- Email: [email protected]
Excess Volatility and Contagion Dynamics in Heterogeneous Agent Models
Copyright Cl IFAC Computation in Economics. Finance and Engineering: Economic Systems. Cambridge. UK. 1998
Excess Volatility and Contagion Dynamics in Heterogeneous Agent Models Taisei Kaizoji Division oJ Social Sciences, International Christian University Osawa, Mitaka, Tokyo, 181-0015, JAPAN E-mail: [email protected] Abstract In this paper we think of a stock market 88 consisting of many traders who follow various feedback strategies, and propose discrete-time models that represent contagion dynamics of the traders using the Synergetic approach. We show excess volatility (chaos) of stock: prices which are generated from the heterogeneous agent models. Copyright
~ 1998 1FAC
Keywords: excess volatility, discrete-time contagion dynamics, the Synergetic approach and chaos
of returns derived from chaotic trajectories of the model exhibit high peaks around the mean as well as fat tails (leptokurtosis). Another important author who developed the theory of non-rational opinion formation further is Aoki (1996, 1998) from a different standpoint. In this paper we propose Synergetic models of speculative price, and show that macro phenomena in the stock market (the speculative bubbles, the speculative chaos, and the speculative crashes) are generated by market sentiment. The basic difference of Lux's models and our model is that Lux's models formalize continuous-time contagion dynamics while our models formalize discrete-time contagion dynamics.
1. INTRODUCTION
The factual movements of stoclc prices in recent yea.ni" have suggested the existence of any systematic deviations of stock prices from their fundamental values. For instance the 1987 stock market crash demonstrated convincingly that the volatility series of stock returns can not be "accounted for by news about fundamental values. The findings of excess volatility give rise to doubts in the overall efficiency of stock markets, and suggest the need for alternative frameworks for thinking about speculative process itself. See, for example, Shiller (1989). The Synergetic approach1 an alternative &Jr proadI to the efficient markets approach which has recently been pursued by Lux (1995, 1998) and Kaizoji (1994, 1998). This approach focuses on fads and sociological or psychological mechanisms rather than rationality of agents. Lux (1995, 1998) formalizes herd behavior or mutual mimetic contagion among speculators in the share market, referring to the concept of Synergetics. Lux (1995) explains the emergence of non-rational bubbles as a self-organizing process of infection among traders leading to equilibrium prices which deviate from fundamental value. In addition Lux (1998) finds that the distributions
2. THE MODEL
Let us think about the stock market which consists of n traders. A subscript j{= 1,2, ... , n) represents the jth trader, and Xj (t) represents the investment attitude of the jth trader. The jth trader's inves~ ment attitudes at period t is defined as follows: if the jth trader is the seller of the stock at period t, then Xj(t) +1, and if the jth trader is the buyer of the stock at period t, then Xj (t) = -1. The aggregate demand for the stock and the aggregate supply of the stock depends on the investment attitudes of all traders. A market-maker compares the buying orders and selling orders, and adjusts the share price. H the aggregate demand for the stock at period t exceeds the aggregate supply of the stock at period t, then the market-maker raises
=
1Synergetics have originally been developed by Haken (1982) and applied to various problems from socia1eciences by Weidlich and Haag (1983).
81
the price of the stock at period t, and if the aggregate demand of the stock at period t falls below the aggregate supply of the stock at period t, then the market-maker reduces the stock price at period t. Hence, the adjustment process of the stock price can be formulated as follows:
2.2 The transition probabilities of the investment attitudes In the foregoing section we formalized the trading strategies of average traders. Of course there are traders who do not follow the positive feedback strategies. Moreover, a trader, who follows the positive feedback strategies at period t, does not always follow the positive feedback strategies at period t + 1. In order to formulate such a state, we introduce the transition probabilities of the investment attitudes. We define the transition probability that a trader changes from the seller to the buyer as 9t and Vice versa, the transition probability that a trader clIanges from the buyer to the seller as 9,J.' It is assumed that the transition probabilities of the investment attitudes depend on X(t), and moreover, are specified as follows:
P(t) - P(t -1) = ..\F(Xl(t),X2(t), ... ,xn (t)), (1)
where F{·) is the excess demand function for the stock that depends on the investment attitudes of all traders, and ..\ > 0 denotes the speed of adjustment of the stock price determined by the market-maker. For convenience of analysis we assume that the trading volume for each trader during one period is the same. Then the adjustment process of the stock price is specified as follows:
P(t) = P(t - 1) +..\[&nX(t)].
