**1. Introduction**

Heterogeneous agent models are present in various fields of economic analysis, such as market maker models, exchange rate models, monetary policy models, overlapping generations models and models of socio-economic behaviour. Yet the field with the most systematic and perhaps most promising nonlinear dynamic approach seems to be asset price modelling. Contributions by Brock and Hommes (1998), LeBaron (2000), Hommes *et al*. (2002), Chiarella and He (2002), Chiarella *et al*. (2003), Gaunersdorfer *et al*. (2003), Brock *et al*. (2005), Hommes *et al*. (2005), and Hommes (2006) thoroughly demonstrate how a simple standard pricing model is able to lead to complex dynamics that makes it extremely hard to predict the evolution of prices in asset markets. The main framework of analysis of such asset pricing models constitutes a financial market application for the evolutionary selection of expectation rules, introduced by Brock and Hommes (1997a) and is called the adaptive belief system.

As a model in which different agents have the ability to switch beliefs, the adaptive belief system in a standard discounted value asset pricing set-up is derived from mean-variance maximization and extended to the case of heterogeneous beliefs (Hommes, 2006, p. 47). It can be formulated in terms of deviations from a benchmark fundamental and therefore used in experimental and empirical testing of deviations from the rational expectations benchmark. Agents are boundedly rational, act independently of each other and select a forecasting or investment strategy based upon its recent relative performance. The key feature of such systems, which often incorporate active learning and adaptation, is endogenous heterogeneity (*cf*. LeBaron, 2002), which means that markets can move through periods that support a diverse population of beliefs, and others in which these beliefs and strategies might collapse down to a very small set.

© 2012 Verbič, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The mixture of different trader types leads to diverse dynamics exhibiting some stylized, qualitative features observed in practice on financial markets (*cf*. Campbell *et al*., 1997; Johnson *et al*., 2003), e.g. persistence in asset prices, unpredictability of returns at daily horizon, mean reversion at long horizons, excess volatility, clustered volatility, and leptokurtosis of asset returns. An important finding so far was that irregular and chaotic behaviour is caused by rational choice of prediction strategies in the bounded-rationality framework, and that this also exhibits quantitative features of asset price fluctuations, observed in financial markets. Namely, due to differences in beliefs these models generate a high and persistent trading volume, which is in sharp contrast to no trade theorems in rational expectations models. Fractions of different trading strategies fluctuate over time and simple technical trading rules can survive evolutionary competition. On average, technical analysts may even earn profits comparable to the profits earned by fundamentalists or value traders.

Memory and Asset Pricing Models with Heterogeneous Beliefs 315

fundamental outcome was that the expectational stability principle, which plays a central role in situations of complete learning, as discussed e.g. in Evans and Honkapohja (2001), retains its importance in the analysis of incomplete learning, though it takes a new form. In the models that were analyzed, expectational stability guaranteed stationary dynamics in

Chiarella *et al*. (2006) proposed a dynamic financial market model in which demand for traded assets had both a fundamentalist and a chartist component in the boundedly rational framework. The chartist demand was governed by the difference between current price and a (long-run) moving average. By examining the price dynamics of the moving average rule they found out that an increase of the window length of the moving average rule can destabilize an otherwise stable system, leading to more complicated, even chaotic behaviour. The analysis of the corresponding stochastic model was able to explain various market price phenomena, including temporary bubbles, sudden market crashes, price

The objective of this chapter is to lay the foundations for a competent and critical theoretical analysis setting the memory assumption in a simple, analytically tractable asset pricing model with heterogeneous beliefs. We shall thus analyze the effects of additional memory in the fitness measure on evolutionary adaptive systems and the nature of consequences for survival of technical trading. In order to examine our research hypothesis adequately, both analytical and numerical analysis will have to be employed and complemented. Therefore, we shall first expand the asset pricing model to include more memory, and then solve it both analytically and numerically. Two cases are going to be analyzed, hopefully sufficiently general to cover some main aspects of financial markets; (1) a two-type case of fundamentalists versus contrarians and (2) a three-type case of fundamentalists versus opposite biased beliefs. Complementing the stability analysis with local bifurcation theory (*cf*. Awrejcewicz, 1991; Palis and Takens, 1993; Kuznetsov, 1995; Awrejcewicz and Lamarque, 2003), we will also be able to analyze numerically the effects of adding different amounts of additional memory to fitness measure on stability of the standard asset pricing model and survival of technical trading. Thus the analysis of both local and global stability

can be performed for different combinations of trader types in the market.

