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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 rationalexpectations 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 cocreate. 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 efficientmarket 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). 4 If, on the other hand, we allow the traders to adapt to new market observations at a more realistic rate, heterogeneous beliefs persist, and the market 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 persistence 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 collections of conditional expectational hypotheses or “predictors” to implement 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 literature, 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. A. 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 arbitrage 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 {dt }, with a risk-free outside asset that pays a constant r units per period. Each agent i may form individual expectations of next period’s dividend and price, Ei dt +1 It and Ei pt +1 It , with [ conditional variance of these combined expectations, 2 σ i,t , ] ∑ w j,t ( E j [ dt +1 It ] + E j [ pt +1 It ]) j ] given current market information It . Assuming perfect arbitrage, the market for the asset clears at the equilibrium price: pt = β [ (1) 5 In other words, the security’s price pt is bid to a value that reflects the current (weighted) average of individuals’ market expectations, discounted by the factor β = 1 (1 + r ) , with weights ( w j,t = 1 σ 2j,t ) ∑ 1 σ k,t2 , the relative “confidence” placed in agent j’s forecast. k Now, assuming intelligent investors, the key question is how the individual dividend and price expectations Ei dt +1 It and Ei pt +1 It might be formed. The standard argument that such expectations can [ ] [ ] be formed rationally (i.e.., using deductive logic) goes as follows. Assume homogeneous investors who (i) use the available information It identically in forming their dividend 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 (1), and (v) that (iii) and (iv) are common knowledge. Then expectations of future dividends Ei dt + k It are by definition [ ] known, shared, and identical. And homogeneity allows us to drop the agent subscript and set the weights to 1/N. It is then a standard exercise (see Diba and Grossman, 1988) to show that by setting up the arbitrage equation (1) for future times t+k, taking expectations across it, and substituting backward repeatedly for E pt + k It , agents can iteratively solve for the current price as 5 [ ] pt = β k ∞ ∑ E [ dt + k I t ] . (2) k =1 If the dividend expectations are unbiased, dividend forecasts will be upheld on average by the market and so the price sequence will be in rational-expectations equilibrium. Thus the price fluctuates as the information {I t} fluctuates over time, and it reflects “correct” or “fundamental” value, so that speculative profits are not consistently available. Of course, rational-expectations models in the literature are typically more elaborate than this. But the point so far is that if we are willing to adopt the above assumptions— which depend heavily on homogeneity—asset pricing becomes deductively determinate, in the sense that agents can, in principle at least, logically derive the current price. Assume now more realistically that traders are intelligent but heterogeneous—each may differ from the others. Now, the available shared information I t consists of past prices, past dividends, trading volumes, economic indicators, rumors, news, and the like. These are merely qualitative information plus data sequences, and there may be many different, perfectly defensible statistical ways, based on different assumptions and different error criteria to use them to predict future dividends (Arthur, 1992; Kurz, 1993). Thus there is no objectively laid-down expectational model that differing agents can coordinate upon, and so there is no objective means for one agent to know other agents’ expectations of future dividends. This is 5 The second, constant-exponential-growth solution is normally ruled out by an appropriate transversality condition. 6 sufficient to bring indeterminacy to the asset price in (1). But worse, the heterogeneous price expectations Ei pt +1 It are also indeterminate. For suppose agent i attempts rationally to deduce this expectation, he [ ] may take expectations across the market clearing equation (1) for time t+1:  Ei pt +1 It = β Ei   [ ]  ∑ {w j,t +1 ( E j [ dt + 2 It ] + E j [ pt + 2 It ])} It  (3)  j This requires that agent i, in forming his expectation of price, take into account his expectations of others’ expectations of dividends and price (and relative market weights) two periods hence. To eliminate in like manner the price expectation E j pt + 2 It requires a further iteration. But this leads agents into [ ] taking into account their expectations of others’ expectations of others’ expectations of future dividends and prices at period t+3—literally, as in Keynes’s (1936) phrase, taking into account “what average opinion expects the average opinion to be.” Now, under homogeneity these expectations of others’ expectations collapsed into single, shared, objectively determined expectations. 