COMPLEXITY INTERNATIONAL |
|
| ISSN 1320-0682 |
| Source: | http://life.csu.edu.au/complex/ci/vol6/kaizoji/ | Received: | 01/07/1998 | ||
| Vol 6: | Copyright 1998 | Accepted for publication: | 15/10/1998 |
Taisei Kaizoji
Division of Social Sciences,
International Christian University
Osawa, Mitaka, Tokyo, 181-8585, Japan.
Email: kaizoji@icu.ac.jp
In this paper we think of the security market as consisting of two types of investors: fundamentalists and bandwagon traders, and propose a heterogeneous agent model that represents speculative dynamics by using the Synergetic approach [17]. We show the characteristic patterns of speculative price (speculative bubbles and speculative chaos) which are generated by trading between the fundamentalists and bandwagon traders.
A recent important development in the nonlinear dynamic theory is the discovery of deterministic chaos ([25], [29]). During the last few decades, the concept of deterministic chaos has received a great deal of attention from very diverse scientific fields.
The stock market crash of October 19, 1987 triggered a revolution of chaos in finance. After the 1987 market crash, a number of economists start thinking seriously about the possibility to apply theories on nonlinear dynamics and chaos to finance. Since chaotic dynamics is able to generate large movements which may look like stochastic processes at a first glance, with greater frequency than linear models, the idea that violent fluctuations of speculative prices are generated by some chaotic process, seems to be intuitively a right solution to bubbles and the crashes in the financial markets(1).
On the other hand, recently a number of structural asset pricing models have been introduced, emphasizing the role of heterogeneous beliefs in financial markets, with different groups of traders having different expectations about future prices. Most of these heterogeneous agent models are composed of two typical agent types. The first type is the fundamentalists or arbitrageurs , who believe that the security price is determined by the market fundamental values. The second type is the noise traders , sometimes called chartists or technical analysts , who may predict the future price using simple technical trading rules, extrapolation of trends and other patterns observed in past prices.
Another distinctive
characteristics of the recent heterogeneous agent models has
emphasized that heterogeneity in beliefs may lead to market
instability and complicated dynamics, including periodic cycles and
chaos in financial markets (e.g.
[5], [6],
[7],
[9], [10], [11], [12],
[13], [15], [16], [19],
[26], [27], [28], [33], [38]).
Among them, the heterogeneous agent models of Lux
([26],
[27], [28])
constitute important examples of recently developed branch of
literature on the heterogeneous market hypothesis. He formalized herd
behavior or mutual mimetic contagion in speculative markets. A basic
feature of the framework adopted by Lux's models is that heterogeneous
agents are treated as a statistical ensemble. His mass-statistical
formalisation of agents' attitudes and behavior follows a tradition in
the so-called `Synergetics' literature. The concept of
synergetics originally developed by
Haken[17], and applied to various problems from social
sciences by [36],
[37], [31](2).
Lux [26] used a Synergetic approach to formalize
the theory of non-rational bubbles and crashes advanced by
[22] who highlights the importance of psychological
factors and irrational factors in explaining historical financial
crises. Lux [27] explained the so-called
leptokurtosis of distribution of returns which
is a basic stylized fact of both exchange rate and share price time
series using the Synergetic model with heterogeneous traders. Lux
[28] extended his models in the earlier paper and found chaotic
attractors within a broad range of parameter values.
He also showed that the distributions of returns derived from chaotic
trajectories of the model exhibit high peaks around the mean as well
as leptokurtosis and become less leptokurtotic under time
aggregation. This paper presents a heterogeneous agent model of
speculative dynamics which is based upon the Synergetic approach to
finance. The basic structure of our model is similar to those of Lux
models ([26],[27],and [28]). The main difference
between Lux models and our model is as follows: in the Lux models the
speculative dynamics is represented by the stochastic
differential equations. In our model
the speculative dynamics, by contrast, is represented by the
stochastic difference equations. We think of the security
market as consisting of two types of traders: fundamentalists
whose demand (or supply) is based on prices relative to the
fundamental value, the so-called fundamentals prices, and
bandwagon traders whose demand (or supply)
is based on positive feedback trading strategies. Namely, the
bandwagon traders involve buying the security when the price rises,
and selling the security when the price falls(3). Therefore, the dynamical properties of the
speculative price depend both on the nature of positive feedback
trading by bandwagon traders and arbitrage by fundamentalists.
