Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

TL;DR: In this paper, the authors propose a theory of asset pricing based on heterogeneous agents who continually adapt their expectations to the market that these expectations aggregatively create, and explore the implications of this theory computationally using Santa Fe artificial stock market.

Abstract: This chapter proposes a theory of asset pricing based on heterogeneous agents who continually adapt their expectations to the market that these expectations aggregatively create. It explores the implications of this theory computationally using Santa Fe artificial stock market. Computer experiments with this endogenous-expectations market explain one of the more striking puzzles in finance: that market traders often believe in such concepts as technical trading, "market psychology," and bandwagon effects, while academic theorists believe in market efficiency and a lack of speculative opportunities. Academic theorists and market traders tend to view financial markets in strikingly different ways. Standard (efficient-market) financial theory assumes identical investors who share rational expectations of an asset's future price, and who instantaneously and rationally discount all market information into this price. While a few academics would be willing to assert that the market has a personality or experiences moods, the standard economic view has in recent years begun to change.

Summary

Introduction

• The authors propose a theory of asset pricing based on heterogeneous agents who continually adapt their expectations to the market that these expectations aggregatively create.
• Many believe that technical trading is profitable2, that something definable as a “market psychology” exists, and that herd effects unrelated to market news can cause bubbles and crashes.
• The natural question is whether these heterogeneous expectations co-evolve into homogeneous rational-expectations beliefs, upholding the efficient-market theory, or whether richer individual and collective behavior emerges, upholding the traders’ viewpoint and explaining the empirical market phenomena mentioned above.

2. Why Inductive Reasoning?

• Before proceeding, the authors show that once they introduce heterogeneity of agents, deductive reasoning on the part of agents fails.
• The authors argue that in the absence of deductive reasoning, agents must resort to inductive reasoning, which is both natural and realistic in financial markets.

A. Forming Expectations by Deductive Reasoning: an Indeterminacy

• The authors make their point about the indeterminacy of deductive logic on the part of agents using a simple arbitrage pricing model, avoiding technical details that will be spelled out later.
• The second, constant-exponential-growth solution is normally ruled out by an appropriate transversality condition.
• Under heterogeneity, however, not only is there no objective means by which others’ dividend expectations can be known, but attempts to eliminate the other unknowns, the price expectations, merely lead to the repeated iteration of subjective expectations of subjective expectations (or equivalently, subjective priors on others’ subjective priors)—an infinite regress in subjectivity.
• The authors can therefore easily imagine swings and swift transitions in investors’ beliefs, based on little more than ephemera—hints and perceived hints of others’ beliefs about others’ beliefs.
• Infinitely intelligent agents cannot form expectations in a determinate way.

B. Forming Expectations by Inductive Reasoning

• They may observe market data, they may contemplate the nature of the market and of their fellow investors.
• In what follows then, the authors assume that each agent acts as a market “statistician.”6.
• It is, in micro-scale, the scientific method.
• Each inductively-rational agent generates multiple expectational models that “compete” for use within his or her mind, and survive or are changed on the basis of their predictive ability.

B. Modeling the Formation of Expectations

• The authors now break from tradition by allowing their agents to form their expectations individually and inductively.
• Each agent therefore has the ability to “recognize” different sets of states of the market, and bring to bear appropriate forecasts, given these market patterns.
• A condition array matches or “recognizes” the current market state if all its 0’s and 1’s match the corresponding bits for the market state with the #’s matching either a 1 or a 0.
• He forecasts next period’s price and dividend by combining statistically the linear forecast of the H most accurate of these active predictors, and given this expectation and its variance, uses (5) to calculate desired stock holdings and generate an appropriate bid or offer.
• They are therefore less likely to survive in the competition among predictors.

