compete for survival against other agents’ ideas or mental models—a world
that is both evolutionary and complex.
REFERENCES
Arthur, W. Brian. “On Learning and Adaptation in the Economy.” Santa Fe Institute (Santa Fe, NM) Paper 92-07-038, 1992.
Bower, Gordon H., and Hilgard, Ernest R. Theories of learning. Englewood Cliffs, NJ: Prentice-Hall, 1981.
De Groot, Adriann. Thought and choice in chess, Psychological Studies, No.
4. Paris: Mouton, 1965.
Feldman, Julian. “Computer Simulation of Cognitive Processes,” in Harold Borko, ed., Computer applications in the behavioral sciences. Englewood Cliffs, NJ: Prentice-Hall, 1962, pp. 336–359.
Grannan, E. R., and Swindle, G. H. “Contrarians and Volatility Clustering.” Santa Fe Institute (Santa Fe, NM) Paper 94-03-010, 1994.
Holland, John H., Holyoak, Keith J., Nisbett, Richard E., and Thagard, Paul R.
Induction. Cambridge, MA: MIT Press, 1986.
Koza, John. Genetic programming. Cambridge, MA: MIT Press, 1992.
Rumelhart, David. “Schemata: The Building Blocks of Cognition,” in R. Spiro, B.
Bruce, and W. Brewer, eds., Theoretical issues in reading comprehension. Hillside, NJ: Erlbaum, 1980, pp. 33–58.
Sargent, Thomas J. Bounded rationality in macroeconomics. New York: Oxford University Press, 1994.
Schank, R., and Abelson, R. P. Scripts, plans, goals, and understanding: An inquiry into human knowledge structures. Hillside, NJ: Erlbaum, 1977.
[ 38 ] Complexity and the Economy
CHAPTER 3
Asset Pricing under Endogenous
Expectations in an Artificial Stock Market
W. BRIAN ARTHUR, JOHN H. HOLL AND, BL AKE
LEBARON, RICHARD PALMER, AND PAUL TAYLER*
This paper grew out of our early experiments with agent-based modeling at Santa Fe in the late 1980s. Standard neoclassical stock-market models at the time assumed identical investors who used identical forecasts (expectations) that were on average correct. But while the theory was elegant, its assumptions were not realistic. Further, it ruled out the possibility of bubbles and crashes, technical trading (using past price patterns to forecast profitably), correlated prices and volatility, and trading in significant volume—all phenomena that appeared in real markets. We created a different model that would play out within the computer, where our investors would be “artificial agents” (small computer programs) that would differ, would have to create their own forecasts, and would have to learn which worked and didn’t work as they gained market experience.
Experiments with our artificial market showed that under realistic parameters it exhibited a “complex regime,” where bubbles and crashes, technical trading, correlated prices and volatility, and realistic trading volume all appeared. Under a narrower—but unrealistic—set of parameters, it exhibited a “neoclassical regime,” and upheld the standard theory. The paper appeared in Economic Notes, 26, 297–330, 1997, and in the volume The Economy as an Evolving Complex System II, edited by David Lane, Steven Durlauf, and myself.1 Variations of our model, the Santa Fe Artificial Stock Market, have been widely used in economics.
* Arthur is Citibank Professor, Santa Fe Institute; Holland is Professor of Computer Science and Engineering, University of Michigan, and with the Santa Fe Institute; LeBaron is Associate Professor of Economics, University of Wisconsin, Madison; Palmer is Professor of Physics, Duke University, and with the Santa Fe Institute; Tayler is with the Department of Computer Science, Brunel University, London.
1. For an earlier version see Palmer et al., 1994.
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.2 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 tem-
porary 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,3 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.4
While few academics would be willing to assert that the market has a per-
sonality or experiences moods, the standard economic view has in recent
years begun to change. The crash of 1987 damaged economists’ beliefs that
sudden price changes reflect rational adjustments to news in the market: sev-
eral studies failed to find significant correlation between the crash and market information issued at the time (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 (see Leroy and Porter, 1981; Shiller, 1981,
1989)—and both show significant autocorrelation (see Bollerslev et al., 1990;
Goodhart et al., 1995). Stock returns also contain small, but significant serial correlations (see Fama and French, 1988; Lo and MacKinlay, 1988; Poterba
2. For the classic statement see Lucas (1978) or Diba and Grossman (1988).
3. For evidence see Frankel and Froot (1990).
4. 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.”