9t(X(t)) = (1- p) + 1 + expt-aX(t))'
(2)
9~(X(t»
where X{t) is the mean value of the investment attitudes Xj(t), that is,
L xj(t)/n,
(3)
j=1
and &> 0 denotes the trading volume for a trader. If the buyers are more than half the number, (X(t) > 0), then the price is raised at period t, and if sellers are more than half the number, (X(t) < 0), then the price is reduced.
1. If X{t) > 0, then 9t(X{t)) increases and 9~(X(t»
decreases.
2. If X{t) < 0, then 9t{X(t)) decreases and 9~(X(t)) increases. See Figure 1-a and Figure 1-b.
2.1 The positive feedback strategies
3. THE CONTAGION DYNAMICS
We assume that average traders follow positive feedback strategies. Namely, average traders try to buy the stock after the price rises, and sell the stock after the price falls. If the majority of the traders, as stated in the foregoing section, are the buyers (X(t) > 0), then the price rises (P(t) P(t+l) > 0), and if the majority ofthetraders are the sellers (X{t) < 0), then the price falls (P{t) P(t-1) < 0). Thus, the average traders' strategies are summarized as follows: 1. When X(t)
(5)
where a > 0 is the bandwa90n coefficient, which denotes the average traders' sentiment and 0 < p < 1. These transition probabilities imply the following:
n
X(t) =
= (1- p) + 1 + ex~aX{t)]'
(4)
With the transition probabilities (4) and (5), the time development ofthe mean value of X(t) becomes: X(t + 1) = X(t) +{[(1- X(t))9t(X(t)) - (1 + X(t))9~(X(t))] (6)
where { denotes the adjustment speed of the transition probabilities, and X(t) denotes the ensemble mean value of X(t). The equation (6) can be derived from the so-called Master equation that represents the movements of the probability distribution of X(t). On details of this derivation see Weidlich and Haag (1983).
> 0, average traders become the
buyers of the stock at period t + 1.
2. When X(t) < 0, average traders become the sellers of the stock at period t + 1. 82
the stock market. See Figure 4-a and Figure 4-b. Figure 4-a is the return map of (6) under a large value of a( = 45), and Figure 4-b is an example of a chaotic orbit of the ensemble mean value of the stock price, pet). Figure 4-b shows a market crash.
Now we have a dynamical system that is formed by the adjustment process of the stock price (2) and the dynamics of the average investment attitude (6). 3.1 Speculative bubbles
4. THE EXTENSION
In this section we investigate the dynamical properties of the system (2) and (6)2. Since gt and g,t. are symmetric, the origin, i.e. X(t) = 0 is always one of the equilibria in the contagion dynamics (6). The origin corresponds to a situation that there exists an equal number of buyers or sellers. As a first step, we investigate the stability of the origin. If the following is satisfied, the equilibrium X(t) =0 is stable. I' + l'a/2 :-. 2 < O.
In this section we 'extend the model of a single stock to the model of multiple stocks. The stocks are indexed by i = 1,2, ... , m. The transition probabilities of the investment attitudes become as follows: .
I'
( )
gt{Y(t)) = (l-p)+ 1 +exp[- E:" ail X, (t))' 8
(7)
.
