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

Complex Dynamics of Speculative Price

Taisei Kaizoji
Division of Social Sciences,
International Christian University
Osawa, Mitaka, Tokyo, 181-8585, Japan.
Email: kaizoji@icu.ac.jp

Abstract:

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.

1    Introduction

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.

2    The Model

We think of a security market where many traders participate in trading. Traders are indexed tex2html_wrap_inline804 . tex2html_wrap_inline806 denotes the investment attitude of trader j . The investment attitude tex2html_wrap_inline806 is defined as follows: if trader j is the buyer of the security at period t , then tex2html_wrap_inline816 . If trader j , in contrast, is the seller of the security at period t , then tex2html_wrap_inline822 . 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,

  equation95

where tex2html_wrap_inline830 denotes the excess demand function for the security and depends on all the traders' investment attitudes, and tex2html_wrap_inline832 denotes the price adjustment speed determined by the market maker. The price change at period t , tex2html_wrap_inline836 becomes plus (minus) if the security market is in excess demand (excess supply) at period t .

Here, we assume that volume of trading per trader is the same quantity. Then the excess demand tex2html_wrap_inline830 depends on the mean value of the investment attitudes tex2html_wrap_inline806

  equation105

If tex2html_wrap_inline844 is 0 , then there exists the same number of the buyers or the sellers. The situation with tex2html_wrap_inline848 exhibit that more than half the number of the traders are the buyers (the sellers). In the extreme cases, tex2html_wrap_inline850 or tex2html_wrap_inline852 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 tex2html_wrap_inline844 :

  equation113

where tex2html_wrap_inline856 denotes the trading volume per trader.

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:

  1. When tex2html_wrap_inline858 , the fundamentalists become the buyer of the security at period t+1 .
  2. When tex2html_wrap_inline862 , the fundamentalists become the seller of the security at period t+1 .
  3. When tex2html_wrap_inline866 , the fundamentalists do not trade the security at period t+1 .
where tex2html_wrap_inline870 denotes the fundamentals price.

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:

  1. When tex2html_wrap_inline872 , the bandwagon traders become the buyer of the security at period t+1 .
  2. When tex2html_wrap_inline876 , the bandwagon traders become the seller of the security at period t+1 .
  3. When tex2html_wrap_inline880 , the bandwagon traders do not trade the security at period t+1 .
As mentioned above, if the majority of the traders are the seller ( tex2html_wrap_inline884 ), then the price rises ( tex2html_wrap_inline872 ), and if the majority of the traders are the buyers ( tex2html_wrap_inline888 ), then the market price falls ( tex2html_wrap_inline876 ). Thus, it is possible that the bandwagon traders' strategies are rewritten as follows:
  1. When tex2html_wrap_inline884 , the bandwagon traders become the buyer of the security at period t+1 .
  2. When tex2html_wrap_inline888 , the bandwagon traders become the seller of the security at period t+1 .
  3. When tex2html_wrap_inline900 , the bandwagon traders do not trade the security at period t+1 .

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 tex2html_wrap_inline836 and tex2html_wrap_inline906 , because the fundamentalists' investment attitude depends upon the deviation of the price from the fundamentals price, tex2html_wrap_inline906 , and the bandwagon traders' investment attitude depends upon the change of the market price, tex2html_wrap_inline836 .

We define the transition probability that a trader changes from the seller to the buyer as tex2html_wrap_inline912 , and, the transition probability that a trader changes from the buyer to the seller as tex2html_wrap_inline914 .

Furthermore, the transition probabilities are specified by the equations (4) and (5):

  equation139

  equation145

where the parameters, tex2html_wrap_inline916 and tex2html_wrap_inline918 are positive. tex2html_wrap_inline916 denotes the strength of the bandwagon traders' reaction upon the price changes, the so-called bandwagon effect, and tex2html_wrap_inline918 denotes the strength of the fundamentalists' reaction upon differences between the actual market price and the fundamental price. We call tex2html_wrap_inline916 the bandwagon coefficient, and tex2html_wrap_inline918 the arbitrage coefficient. These transition probabilities imply the following:

  1. If tex2html_wrap_inline928 , then tex2html_wrap_inline912 increases (decreases), and simultaneously tex2html_wrap_inline914 decreases (increases).
  2. If tex2html_wrap_inline934 , then tex2html_wrap_inline912 decreases (increases), and simultaneously tex2html_wrap_inline914 increases (decreases).

