**2. The heterogeneous agents model**

314 Nonlinearity, Bifurcation and Chaos – Theory and Applications

traders.

market.

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

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

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 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 asset pricing model as constructed by Brock and Hommes (1998). The model can be viewed as composed of two simultaneous parts; present value asset pricing and the evolutionary selection of strategies, resulting in equilibrium pricing equation and fractions of belief types equation. We shall also make an indication of where memory in the fitness measure (and in expectation rules) enters the model and how it might affect the analysis.

### **2.1. Present value asset pricing**

The model incorporates one risky asset and one risk free asset. The latter is perfectly elastically supplied at given gross return *R*, where *R* = 1 + *r*. Investors of different types *h* have different beliefs about the conditional expectation and the conditional variance of modelling variables based on a publicly available information set consisting of past prices and dividends. The present value asset pricing part of the adaptive demand system is used to model each investor type as a myopic mean variance maximizer of expected wealth demand, *Eh,tWt*, for the risky asset:

$$RE\_{h,t} \mathcal{W}\_{t+1} = RE\_{h,t} \mathcal{W}\_t + (p\_{t+1} + \mathcal{y}\_{t+1} - \mathcal{R}p\_{t})z\_{h,t} \tag{1}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 317

, (4)

2

, (5)

2

*<sup>t</sup>* is included, which represents random fluctuations in the supply of

. (7)

. (8)

, (6)

2

Solving this optimization problem produces quantities of shares purchased by agents of different types, which enables us to seek for the equilibrium between the constant supply of

> *<sup>H</sup> <sup>s</sup> ht ht t t t*

*n kE p y Rp z*

where the fraction of traders of type *h* out of altogether *H* types at time *t* is denoted by *nh,t*,

,, 1 1

holding the risky asset. Traders will only purchase the risky asset if its expected value is equal or higher than the expected value of the risk-free asset. Since the outcome of the risky

In the simplest case of IID dividends with mean *y* and with traders having correct beliefs about dividends, i.e. *ht t* , 1 *Ey y* , the market price of the risky asset *pt* at time *t* is

> *<sup>H</sup> <sup>s</sup> t ht ht t t*

risky shares. Considering a special case with a constant zero supply of outside shares, i.e. *zs*

,, 1

If we instead consider for a moment the case of homogeneous beliefs with no noise and all

1 *<sup>s</sup> Rp E p y a z t tt*

In equilibrium the expectations of the price will be the same and equal to the fundamental price. The constant fundamental value of the price of the risky asset *p*\* in the case of

<sup>2</sup> \* \* *<sup>s</sup> Rp p y a z*

By imposing a transversality condition on expression (7) with infinitely many solutions we exclude bubble solutions (*cf*. Cuthbertson, 1996) and expression (8) now has only one

*Rp n E p y*

*t ht ht t t*

.

,, 1

1

*H*

*h*

*Rp n E p y a z*

*Rp n E p y a z*

*t ht ht t t*

*<sup>H</sup> <sup>s</sup>*

is the risk premium. The latter is an extra amount of money that traders get for

*<sup>h</sup> h t <sup>n</sup>* . The price of the risky asset is determined by market clearing, which can

and the sum of demands:

1

1

1

*h*

*h*

*h*

be seen by rewriting expression (4) in the form:

asset is uncertain, a risk premium is associated with it.

traders being rational, the pricing equation simplifies to:

homogeneous beliefs is derived from the expression:

,, 1 1

the risky asset per trader *zs*

where 1 , <sup>1</sup> *<sup>H</sup>*

where <sup>2</sup> *<sup>s</sup> a z* 

determined by:

where a noise term

= 0, we obtain:

where *pt* is the price (ex dividend) at time *t* per share of risky asset, *yt* is an IID dividend process at time *t* of the risky asset, *zh,t* is number of shares purchased at date *t* by agent of type *h*, and *R p y Rp ttt t* <sup>111</sup> is the excess return.