P
gl{Y{t)) = (1- p) + 1 +exp[E:" ail X, (t))' (9)
For a > {2 -1')/1', the origin becomes unstable, and two new fixed points, that is, 'a bull market equilibrium, A and a bear market equilibrium, B, are created by a pitchfork bifurcation (see Figure 2). At the bull market equilibrium the ensemble mean values of the stock price Pet) rises continuously. Since pet) is caused by changes of market sentiment that ' are not justified by information about fundamentals, that is, increases of the bandwagon coefficient a, it seems reasonable to suppose that speculative bubbles occur in the stock market.
where X,{t) = E'j X,j(t)/n and I = 1,2, ..... , m. The time development of the mean value of Xi(t) becomes:
Xi{t + 1) = Xi(t) +{iFi(Y(t)),
(10)
where
The contagion dynamics are written as follows:
yet + 1) = yet)
3.2 Speculative chaos and speculative crash
+ SF(Y(t))
(11)
where Y(t) = (Xl{t), .....'Xm{t))' and S denotes the diagonal matrix of {i, and F(Y(t)) denotes the · column vectors of F;(Y{t)). To obtain more insights into the contagion dynamics (11), we make the following assumptions:
Figure 3 is the bifurcation diagram for the map of (6) where the bandwagon coefficient a varies smoothly from 10 to 45. The figure shows that the Feigenbaum route to chaos is generated by an infinite sequence of pitchfork bifurcations. As the bandwagon effect becomes strong in the stock market, the attractor is growing within the range of the bull market, and at a certain value of a the strange attractor extends the range to the range of the bear market. Such a bifurcation is called a symmetry increasing bifurcation. From another point of view the symmetry increasing bifurcation implies that market crashes occur in
Assumption I:
1. The functions Fi(Y(t)) are continuously differentiable functions of all Xi{t). 2. The state space of the contagion dynamics (11) is a compact convex multidimensional smooth manifold with boundary which we will denote by M, and F(Y(t)) is a continuous vector field on M.
2The contagion dynamics (6) depend on the only X(t), and 80 it follows that the properties of the whole dynamical system are understood by analyzing the contagion dynamics.
3. The vector field F(Y(t)) points inward on the boundary of M. 83
4.
Th~ mechanism of the contagion defined by F(Y(t)) is regular in the sense that Jacobian matrix DF(Y(t)) is nonsingular.
where Y(t) denotes the column vector of Xi (t),i = 2,3, ...... , m. Equation (12) is the contagion dynamics .of speculative s4ares and equation (13) that of non-speculative dynamics. Letting B = 0 and C = 0, the contagion process (11) is separated into two independent dynamics.
Assumption I states that the contagion dynamics (11) are eventually sell-correcting: as Xi(t) goes to the lower (upper) boundary of M, Xi(t + 1) increases (decreases). We will use a result from the Poincare-Hopf theorem3 which gives a criterion for the existence of multi equilibria.
Xl (t + 1) = Xl (t) + elFI (Xl (t)), Y(t + 1) = Y(t)
Lemma I: Suppose Assumption 1. H the Jacobian of - F, evaluated at the equilibrium is negative, then there must exist at least two equilibria except for the origin.
Proposition 11 Suppose Assumption I, and that the indepen-. dent contagion dynamics of speculative share (13) has a snapback repeller. Then there exists positive constant E such that for all IBI,ICI < E the contagion dynamics (12) and (13) are chaotic. This proposition is proved by using the theorems of Marotto(1978). As demonstrated in the foregoing section, for the large values of the bandwagon coefficient the contagion dynamics .of speculative stock (14) are chaotic. It is demonstrated easily that (14) has a snapback repeller for large values of the bandwagon coefficient. Therefore, an excess volatility of stock prices is able to be caused by the existence of a single speculative stock.
I: Suppose that Assumption I holds~ Then there exists positive constant and E such that for any ~ > the origin, Y(t) = 0 is a snap-back repeller.
