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where a is the risk aversion constant. For the constant conditional

variance Vi, t Â¼ s2 , the CARA function yields the demand

Ei, t [rtÃ¾1 ] Ã€ r

pi, t Â¼ (12:2:9)

as2

The number of shares of the risky asset that corresponds to demand

pi, t equals

Ni, t Â¼ pi, t Wi, t =pt (12:2:10)

Since the total number of shares assumed to be fixed

P

Ni, t Â¼ N Â¼ const , the market-clearing price equals

i

1X

pt Â¼ pi, t Wi, t (12:2:11)

Ni

The adaptive equilibrium model described so far does not contradict

the classical asset pricing theory. The new concept in this model is the

heterogeneous beliefs. In its general form [7, 10]

Ei, t [rtÃ¾1 ] Â¼ fi (rtÃ€1 , . . . , rtÃ€Li ), (12:2:12)

Vi, t [rtÃ¾1 ] Â¼ gi (rtÃ€1 , . . . , rtÃ€Li ) (12:2:13)

The deterministic functions fi and gi depend on past returns with lags

up to Li and may vary for different agents.3

While variance is usually assumed to be constant (gi Â¼ s2 ), several

trading strategies fi are discussed in the literature. First, there are

fundamentalists who use analysis of the business fundamentals to

make their forecasts on the risk premium dF

EF, t [rtÃ¾1 ] Â¼ r Ã¾ dF (12:2:14)

In simple models, the risk premium dF > 0 is a constant but it can be a

function of time and/or variance in the general case. Another major

strategy is momentum trading (traders who use it are often called

chartists). Momentum traders use history of past returns to make

their forecasts. Namely, their strategy can be described as

133

Agent-Based Modeling of Financial Markets

X

L

EM, t [rtÃ¾1 ] Â¼ r Ã¾ dM Ã¾ ak rtÃ€k (12:2:15)

kÂ¼1

where dM > 0 is the constant component of the momentum risk

premium and ak > 0 are the weights of past returns rtÃ€k . Finally,

contrarians employ the strategy that is formally similar to the momen-

tum strategy

X

L

EC, t [rtÃ¾1 ] Â¼ r Ã¾ dC Ã¾ bk rtÃ€k (12:2:16)

kÂ¼1

with the principal difference that all bk are negative. This implies that

contrarians expect the market to turn around (e.g., from bull market

to bear market).

An important feature of adaptive equilibrium models is that agents

are able to analyze performance of different strategies and choose the

most efficient one. Since these strategies have limited accuracy, such

adaptability is called bounded rationality.

In the limit of infinite number of agents, Brock and Hommes offer

a discrete analog of the Gibbs probability distribution for the fraction

of traders with the strategy i [7]

X

nit Â¼ exp [b(Fi, tÃ€1 Ã€ Ci )]=Zt , Zt Â¼ exp [b(Fi, tÃ€1 Ã€ Ci )] (12:2:17)

i

In (12.2.17), Ci ! 0 is the cost of the strategy i, the parameter b is

called the intensity of choice, and Fi, t is the fitness function that

characterizes the efficiency of strategy i. The natural choice for the

fitness function is

Fi, t Â¼ gFi, tÃ€1 Ã¾ wi, t , wi, t Â¼ pi, t (Wi, t Ã€ Wi, tÃ€1 )=Wi, tÃ€1 (12:2:18)

where 0 g 1 is the memory parameter that retains part of past

performance in the current strategy.

Adaptive equilibrium models have been studied in several direc-

tions. Some work has focused on analytic analysis of simpler models.

In particular, the system stability and routes to chaos have been

discussed in [7, 10]. In the meantime, extensive computational model-

ing has been performed in [9] and particularly for the so-called Santa

Fe artificial market, in which a significant number of trading strat-

egies were implemented [8].

134 Agent-Based Modeling of Financial Markets

12.3 NON-EQUILIBRIUM PRICE MODELS

The concept of market clearing that is used in determining price of

the risky asset in the adaptive equilibrium models does not accurately

reflect the way real markets work. In fact, the number of shares

involved in trading varies with time, and price is essentially a dynamic

variable. A simple yet reasonable alternative to the price-clearing

paradigm is the equation of price formation that is based on the

empirical relation between price change and excess demand [4].