Following [21], we introduce the stochastic transition
between the seller and the buyer. Under the circumstance that one
cannot get information on the expectations formations and
decision-making of all the traders, a probabilistic setting may be one
of best means to formalize the behavior of a large number of
heterogeneous traders. We will show that the characteristic patterns
of speculative price (speculative bubbles and speculative
chaos) might be generated from the Synergetic model with
fundamentalists and bandwagon traders. We investigate three cases:
(i) the case of the fundamentalists that the only fundamentalists
exist in the security market, (ii) the case of the bandwagon traders
that the only bandwagon traders exist in the security market, and
(iii) the case of the coexistence that both the typical trader types
participate in exchanges. To sum up the major results, we show that
(i) arbitrage by the fundamentalists tends to stabilize the price and
tends to converge its price into the fundamental price, and that (ii)
the positive-feedback trading by the bandwagon traders tends to cause
the instability of speculative dynamics, and particularly positive
feedback trading reinforced by the bandwagon effect creates
bubble-like price patterns and chaos, and that (iii) the combination
of the arbitrage by the fundamentalists and the positive-feedback
trading by the bandwagon traders generate various patterns of
speculative price including speculative chaos, and large slowly
decaying swings away from fundamentals price. Section 2 describes
the trading strategies of the fundamentalists and the bandwagon
traders, and the transition probabilities of the investment
attitude. Section 3 analyzes the characteristics of the speculative
dynamics. Section 4 gives a brief summary and a remark on limitations
of our model.
We think of a security market where many traders participate
in trading.
Traders are indexed
where
Here, we assume that volume of trading per trader
is the same quantity. Then the excess demand
If
where
2.1 Fundamentalists and bandwagon traders
As mentioned in the introduction, the security market
is composed of two groups of traders having the different
trading strategies; that is,
fundamentalists and bandwagon traders.
The trading strategies of fundamentalists are described as follows:
if the security price is below the fundamentals price, then
they try to buy the security until the price is equal to the
fundamentals price because they think that the security is
undervalued. In contrast, if the security price is above
the fundamentals price,
then they try to sell the security until the price is equal to
the fundamentals price because they think that
the security is overvalued.
To sum up, the fundamentalists' strategies are described as follows:
On the other hand, the trading strategies of the bandwagon traders are
described as follows:
the bandwagon traders try to buy the security after the price rises,
and sell the security after
the price falls, that is, they follow positive feedback strategies.
Thus, the bandwagon traders' strategies are summarized
as follows:
2.2 The transition probability of the investment attitudes
In this subsection we formalize the transition probabilities of
the investment attitudes.
It is assumed that the transition probabilities of the investment
attitude depends on
We define the transition probability that
a trader changes from the seller to the buyer
as
Furthermore, the transition probabilities are specified by
the equations (4) and (5):
where the parameters,
With the transition probabilities (4) and (5),
the time development of the mean values of the investment attitude
where
Now we have a dynamical system that is formed by
the adjustment process of the price (6) and
the dynamics of the investment attitudes (7).
We investigate the dynamical properties of
the system (6) and (7) below.
We consider three typical cases: the case of the
fundamentalists that the only fundamentalists exist in the security market,
the case of the bandwagon traders that the only bandwagon traders exist
in the market and the case that the two typical trader types coexist
in the security market.
3.1 The case of the fundamentalists :
First, we consider the case that all the traders are the fundamentalist.
In this case the dynamical system (6) and (7)
rewritten by the following
For simplicity of analysis we specify the parameters
as follows:
The dynamical system has a unique fixed point,
If the arbitrage coefficient
The proof is trivial and omitted.