e,
e
It follows from Marotto (1978) that the contagion dynamics (11) are chaotic. The proposition can be proven easily by utilizing Hata's Theorem. [See Hata (1982)]. Proposition I demonstrates that excess volatility (chaos) of the contagion dynamics is observed whenever the system of contagion (11) has the multiple equilibria and the adjustment speed of the transition probabilities for all shares are sufficiently fast. However the assumptions seems quite str~ng. Even more important is that for any large Xi(t) might be greater than 1, orless than -I". As another example of the extension, let us consider that there are two types of many stocks, that is, a speculative stock and a lot of non-speculative stocks in order to explore whether the contagion dynamics (11) are chaotic. Then the total contagion dynamics (11) are rewritten as follows :
5. CONCLUDING REMARK In this paper we have represented models of
discrete-time contagion dynamics in the stock market. We show characteristic patterns of stock prices (the speculative bubbles, the speculative chaos and the specwative crashes) which are generated by changes .of the market sentiment. In this paper we assume implicitly that the traders have heterogeneous belief. Such a hypothesis makes it pnssible to interpret the existence of excess volatility of speculative prices. It seems that the phenomenon admits of no other explanation. We may, therefore, reasonably conclude that the Synergetic approach helps account for the finding of excess volatility, while the approach should be developed further to be closer to the real explanation of herd behaviors of human beings.
ei
Xl (t + 1) = Xl (t)
+elFI (Xl (t), BY(t»,
Y(t + 1) = Y(t) +3F2 (CX I (t), Y(t)).
(15)
Then the following are demonstrated:
Using Lemma I we can demonstrate sufficient conditions for chaos (excess volatility) in the contagion dynamics. It is shown that the contagion dynamics (11) are chaotic if the adjustment speed of transition probabilities ei is sufficiently fast. Pr~position
+ 3F2(Y(t».
(14)
(12)
(13)
30n the Poincare-Hopf theorem see Milnor {1965}. 4There is room for further investigation on this problem. 84
References
Y1pf81-a: The tnmsitionprobahility
(1] Aoki, M., (1996) New Approaches to Macroeconomic Modelling: EtJolutionary Stochastic Dynamics, Multiple Equilibria, and Expternalities as Field Effects, Cambridge University Press, New York.
It(t)
(2] Aoki, M., (1998) A stochastic model of prices and volumes in a share market with two types of participants, mimeo. (3] Haken, H.,(1982) Synergetics, An Introduction, 2nd ed., Springer-Verlag.
-0.5
-1
[4] Hata, M., (1982), Euler's Finite Difference Scheme and Chaos in Rn, Proceedings 0/ Japan Academy 58, Series A, 178 - 181.
o
0.5
1
(X(t)
Ylpn! I-b: The transition probability
(5] Kaizoji, T., (1994) Dynamics of stock prices and chaos, Simulation, 12, 2, 155-162 (in Japanese).
(6] Kaizoji, T., (1998) A theory 'of speculative bubble, chaos and crash, Simulation, 17, 2, 141-152 (in Japanese).
[7] Lux, T., (1995) Herd behavior, bubbles and crashes, Economic Journal, vol. 105(July), pp. 881-896. [8] Lux, T., (1998) The sodo-economic dynamics of speculative markets: interacting agents, . chaos, and the fat tails of return distributions, Journal 0/ Economic BehatJior and Organization, 33, 143-165.
(9] Marotto, F. R., (1978) Snap-back repellers imply chaos in Rn, Journal 0/ Mathematical
-1
-0.5
0
0.5
1
(x(tD
Ylpn! 2: The return map
(x(tt1D
Analysis and Applications, 63, 199 - 223.
[10] Marotto, F. R., (1979) Perturbations of Sta-
0.4
ble Chaotic Difference Equations, Journal 0/ Mathematical Analysis and Applications, 72, 716 - 29. 0.8
[11] Milnor, J., (1965) Topology from the Differentiable Viewpoin~ University of Virginia Press, Charlottesville.
\
[12] Weidlich,W. a,nd G. Haag, (1983) Concepts and Models 0/ a QuantitatitJe Sociology, Springer-Verlag.
