Heterogeneous Fundamentalists in a Continuous Time Model with Delays

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Research Article

Heterogeneous Fundamentalists in a Continuous

Time Model with Delays

Luca Gori,

1

Luca Guerrini,

2

and Mauro Sodini

3

1_{Department of Law, University of Genoa, Via Balbi 30/19, 16126 Genoa, Italy}

2_{Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121 Ancona, Italy}

3_{Department of Economics and Management, University of Pisa, Via Cosimo Ridol 10, 56124 Pisa, Italy}

Correspondence should be addressed to Luca Guerrini; luca.guerrini@univpm.it Received 14 March 2014; Accepted 17 July 2014; Published 20 August 2014 Academic Editor: Jianping Li

Copyright © 2014 Luca Gori 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. We develop a continuous time model with heterogeneous fundamentalists, imitators, and discrete time delays. We show that nonlinear dynamic phenomena, such as coexistence of attractors and local and global bifurcations, occur due to the existence of a time gap in the process of adjustment of market prices.

1. Introduction

Starting from the seminal paper by Brock and Hommes [1], attention has been placed in building on models that try to mimic the behaviours of financial markets also in the absence of stochastic shocks. A mechanism that has proved to be fruitful in this direction has been the introduction of heterogeneous expectations of financial markets operators [2]. This literature has essentially been developed through the study of discrete time models [3]. For instance, Chiarella et al. [4] deepen the mathematical properties of a two-dimensional dynamic model with fundamentalists and chartists, while Tramontana et al. [5] develop a three-dimensional nonlinear dynamic model of internationally connected financial mar-kets.

By pursuing the line of research of He et al. [6], He and Zheng [7], and He and Li [8], the aim of this paper is to inquire whether complex dynamic phenomena obtained in discrete time models also hold in continuous time models with discrete time delays. To this purpose, we use a model sim-ilar to that adopted by He and Westerhoff [9] and Naimzada and Ricchiuti [10]. We assume two experts that have different beliefs on the fundamental of an asset and choose their allocations by using the mean-variance criterion in every moment in time [2]. Financial operators are imitators and

select allocations established by the two experts depending on performances obtained.

We show that this can generate nonlinear dynamics also in a continuous time framework with time delays. In particular, coexistence of attractors as well as local and global bifurcations can occur.

The rest of the paper is organised as follows.Section 2sets up the model.Section 3characterises local stability properties and local bifurcations of equilibria.Section 4provides some numerical experiments to validate the theoretical results established inSection 3, while also showing the emergence of global bifurcations.Section 5outlines the conclusions.

2. The Model

We consider a continuous time version of the model devel-oped by Naimzada and Ricchiuti [10] augmented with discrete time delays. Specifically, we set up a model with a risk-free asset, characterised by a perfectly elastic supply and an instantaneous rate of return𝑟 > 0, and a risky asset, with a price per share𝑥(𝑡) and a (stochastic) dividend process 𝑦(𝑡). There are two types of market operators (fundamentalists and imitators) and a market maker that behaves as a Walrasian auctioneer. We assume there exist two types (Type 1 and Type 2) of fundamentalists with different beliefs on the Volume 2014, Article ID 959514, 6 pages

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fundamental of the traded assets. They are myopic and behave on the basis of the mean-variance criterion at each moment in time. By considering (1) in Naimzada and Ricchiuti [10, page 173], we get the following continuous time version of wealth dynamics of the Type𝑖 = 1, 2 fundamentalist:

̇𝑤𝑖(𝑡) = 𝑟𝑤𝑖(𝑡) + [ ̇𝑥 (𝑡) + ̇𝑦 (𝑡) + 𝑦 (𝑡) − 𝑟𝑥 (𝑡)] 𝑞𝑖(𝑡) , (1)

where𝑤_𝑖(𝑡), ̇𝑤_𝑖(𝑡), and 𝑞_𝑖(𝑡) are the wealth, its time derivative, and the share of risky asset of fundamentalist 𝑖 at time 𝑡, respectively.

