Asset Price Dynamics in a Chartist-Fundamentalist Model with Time Delays: A Bifurcation Analysis

Dobrescu, Loretti I.; Neamtu, Mihaela; Mircea, Gabriela

doi:https://doi.org/10.1155/2016/4907468

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2016 | Article ID 4907468 | https://doi.org/10.1155/2016/4907468

Asset Price Dynamics in a Chartist-Fundamentalist Model with Time Delays: A Bifurcation Analysis

Loretti I. Dobrescu,¹Mihaela Neamtu,²and Gabriela Mircea³

Academic Editor: Zhengqiu Zhang

Received25 Nov 2015

Accepted11 Jan 2016

Published22 Feb 2016

Abstract

This paper studies the dynamic behavior of asset prices using a chartist-fundamentalist model with two speculative markets. To this effect, we employ a differential system with delays à la Dibeh (2007) to describe the price dynamics and we assume that the two markets are coupled via diffusive coupling terms. We study two different time delay cases, namely, when both markets experience the same time delay and when the time delay is different across markets. First, we theoretically determine that the equilibrium exists and investigate its stability. Second, we establish the general conditions for the existence of local Hopf bifurcations and analyze their direction and stability. The common conclusion from both the delay scenarios we consider is that coupled speculative markets with heterogeneous agents in each, but with different price dynamics, can be synchronized through diffusive coupling. Finally, we provide some numerical illustrations to confirm our theoretical findings.

1. Introduction

When it comes to financial markets dynamics, there is wide consensus that the Efficient Market Hypothesis (EMH) is the standard theory [1]. This theory however fails to explain or predict situations when, for instance, speculative booms are followed by severe crashes. In other words, it cannot explain the excess volatility in these financial markets. As a result, several models have been developed in the last decade to describe markets fluctuations [1–12]. For instance, Dibeh [5] considers a market with two types of participants, namely, fundamentalists and chartists. The first categories of agents follow the EMH theory and base their decisions (and hence their demand formation) on the difference between the actual asset price and the fundamental asset price. On the other hand, the chartists base their market participation decision on the price trend of an asset. Thus, they attempt to exploit past price information when deciding whether to purchase or sell an asset. Specifically, Dibeh [5] used the following delay-differential equation to describe the dynamics of asset prices:where is the actual asset price, is the fundamental asset price, and is the market fraction of fundamentalists. For chartists, the time delay was introduced to capture the fact that they base the slope estimation of their asset price trend on adaptive expectations that consider the past values of the price trend slope. Simulation results showed that there may exist limit cycles for (1), which could explain the persistence of deviations from the fundamental price in speculative markets. These findings are crucial, as Bouchaud and Potters [13] found that, in models considering one asset (i.e., one market), feedback mechanisms are the main driver of market fluctuations, booms, and crashes.

Using Hassard et al. [14], Qu and Wei [1] provided a theoretical justification for the results in Dibeh [5]. Applying the local Hopf bifurcation theory, they analytically investigated the existence of periodic oscillations for (1), which depends on both the time delay and the market fraction of fundamentalists . Using the normal form theory and center manifold theorem (see also [15, 16]), they derived the sufficient conditions to determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions.

In modern finance theory however one of the key ideas is portfolio diversification. And once two or more risky assets (or markets) are available to investors (beliefs on) correlation in price/returns also becomes an investment decision factor. But could agents’ beliefs about returns correlation generate comovements in risky assets prices? And how can markets become interdependent? For instance, to what extent may the price dynamics in one market be affected by changes in agents’ behavior and beliefs in the alternative market?

In one of the first attempts to tackle these questions, Chiarella et al. [17] developed a discrete time model that combines the chartists-fundamentalists setup with the classical model of diversification between one risk-free and two risky assets. Their analysis showed how chartists’ beliefs and behavior may cause increasingly irregular price fluctuations that can be transmitted from one market to another. Furthermore, extending his previous work on models involving one asset (or one market), Dibeh [18] studied the role of feedback mechanism in synchronization and contagion effects (i.e., high asset prices correlation) between different markets. To this effect, the author used a nonlinear chartist-fundamentalist model with two markets coupled through a position feedback mechanism given by the price differential between the asset prices.

Drawing on the work in Dibeh [18], in this paper we consider the following system:where , are the market shares of fundamentalists, , , are time delays, , , denote the fundamental asset prices, , , represent the coupling strength between the two markets, and denotes an asset’s market index. Note that the expectations function of the chartists is nonlinear, which is conveyed through the hyperbolic tangent function. This formulation represents the chartists’ belief that price growth rates are bounded and so it introduces a saturation effect into the chartists demand function. Previous results showed that the synchronization of coupled speculative markets with different dynamics can occur through diffusive coupling. When going from an uncoupled to a coupled market model, the stable converging dynamics is replaced with limit cycle oscillations around the fundamental prices. Thus, coupling and contagion between financial markets can be responsible for the transmission of fluctuations across these markets and appear crucial for their stability [19].