85
(x(t~
Figure 3: The hiftll'tation diagram
0.6
(xet»
0 J-1--L-..I-L-L.....L.-L-L.....L.-.J-L......L.= 1O -0.2
-0.6
a
FIgUre 4-a: Chaos and market traSh
(X(t+l»
0.2
0.4
O.B
(X(t»
Figure 4-b: Chaos and Market crashes 5r-----O;;;;=,;;;;;;.-------, 4 3
2 1
0"---'---'-....1..--..4.._"---'----1._....1..-........ 1
101 201 .301 401 561 601 701 801 901 . period
86
Excess Volatility and Contagion Dynamics in Heterogeneous Agent Models Taisei Kaizoji Division oJ Social Sciences, International Christian University Osawa, Mitaka, Tokyo, 181-0015, JAPAN E-mail: [email protected] Abstract In this paper we think of a stock market 88 consisting of many traders who follow various feedback strategies, and propose discrete-time models that represent contagion dynamics of the traders using the Synergetic approach. We show excess volatility (chaos) of stock: prices which are generated from the heterogeneous agent models. Copyright
~ 1998 1FAC
Keywords: excess volatility, discrete-time contagion dynamics, the Synergetic approach and chaos
of returns derived from chaotic trajectories of the model exhibit high peaks around the mean as well as fat tails (leptokurtosis). Another important author who developed the theory of non-rational opinion formation further is Aoki (1996, 1998) from a different standpoint. In this paper we propose Synergetic models of speculative price, and show that macro phenomena in the stock market (the speculative bubbles, the speculative chaos, and the speculative crashes) are generated by market sentiment. The basic difference of Lux's models and our model is that Lux's models formalize continuous-time contagion dynamics while our models formalize discrete-time contagion dynamics.
1. INTRODUCTION
The factual movements of stoclc prices in recent yea.ni" have suggested the existence of any systematic deviations of stock prices from their fundamental values. For instance the 1987 stock market crash demonstrated convincingly that the volatility series of stock returns can not be "accounted for by news about fundamental values. The findings of excess volatility give rise to doubts in the overall efficiency of stock markets, and suggest the need for alternative frameworks for thinking about speculative process itself. See, for example, Shiller (1989). The Synergetic approach1 an alternative &Jr proadI to the efficient markets approach which has recently been pursued by Lux (1995, 1998) and Kaizoji (1994, 1998). This approach focuses on fads and sociological or psychological mechanisms rather than rationality of agents. Lux (1995, 1998) formalizes herd behavior or mutual mimetic contagion among speculators in the share market, referring to the concept of Synergetics. Lux (1995) explains the emergence of non-rational bubbles as a self-organizing process of infection among traders leading to equilibrium prices which deviate from fundamental value. In addition Lux (1998) finds that the distributions
2. THE MODEL
Let us think about the stock market which consists of n traders. A subscript j{= 1,2, ... , n) represents the jth trader, and Xj (t) represents the investment attitude of the jth trader. The jth trader's inves~ ment attitudes at period t is defined as follows: if the jth trader is the seller of the stock at period t, then Xj(t) +1, and if the jth trader is the buyer of the stock at period t, then Xj (t) = -1. The aggregate demand for the stock and the aggregate supply of the stock depends on the investment attitudes of all traders. A market-maker compares the buying orders and selling orders, and adjusts the share price. H the aggregate demand for the stock at period t exceeds the aggregate supply of the stock at period t, then the market-maker raises
=
1Synergetics have originally been developed by Haken (1982) and applied to various problems from socia1eciences by Weidlich and Haag (1983).
81
the price of the stock at period t, and if the aggregate demand of the stock at period t falls below the aggregate supply of the stock at period t, then the market-maker reduces the stock price at period t. Hence, the adjustment process of the stock price can be formulated as follows:
2.2 The transition probabilities of the investment attitudes In the foregoing section we formalized the trading strategies of average traders. Of course there are traders who do not follow the positive feedback strategies. Moreover, a trader, who follows the positive feedback strategies at period t, does not always follow the positive feedback strategies at period t + 1. In order to formulate such a state, we introduce the transition probabilities of the investment attitudes. We define the transition probability that a trader changes from the seller to the buyer as 9t and Vice versa, the transition probability that a trader clIanges from the buyer to the seller as 9,J.' It is assumed that the transition probabilities of the investment attitudes depend on X(t), and moreover, are specified as follows:
P(t) - P(t -1) = ..\F(Xl(t),X2(t), ... ,xn (t)), (1)
where F{·) is the excess demand function for the stock that depends on the investment attitudes of all traders, and ..\ > 0 denotes the speed of adjustment of the stock price determined by the market-maker. For convenience of analysis we assume that the trading volume for each trader during one period is the same. Then the adjustment process of the stock price is specified as follows:
P(t) = P(t - 1) +..\[&nX(t)].
