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
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