**4.3. Numerical analysis**

336 Nonlinearity, Bifurcation and Chaos – Theory and Applications

interest in our case, we can observe that a saddle-node bifurcation would occur for:

2

4,5 are complex for

while 0 1 *w* and the right-hand side is either negative or not defined.

2 3 2 (1 ) *R bk R*

and the right-hand side is always negative. On the other hand, a period doubling bifurcation

3 ( 1) 2 ( 1)( 1) *R w bkR w*

The remaining qualitative change of the three discussed in Section 4.3 is the Hopf bifurcation. For this to occur, a complex conjugate pair of eigenvalues has to cross the unit

 2 2

**Lemma 6:** There exists an intensity of choice value \* such that the fundamental steady state,

the system exhibits a Hopf bifurcation. Memory affects the emergence of

As we have just established, in the case of fundamentalists versus opposite biased beliefs increasing intensity of choice to switch predictors destabilizes the fundamental steady state. This happens through a Hopf bifurcation. We can thus conclude, as did Brock and Hommes (1998) for the simpler version of the model, that in the presence of biased agents the first step towards complicated price fluctuations is different from that in the presence of

 *in the expressions for the eigenvalues* 

*have g*(1) 0  *and g*( 1) 0 *, a bifurcation has to occur. At the moment of the bifurcation the* 

*thus becomes unstable. Since it is obvious from the characteristic equation (62) that for all* 

3 6 2( ) 3 6 2( ) 2 ( 1) 2 ( 1) *R w R RR w R w R RR w bkw bkw* 

4,5 of the characteristic equation (62), which are of

. (63)

. (64)

0 and the left-hand side is again always non-negative,

<sup>2</sup> <sup>2</sup> 2 2 2 ( 1) 3 24 (1 ) 0 *b k w Rw b k w R*

. (65)

 

, becomes unstable and remains such for \*

 , which

> . For

*4,5 increase as well, and one of* 

0 *we* 

\* *. The fundamental steady state* 

0 and the left-hand side is always non-negative, while *R* 1

If we now take a look at the eigenvalues

This can never hold, since

This can never hold either, since

produces the following interval of values:

We therefore state the following lemma.

 

*the eigenvalues has to cross the unit circle at some critical* 

this bifurcation, viz. with more memory the bifurcation occurs later.

contrarians. This fact does not change when we take memory into account.

which is stable for 0 \*

2 <sup>3</sup> \* 2 ( 1) *kb w*

**Proof of Lemma 6:** 

 *increases, terms with* 

*When* 

would occur for:

circle. Eigenvalues

Our numerical analysis in the case of fundamentalists and opposite biased beliefs will be conducted for fixed values of parameters *R* = 1.1, *k* = 1.0, *b*2 = 0.2 and *b*3 = –0.2. We shall thus vary the memory strength parameter *w* and the intensity of choice parameter . The same four analytical tools will be used than in Section 3.4.

Dynamic behaviour of the system can again first and foremost be determined by investigating bifurcation diagrams. From Figure 5 we can observe that for low values of we have a stable steady state, i.e. the fundamental steady state. As has been proven in Lemma 4, the position of this steady state, i.e. *xeq* = 0, is independent of the memory, which is clearly demonstrated by the simulations. For increasing a bifurcation occurs at = \*, which is a Hopf bifurcation; the steady state becomes unstable and an attractor appears, consisting of an invariant circle around the (unstable) steady state. It is again a supercritical Hopf bifurcation, where the steady state gradually changes either into an unstable equilibrium or into an attractor.

The bifurcation value varies with changing memory strength parameter, as given by expression in Lemma 6. As can also be seen from Figure 5 at higher memory strength the bifurcation occurs later. For > \* complex dynamical behaviour appears, which is interspersed with stable cycles. As we have already discovered in Section 4.2, irrespective of the amount of additional memory that is taken into account such a (bifurcation) route to complicated dynamics is different from that in the presence of contrarians, where we observed period doubling route to chaos (rational route to randomness).

