How to grow a bubble: A model of myopic adapting agents

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Abstract

We present a simple agent-based model to study the development of a bubble and the consequential crash and investigate how their proximate triggering factor might relate to their fundamental mechanism, and vice versa. Our agents invest according to their opinion on future price movements, which is based on three sources of information, (i) public information, i.e. news, (ii) information from their “friendship” network and (iii) private information. Our bounded rational agents continuously adapt their trading strategy to the current market regime by weighting each of these sources of information in their trading decision according to its recent predicting performance. We find that bubbles originate from a random lucky streak of positive news, which, due to a feedback mechanism of these news on the agents’ strategies develop into a transient collective herding regime. After this self-amplified exuberance, the price has reached an unsustainable high value, being corrected by a crash, which brings the price even below its fundamental value. These ingredients provide a simple mechanism for the excess volatility documented in financial markets. Paradoxically, it is the attempt for investors to adapt to the current market regime which leads to a dramatic amplification of the price volatility. A positive feedback loop is created by the two dominating mechanisms (adaptation and imitation) which, by reinforcing each other, result in bubbles and crashes. The model offers a simple reconciliation of the two opposite (herding versus fundamental) proposals for the origin of crashes within a single framework and justifies the existence of two populations in the distribution of returns, exemplifying the concept that crashes are qualitatively different from the rest of the price moves.

Highlights

► We model a financial market, which can experience bubbles and subsequent crashes. ► Positive news, which increase the price, initiate a bubble. ► Local adaptation of investors’ strategy amplifies the price appreciation. ► A bubble is ended by a crash once the price reaches an unsustainable high level.

Introduction

Bubbles and crashes in financial markets are events that are fascinating to academics and practitioners alike. According to the consecrated academic view that markets are efficient, bubbles, being temporally persistent, self-reenforcement deviations of the price from the fundamental value, are impossible. And crashes should only result from the revelation of a dramatic piece of information. Yet in reality, there is a large consensus both from professionals (Dudley, 2010, Trichet, 2010) and academia (Shiller, 2000, Abreu and Brunnermeier, 2003) that bubbles do exist, and even the most thorough post-mortem analyses are typically inconclusive as to what piece of information might have triggered the observed crash (Barro et al., 1989).

It is often observed that crashes occur soon after a long run-up of prices, referred to as a bubble. A crash is thus often the burst of the bubble. There is a vast amount of literature aiming at characterizing the underlying origin(s) and mechanism(s) of financial bubbles (Abreu and Brunnermeier, 2003, Kaufman, 2001, Sheffrin, 2005, Shiller, 2000, Sornette, 2003a) but there is still no consensus in the academic community on what is really a bubble and what are its characteristic properties. Bubbles do not seem to be fully explained by bounded rationality (Levine and Zajac, 2007), speculation (Lei et al., 2001) or the uncertainty in the market (Smith et al., 1988). Finally, there is no really satisfactory theory of bubbles, which both encompasses its different possible mechanisms and adheres to reasonable economic principles (no arbitrage, equilibrium, bounded rationality, etc.).

Most approaches to explain crashes search for possible mechanism or effects that operate at very short time scales (hours, days, or weeks at most). Other mechanisms concentrate on learning an exogenously given crash rate (Sandroni, 1998). Here, we build on the radically different hypotheses summarized in (Sornette, 2003a) that the underlying cause of the crash should be found in the preceding months and years, in the progressively increasing build-up of a characteristic that we refer to as ‘market cooperation’, which expresses the growth of the correlation between investors’ decisions leading to stronger effective interactions between them as a result of several positive feedback mechanisms. According to this point of view, the proximal triggering factor for price collapse should be clearly distinguished from the fundamental factor. A crash occurs because the market has entered an unstable phase towards the culmination of a bubble and any small disturbance or process may reveal the existence of the instability. Think of a ruler held up vertically on your finger: this very unstable position will lead eventually to its collapse, as a result of a small (or an absence of adequate) motion of your hand or due to any tiny whiff of air. This is the proximal cause of the collapse. But the fundamental cause should be attributed to the intrinsically unstable position.

What is then the origin of the maturing instability? Many studies have suggested that bubbles result from the over-optimistic expectation of future earnings and history provides a significant number of examples of bubbles driven by such unrealistic expectations (Kindleberger and Aliber, 2005, Sheffrin, 2005, Sornette, 2003a). These studies and many others show that bubbles are initially nucleated at times of burgeoning economic fundamentals in so-called “new economy” climates. This vocable refers to new opportunities and/or new technological innovations. But, because there are large uncertainties concerning present values of the economies that will result from the present innovations, investors are more prone to influences from their peers (Hong et al., 2005), the media, and other channels that combine to build a self-reflexive climate of (over-)optimism (Umpleby, 2007). In particular, these interactions may lead to significant imitation, herding and collective behaviors. Herding due to technical as well as behavioral mechanisms creates positive feedback mechanisms, which lead to self-organized cooperation and the development of possible instabilities or to the “building of castles in the air”, to paraphrase Malkiel (1990). This idea is probably best exemplified in the context of the Internet bubble culminating in 2000 or the recent the CDO bubble in the USA peaking in 2007, where the new economies where the Internet or complex derivatives on sub-prime mortgages building on accelerating real-estate valuations.