9t(X(t)) = (1- p) + 1 + expt-aX(t))'
(2)
9~(X(t»
where X{t) is the mean value of the investment attitudes Xj(t), that is,
L xj(t)/n,
(3)
j=1
and &> 0 denotes the trading volume for a trader. If the buyers are more than half the number, (X(t) > 0), then the price is raised at period t, and if sellers are more than half the number, (X(t) < 0), then the price is reduced.
1. If X{t) > 0, then 9t(X{t)) increases and 9~(X(t»
decreases.
2. If X{t) < 0, then 9t{X(t)) decreases and 9~(X(t)) increases. See Figure 1-a and Figure 1-b.
2.1 The positive feedback strategies
3. THE CONTAGION DYNAMICS
We assume that average traders follow positive feedback strategies. Namely, average traders try to buy the stock after the price rises, and sell the stock after the price falls. If the majority of the traders, as stated in the foregoing section, are the buyers (X(t) > 0), then the price rises (P(t) P(t+l) > 0), and if the majority ofthetraders are the sellers (X{t) < 0), then the price falls (P{t) P(t-1) < 0). Thus, the average traders' strategies are summarized as follows: 1. When X(t)
(5)
where a > 0 is the bandwa90n coefficient, which denotes the average traders' sentiment and 0 < p < 1. These transition probabilities imply the following:
n
X(t) =
= (1- p) + 1 + ex~aX{t)]'
(4)
With the transition probabilities (4) and (5), the time development ofthe mean value of X(t) becomes: X(t + 1) = X(t) +{[(1- X(t))9t(X(t)) - (1 + X(t))9~(X(t))] (6)
where { denotes the adjustment speed of the transition probabilities, and X(t) denotes the ensemble mean value of X(t). The equation (6) can be derived from the so-called Master equation that represents the movements of the probability distribution of X(t). On details of this derivation see Weidlich and Haag (1983).
> 0, average traders become the
buyers of the stock at period t + 1.
2. When X(t) < 0, average traders become the sellers of the stock at period t + 1. 82
the stock market. See Figure 4-a and Figure 4-b. Figure 4-a is the return map of (6) under a large value of a( = 45), and Figure 4-b is an example of a chaotic orbit of the ensemble mean value of the stock price, pet). Figure 4-b shows a market crash.
Now we have a dynamical system that is formed by the adjustment process of the stock price (2) and the dynamics of the average investment attitude (6). 3.1 Speculative bubbles
4. THE EXTENSION
In this section we investigate the dynamical properties of the system (2) and (6)2. Since gt and g,t. are symmetric, the origin, i.e. X(t) = 0 is always one of the equilibria in the contagion dynamics (6). The origin corresponds to a situation that there exists an equal number of buyers or sellers. As a first step, we investigate the stability of the origin. If the following is satisfied, the equilibrium X(t) =0 is stable. I' + l'a/2 :-. 2 < O.
In this section we 'extend the model of a single stock to the model of multiple stocks. The stocks are indexed by i = 1,2, ... , m. The transition probabilities of the investment attitudes become as follows: .
I'
( )
gt{Y(t)) = (l-p)+ 1 +exp[- E:" ail X, (t))' 8
(7)
.
