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Forfatter:Simon Toft Nielsen
Studienummer:SN70199
Vejleder: Stefan Hirth
A game-theoretic analysis of price bubbles in financial markets
Abstract
Price bubbles have had profound influence on economies in the latest centuries. Studying the phenomenon has been greatly restricted by the assumption that the efficient market hypothesis holds. This has incurred a basic lack of understanding investors behavior in price bubbles.
Accepting that the efficient market hypothesis does not hold implies, however, that irrational investors are directing the market. This raises the issue of how rational investors behavior can be modeled in an irrational environment.
Game theory is found to be a very efficient tool for handling this issue. First of all, irrational investors can be modeled quite well. Second, it allows investors to make strategic choices, depending on their expectations of other investors. The strategic aspect of investing has formerly been ignored by assuming that rational investors only based decisions on the fundamental value of stocks.
Furthermore, game theory is in general designed with the purpose of determining players behavior. Logically, it gives a great framework to analyze investors behavior in price bubbles.
To determine optimal behavior in price bubbles, Abreu & Brunnermeier have presented a comprehensive model, based on game theory.
This model represents an environment, encompassing both rational and irrational investors. Additionally, heterogeneity among rational investors leads to a situation, where they become sequentially aware of the price bubble.
The role of irrational investors is to participate in creating the price bubble, while simultaneously preventing rational investors from identifying other investors strategies. However, there is a limit of the amount of selling pressure, irrational investors are able to absorb. When a sufficient amount of selling pressure is generated, the prices will drop and the bubble burst.
Because of dispersions of opinion, and the absorption capacity of irrational investors, it becomes rationally optimal to ride the bubble. Until the risk of the bubble bursting exceeds the costs compared to the benefits of attacking it, investors will stay in the market.
The model additionally presents a broad range of assumptions about the reality.
Violation of these assumptions, logically has major impact on the optimal equilibrium. Discussing them reveals that they generally are well supported.
Furthermore, presenting a few investigations of actual behavior of rational investors in the dot.com bubble also supports the results of the model.
Content
1)Introduction5.
1.1) Method6.
1.2) Limitations7.
2)Definition of a price bubble8.
3)Principles of game theory10.
4)Game theory in price bubbles15.
5)Rules of the game18.
6)The Abreu/Brunnermeier model19.
6.1) Game theoretical background25.
6.2) Preliminary analysis26.
6.3) Identifying the optimal behavior of investors31.
6.3.1) Exogenous crashes32.
6.3.2) Endogenous crashes35.
6.4) Optimal behavior38.
7)Assumptions40.
8)Supporting results of the model49.
8.1) “Hedge funds and the technology bubble”50.
8.2) “Who drove and burst the tech bubble”50.
9)Conclusion51.
10)References53.
1. Introduction
Analysing modern financial market behavior by game theory is still considered unconventional compared to traditional asset pricing theories.
However, game theory points out important aspects of investor behavior which contradicts the fundamental assumptions of traditional theories. Traditional theories are typically founded on the efficient market hypothesis which constitutes that asset prices are based on every known information, and that all investors is able to access and benefit from this information[1]. This implies that the development of asset prices is unpredictable, since information and news in the future by nature are unpredictable. As a consequence of unpredictability, the efficient market hypothesis implies that no investor can consistently outperform the market.
This leads to the idea of the random walk of Wall Street[2] where any portfolio of stocks in the long run will be as good as any other portfolios.
It also leads to the “no-trade theorem”[3], where investors cannot benefit from private information. This means that using private information will have the effect of a signal to other investors. This will therefore become known information.
Furthermore, if the efficient market hypothesis holds, it implies that a price bubble cannot develop in the market. The argumentation is, that investors through interpretation of news, immediately will become aware of the over-valued stocks, and sell them to collect the instant profit, before other rational investors become aware.
Several papers have proven the weaknesses of the efficient market hypothesis[4], and how the development of asset prices cannot completely be explained by news and information. This shows, that the asset prices are influenced by other factors, such as historical development, psychology of investors and other behavioral factors.
The most important contribution of game theory to the analysis of financial markets is, that it can be extended to cover a market situation which includes both rational and behavioral traders. In a price bubble situation, which is the focus of this paper, game theory additionally lays a unique framework to analyze the optimal behavior of rational investors that hold private information.
This paper describes how game theory can be applied to a price bubble situation. Including a thorough discussion of the assumptions made especially in relation to the roles of rationality and information.