The adaptive belief system employs a mechanism dealing with interaction between fractions of market traders of different types, and the distance between the fundamental and the actual price. Financial markets are thus viewed as an evolutionary system, where price fluctuations are driven by an evolutionary dynamics between different expectation schemes. Pioneering work in this field has been done by Brock and Hommes (1997a), who attempted to conciliate the two main perspectives concerning economic fluctuations, i.e. the new classical and the Keynesian view (*cf*. Hommes, 2006, pp. 1-5), and the underlying rules relating to the formation of expectations. In order to get some insight into possible ways of theoretical analysis to follow, we shall describe a simple, analytically tractable version of the

**2. The heterogeneous agents model** 

the learning economy and unbiased forecasts.

resistance and price switching between different levels.

While recent literature on asset price modelling focuses mainly on impacts of heterogeneity of beliefs in the standard adaptive belief system as set up by Brock and Hommes (1997a) on market dynamics and stability on one hand, and the possibility of the survival of such 'irrational' and speculative traders in the market on the other, several crucial issues regarding the foundations of asset price modelling and its underlying theoretical findings remain open and indeterminate. One of those issues is related to heterogeneity in investors' time horizon; both their planning and their evaluation perspective. Namely, it has been scarcely addressed so far how memory in the fitness measure, i.e. the share of past information that boundedly rational economic agents take into account as decision makers, affects stability of evolutionary adaptive systems and survival of technical trading.

LeBaron (2002) was using simulated agent-based financial markets of individuals following relatively simple behavioural rules that are updated over time. Actually, time was an essential and critical feature of the model. It has been argued that someone believing that the world is stationary should use all available information in forming his or her beliefs, while if one views the world as constantly in a state of change, then it will be better to use time series reaching a shorter length into the past. The dilemma is thus seen as an evolutionary challenge where long-memory agents, using lots of past data, are pitted against shortmemory agents to see who takes over the market. Agents with a short-term perspective appear to both influence the market in terms of increasing volatility and create an evolutionary space where they are able to prosper. Changing the population to more longmemory types has led to a reliable convergence in strategies. Memory or perhaps the lack of it therefore appeared to be an important aspect of the market that is likely to keep it from converging and prevent the elimination of 'irrational', speculative strategies from the market.

Honkapohja and Mitra (2003) provided basic analytical results for dynamics of adaptive learning when the learning rule had finite memory and the presence of random shocks precluded exact convergence to the rational expectations equilibrium. The authors focused on the case of learning a stochastic steady state. Even though their work is not done in the heterogeneous agent setting, the results they obtained are interesting for our analysis. Their fundamental outcome was that the expectational stability principle, which plays a central role in situations of complete learning, as discussed e.g. in Evans and Honkapohja (2001), retains its importance in the analysis of incomplete learning, though it takes a new form. In the models that were analyzed, expectational stability guaranteed stationary dynamics in the learning economy and unbiased forecasts.

Chiarella *et al*. (2006) proposed a dynamic financial market model in which demand for traded assets had both a fundamentalist and a chartist component in the boundedly rational framework. The chartist demand was governed by the difference between current price and a (long-run) moving average. By examining the price dynamics of the moving average rule they found out that an increase of the window length of the moving average rule can destabilize an otherwise stable system, leading to more complicated, even chaotic behaviour. The analysis of the corresponding stochastic model was able to explain various market price phenomena, including temporary bubbles, sudden market crashes, price resistance and price switching between different levels.

The objective of this chapter is to lay the foundations for a competent and critical theoretical analysis setting the memory assumption in a simple, analytically tractable asset pricing model with heterogeneous beliefs. We shall thus analyze the effects of additional memory in the fitness measure on evolutionary adaptive systems and the nature of consequences for survival of technical trading. In order to examine our research hypothesis adequately, both analytical and numerical analysis will have to be employed and complemented. Therefore, we shall first expand the asset pricing model to include more memory, and then solve it both analytically and numerically. Two cases are going to be analyzed, hopefully sufficiently general to cover some main aspects of financial markets; (1) a two-type case of fundamentalists versus contrarians and (2) a three-type case of fundamentalists versus opposite biased beliefs. Complementing the stability analysis with local bifurcation theory (*cf*. Awrejcewicz, 1991; Palis and Takens, 1993; Kuznetsov, 1995; Awrejcewicz and Lamarque, 2003), we will also be able to analyze numerically the effects of adding different amounts of additional memory to fitness measure on stability of the standard asset pricing model and survival of technical trading. Thus the analysis of both local and global stability can be performed for different combinations of trader types in the market.
