Technical Report: CES-497 A summary for the Brock and Hommes Heterogeneous beliefs and routes to chaos in a simple asset pricing model 1998 JEDC paper Michael Kampouridis, Shu-Heng Chen, Edward P.K. Tsang June 25, 2009 Abstract The purpose of this paper is to summarise the 1998 paper by William A. Brock and Cars H. Hommes, Heterogeneous beliefs and routes to chaos in a simple asset pricing model, in Journal of Economic Dynamics and Control, 22, 1235-1274 (Brock and Hommes 1998). 1 Introduction In their paper, Brock and Hommes presented an asset pricing model with heterogeneous beliefs. The significance of the paper is that it presents numerical evidence of the existence of chaotic price fluctuations, when the intensity of choice among different trading strategies is high. The way this report is organised is as follows: section 2 presents the asset pricing model used in their paper, section 3 briefly presents the numerical tools used for their analysis, section 4 focusses on the dynamics of the equilibrium of asset prices with a small (two, three and four) number of trading strategies and finally, section 5 concludes this report. 2 The Model 2.1 Model description This paper presents a simple Present Discounted Value (PDV) asset pricing model with heterogeneous beliefs. The model has one risky and one risk-free University of Essex, UK National Cheng Chi University, Taiwan University of Essex, UK. 1
asset and each agent has to its disposal a finite set of strategies 1. At any time t, an agent can adopt a strategy h. This strategy h that an agent adopts reflects his beliefs 2 f h,t about the price x t of the risky asset at time t. All agent beliefs are of the form f h,t = g h x t 1 + b h (1) where g h is the trend and b h is the bias of trader type h. Details about these parameters are given below, in Section 2.3. These beliefs are updated based on the performance measure of accumulated past profits U h,t 1 of the previous period t 1: U h,t 1 = π h,t 1 + ηu h,t 2 (2) where π h,t is the past realized profits and η is the memory strength. Again, details about the parameter η will follow in Section 2.3. Profits π h,t 1 are equal to: π h,t 1 = 1 ασ 2 (x t Rx t 1 )(g h x t 2 + b h Rx t 1 ) (3) where α denotes the risk aversion and is assumed equal for all agents and σ 2 is the beliefs about the conditional variance of excess returns, and is also assumed to be a constant and equal for all types of traders. The equilibrium price is Rx t = Σn h,t 1 f h,t (4) where R is the excess return, and n h,t 1 is the fraction of agent type h at the beginning of period t, before the equilibrium price x t has been observed. The fractions of the strategies are equal to: where n h,t = exp[βu h,t 1 ]/Z t (5) Z t = Σexp[βU h,t 1 ] (6) Parameter β denotes the intensity agents can switch strategies and is called intensity of choice. Intensity of choice plays an important role at this model and is also described at the next section. 2.2 How the fractions change As we said earlier, n h,t 1 is the fraction of agent type h at the beginning of period t, before the equilibrium price x t has been observed. These old fractions are therefore used in the equilibrium equation (4) in order to calculate the new 1 In Brock and Hommes paper, each strategy represents a strategy type, such as fundamentalist, technical trader, rational and contrarian 2 A belief is also referred to as predictor in Brock and Hommes 2
equilibrium price x t. After this price has been revealed, it is used in equations (3) and (2) in order to to update the fractions n h,t of equation (5). This procedure is repeated for t number of times. Table 1 is given as a reference for this model s components, parameters and constants. Components, Parameters and Constants of the Brock-Hommes model Description Model Components Model Parameters Model Constants Strategy of an agent h Time t Price of the risky asset x t Agent belief f h,t Trend g h Bias b h Accumulated past profits U h,t Memory strength η Past realized profits π h,t Risk aversion α Beliefs about the conditional σ 2 variance of excess returns Gross return R Fraction of a strategy n Intensity of choice β Table 1: Components, Parameters and Constants of the Brock-Hommes asset pricing model 2.3 Parameters of the model In this subsection we present the main parameters that this model uses. 2.3.1 Trend g h and Bias b h Depending on the value of these two parameters, agents are characterised differently. In total, we can have five different cases: 1. If b h = 0 and (a) if g h > 0, then agent h is called a pure trend chaser (b) if g h < 0, then agent h is called a contrarian 2. If g h = 0 and (a) if b h > 0, then agent h is upward biased (b) if b h < 0, then agent h is downward biased 3. If g h = b h = 0, then agent h is a fundamentalist and thus believes that the prices will return to their fundamental value. 3