Therefore in the case of the fundamentalists the price
is stabilized by the fundamentalists' arbitrage and
converge to the fundamental equilibrium,
provided that the arbitrage coefficient
3.2 The case of the bandwagon traders :
Next, we consider the case that all the traders are the bandwagon traders.
Substituting the equation (6) to the equation (7),
the dynamical system is rewritten by the following equations
Thus the map of
If the bandwagon coefficient
The proof is trivial and omitted.
The parameters are specified as follows:
Figure 1, Figure 2 and Figure 3 illustrate the map (11)
with the different values of
the bandwagon coefficient
Figure 5 and Figure 6 illustrate the time paths of
It follows from the numerical analysis that the existence of
the bandwagon traders tends to destabilize the price, and
positive feedback trading reinforced by the bandwagon effect
give cause to bubble-like price patterns. When the bandwagon
effect is further strong, the instability of the speculative
dynamics is amplified, so that chaos of speculative price is caused.
3.3 The case of coexistence of fundamentalists and bandwagon traders
:
Finally, we consider the case that both typical trader types exist
in the security market.
In this case the dynamical system is described by
the original dynamical system (6) and (7).
We specify the parameters
as follows:
We observed that in the case of the bandwagon traders,
the route to chaos
passes through a cascade of period doublings.
In the case of coexistence
of both trader types we will see the existence of quasi-periodic
transitions to chaos.
First of all we consider the stability conditions for the
fundamentals equilibrium
If
The proof is trivial and omitted.
We see from the above stability conditions
that if the bandwagon coefficient
To understand the global characteristics of the speculative dynamics
in detail, we consider two cases: the cases with a large arbitrage
coefficient and a small arbitrage coefficient. First, we
investigate the case with the large arbitrage coefficient (
Second, we investigate the case with the small arbitrage coefficient
(
This paper has represented a Synergetic model that stresses the role
of irrational sentiment of heterogeneous traders.
In our model, price fluctuations are caused by
an endogenous mechanism relating the fraction of the
fundamentalists and the bandwagon traders to the strength of
the arbitrage by the fundamentalists,
and that of positive-feedback trading by
the bandwagon traders.
The important points as regards the endogenous mechanism
that generates speculative dynamics are that (i)
a large fraction of the fundamentalists, or
the increasing strength
of the fundamentalists' reaction upon difference between actual
price and the fundamentals price, tends to stabilize the speculative
price, and in contrast, (ii) a large fraction of the bandwagon traders, or
the increasing strength of the bandwagon traders' reaction upon
difference between the price at the present date and the price
at the previous date, tends to destabilize the security price, and
(iii) trading between the fundamentalists and the bandwagon traders
generates various patterns of
speculative dynamics including speculative bubble, speculative chaos, and
large slowly decaying swings away from fundamental price.
These results seem to provide a useful analytic foundation to experiments
on stock market behavior(4).
However, the characterization of the traders is oversimplified and stylized
in our model.
We employ implicitly the extreme
assumptions on the behavior of the traders: (i) the fundamentalists
can know the exact value of the fundamentals price,
and (ii) the bandwagon traders decide their
investment attitude on the basis of the price change from
the preceding period to the current period. The latter assumption
implies that the bandwagon traders use only the data of the
price at the present period and the preceding period in order
to forecast the future price.
As the result our model of speculative dynamics are formalized by two
dimensional difference equations.
On the other hand standard finance models based on the efficient
market hypothesis
([14]) assume that fundamentalists use
all information available to them at present in order to perceive
the fundamentals price. Similarly,
in most of the heterogeneous
agent models typical noise traders such as chartists or
technical analysts are assumed to find price trends and
other patterns observed in past prices from the long-term data of
the prices, and then predict the future price using their technical
trading rules.
Therefore it is reasonable to suppose that
the trading strategies of the traders will, at least,
depend upon the long-run data of the prices.
Whereas we recognize the importance of the problem on the time
horizon in our model,
it seems to us that the essential nature of speculative dynamics
remains unchanged in more general frameworks. We leave it for future work
to see whether this conjecture is true.