Following questions will be answered with a game-theoretic approach:
- What is a price bubble
- How does a price bubble origin and evolve
- What is the optimal behavior of investors in a price bubble
The problems above will be explained using a comprehensive model presented by Abreu and Brunnermeier in their article “bubbles and Crashes”. Finally, the paper aims to show which behavior rational investors had during the dot.com bubble, compared to the behavior which the game theoretic model constitutes as optimal.
1.1 Method
As mentioned, the paper is founded on the model presented by Abreu and Brunnermeier. The assumptions made, however, will be discussed thoroughly and will draw insights from a broad range of relevant literature.
The presentation of the model is following the same structure which Abreu & Brunnermeier had used in their paper Bubbles and Crashes (2003). First presenting their model, then develop some useful propositions in a preliminary analysis, before finally identifying equilibrium. However, trying to grasp the ideas of the model, did not at first seem straight forward. I have felt it necessary to clarify some of the issues and interpret the mathematical expressions. Therefore I include a few more steps in the explanation.
The model explains how a price bubble evolves, and determines the optimal behavior. Each step of the model will be presented, and assumptions will be discussed, so the implications of the optimal strategy can be analyzed.
In order to compare the behavior of rational investors during the dot.com, with the optimal behavior identified in Abreu & Brunnermeiers model, I initially wanted to do an investigation on Hedge funds.
The action ability and the structural design of a hedge fund means, that it is a good representative of a rational investor in the model.
Gathering data, however, turned out to be a problem that was not possible to overcome. Access to databases that hold the necessary information was very expensive.
The investigation of rational investors behavior in the dot.com bubble will, therefore, consist of a short presentation of 2 papers addressing the issue.
(1)“Hedge funds and the technology bubble” by Brunnermeier and Nagel
(2)“Who drove and burst the tech bubble” by Griffin, Harris and Topaloglu
1.2 Limitations:
Abreu & Brunnermeiers model include some further theories especially about synchronizing events. This part of the model will not be included in this paper.
Additionally the special case where the price equals the fundamental value at the time , will be ignored, since this paper focus its attention on broader aspects of the model.
2. Definition of a price bubble
Price bubbles have had a significant impact in the global economy at numerous times during the latest centuries. Naming the Tulip mania, the South Sea bubble and the dot.com bubble as a few[5].
Kindleberger (1978)[6], originally defined price bubbles by the following; “A bubble is an upward price movement over an extended range that then implodes”. However, this definition seems a little too simplified, so further explanation is necessary.
A price bubble origins, when a price of an asset rises over its fundamental value. However, it is complicated to define precisely when this happens.
In the stock market assets are valued as any other goods by the limitations of supply and the extent of demand. Hence, the price is naturally set by the market, and the price of the asset reflects its value to the market. The fundamental value of an asset reflects the collected value of the particular company, including future dividends. Future dividends will naturally depend on an assessment of how profitable the company will be in the future.
This assessment of the companies profitability in the future is exactly the source of price bubbles in the financial markets.
Many of the well-known financial bubbles have originated around major technological innovations[7]. Companies, that are closely related to or are able to exploit the new technology, experience exponential increases in stock prices, because investors believe that their future growth will be higher than the market growth. Under these structural changes unsophisticated investors tend to believe that the economy has changed as well. They believe that the technological innovations have created permanently higher growth rates in the particular sector of the economy.
The expectations of permanently higher growth rates will make unsophisticated investors overly-optimistic in relation to a company’s future dividends, and the intensity of demand will increase and inflate prices significantly higher than the realfundamental value.
Some literature[8] also suggests that the amount of unsophisticated investors compared to the amount of sophisticated investors will increase during a bubble situation. It is argued that the high growth rates, attracts private investors who have not yet experienced a downturn in the stock market. They will for this reason behave over-optimistically and hence, support the development of the bubble.
As time goes by and the investors’ expectations are revealed to be overly-optimistic, the bubble crashes and leaves the stock prices at very low levels.
The key for a sophisticated investor is to realize when other investors’ expectations of the companies future profitability is unrealistic. This will inevitably mean that the stock price will rise over the fundamental value and create the bubble effect.
This definition of a price bubble implies that it, in the situation virtually, is impossible to be certain whether or not it is a bubble. Only the future will prove whether or not the expectations to the companies profitability where realistic. However, it is safe to say that all rational investors, during a period of rapid price increases, will develop a belief from which they act.
The model of Abreu & Brunnermeier corresponds with this perception of a price bubble. Although, it is not clearly explained, exactly how they define a price bubble, the overall structure of the model reveals their perspective[9].