P
gl{Y{t)) = (1- p) + 1 +exp[E:" ail X, (t))' (9)
For a > {2 -1')/1', the origin becomes unstable, and two new fixed points, that is, 'a bull market equilibrium, A and a bear market equilibrium, B, are created by a pitchfork bifurcation (see Figure 2). At the bull market equilibrium the ensemble mean values of the stock price Pet) rises continuously. Since pet) is caused by changes of market sentiment that ' are not justified by information about fundamentals, that is, increases of the bandwagon coefficient a, it seems reasonable to suppose that speculative bubbles occur in the stock market.
where X,{t) = E'j X,j(t)/n and I = 1,2, ..... , m. The time development of the mean value of Xi(t) becomes:
Xi{t + 1) = Xi(t) +{iFi(Y(t)),
(10)
where
The contagion dynamics are written as follows:
yet + 1) = yet)
3.2 Speculative chaos and speculative crash
+ SF(Y(t))
(11)
where Y(t) = (Xl{t), .....'Xm{t))' and S denotes the diagonal matrix of {i, and F(Y(t)) denotes the · column vectors of F;(Y{t)). To obtain more insights into the contagion dynamics (11), we make the following assumptions:
Figure 3 is the bifurcation diagram for the map of (6) where the bandwagon coefficient a varies smoothly from 10 to 45. The figure shows that the Feigenbaum route to chaos is generated by an infinite sequence of pitchfork bifurcations. As the bandwagon effect becomes strong in the stock market, the attractor is growing within the range of the bull market, and at a certain value of a the strange attractor extends the range to the range of the bear market. Such a bifurcation is called a symmetry increasing bifurcation. From another point of view the symmetry increasing bifurcation implies that market crashes occur in
Assumption I:
1. The functions Fi(Y(t)) are continuously differentiable functions of all Xi{t). 2. The state space of the contagion dynamics (11) is a compact convex multidimensional smooth manifold with boundary which we will denote by M, and F(Y(t)) is a continuous vector field on M.
2The contagion dynamics (6) depend on the only X(t), and 80 it follows that the properties of the whole dynamical system are understood by analyzing the contagion dynamics.
3. The vector field F(Y(t)) points inward on the boundary of M. 83
4.
Th~ mechanism of the contagion defined by F(Y(t)) is regular in the sense that Jacobian matrix DF(Y(t)) is nonsingular.
where Y(t) denotes the column vector of Xi (t),i = 2,3, ...... , m. Equation (12) is the contagion dynamics .of speculative s4ares and equation (13) that of non-speculative dynamics. Letting B = 0 and C = 0, the contagion process (11) is separated into two independent dynamics.
Assumption I states that the contagion dynamics (11) are eventually sell-correcting: as Xi(t) goes to the lower (upper) boundary of M, Xi(t + 1) increases (decreases). We will use a result from the Poincare-Hopf theorem3 which gives a criterion for the existence of multi equilibria.
Xl (t + 1) = Xl (t) + elFI (Xl (t)), Y(t + 1) = Y(t)
Lemma I: Suppose Assumption 1. H the Jacobian of - F, evaluated at the equilibrium is negative, then there must exist at least two equilibria except for the origin.
Proposition 11 Suppose Assumption I, and that the indepen-. dent contagion dynamics of speculative share (13) has a snapback repeller. Then there exists positive constant E such that for all IBI,ICI < E the contagion dynamics (12) and (13) are chaotic. This proposition is proved by using the theorems of Marotto(1978). As demonstrated in the foregoing section, for the large values of the bandwagon coefficient the contagion dynamics .of speculative stock (14) are chaotic. It is demonstrated easily that (14) has a snapback repeller for large values of the bandwagon coefficient. Therefore, an excess volatility of stock prices is able to be caused by the existence of a single speculative stock.
I: Suppose that Assumption I holds~ Then there exists positive constant and E such that for any ~ > the origin, Y(t) = 0 is a snap-back repeller.
