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In the model [11], the excess demand equals

Dex Â¼ tc (nÃ¾ Ã€ nÃ€ ) Ã¾ gnf (Pf Ã€ 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, tc and g are parameters.

The system (12.3.1)â€“(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

136 Agent-Based Modeling of Financial Markets

model [11] was modified to describe the arrival of news in the market,

which affects the fundamental price. This process was modeled with

the Gaussian random variable e(t) so that

ln Pf (t) Ã€ ln Pf (t Ã€ 1) Â¼ e(t) (12:3:9)

The modeling results exhibited the power-law scaling and temporal

volatility dependence in the price distributions.

12.4 THE OBSERVABLE VARIABLES MODEL

12.4.1 THE FRAMEWORK

The models discussed so far are capable of reproducing important

features of financial market dynamics. Yet, one may notice a degree

of arbitrariness in this field. The number of different agent types and

the rules of their transition and adaptation vary from one model to

another. Also, little is known about optimal choice of the model

parameters [15, 16]. As a result, many interesting properties, such as

deterministic chaos, may be the model artifacts rather than reflections

of the real world.5

A parsimonious approach to choosing variables in the agent-based

modeling of financial markets was offered in [17]. Namely, it was

suggested to derive agent-based models exclusively in terms of observ-

able variables. Note that the notion of observable data in finance

should be discerned from the notion of publicly available data. While

the transaction prices in regulated markets are publicly available, the

market microstructure is not (see Section 2.1). Still, every event in the

financial markets that affects the market microstructure (such as

quote submission, quote cancellation, transactions, etc.) is recorded

and stored for business and legal purposes. This information allows

one to reconstruct the market microstructure at every moment. We

define observable variables in finance as those that can be retrieved or

calculated from the records of market events. Whether these records

are publicly available at present is a secondary issue. More import-

antly, these data exist and can therefore potentially be used for

calibrating and testing the theoretical models.

The numbers of agents of different types generally are not observ-

able. Indeed, consider a market analog of â€˜â€˜Maxwellâ€™s Demonâ€™â€™ who is

137

Agent-Based Modeling of Financial Markets

able to instantly parse all market events. The Demon cannot discern

â€˜â€˜chartistsâ€™â€™ and â€˜â€˜fundamentalistsâ€™â€™ in typical situations, such as when

the current price, being lower than the fundamental price, is growing.

In this case, all traders buy rather than sell. Similarly, when the

current price, being higher than the fundamental price, is falling, all

traders sell rather than buy.

Only price, the total number of buyers, and the total number of

sellers are always observable. Whether a trader becomes a buyer or

seller can be defined by mixing different behavior patterns in the

trader decision-making rule. Let us describe a simple non-equilibrium

price model derived along these lines [17]. We discern â€˜â€˜buyersâ€™â€™ (Ã¾)

and â€˜â€˜sellersâ€™â€™ (Ã€). Total number of traders is N

NÃ¾ (t) Ã¾ NÃ€ (t) Â¼ N (12:4:1)

The scaled numbers of buyers, nÃ¾ (t) Â¼ NÃ¾ (t)=N, and sellers, nÃ€ (t)

Â¼ NÃ€ (t)=N, are described with equations

dnÃ¾ =dt Â¼ vÃ¾Ã€ nÃ€ Ã€ vÃ€Ã¾ nÃ¾ (12:4:2)

dnÃ€ =dt Â¼ vÃ€Ã¾ nÃ¾ Ã€ vÃ¾Ã€ nÃ€ (12:4:3)

The factors vÃ¾Ã€ and vÃ€Ã¾ characterize the probabilities for transfer

from seller to buyer and back, respectively

vÃ¾Ã€ Â¼ 1=vÃ€Ã¾ Â¼ n exp (U), U Â¼ apÃ€1 dp=dt Ã¾ b(1 Ã€ p) (12:4:4)

Price p(t) is given in units of its fundamental value. The first term in

the utility function, U, characterizes the â€˜â€˜chartistâ€™â€™ behavior while the

second term describes the â€˜â€˜fundamentalistâ€™â€™ pattern. The factor n has

the sense of the frequency of transitions between seller and buyer

behavior. Since nÃ¾ (t) Â¼ 1 Ã€ nÃ€ (t), the system (12.4.1)â€“(12.4.3) is re-

duced to the equation

dnÃ¾ =dt Â¼ vÃ¾Ã€ (1 Ã€ nÃ¾ ) Ã€ vÃ€Ã¾ nÃ¾ (12:4:5)

The price formation equation is assumed to have the following

form

dp=dt Â¼ gDex (12:4:6)

where the excess demand, Dex , is proportional to the excess number of

buyers

Dex Â¼ d(nÃ¾ Ã€ nÃ€ ) Â¼ d(2nÃ¾ Ã€ 1) (12:4:7)

138 Agent-Based Modeling of Financial Markets

12.4.2 PRICE-DEMAND RELATIONS

The model described above is defined with two observable vari-

ables, nÃ¾ (t) and p(t). In equilibrium, its solution is nÃ¾ Â¼ 0:5 and

p Â¼ 1. The necessary stability condition for this model is

1 (12:4:8)

adgn

The typical stable solution for this model (relaxation of the initially

perturbed values of nÃ¾ and p) is given in Figure 12.1. Lower values of

a and g suppress oscillations and facilitate relaxation of the initial

perturbations. Thus, the rise of the â€˜â€˜chartistâ€™â€™ component in the utility

function increases the price volatility. Numerical solutions with the

values of a and g that slightly violate the condition (12.4.8) can lead to

the limit cycle providing that the initial conditions are very close to

the equilibrium values (see Figure 12.2). Otherwise, violation of the

condition (12.4.8) leads to system instability, which can be interpreted

as a market crash.

The basic model (12.4.1)â€“(12.4.7) can be extended in several

ways. First, the condition of the constant number of traders (12.4.1)

1.2 0.4

1.15 0.3

1.1 0.2

1.05 0.1

Price

Dex

1 0

âˆ’0.1

0.95

âˆ’0.2

0.9

âˆ’0.3

0.85

Price

Time Dex

âˆ’0.4

0.8

0 5 10 15 20 25 30 35 40 45

Figure 12.1 Dynamics of excess demand (Dex) and price for the model

(12.4.5)â€“(12.4.7) with a Â¼ b Â¼ g Â¼ 1, nÃ¾(0) Â¼ 0.4 and p(0) Â¼ 1.05.

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