Speculative Bubbles and Excess Demand Functions.

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Now,  we shall indicate how local and global properties of the deterministic dynamics vary when the parameter α is allowed to vary. α measures the strength of the non-linearity of the excess demand functions

pt+1 = pt + θn[(1−κt−ξ)(exp(αν(p − pt))−1) + κt(exp(α(1−µ)(pct − pt)) − 1) + ξγεt],
pft+1 = pt + ν(p − pt), pct+1 = pct + µ(pt − pct)  —– (1)

where κt = 1/(1 + exp(ψ((pt − pct)2 − (pt − pft)2)) .

The first of the equations above represents the adjustment of the price performed by the market maker. The second and third equations show the formation of the expectations of fundamentalists and chartists, respectively. The fourth equation represents the movement of the chartist fraction. Throughout this section, we assume that there exists no noise traders (ξ = 0). Thus, the dynamics of the system is deterministic. As can be checked easily, the dynamic system has as an unique fixed point: P ≡ (p¯, p¯f, p¯c) = (p, p, p).

The local stability conditions:

To investigate the dynamics of the model, we shall first determine the local stability region of the unique equilibrium point P. The local stability analysis of the equilibrium point P is performed via evaluation of the three eigenvalues of the Jacobian matrix at P. Let us denote by

c(λ) = λ3 − Tλ2 + Dλ —– (2)

the associated characteristic polynomial of the Jacobian matrix at P, where

D = (2−µ−0.5θnα(1−µ+ν))

and

T = (1−µ−0.5θnα(1+ν)(1−µ)).

An eigenvalue of the Jacobian is 0, and the other roots λ1 and λ2, satisfy the relation

λ2 − Tλ + D = 0 —– (3)

Thus, the stability of the equilibrium point is determined by the absolute values of λ1 and λ2. The eigenvalues λ1, λ2 are λ1,2 = T/2 + ± √ ∆/2 where ∆ ≡ T2 − 4D. As is well known, a sufficient condition for local stability consists of the following inequalities:

(i) 1−T +D > 0,

(ii) 1+T +D > 0, and

(iii) D < 1,

giving necessary and sufficient conditions for the two eigenvalues to be inside the unit circle of the complex plane. Elementary computations lead to the condition:

α < (3 − 2µ) / (θn(1 + ν − µ(1 + 0.5ν))) —– (4)

The above local conditions demonstrate that while increasing α starting from a sufficiently low value inside the stability region, a loss of local stability of the equilibrium point P may occur via a flip bifurcation, when crossing the curve

α = (3 − 2µ) / (nθ(1 + ν − µ(1 + 0.5ν))) —– (5)

Global bifurcations:

What happens as α is further increased? We will see that first, a stable 2-cycle appears; secondly, the time path of the price diverges; and thirdly, the time evolution of pt displays a remarkable transition from regular to chaotic behavior around an upward time trend. It is important that the nonstationary chaos can be transformed to a stationary series by differentiation once ∆pt = pt−pt−1. The price series fluctuates irregularly around an upward price trend, and the series of the price change fluctuates chaotically within a finite interval. Below, this non-stationary chaotic pricing will be referred to as a speculative bubble.

Speculative bubbles:

Why does a speculative bubble or non-stationary chaos appear when α increases? Unfortunately, it is difficult to prove mathematically the existence of the strange attractor. However, as we shall see, it is possible to give an intuitive interpretation of speculative bubbles.  On a phase plot, the stretching and folding is the root of the chaotic price behavior. The next question is why speculative bubbles occur when α is large. The price adjustment equation suggests that a price trend can occur only when excess demand aggregated over two types of traders is positive on average. Why would average excess demand be positive as α become large? For an answer, let us recall the traders’ excess-demand functions. xt denotes the traders’ excess demand, and pet+1 denotes the expected price. The partial derivative of the excess-demand function with respect to the parameter α is obtained as