[ 40 ] Complexity and the Economy
and Summers, 1988; Summers, 1986). Certain technical-trading rules produce statistically significant, if modest, long-run profits (see Brock et al., 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 (see
Grossman, 1976; Grossman and Stiglitz, 1980; Milgrom and Stokey, 1982). 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 per-
sist, 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 nonrational traders in the market to prime these strategies (see 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 that their forecasts are erroneous; and the existence 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 formu-
late 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 mar-
ket. 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 nonidentical, 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
a sse t Pr icing under endogenous exPectat ion s [ 41 ]
the market with an eye to discovering 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 cocreate 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 expec-
tations 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 mar-
ket,” 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 coevolve in a world they
co-create.
The natural question is whether these heterogeneous expectations
coevolve into homogeneous rational-expectations beliefs, upholding the
efficient-market theory, or whether richer individual and collective behav-
ior 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, is too complicated to allow
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.5
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 cir-
cumstances, we initiate our traders with heterogeneous beliefs clustered ran-
domly in an interval near homogeneous rational expectations. We find that if
our agents very slowly adapt their forecasts to new observations of the mar-
ket’s behavior, the market converges to a rational-expectations regime. Here
“mutant” expectations cannot get a profitable footing; and technical trading,
5. For an earlier report on the SFI artificial stock market, see Palmer et al. (1994).
[ 42 ] Complexity and the Economy
bubbles, crashes, and autocorrelative behavior do not emerge. Trading volume remains low. The efficient-market theory prevails.
If, on the other hand, we allow the traders to adapt to new market obser-
vations at a more realistic rate, heterogeneous beliefs persist, and the mar-
ket self-organizes into a complex regime. A rich “market psychology”—a
rich set of expectations—becomes observable. Technical trading emerges
as a profitable activity, and temporary bubbles and crashes occur from
time to time. Trading volume is high, with times of quiescence alternating
with times of intense market activity. The price time series shows persis-
tence in volatility, the characteristic GARCH signature of price series from
actual financial markets. And it shows persistence in trading volume. And
over the period of our experiments, at least, individual behavior evolves
continually and does not settle down. In this regime, the traders’ view is
upheld.
In what follows, we discuss first the rationale for our endogenous-
expectations approach to market behavior; and introduce the idea of col-
lections of conditional expectational hypotheses or “predictors” to imple-
ment this. We next set up the computational model that will form the basic
framework. We are then in a position to carry out and describe the computer
experiments with the model. Two final sections discuss the results of the
experiments, compare our findings with other modern approaches in the lit-
erature, and summarize our conclusions.
2. WHY INDUCTIVE REASONING?
Before proceeding, we show that once we introduce heterogeneity of agents,
deductive reasoning on the part of agents fails. We argue that in the absence
of deductive reasoning, agents must resort to inductive reasoning, which is both natural and realistic in financial markets.
Forming Expectations by Deductive Reasoning: An Indeterminacy
We make our point about the indeterminacy of deductive logic on the part
of agents using a simple arbitr
age pricing model, avoiding technical details
that will be spelled out later. (This pricing model is a special case of our model in Section 3, assuming risk coefficient λ arbitrarily close to 0, and gaussian expectational distributions). Consider a market with a single security that
provides a stochastic payoff or dividend sequence { d }, with a risk-free outside t
asset that pays a constant r units per period. Each agent i may form individual expectations of next period’s dividend and price, E [ d | I ] and E [ p | I ], i
t+1
t
i
t+1
t
with conditional variance of these combined expectations, σ 2 , given current i t
,
a sse t Pr icing under endogenous exPectat ion s [ 43 ]
market information I . Assuming perfect arbitrage, the market for the asset t
clears at the equilibrium price:
p = β∑
( [
(1)
1 |
]
[ 1| ])
,
+
+
t
wj t Ej dt
It
Ej pt+ It
j
In other words, the security’s price p is bid to a value that reflects the cur-t
rent (weighted) average of individuals’ market expectations, discounted by
the factor β = 1 / (1 + r), with weights w = (1 / 2 ) / ∑ 1 / 2
σ
σ
j, t
j, t
k
k, t the relative
“confidence” placed in agent j’s forecast.
Now, assuming intelligent investors, the key question is how the individual
dividend and price expectations E [ d | I ] and E [ p | I ], respectively, might i
t+1
t
i
t+1
t
be formed. The standard argument that such expectations can be formed ratio-
nally (i.e., using deductive logic) goes as follows. Assume homogeneous investors who (i) use the available information I identically in forming their dividend t
expectations, and (ii) know that others use the same expectations. Assume
further that the agents (iii) are perfectly rational (can make arbitrarily difficult logical inferences), (iv) know that price each time will be formed by arbitrage as in Eq. (1), and (v) that (iii) and (iv) are common knowledge. Then, expectations of future dividends E [ d | I ] are by definition known, shared, and identical.
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