In order to perform myopic mean variance maximization of expected wealth demand for risky asset of type *h*, we seek for *zh,t* that solves:

$$\max\_{\mathbf{x}\_{k,t}} \left\{ E\_{\mathbf{h},t} W\_{t+1} - \frac{1}{2} a V\_{\mathbf{h},t} W\_{t+1} \right\} \tag{2}$$

and thus:

$$z\_{h,t} = \frac{E\_{h,t}\left[p\_{t+1} + y\_{t+1} - Rp\_t\right]}{aV\_{h,t}\left[p\_{t+1} + y\_{t+1} - Rp\_t\right]} = \frac{1}{a\sigma^2}E\_{h,t}\left[p\_{t+1} + y\_{t+1} - Rp\_t\right],\tag{3}$$

where the belief about expected value of wealth at time *t* + 1, conditional on all publicly available information at time *t*, for a trader of type *h* is *ht t* , <sup>1</sup> *E W* , the belief about conditional variance is *V Wht t* , <sup>1</sup> , and there is a risk factor 2 <sup>1</sup> *<sup>k</sup> a* present. Beliefs about the conditional variance of excess return are assumed to be constant and the same for all types of investors, i.e. <sup>2</sup> *Vh t*, . All traders are assumed to be equally risk averse with a given risk aversion parameter *a*, which is constant over time1.

<sup>1</sup> Gaunersdorfer (2000) investigated the case of time varying variance and supported the assumption of a constant and homogeneous variance term.

Solving this optimization problem produces quantities of shares purchased by agents of different types, which enables us to seek for the equilibrium between the constant supply of the risky asset per trader *zs* and the sum of demands:

316 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**2.1. Present value asset pricing** 

demand, *Eh,tWt*, for the risky asset:

and thus:

i.e. <sup>2</sup> *Vh t*, 

homogeneous variance term.

type *h*, and *R p y Rp ttt t* <sup>111</sup> is the excess return.

max

,1 1

*E p y Rp*

*ht t t t* 1

, 11

variance is *V Wht t* , <sup>1</sup> , and there is a risk factor 2

parameter *a*, which is constant over time1.

*ht t t t*

risky asset of type *h*, we seek for *zh,t* that solves:

asset pricing model as constructed by Brock and Hommes (1998). The model can be viewed as composed of two simultaneous parts; present value asset pricing and the evolutionary selection of strategies, resulting in equilibrium pricing equation and fractions of belief types equation. We shall also make an indication of where memory in the fitness measure (and in

The model incorporates one risky asset and one risk free asset. The latter is perfectly elastically supplied at given gross return *R*, where *R* = 1 + *r*. Investors of different types *h* have different beliefs about the conditional expectation and the conditional variance of modelling variables based on a publicly available information set consisting of past prices and dividends. The present value asset pricing part of the adaptive demand system is used to model each investor type as a myopic mean variance maximizer of expected wealth

where *pt* is the price (ex dividend) at time *t* per share of risky asset, *yt* is an IID dividend process at time *t* of the risky asset, *zh,t* is number of shares purchased at date *t* by agent of

In order to perform myopic mean variance maximization of expected wealth demand for

, , 1 ,1 1

<sup>2</sup> *h t z ht t ht t E W aV W*

, 2 ,1 1

*h t ht t t t*

where the belief about expected value of wealth at time *t* + 1, conditional on all publicly available information at time *t*, for a trader of type *h* is *ht t* , <sup>1</sup> *E W* , the belief about conditional

variance of excess return are assumed to be constant and the same for all types of investors,

1 Gaunersdorfer (2000) investigated the case of time varying variance and supported the assumption of a constant and

<sup>1</sup> *<sup>k</sup> a*

. All traders are assumed to be equally risk averse with a given risk aversion

*<sup>z</sup> E p y Rp aV p y Rp <sup>a</sup>*

,1 , 1 1 , ( ) *ht t ht t t t t ht E W RE W p y Rp z* , (1)

(2)

present. Beliefs about the conditional

, (3)

expectation rules) enters the model and how it might affect the analysis.