I would like to thank Yoko Yamaguchi, Nobuharu
Miyatake, Thomas Lux for helpful comments on an earlier draft of this
paper. Financial support by Japan productivity center for socio-economic
development is gratefully acknowledged.
2 The Model
.
denotes the investment attitude of trader j .
The investment attitude
is defined as follows:
if trader j is the buyer of the security at period t ,
then
. If trader j , in contrast,
is the seller of the security at period t ,
then
.
There is a market-maker, such as the specialists in
the New York Stock Exchange, and he/she compares the buying and
selling orders by traders, and executes trading.
If the aggregate demand for the security at period t exceeds
the aggregate supply of the security at period t ,
then the market-maker raises the price of the security
at period t , and vice versa.
Hence, an adjustment process of the security price can
be described as follows,
denotes the excess demand function for the security
and depends on all the traders' investment attitudes, and
denotes the price adjustment speed determined by the
market maker.
The price change at period t ,
becomes plus
(minus) if the security market is in excess demand (excess supply)
at period t .
depends on the mean value of the investment attitudes
is 0 , then there exists
the same number of the buyers or the sellers.
The situation with
exhibit that
more than half the number of the traders are the buyers (the sellers).
In the extreme cases,
or
all the traders are the buyers or the sellers.
For convenience of analysis, we assume that
the excess demand function (2)
is specified by the following linear function with respect to
:
denotes the trading volume per trader.
where
, the fundamentalists become the buyer of
the security at period t+1 .
, the fundamentalists become the seller
of the security at period t+1 .
, the fundamentalists do not trade
the security at period t+1 .
denotes the fundamentals price.
As mentioned above,
if the majority of the traders are the seller
(
, the bandwagon traders become the buyer
of the security
at period t+1 .
, the bandwagon traders become the seller
of the security at period t+1 .
, the bandwagon traders do not trade
the security at period t+1 .
), then the price rises
(
), and if the majority of the
traders are the buyers (
), then the market price falls
(
).
Thus, it is possible that the bandwagon traders' strategies
are rewritten as follows:
, the bandwagon traders become the buyer
of the security at period t+1 .
, the bandwagon traders become the seller
of the security at period t+1 .
, the bandwagon traders do not trade
the security at period t+1 .
and
,
because the fundamentalists' investment attitude depends upon
the deviation of the price from the fundamentals price,
, and the bandwagon traders' investment attitude
depends upon the change of the market price,
.
,
and, the transition probability
that a trader changes from the buyer to the seller as
.
and
are positive.
denotes the strength of the bandwagon traders'
reaction upon the price changes, the so-called
bandwagon effect, and
denotes the strength of
the fundamentalists' reaction upon differences between
the actual market price and the fundamental price.
We call
the bandwagon coefficient, and
the arbitrage coefficient.
These transition probabilities imply the following:
, then
increases (decreases), and simultaneously
decreases (increases).
,
then
decreases (increases), and simultaneously
increases (decreases).
3 Speculative dynamics
and the price P(t) becomes
the equations (6) and (7):
and
denote
the ensemble mean values of
and
.
The equation (7) can be derived from
the original stochastic system using the Master equation.
On details of this derivation see Weidlich and Haag[36].
,
,
,
and
.
.
We call the fixed point the fundamental equilibrium.
We can demonstrate the following stability condition:
,
then the desired market equilibrium is locally stable.
is selected
suitably.
and
, (11) is one-dimensional.
We can demonstrate the following stability condition:
,
then the origin is the unique equilibrium and locally stable.
,
,
,
and
.
.
These figures show that (i) for
the origin is
a unique equilibrium and is stable (Figure 1), and
that (ii) for
a pitchfork
bifurcation at the origin occurs (Figure 2),
and that (iii) for
the origin becomes
unstable, and simultaneously the two new equilibria,
the bull market equilibrium and
the bear market equilibrium are created, one above and
one below the origin (Figure 3).
Figure 4 is the bifurcation diagram for the map (11)
where the bandwagon coefficient
varies smoothly from 7 to 8.2.
Note that Figure 4 is created for some positive initial values of the
the investment attitude index, (
).
