from the fundamental value of an asset. Price distribution is ; distribution of the fundamental value (growth rate of price index is higher than growth of fundamental value - risk-free rate). As the fundamental value is equal to

We are interested in expected exercise price. The price of an asset with bubble changes over time with growth . But it also depends on behavior of agents. Informed traders have the signaling price which is based on some knowledge about deviation from fundamental value. Uninformed players make their decisions only with actual price. So, the expected price has the form

,

where is a pre-crash price and is a post-crash price. The post-crash can be represented through the pre-cash price:

This is an expected exercise price depended only from the pre-cash price. It means that we can look only at the level of actual market price. If it is higher than zero than it means that selling pressure has ability to create an endogenous burst. If selling pressure is relatively small than alpha is equal to zero. For simplicity selling pressure is distributed as

Deals will be distributed between 0 and 1 where 0 is a maximum long position and full investment in the market and 1 is a maximum short position. This will reflect the selling pressure: if the share of players who short their assets increasing than the selling pressure from short positions on price will tends to 1.

The bursting time depends on selling pressure. It can be exogenous if there is only end of bubble life-cycle. The function of bursting for a given is represented by formula:

So, it is a minimum value of t in the case of exogenous or endogenous burst. Agents believe that the time when they leave the market is less than the bursting time. This can be interpreted as the probability of beliefs about bursting time: . The probability of fail in expectations is: .

The demand on asset of informed traders depends on the known component of bubble and the price of an asset. Uninformed traders demand depends only on the price but they learn the relationship between return and price. The more individuals who purchased inside information the more informative the price of an asset is. This logic helps to see that with the increasing the part of insiders the ratio of utilities of informed to uninformed decreasing. Also the ratio of insiders decreases with high information fee. By the paper of Grossman and Stiglitz if the price has a high level of noise than the expected utility of uninformed agents is low. Let be the share of traders who purchased for inside information . Assume that price can be interpreted as known from inside information component plus noise process (white noise):

To find the equilibrium quantity of insiders the condition of equality of utilities from buying and not buying extra information should be satisfied. If utility of insiders higher than utility of uninformed then all players buy the information. All players will have full information and bubble cannot exist. In another situation then utility from buying is less than from not buying all traders will wait until someone buy information and equilibrium will be not sustainable. Recall that equivalency of utilities condition is

With the increasing of (share of insiders) the utility of informed traders goes down relative to utility of uninformed. Utility from purchasing the inside does not change with increasing of people who willing to pay too, but the informativeness of price index loss some noise and uninformed agents have positive return. It means that:

If the variance of price decreasing then demand of an asset is very responsive to changes in . The variance decreases because of less noise in the price. Uninformed traders are able to confidently sure that with rising of the price increasing (positive dependence). It leads to transfer of inside information to uninformed traders.

Information fee is a function of . Each investor decides to buy information or not at the exact time she takes a signal. The price witch agencies charge for inside will growth with the risk-free rate. It means that costs for arbitrageurs increases proportionally with increasing of arbitrage revenue. If agent was one of the first who became aware of bubble and decide to purchase for inside then his informational fee will be relatively less than for all who would take inside later. We do not need all function of informational fee. At the period when agents decide to buy extra information from agencies the price of an asset started to absorb the purchased information and at price system reflects the inside partly. There is still some noise because of impossibility of full reflection of inside information into the price. As in the paper of Laura Veldkamp, suppliers of information make prices on information on glamour subsets smaller because of high demand on this kind of subset and low costs of information collection and development of forecasts. Laura Veldkamp INFORMATION MARKETS AND THE COMOVEMENT OF ASSET PRICES This creates coordination between players. They buy the same subset of information from agencies

3.2 ANALYSIS

Insiders before others know the beginning of bubble formation and his distribution of is the lower bound of the support . For all traders this function of support exists.

