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, n

c

(t), and fundamentalists, n

f

(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

n

þ

(t) and n

(t), respectively

n

c

(t) þ n

f

(t) ¼ N, n

þ

(t) þ n

(t) ¼ n

c

(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:

dn

þ

=dt ¼ (n

p

þ

n

þ

p

þ

)(1 n

f

=N) þ mimetic contagion

n

f

n

þ

(p

þf

p

fþ

)=N þ changes of strategy

(b a)n

þ

market entry and exit (12:3:2)

Here, p

ab

denotes the probability of transition from group b to group

a. Similarly, the bearish chartist dynamics is given by

dn

=dt ¼ (n

þ

p

þ

n

p

þ

)(1 n

f

=N) þ mimetic contagion

n

f

n

(p

f

p

f

)=N þ changes of strategy

(b a)n

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

134 Agent-Based Modeling of Financial Markets

relation b ¼ aN=n

c

. Equations (12.3.1)–(12.3.3) describe the dynam-

ics of three trader groups (n

f

,n

þ

,n

) assuming that all transfer

probabilities p

ab

are determined. The change between the chartist

bullish and bearish mood is given by

p

þ

¼ 1=p

þ

¼ n

1

exp(U

1

),

U

1

¼ a

1

(n

þ

n

)=n

c

þ (a

2

=n

1

)dP=dt (12:3:4)

where n

1

, a

1

and a

2

are parameters and P is price. Conversion of

fundamentalists into bullish chartists and back is described with

p

þf

¼ 1=p

fþ

¼ n

2

exp(U

21

),

U

21

¼ a

3

((r þ n

1

2

dP=dt)=P R sj(P

f

P)=Pj) (12:3:5)

where n

2

and a

3

are parameters, r is the stock dividend, R is the

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

P

f

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

p

f

¼ 1=p

f

¼ n

2

exp(U

22

),

U

22

¼ a

3

(R (r þ n

1

2

dP=dt)=P sj(P

f

P)=Pj) (12:3:6)

Price P in (12.3.4)–(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)–(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 models

4

dP=dt ¼ bD

ex

(12:3:7)

In the model [11], the excess demand equals

D

ex

¼ t

c

(n

þ

n

) þ gn

f

(P

f

P) (12:3:8)

The first and second terms in the right-hand side of (12.3.8) are the

excess demands of the chartists and fundamentalists, respectively;

b,t

c

and g are parameters.

The system (12.3.1)–(12.3.8) has rich dynamic properties deter-

mined by its input parameters. The system solutions include stable

equilibrium, periodic patterns, and chaotic attractors. Interestingly,

the distributions of returns derived from the chaotic trajectories

may have fat tails typical for empirical data. Particularly in [14], the

Agent-Based Modeling of Financial Markets 135

Get *Quantitative Finance for Physicists* now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.