The objective of Type 𝑖 fundamentalist is therefore the following:

max

{𝑞𝑖(𝑡)}

𝐸_𝑖[ ̇𝑤_𝑖(𝑡) + 𝑤 (𝑡)] −𝑎

2𝑉𝑖[ ̇𝑤𝑖(𝑡) + 𝑤 (𝑡)] , (2) where 𝑎 > 0 is a parameter that measures the degree of risk aversion of both agents. By assuming that the variance is constant and is given by𝜎2, the maximisation programme (2) gives the following market demands of the risky asset:

𝑞_𝑖(𝑡) = 𝐸𝑖[ ̇𝑥 (𝑡) + ̇𝑦 (𝑡) + 𝑦 (𝑡) − 𝑟𝑥 (𝑡)]

𝑎𝜎2 . (3)

Similarly with Naimzada and Ricchiuti [10], we assume that both fundamentalists have the same correct expectations on dividend dynamics, that is,𝐸_𝑖[ ̇𝑦(𝑡)] = ̇𝑦(𝑡), but hetero-geneous expectations on price per share dynamics, that is, 𝐸_𝑖[ ̇𝑥(𝑡)+𝑥(𝑡)] = 𝐹_𝑖> 0. This means that every fundamentalist expects that the value of the risky asset tends to a level believed as being its fundamental. From (3), it follows that the share of risky asset of fundamentalist𝑖 is given by

𝑞_𝑖(𝑡) = 𝛼 [𝐹_𝑖− 𝑃 (𝑡)] , (4)

where𝑃(𝑡) = (𝑟 + 1)𝑥(𝑡) − ̇𝑦(𝑡) − 𝑦(𝑡) and 𝛼 = 1/(𝑎𝜎2). In addition, we set𝐹₁< 𝐹₂without loss of generality.

In this model, fundamentalists play the role of experts in the market and other agents follow fundamentalists’ choices depending on a mechanism that rewards the portfolio alloca-tions based on the fundamental closer to the realised market price. Specifically, we assume the following adjustment rule:

̇𝐿 (𝑡) = [1 − (𝐹1− 𝑃 (𝑡))2

(𝐹₁− 𝑃 (𝑡))2+ (𝐹₂− 𝑃 (𝑡))2] − 𝐿 (𝑡) , (5) where𝐿(𝑡) is the share of agents that follow Type 1 funda-mentalist, ̇𝐿(𝑡) is the time derivative of 𝐿(𝑡), and (𝐹_𝑖− 𝑃(𝑡))2is the squared error related to expert𝑖. Then, this rule stimulates agents to adopt portfolio choices of the fundamentalist whose fundamental value deviates less from the price actually realised on the market.

In addition, we assume the existence of a Walrasian auctioneer that fixes the price of the risky asset according to the following mechanism:

̇𝑃 (𝑡) = 𝛽 {[𝐿 (𝑡 − 𝜏) + ̇𝐿 (𝑡 − 𝜏)] 𝑞1(𝑡 − 𝜏)

+ [1 − 𝐿 (𝑡 − 𝜏) − ̇𝐿 (𝑡 − 𝜏)] 𝑞2(𝑡 − 𝜏)} ,

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where 𝛽 > 0 is the speed of adjustment of prices over time. The dynamics of price defined in (6) determines the variation of the market price of the risky asset depending on the price and allocation choices of imitators (followers) made at𝑡 − 𝜏. This is a quite realistic assumption with regard to the adjustment rule of prices, given that the time gap𝜏 can be referred to small time intervals (e.g., minutes or hours) as those observed in actual financial markets.

In order to simplify notation, in what follows we omit the time dependence for variables and derivatives referred at time 𝑡 and use V_𝑑 to indicate the state of a generic variableV at time𝑡 − 𝜏. Now, by using (4) and (5) to substitute into (6) to eliminate𝑞_𝑖,𝑑and𝐿_𝑑+ ̇𝐿_𝑑, respectively, we get

̇𝑃 = 𝛾 [(𝐹1− 𝑃𝑑) (𝐹2− 𝑃𝑑)2+ (𝐹2− 𝑃𝑑) (𝐹1− 𝑃𝑑)2

(𝐹₁− 𝑃_𝑑)2+ (𝐹₂− 𝑃_𝑑)2 ] , (7) where𝛾 := 𝛼𝛽 > 0.

3. Existence of Equilibria and

Local Bifurcations

In this section we perform the analysis of the delay differential equation defined in (7) (see, e.g., [11]).

Equilibria of (7) are obtained by setting ̇𝑃 = 0 and 𝑃_𝑑 = 𝑃 = 𝑃_∗for all𝑡. By doing this, one finds that the nontrivial equilibria𝑃_∗of (7) must satisfy the following equation:

(𝐹₁− 𝑃_∗) (𝐹₂− 𝑃_∗) (𝐹₁+ 𝐹₂− 2𝑃_∗) = 0. (8) Consequently, (7) has three positive equilibria:𝑃_∗= 𝐹₁, 𝑃_∗= 𝐹₂, and𝑃_∗= (𝐹₁+ 𝐹₂)/2.