∂xt/∂α = (pet+1 − pt) exp(α(pet+1 − pt)), —– (7)

where we assume β = 1. This equation suggests a rise in α increasing excess demand exponentially, given the expected price change (pet+1 − pt) is positive. As shown in the preceding paragraph, the maximum value of the excess demand increases freely when α increases, while the maximum value of the excess supply never can be lower than −1/β, regardless of the value of α. Assume that, in the initial state, the price pt exceeds the fundamental price p in period t, and that the chartists forecast a rise over the next period. The fundamentalists predict a falling price. Then, chartists try to buy and fundamentalists try to sell stock. When α is sufficiently large, one can safely state that chartist excess demand will exceed fundamentalist excess supply. The market maker observes excess demand, and raises the price for the following period. If the rise is strong, chartists may predict a new rise, which provokes yet another reaction of the market maker. Once the price deviates strongly from the fundamental price, the fundamentalists become perpetual sellers of stock, and when α is sufficiently large, the chartists’ excess demand may begin to exceed the fundamentalists’ excess supply on average. Furthermore, fundamentalists may be driven out of the market by evolutionary competition. It follows that when the value of α is sufficiently large, average excess demand is positive and speculative bubbles occur.

Price Dynamics for Fundamentalists – Risky Asset – Chartists via Modeling

Substituting (1), (2) and (3) to (4) from here, the dynamical system can be obtained as

pt+1 − pt = θN[(1 − κ)(exp(α(p − pt)) − 1) + κ(exp(β(1 − µ)(pet − pt)) − 1)] pet+1 − pet = µ(pt − pet ) —– (5)

In the following discussion we highlight the impact of increases in the total number of traders n on the price fluctuation that is defined as the price increment,

rt = pt+1 − pt

We first restrict ourselves to investigating the following set of parameters:

α = 3, β = 1, µ = 0.5, κ = 0.5, θ = 0.001 —– (6)

It is clear that the two-dimensional map (5) has a unique equilibrium with pet = pt = p, namely (p¯e , p¯) = (p, p), given the above conditions. Elementary computations show that for our map (5) the sufficient condition for the local stability of the fixed point p is given as

N < (2(2 − µ))/(θ[α(2 − µ)(1 − κ) + 2β(1 − µ)κ]) —– (7)

From (7) it follows that, starting from a small number of traders N inside the stability region, when the number of traders N increases, a loss of stability may occur via a flip bifurcation. We shall now look more globally into the effect of increases in the number of traders on the price dynamics. Figure 1 shows a bifurcation diagram of the price increments rt with N as the bifurcation parameter under the set of parameters (6). For the convenience of illustration, Figure 1 is drawn using θN as the bifurcation parameter. This figure suggests the following bifurcation scenario.

0312065v1.fig1

The price increments rt converge to 0 when the number of traders N is small. In other words, the price converges to the fundamental price p when the active traders are few. However the price dynamics become unstable when the number of traders N exceeds about 1000, and chaotic behavior of the price increments occurs after infinitely many period-doubling bifurcations. If N is further increased, then the price increments rt become more regular again after infinitely many period-halving bifurcations. A stable 2 orbit occurs for an interval of N-values. However as N is further increased, the behavior of the price increment rt becomes once again chaotic, and the prices diverge. Let us investigate closely the characteristics of chaos that are observed in the parameter interval (2000 < N < 4000). Figure 2 shows a series of price increments rt with N = 4000 and the set of parameters (6). The figure shows apparently the characteristic of intermittent chaos, that is, a long laminar phase, where the price fluctuations behave regularly, is interrupted from time to time by chaotic bursts. 

