FINANCIAL MARKETS AS NONLINEAR ADAPTIVE EVOLUTIONARY SYSTEMS

Cars H. Hommes

Quantitative Finance, Center for Nonlinear Dynamics in Economics and Finance (CeNDEF)

Introduction

  The key difference between economics and the natural sciences is perhaps the fact that decisions of economic agents today depend upon their expectations or beliefs about the future. For example, after a couple of weeks of bad weather in the Netherlands and Western Europe in July 2000, the dreams and hopes of the Dutch about nice weather for summer holidays will not affect the weather in August. In contrast, the dreams and hopes of Dutch investors for excessive high returns on their investments in tulip bulbs in the seventeenth century may have contributed to or even caused what is nowadays known as the Dutch "tulip mania", when the price of tulip bulbs exploded by a factor of more than 20 in the beginning of 1636 but ‘crashed’ back to its original level by the end of the year. Nowadays, in financial markets an over-optimistic estimate of future growth of ICT industries may contribute to an excessively rapid growth of stock prices and indices and might lead to over valuation of stock marketsworldwide. Any dynamic economic system is in fact an expectations feedback system. A theory of expectation formation is therefore a crucial part of any economic model or theory.

  Since its introduction in the sixties by Muth (1961) and its popularization in macroeconomics by Lucas (1971), the rational expectations hypothesis (REH) has become the dominating expectation formation paradigm in economic theory. According to the REH all agents are rational and take as their subjective expectation of future variables the objective prediction by economic theory. In a rational expectations model agents have perfect knowledge about the (linear) market equilibrium equations and use these to derive their expectations. Although many economists nowadays view rational expectations as something unrealistic, it is still viewed as an important benchmark. Despite a rapidly growing literature on bounded rationality, where agents use learning models for their expectations, it seems fair to say that at this point no generally accepted alternative theory of expectations is available.

  In finance, the REH is intimately related to the efficient market hypothesis (EMH). There are weak and strong forms of the EMH, but when economists speak of financial marketsas being efficient, they usually mean that they view asset prices and returns as the outcome of a competitive market consisting of rational traders, who are trying to maximize their expected returns. The main reason why financial markets must be efficient is based upon an arbitrage argument (e.g. Fama (1970)). If markets were not efficient, then there would be unexploited profit opportunities, that could and would be exploited by rational traders. For example, rational traders would buy (sell) an underpriced (overpriced) asset, thus driving its price back to the correct, fundamental value. In an efficient market, there can be no forecastable structure in asset returns, since any such structure would be exploited by rational traders and therefore would be doomed to disappear. Rational agents thus process information quickly and this is reflected immediately in asset prices. The value of a risky asset is completely determined by its fundamental price, equal to the present discounted value of the expected stream of future dividends. In an efficient market, all traders are rational and changes in asset prices are completely random, solely driven by unexpected ‘news’ about changes in economic fundamentals.

  In contrast, Keynes (1936) already questioned a completely rational valuation of assets, arguing that investors sentiment and mass psychology ("animal spirits") play a significant role in financial markets. Keynes used his famous beauty contest as a parable about financial markets. In order to predict the winner of a beauty contest, objective beauty is not all that important, but knowledge or prediction of others" perceptions of beauty is much more relevant. Keynes argued that the same may be true for the fundamental price of an asset: "Investment based on genuine long-term expectation is so difficult as to be scarcely practicable. He who attempts it must surely lead much more laborious days and run greater risks than he who tries to guess better than the crowd how the crowd will behave; and, given equal intelligence, he may make more disastrous mistakes’ (Keynes (1936) p 157). In Keynes view, stock prices are thus not governed by an objective viewof "fundamentals", but by ‘what average opinion expects average opinion to be".

  New classical economists have viewed ‘market psychology’ and "investors sentiment" as being irrational however, and therefore inconsistent with the REH. For example, Friedman (1953) argued that irrational speculative traders would be driven out of the market by rational traders, who would trade against them by taking long opposite positions, thus driving prices back to fundamentals. In an efficient market, ‘irrational’ speculators would simply lose money and therefore fail to survive evolutionary competition.

