A simple model of a speculative housing market

We develop a simple model of a speculative housing market in which the demand for houses is influenced by expectations about future housing prices. Guided by empirical evidence, agents rely on extrapolative and regressive forecasting rules to form their expectations. The relative importance of these competing views evolves over time, subject to market circumstances. As it turns out, the dynamics of our model is driven by a two-dimensional nonlinear map which may display irregular boom and bust housing price cycles, as repeatedly observed in many actual markets. However, we also find that speculation may be a source of both stability and instability. --Housing markets,Speculation,Boom and bust cycles,Nonlinear Dynamics

On the inherent instability of international financial markets: Natural nonlinear interactions between stock and foreign exchange markets

We develop a novel financial market model in which the stock markets of two countries are linked via and with the foreign exchange market. To be precise, there are domestic and foreign speculators in each of the two stock markets which rely either on linear technical or linear fundamental trading strategies to determine their orders. Since foreign stock market speculators require foreign currency to conduct their trades, all three markets are connected. Our setup entails a natural nonlinearity which may cause persistent endogenous price dynamics. Moreover, we analytically show that market interactions can destabilize the model's fundamental steady state. --Stock prices,exchange rates,market stability,technical and fundamental analysis,nonlinear market interactions,endogenous dynamics

Asset Price and Wealth Dynamics in a Financial Market with Heterogeneous Agents

This paper considers a discrete-time model of a financial market with one risky asset and one risk-free asset, where the asset price and wealth dynamics are determined by the interaction of two groups of agents, fundamentalits and chartists. In each period each group allocates its wealth between the risky asset and the safe asset according to myopic expected utility maximization, but the two groups have heterogeneous beliefs about the price change over the next period: the chartists are trend extrapolators, while the fundamentalists expect that the price will return to the fundamental. We assume that investors have CRRA utility, so that their optimal demand for the risky asset depends on wealth. A market maker is assumed to adjust the price at the end of each trading period, on the basis of the excess demand and according to particular stabilization policies. The model results in a three-dimensional nonlinear discrete-time dynamical system, with growing price and wealth processes, but it is reduced to a stationary system in terms of asset returns and wealth shares of the two groups. It is shown that the long-run market dynamics are highly dependent on the parameters which characterize agents' behavior (in particular the risk aversion coefficient and the chartist extrapolation parameter) as well as on the initial condition (in particular the initial wealth shares of fundamentalists and chartists). It is also shown that the for wide ranges of the parameters a (locally) stable fundamental steady state may coexist with a stable "nonfundamental" steady state, where price grows faster than the fundamental and only chartists survive in the long-run. In such cases, the role played by the initial condition is analysed by means of numerical investigations and graphical representation of the basins of attraction. Other dynamic scenarios include limit cycles, periodic orbits or more complex attractors, where in general both types of agents survive in the long run, with time varying wealth fractions.heterogeneous agents; financial market dynamics; wealth dynamics; coexisting attractors

A Dynamic Analysis of Speculation Across Two Markets

A discrete time model of a financial market is proposed, where the time evolution of asset prices and wealth arises from the interaction of two groups of agents, fundamentalists and chartists. Each group allocates its wealth between a risky asset (stock) and an alternative asset (bond), and the two groups have heterogeneous expectations about returns. We assume that chartists compute expected returns by extrapolating past price changes, while fundamentalists form their expectations on the basis of their superior knowledge of fundamentals. Under the assumption that agents have CRRA utility, investors' optimal demand for each asset depends on their wealth, and this results in growing price and wealth processes. The time evolution of the prices is modeled by assuming the existence of a market maker, who sets excess demand of each asset to zero at the end of each trading period by taking an off-setting long or short position. The market maker is assumed to adjust the price, in each period, partly on the basis of the excess demand and partly according to a particular market stabilization policy. The model is reduced to a high dimensional nonlinear discrete-time dynamical system with growing prices and wealth. Although the model is nonstationary, suitable changes of variables lead to a stationary model where the dynamic variables are actual and expected returns, fundamental/price ratios, and wealth proportions of chartists and fundamentalists. The steady states and other invariant sets of the model are determined, and important global dynamic phenomena are studied via numerical techniques. Stochastic simulations are also performed, that show the ability of the model to generate some of the characteristic features of financial time series.