0312065v1.fig2

Fundamentalists – Risky Asset – Chartists

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Let us consider the market for a risky asset, composed of two groups of traders having different trading strategies: fundamentalists and chartists. Fundamentalists are assumed to have a reasonable knowledge of the fundamental value of the risky asset. The fundamentalist’s strategy can be described as follows: if the price pt is below the fundamental value p, then the fundamentalist tries to buy the risky asset because it is undervalued; if the price pt is above the fundamental value p, then he tries to sell the risky asset because it is overvalued. The excess demand for the risky asset is given by:

xft = exp(α(p − pt)) − 1, α > 0 —– (1)

where α is the parameter that denotes the strength of the non-linearity of the fundamentalist excess demand function (1). The fundamentalist’s excess demand function (1) is derived in a one-period utility optimizing framework. The technical details of the derivation of (1) within a utility maximizing framework are given in Kaizoji. We can see that Equation (1) has captured the distinctive features of the fundamentalist’s strategy. While fundamentalists calculate the fundamental value, chartists estimate a trend in the price change. Chartists can be assumed to form their expectation of the price of the risky asset according to the simple adaptive scheme:

pet+1 = pet + µ(pt − pet), 0 < µ ≥ 1 —– (2)

where pet denotes the price at period t expected by chartists, and the parameter µ is the error correction coefficient. As above, the chartist’s excess demand function is given by:

xct = exp(β(pet+1 − pt)) − 1, β > 0 —– (3)

where β is the parameter that denotes the strength of the non-linearity of the fundamentalist excess demand function (1). The chartist’s excess demand function (3) means that chartists try to buy the risky asset when they anticipate that the price will rise within the next period; inversely, that they try to sell the risky asset when they expect the risky asset price to fall within the next period. Let us now consider the adjustment process of the price in the market. We assume the existence of a market-maker who mediates the trading. If the excess demand in period t is positive (negative), the market maker raises (reduces) the price for the next period t+1. Let κ be the fraction of chartists in the total number of traders. Then the process of price adjustment can bewritten as

pt+1 − pt = θN[(1 − κ)xft + κxct] —– (4)

where θ denotes the speed of the adjustment of the price, and N the total number of traders…..

Conjuncted Again: Noise Traders, Chartists and Fundamentalists: The In-Betweeners

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Interesting questions are whether the rational traders (fundamentalists) will drive irrational traders (chartists and noise traders) out of the market, or whether the irrational traders (chartists and noise traders) will derive the rational traders (fundamentalists) out of the market. As in many studies on heterogeneous interacting agent models, it may seem natural that switching between different trading strategies plays an important role. Now the question is: how a trader makes his choice between the fundamentalist and chartist strategies. The basic idea is that he chooses according to the accuracy of prediction. More precisely, the proportion of chartists κt is updated according to the difference between the squared prediction errors of each strategy. Formally, we write the dynamics of the proportion of chartists κ as

κt = (1 – ξ)/(1 + exp(Ψ(Ect – Eft)) —– (1)

Ect = (pt – pct)2>, Eft = (pt – pft)2

where ψ measures how sensitively the mass of traders selects the optimal prediction strategy at period t. This parameter was introduced as the intensity of choice to switch trading strategies. Equation 1 shows that if the chartists’ squared prediction error Ect is smaller than that of fundamentalists Eft , some fraction of fundamentalists will become chartists, and visa versa.

Chartists

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Chartists are assumed to have the same utility function as the fundamentalists. Their behavior is formalized as maximizing the utility function

v = α(yt + pt+1cxtc) + βxtc – (1+ βxtc) log (1+ βxtc) —– (1)

subject to the budget constraint

ytc + ptxtc = 0 —– (2)

where xtc and ytc represent the chartist’s excess demand for stock and for money at period t, and pt+1c denotes the price expected by him. The chartist’s excess demand function for the stock is given by

xtc = 1/β (exp α (pt+1– pt)/β – 1) —– (3)

His expectation formation is as follows: He is assumed to forecast the future price pt+1c using adaptive expectations,

pt+1c  = pt + μ (p– ptc) —– (4)

where the parameter µ(0 < µ < 1) is a so-called error correction coefficient. Chartists’ decisions are based on observation of the past price-data. This type of trader, who simply extrapolates patterns of past prices, is a common stylized example, currently in popular use in heterogeneous agent models. It follows that chartists try to buy stock when they anticipate a rising price for the next period, and, in contrast, try to sell stock when they expect a falling price.