A. Experimental Design

• The authors now explore computationally the behavior of their endogenous-expectations market in a series of experiments.
• Bits 7-10 are “technical trading” bits which indicate whether a trend in the price is under way.
• They convey no useful market information, but can tell us the degree to which agents act upon useless information at any time.
• The authors find indeed that such predictions are upheld—that the model indeed reproduces the homogeneous rational expectations equilibrium—which 13 assures us that the computerized model, with its expectations, demand functions, aggregation, market clearing, and timing sequence, is working correctly.
• In the second test, the authors show the agents a given dividend sequence and a calculated h.r.e.e. price series that corresponds to it, and test whether they individually learn the correct forecasting parameters.

B. The Experiments

• The authors now run two sets of fundamental experiments with the computerized model, corresponding respectively to slow and medium rates of exploration by agents of alternative expectations.
• The authors now describe these two sets of experiments and the two regimes or phases of the market they induce.
• Thus technical analysis can emerge if trendfollowing (or mean-reversion) beliefs are by chance generated in the population, and if random perturbations in the dividend sequence activate them and subsequently validate them.
• One of the striking characteristics of actual financial markets is that both their price volatility and trading volume show persistence or autocorrelation.
• (Of course, on this very short time-lag scale, these avalanches occur not through the genetic algorithm but by agents changing their active predictors.).

6. Discussion

• The authors find experimentally by varying both the model’s parameters and the expectational-learning 13 For a discussion of volatility clustering in a different model, see Youssefmir and Huberman, 1995; and also Grannan and Swindle, 1994.
• If a clever meta-expectational model was “out there” that might exploit others’ expectations, such a model would, by aggregation of others’ expectations, be a complicated nonlinear function of current market information.
• Thus market signals must be of value to be used, and technical trading emerges only because such market signals induce mutually supporting expectations that condition themselves on these market signals.
• It might appear that, because their agents have equal abilities as statisticians, they are irrational to trade at all.
• In actual financial markets, investors do not perfectly optimize portfolios, nor is full market clearing achieved each period.

7. Conclusion

• In asset markets, agents’ forecasts create the world agents are trying to forecast.
• The market becomes driven by expectations that adapt endogenously to the ecology these expectations cocreate.
• Experiments with a computerized version of this endogenous-expectations market explain one of the more striking puzzles in finance: Standard theory tends to see markets as efficient, with no rationale for herd effects, and no possibility of systematic speculative profit, whereas traders tend to view the market as exhibiting a “psychology,” bandwagon effects, and opportunities for speculative profit.
• And prices show statistical features—in particular, GARCH behavior—characteristic of actual market data.
• 15 This point was also made by Soros (1992) whose term r flexivity the authors adopt.

Figures

Figure 6. Crosscorrelation of trading volume with volatility, in the rational-expectations and complex regimes, and in IBM daily returns

Table 1. Returns and volume statistics (medians) in the two regimes collected for 25 experiments after 250,000 periods.

Figure 3. Number of technical-trading bits that become set as the market evolves, (median over 25 experiments in the two regimes).

Figs 2. Deviations of the price series in the complex regime from fundamental value. The bottom graph shows the difference between the two price series in Fig 1. (with the complex series rescaled to match the r.e.e. one and the difference between the two

Fig. 4. Autocorrelation of volatility in rational-expectations and complex regimes, and in IBM daily returns

Fig. 5. Autocorrelation of trading volume in the rational-expectations and complex regimes, and in IBM daily returns

Figs 1a and 1b. Rational-expectations price vs. price in the rich psychological regime. The two price series are generated on the same random dividend series. The upper is the homogeneous r.e.e. price, the lower is the price in the complex regime. The higher variance in the latter case causes the lower price through risk aversion.

Full text

Asset Pricing Under Endogenous Expectations
in an Artificial Stock Market
by
W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler
*
Dec 12, 1996
*
All authors are affiliated with the Santa Fe Institute, where Arthur is Citibank Professor. In addition, Holland is
Professor of Computer Science and Engineering, University of Michigan, Ann Arbor; LeBaron is Associate Professor
of Economics, University of Wisconsin; Palmer is Professor of Physics, Duke University; and Tayler is with the Dept.
of Computer Science, Brunel University, London.

