**4. Fundamentalists versus opposite biased beliefs**

The second case we are going to examine is a three-type heterogeneous agents model with fundamentalists and opposite biased beliefs as market participants. Fundamentalists again exhibit a deterministic function of the form:

$$f\_{1,t} \equiv 0 \,, \tag{41}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 333

 

> 

, (49)

1 1 , 1 , 1

*U U*

*i t i t i i*

 

*t t*

2, 1 1, 1 *tt t xx x* , (50)

. (54)

(55)

exp exp

*b b*

*u U wu w kRx x Rx* 1, 1, 1 1, 1 *tt t* (1 ) 2, 1 1, 1 2, 1 *tt t* , (51)

*u U wu w k x Rx b Rx* 2, 2, 1 2, 1 *tt t* (1 ) 1, 1 2, 1 2 2, 1 *tt t* , (52)

*u U wu w k x Rx b Rx* 3, 3, 1 3, 1 *tt t* (1 ) 1, 1 2, 1 3 2, 1 *tt t* . (53)

 

where *X xxuuu t t tttt* 1 1, 1 2, 1 1, 1 2, 1 3, 1 ,,,, is a vector of new variables, defined as:

 2, 1 3, 1 1, 2, 2 3, 3 3 3 2 3

2, 3, 3 3 2 3 1 1 , ,

*u u*

*t t*

exp exp

 

Our three-type heterogeneous agents model of fundamentalists versus biased beliefs in

We obtain the fundamental steady state for 2 3 *b bb* 0 (opposite biased beliefs), where <sup>0</sup> *eq <sup>x</sup>* . This is implied by 123 <sup>0</sup> *eq eq eq uuu* when *<sup>w</sup>* 1 and consequently by

**Lemma 4:** *The fundamental steady state in the case of fundamentalists versus opposite biased beliefs* 

*We will prove a more general result for the case with h = 1, …, H purely biased types bh (including* 

*H t ht h h Rx n b* 

, 1

,

*is a unique steady state of the system. Memory does not affect the position of this steady state.* 

*fundamentalists with b1 = 0). Proceeding from the non-transformed variables the system is:* 

 22 33 <sup>1</sup> *x nb nb R*

*R u u*

*i t i t i i*

1, 1 1 : *t t x x* , 2, 1 2 : *t t x x* , 1, 1 1, 2 : *t t u U* , 2, 1 2, 2 : *t t u U* and 3, 1 3, 2 : *t t u U* .

*tt t t*

We therefore obtain the following 5-dimensional first-order difference equation:

1 exp exp

general can have the following steady state price deviations:

*eq eq eq nnn* , originating from the rewritten expression (44).

By performing a generalization we can state the following lemma.

123

**Proof of Lemma 4:** 

1 3 1 1 exp exp

*x x nb nb b b R R U U* 

though this time with no information gathering costs, i.e. *C* = 0. Biased beliefs exhibit deterministic functions:

$$f\_{2,t} = b\_2; \quad b\_2 > 0 \, , \tag{42}$$

$$f\_{3,t} = b\_3; \quad b\_3 < 0 \; \text{s} \tag{43}$$

for optimist and pessimist biases, respectively6. Biases also exhibit zero information gathering costs. We have the following fractions of belief types equation:

$$m\_{h,t} = \frac{\exp\left[\beta \mathcal{U}\_{h,t-1}\right]}{\sum\_{i=1}^{3} \exp\left[\beta \mathcal{U}\_{i,t-1}\right]}; \quad h = 1,2,3\ \text{s.} \tag{44}$$

Finally, we have the fitness measures of each type:

$$
\Delta I\_{1,t} = w \, \mathcal{U}\_{1,t-1} + (1 - w) \left[ -k \mathcal{R} \mathbf{x}\_{t-1} \left( \mathbf{x}\_t - \mathcal{R} \mathbf{x}\_{t-1} \right) \right], \tag{45}
$$

$$\mathbf{U}L\_{2,t} = w\mathbf{U}L\_{2,t-1} + (\mathbf{1} - w) \left[ k \left( \mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1} \right) \left( \mathbf{b}\_2 - \mathbf{R}\mathbf{x}\_{t-1} \right) \right],\tag{46}$$

$$\mathbb{L}L\_{3,t} = w\mathbb{L}I\_{3,t-1} + (1-w)\left[k\left(\mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1}\right)\left(\mathbf{b}\_3 - \mathbf{R}\mathbf{x}\_{t-1}\right)\right].\tag{47}$$

In order to analyze memory in our heterogeneous asset pricing model, we shall first determine the position and stability of the steady state, and then examine the possible qualitative changes in dynamics; all in relation to the memory strength parameter. Then we shall perform some numerical simulations to combine global stability analysis with local stability analysis.