2.3.2 Memory strength η In most of the strategies, agents have no memory regarding their past profits, so η = 0. There is only one case that is examined where η > 0 and that is the case of traders with rational expectations/perfect foresight. 2.3.3 Intensity of choice β This is a very important parameter of the model. It measures how fast agents switch between different prediction strategies. When β traders use the strategy with the highest fitness. When β = 0, then traders distribute themselves evenly across the set of available strategies. The paper shows that an increase to the intensity of choice β to switch predictors can lead to market instability. When β is high, asset price fluctuations are characterized by an irregular switching between phases where prices are close to the EMH fundamental, phases of optimism, where traders become excitepd and extrapolate upwards trends, and phases of pessimism, where traders become nervous, causing a sharp decline in asset prices. 3 Numerical Analysis in nonlinear dynamics This section briefly presents the numerical tools Brock and Hommes use in order to detect the existence of chaos. In their paper, Brock and Hommes use the following tools: 1. Bifurcation diagrams 2. Phase diagrams 3. Lyapunov characteristic exponents (LCEs) 3.1 Bifurcation diagrams A bifurcation is a qualitative change in the dynamics, for instance concerning the existence or stability of a steady state or a cycle. Bifurcation diagrams therefore present the long-run behaviour of the model as a function of a parameter (in Brock and Hommes paper this parameter is the intensity of choice β). Observing a qualitative change in the dynamics of the model could be a possible route to chaos. 3.2 Phase diagrams In addition to bifurcation diagrams, Brock and Hommes plot phase diagrams for the price x t and the fractions n h,t. Their goal is to observe whether in the phase space strange attractors 3 can be observed. Existence of strange attractors is another indication of chaos. Figure 2 gives an example of a phase diagram. 3 An attracting set for a dynamical system is a closed subset A of its phase space, such that for many choices of initial point the system will evolve towards A. Attractor is therefore 4
Figure 1: An example of a Bifurcation diagram. This diagram presents how x changes when β increases. As we can observe, for high values of β, x does not take only one value. This is an indication of chaos. 3.3 Lyapunov characteristic exponents (LCEs) Bifurcation and phase diagrams are simple ways of getting information about the global dynamics. However, these diagrams provide just an idication for chaos and not a proof. A numerical analysis is therefore needed, in order to investigate whether indeed the model exhibits chaos. This is done by the numerical computation of the Lyapunov characteristic exponents (LCEs). LCEs measure the average rate of divergence (or convergence) of nearby to the equilibrium initial states, along an atrractor in several directions. When the largest LCE λ > 0, then this is numerical evidence for chaos. For a more detailed description of the above tools Brock and Hommes refer the readers to (Guckehneimer and Holmes 1983; Arrowsmith and Place 1994; Eckmann and Ruelle 1985; Brock 1986). To summarise this section, Brock and Hommes draw a conclusion whether chaos exists or not in the following way: first of all, with bifurcation and phase diagrams observe whether there are any indications for chaos. If there are indicaan attracting set which satisfies some supplementary condition, so that it cannot be split into smaller pieces. An attractor is called strange when it is a complicated set with a fractal structure. For further details on attractors and strange attractors the reader is referred to (Milnor 1985; Ruelle 2006) 5
Figure 2: An example of a Phase diagram for the four-type belief system from Brock and Hommes. The diagrams present evidence of the existence of strange attractors, as the intensity of choice β increases tions indeed, they then calculate the LCEs. If they also find that the maximum λ > 0, they then conclude that chaos exists. 4 Simple Belief Types Brock and Hommes examine their model under two, three and four belief types. Furthermore, especially for the case of two beliefs, they examine three different cases, with different types of traders. The following sections present the different belief types examined and their results, i.e. whereas the model exhibits chaos, under a certain type of agents. 4.1 Two Belief Types 4.1.1 Perfect foresight (rational agents) vs Trend chasers 1. Type 1 - Rational agents (g h = b h = 0 and memory strength η > 0) 2. Type 2 - Trend chasers with upward bias (g h = 0) Rational agents have perfect foresight. They know at each date all past prices and dividends, and the market equilibrium with all fractions n h,t for other belief types. Their belief equation is: 6