2
Asset Pricing Under Endogenous Expectations in an Artificial Stock Market
Abstract
We propose a theory of asset pricing based on heterogeneous agents who continually adapt their
expectations to the market that these expectations aggregatively create. And we explore the implications of
this theory computationally using our Santa Fe artificial stock market.
Asset markets, we argue, have a recursive nature in that agents’ expectations are formed on the basis
of their anticipations of other agents’ expectations, which precludes expectations being formed by
deductive means. Instead traders continually hypothesize—continually explore—expectational models, buy
or sell on the basis of those that perform best, and confirm or discard these according to their performance.
Thus individual beliefs or expectations become endogenous to the market, and constantly compete within
an ecology of others’ beliefs or expectations. The ecology of beliefs co-evolves over time.
Computer experiments with this endogenous-expectations market explain one of the more striking
puzzles in finance: that market traders often believe in such concepts as technical trading, “market
psychology, ” and bandwagon effects, while academic theorists believe in market efficiency and a lack of
speculative opportunities. Both views, we show, are correct, but within different regimes. Within a regime
where investors explore alternative expectational models at a low rate, the market settles into the rational-
expectations equilibrium of the efficient-market literature. Within a regime where the rate of exploration of
alternative expectations is higher, the market self-organizes into a complex pattern. It acquires a rich
psychology, technical trading emerges, temporary bubbles and crashes occur, and asset prices and trading
volume show statistical features—in particular, GARCH behavior—characteristic of actual market data.
Acknowledgments
We are grateful to Kenneth Arrow, Larry Blume, Buz Brock, John Casti, Steven Durlauf, David Easley,
David Lane, Ramon Marimon, Tom Sargent, and Martin Shubik for discussions of the arguments in this
paper, and of the design of the artificial market. All errors are our own.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market
by
W. Brian Arthur, John H. Holland, Blake LeBaron, Richard Palmer, and Paul Tayler
Introduction
Academic theorists and market traders tend to view financial markets in strikingly different ways.
Standard (efficient-market) financial theory assumes identical investors who share rational expectations of
an asset’s future price, and who instantaneously and rationally discount all market information into this
price.
1
It follows that no opportunities are left open for consistent speculative profit, that technical trading
(using patterns in past prices to forecast future ones) cannot be profitable except by luck, that temporary
price overreactions—bubbles and crashes—reflect rational changes in assets’ valuations rather than sudden
shifts in investor sentiment. It follows too that trading volume is low or zero, and that indices of trading
volume and price volatility are not serially correlated in any way. The market, in this standard theoretical
view, is rational, mechanistic, and efficient. Traders, by contrast, often see markets as offering speculative
opportunities. Many believe that technical trading is profitable
2
, that something definable as a “market
psychology” exists, and that herd effects unrelated to market news can cause bubbles and crashes. Some
traders and financial writers even see the market itself as possessing its own moods and personality,
sometimes describing the market as “nervous” or “sluggish” or “jittery.” The market in this view is
psychological, organic, and imperfectly efficient. From the academic viewpoint traders with such beliefs—
embarrassingly the very agents assumed rational by the theory—are irrational and superstitious. From the
traders’ viewpoint, the standard academic theory is unrealistic and not borne out by their own perceptions.
3
While few academics would be willing to assert that the market has a personality or experiences
moods, the standard economic view has in recent years begun to change. The crash of 1987 damaged
economists’ beliefs that sudden prices changes reflect rational adjustments to news in the market: several
studies failed to find significant correlation between the crash and market information issued at the time
1
For the classic statement see Lucas (1978), or Diba and Grossman (1988).
2
For evidence see Frankel and Froot (1990).
3
To quote one of the most successful traders, George Soros (1994): “this [efficient market theory]
interpretation of the way financial markets operate is severely distorted. … It may seem strange that a
patently false theory should gain such widespread acceptance.”