Different agent decision-making rules may be implemented within

this approach. Here the elaborated model offered by Lux [11] is

described. In this model, two groups of agents, namely chartists and

fundamentalists, are considered. Agents can compare the efficiency of

different trading strategies and switch from one strategy to another.

Therefore, the numbers of chartists, nc (t), and fundamentalists, nf (t),

vary with time while the total number of agents in the market N is

assumed constant. The chartist group in turn is sub-divided into

optimistic (bullish) and pessimistic (bearish) traders with the numbers

nÃ¾ (t) and nÃ€ (t), respectively

nc (t) Ã¾ nf (t) Â¼ N, nÃ¾ (t) Ã¾ nÃ€ (t) Â¼ nc (t) (12:3:1)

Several aspects of trader behavior are considered. First, the chartist

decisions are affected by the peer opinion (so-called mimetic conta-

gion). Secondly, traders change strategy while seeking optimal per-

formance. Finally, traders may exit and enter markets. The bullish

chartist dynamics is formalized in the following way:

dnÃ¾ =dt Â¼ (nÃ€ pÃ¾Ã€ Ã€ nÃ¾ pÃ€Ã¾ )(1 Ã€ nf =N) Ã¾ mimetic contagion

nf nÃ¾ (pÃ¾f Ã€ pfÃ¾ )=N Ã¾ changes of strategy

(b Ã€ a)nÃ¾ market entry and exit (12:3:2)

Here, pab denotes the probability of transition from group b to group

a. Similarly, the bearish chartist dynamics is given by

dnÃ€ =dt Â¼ (nÃ¾ pÃ€Ã¾ Ã€ nÃ€ pÃ¾Ã€ )(1 Ã€ nf =N) Ã¾ mimetic contagion

nf nÃ€ (pÃ€f Ã€ pfÃ€ )=N Ã¾ changes of strategy

(b Ã€ a)nÃ€ market entry and exit (12:3:3)

It is assumed that traders entering the market start with the chartist

strategy. Therefore, constant total number of traders yields the

135

Agent-Based Modeling of Financial Markets

relation b Â¼ aN=nc . Equations (12.3.1)â€“(12.3.3) describe the dynam-

ics of three trader groups (nf , nÃ¾ , nÃ€ ) assuming that all transfer

probabilities pab are determined. The change between the chartist

bullish and bearish mood is given by

pÃ¾Ã€ Â¼ 1=pÃ€Ã¾ Â¼ n1 exp(Ã€U1 ),

U1 Â¼ a1 (nÃ¾ Ã€ nÃ€ )=nc Ã¾ (a2 =n1 )dP=dt (12:3:4)

where n1 , a1 and a2 are parameters and P is price. Conversion of

fundamentalists into bullish chartists and back is described with

pÃ¾f Â¼ 1=pfÃ¾ Â¼ n2 exp(Ã€U21 ),

U21 Â¼ a3 ((r Ã¾ nÃ€1 dP=dt)=P Ã€ R Ã€ sj(Pf Ã€ P)=Pj) (12:3:5)

2

where n2 and a3 are parameters, r is the stock dividend, R is the

average revenue of economy, s is a discounting factor 0 < s < 1, and

Pf is the fundamental price of the risky asset assumed to be an input

parameter. Similarly, conversion of fundamentalists into bearish

chartists and back is given by

pÃ€f Â¼ 1=pfÃ€ Â¼ n2 exp(Ã€U22 ),

U22 Â¼ a3 (R Ã€ (r Ã¾ nÃ€1 dP=dt)=P Ã€ sj(Pf Ã€ P)=Pj) (12:3:6)

2

Price P in (12.3.4)â€“(12.3.6) is a variable that still must be defined.

Hence, an additional equation is needed in order to close the system

(12.3.1)â€“(12.3.6). As it was noted previously, an empirical relation

between the price change and the excess demand constitutes the

specific of the non-equilibrium price models4

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