e,
e
It follows from Marotto (1978) that the contagion dynamics (11) are chaotic. The proposition can be proven easily by utilizing Hata's Theorem. [See Hata (1982)]. Proposition I demonstrates that excess volatility (chaos) of the contagion dynamics is observed whenever the system of contagion (11) has the multiple equilibria and the adjustment speed of the transition probabilities for all shares are sufficiently fast. However the assumptions seems quite str~ng. Even more important is that for any large Xi(t) might be greater than 1, orless than -I". As another example of the extension, let us consider that there are two types of many stocks, that is, a speculative stock and a lot of non-speculative stocks in order to explore whether the contagion dynamics (11) are chaotic. Then the total contagion dynamics (11) are rewritten as follows :
5. CONCLUDING REMARK In this paper we have represented models of
discrete-time contagion dynamics in the stock market. We show characteristic patterns of stock prices (the speculative bubbles, the speculative chaos and the specwative crashes) which are generated by changes .of the market sentiment. In this paper we assume implicitly that the traders have heterogeneous belief. Such a hypothesis makes it pnssible to interpret the existence of excess volatility of speculative prices. It seems that the phenomenon admits of no other explanation. We may, therefore, reasonably conclude that the Synergetic approach helps account for the finding of excess volatility, while the approach should be developed further to be closer to the real explanation of herd behaviors of human beings.
ei
Xl (t + 1) = Xl (t)
+elFI (Xl (t), BY(t»,
Y(t + 1) = Y(t) +3F2 (CX I (t), Y(t)).
(15)
Then the following are demonstrated:
Using Lemma I we can demonstrate sufficient conditions for chaos (excess volatility) in the contagion dynamics. It is shown that the contagion dynamics (11) are chaotic if the adjustment speed of transition probabilities ei is sufficiently fast. Pr~position
+ 3F2(Y(t».
(14)
(12)
(13)
30n the Poincare-Hopf theorem see Milnor {1965}. 4There is room for further investigation on this problem. 84
References
Y1pf81-a: The tnmsitionprobahility
(1] Aoki, M., (1996) New Approaches to Macroeconomic Modelling: EtJolutionary Stochastic Dynamics, Multiple Equilibria, and Expternalities as Field Effects, Cambridge University Press, New York.
It(t)
(2] Aoki, M., (1998) A stochastic model of prices and volumes in a share market with two types of participants, mimeo. (3] Haken, H.,(1982) Synergetics, An Introduction, 2nd ed., Springer-Verlag.
-0.5
-1
[4] Hata, M., (1982), Euler's Finite Difference Scheme and Chaos in Rn, Proceedings 0/ Japan Academy 58, Series A, 178 - 181.
o
0.5
1
(X(t)
Ylpn! I-b: The transition probability
(5] Kaizoji, T., (1994) Dynamics of stock prices and chaos, Simulation, 12, 2, 155-162 (in Japanese).
(6] Kaizoji, T., (1998) A theory 'of speculative bubble, chaos and crash, Simulation, 17, 2, 141-152 (in Japanese).
[7] Lux, T., (1995) Herd behavior, bubbles and crashes, Economic Journal, vol. 105(July), pp. 881-896. [8] Lux, T., (1998) The sodo-economic dynamics of speculative markets: interacting agents, . chaos, and the fat tails of return distributions, Journal 0/ Economic BehatJior and Organization, 33, 143-165.
(9] Marotto, F. R., (1978) Snap-back repellers imply chaos in Rn, Journal 0/ Mathematical
-1
-0.5
0
0.5
1
(x(tD
Ylpn! 2: The return map
(x(tt1D
Analysis and Applications, 63, 199 - 223.
[10] Marotto, F. R., (1979) Perturbations of Sta-
0.4
ble Chaotic Difference Equations, Journal 0/ Mathematical Analysis and Applications, 72, 716 - 29. 0.8
[11] Milnor, J., (1965) Topology from the Differentiable Viewpoin~ University of Virginia Press, Charlottesville.
\
[12] Weidlich,W. a,nd G. Haag, (1983) Concepts and Models 0/ a QuantitatitJe Sociology, Springer-Verlag.
85
(x(t~
Figure 3: The hiftll'tation diagram
0.6
(xet»
0 J-1--L-..I-L-L.....L.-L-L.....L.-.J-L......L.= 1O -0.2
-0.6
a
FIgUre 4-a: Chaos and market traSh
(X(t+l»
0.2
0.4
O.B
(X(t»
Figure 4-b: Chaos and Market crashes 5r-----O;;;;=,;;;;;;.-------, 4 3
2 1
0"---'---'-....1..--..4.._"---'----1._....1..-........ 1
101 201 .301 401 561 601 701 801 901 . period
86