LEMMA: In equilibrium, agent believes that at most mass of arbitrageurs became aware of the bubble prior to her. That is . D. Abreu and M. Brunnermeier (2003) BUBBLES AND CRASHES

In equilibrium by the strategy of taking maximum available arbitrage profit arbitrageur maintains the maximum short position and this decision does not depend on the category of agent (insider or uninformed). D. Abreu and M. Brunnermeier (2003 )BUBBLES AND CRASHES

PROOF: suppose that can re-enter the market. Given that there is a transaction costs and the lemma, in equilibrium must leave the market for a positive interval of time. Trader will stay out of the market until next player exits the market (someone who receive the signal later and independent of ). When arbitrageur sells out share, all arbitrageurs who receive the signal early already sold. Arbitrageur cannot re-enter the market before the next player enter too. The same reasoning applies to next players with the respect to agents who stay on the line after them. This way we conclude that arbitrageur stays out the market until the bubble burst or the next player exit and re-enter too. The latest possible date at which the bubble bursts from view is when player leaves the market. This lemma was taken from paper of Abreu &Brunnermeier. I need it because my analysis of equilibrium on the bubble based on `trigger strategy' at which all players in the face of crash want to maximize their abnormal return. This goal cannot be achieved if does not take maximum short and still holds a part of his wealth in the asset. This is taken from CUT-OFF property. It tells that at the moment of selling asset holds by all agents who receive this signal early already sold at the maximum amount. Another word, all players will close fully their positions in the equilibrium.

The optimal selling pressure can take only two possible values: one, if arbitrageur sells all assets; zero, if he holds all his available money in the asset. This is a trigger strategy. He will `push the trigger' and sell his asset fully when he think that the next period will be the bursting time. The maximum profit he can create is at the time right before the crash. In this case he will close on a pre-crash price and leave the market without losses. It is not rational to hold a part of money in asset before the crash because next closing will be at smaller price (after-crash) and profit would be less. In the analysis of a game process all players will stand by this (trigger) strategy.

3.3 PERSISTENCE OF BUBBLES

All players want to maximize their arbitrage payoffs. For achieving this goal they will stand by their dominant strategy. For insiders this strategy is straightforward, because they know about their closing position (derive it through and known from purchased information). The second type of players is uninformed. They know about existence of insiders and know that their own return depends on return of insiders. The strategy of uninformed will be `mimic', they will do the same action as informed traders. If someone start to short an asset than bubble ultimately bursts at exogenous bursting time. If insiders will not short their assets then uninformed players also holds their positions and bursting date shifts to the right. So, is a function of . There is a positive dependence between length of the bubble and actions of insiders. If confidence in purchased information is high then insiders can generate higher profit and helps uninformed to increase their payoffs too.

Given the structure of extension from my game, the actions of traders affect payoffs of others. In the paper of Abreu and Brunnermeier agents affected by actions of others only in the case of bubble bursts.

4. EQUILIBRIUM

To derive an equilibrium on the bubble, we need to look at the choices which available for agents. Then they receive the signal they have opportunity to buy inside information or not. From this point, two groups of agents will have the different information sets. The strategy for each player is the same but the common knowledge is different ex-post. Here we can define the length of time for game of each agent:

This is the time line between the boarders of which the player takes an abnormal profit from arbitrage strategy. The category of agent plays the role in the equation of expected payoff of agent. If it is an insider than he exactly know the right time of leaving the market and closing position at the highest possible price. For uninformed traders there is a probability of fail in the forecasted exogenous bursting time. Expected payoff of insider will have the form:

As insider can exactly know the time of exogenous burst than he closes the position at pre-crash price with 100% probability and

For uninformed agent there exists probability more than zero to close the position at post-crash price

The difference between two equations of expected profit is the premia from purchasing the informational fee. If agent pays for inside information then he is absolutely sure in the exercise pre-crash price. Uninformed trader increases his expected payoff by avoiding the informational fee but in this case he cannot be sure in exercise price and it is decreasing his expected payoff. Recall that actions of insiders reflected in the price index. So, expected closing price of uninformed player depends positively on the inside information which is reflected in the price absorbs the extra information

As insiders do not act in the period of post-crash has no effect on . Than we can assume that