### **4.1. Position of the steady state**

In our three-type heterogeneous agents model of fundamentalists versus biased beliefs, we shall again start by rewriting our system as a difference equation:

$$X\_t = F\_2\left(X\_{t-1}\right),\tag{48}$$

<sup>6</sup> In this chapter we will mainly focus on the symmetric case.

where *X xxuuu t t tttt* 1 1, 1 2, 1 1, 1 2, 1 3, 1 ,,,, is a vector of new variables, defined as: 1, 1 1 : *t t x x* , 2, 1 2 : *t t x x* , 1, 1 1, 2 : *t t u U* , 2, 1 2, 2 : *t t u U* and 3, 1 3, 2 : *t t u U* .

We therefore obtain the following 5-dimensional first-order difference equation:

$$\mathbf{x}\_{1,t} = \mathbf{x}\_t = \frac{1}{R} \left( n\_{2,t} b\_2 + n\_{3,t} b\_3 \right) = \frac{1}{R} \left( \frac{\exp\left[\beta \mathcal{U} I\_{2,t-1}\right]}{\sum\_{i=1}^3 \exp\left[\beta \mathcal{U} I\_{i,t-1}\right]} b\_2 + \frac{\exp\left[\beta \mathcal{U} I\_{3,t-1}\right]}{\sum\_{i=1}^3 \exp\left[\beta \mathcal{U} I\_{i,t-1}\right]} b\_3 \right) = \frac{1}{R} \left( \frac{\exp\left[\beta \mathcal{U} I\_{2,t-1}\right]}{\sum\_{i=1}^3 \exp\left[\beta \mathcal{U}\_{i,t}\right]} b\_2 \right), \tag{49}$$

$$= \frac{1}{R} \left( \frac{\exp\left[\beta \mathcal{U}\_{2,t}\right]}{\sum\_{i=1}^3 \exp\left[\beta \mathcal{U}\_{i,t}\right]} b\_2 + \frac{\exp\left[\beta \mathcal{U}\_{3,t}\right]}{\sum\_{i=1}^3 \exp\left[\beta \mathcal{U}\_{i,t}\right]} b\_3 \right), \tag{40}$$

$$\mathbf{x}\_{2,t} = \mathbf{x}\_{t-1} = \mathbf{x}\_{1,t-1} \,\prime \tag{50}$$

$$\mathbf{u}\_{1,t} = \mathbf{U}\_{1,t-1} = \mathbf{w}u\_{1,t-1} + (\mathbf{1} - \mathbf{w}) \left[ -k \mathbf{R} \mathbf{x}\_{2,t-1} (\mathbf{x}\_{1,t-1} - \mathbf{R} \mathbf{x}\_{2,t-1}) \right] \tag{51}$$

$$\mathbf{u}\_{2,t} = \mathbf{L}\mathbf{I}\_{2,t-1} = w\mathbf{u}\_{2,t-1} + (\mathbf{1} - \mathbf{w}) \left[ k \left( \mathbf{x}\_{1,t-1} - \mathbf{R}\mathbf{x}\_{2,t-1} \right) \left( \mathbf{b}\_2 - \mathbf{R}\mathbf{x}\_{2,t-1} \right) \right] \tag{52}$$

$$\mathbf{u}\_{3,t} = \mathbf{L}\mathbf{I}\_{3,t-1} = \mathbf{w}\mathbf{u}\_{3,t-1} + (\mathbf{1} - \mathbf{w}) \left[ k \left( \mathbf{x}\_{1,t-1} - \mathbf{R}\mathbf{x}\_{2,t-1} \right) \left( \mathbf{b}\_3 - \mathbf{R}\mathbf{x}\_{2,t-1} \right) \right]. \tag{53}$$

Our three-type heterogeneous agents model of fundamentalists versus biased beliefs in general can have the following steady state price deviations:

$$\propto = \frac{1}{R} \left( n\_2 b\_2 + n\_3 b\_3 \right). \tag{54}$$

We obtain the fundamental steady state for 2 3 *b bb* 0 (opposite biased beliefs), where <sup>0</sup> *eq <sup>x</sup>* . This is implied by 123 <sup>0</sup> *eq eq eq uuu* when *<sup>w</sup>* 1 and consequently by 123 1 3 *eq eq eq nnn* , originating from the rewritten expression (44).

By performing a generalization we can state the following lemma.

**Lemma 4:** *The fundamental steady state in the case of fundamentalists versus opposite biased beliefs is a unique steady state of the system. Memory does not affect the position of this steady state.* 

### **Proof of Lemma 4:**

332 Nonlinearity, Bifurcation and Chaos – Theory and Applications

exhibit a deterministic function of the form:

deterministic functions:

stability analysis.