Aggregation of Heterogeneous Beliefs and Asset Pricing: A Mean-Variance Analysis

The aim of this paper is to show, within the mean-variance framework, how the market belief can be constructed as the result of the aggregation of heterogeneous beliefs and how the market equilibrium prices of risky assets can thus be determined. The heterogeneous beliefs are defined in terms of not only the means but also variances and covariances. By constructing the mean and variance of the market belief, we analyze the impact of the heterogeneous beliefs on the market equilibrium asset pricing relation. In particular, we extend the standard CAPM under homogenous beliefs to the one under the heterogeneous beliefs.Mean variance analysis, heterogeneous beliefs, aggregation, asset pricing

A Framework for CAPM with Heterogenous Beliefs

We introduce heterogeneous beliefs in to the mean-variance framework of the standard CAPM, in contrast to the standard approach which assumes homogeneous beliefs. By assuming that agents form optimal portfolios based upon their heterogeneous beliefs about conditional means and covariances of the risky asset returns, we set up a framework for the CAPM that incorporates the heterogeneous beliefs when the market is in equilibrium. In this framework we first construct a consensus belief (with respect to the means and covariances of the risky asset returns) to represent the aggregate market belief when the market is in equilibrium. We then extend the analysis to a repeated one-period set-up and establish a framework for a dynamic CAPM using a market fraction model in which agents are grouped according to their beliefs. The exact relation between heterogeneous beliefs, the market equilibrium returns and the ex-ante beta-coeffcients is obtained. CAPM and Heterogeneous beliefs.

Aggregation of Heterogeneous Beliefs and Asset Pricing Theory: A Mean-Variance Analysis

Within the standard mean-variance framework, this paper provides a procedure to aggregate the heterogeneous beliefs in not only risk preferences and expected payoffs but also variances/covariances into a market consensus belief. Consequently, an asset equilibrium price under heterogeneous beliefs is derived. We show that the market aggregate behavior is in principle a weighted average of heterogeneous individual behaviors. The CAPM-like equilibrium price and return relationships under heterogeneous beliefs are obtained. The impact of diversity of heterogeneous beliefs on the market aggregate risk preference, asset volatility, equilibrium price and optimal demands of investors is examined. As a special case, our result provides a simple explanation for the empirical relation between cross-sectional volatility and expected returns.

Fundamental and Non-Fundamental Equilibria in the Foreign Exchange Market. A Behavioural Finance Framework

We develop a simple model of the exchange rate in which agents optimize their portfolio and use different forecasting rules. They check the profitability of these rules ex post and select the more profitable one. This model produces two kinds of equilibria, a fundamental and a bubble one. In a stochastic environment the model generates a complex dynamics in which bubbles and crashes occur at unpredictable moments. We contrast these ”behavioural” bubbles with ”rational” bubbles.exchange rate, bounded rationality, heterogeneous agents, bubbles and crashes, complex dynamics, basins of attraction

Time-Varying Beta: A Boundedly Rational Equilibrium Approach

By taking into account conditional expectations and the dependence of the systematic risk of asset returns on micro- and macro-economic factors, the conditional CAPM with time-varying betas displays superiority in explaining the cross-section of returns and anomalies in a number of empirical studies. Most of the literature on time-varying beta is motivated by econometric estimation rather than explicit modelling of the stochastic behaviour of betas through agents' behaviour. Within the mean-variance framework of repeated one-period optimisation, we set up a boundedly rational dynamic equilibrium model of a financial market with heterogeneous agents and obtain an explicit dynamic CAPM relation between the expectede quilibrium returns and time-varying betas. By incorporating the three most popular types of investors, fundamentalists, chartists and noise traders, into the model, we show that, independent of the fundamentals, there is a systematic change in the market portfolio, risk-return relationships, and time varying betas when investors change their behaviour, such as the chartists acting as momentum traders. In particular, we demonstrate the stochastic nature of time-varying betas and show that the commonly used rolling window estimates of time-varying betas may not be consistent with the ex-ante betas implied by the equilibrium model. The results provide a number of insights into an understanding o ftime-varying beta.equilibrium asset prices; CAPM; time-varying betas, heterogeneous expectations

Heterogeneity, Market Mechanisms, and Asset Price Dynamics

This chapter surveys the boundedly rational heterogeneous agent (BRHA) models of financial markets, to the development of which the authors and several co-authors have contributed in various papers. We give particular emphasis to role of the market clearing mechanism used, the utility function of the investors, the interaction of price and wealth dynamics, portfolio implications, the impact of stochastic elements on the markets dynamics, and calibration of this class of models. Due to agents’ behavioural features and market noise, the BRHA models are both nonlinear and stochastic. We show that the BRHA models produce both a locally stable fundamental equilibrium corresponding to that of standard paradigm, as well as instability with a consequent rich range of possible complex behaviours characterised both indirectly by simulation and directly by stochastic bifurcations. A calibrated model is able to reproduce quite well the stylized facts of financial markets. The BRHA framework is thus able to accommodate market features that seem not easily reconcilable for the standard financial market paradigm, such as fat tail, volatility clustering, large excursions from the fundamental and bubbles.Bounded rationality; interacting heterogeneous agents; behavioural finance; nonlinear economic dynamics; complexity