Financial markets as nonlinear evolutionary adaptive systems

  In a perfectly rational EMH world all traders are rational and it is common knowledge that all traders are rational. In real financial markets however, traders are different, especially with respect to their expectations about future prices and dividends. Aquick glance at the financial pages of newspapers is sufficient to observe that difference of opinions among financial analysts is the rule rather than the exception. In the last decade, a rapidly increasing number of structural heterogeneous agent models have been introduced in the finance literature, see for example Arthur et al (1997), Brock (1993, 1997), Brock and Hommes (1997a, b, 1998), Brock and LeBaron (1996), Chiarella (1992), Chiarella and He (2000), Dacorogna et al (1995), DeGrauwe et al (1993), De Long et al (1990), Farmer (1998), Farmer and Joshi (2000), Frankel and Froot (1988), Gaunersdorfer (2000), Gaunersdorfer and Hommes (2000), Kirman (1991), Kirman and Teyssi`ere (2000), Kurz (1997), LeBaron (2000), LeBaron et al (1999), Lux (1995), Lux and Marchesi (1999a, b), Wang (1994) and Zeeman (1974), as well as many more references in these papers. Some authors even talk about a heterogeneous market hypothesis, as a new alternative to the efficient market hypothesis. In all these heterogeneous agent models different groups of traders, having different beliefs or expectations, coexist. Two typical trader types can be distinguished. The first are rational, ‘smart money’ traders or fundamentalists, believing that the price of an asset is determined completely by economic fundamentals. The second typical trader type are ‘noise traders’, sometimes called chartists or technical analysts, believing that asset prices are not determined by fundamentals, but that they can be predicted by simple technical trading rules based upon patterns in past prices, such as trends or cycles.

  In a series of papers, Brock and Hommes (1997a, b, 1998, 1999), henceforth BH, propose to model economic and financial markets as adaptive belief systems (ABS). The present paper reviews the main features of ABS and discusses a recent extension by Gaunersdorfer and Hommes (2000) as well as some recent experimental testing, jointly with my colleagues Joep Sonnemans, Jan Tuinstra and Henk van de Velden (Hommes et al 2000b) at CeNDEF. An ABS is an evolutionary competition between trading strategies. Different groups of traders have different expectations about future prices and future dividends. For example, one group might be fundamentalists, believing that asset prices return to their fundamental equilibrium price, whereas another group might be chartists, extrapolating patterns in past prices. Traders choose their trading strategy according to an evolutionary ‘fitness measure’, such as accumulated past profits. Agents are boundedly rational, in the sense that most traders choose strategies with higher fitness. BH introduce the notion of adaptive rational equilibrium dynamics (ARED), an endogenous coupling between market equilibrium dynamics and evolutionary updating of beliefs. Current beliefs determine new equilibrium prices, generating adapted beliefs which in turn lead to new equilibrium prices again, etc. In an ARED, equilibrium prices and beliefs coevolve over time.

  Most of the heterogeneous agent literature is computationally oriented. An ABS may be seen as a tractable theoretical framework for the computationally oriented "artificial stock market" literature, such as the Santa Fe artificial stock market of Arthur et al (1997) and LeBaron et al (1999). A convenient feature of an ABS is that the model can be formulated in terms of deviations from a benchmark fundamental. In fact, the perfectly rational EMH benchmark is nested within an ABS as a special case. An ABS may thus be used for experimental and empirical testing whether deviations from a suitable RE benchmark are significant.

  The heterogeneity of expectations among traders introduces an important nonlinearity into the market. In an ABS there are also two important sources of noise: model approximation error and intrinsic uncertainty about economic fundamentals. Asset price fluctuations in an ABS are characterized by an irregular switching between phases of close-to-the-fundamental-price fluctuations, phases of optimism where most agents follow an upward price trend, and phases of pessimism with small or large market crashes. Temporary speculative bubbles (rational animal spirits) can occur, triggered by noise and amplified by evolutionary forces. An ABS is able to generate some of the important stylized facts in many financial series, such as unpredictable returns, fat tails and volatility clustering.