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. Further, this regress may lead to instability: If investor i believes that others believe future prices will increase, he may revise his expectations to expect upward-moving prices. If he believes that others believe a reversion to lower values is likely, he may revise his expectations to expect a reversion. We 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. Under heterogeneity then, deductive logic leads to expectations that are not determinable. Notice the argument here depends in no way on agents having limits to their reasoning powers. It merely says that given differences in agent expectations, there is no logical means by which to arrive at expectations. And so, perfect rationality in the market can not be well-defined. Infinitely intelligent agents cannot form expectations in a determinate way. B. Forming Expectations by Inductive Reasoning If heterogeneous agents cannot deduce their expectations, how then do they form expectations? They may observe market data, they may contemplate the nature of the market and of their fellow investors. They may derive expectational models by sophisticated, subjective reasoning. But in the end all such models will 7 be—can only be—hypotheses. There is no objective way to verify them, except by observing their performance in practice. Thus agents, in facing the problem of choosing appropriate predictive models, face the same problem that statisticians face when choosing appropriate predictive models given a specific data set, but no objective means by which to choose a functional form. (Of course, the situation here is made more difficult by the fact that the expectational models investors choose affect the price sequence, so that our statisticians’ very choices of model affect their data and so their choices of model.) In what follows then, we assume that each agent acts as a market “statistician.”6 Each continually creates multiple “market hypotheses”—subjective, expectational models—of what moves the market price and dividend. And each simultaneously tests several such models. Some of these will perform well in predicting market movements. These will gain the agent’s confidence and be retained and acted upon in buying and selling decisions. Others will perform badly. They will be dropped. Still others will be generated from time to time and tested for accuracy in the market. As it becomes clear which expectational models predict well, and as poorly predicting ones are replaced by better ones, the agent learns and adapts. This type of behavior—coming up with appropriate hypothetical models to act upon, strengthening confidence in those that are validated, and discarding those that are not—is called inductive reasoning. 7 It makes excellent sense where problems are ill-defined. It is, in micro-scale, the scientific method. Agents that act use inductive reasoning we will call inductively rational.8 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. The agents’ hypotheses and expectations adapt to the current pattern of prices and dividends; and the pattern of prices changes to reflect the current hypotheses and expectations of the agents. We see immediately that the market possesses a psychology. We define this as the collection of market hypotheses or (expectational models or mental beliefs) that are being acted upon at a given time. If there were some attractor inherent in the price-and-expectation-formation process, this market psychology might converge to a stable unchanging set of heterogeneous (or homogeneous) beliefs. Such a set would be statistically validated, and would therefore constitute a rational-expectations equilibrium. We investigate whether the market converges to such an equilibrium below. 6 The phrase is Tom Sargent’s (1993). Sargent argues similarly, within a macroeconomic context, that to form expectations agents need to act as market statisticians. 7 For earlier versions of induction applied to asset pricing and to decision problems, see Arthur (1992) and (1994: the El Farol problem), and Sargent, op. cit. For accounts of inductive reasoning in the psychological and adaptation literature, see Holland et al. (1986), Rumelhart (1980), and Schank and Abelson (1977). 8 In the sense that they use available market data to learn—and switch among—appropriate expectational models. Perfect inductive rationality, or course, is indeterminate. Learning agents can be arbitrarily intelligent, but without knowing other’s learning methods cannot tell a-priori that their learning methods are maximally efficient . They can only discover the efficacy of their methods by testing them against data. 8 3. A Market with Induced Expectations A. The Model We now set up a simple model of an asset market along the lines of Bray (1982) or Grossman and Stiglitz (1980). The model will be neoclassical in structure, but will depart from standard models by assuming heterogeneous agents who form their expectations inductively by the process outlined above. Consider a market in which N heterogeneous agents decide on their desired asset composition between a risky stock paying a stochastic dividend, and a risk-free bond. These agents formulate their expectations separately, but are identical in other respects. They possess a constant absolute risk aversion (CARA) utility function, U ( c ) = − exp( − λ c ) . They communicate neither their expectations nor their buying or selling intentions to each other. Time is discrete and is indexed by t; the horizon is indefinite. The risk-free bond is in infinite supply and pays a constant interest rate r. The stock is issued in N units, and pays a dividend, dt , which follows a given exogenous stochastic process {dt } not known to the agents. The dividend process, thus far, is arbitrary. In the experiments we carry out below, we specialize it to an AR(1) process ( ) dt = d + ρ dt −1 − d + ε t , (4) where ε t is gaussian, i. i. d., and has zero mean, and variance σ e2 . Each agent attempts at each period to optimize his allocation between the risk-free asset and the stock. Assume for the moment that agent i ’s predictions at time t of the next period’s price and dividend are 2 normally distributed with (conditional) mean and variance, Ei,t [ pt +1 + dt +1 ], and σ t,i, p+ d . (We say presently how such expectations are arrived at.) It is well known that under CARA utility and gaussian distributions for forecasts, agent i’s demand, xi,t , for holding shares of the risky asset is given by: xi,t = Ei,t ( pt +1 + dt +1 ) − p(1 + r) 2 λσ i,t, p+ d (5) where pt is the price of the risky asset at t, and λ is the degree of relative risk aversion. Total demand must equal the number of shares issued: