De la finance informelle à la microfinance

Documents de Travail du
Market Efficiencies and Market Risks
Pierre-André MAUGIS
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http://centredeconomiesorbonne.univ-paris1.fr/bandeau-haut/documents-de-travail/
November 22, 2010
Abstract
markets at an agent level, aiming to explain the non-stationarity of price
processes. All such papers agree that the heterogeneity of agents and of
pricing models creates a dynamics in terms of pricing models used that
explains not only the non-stationarity of price processes, but also stylised
facts such as bubbles and fat tails. However, all these results issue from
it, there is no proof of the aforementioned results outside of such spec-
that they transform into an output information, information according to
of agents is directly associated to the resulting quality of the information
propagation but poor information aggregation by the price, while het-
erogenous agents lead to good information aggregation but poor informa-
JEL: JELG14, JELG10, JELG01 and JELC73.
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Introduction
Many approaches have been considered to explain the randomness and unpre-
computer intensive by nature, consists of simulating the behaviour of many
[1] showed that such a market creates price paths similar to those issuing from
GARCH-type processes: non stationary, heavy tailed processes. More recently,
[24] presented a model where all the agents have access to the past values of
the price of a given asset, and trade according to one of two trading strategies:
a naive cost free strategy, or a costly rational strategy. At any point in time
agents may decide to use one or the other strategy. Through simulation, and
also through analytic demonstration relying on chaos theory, they show that the
added degree of freedom of allowing agents to select their model generates highly
complex, completely unpredictable stock prices dynamics. Even so, within the
in term of pricing model usage: the system oscillates between a state where
all the agents use the naive strategy and a state where all the agents use the
stock markets, but also in social experiments [22, 23, 37, 38]. In this light, it is
compete to choose the best strategy rather than a method to build lifelike price
processes.
From this point of view, heterogenous agent models resemble cascade models
up where agents have private information on the quality of the strategies, but
also knowledge of the choice made by the people before them in the sequence.
[2, 3, 13] showed that in such a situation, after enough time, all the agents will
choose the same strategy, hence the name of cascade. Moreover, this equilibrium
Hence, even if heterogenous agents models and cascade models share the
agents models, the state of all the agents using the same strategy is unstable,
in the heterogenous agents model, agents do not have perfect knowledge of the
strategies employed by the other agents, but only an approximation of it through
an educated guess made from the past realizations of the price, while they do
have this knowledge in the cascade model. This lack of precise information
causes the instability of the state where all agents use the same model in the
heterogenous agents model.
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Interestingly, both heterogenous agents and cascade models, beside sharing
the quality of the information is the central parameter of both the heterogenous
agents and the cascade models: this at both the agent level and the market
level. There are two types of information to consider: the information each
contains. So the key concept to be studied is the propagation of the information:
from the agent level to the market level, from the market level to the agent level
and in-between agents.
This was already noted by R. Shiller; in [35] he went even further stating
be summarised as follows: it is the means agents use to transform and interpret
the information they receive as well as communicate it to other agents, much
as language is our means of expressing and interpreting ideas and concepts.
to discuss at the most general level the quality of a system where agents in a
in order to acquire a point of view regarding the corresponding share price. We
from the aggregate of all the input information. This is what we referred to
The equivalence of the input
and output information could be claimed only if agents processed information
perfectly. This means that agents would not only have to be perfectly rational
assumption we do not make.
The second consequence is that each agent possesses a black box that trans-
chology. The second is how the agent processes the information: for instance
a set of formulas or a mathematical model. It is however best understood as
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a mathematical function, that takes as a parameter all the input information,
and that returns the output information: a predicted distribution of the price.
the current price and infer the sum of all the input information. The former is
information to all the agents. Mirroring these two concepts, we also use the
terms information risk and language risk to denote, respectively, cases where
tween the two becomes apparent in our framework only because we do not make
assumptions regarding the models used. We conduct our analysis by studying
two cases of our framework: the case where all the agents use the same model
and the case where multiple models exist.
First we present the mathematical framework used to conduct our analysis.
We then present the properties of the constructed framework and conclude with
a simulation study.
The Framework
be an increasing family of sets containing for each u all the
possible values concerning A at that time.
be an increasing family of sets containing for each u all the
possible values of the parameters concerning the economy at that time.
The following further explains our premise with examples of these mathematical
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Examples
t and Ft could be equal to R
, dt being the number of relevant economic
variables at time t, concerning the whole economy for Wt, and concerning
Ft is increasing with t because it contains the past realisations of the price
among other possible new variables of interest.
over S and as such are information on the price of the asset.
We will now put the framework into action. As our analysis is mostly static we
will omit the index t until part 4.2.
Price Construction
distribution representative of all of them. We make no assumptions regarding
hypothesis.