2
(e.g. Cutler et al. 1989). Trading volume and price volatility in real markets are large—not zero or small,
respectively, as the standard theory would predict (Shiller, 1981, 1989; Leroy and Porter, 1981)—and both
show significant autocorrelation (Bollerslev et al., 1990; Goodhart and O’Hara, 1995). Stock returns also
contain small, but significant serial correlations (Fama and French, 1988; Lo and Mackinlay, 1988;
Summers, 1986; Poterba and Summers, 1988). Certain technical-trading rules produce statistically
significant, if modest, long-run profits (Brock, Lakonishok, and LeBaron, 1991). And it has long been
known that when investors apply full rationality to the market, they lack incentives both to trade and to
gather information (Milgrom and Stokey, 1982; Grossman 1976; Grossman and Stiglitz, 1980). By now,
enough statistical evidence has accumulated to question efficient-market theories and to show that the
traders’ viewpoint cannot be entirely dismissed. As a result, the modern finance literature has been
searching for alternative theories that can explain these market realities.
One promising modern alternative, the noise-trader approach, observes that when there are “noise
traders” in the market—investors who possess expectations different from those of the rational-expectations
traders—technical-trading strategies such as trend chasing may become rational. For example, if noise
traders believe that an upswing in a stock’s price will persist, rational traders can exploit this by buying into
the uptrend thereby exacerbating the trend. In this way positive-feedback trading strategies—and other
technical-trading strategies—can be seen as rational, as long as there are non-rational traders in the market
to prime these strategies (De Long et al. 1990a, 1990b, 1991; Shleifer and Summers, 1990). This
“behavioral” noise-trader literature moves some way toward justifying the traders’ view. But it is built on
two less-than-realistic assumptions: the existence of unintelligent noise traders who do not learn over time
their forecasts are erroneous; and of rational players who possess, by some unspecified means, full
knowledge of both the noise traders’ expectations and their own class’s. Neither assumption is likely to
hold up in real markets. Suppose for a moment an actual market with minimally intelligent noise traders.
Over time, in all likelihood, some would discover their errors and begin to formulate more intelligent (or at
least different) expectations. This would change the market, which means that the perfectly intelligent
players would need to readjust their expectations. But there is no reason these latter would know the new
expectations of the noise-trader deviants; they would have to derive their expectations by some means such
as guessing or observation of the market. As the rational players changed, the market would change again.
And so the noise traders might again further deviate, forcing further readjustments for the rational traders.
Actual noise-trader markets, assumed stationary in theory, would start to unravel; and the perfectly rational
traders would be left at each turn guessing the changed expectations by observing the market.
Thus noise-trader theories, while they explain much, are not robust. But in questioning such theories
we are led to an interesting sequence of thought. Suppose we were to assume “rational,” but non-identical,
agents who do not find themselves in a market with rational expectations, or with publicly-known
expectations. Suppose we allowed each agent continually to observe the market with an eye to discovering

3
profitable expectations. Suppose further we allowed each agent to adopt these when discovered and to
discard the less profitable as time progressed. In this situation, agents’ expectations would become
endogenous—individually adapted to the current state of the market—and they would co-create the market
they were designed to exploit. How would such a market work? How would it act to price assets? Would it
converge to a rational-expectations equilibrium—or would it uphold the traders’ viewpoint?
In this paper we propose a theory of asset pricing that assumes fully heterogeneous agents whose
expectations continually adapt to the market these expectations aggregatively create. We argue that under
heterogeneity, expectations have a recursive character: agents have to form their expectations from their
anticipations of other agents’ expectations, and this self-reference precludes expectations being formed by
deductive means. So, in the absence of being able to deduce expectations, agents—no matter how
rational—are forced to hypothesize them. Agents therefore continually form individual, hypothetical,
expectational models or “theories of the market,” test these, and trade on the ones that predict best. From
time to time they drop hypotheses that perform badly, and introduce new ones to test. Prices are driven
endogenously by these induced expectations. Individuals’ expectations therefore evolve and “compete” in a
market formed by others’ expectations. In other words, agents’ expectations co-evolve in a world they co-
create.
The natural question is whether these heterogeneous expectations co-evolve into homogeneous
rational-expectations beliefs, upholding the efficient-market theory, or whether richer individual and
collective behavior emerges, upholding the traders’ viewpoint and explaining the empirical market
phenomena mentioned above. We answer this not analytically—our model with its fully heterogeneous
expectations it is too complicated to admit of analytical solutions—but computationally. To investigate
price dynamics, investment strategies, and market statistics in our endogenous-expectations market, we
perform carefully-controlled experiments within a computer-based market we have constructed, the SFI
Artificial Stock Market.
4
The picture of the market that results from our experiments, surprisingly, confirms both the efficient-
market academic view and the traders’ view. But each is valid under different circumstances—in different
regimes. In both circumstances, we initiate our traders with heterogeneous beliefs clustered randomly in an
interval near homogeneous rational expectations. We find that if our agents adapt their forecasts very
slowly to new observations of the market’s behavior, the market converges to a rational-expectations
regimes. Here “mutant” expectations cannot get a profitable footing; and technical trading, bubbles,
crashes, and autocorrelative behavior do not emerge. Trading volume remains low. The efficient-market
theory prevails.
4
For an earlier report on the SFI artificial stock market, see Palmer et al. (1994).