**4.1. Position of the steady state** 

6 In this chapter we will mainly focus on the symmetric case.

**4. Fundamentalists versus opposite biased beliefs** 

The second case we are going to examine is a three-type heterogeneous agents model with fundamentalists and opposite biased beliefs as market participants. Fundamentalists again

though this time with no information gathering costs, i.e. *C* = 0. Biased beliefs exhibit

for optimist and pessimist biases, respectively6. Biases also exhibit zero information

, 1

*U*

In order to analyze memory in our heterogeneous asset pricing model, we shall first determine the position and stability of the steady state, and then examine the possible qualitative changes in dynamics; all in relation to the memory strength parameter. Then we shall perform some numerical simulations to combine global stability analysis with local

In our three-type heterogeneous agents model of fundamentalists versus biased beliefs, we

*h t*

 

exp ; 1,2,3

1, 1, 1 <sup>1</sup> <sup>1</sup> (1 ) *U wU w kRx x Rx t t tt t* , (45)

2, 2, 1 12 1 (1 ) *U wU w k x Rx b Rx t t tt t* , (46)

3, 3, 1 13 1 (1 ) *U wU w k x Rx b Rx t t tt t* . (47)

*X FX t t* 2 1 , (48)

1 , 1

*i t i*

*U n h*

exp

gathering costs. We have the following fractions of belief types equation:

, 3

shall again start by rewriting our system as a difference equation:

*h t*

Finally, we have the fitness measures of each type:

1, 0 *<sup>t</sup> f* , (41)

2, 2 2 ; 0 *<sup>t</sup> f bb* , (42)

3, 3 3 ; 0 *<sup>t</sup> f bb* , (43)

. (44)

*We will prove a more general result for the case with h = 1, …, H purely biased types bh (including fundamentalists with b1 = 0). Proceeding from the non-transformed variables the system is:* 

$$R\boldsymbol{\infty}\_t = \sum\_{h=1}^H \boldsymbol{n}\_{h,t} \boldsymbol{b}\_{h'} \tag{55}$$

$$m\_{h,t} = \frac{\exp\left[\beta \left(wL\_{h,t-2} + (1-w)\left[k\left(\mathbf{x}\_{t-1} - \mathbf{R}\mathbf{x}\_{t-2}\right)\left(b\_h - \mathbf{R}\mathbf{x}\_{t-2}\right)\right]\right)\right]}{\sum\_{i=1}^{H} \exp\left[\beta \left(wL\_{i,t-2} + (1-w)\left[k\left(\mathbf{x}\_{t-1} - \mathbf{R}\mathbf{x}\_{t-2}\right)\left(b\_i - \mathbf{R}\mathbf{x}\_{t-2}\right)\right]\right)\right]}; \quad 1 \le h \le H. \tag{56}$$

*After subtracting off identical terms from the exponents of both numerator and denominator in expression (56) we obtain a new expression for the fractions:* 

$$m\_{h,t} = \frac{\exp\left[\beta \left(wU\_{h,t-2}^\circ + (1-w)k\left(\mathbf{x}\_{t-1} - \mathbf{R}\mathbf{x}\_{t-2}\right)b\_h\right)\right]}{\sum\_{i=1}^H \exp\left[\beta \left(wU\_{i,t-2}^\circ + (1-w)k\left(\mathbf{x}\_{t-1} - \mathbf{R}\mathbf{x}\_{t-2}\right)b\_i\right)\right]}; \quad 1 \le h \le H\_\prime \tag{57}$$

*where Uh t*,  *is the fitness of trader type h, adjusted by subtracting off identical terms as above. The dynamic system defined by (55) and (57) is thus of the form:* 

$$R\mathbf{x}\_t = V\_{\beta k}(\mathbf{x}\_{t-1} - R\mathbf{x}\_{t-2}),\tag{58}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 335

*H*

 

 

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

*. Memory affects the stability of this steady state by restricting* 

. Similarly, the condition for stability in terms of w indicates 2

, while in terms of the memory strength this is

0 ,

1 , which in our case is limited to the product

.

. In terms of the intensity

*, so that x\* equals the* 

*<sup>k</sup> ,* 

, (62)

2,3 *w* and

<sup>3</sup> <sup>1</sup> 2

.

*k b* 

*w*

*k h <sup>b</sup> V b*

*fundamental steady state ifequation reference goes here and only if b* 0 *, i.e. when all biases are exactly balanced. Since there is no memory strength parameter left in expressions (60) and* (0) *V*

*memory does not affect the position of this steady state. It has to be mentioned though, that our* 

The local stability of a steady state is again determined by the eigenvalues of the Jacobian matrix. At the fundamental steady state *Xeq* = (0, 0, 0, 0, 0) the Jacobian matrix exhibits the

<sup>2</sup> 22 2 2 2 ( ) ( 1) ( 1) <sup>0</sup>

 