The mathematical framework we just presented has the advantage of being
and behavioural literatures, and also allows for agents to transform the infor-
the information, mh(ih) is then a prediction of the value of the price that the
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predictions and the price is the mean of this distribution. In the same fashion,
our framework is coherent with all of the works we cite.
By making the simplifying assumptions that the information is homogeneously
them as random variables of parameter the input information is also possible, for instance to
2A more general construction of the price is possible; for instance, it could be a ran-
can rewrite formula (1):
precisely to E [P I ]), one can see that in our framework the market can only be
to the actual information possessed by agents, I or input information. Instead
processing is discussed in [35, 36]. These works argue that our communication
A situation where a perfect model is used by all agents would allow for
in our framework is paradoxical. To say that a model is better than another
paradox. Indeed, in our framework the link between the information and the
standards. As such, all models are neither perfect nor imperfect: they represent
a relationship between the input information and the price.
that cannot be circumvented, we will focus on the quality of the subsequent
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the behaviour of the market, we will study the following two cases: the one
The One Model Case
If there exist only one model, formula (2) can be rewritten as:
Formula (3) shows that if the considered model was to change so would the
economy, most likely leading to a severe economic crisis [20, 30]. We distinguish
three risks associated with a reliance on only one model. A simple example
illustrates each.
Information Decay
of information decay. Consider the following:
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will lose all their capital as A fails to pay its debt. Other more subtle cases
gambling behaviour . . . [35, 36].
Information Misuses
where the model m regards a variable as relevant when it should not. Consider
the following:
R . A is a corn related company. Agents possess information concern-
model is that an increase in the number of sunspots n will increase the
quality of the corn. However, this is untrue.
In this example, the share price depends on n although the actual quality of the
corn does not. Hence, the price possesses a random component when it should
not. If the number of expected sunspots dramatically increases, mechanically
the price of the corn will increase even if there is no excessive demand for it,
creating a bubble. More subtle cases of information misuse can be found in
either Sunspot literature or in studies on magical thinking [9, 14, 26, 41].
Systemic Cascades
Thirdly, structural problems may occur when all agents
rely on a single model based on other agents behaviour. Indeed, such situation
can lead to self sustaining loops. Consider the following example:
}. m1 and m2 are two sub-
functions such that m2 is increasing in k, and m1 is positive.
A the price remains forever at 1 despite the other parameters. Similar results
These arguments show that the one model case is economically unstable.
Even though the information is correctly aggregated, the information according
to which the price is set is unreliable.
As described above, using only one
model m presents real risks, associated with the quality of its transformation
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From Model To Language
As previously shown, the capacity of agents to infer from the price information
present in the market is a possible escape from the no-trade theorems [7, 31]
and also insures liquidity [6, 10, 33]4. In the one model case, each agent can use
by any other agent in a trade. Consequently, m becomes a language5 in the
sense that it transmits meaning, ih, through a representation m(ih) that can be
information to all agents and vice versa.
tion also leads to bubbles caused by informational cascades. We consider this to be explained
by the arguments exposed in 4.1.1 and not by the readability itself.
5m is not strictly speaking a language because it does not possess a semantic structure or
grammar [28].
In the one model case, we consider that the use of m as a language and
the ensuing readability of the market indicates that there is proper information
propagation between agents, and that it returns to the market price its value as
an economical indicator. Consequently, we say that in the one model case the
The Multiple Model Case
This section studies the multiple model case. We consider this case not only
for the reasons explained above, but also because each agent may possess a
inside a market: for example, an agent may buy a share because he expects
its price to rise or it presents a good diversifying property for his portfolio.
He might also be looking for ownership of the company. More generally, the
purchase of an asset can be motivated by varied strategies each implying a
Here the price is set by formula (1):
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multiplicity of models make the risk described in 4.1.1 less likely. For the system
same failing. The probability of such an event decreases with the number of
models [12, 25]. The same can be said about the three risks in 4.1.1: information
decay, information misuses and systemic cascades. In this case the market is
information risk.
Readability
However, in the presence of multiple models using m as a language becomes less
only provides
Fixed Model
a parametric representation of the models in a framework similar to ours, the
[17, 18, 21, 25]. Consider the following example:
and p the past realisations of
holds a model mh as true, and he estimates this model with the estimator
as follows:
h,t are linear forms such that mh
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Assuming that the it are identically distributed of law I and mean i,
asymptotically we have:
1. In such cases a small variation in parameter can have large and complex
the dimension of the phase space, making the equilibrium path less and less
predictable.
6We use the shorthand: 1
results.