$$\sum\_{h=1}^{H} n\_{h,t} k E\_{h,t} \left[ p\_{t+1} + y\_{t+1} - R p\_t \right] = z^s \tag{4}$$

where the fraction of traders of type *h* out of altogether *H* types at time *t* is denoted by *nh,t*, where 1 , <sup>1</sup> *<sup>H</sup> <sup>h</sup> h t <sup>n</sup>* . The price of the risky asset is determined by market clearing, which can be seen by rewriting expression (4) in the form:

$$Rp\_t = \sum\_{h=1}^{H} n\_{h,t} E\_{h,t} \left[ p\_{t+1} + y\_{t+1} \right] - a\sigma^2 z^s \,\,\,\,\tag{5}$$

where <sup>2</sup> *<sup>s</sup> a z* is the risk premium. The latter is an extra amount of money that traders get for holding the risky asset. Traders will only purchase the risky asset if its expected value is equal or higher than the expected value of the risk-free asset. Since the outcome of the risky asset is uncertain, a risk premium is associated with it.

In the simplest case of IID dividends with mean *y* and with traders having correct beliefs about dividends, i.e. *ht t* , 1 *Ey y* , the market price of the risky asset *pt* at time *t* is determined by:

$$Rp\_t = \sum\_{h=1}^{H} n\_{h,t} E\_{h,t} \left[ p\_{t+1} \right] + \overline{y} - a\sigma^2 z^s + \varepsilon\_{t'} \tag{6}$$

where a noise term *<sup>t</sup>* is included, which represents random fluctuations in the supply of risky shares. Considering a special case with a constant zero supply of outside shares, i.e. *zs* = 0, we obtain:

$$Rp\_t = \sum\_{h=1}^{H} n\_{h,t} E\_{h,t} \left[ p\_{t+1} \right] + \overline{y} + \varepsilon\_t \ . $$

If we instead consider for a moment the case of homogeneous beliefs with no noise and all traders being rational, the pricing equation simplifies to:

$$R p\_t = E\_t \left[ p\_{t+1} \right] + \overline{y} - a \sigma^2 z^s. \tag{7}$$

In equilibrium the expectations of the price will be the same and equal to the fundamental price. The constant fundamental value of the price of the risky asset *p*\* in the case of homogeneous beliefs is derived from the expression:

$$Rp^\* = p^\* \overline{+y} - a\sigma^2 z^s. \tag{8}$$

By imposing a transversality condition on expression (7) with infinitely many solutions we exclude bubble solutions (*cf*. Cuthbertson, 1996) and expression (8) now has only one solution. We are thus able to derive the fundamental price as the discounted sum of expected future dividends:

$$p^\* = \frac{1}{R-1} \left[ \overline{y} - a\sigma^2 z^s \right]. \tag{9}$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 319

*x Rx kE x Rx C* . (13)

*k x Rx f Rx C* . (15)

, (14)

*ht t ht h t t t ht t t t h R z C p y Rp kE p y Rp C* , (12)

with constant conditional

, we are thus able to

Technical analysts or chartists, on the other hand, believe that asset prices are not completely determined by fundamentals, but may be predicted by inferences on past prices. Depending on the purpose of analysis, it is possible to distinguish between (pure) trend chasers with expectation rule , 1 ; 0 *ht h t h f gx g* , (pure) contrarians with expectation rule , 1 ; 0 *ht h t h f gx g* , and (pure) biased beliefs with expectation rule *ht h* , *f b* , where *gh* is the trend and *bh* is the bias (difference between *p*\* and trader's belief of *p*\*) of the trader of type *h*.

In order to be able to understand the dynamics of fractions of different trader types, we consider the appropriate formulations of realized excess return *Rt* from expression (1), and demand of different types of market traders, *h t*, <sup>1</sup> *z* , defined by expression (3). Taking again

formulate profits for a particular type of traders in each period as the product of realized

where *Ch* represents the costs traders have to pay to use strategy *h*. Albeit introducing additional analytical complexity, we usually take into account the costs for predictor of particular trader type, since more information-intense predictors are evidently more costly. It is of course convenient to rewrite profits of different types of traders in terms of

, 1 ,1 <sup>1</sup> ( ) *ht t t t ht t t h*

The fitness function or performance measure of each trader type can now be defined in terms of its realized profits. In fact, it can be expressed as the weighted sum of realized profits, i.e. as the sum of current realized profits and a share of past fitness, which is in turn