Here is the rationale. The global financial system is formed of a multitude of very diverse markets, located all around the world, and those trade wide-ranging classes of assets. One of their common denominators is that assets price changes often respond to the same economic information and market news [20–22]. This dependency on the same signals leads to price variations that are often correlated. In other words, the price time series can exhibit similar characteristics, which implies that there is “coupling” between markets. Moreover, Fenn et al. [19, 23] found that the strength of correlations between many different assets increases following a credit crisis (i.e., 2007-2008 one), as financial institutions hold similar portfolios of assets [24]. This has important implications for the robustness of financial markets [25]. If many assets are correlated and prices fall, this can cause several financial institutions to write down the value of their assets. And these write-downs can then significantly impact the credit relationships between different institutions [26].

The aim of this paper is to provide a detailed theoretical analysis of the phenomenon of coupling and contagion between markets from the bifurcation point of view. First, we will analytically prove that a model involving two financial markets with correlated prices has equilibrium points that are locally asymptotically stable. Second, for these equilibrium points, we will determine the specific values and (i.e., the time delay parameters of the two markets) for which a Hopf bifurcation occurs. Finally, we will study the direction of the Hopf bifurcation, as well as the stability and period of the bifurcating periodic solutions.

We choose the time delays and as bifurcation parameters as they represent the “memory” of financial markets. Frank [27] discusses extensively the importance of time delays in modeling financial markets memory and their effect on the qualitative behavior of asset prices. Relatedly, in a 2001 paper, LeBaron [28] also discusses the importance of time horizons in agent-based computational economics. Other works that use time delays as parameters of bifurcations include Hale and Lunel [29], Hassard et al. [14], Mircea et al. [10], Qu and Wei [1], and Xu and Li [30].

The paper proceeds as follows. In Section 2 we determine the equilibrium points of differential system (2) for . Section 3 investigates the local asymptotic stability of the equilibrium points and establishes the existence of the Hopf bifurcation for these points. Section 4 deals with the direction and the stability of the Hopf bifurcation. In Section 5 we use numerical simulations to illustrate the validity of our main results. Section 6 concludes the paper.

2. Existence and Stability of the Equilibrium Points

Consider the model given by system (2). Since time delays do not change, the equilibrium points of this model are given by the solutions of the system:where and and , for The solution to system (3) is . To find the solution of system (3) with and , , we proceed as follows.

First, from (3) it follows thatwhich yields

Since , from (5) it follows that the coupling coefficients , , satisfy the conditions:

From (4) it follows that satisfies the following equation:whereLetand let be the roots of

Lemma 1. Let , and is fixed. If(i), then ;(ii), then ;(iii), then .

Proof. The proof follows from and , for any and , and .

Proposition 2. Letwhere , and are given in (8). The equilibrium point of system (2) is , whereFor and fixed, the coordinates of are positive.

Proof. The proof follows from Lemma 1 and Cardan’s formula, which give the positive solution of a 3rd-degree equation.

Let be an equilibrium point, different from (0,0), and let , and . System (2) then becomeswhereThe terms are given bywith , being given by

The characteristic function for the linear part of (13) is given bywhere

In order to study the distribution of the roots of the transcendental equation , we use the lemma in Ruan and Wei [31] (see Appendix A) and analyze (17) considering four cases: (i) , (ii) , (iii) , and (iv) .

We note that, generally, the analysis of an equation with two delay parameters implies first studying the case when one delay parameter is zero and the other one is different from zero. For this purpose, we set . The next step is to set as a variable and determine the value (of ) for which a Hopf bifurcation occurs. From a practical point of view, only the cases and are relevant. The other two cases (that involve ) are however needed methodologically, as they allow us to derive some of the results that we need when analyzing the economically relevant cases.

Case 1. If , thenUsing (14) and (18), the equation becomesThe necessary and sufficient conditions for (20) to admit roots with a negative real part areIf the conditions in (21) hold, then the equilibrium point is locally asymptotically stable.

3. The Local Stability of the Hopf Bifurcation

In the analysis of the Hopf bifurcation, we consider and as parameters and as an equilibrium point for which one of the situations presented in Proposition 2 is verified.

Case 2. If , the characteristic function (17) becomeswhereWe now rewrite (22) asnoting that when is a root of (24) if and only if satisfies By direct computation, we have Using , from (26), it follows thatDenoting , (27) yieldsLetand defineThen, from the sign of , we have thatwhereand is one of the cubic roots of the complex number , and
Assuming that the following conditions hold:the equation has at least one positive root We denote and definewhere

From the above relations, we have the following result.