Letℎ = 𝑃 − 𝑃_∗. Then (7) can be transformed into ̇ℎ = 𝛾 { ([𝐹₁− (ℎ + 𝑃_∗)] [𝐹₂− (ℎ_𝑑+ 𝑃_∗)]2 + [𝐹₂− (ℎ + 𝑃_∗)] [𝐹₁− (ℎ_𝑑+ 𝑃_∗)]2) × ([𝐹1− (ℎ𝑑+ 𝑃∗)]2+ [𝐹2− (ℎ𝑑+ 𝑃∗)]2) −1 } . (9)

The characteristic equation of the linear part of (9) is given by 𝑃 (𝜆, 𝜏) = 𝜆 − 𝑀𝑒−𝜆𝜏= 0, (10) where 𝑀 ={{_{_{ { −𝛾, if 𝑃_∗ = 𝐹₁ or𝑃_∗= 𝐹₂, 𝛾, if 𝑃_∗ =𝐹1+ 𝐹2 2 . (11)

When there is no delay, that is,𝜏 = 0 in (9), the characteristic equation becomes𝑃(𝜆, 0) = 𝜆 − 𝑀 = 0. Then, 𝑃_∗ = 𝐹₁and 𝑃_∗= 𝐹₂are locally asymptotically stable (𝜆 = −𝛾 < 0), while 𝑃_∗= (𝐹₁+ 𝐹₂)/2 is unstable (𝜆 = 𝛾 > 0).

Now, assume that𝜏 > 0 in (9). We will investigate the location of the roots of the transcendental equation. First, it is immediate that (10) has no zero root. Next, we examine when this equation has pure imaginary roots.

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9 8 7 6 5 4 3 2 1 3 2 1 4 5 6 7 8 9 P(t) P( t− 𝜏) (a) 10 9 8 7 6 5 4 3 1 2 0 80 40 60 0 20 100 120 140 160 180 200 t P( t) (b)

Figure 1:𝜏 = 6. (a) Two trajectories (depicted in blue and red) generated by two different initial conditions 𝑃(𝑡) = 4.8 (blue line) and 𝑃(𝑡) = 5.2 (red line), 𝑡 ∈ (−6, 0). (b) Corresponding time series.

10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 P(t) P( t− 𝜏) (a) 600 500 400 300 200 100 0 10 9 8 7 6 5 4 3 2 1 0 t P( t) (b)

Figure 2:𝜏 = 9. (a) Two attracting closed invariant curves. (b) Corresponding time series with unique maximum and minimum.

Proposition 1. Characteristic equation (10) has a pair of

purely imaginary conjugate roots𝜆 = ±𝑖𝜔₀when𝜏 = 𝜏_𝑗 (𝑗 =

0, 1, 2, . . .), where 𝜔₀= 𝛾, 𝜏_𝑗= { { { { { { { 1 𝛾( 𝜋 2 + 2𝜋𝑗) , 𝑖𝑓 𝑃∗ = 𝐹1 𝑜𝑟 𝑃∗= 𝐹2, 1 𝛾( 3𝜋 2 + 2𝜋𝑗) , 𝑖𝑓 𝑃∗= 𝐹₁+ 𝐹₂ 2 . (12)

Proof. For𝜔 > 0, 𝜆 = 𝑖𝜔 is a root of (10) if and only if

𝑖𝜔 − 𝑀𝑒−𝑖𝜔𝜏= 0. (13)

Separating the real and imaginary parts, we obtain

𝜔 = −𝑀 sin 𝜔𝜏, 0 = 𝑀 cos 𝜔𝜏. (14)

If𝑃_∗ = 𝐹₁or𝑃_∗ = 𝐹₂, then (14) yield𝜔𝜏 = 𝜋/2 and 𝜔 = 𝛾; on the other hand, if𝑃_∗ = (𝐹₁+ 𝐹₂)/2, then 𝜔𝜏 = 3𝜋/2 and 𝜔 = 𝛾. The conclusion is immediate.

From (12), we note that there exists an inverse relationship between the value of𝛾 (and thus of the speed of adjustment of prices over time𝛽) and the bifurcation value of 𝜏. This implies that the speed of adjustment of prices over time𝛽 plays a destabilising role in the model (i.e., for a high value of 𝛽 the Hopf bifurcations occur for a low value of 𝜏), while the variance𝜎2and the degree of risk aversion𝑎 play a stabilising role (i.e., for high values of𝜎2 and𝑎 the Hopf bifurcations occur for a high value of𝜏).