, , 1 , (1 ) *U wU w ht ht h t*

, 1 ,1 1 ( )( ) *ht t t ht t h*

The fitness function can for *Uh*,0 = 0 also be rewritten in the following expanded form with

, ,1 ,2 , 1 , (1 ) (1 ) ... (1 ) (1 ) *t t U w w w w ww w h t <sup>h</sup> <sup>h</sup> h t h t*

In case of the equilibrium pricing equation, herein formulated as the sum over trader types of products of a fraction of particular trader type and its deterministic function, the fitnesses

 .

 

, ,1 1 ,1 <sup>1</sup> ( )

**2.2. Evolutionary selection of strategies** 

deviations from the benchmark fundamental:

defined as past realized profits:

exponentially declining weights:

into account the nature of the dividend process *t t y y*

expectation, *y Ey <sup>t</sup>* <sup>1</sup> , and assumed distribution <sup>2</sup> IIDN(0, ) *<sup>t</sup>*

excess return and number of shares purchased by traders of that type:

where current realized profits are defined in the following final form:

1 2

By simplification of the fundamental price equation for the case of the IID dividend process with constant conditional expectation we thus obtain the standard benchmark notion of the

'fundamental', i.e. \**<sup>t</sup> <sup>y</sup> <sup>p</sup> <sup>r</sup>* , to be used in the model hereinafter.

Taking into account the appropriate form of heterogeneous beliefs of future prices, i.e. including some deterministic function *fh,t*, which can differ across trader types:

$$E\_{h,t}\left[\boldsymbol{p}\_{t+1}\right] = E\_t\left[\boldsymbol{p}\_{t+1}^\*\right] + E\_{h,t}\left[\boldsymbol{\chi}\_{t+1}\right] = \boldsymbol{p}\_{t+1}^\* + f\_h(\boldsymbol{\chi}\_{t-1}, \dots, \boldsymbol{\chi}\_{t-L}) \; ,$$

we restrict beliefs about the next deviation of the actual from the fundamental price, *xt*, to deterministic functions of past deviations from the fundamental:

$$E\_{h,t}\left[\underline{p}\_{t+1}\right] = \underline{p}^\* + f\_h(\mathbf{x}\_{t-1}, \dots, \mathbf{x}\_{t-L}) \; , \tag{10}$$

where *L* is the number of lags of past information, taken into account. Since the deterministic function in the expectation rule depends on preceding price deviations, it can also be seen as including memory. However, due to rapidly increasing analytical complexity, *viz*. including more preceding price deviations rapidly increases the dimension of the system, this issue has so far mainly been neglected. In this chapter we are focusing on the memory in the fitness measure and will thus include only one lag in the memory in the expectation rule, i.e. 1 ( ) *h t f x* .

Taking into account that \**<sup>t</sup> <sup>y</sup> <sup>p</sup> <sup>r</sup>* , the equilibrium pricing equation (5) can thus finally be rewritten in terms of deviations from the fundamental price, *xt* = *pt* – *p*\*:

$$R\mathbf{x}\_t = \sum\_{h=1}^H n\_{h,t} E\_{h,t} \left[ \mathbf{x}\_{t+1} \right] = \sum\_{h=1}^H n\_{h,t} f\_{h,t} \ . \tag{11}$$

The particular form of deterministic function in the forecasting or expectation rule is thus what determines different types of heterogeneous agents in an adaptive belief system. In general, we distinguish between two typical investor types; fundamentalists and 'noise traders' or technical analysts. Fundamentalists believe that the price of an asset is defined solely by its efficient market hypothesis fundamental value (Fama, 1991), i.e. the present value of the stream of future dividends. Since they have no knowledge about other beliefs and fractions, *fh,t* 0. Actual financial data show that fundamentalists have a stabilizing effect on prices (De Grauwe and Grimaldi, 2006).