Proposition 3. Let , and be given by (18), (28), and (36). Suppose that conditions (34) are satisfied and that the positive root of (29) satisfies Then, the equilibrium point is asymptotically stable, when , and it is unstable, when Moreover, at is a pair of simple imaginary roots of (29) and system (2) undergoes Hopf bifurcation near If (29) does not have a positive root, then the equilibrium point is locally asymptotically stable for all

Case 3. If and , then becomesFor , let be a root of (37). It then follows thatwhich leads toIt is easy to see that if the conditionshold, then (39) has no positive roots. Hence, all the roots of (37) have negative real parts when .
If , then (39) has a unique positive root . Substituting into (38), we obtainwith . If the conditions hold, then (39) has two positive roots and . Substituting into (38), we obtainwith .
Let be a root of (37) near and . According to the functional differential equation theory, for every , there exists such that is continuously differentiable in for Substituting into the LHS of (37) and taking derivative with respect to yield which leads to Since , we have thatSimilarly, we can obtain

From the above analysis and Ruan and Wei [31, Corollary ], we have the following result.

Proposition 4. For , assume that either condition (41) or condition (43) is satisfied. Then, the following results hold.(i)If (41) holds, the equilibrium point is asymptotically stable for all .(ii)If (43) holds, then there is a positive integer such that the equilibrium point is asymptotically stable when , and it is unstable when Moreover, at , system (2) undergoes Hopf bifurcation at , where are given by (44).

Case 4. Let and . We consider (17), with in its stable interval , and regard as a parameter. Without loss of generality, we assume that and let () be a root of (17). Then, we can obtain which leads towhereDenoteWe assume that there exists such that . For , there exists a sequence such that (50) holds.
Let When , (17) has a pair of purely imaginary roots , for
In the following we assume that Thus, by the general Hopf bifurcation theorem for FDEs in Hale and Lunel [29], we have the following result on stability and Hopf bifurcation for system (2).

Proposition 5. Assume that the above condition, , is satisfied and that Then, the equilibrium point is locally asymptotically stable when Moreover, when , system (2) undergoes Hopf bifurcation at

If is a positive root of (50), then is given bywhere

4. The Direction and the Stability of the Hopf Bifurcation

In the previous section, we have obtained the condition for the Hopf bifurcation to occur when , where is given by (42). Using the techniques from normal form and center manifold theory introduced by Hassard et al. [14], we now derive the explicit formulae determining the direction, stability, and periodicity of these solutions bifurcating from the positive equilibrium at the critical value . To this effect, we will use the first Lyapunov coefficient.

Throughout this section, we assume that differential system (2) undergoes a Hopf bifurcation at the positive equilibrium for and that are the corresponding purely imaginary roots of the characteristic equation at . Without loss of generality, we assume that , where and is given in (54). For convenience, let , and , where is given by (42) and Differential system (2) can be written as a FDE in as follows:where and The terms and are, respectively, given bywhere and with and being given by (13).

From Section 3 we know that if , then differential system (2) undergoes a Hopf bifurcation at and the characteristic equation associated with system (2) has a pair of simple imaginary roots By the Reisz representation theorem, there is a matrix function with bounded variation components such that We can choose For , we define Then, (57) is equivalent to the abstract differential equation:where , for

For , we define For and , we define the bilinear form where , , and are adjoint operators. Following from Section 3 discussion, we also know that are eigenvalues of , but they are also eigenvalues of

Proposition 6. (i) If the eigenvector of is given by(ii) If the eigenvector of is given byThe proof is provided in Appendix A.

Using , we can obtain as follows: Hence, we haveSimilarly, we can verify that

We now turn to studying the stability of the bifurcating periodic solution. As in Xu and Li [30], the bifurcating periodic solution has amplitude and nonzero Floquet exponent , with Under these assumptions, and are given by The sign of indicates the direction of the bifurcation, while determines the stability of , with being stable if and unstable if

In what follows, we will show how to derive these coefficients (i.e., and ). To this effect, we first construct the coordinates needed to describe the center manifold near , which is a local invariant, attracting a two-dimensional manifold [30].

Let be the solution of (63) when , and defineOn the center manifold , we have , whereand and are local coordinates for the center manifold in the direction of and Note that is real if is real (i.e., we consider only the real solutions). For the solution , since , it follows thatWe can rewrite this equation as In Appendix B, we show how to compute the , , , and coefficients explicitly. Using these coefficients, we can therefore computeFrom (77), it is easy to get the values of , and , as follows:

Using these results, we establish the following theorem on the system trajectories properties.