f 1,t = x t+1 The equilibrium equation (4) thus becomes: Rx t = Σn h,t 1 f h,t = n 1,t 1 x t+1 + n 2,t 1 gx t 1 Because of the assumtion of perfect foresight, it is not possible to do numerical analysis of the global dynamics. The reason for this is because the equilibrium equation has a forward looking element (x t+1 ). 4.1.2 Fundamentalists vs Trend chasers 1. Type 1 - Fundamentalists (g h = b h = 0) 2. Type 2 - Trend chasers (g h = 0) Results show that high intensity of choice leads to weakly 4 chaotic asset price fluctuations. Prices switch irregularly between close to the EMH fundamental prices and upward and downward trends. 4.1.3 Fundamentalists vs Contrarians 1. Type 1 - Fundamentalists (g h = b h = 0) 2. Type 2 - Contrarians (b h = 0, g h < 0) Prices are characterised by an irregular switching between a stable phase with prices close to their fundamental value and an unstable phase of up and down price oscillations of increasing amplitude. Once again, a high intensity of choice produces chaotic asset price dynamics with irregular fluctuations around the EMH fundamental. 4.2 Three Belief types 4.2.1 Fundamentalists vs Opposite biases 1. Type 1 - Fundamentalists (g h = b h = 0) 2. Type 2 - Trend chasers with upward bias (g h = 0, b h > 0) 3. Type 3 - Trend chasers with downward biases (g h = 0, b h < 0) As we can see, type 1 agents are fundamentalists, whereas types 2 and 3 are opposite biased traders. Type 2 traders are upward biased, whereas type 3 are downward biased, under the assumption that the biases are exactly opposite. Results show that only regular periodic fluctuations around the unstable fundamental steady state occur. Opposite biases can cause perpetual oscillations around the fundamental, but cannot lead to chaotic movements. 4 These chaotic price fluctuations are considered to be weak, because the largest LCE λ 1 is only slightly greater than 0 7
4.3 Four Belief types 4.3.1 Fundamentalists vs Trend chasers vs Bias The four types here are the following: 1. Type 1 - Fundamentalists (g h = b h = 0) 2. Type 2 - Trend chasers with upward bias (g h = 0, b h > 0) 3. Type 3 - Trend chasers with downward biases (g h = 0, b h < 0) 4. Type 4 - Strong, pure, trend chasers (b h = 0, g h > R) Results suggest again the existence of chaos, with switching between close to the EMH prices, oscillations around the fundamental and trends at irregular intervals. Table 2 summarises the results presented in this section. Summary Results Agent type Two-type: Perfect foresight vs Trend chasers Fundamentalists vs Trend chasers Fundamentalists vs Contrarians Three-type: Fundamentalists vs Opposite biases Four-type: Fundamentalists vs Trend vs Bias Exhibits chaos? NO YES YES NO YES Table 2: Summary results regarding the existence of chaos in the different belieftype models 5 Conclusion This report gave a brief summary of the 1998 Brock and Hommes paper regarding heterogeneous beliefs and routes to chaos, in a simple asset pricing model. In this report, we described the asset pricing model Brock and Hommes used, its parameters, the numerical tools they used for analysis and finally, we presented their results, which were conditions in which chaotic price fluctuations appear. According to their results, chaos was observed in 2 out of 3 two-belief types they examined (Fundamentalists vs Trend chasers and Fundamentalists vs Contrarians) and in the four belief type (Fundamentalists vs Trend vs Bias). Chaos was not observed in the Perfect foresight vs Trend chasers two type belief and in the Fundamentalists vs Opposite biases three type belief models. 8
References Arrowsmith D, Place C (1994) An Introduction to Dynamic Systems. Cambridge University Press, Cambridge Brock W (1986) Distinguishing random and deterministic systems. abridged version. Journal of Economic Theory 40:168 195 Brock W, Hommes C (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control 22:1235 1274 Eckmann JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57:617 656 Guckehneimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York Milnor J (1985) On the concept of attractor. Communications of Mathematical Physics 99:177 195 Ruelle D (2006) What is a strange attractor? Notices of the American Mathematical Society 53(7):764 765 9