Lemma 2. 𝜆 = 𝑖𝜔0is a simple purely imaginary root of (10),

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10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 P(t) P( t− 𝜏) (a) 600 500 400 300 200 100 0 10 9 8 7 6 5 4 3 2 1 0 t P( t) (b)

Figure 3:𝜏 = 12. (a) Two attracting closed invariant curves. (b) Corresponding time series with two local maxima and minima.

Proof. If 𝜆 = 𝑖𝜔₀ is not simple, that is, 𝑃(𝑖𝜔₀, 𝜏_𝑗) = 𝑃󸀠(𝑖𝜔0, 𝜏𝑗) = 0, then one has 1 + 𝑖𝜔0𝜏𝑗 = 0, which is a

contradiction. Let us assume that there exists a root𝜆_𝑛such that𝑃(𝜆_𝑛, 𝜏₀) = 0 and 𝜆_𝑛 = 𝑖𝑛𝜔₀for some𝑛 ̸= 0, ±1. From (14), we get𝑛2= 𝑀2/𝜔2₀= 1. Therefore, the statement follows immediately.

Let𝜆_𝑗= ]_𝑗(𝜏) + 𝑖𝜔_𝑗(𝜏) be the roots of (10) close to𝜏 = 𝜏_𝑗 that satisfy]_𝑗(𝜏_𝑗) = 0 and 𝜔_𝑗(𝜏_𝑗) = 𝜔₀, where𝜔₀and𝜏_𝑗are defined by (12). The next result indicates that each crossing of the real part of characteristic roots at𝜏_𝑗must be from left to right; that is, stability is lost at the smallest stability switch and it cannot be regarded later.

Lemma 3. The following transversality condition ]󸀠_𝑗(𝜏_𝑗) > 0

holds.

Proof. Differentiating both sides of (10) with respect to𝜏 gives 𝑑𝜆 𝑑𝜏 = − 𝜆2 1 + 𝜆𝜏. (15) Then, we have 𝑑𝜆 𝑑𝜏󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜏=𝜏𝑗 = _{1 + 𝑖𝜔}𝜔20 0𝜏𝑗, (16) which implies ]󸀠_𝑗(𝜏_𝑗) =_{1 + 𝜔}𝜔02₂ 0𝜏𝑗2, (17)

concluding the proof.

Lemma 4. Let 𝑃∗= 𝐹1or𝑃∗ = 𝐹2. If𝜏 ∈ [0, 𝜏0), then all roots

of (10) have negative real parts; if𝜏 = 𝜏₀, then all roots of (10)

except𝜆 = ±𝑖𝜔₀have negative real parts; if𝜏 ∈ (𝜏_𝑗, 𝜏_𝑗+1) for

𝑗 = 1, 2, . . ., then (10) has2(𝑗+1) roots with positive real parts.

Proof. The first part follows noting that the equilibrium𝑃_∗is locally asymptotically stable when𝜏 = 0 and so its stability can only be lost if eigenvalues cross the imaginary axis from left to right. Let𝜆(𝜏₀) = ]+𝑖𝜔 be a root of (10) with] > 0. Then ] = −𝛾𝑒−]𝜏0_cos𝜔𝜏

0,𝜔 = 𝛾𝑒−]𝜏0sin𝜔𝜏0, and thus, we derive

(1/𝜔)tan−1(−𝜔/]) = 𝜏0 > 0, which is a contradiction the

left-hand side term of this equation being a negative number. Since the rate of change of the real part of an eigenvalue with respect to𝜏 when 𝜏 = 𝜏_𝑗 is positive, then the number of roots with positive real parts is increasing. Due to the fact above, the number of roots of the characteristic equation with positive real part will be constant for0 ≤ 𝜏 < 𝜏₀and equal to the number of eigenvalues with positive real parts when 𝜏 = 0 (i.e., zero being 𝑃_∗stable). For each subsequent interval 𝜏_𝑗 < 𝜏 < 𝜏_𝑗+1, the number can be determined from the number in the previous interval𝜏_𝑗−1< 𝜏 < 𝜏_𝑗and the number of roots with zero real part at𝜏_𝑗.

Spectral properties in the previous lemma immediately lead to stability properties of the zero solution of (9) and, equivalently, of the positive equilibrium𝑃_∗of (7).

Theorem 5. Let 𝜏_𝑗be defined as in (12).

(1) The positive equilibrium𝑃_∗ = (𝐹₁+ 𝐹₂)/2 is unstable

for all𝜏 ≥ 0.