Stocks and Fundamentalists’ Behavior

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Let us consider a simple stock market with the following characteristics. A large amount of stock is traded. In the market, there are three typical groups of traders with different strategies: fundamentalists, chartists, and noise traders. Traders can invest either in money or in stock. Since the model is designed to describe stock price movements over short periods, such as one day, the dividend from stock and the interest rate for the risk-free asset will be omitted for simplicity. Traders are myopic and bent on maximizing utility. Their utility depends on the price change they expect, and on their excess demand for stock rather than simply their demand. Their excess demand is derived from utility maximization.

Let Ytf be the amount of money that a fundamentalist holds at a time t and Xtf be the number of shares purchased by a fundamentalist at time t. Let pt be the price per share at time t. The fundamentalist’s budget constrain is given by

Ytf + ptXtf = Yt-1f + ptXt-1f —– (1)

or equivalently

ytf + ptxtf = 0 —– (2)

where

ytf = Ytf – Yt-1f

denotes the fundamentalist’s excess demand for money, and

xtf = Xtf – Xt-1f

his excess demand for stock. Suppose that the fundamentalist’s preferences are represented by the utility function,

u = α(ytf + pt+1fxtf + βxtf – (1 + βxtf) log (1 + βxtf) —– (3)

where pt+1f denotes the fundamentalist’s expectation in period t about the price in the following period t + 1. The parameters α and β are assumed to be positive. Inserting (2) into (3) the fundamentalist’s utility maximization problem becomes:

maxxtf  u =  α(pt+1– pt)xtf  βxtf – (1 + βxtf) log (1 + βxtf) —– (4)

The utility function u satisfies the standard properties: u′ (|xtf|) > 0, u′′(|xtf|) < 0 ∀ |xf|t ≤ |xf*|, where |xf*| denotes the absolute value of xf producing a maximum utility. Thus, the utility function is strictly concave. It depends on the price change expected by fundamentalists (pt+1– pt) as well as fundamentalist’s excess demand for stock xtf. The first part α(pt+1– pt)xtf implies that a rise in the expected price change increases his utility. The remaining part expresses his attitude toward risk. Even if the expected price change is positive, he does not want to invest his total wealth in the stock, and vice versa. In this sense, fundamentalists are risk averse. β is the parameter that sets the lower limitation on excess demand. All excess demand for stock derived from the utility maximization is limited to -1/β. When the expected price change (pt+1– pt) is positive, the maximum value of the utility function is also positive. This means that fundamentalists try to buy stock. By analogy, when the expected price change (pt+1– pt) is negative, the maximum value of the utility function is negative, which means that they try to sell. The utility maximization problem (4) is solved for the fundamentalist’s excess demand,

xtf = 1/β(exp(α(pt+1– pt)/β) – 1) —– (5)

Excess demand increases as the expected price change (pt+1– pt) increases. It should be noticed that the optimal value of excess supply is limited to -1/β, while the optimal value of excess demand is not restricted. Since there is little loss of generality in fixing the parameter β at unity, below, we will assume β to be constant and equal to 1. Then let us think of the fundamentalist’s expectation formation. We assume that he form his price expectation according to a simple adaptive scheme:

pt+1f = p+ ν(p* – pt) —– (6)

We see from Equation (6) that fundamentalists believe that the price moves towards the fundamental price p* by factor ν. To sum up fundamentalists’ behavior: if the price pt is below their expected price, they will try to buy stock, because they consider the stock to be undervalued. On the contrary, if the price is above the expected value, they will try to sell, because they consider the stock to be overvalued.