Frequently asked questions

Q1. What are the contributions in "Asset pricing under endogenous expectations in an artificial stock market" ?

The authors propose a theory of asset pricing based on heterogeneous agents who continually adapt their expectations to the market that these expectations aggregatively create. Both views, the authors show, are correct, but within different regimes. 

Q2. What are the future works in "Asset pricing under endogenous expectations in an artificial stock market" ?

Experiments with a computerized version of this endogenous-expectations market explain one of the more striking puzzles in finance: Standard theory tends to see markets as efficient, with no rationale for herd effects, and no possibility of systematic speculative profit, whereas traders tend to view the market as exhibiting a “ psychology, ” bandwagon effects, and opportunities for speculative profit. The authors show, without behavioral assumptions, that both views can be correct. Their endogenous-expectations market shows that heterogeneity of beliefs, deviations from fundamental trading, and persistence in time series can be maintained indefinitely in actual markets with inductively rational traders. 

Q3. What is the characteristic GARCH signature of price series from actual financial markets?

The price time series shows persistence in volatility, the characteristic GARCH signature of price series from actual financial markets. 

Q4. How is the genetic algorithm used in the slow-exploration-rate experiments?

In the medium-exploration-rate experiments, the genetic algorithm is invoked every 250 periods on average, crossover occurs with probability 0.1, and the predictors’ accuracy-updating parameter θ is set to 1/75.9 

Q5. What is the effect of homogeneous rational expectations on the market price?

The market price, in these experiments, converges rapidly to homogeneous rational expectations value adjusted for risk, even though the agents start with non rational expectations. 

Q6. What is the significance of the trend indicator in the complex regime?

In the complex regime, the trend indicator is significant (with t-value of 5.1 for the mean of the sample of 25 experiments), showing that the indicator does indeed carry useful market information. 

Q7. What is the way to test this conjecture?

One way to test this conjecture is to see whether autocorrelations increase as the predictor accuracy-updating parameter θ in (7) in Appendix A is increased. 

Q8. What is the significance of the Chi-square statistic in the Engle GARCH Test?

Their inductive market also shows persistence in volatility or GARCH behavior in the complex regime, Fig. 4, (with the Chi-square statistic in the Engle GARCH Test significant at the 95%18level). 

Q9. What are the phenomena that arise when individual expectations that involve trend following or mean reversion become?

These phenomena arise when individual expectations that involve trend following or mean reversion become mutually reinforcing in the population of expectations, and when market indicators become used as signaling devices that coordinate these sets of mutuallyreinforcing beliefs. 

Q10. What are the 12 binary descriptors that summarize the state of the market?

The 12 binary descriptors that summarize the state of the market are the following:1-6 Current price × interest rate/dividend > 0.25, 0.5, 0.75, 0.875, 1.0, 1.1257-10 Current price >