Uninformed arbitrageur will act as in the model of Abreu and Brunnermeier. He will stay at the market as long as his willingness to ride the bubble is less or equal to the cost-benefit ratio which is the constant horizontal line with the formula

where is the size of the bubble at the point of interception with fear function. For insiders this rate is workless their rate of fear is equal to 0 + some noise from the bias of information. The uninformed players will construct their strategy based on the hazard rate. They start to leave the market then the fear of the bubble exceeds the hazard rate:

The conditional cumulative distribution function of the bursting date is . The trigger strategy simplifies the analysis of equilibrium. We need to derive the expected profit of trader at the selling time. This is a difference between the price level and fundamental value of an asset at this time minus transaction costs and informational fee if it exists. The gross profit will be the integral of expected profit from closing position before the bubble and closing after bubble:

where the probability of closing time is greater than exogenous bursting time and .

If arbitrageur purchased for the inside then the probability that she will close at pre-crash price is one. For uninformed this probability will be less than one.

The beliefs of players about bursting date are truncated distributed

If we differentiate the payoff respect to yields to holding the maximum long position if fear of burst is less than cost-benefit ratio we can derive the hazard rate. Each trader views the market from his own relative perspective which is depending on time of his awareness of bubble. The corresponding hazard rate is:

 D. Abreu and M. Brunnermeier (2003) BUBBLES AND CRASHES. The formula of payoff was taken from the model of Abreu &Brunnermeier to show the connection between the trigger strategy on the bubble equilibrium and the expected profits of two types of arbitrageurs. Also the formula of hazard rate was taken as a standard form often used by bubble researches.

Exogenous bursting time is independent of selling time which can be found through interception point of cost-benefit ratio and the hazard rate function:

This point will be the starting selling date of uninformed traders because insiders exactly know the time at which they leave the market.

Figure 4

Then the moment of time exceeds the then all uninformed start to short asset. If there is a high confidence in the asset then the panic can start not at the moment . We need to find the optimal time of closing position after the at the period at which the fear of the bubble exceeds the cost-benefit ratio. This will be not only the effect on uninformed traders but also for informed. For players there is no confidence in their own forecasts because of the lack of knowledge and skills for working with this type of asset. All their decisions will be based on purchased information (analysis of agencies). The main goal of this part of paper is derivation of optimal strategies for both types of players with exogenous burst. This can help us to determine the time at which the last period of bubble life cycle (panic) starting. The optimal closing time will be more or equal to the point of interception of hazard rate and cost-benefit ratio but it will be less than time of exogenous burst. Rational traders do not want to close after crash.

The fundamental value of an asset is known by arbitrageurs and does not volatile over the time. Every market maker knows the real value (not exactly know) if he already receive the signal about bubble. The behavior of uninformed traders can be less depended on the insider's information because all players have their opinions, strategies and forecasts. In this case the insider's information will be less reflected in the actual price index. The optimal choice in the equilibrium will be the same as in general model: trigger strategy with closing at time when fear exceeds the `cost-benefit' ratio.

Recall that is a price depended on the share of informed traders ( - share of investors who prefer to purchase for information and now his actions absorbed in price).

When takes a signal about bubble and buy inside information his expectations change because of exact knowledge of crashing time. The expected profit of insiders stays unchanged. There are changes in expected profit of uninformed agent.

The exogenous burst takes place if players start to short their assets at the time when hazard rate greater or equal to `cost-benefit' ratio. We need to define the trigger strategies. Firstly, it will be optimal expected profit maximization strategy. For risk neutral players it is just a solving maximization problem. Let's start from insiders strategies.

Now, let's find the point of optimal leaving the market for uninformed traders. is a time when arbitrageur i assumes that bubble will burst and she has a chance to close position at the pre-crash price. From this we can see the distribution of which is exponential. Conditional distribution of crashing date is exponential truncated. For D. Abreu and M. Brunnermeier (2003) BUBBLES AND CRASHES. This is a truncated exponential distribution. In the given paper Abreu and Brunnermeier use this distribution to show the beginning of bubble formation and with the same technique derive the function: simplicity, normalize this distribution. The area below the graph is equal to one. So, we have the next equation:

With the help of CDF of with respect to we can derive the time of optimal position closing. This is the equation for the expected profit of uninformed traders:

,

where is exogenous bursting date.

can be re-written as:

The next part of maximization problem for uninformed traders can be also re-written:

Derive an expected value of bubble component:

This is a function of optimal closing positions time for uninformed arbitrageurs. If they do not purchase for inside information they are optimally short at . The optimal time of shorting asset is a solution of maximization problem by :

F.O.C.