3 3 *<sup>g</sup> w kb w kb w w*

<sup>2</sup> <sup>2</sup> <sup>2</sup> 2 2

**Lemma 5:** *The fundamental steady state in case fundamentalists versus opposite biased beliefs is* 

From the characteristic equation (62) we can observe five eigenvalues. The first three eigenvalues always assure stability, while the last two eigenvalues limit stability. Given k > 0, b > 0, 0, R > 1 and 0 w 1, the condition for stability in terms of implies

 

*R*

 

which has the following three solutions, two of them being double: 1

<sup>1</sup> 2 (1 ) 3 2 ( 1) 3 24 (1 ) <sup>6</sup> *b k w Rw b k w Rw b k w R*

being smaller than one, i.e. <sup>2</sup> <sup>2</sup>

3 2 ( 1) *kb w*

 

*state x\* has to be unique. From expression (59) we obtain* <sup>1</sup> (0) *<sup>H</sup> <sup>h</sup>*

**4.2. Stability of the steady state and bifurcations** 

characteristic equation that is in our case given by:

 

The fundamental steady state is stable for

3 2 ( 1) *kb w*

Memory therefore affects the stability of the steady state as shown.

guaranteed for 2

*globally stable for* <sup>2</sup>

**Proof of Lemma 5:** 

2 3 2 ( 1) *kb w*

*w*

of choice this happens for 2

Thus we can state the following lemma.

*it to the given interval of the parameter value.* 

<sup>3</sup> <sup>1</sup> 2

.

*k b* 

4,5

*R* 

of eigenvalues 4,5

*derivation holds for finite intensity of choice, since fractions are only then all positive.* 

*where the right-hand side function is defined as:* 

$$W\_{\beta k}(y\_t) = \frac{\exp\left[\beta \left(w \wr I\_{h, t-2}^\circ(y\_{t-1}) + (1 - w)kb\_h y\_t\right)\right]}{\sum\_{i=1}^H \exp\left[\beta \left(w \wr I\_{i, t-2}^\circ(y\_{t-1}) + (1 - w)kb\_i y\_t\right)\right]} = \sum\_{h=1}^H b\_h n\_h = \left\langle b\_h \right\rangle. \tag{59}$$

*Since it follows from (52) and (53) that*  \* \* \* <sup>1</sup> *U kx R b Rx h h , steady states of expressions (55) and (57) or expression (58) are determined by:* 

$$\text{Rx}^\* = V\_{\beta k} (\text{x}^\* - \text{Rx}^\*) = V\_{\beta k} (-r\text{x}^\*) \tag{60}$$

*where r = R – 1. Since a steady state has to satisfy expression (60), following Brock and Hommes (1998, p. 1271), a straightforward computation shows that:* 

$$\frac{\mathbf{d}}{\mathbf{d}\mathbf{y}} V\_{\beta\mathbf{k}}(\mathbf{y}) = \sum\_{h=1}^{H} \left( \frac{\beta k b\_h \exp\left[\beta k b\_h \mathbf{y}\right]}{\sum\_{i=1}^{H} \exp\left[\beta k b\_i \mathbf{y}\right]} - \frac{\exp\left[\beta k b\_h \mathbf{y}\right]}{\left(\sum\_{i=1}^{H} \exp\left[\beta k b\_i \mathbf{y}\right]\right)^2} \cdot \frac{\mathbf{d}}{\mathbf{d}\mathbf{y}} \left(\sum\_{i=1}^{H} \exp\left[\beta k b\_i \mathbf{y}\right]\right) \bigg|\_{\mathbf{b}} \mathbf{b}\_h = \mathbf{0}$$

$$= \sum\_{h=1}^{H} \left(\beta k n\_h b\_h^2 - \beta k n\_h b\_h \sum\_{h=1}^{H} n\_h b\_h\right) = \sum\_{h=1}^{H} \left(\beta k n\_h b\_h^2 - \beta k n\_h b\_h \left\{b\_h\right\}\right) =$$

$$= \beta k \left[\left\langle b\_h^2 \right\rangle - \left\langle b\_h \right\rangle^2\right] > 0,\tag{61}$$

*where the inequality follows from the fact that the term between square brackets can be interpreted as the variance of the stochastic process, where each bh is drawn with probability nh. Therefore,* ( ) *V y k is increasing and* \* ( ) *V rx <sup>k</sup> decreasing in x\*. It then follows from expression (60) that the steady*  *state x\* has to be unique. From expression (59) we obtain* <sup>1</sup> (0) *<sup>H</sup> <sup>h</sup> k h <sup>b</sup> V b H , so that x\* equals the fundamental steady state ifequation reference goes here and only if b* 0 *, i.e. when all biases are* 

*exactly balanced. Since there is no memory strength parameter left in expressions (60) and* (0) *V<sup>k</sup> ,* 

*memory does not affect the position of this steady state. It has to be mentioned though, that our derivation holds for finite intensity of choice, since fractions are only then all positive.* 