By examining largest Lyapunov characteristic exponent (LCE) plots of we arrive at more precise conclusions about the dynamic behaviour of the system. It can be seen from Figure 5 that the largest LCE is smaller than 0 and the system is thus stable until the bifurcation. At the bifurcation value a qualitative change in dynamics occurs, i.e. a Hopf bifurcation. The dynamics is somewhat more complicated. Namely, we can observe that the largest LCE after = \* is non-positive, but mainly close to 0, which implies periodic and quasi-periodic dynamics, i.e. for high values of the intensity of choice only regular (quasi-)periodic fluctuations around the unstable fundamental steady state occur. An important finding is that the predominating quasi-periodic dynamics does not seem to evolve to chaotic dynamics and the route to complex dynamics is indeed different from the routes examined so far.

Memory and Asset Pricing Models with Heterogeneous Beliefs 339

= 1500

= 35000

= 1500

= 35000

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents lagged deviations of the price from the fundamental value (*xt*–1). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

= 450

= 10000

= 450

= 10000

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The groups of four diagrams differ with respect to the memory strength parameter *w*;

the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

= 5000

= 450

**Figure 7.** Phase plots of (*xt*, *n*1,*<sup>t</sup>*) in case of fundamentalists versus opposite biases

**Figure 6.** Phase plots of (*xt*, *xt*–1) in case of fundamentalists versus opposite biases

= 450

= 5000

= 100

= 1500

= 100

= 1500

fundamental value (*x*) in the upper two diagrams and the value of the largest LCE in the lower two diagrams, respectively. The diagrams differ with respect to the memory strength parameter *w*; the left one corresponds to *w* = 0.3, while the right one corresponds to *w* = 0.9.

**Figure 5.** Bifurcation diagrams and Largest LCE plots of in case of fundamentalists versus opposite biased beliefs

Next, we shall examine plots of the attractors in the planes, determined by (*xt*, *xt*–1) and (*xt*, *n*1,*<sup>t</sup>*). In the upper left plot of each of the two parts of Figure 6 we can first observe the appearance of an attractor for the intensity of choice beyond the bifurcation value. The orbits converge to such an attractor consisting of an invariant 'circle' around the (unstable) fundamental steady state. The attractor obtained in the (*xt*, *n*1,*<sup>t</sup>*) plane is somewhat different. Namely, the unstable steady state dissipates into numerous points and evolves into a 'loop' shape, as shown in Figure 7.

As the intensity of choice increases, the dynamics remains periodic or quasi-periodic; in case of past deviations of prices from the fundamental value and fractions of biased beliefs the invariant circle slowly changes its shape into a '(full) square' (see Figure 6), while in case of fractions of fundamentalists the loop slowly changes into a 'three-sided square' (see Figure 7). For high values of intensity of choice we seem to obtain (stable) higher period cycles; in the case of past deviations of prices from the fundamental value and fractions of biased beliefs we seem to attain a stable period four-cycle, while in the case of fractions of fundamentalists it is difficult to obtain any solid indications based solely on numerical simulations due to

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents lagged deviations of the price from the fundamental value (*xt*–1). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

**Figure 6.** Phase plots of (*xt*, *xt*–1) in case of fundamentalists versus opposite biases

338 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Notes:** Horizontal axis represents the intensity of choice (

**Figure 5.** Bifurcation diagrams and Largest LCE plots of

while the right one corresponds to *w* = 0.9.

biased beliefs

Next, we shall examine plots of the attractors in the planes, determined by (*xt*, *xt*–1) and (*xt*, *n*1,*<sup>t</sup>*). In the upper left plot of each of the two parts of Figure 6 we can first observe the appearance of an attractor for the intensity of choice beyond the bifurcation value. The orbits converge to such an attractor consisting of an invariant 'circle' around the (unstable) fundamental steady state. The attractor obtained in the (*xt*, *n*1,*<sup>t</sup>*) plane is somewhat different. Namely, the unstable steady state dissipates into numerous points and evolves into a 'loop' shape, as shown in Figure 7.