This is the case when agents can freely change models. For
suming a parametric form for the used models in a framework similar to ours, the
R . At time t agents know pd, the d past realisations of the price p
each model k is associated a performance measure Uk, which is a function
of pd. At time t, agent h measures U
) with noise, and we denote U
as this measure:
Where the noises h
follow the Gumbel type I extreme value distribu-
tion7 and are independent between agents. According to classical multi-
nomial choice theory [29]:
suming that H is large enough:
can be described by the following set of equations:
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System that can be re-written:
resulting time series are highly unpredictable. In practice, the price trajec-
tory becomes less and less predictable as the number of models increases, and
In both cases the market price becomes unpredictable and less informative to
the agents as the number of models increases because its interpretation becomes
ambiguous [16]. This causes multiple market issues: cascades, bubbles, crashes
and fat-tails. In the multiple model case, the market has a large language risk
because it is not readable by the agents and the price loses its use an economic
indicator.
Discussion
Our results can be summarised as follows:
Single Model
Multiple Models
more information is present in the multiple model case it is not accessible to
the agents. Whereas, in the one model case, information can be inferred by
agents but this information is unreliable. Overall there is an incompressible
second part of this paper where we will estimate how a group of agents would
Here we will present a simulation of the above described framework in the
multiple free model case. We are interested in how a group of agents naturally
assume that agents are unaware of the issue we are discussing and are looking
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for a natural equilibrium.
Agents possess ik with probability pk. Each bit of information predicts
that an event will occur with probability one. At the end of their turn,
agents can verify the validity of these bits of information.
model the other agents are using.
relative position. They trade only once.
price is the one that maximise the number of exchanges.
formation and still predicted the wrong price. They draw uniformly from
the set of models.
the information nor the models used here, we draw one permutation uniformly
sets the equilibrium price.
After the price has been selected, agents can verify whether they had true
because they have no reason to. Agents with good information but a wrong price
change models, since they have proof that their model predicted the wrong price
with good information. We then iterate the process.
Analysis
We use permutations to set the equilibrium because it permits to avoid giving
The aim in using Dirac masses is simplicity. It permits to make agent change
models after failing once instead of requiring them to fail enough times to es-
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analysis is not concerned with the intermediary steps.
This model and subsequent behaviour is similar to that of complex sys-
tems and evolutionary games [27, 39, 40]. It is also a Markov process. How-
ever, due to the use of random permutations, the transition matrix dimen-
k so that its computation
though closed formula can be produced, to compute the distribution of M at
Results and Discussion
Figure 1: Proportion of agents using each model in an example of 1000 steps
dynamic with 4 models, 3 bits of information, 500 agents and a probability of
having true information of 0.6.
state model, which is similar to a two state space Markov proces. This system is
entirely determined by the following two probabilities: pChaos, the probability
of remaining in the chaos state, and pCascade the probability of remaining in the
cascade state. We plot estimations of pCascade and pChaos in Figure 2.
According to 4.3, these results imply that the market oscillates between an
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present. In our simulation the higher the probability of having correct informa-
the number of agents (N ) or the number of bits of information (l) does not alter
this. Still, according to 4.3, this would indicate that the gain of information
automatically compensated by an increased information risk through a more
frequent and stable cascades toward a single model. Overall the market tends
This answers our question on how the number of models is naturally reg-
constant. The fact that, unknown to the agents, the market stabilises itself to
pCascade
Probobility of true information
Probobility of true information
N: Number of agents
N: Number of agents
with 500 simulations of 1000 steps.
Conclusion
We conclude that if one assumes that agents transform the input information
the number of models present. If given more reliable information, a group of
risk neutral agents tend to reduce the number of models used to increase the
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Within real markets there are none that fall strictly within the single model
stable and readable but prone to extreme variation, which is the case. Con-
versely, there is much more heterogeneity within short term models so that
the short term market should appear random. This is a widely accepted fact.
terpretation: agents assumed the information present to be reliable, which led
to the widespread use of a single model that weakened the economy in the face
of wrong information (subprimes) and contributed to systemic issues (the credit
crunch).
behaviour of the market, however, in practice these models intend to accurately
represent the market. This paper presents a new perspective where one per-
fect model or underlying mechanism to be modelled does not exist but instead
lows for the existence of a set of models that would make the economy behave as
the market while inducing a price-setting process coherent with this objective
behaviour. Lacking this coherence, the model would generate instability most
likely leading to a crisis and its abandonment.
However, this raises the question of how we want the markets to behave.
Should gains be possible only if one has informational advantage? Or should it
be a place where speculation is possible? We should note that in our framework
people in favour of speculation have the advantage. If they wish for the market
Further perspectives would be concerned by the following aspects of the
how the adoption of models propagate within a group. Both would need to rep-
resent the agent-to-agent relationship and would most likely rely on a network
Acknowledgments
with Miriam Maugis, Paul Daladier, Arun Chandrasekhar, Fabien Gensbittel,
Vincent Pons. Of course, any remaining errors are entirely ours.