3    Speculative dynamics

With the transition probabilities (4) and (5), the time development of the mean values of the investment attitude tex2html_wrap_inline844 and the price P(t) becomes the equations (6) and (7):

  equation184

  eqnarray190

where tex2html_wrap_inline944 and tex2html_wrap_inline946 denote the ensemble mean values of tex2html_wrap_inline844 and tex2html_wrap_inline950 . 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].

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 : tex2html_wrap_inline952 , tex2html_wrap_inline954

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

  equation219

  eqnarray223

For simplicity of analysis we specify the parameters as follows: tex2html_wrap_inline956 , tex2html_wrap_inline958 , and tex2html_wrap_inline960 .

The dynamical system has a unique fixed point, tex2html_wrap_inline962 . We call the fixed point the fundamental equilibrium.  We can demonstrate the following stability condition:

If the arbitrage coefficient tex2html_wrap_inline964 , then the desired market equilibrium is locally stable.

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 tex2html_wrap_inline918 is selected suitably.

3.2 The case of the bandwagon traders : tex2html_wrap_inline968 and tex2html_wrap_inline970

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

  equation232

  eqnarray236

Thus the map of tex2html_wrap_inline946 , (11) is one-dimensional. We can demonstrate the following stability condition:

If the bandwagon coefficient tex2html_wrap_inline974 , then the origin is the unique equilibrium and locally stable.

The proof is trivial and omitted.

The parameters are specified as follows: tex2html_wrap_inline976 , tex2html_wrap_inline956 , tex2html_wrap_inline958 , and tex2html_wrap_inline960 .

Figure 1, Figure 2 and Figure 3 illustrate the map (11) with the different values of the bandwagon coefficient tex2html_wrap_inline916 . These figures show that (i) for tex2html_wrap_inline986 the origin is a unique equilibrium and is stable (Figure 1), and that (ii) for tex2html_wrap_inline988 a pitchfork bifurcation at the origin occurs (Figure 2), and that (iii) for tex2html_wrap_inline990 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 tex2html_wrap_inline916 varies smoothly from 7 to 8.2. Note that Figure 4 is created for some positive initial values of the the investment attitude index, ( tex2html_wrap_inline994 ). This figure suggests the following bifurcation scenario with respect to the bandwagon coefficient tex2html_wrap_inline916 . If the bandwagon coefficient tex2html_wrap_inline916 is small, then the bull market equilibrium is stable for any positive initial values. If tex2html_wrap_inline916 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 tex2html_wrap_inline916 is further increased, the symmetry- breaking bifurcation [8] occurs, and the sudden increase in symmetry of the chaotic attractor.

tex2html_wrap1160

Figure 1: The map (11) with tex2html_wrap_inline1004 .

tex2html_wrap1162

Figure 2: The map (11) with tex2html_wrap_inline988 .

tex2html_wrap1164

Figure 3: The map (11) with tex2html_wrap_inline1008 .

tex2html_wrap1166

Figure 4: The bifurcation diagram for the map (11) with respect to tex2html_wrap_inline916 .

Figure 5 and Figure 6 illustrate the time paths of tex2html_wrap_inline946 and tex2html_wrap_inline944 with tex2html_wrap_inline1016 . The mean value of the investment attitude tex2html_wrap_inline946 fluctuates within the range of the bull market, [ tex2html_wrap_inline1020 ], so that the mean value of the price tex2html_wrap_inline944 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 tex2html_wrap_inline946 and tex2html_wrap_inline944 with tex2html_wrap_inline1028 after the symmetry-breaking bifurcation. These figures show that the mean value of the investment attitude tex2html_wrap_inline946 fluctuates chaotically in the broad range of [ tex2html_wrap_inline1032 ], and various rise and fall patterns of the mean value of the price tex2html_wrap_inline944 are created by the positive feedback trading of the bandwagon traders.

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.

tex2html_wrap1168

Figure 5: The time path of tex2html_wrap_inline946 with tex2html_wrap_inline1016 .

tex2html_wrap1170

Figure 6: The time path of tex2html_wrap_inline944 with tex2html_wrap_inline1016 : Speculative Bubble.

tex2html_wrap1172

Figure 7: The time path of tex2html_wrap_inline946 with tex2html_wrap_inline1028 .

tex2html_wrap1174

Figure 8: The time path of tex2html_wrap_inline944 with tex2html_wrap_inline1028 : Speculative Chaos.