As the intensity of choice increases, the dynamics remains periodic or quasi-periodic; in case of past deviations of prices from the fundamental value and fractions of biased beliefs the invariant circle slowly changes its shape into a '(full) square' (see Figure 6), while in case of fractions of fundamentalists the loop slowly changes into a 'three-sided square' (see Figure 7). For high values of intensity of choice we seem to obtain (stable) higher period cycles; in the case of past deviations of prices from the fundamental value and fractions of biased beliefs we seem to attain a stable period four-cycle, while in the case of fractions of fundamentalists it is difficult to obtain any solid indications based solely on numerical simulations due to

fundamental value (*x*) in the upper two diagrams and the value of the largest LCE in the lower two diagrams, respectively. The diagrams differ with respect to the memory strength parameter *w*; the left one corresponds to *w* = 0.3,

). Vertical axis represents deviations of the price from the

in case of fundamentalists versus opposite

**Notes:** Horizontal axis represents deviations of the price from the fundamental value (*xt*). Vertical axis represents the fraction of fundamentalists (*n*1,*<sup>t</sup>*). The groups of four diagrams differ with respect to the memory strength parameter *w*; the left group corresponds to *w* = 0.3, while the right group corresponds to *w* = 0.9.

**Figure 7.** Phase plots of (*xt*, *n*1,*<sup>t</sup>*) in case of fundamentalists versus opposite biases

convergence problems for very high values of intensity of choice. In the latter case we can observe stable period four- and six-cycles, however (see lower right plot of each of the two parts of Figure 7). Indeed, Brock and Hommes (1998) proved for the case of exactly opposite biased beliefs and infinite intensity of choice in their simpler version of the model without additional memory that the system has a stable four-cycle attracting all orbits, except for hairline cases converging on the unstable fundamental steady state. Additionally, they discovered that for all three trader types average profits along the four-cycle equal *b*2.

Memory and Asset Pricing Models with Heterogeneous Beliefs 341

**Notes:** Horizontal axis represents the time (*t*). Vertical axis in each set of time series plots represents deviations of the price from the fundamental value (*xt*), and the fractions of fundamentalists (*n*1,*<sup>t</sup>*), optimistic biased beliefs (*n*2,*<sup>t</sup>*) and pessimistic biased beliefs (*n*3,*<sup>t</sup>*). The plots on the left-hand side and the right-hand side of the figure differ with respect to the memory strength parameter *w*; the ones on the left correspond to *w* = 0.3, while the ones on the right to *w* = 0.9.

**Figure 8.** Time series of prices and fractions in case of fundamentalists versus opposite biases

Again, we can observe that the memory has an impact on the dynamics of the system. Namely, both the convergence of the system on an attractor and the further development of such an attractor seem to be dependent on the value of the memory strength parameter. The precise impact of memory is somewhat more difficult to establish due to the dependence of the bifurcation value on memory strength and the subsequent need to choose higher intensities of choice with higher memory strength in order to demonstrate different nature of attractors of the system. However, we can still establish that at the same intensity of choice (after the bifurcation value) the system apparently needs less additional memory in order to develop a specific stage of an attractor or even a (stable) higher period cycle.

Finally, we shall examine time series plots of deviations of the price from the fundamental value and of the fractions of all three types of traders. Figure 8 shows some time series corresponding to the attractors in Figures 6 and 7. We can observe that opposite biases may cause perpetual oscillations around the fundamental, even when there are no costs for fundamentalists, but can not lead to chaotic movements. Furthermore, as has already been indicated by the appearance of stable higher period cycles for high intensities of choice, in a three-type world, even when there are no costs and memory is infinite, fundamentalist beliefs can not drive out opposite purely biased beliefs, when the intensity of choice to switch strategies is high.

Hence, according to the argumentation of Brock and Hommes (1998, p. 1260), the market can protect a biased trader from his own folly if he is part of a group of traders whose biases are 'balanced' in the sense that they average out to zero over the set of types. Centralized market institutions can make it difficult for unbiased traders to prey on a set of biased traders provided they remain 'balanced' at zero. On the other hand, in a pit trading situation unbiased traders could learn which types are balanced and simply take the opposite side of the trade. In such situations biased traders would be eliminated, whereas a centralized trading institution could 'protect' them.