This figure suggests the following
bifurcation scenario with respect to the bandwagon coefficient
.
If the bandwagon coefficient
is small, then the bull market equilibrium is stable for any
positive initial values.
If
is increased, then the bull market equilibrium
becomes unstable and period doubling bifurcations occur.
After infinitely many period doubling bifurcations the dynamics
becomes chaotic. When
is further increased, the symmetry-
breaking bifurcation [8] occurs, and the sudden
increase in symmetry of the chaotic attractor.
Figure 1: The map (11) with
.
Figure 2: The map (11) with
.
Figure 3: The map (11) with
.
Figure 4: The bifurcation diagram for the map (11)
with respect to
.
and
with
.
The mean value of the investment attitude
fluctuates within the range of the bull market,
[
],
so that the mean value of the price
keeps rising over time.
Since the rise in the mean value of the price is caused by
the bandwagon effect that are not justified by
fundamentals, it seems reasonable to suppose that speculative
bubbles occur in the security market.
Figure 7 and Figure 8 illustrate the time paths of
and
with
after the symmetry-breaking bifurcation.
These figures show that the mean value of the investment attitude
fluctuates
chaotically in the broad range of [
],
and various rise and fall
patterns of the mean value of the price
are created by the positive feedback trading of
the bandwagon traders.
Figure 5: The time path of
with
.
Figure 6: The time path of
with
: Speculative Bubble.
Figure 7: The time path of
with
.
Figure 8: The time path of
with
: Speculative Chaos.
,
,
,
and
.
. We can demonstrate the following stability conditions:
and
,
then the fundamental equilibrium is locally stable.
is small,
then the speculative dynamics is stable and the mean value of the price
is converged into the fundamentals equilibrium by
the arbitrage of fundamentalists. In other words, if the bandwagon
effect is weak in the market,
then the speculative dynamics is stabilized by the arbitrage
of the fundamentalists.
). In this case if the bandwagon coefficient
is above 7
in this case, then a Hopf bifurcation occurs at the fundamental
equilibrium, and then a quasi-periodic orbit starting from the market
fundamental equilibrium appears. Figure 9 and Figure 10 show two
attractors in the
plane and
plane with
. In both the figures, the orbits
converges to attracting invariant `circle' created in the Hopf
bifurcation. When the bandwagon coefficient
is increased
from 7 to 7.4, the invariant circles break up the into strange
attractors (Figure 11 and Figure 12). It
follows from these figures that the transition occurs from
quasiperiodicity to chaos when
is increased from 7 to
7.4 , under
. Figure 13 and Figure 14 show the time
paths of
and
with
. The time series of the mean value of
investment attitude
fluctuate irregularly
within the broad range
. The corresponding time series of
is also chaotic.
Figure 9: bf The quasi-periodic attractor with
.
Figure 10: The quasi-periodic attractor with
.
Figure 11: The strange attractor with
.
Figure 12: The strange attractor with
.
Figure 13: The time path of
with
.
Figure 14: The time path of
with
.
).
If the bandwagon coefficient
is above
7 in the case with
, then a Hopf bifurcation
occurs at the fundamental equilibrium, and
the orbits converges to attracting invariant `square' created in
the Hopf bifurcation (Figure 15 and Figure 16).
As the bandwagon coefficient
is further increased,
the invariant squares break up the into
strange attractors (Figure 17 and Figure 18).
Figure 19 and Figure 20 show the time series of
the time paths of
and
with
.
The mean value of the investment attitude
fluctuates irregularly within the range of [
]
and suddenly falls into the range of [
] ,
and then fluctuates erratically within
the range, and again suddenly jumps up the range of
[
], and the same process
is repeated. The corresponding time series of the mean value of the price
repeats large slowly decaying swings away from fundamental price.
Figure 15: The quasi-periodic attractor with
.
Figure 16: The quasi-periodic attractor with
.
Figure 17: The strange attractor with
.
Figure 18: The strange attractor with
.
Figure 19: The time path of
with
.
Figure 20: The time path of
with
.4 Conclusion
5 Acknowledgements



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