3.3 The case of coexistence of fundamentalists and bandwagon traders : tex2html_wrap_inline1052

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: tex2html_wrap_inline976 , tex2html_wrap_inline956 , tex2html_wrap_inline958 , and tex2html_wrap_inline960 .

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 tex2html_wrap_inline962 . We can demonstrate the following stability conditions:

If tex2html_wrap_inline986 and tex2html_wrap_inline1066 , then the fundamental equilibrium is locally stable.

The proof is trivial and omitted. We see from the above stability conditions that if the bandwagon coefficient tex2html_wrap_inline916 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.

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 ( tex2html_wrap_inline1070 ). In this case if the bandwagon coefficient tex2html_wrap_inline916 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 tex2html_wrap_inline1076 plane and tex2html_wrap_inline1078 plane with tex2html_wrap_inline1080 . In both the figures, the orbits converges to attracting invariant `circle' created in the Hopf bifurcation. When the bandwagon coefficient tex2html_wrap_inline916 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 tex2html_wrap_inline916 is increased from 7 to 7.4 , under tex2html_wrap_inline1070 . Figure 13 and Figure 14 show the time paths of tex2html_wrap_inline946 and tex2html_wrap_inline944 with tex2html_wrap_inline1096 . The time series of the mean value of investment attitude tex2html_wrap_inline946 fluctuate irregularly within the broad range tex2html_wrap_inline1100 . The corresponding time series of tex2html_wrap_inline944 is also chaotic.

tex2html_wrap1176

Figure 9: bf The quasi-periodic attractor with tex2html_wrap_inline1080 .

tex2html_wrap1178

Figure 10: The quasi-periodic attractor with tex2html_wrap_inline1080 .

tex2html_wrap1180

Figure 11: The strange attractor with tex2html_wrap_inline1108 .

tex2html_wrap1182

Figure 12: The strange attractor with tex2html_wrap_inline1108 .

tex2html_wrap1184

Figure 13: The time path of tex2html_wrap_inline946 with tex2html_wrap_inline1096 .

tex2html_wrap1186

Figure 14: The time path of tex2html_wrap_inline944 with tex2html_wrap_inline1096 .

Second, we investigate the case with the small arbitrage coefficient ( tex2html_wrap_inline1120 ). If the bandwagon coefficient tex2html_wrap_inline916 is above 7 in the case with tex2html_wrap_inline1126 , 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 tex2html_wrap_inline916 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 tex2html_wrap_inline946 and tex2html_wrap_inline944 with tex2html_wrap_inline1134 . The mean value of the investment attitude tex2html_wrap_inline946 fluctuates irregularly within the range of [ tex2html_wrap_inline1020 ] and suddenly falls into the range of [ tex2html_wrap_inline1140 ] , and then fluctuates erratically within the range, and again suddenly jumps up the range of [ tex2html_wrap_inline1142 ], 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.

tex2html_wrap1188

Figure 15: The quasi-periodic attractor with tex2html_wrap_inline1144 .

tex2html_wrap1190

Figure 16: The quasi-periodic attractor with tex2html_wrap_inline1144 .

tex2html_wrap1192

Figure 17: The strange attractor with tex2html_wrap_inline1134 .

tex2html_wrap1194

Figure 18: The strange attractor with tex2html_wrap_inline1134 .

tex2html_wrap1196

Figure 19: The time path of tex2html_wrap_inline946 with tex2html_wrap_inline1134 .

tex2html_wrap1198

Figure 20: The time path of tex2html_wrap_inline944 with tex2html_wrap_inline1134 .

4    Conclusion

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.

5    Acknowledgements

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.

Footnotes

(1)
It has been greatly debated, whether price fluctuations in the financial markets is random walk  or chaos ([32], [20], [18]).
(2)
Similar statistical approaches have developed to study various problems of social interactions among heterogeneous agents by [1], [2], [3], [4], [23] and [30].
(3)
For implications of positive feedback trading see [34].
(4)
In the last decade a growing literature on experimental asset markets has emerged. A nice survey of this literature is given in [35].
 

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