References
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[1] Arthur, B., Holland, J., LeBaron, B., Palmer, R., Tayler, P., 1996. Asset
Santa Fe Institute Publication.
[2] Avery, C., Zemsky, P., 1998. Multidimensional uncertainty and herd behav-
[3] Banerjee, A. V., 1992. A simple model of herd behavior. The Quarterly
[4] Bikhchandani, S., 1992. A theory of fads, fashion, custom, and cultural
change as informational cascades. The Journal of Political Economy 100 (5),
[5] Bikhchandani, S., Hirshleifer, D., Welch, I., 1998. Learning from the behav-
ior of others: Conformity, fads, and informational cascades. The Journal of
[6] Brockman, P., Chung, D. Y., 2000. An empirical investigation of trading
on asymmetric information and heterogeneous prior beliefs. Journal of Em-
[7] Brunnermeier, M. K., 2003. Asset Pricing under Asymmetric Information.
Oxford Scholarship.
[8] Canina, L., Figlewski, S., 1993. The Informational Content of Implied
[9] Cass, D., Shell, K., 1983. Do Sunspots Matter? Journal of Political Econ-
[10] Chen, Q., Goldstein, I., Jiang, W., 2003. Price Informativeness and Invest-
ment Sensitivity to Stock Price. SSRN eLibrary.
[11] Christensen, B. J., Prabhala, N. R., 1998. The relation between implied
[12] Cremer, J., Garicano, L., Prat, A., 2007. Language and the Theory of the
[15] Dumas, B., Fleming, J., Whaley, R. E., 1998. Implied Volatility Functions:
[16] Easley, D., Maureen, O., 2010. Microstrutcture and ambiguity. The Journal
halshs-00544324, version 1 - 7 Dec 2010
of Finance.
[17] Evans, G., 1985. Expectational Stability and the Multiple Equilibria Prob-
lem in Linear Rational Expectations Models. Quarterly Journal of Eco-
[18] Evans, G., Honkapohja, S., 2001. Learning and Expectations in Macroe-
conomics. Princeton University Press.
[19] Fabozzi, F. J. (Ed.), 2008. Handbook of Finance: Financial Markets and
Instruments. J. Wiley Press.
[21] Guesnerie, R., 2002. Anchoring Economic Predictions in Common Knowl-
[22] Hommes, C., 1996. Financial markets as nonlinear adaptive evolutionary
nonlinear adaptive systems. Proceedings of the National Academy of Sci-
[24] Huang, H., Wang, D., Hommes, C., 2005. A robust rational route to ran-
domness in a simple asset pricing model. Journal of Economic Dynamics
[25] Huberman, B. A., Glance, N. S., 1993. Diversity and Collective Action.
[27] Johnson, N. F., Smith, D. M., Hui, P. M., 2006. Multi-Agent Complex
[28] Kello, C. T., 2004. Characterizing the evolutionary dynamics of language.
[29] Kenneth, T., 2003. Discrete choice methods with simulation. Cambridge
University Press.
[30] Merton, R., 1968. Social Theory and Social Structure. New York: Free
[31] Milgrom, P., Stokey, N., 1982. Information, Trade and Common Knowl-
[32] Morris, S., Song-Shin, H., 2002. Social Value of Public Information. The
[33] Romer, D., 2006. Rational Asset Price Movements Without News. NBER
Working Paper No. 4121.
halshs-00544324, version 1 - 7 Dec 2010
[34] Sethi, R., Franke, R., 1995. Behavioural Heterogeneity Under Evolution-
ary Pressure: Macroeconomic Implications of Costly Optimisation . The
[35] Shiller, R. J., 1995. Conversation, information, and herd behavior. The
[37] Sonnemans, J., Tuinstra, J., van de Velden, H., Homes, C., 2005. Coor-
dination of expectations in asset pricing experiments. Review of Financial
[38] Sonnemans, J., Tuinstra, J., van de Velden, H., Homes, C., 2008. Ex-
pectations and bubbles in asset pricing experiments. Journal of Economic
Behavior and Organization in press.
[39] Tarnita, C. E., Antal, T., Ohtsuki, H., Nowak, M. A., 2009. Evolution-
ary dynamics in set structured populations. Proceedings of the National
[40] Traulsena, A., Hauert, C., De Silva, H., Nowak, M., Sigmund, K., 2009.
Exploration dynamics in evolutionary games. Proceedings of the National
[41] Woodford, M., 1990. Learning to Believe in Sunspots. Econometrica 58 (2),
halshs-00544324, version 1 - 7 Dec 2010