Based on these ideas, the present paper adds to the literature by providing a detailed analysis of how the proximate triggering factor of a crash might relate to its fundamental mechanism in terms of a global cooperative herding mechanism. In particular, we rationalize the finding of Cutler et al. (1989) that exogenous news are responsible for no more than a third of the variance of the returns and that major financial crises are not preceded by any particular dramatic news.

In a nutshell, our multi-period many agent-based model is designed as follows. At each time step t, each investor forms an opinion on the next-period value of a single stock traded on the market. This opinion is shaped by weighting and combining three sources of information available at time t: (i) public information, i.e. news, (ii) information from their “friendship” network, promoting imitation and (iii) private information. In addition, we assume that the agents adapt their strategy, i.e., the relative importance of these different sources of information according to how well they performed in the past in predicting the next-time step valuation.

The a priori sensible qualities of our agents to gather all possible information and adapt to the recent past turn out to backfire. As their decisions are aggregated in the market, their collective impact leads to the nucleation of transient phases of herding with positive feedbacks. These nucleations occur as a result of random occurrences of short runs of same signed news. Our main findings can thus be summarized as follows: rallies and crashes occur due to random lucky or unlucky streaks of news that are amplified by the feedback of the news on the agents’ strategies into collective transient herding regimes. In addition to providing a convincing mechanism for bubbles and crashes, our model also provides a simple explanation for the excess volatility puzzle (Shiller, 1981).

Before presenting the model and its results, it is useful to compare it with the relevant literature and related models. A related line of research aims at developing a theory of “convention” (Orléan, 1984, Orléan, 1986, Orléan, 1989a, Orléan, 1989b, Orléan, 1991, Orléan, 1995), which emphasizes that even the concept of “fundamental value” may be a convention established by positive and negative feedbacks in a social system. A first notable implementation by Topol (1991) proposes a model with an additive learning process between an ‘agent-efficient’ price dynamics and a mimetic contagion dynamics. Similar to our own set-up, the agents of Topol (1991) adjust their bid-ask prices by combining the information from the other buyers’ bid prices, the other sellers’ ask prices and the agent's own efficient price corresponding to his knowledge of the economic fundamentals. Topol (1991) shows that mimetic contagion provides a mechanism for excess volatility. Another implementation of the concept of convention by Wyart and Bouchaud (2007) shows that agents who use strategies based on the past correlations between some news and returns may actually produce by their trading decisions the very correlation that they postulated, even when there is no a priori economic basis for such correlation. The fact that agents trade on the basis of how the information forecasts the return is reminiscent of our model, with however several important differences. The first important conceptual change is that Wyart and Bouchaud (2007) use a representative agent approach (in contrast with our heterogeneous agent framework), so that effect of imitation through the social network is neglected. The second difference is in the agent's calculation of the correlation to adapt their strategies. In Wyart and Bouchaud (2007), agents’ strategies are controlled by the correlation between the news and the return resulting immediately from their aggregate action based on those news (taking into account the agents’ own impact). Our agents’ strategies are determined by the correlation between their information and the return one time step later, which embodies the more realistic situation, in which a postion first has to be open and then closed a time step later for the trade's payoff to be observed.

Another closely related line of research is known as “information cascades”. According to (Bikhchandani et al., 1992), “an informational cascade occurs when it is optimal for an individual having observed the action of those ahead of him, to follow the behavior of the preceding individual without regard to his own information”. In these models, agents know that they have only limited information and use their neighbors actions in order to complement their information set. Bikhchandani et al. (1992) showed that the fact that agents use the decisions of other agents to make their own decision will lead with probability 1 to an informational cascade under conditions where the decisions are sequential and irreversible. This model was later generalized by Orléan (1995) into a non-sequential version, where informational cascades were still found to be possible.

The concept of information cascades is not new in modeling bubbles. Chari and Kehoe (2003) developed a model where agents try to compensate their uncertainty about the a priori fixed payoff of an asset by observing all other agents’ actions. In our model, agents are also using the opinions of their neighbors to determine how to act but the reason behind this is different. Our agents are not so much interested in the fundamental value of the stock, but more in its future directions. They try to buy the asset before its price rises and sell before it falls, making profit from the difference in the price. The true underlying equilibrium value is not the only important information to them, and they are more clever than purely fundamental value investors. They recognize that fundamental value is just one component among others that will set the market price. They include the possibility that the price may deviate from fundamental value, due to other behavioral factors. And they try to learn and adapt to determine what are the dominant factors. In principle, they should be able to discover the fundamental value and converge to its equilibrium. But it is a fact that they do not in some circumstances, due to the amplification of runs of positive or negative news in the presence of their collective behavior when sufficiently strong. In the “information cascade” set-up, one assume that the “truth” exists, that there is a true fundamental price or a correct choice to be made which is exogenously given, and agents have no influence on the outcome. In our model however the outcome, whether selling or buying a stock was the right choice, is endogenously emerging from the aggregated choices of all agents. There is no a priori right or wrong answer, it is decided during the process. Moreover, the strength of the influence of her neighbors onto a given agent is not constant in time. This influence by the social environment evolves in time according to its past relevance and success.