Technical analysts or chartists, on the other hand, believe that asset prices are not completely determined by fundamentals, but may be predicted by inferences on past prices. Depending on the purpose of analysis, it is possible to distinguish between (pure) trend chasers with expectation rule , 1 ; 0 *ht h t h f gx g* , (pure) contrarians with expectation rule , 1 ; 0 *ht h t h f gx g* , and (pure) biased beliefs with expectation rule *ht h* , *f b* , where *gh* is the trend and *bh* is the bias (difference between *p*\* and trader's belief of *p*\*) of the trader of type *h*.

### **2.2. Evolutionary selection of strategies**

318 Nonlinearity, Bifurcation and Chaos – Theory and Applications

*<sup>y</sup> <sup>p</sup>*

expected future dividends:

'fundamental', i.e. \**<sup>t</sup>*

expectation rule, i.e. 1 ( ) *h t*

Taking into account that \**<sup>t</sup>*

*f x* .

effect on prices (De Grauwe and Grimaldi, 2006).

*<sup>y</sup> <sup>p</sup>*

rewritten in terms of deviations from the fundamental price, *xt* = *pt* – *p*\*:

solution. We are thus able to derive the fundamental price as the discounted sum of

<sup>1</sup> <sup>2</sup> \* <sup>1</sup> *<sup>s</sup> p ya z <sup>R</sup>*

*<sup>r</sup>* , to be used in the model hereinafter.

including some deterministic function *fh,t*, which can differ across trader types:

deterministic functions of past deviations from the fundamental:

By simplification of the fundamental price equation for the case of the IID dividend process with constant conditional expectation we thus obtain the standard benchmark notion of the

Taking into account the appropriate form of heterogeneous beliefs of future prices, i.e.

\* \* ,1 1 ,1 1 1 ( ,..., ) *ht t t t ht t t h t t L E p Ep E x p fx x* ,

we restrict beliefs about the next deviation of the actual from the fundamental price, *xt*, to

where *L* is the number of lags of past information, taken into account. Since the deterministic function in the expectation rule depends on preceding price deviations, it can also be seen as including memory. However, due to rapidly increasing analytical complexity, *viz*. including more preceding price deviations rapidly increases the dimension of the system, this issue has so far mainly been neglected. In this chapter we are focusing on the memory in the fitness measure and will thus include only one lag in the memory in the

> , , 1 ,, 1 1

*H H t ht ht t ht ht h h Rx n E x n f* 

The particular form of deterministic function in the forecasting or expectation rule is thus what determines different types of heterogeneous agents in an adaptive belief system. In general, we distinguish between two typical investor types; fundamentalists and 'noise traders' or technical analysts. Fundamentalists believe that the price of an asset is defined solely by its efficient market hypothesis fundamental value (Fama, 1991), i.e. the present value of the stream of future dividends. Since they have no knowledge about other beliefs and fractions, *fh,t* 0. Actual financial data show that fundamentalists have a stabilizing

. (9)

, 1 <sup>1</sup> \* ( ,..., ) *ht t h t tL E p p fx x* , (10)

*<sup>r</sup>* , the equilibrium pricing equation (5) can thus finally be

. (11)

In order to be able to understand the dynamics of fractions of different trader types, we consider the appropriate formulations of realized excess return *Rt* from expression (1), and demand of different types of market traders, *h t*, <sup>1</sup> *z* , defined by expression (3). Taking again into account the nature of the dividend process *t t y y* with constant conditional expectation, *y Ey <sup>t</sup>* <sup>1</sup> , and assumed distribution <sup>2</sup> IIDN(0, ) *<sup>t</sup>* , we are thus able to formulate profits for a particular type of traders in each period as the product of realized excess return and number of shares purchased by traders of that type:

$$\boldsymbol{\pi}\_{h,t} = \boldsymbol{R}\_t \boldsymbol{z}\_{h,t-1} - \boldsymbol{C}\_h = (\boldsymbol{p}\_t + \boldsymbol{y}\_t - \boldsymbol{R}\boldsymbol{p}\_{t-1})\boldsymbol{k}\boldsymbol{E}\_{h,t-1} \left[\boldsymbol{p}\_t + \boldsymbol{y}\_t - \boldsymbol{R}\boldsymbol{p}\_{t-1}\right] - \boldsymbol{C}\_h\tag{12}$$

where *Ch* represents the costs traders have to pay to use strategy *h*. Albeit introducing additional analytical complexity, we usually take into account the costs for predictor of particular trader type, since more information-intense predictors are evidently more costly. It is of course convenient to rewrite profits of different types of traders in terms of deviations from the benchmark fundamental:

$$
\Delta \pi\_{h,t} = (\mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1} + \boldsymbol{\delta}\_t) k E\_{h,t-1} \left[ \mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1} \right] - \mathbf{C}\_h \tag{13}
$$