To find the equilibrium share of insiders in the economy we need to derive optimal closing date for uninformed trader which is defined implicitly from the F.O.C. above.

This is an optimal choice to leave the market for uninformed traders. Their expected profit will be the function depended on this time:

Recall that is a share of insiders, - number of arbitrageurs, - growth of price index, - growth of fundamental value after the bubble formation and is a time of taking a signal by arbitrageur.

For agents who want to buy an inside information the expected profit has the form:

Insiders have information about start of bubble formation. We can find expected profit of insider with respect to common . Another word, we only need to integrate in the equation of expected profit.



This is an expected profit for insiders. The next step of deriving the bubble equilibrium is to find optimal (share of insiders in the economy).

From this we can define the optimal inside information at which the share of insiders will be at equilibrium point and the whole economy at the bubble will be at equilibrium.

There are no real roots for this equation and we cannot continue solution for this problem in the case of Abreu &Brunnermeier model. We can try to make the model simpler to look at the result and comparative statics. Derivation of expected profits and the function of equilibrium information fee is in APPENDIX (4,5,6) This fee will satisfy the equivalency condition and IR&IC (individual rationality and incentive compatibility).

Other words, this model can work and show the equilibrium of the bubble if we can solve this implicit equation for price information. I assume that informational fee positively depends on the time at which an arbitrageur becomes aware of bubble because of constant growth exponential function. Also, the size of the bubble increases the demand on information about known component . The share of informed agents depends on time at which uninformed agents expect to short. will be the function of - the moment of awareness, - time of closing positions, the size of awareness window (as it was in the original model) and the value of information fee.

CONCLUSION

This paper argues that bubbles can exist with rational decisions of agents even if they know about future crash of the market. The bubble will burst but some agents can generate an abnormal return on this asset. In this work we derived an equilibrium share of insiders in the economy. All players have identical preferences and they have choice to be informed and base strategy on full information or be uninformed and base strategy on their own forecasts. The main point of this paper is derivation of arbitrage strategies of two groups of agents. We derived the point of time when panic between uninformed traders begins (interception of fear of the bubble and `cost-benefit' ratio). Also, we showed the distribution of starting point of bubble formation for players who did not purchase for inside and do not know it exactly. This work shows the equilibrium at which bubble will burst only because of exogenous reasons and not endogenous. Other words, if all conditions satisfied the equilibrium then crash will be only at the point of maximum possible value of the bubble.

In the Section 3 we showed strategies of traders and derive the optimal informational fee which create an equilibrium proportion of informed traders. In the Section 4 we tried to achieve same results with more complex model. As a result, we have optimal strategies, time at which panic begins. Unfortunately, it was not possible to withdraw the equilibrium number of informed players.

REFERENCES

1. Dilip Abreu and Markus K. Brunnermeier (January, 2003). BUBBLES AND CRASHES.Econometrica, Vol. 71, No 1,pp. 173-204.

2. Antonio Doblas-Madrid (September, 2012). A ROBUST MODEL OF BUBBLES WITH MULTIDIMENSIONAL UNCERTAINTY.Econometrica, Vol. 80, No 5, pp. 1845-1893.

3. Sanford J. Grossman and Joseph E. Stiglitz (June, 1980). ON IMPOSSIBILITY OF INFORMATIONALLY EFFICIENT MARKETS. The American Economic Review, Vol. 70, No 3, pp. 393-408.

4. Laura L. Veldkamp (July, 2004). INFORMATION MARKETS AND THE COMOVEMENT OF ASSET PRICES. The Review of Economic Studies (2006) 73, pp. 823-845.

5. Christophe P. Chamley (2004). RATIONAL HERDS.Cambridge University Press 2004.

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