  In our discussion of ABS we will focus on the following questions central to the SFI workshop:

1. Q1 Can technical analysts or habitual rule-of-thumb trading strategies survive evolutionary competition against rational or fundamental traders?
2. Q2 Is an evolutionary adaptive financial market with competing heterogeneous agents efficient?
3. Q3 Does heterogeneity in beliefs lead to excess volatility?

  The paper is organized as follows. In section 2 we discuss the modelling philosophy emphasizing recent developments in nonlinear dynamics and their relevance to economics and finance. Section 3 presents ABS in a general mean-variance framework. In section 4 we present simple, but typical examples. Although the ABS are very simple, subsection 4.4 presents an example where the autocorrelations of returns, squared returns and absolute returns closely resemble those of 40 years of S&P 500 data. Section 5 briefly discusses some first experimental testing of ABS. Finally, section 6 sketches a future perspective of the research program proposed here.

Philosophy of nonlinear dynamics

  The past 25 years have witnessed an explosion of interest in nonlinear dynamical systems, in mathematics as well as in applied sciences. In particular, the fact that simple deterministic nonlinear systems exhibit bifurcation routes to chaos and strange attractors, with "random looking" dynamical behaviour, has received much attention. This section discusses some important features of nonlinear systems, emphasizing their relevance to economics and finance. Let us start by stating the main goal of our research program, namely to explain the most important "stylized facts" in financial series, such as:

1. S1 Asset prices are persistent and have, or are close to having, a unit root.
2. S2 Asset returns are fairly unpredictable, and typically have little or no autocorrelations.
3. S3 Asset returns have fat tails and exhibit volatility clustering and long memory. Autocorrelations of squared returns and absolute returns are significantly positive, even at high order lags, and decay slowly possibly following a scaling law.
4. S4 Trading volume is persistent and there is positive cross correlation between volatility and volume.

  In this paper we will be mainly concerned with stylized facts S2 and S32. The adaptive belief system introduced in the next section will be a nonlinear stochastic system of the form

Xt+1 = F(Xt; n1t, . . . , nHt; λ; δt ; Et ),

  where F is a nonlinear mapping, Xt is a vector of prices (or lagged prices), njt is the fraction or weight of investors of type h, 1 <= h <= H, λ is a vector of parameters and δt and Et are noise terms. In an ABS there are two types of noise terms which are relevant for financial markets. The noise term Et is the model approximation error representing the fact that a model can only be an approximation of the real world. Approximation errors will also be present in a physical model, although the corresponding noise terms might be of smaller magnitude than in economics. In contrast to physical models however, in economic and financial models one almost always has to deal with intrinsic uncertainty represented here by the noise term Et . In finance, for example, one typically deals with investors’ uncertainty about economic fundamentals. In the ABS there will be uncertainty about future dividends and the noise term Et represents unexpected random news about dividends. An important goal of our research program is to match the nonlinear stochastic model (1) to the statistical properties of the data, as closely as possible, and in particular to first match the most important stylized facts in the data.

  Aspecial case of the nonlinear stochastic system (1) arises when all noise terms are set to zero. We will refer to this system as the (deterministic) skeleton denoted by

Xt+1 = F(Xt ; n1t, . . . , nHt; λ).

  In order to understand the properties of the general stochastic model (1) it is important to understand the properties of the deterministic skeleton. In particular, one would like to impose as little structure on the noise process as possible, and relate the stylized facts of the general stochastic model (1) directly to generic properties of the underlying deterministic skeleton. There are three important, generic features of nonlinear deterministic systems which may play an important role in generating some of the stylized facts in finance and may in particular cause volatility clustering:

1. F1 Chaos and strange attractors due to homoclinic bifurcations.
2. F2 Simultaneous coexistence of different attractors.
3. F3 Local bifurcations of steady states.