The fitness function or performance measure of each trader type can now be defined in terms of its realized profits. In fact, it can be expressed as the weighted sum of realized profits, i.e. as the sum of current realized profits and a share of past fitness, which is in turn defined as past realized profits:

$$\mathcal{L}I\_{h,t} = \text{\textquotedblleft} \mathcal{L}I\_{h,t-1} + (1 - \text{\textquotedblright})\pi\_{h,t} \text{\textquotedblright} \tag{14}$$

where current realized profits are defined in the following final form:

$$\mathbf{x}\_{h,t} = \mathbf{k}(\mathbf{x}\_t - \mathbf{R}\mathbf{x}\_{t-1})(f\_{h,t-1} - \mathbf{R}\mathbf{x}\_{t-1}) - \mathbf{C}\_h \,. \tag{15}$$

The fitness function can for *Uh*,0 = 0 also be rewritten in the following expanded form with exponentially declining weights:

$$dL\_{h,t} = w^{t-1}(1 - w)\pi\_{h,1} + w^{t-2}(1 - w)\pi\_{h,2} + \dots + w(1 - w)\pi\_{h,t-1} + (1 - w)\pi\_{h,t} \cdot \frac{1}{1 - w}$$

In case of the equilibrium pricing equation, herein formulated as the sum over trader types of products of a fraction of particular trader type and its deterministic function, the fitnesses

enter the adaptive belief system before the equilibrium price is observed. This is suitable for analyzing the asset pricing model as an explicit nonlinear difference equation. Even though nonlinear asset pricing dynamics can be modelled either as a deterministic or a stochastic process, only the latter enables investigation of the effects of noise upon the asset pricing dynamics.

The share of past fitness in the performance measure is expressed by the parameter *w*; 0 *w* 1, called memory strength. When the value of this parameter is zero (*w* = 0), the fitness is given by most recent net realized profit. Due to analytical tractability this is at present, y for the most part, the case in the existing literature on asset pricing models with heterogeneous agents, though not in this chapter. The main contribution of this chapter is that it analyzes the case of nonzero memory in the fitness measure. When the memory strength parameter takes a positive value, some share of current realized profits in any given period is taken into account when calculating the performance measure in the next time period. If the value of memory strength parameter amounts to one then of course the entire accumulated wealth is taken into account.

The expression (14) for the fitness function is somewhat different that the one used in Brock and Hommes (1998), where the coefficient of the current realized profits was fixed to 1. Namely, if we rewrite the memory strength parameter as <sup>1</sup> *w* 1 *<sup>T</sup>* , where *T* is considered to be a specific number of time periods, we obtain the following expression for the fitness function:

$$
\mathcal{U}I\_{h,t} = \left(1 - \frac{1}{T}\right) \mathcal{U}\_{h,t-1} + \frac{1}{T} \pi\_{h,t} \, \prime \tag{16}
$$

Memory and Asset Pricing Models with Heterogeneous Beliefs 321

1, <sup>0</sup> *<sup>t</sup> <sup>f</sup>* (18)

2, 1; 0 *t t f gx g* (19)

. (20)

. (21)

Trader fractions are therefore determined by fitness and intensity of choice. Rationality in the asset pricing model is evidently bounded, since fractions are ranked according to fitness, but not all agents choose the best predictor. To ensure that fractions of belief types depend only upon observable deviations from the fundamental at any given time period, fitness function in the fractions of belief types equation may only depend on past fitness and past return. This indeed ensures that past realized profits are observable quantities that can be