A model for the formation of a boom followed by a crash was also developed by Veldkamp (2005), where the price of an unknown company can rise only slowly due to infrequent news coverage. If the company performs well resulting in a slow boom, its susceptibility towards news increases as the media become more aware of the successful company so that, eventually, a single piece of bad news can induce a sudden crash. Although the subject of research is the same, we show how a boom can also be formed with news not being constantly positive and that a single piece of bad news does not necessarily lead to the burst a bubble.

The endogenization of the sources of information onto the decisions of the agents is inspired by the model of Zhou and Sornette (2007), which focuses on herding and on the role of “irrational” mis-attribution of price moves to generate most of the stylized facts observed in financial time series. Similarly to their model as well as many other artificial financial market models investigating the interaction between trading agents, our model is based on the Ising model, one of the simplest models describing the competition between the ordering force of imitation or contagion and the disordering impact of private information or idiosyncratic noise that promotes heterogeneous decisions (McCoy and Wu, 1973).

Our paper is organized into four sections. In Section 2, the detailed working of the model is presented. The results are shown and discussed in Section 3 and Section 4 concludes.

Section snippets

General set-up

We consider a fixed universe of N agents who are trading (buying or selling) a single asset, which can be seen as a stock, the market portfolio or any other exchange traded asset. This asset is traded on an organized market, coordinated by a market maker. At each time step, agents have the possibility to either trade or to remain passive. The trading decision of a given agent is based on her opinion on the future price development.

To form their opinion, agents use information from three

General properties

Our model is an idealized “test tube” representation of a financial market and given the simplifications put into the model, we do not aim at reproducing faithful statistical characteristics of realistic price dynamics. Our objective is to obtain an understanding of how the interplay of news, herding and private information can lead to the formation of bubbles and crashes. We first point out a few properties of the model, that derive straightforwardly from our set-up.

Because we model a closed

Conclusion

In this paper, we have addressed two major questions:

  • -

    Why do bubbles and crashes exist?

  • -

    How to they emerge?

We approached these questions by constructing a model of bounded rational, locally optimizing agents, trading a single asset with a very parsimonious strategy. The actions of the agents are determined by their anticipation of the future price changes, which is based on three different sources of information: private information, public information (news) and information from their neighbors

Acknowledgments

We would like to thank Wei-Xing Zhou for invaluable discussions during the course of the project and Gilles Daniel and Ryan Woodard for a critical reading of the manuscript.

References (61)

  • D. Abreu et al.

    Bubbles and crashes

    Econometrica

    (2003)
  • R. Barro et al.
  • A. Beja et al.

    On the dynamic behavior of prices in disequilibrium

    The Journal of Finance

    (1980)
  • S. Bikhchandani et al.

    A theory of fads, fashion, custom, and cultural change as informational cascades

    The Journal of Political Economy

    (1992)
  • G.-I. Bischi et al.

    Herd behavior and nonfundamental asset price fluctuations in financial markets

    Macroeconomic Dynamics

    (2006)
  • W.A. Brock et al.

    Discrete choice with social interactions

    The Review of Economic Studies

    (2001)
  • G. Caginalp et al.

    Financial bubbles: excess cash, momentum, and incomplete information

    The Journal of Psychology and Financial Markets

    (2001)
  • E. Callen et al.

    A theory of social imitation

    Physics Today

    (1974)
  • G.J. Chaitin

    Algorithmic Information Theory, Cambridge Tracts in Theoretical Computer Science

    (1987)
  • V.V. Chari et al.

    Hot money

    Journal of Political Economy

    (2003)
  • R. Crane et al.

    New power law signature of media exposure in human response waiting time distributions

    Physical Review E

    (2010)
  • D.M. Cutler et al.

    What Moves Stock Prices? Tech. Rep. 2538

    (July 1989)
  • W.C. Dudley
  • J. Galbraith

    The Great Crash, 1929

    (1997)
  • H. Hong et al.

    Thy neighbor's portfolio: word-of-mouth effects in the holdings and trades of money managers

    Journal of Finance

    (2005)
  • R.N. Hussam et al.

    Thar she blows: can bubbles be rekindled with experienced subjects?

    American Economic Review

    (2008)
  • A. Johansen et al.

    Stock market crashes are outliers

    European Physical Journal B

    (1998)
  • A. Johansen et al.

    Large stock market price drawdowns are outliers

    Journal of Risk

    (2001)
  • G. Kaminsky

    Currency and Banking Crises: The Early Warnings of Distress. Social Science Research Network Working Paper Series

    (December 1998)
  • G.G. Kaufman

    Asset Price Bubbles: Implications Monetary and Regulatory Policies (Research in Financial Services: Private and Public Policy)

    (2001)
  • Cited by (0)

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