Additional memory does not change the pattern of asset prices and trader fractions *per se*, but it does affect its period. Namely, at the same intensity of choice and higher memory strength the period of these cycles appears to be elongated on average, in a way that both the negative and the positive deviation of the price from the fundamental value last longer. The same is valid for fractions ob biased traders, while in the case of fractions of fundamentalists the prolongation of the period of the irregular cycle appears in the form of less frequent 'spikes', which is understandable, since more persistent deviations of prices from the fundamental imply more space for biased traders and less chance for appearance of the fundamentalists. More memory causes the traders to stick longer to the strategy that has been profitable in the past, but might not be so profitable in the recent periods; therefore the system approaches purely quasiperiodic dynamics when the memory strength increases at given intensity of choice.

convergence problems for very high values of intensity of choice. In the latter case we can observe stable period four- and six-cycles, however (see lower right plot of each of the two parts of Figure 7). Indeed, Brock and Hommes (1998) proved for the case of exactly opposite biased beliefs and infinite intensity of choice in their simpler version of the model without additional memory that the system has a stable four-cycle attracting all orbits, except for hairline cases converging on the unstable fundamental steady state. Additionally, they

Again, we can observe that the memory has an impact on the dynamics of the system. Namely, both the convergence of the system on an attractor and the further development of such an attractor seem to be dependent on the value of the memory strength parameter. The precise impact of memory is somewhat more difficult to establish due to the dependence of the bifurcation value on memory strength and the subsequent need to choose higher intensities of choice with higher memory strength in order to demonstrate different nature of attractors of the system. However, we can still establish that at the same intensity of choice (after the bifurcation value) the system apparently needs less additional memory in

discovered that for all three trader types average profits along the four-cycle equal *b*2.

order to develop a specific stage of an attractor or even a (stable) higher period cycle.

purely biased beliefs, when the intensity of choice to switch strategies is high.

trading institution could 'protect' them.

Finally, we shall examine time series plots of deviations of the price from the fundamental value and of the fractions of all three types of traders. Figure 8 shows some time series corresponding to the attractors in Figures 6 and 7. We can observe that opposite biases may cause perpetual oscillations around the fundamental, even when there are no costs for fundamentalists, but can not lead to chaotic movements. Furthermore, as has already been indicated by the appearance of stable higher period cycles for high intensities of choice, in a three-type world, even when there are no costs and memory is infinite, fundamentalist beliefs can not drive out opposite

Hence, according to the argumentation of Brock and Hommes (1998, p. 1260), the market can protect a biased trader from his own folly if he is part of a group of traders whose biases are 'balanced' in the sense that they average out to zero over the set of types. Centralized market institutions can make it difficult for unbiased traders to prey on a set of biased traders provided they remain 'balanced' at zero. On the other hand, in a pit trading situation unbiased traders could learn which types are balanced and simply take the opposite side of the trade. In such situations biased traders would be eliminated, whereas a centralized

Additional memory does not change the pattern of asset prices and trader fractions *per se*, but it does affect its period. Namely, at the same intensity of choice and higher memory strength the period of these cycles appears to be elongated on average, in a way that both the negative and the positive deviation of the price from the fundamental value last longer. The same is valid for fractions ob biased traders, while in the case of fractions of fundamentalists the prolongation of the period of the irregular cycle appears in the form of less frequent 'spikes', which is understandable, since more persistent deviations of prices from the fundamental imply more space for biased traders and less chance for appearance of the fundamentalists. More memory causes the traders to stick longer to the strategy that has been profitable in the past, but might not be so profitable in the recent periods; therefore the system approaches purely quasi-

periodic dynamics when the memory strength increases at given intensity of choice.

**Notes:** Horizontal axis represents the time (*t*). Vertical axis in each set of time series plots represents deviations of the price from the fundamental value (*xt*), and the fractions of fundamentalists (*n*1,*<sup>t</sup>*), optimistic biased beliefs (*n*2,*<sup>t</sup>*) and pessimistic biased beliefs (*n*3,*<sup>t</sup>*). The plots on the left-hand side and the right-hand side of the figure differ with respect to the memory strength parameter *w*; the ones on the left correspond to *w* = 0.3, while the ones on the right to *w* = 0.9.

**Figure 8.** Time series of prices and fractions in case of fundamentalists versus opposite biases