(2) The positive equilibria𝑃_∗ = 𝐹₁and𝑃_∗ = 𝐹₂are locally asymptotically stable for𝜏 ∈ [0, 𝜏₀) and unstable for

𝜏 > 𝜏₀. Furthermore, (7) undergoes a Hopf bifurcation

at𝑃_∗when𝜏 = 𝜏_𝑗 (𝑗 = 0, 1, 2, . . .).

Remark 6. Since when 𝜏 = 0 (7) collapses into a one-dimensional autonomous differential equation, then in the continuous time version of the model by Naimzada and Ricchiuti [10] no persistent oscillations can occur, and the dynamics are therefore monotonic and convergent towards one of the equilibria. This result stresses the importance of

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10 9 8 7 6 5 4 3 2 1 0 10 9 8 7 6 5 4 3 2 1 0 P(t) P( t− 𝜏) (a) 600 500 400 300 200 100 0 10 9 8 7 6 5 4 3 2 1 0 t P( t) (b)

Figure 4:𝜏 = 14.3. (a) A unique attractor generated by the global bifurcation. (b) Corresponding time series with large oscillations.

time delays in generating endogenous fluctuations in this kind of models.

4. Numerical Simulations

In this section we provide some numerical simulations to validate the theoretical results on local bifurcations stated in

Section 3, and we show the occurrence of global bifurcations as well. For this purpose, we fix the following parameter set, 𝛾 = 0.2, 𝐹₁ = 2, and 𝐹₂ = 8, and let 𝜏 vary. For this parameter values the Hopf bifurcation occurs at𝜏₀ ≃ 7.854. For 𝜏 < 𝜏₀ (but sufficiently high), the dynamics of the model are oscillatory and convergent towards one of the equilibria depending on the initial conditions (as shown in Figure 1). Just after the Hopf bifurcation value there exist two attracting closed invariant curves (each of which with its basin of attraction), and then the dynamics of prices (and also the shares of imitators of Type 1 and

7 12 10 8 6 4 2 0 −2 ₈ ₉ ₁₀ ₁₁ ₁₂ ₁₃ ₁₄ ₁₅ P( t) 𝜏

Figure 5: Bifurcation diagram for𝜏 showing local maxima and minima of𝑃(𝑡) for a given value of 𝜏. Given the initial condition, for𝜏 < 𝜏₀ the dynamics converges to the equilibrium𝐹₂ = 8. For𝜏 = 𝜏₀, the equilibrium undergoes a Hopf bifurcation and the dynamics are characterised by oscillations with a maximum and a minimum for𝜏 ∈ (𝜏₀, 𝜏₁), with 𝜏₁ ≃ 11.6. Just after 𝜏₁it is possible to observe the birth of another local extremum. For𝜏 = 𝜏₂, a global bifurcation occurs and the dynamics lies in a larger portion of the phase plane. We note that for𝜏 > 𝜏₂ there are several windows in which many maxima and minima coexist, so that fluctuations become more complicated.

Type 2 fundamentalists) show persistent oscillations (see Figures2(a)and2(b)) characterised by unique maximum and minimum points.

By increasing the value of 𝜏, other phenomena are possible. In particular, we observe an increase in the number of local maxima and minima that resembles the sequence of period-doubling bifurcations shown by Naimzada and Ricchiuti [10] in a discrete time model (see Figures3(a)and

3(b)). When𝜏 increases further, a global bifurcation occurs and the two attractors merge each other. Then, a unique attractor is born (as shown in Figures 4(a) and 4(b)). At this stage, the dynamics of prices are characterised by large oscillations and possibly a high degree of unpredictability as suggested by the bifurcation diagram shown inFigure 5.

5. Conclusion

The debate on whether it is better to adopt continuous time models or discrete time models to describe the behaviours of financial markets operators is still open, as pointed out by He and Li [8, page 974]: “The [discrete time] set up facili-tates economic understanding of the role of heterogeneous expectations and mathematical analysis, it, however, faces a limitation when dealing with expectations formed from the lagged prices over different time horizons and a challenge to characterise the adaptive behaviour in a continuous-time.” Generally speaking, in the absence of stochastic shocks the continuous time framework tends to generate regular dynam-ics. Nevertheless, when one assumes that some economic

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processes (e.g., the adjustment of prices) react to changes occurred at a certain time gap, the dynamics generated by continuous time models tend to mimic those generated by discrete time models. This paper has confirmed the existence of nonlinear asset price dynamics in a continuous time version (with discrete time delays) of the model proposed by Naimzada and Ricchiuti [10], characterised by the existence of fundamentalists with heterogeneous expectations on the value of a risky asset.

Conflict of Interests

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

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Heterogeneous Fundamentalists in a Continuous Time Model with Delays

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