Theorem 7 (see [14]). If all the conditions above are satisfied (i.e., differential system (2) undergoes a Hopf bifurcation at the positive equilibrium for ), then(i) is the Hopf bifurcation value of system (57);(ii) establishes the direction of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for ;(iii) determines the stability of the bifurcating periodic solutions: if , then the bifurcating periodic solutions are stable (unstable);(iv) determines the period of the bifurcating periodic solutions: the period increases (decreases) if

5. Numerical Illustrations

To illustrate our results, we employ several numerical simulations. To this purpose, we set and (with ) and choose according to Lemma 1 and Proposition 2. Specifically, let , and . Using Lemma 1 and Proposition 2, we set . From Proposition 2, we then get , and . Using (9), we have that and . With this set of parameters, we obtain the following results:(1)If , from (20) it follows that and . The system is therefore locally asymptotically stable.(2)If , from (27) it follows that and from (35) it follows that . For , the system is asymptotically stable; for , it has a Hopf bifurcation; and for , it is unstable.

The orbits (in black) and (in blue) are given in Figures 1, 3, and 5 for , , and , respectively. Similarly, Figures 2, 4, and 6 plot the market index , where , for each of the three cases.(3)If ; from (39) it follows that . From (42) it follows that For , the system is locally asymptotically stable. For , the system has a Hopf bifurcation. Finally, for , the system is unstable.(4)If , let From (50), using the plot of the function , it follows that , and from (55) we get that For and , the system is locally asymptotically stable. If , the system has a Hopf bifurcation. Finally, if , the system is unstable.

The Lyapunov coefficients are , , and . Hence, the Hopf bifurcation is supercritical and with increasing period, while the bifurcating periodic solutions are unstable.

As before, for , the orbits (in black) and (in blue) are given in Figure 7, while Figure 8 plots , where denotes the market index.

We thus can conclude that all numerical simulations verify our previous theoretical findings.

6. Conclusions

This paper considers a deterministic dynamic model with delay for two financial markets that trade one asset. The deterministic system is analyzed both analytically and numerically. We study two different cases of time delay, namely, when both markets experience the same time delay and when the time delay is different across markets. The common conclusion from these two scenarios is that coupled speculative markets with chartists and fundamentalists in each, but with different dynamics, can be synchronized through diffusive coupling. The stable converging dynamics is replaced with limit cycle oscillations around the fundamental price. This qualitative change in the dynamic behavior shows that coupling and contagion between different markets can indeed lead to transmission of fluctuations across financial markets. Since the delay can lead to instability, the next natural step would be to consider the stochastic approach. We leave this issue for future research.

In terms of contribution, our work ultimately provides a real-world application of bifurcation concepts to a situation widely observed in the financial world. To this effect, we provide a rigorous analysis of the dynamic properties of a model that can successfully explain stylized facts related to financial market behavior. Once established, the same methodology could be easily implemented in economics, demography, and social sciences in general, allowing further literature advances in these fields. For instance, this application fits well and can be quickly integrated in the structural work done in these fields.

Appendices

A. Data of Lemma A.1

Lemma A.1 (see [31, Corollary , page 867]). For the transcendental equationwhereas vary, the sum of the orders of the zeros of in the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Proof of Proposition 6. (i) Let be given by (66). It follows from above thatimplyingAs a result, (67) follows.
(ii) Let be given by (68). Fromit follows that is given by (70).

B. Coefficients of Center Manifold

Following Hassard et al. [14], by substituting (63) in (75) we have Hence,whereTaking the derivative of with respect to in (74), we haveFrom (74), (76), and (B.4), we obtainand substituting (74), (B.3), and (B.2) yieldsComparing the coefficients of (B.5) and (B.6), the following equalities hold:where is a unity matrix.

From (75) and (76), we obtainwhere

Proposition B.1. The coefficients , , , and are given by

Proof. Comparing the coefficients of (77) and (B.9), (B.10) follows.

Proposition B.2. The vectors, and , are given bywhere

Proof. First, we use (B.2) to obtain Comparing the coefficients with (B.3), for , we obtain Substituting the above equalities into (B.7) and (B.8), respectively, and with being given by (62), we get thatSolving (B.16) leads to (B.12). To obtain the appropriate and , we proceed as follows.
From (B.7) and (B.8), it follows thatFrom the definition of in (62), we obtain From (B.15) and the definition of , we have that Substituting (B.12) and (B.19) into (B.18) and using the fact that we obtain which leads to and soSimilarly, substituting (B.12) and (B.19) into (B.17), we have that

We can therefore derive and , with in (B.10) being determined by the parameters and the time delays in system (2).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are especially grateful to D. Opris, the editor, for helpful comments and suggestions.

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Copyright © 2016 Loretti I. Dobrescu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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