One might wonder whether the traders' myopic mean-variance maximization is a reasonable assumption, especially when we allow for traders with a longer memory span. This assumption is widely used in modelling in economics and finance, though it would certainly be interesting to let traders plan longer ahead, even with an infinite planning horizon, as in the Lucas (1978) asset pricing model. However, in this kind of model one usually assumes perfect rationality to keep the analysis tractable. So far very little work has been done on infinite horizon models with bounded rationality and heterogeneous beliefs. Furthermore, one can also discuss whether individuals are really able to plan over a long horizon, or whether they might use simple heuristics over a short horizon and occasionally adapt them. After all, memory in the fitness measure is not equivalent to the planning horizon, but rather an "evaluation horizon" used to decide whether or not to switch strategies. There is empirical and experimental evidence that humans give more weight to

The first case we are going to examine is a two-type heterogeneous agents model with fundamentalists and contrarians as market participants. Fundamentalists exhibit deterministic

and have some positive information gathering costs *C*, i.e. *C* > 0. Contrarians exhibit a

and zero information gathering costs. It is thus a case of fundamentalists versus pure

, 1

 

*h t*

*U n h U U* 

exp ; 1,2

1, 1 2, 1

1, 1 2, 1

1, 2, 1, 1 2, 1 1, 1 2, 1 exp exp tanh exp exp 2 *t t ttt t t t t*

 

 

   

*t t*

contrarians. We have the following fractions of belief types equation:

For convenience we shall also introduce a difference in fractions *mt*:

exp exp

*U U mn n U U U U*

,

*h t*

the recent past than the far distant past, and this is formalized in our model.

**3. Fundamentalists versus Contrarians** 

used in predictor selection.

function of the form:

deterministic function:

which is equivalent to taking the last *T* observations into account with equal weight (as benchmark). When *T* approaches infinity, the memory parameter approaches 1 and the entire accumulated wealth is taken into account. We thus believe the expression (14) to be a more suitable formulation of the fitness measure than the one used in Brock and Hommes (1998), and in several other contributions.

Finally, we can express fractions of belief types, *nh,t*, which are updated in each period, as a discrete choice probability by a multinomial logit model:

$$m\_{h,t} = \frac{\exp\left[\beta \mathcal{U} I\_{h,t-1}\right]}{\sum\_{i=1}^{H} \exp\left[\beta \mathcal{U} I\_{i,t-1}\right]} \text{ \,\,\,\tag{17}$$

by using parameter , determining the intensity of choice. The latter measures how fast economic agents switch between different prediction strategies; if the value of intensity of choice is zero, then all trader types have equal weight and the mass of traders distributes itself evenly across the set of available strategies, while on the other hand the entire mass of traders tends to use the best predictor, i.e. the strategy with the highest fitness, when the intensity of choice approaches infinity (the neoclassical limit).

Trader fractions are therefore determined by fitness and intensity of choice. Rationality in the asset pricing model is evidently bounded, since fractions are ranked according to fitness, but not all agents choose the best predictor. To ensure that fractions of belief types depend only upon observable deviations from the fundamental at any given time period, fitness function in the fractions of belief types equation may only depend on past fitness and past return. This indeed ensures that past realized profits are observable quantities that can be used in predictor selection.

One might wonder whether the traders' myopic mean-variance maximization is a reasonable assumption, especially when we allow for traders with a longer memory span. This assumption is widely used in modelling in economics and finance, though it would certainly be interesting to let traders plan longer ahead, even with an infinite planning horizon, as in the Lucas (1978) asset pricing model. However, in this kind of model one usually assumes perfect rationality to keep the analysis tractable. So far very little work has been done on infinite horizon models with bounded rationality and heterogeneous beliefs. Furthermore, one can also discuss whether individuals are really able to plan over a long horizon, or whether they might use simple heuristics over a short horizon and occasionally adapt them. After all, memory in the fitness measure is not equivalent to the planning horizon, but rather an "evaluation horizon" used to decide whether or not to switch strategies. There is empirical and experimental evidence that humans give more weight to the recent past than the far distant past, and this is formalized in our model.