From their point of view a mispricing is required to be sustained until a certain amount of investors are aware before it can be regarded as a price bubble.
As the definition above, a price bubble starts to origin, when the price is separating from the fundamental value. However, a bubble is not established, until a sufficient number of investors are aware of the mispricing to burst the bubble, but don’t.
It can be argued that as long as investors actually believe in higher future dividends they also believe that the fundamental value is represented by the price. If enough investors share this opinion then the mispricing has not established itself as a bubble. In order for a mispricing to become a bubble situation at least some investors is required to be aware.
The reasoning behind this is to separate minor mispricing from bubble situations.
Summing up, a price bubble is defined as a situation where the price departs from the fundamental value, because of overly-optimistic expectations to future dividends. When enough investors have become aware to correct the mispricing, but don’t, the bubble situation is established.
3. Principles of Game theory
Game theory will, in this paper, outline the framework of the analysis. The following will give the basic ideas of how game theory works in order to understand how it can be applied to price bubbles.
Game theory was created and first published by Von Neumann and Morgenstern in 1944[10]. Recently their ideas have gained increasing success in analyzing economic problems. Experts believe that the general methodology of economics has changed, and that the principles of game theory fit in well with the new paradigm[11].
Comparing game theory to traditional economic analysis tools, shows a distinctive characteristic of game theory. The fundamental assumptions are very primitive. This is because game theories set of point are the actors which essentially are the most basic units in a given economy. Previously most economic theories started out with higher-level assumptions about actors’ behavior[12]. Macroeconomists were typically assuming different behavioral relationships, like the consumption function (the relationship between actors’ income and their consumption). Assuming such relationships, makes it harder for economists to evaluate results on real-life basis.
The reason is that actors’ behavior is conditioned of these assumptions. This implies, that if the assumptions do not hold, then results do not hold.
Correspondingly, micro economists often used assumption of sales maximization is problematic. Sales maximization does not include any cost terms, which indicates that sales maximization can be viewed as irrational.
Game theory is applying much more simple assumptions. The most important and most general being that, every actor in a game wishes to maximize his utility function. The actors seek to do this, given the constraints they are exposed to during the game. Essentially, the assumption implies, that actors behave rationally[13]. Traditional economic theories have in recent years adopted game theories simplicity of assumptions. The paradigms of game theory and traditional theories have been converging, which is suspected to be the reason of the latest success of game theory.
The basic idea of game theory is to analyze how players determine their optimal behavior in a given game.
The first condition that has to be met to use game theory is, that the players strategies must have an effect on other players strategies. And all players must have an understanding of that relation. In order to analyze the stock market as a game, this raises a problem, which will be discussed later.
A game consists of players, actions, payoffs and information. All together they form the rules of the game. The principle of game theory is to take an economic situation and model it, as a game. The model assigns pay off functions and strategies to each player, and analyzes what happens in the equilibrium, when every player chooses the strategy, that maximizes his utility pay off.
First of all, the players need to be defined.
A player in a game is an individual, who makes decisions in order to maximize his utility. This, however, has an exception. In order to structure a game, where payoffs are dependent on the state of the world, it is appropriate to create a pseudo-player, normally called nature. This player can given specific probabilities, randomly decide the state of the world. The move of nature can be either before the players move, or after the players move. If nature moves before the players, the concept of the Harsanyi transformation and Baye’s rule becomes relevant.
Three types of information exist in games. The first two types are information about natures move, and information about other players’ moves. Both are highly influential to which strategies the players will follow. The third type of information is common knowledge which also has an important role in the fundamental construction of the game. Common knowledge is the term used to describe information that are known to all players, and that all players know that all players know, and that all players know that all players knows that all players know... (ad infinitum). Common knowledge is the information which the players share at the beginning of the game. Including beliefs concerning the state of the world and the probabilities assigned to natures moves.
Common knowledge is often named as common priors. This expression will be used further on in this paper, because the game analyzed involves updating these common prior beliefs.
Information about nature are said to be certain, if nature moves before the players. In these cases the players are able to adapt their actions to the moves of nature. If the nature moves after the players, information is said to be uncertain. The uncertainty reflects in the pay off functions. Normally, players select strategies in order to maximize their utility pay off. When natures moves are uncertain, the pay off becomes uncertain, which makes it difficult for players to chose the optimal strategy. This problem can be overcome by rationalizing that, when pay off is uncertain, players will chose the strategies, which maximizes their expected utility. In these cases, players are said to have Von Neumann-Morgenstern utility functions[14].