*4.4. Fundamental and Chartist Demand*

Let us now consider the dynamics with the fundamental and chartist demands. In Figure 10 (lhs), *a*, *b*, *c* are the same as those in the bottom of Figure 5, while *e* ∈ [0, 7.6]. The bifurcation diagrams show that low values of *e* do not change the dynamics, then the presence of chartist investors stabilizes the price pushing it to converge to a slightly overvalued price. Increasing *e* the dynamics becomes chaotic with small intervals in which the contribution of chartist demand pushes prices to converge or to oscillate around the equilibrium, not far from the fundamental value. How and why chartist demand can do this is not straightforward to understand and appears counterintuitive. Note that when *a* = 1.1, *b* = 0.8 and *c* < <sup>1</sup> *<sup>b</sup>* the fundamental map converges and chartist investors do

not affect the convergence, even a large amount of chartists do not change the dynamics. Similarly when *a* = 1 and *b* = 1.9 and *c* = <sup>1</sup> *<sup>b</sup>* chartist demand has no effect on the periodic dynamics of fundamental map. Differently, when *a* < 1, *b* < 1 and *c* < <sup>1</sup> *<sup>b</sup>* , such that the fundamental map dynamics is asymptotically stable, a small presence of chartist has no effect, while high values of *e* move the dynamics into chaos. Note that in Figure 10 (rhs) there is a range of *e* in which the price oscillates chaotic into the range (*v* − *δ*, *v* + *δ*), while for *e* approaching values near 7 prices move chaotic in a very large range. This fact suggests the following picture: prices oscillate in a reasonable range under the action of fundamental and chartist forces, for some reason, chartist investors increase in number till to surpass a certain threshold, after that, prices start to oscillate widely as it happens in case of market turbulence, with prices going up and down crazily. This confirms the belief that chartist investors participate in instability and bubble inflation. However, it appears that the responsibility of instability stays more in the unequal proportion of force pressures on the market than the specific action of an investment style itself. In other words, it is not the adoption of the technical analysis in itself the problem but their dominance in proportion.

Figures 11 and 12 propose three dynamics with comparable parameters. On the left-hand side, only fundamental demand is considered, while in the center fundamental demand is coupled with chartist demand and on the right-hand side, all the components are effective. The dynamics changes according to the demand: with only fundamental demand, the price stays in a narrow neighborhood of O2, when chartist demand intervenes, the regular dynamics moves into a chaotic elliptic attractor (see the plot in the center), while the intervention of market maker regularizes and pushes the price to converge fast (see rhs). In this case, chartist demand moves prices to chaos, and market makers restore the convergence. Differently in Figure 12, fundamental parameters are set to generate a periodic elliptic attractor (lhs), chartist demand has the effect to shrink the price to the fixed point O<sup>2</sup> (see in the center) and again the market makers accelerate the convergence (rhs). In this example, chartist demand has the effect to push prices to equilibrium.

**Figure 10.** Bifurcation diagrams for the map that considers together fundamentalist, and chartist demands. The diagrams are presented with respect to *e*. (**left**) shows the diagram with *a* = 1.1, *b* = 0.8, *c* = <sup>1</sup> *<sup>b</sup>* and *<sup>e</sup>* <sup>∈</sup> [0, 7.6]. (**right**) shows the diagram with *<sup>a</sup>* = 0.9, *<sup>b</sup>* = 0.8, *<sup>c</sup>* <sup>=</sup> <sup>1</sup> *<sup>b</sup>* and *e* ∈ [0, 7.6].

**Figure 11.** (**left**) shows the phase space of the fundamental map with *a* = 0.8, *b* = 0.9 and *c* = 1.1, (**middle**) shows the phase space of fundamental and chartist map with *a* = 0.8, *b* = 0.9 and *c* = 1.1 and *e* = 2.0, and (**right**) shows the phase space of the complete map with the same values of previous plot and *dm* = 0.5.

**Figure 12.** (**left**) shows the phase space of the fundamental map with *a* = 1.1, *b* = 0.9 and *c* = <sup>1</sup> *b* (periodic dynamics), (**middle**) shows the phase space of fundamental and chartist map with *a* = 1.1, *b* = 0.9 and *c* = <sup>1</sup> *<sup>b</sup>* and *e* = 2.0, (**right**) shows the phase space of the complete map with the same values of previous plot and *dm* = 0.5.

#### **5. Further Discussions**

The present section addresses a general discussion about the model, its limits, and some possible future evolutions. The presented model is deterministic and pointed toward considering how market participants, their styles, and their attitudes have an impact on prices and market instability. The model considers three rather standard strategies and provides stylized explanations of empirical market features. Thanks to its simplicity the model is flexible with respect to many future variations. One could include the possibility to have fundamental value varying over time, following for example a random walk as in [4]. This can be done substituting in Equation (10) *v* by

$$
\upsilon\_t = \upsilon\_{t-1} + \epsilon\_\prime
$$

where <sup>∼</sup> *<sup>N</sup>*(0, *<sup>σ</sup>*2).

Another variation could consider the switching between investment styles; this can be done by allowing the reaction parameters *c*, *d*, *e*, and *dm* also including the tolerance *δ* to change in time as in [19]. Indeed, in the full demand, we can see the contribution of investment styles accounted by traders simultaneously in the proportion given by the reaction parameters. Traders can consider differently the misalignment by fundamentals or the trend, thus reaction parameters can vary according to the strength of the investors' belief in each specific investment style. In other words, there could be phases in the market in which investors believe and adopt more fundamental analysis than technical one or vice versa. For doing so a mechanism of imitation and profitability evaluation must be

implemented. Alternatively, although more simplistic it could be done by considering parameters varying randomly in time. A stochastic version of the model can also be considered, including a random perturbation of prices to incorporate the effect of traders moved by liquidity needs or the effect of exogenous news arrival on the market. The stochastic version has the advantage to replicate more realistically the market features. Indeed, a deterministic model has mainly the aim to offer a stylized explanation of market features and not the presumption to be completely realistic as it is needed to be used for forecasting purposes for example. The stochastic version could be also subjected to a calibration procedure following the simulated method of moments as in [20,21]. Although the undeniable value of work pointed towards the reproduction of real market dynamics in its entirety, such an issue is outside the aims of the present paper and left for future investigations.

#### **6. Conclusions**

In the present model, a simple financial market model is considered with three rather standard market forces contributing simultaneously to the total demand: fundamental, chartist, and market makers. The model is given by a discrete difference equation system with a piecewise linear map of a high order. Each component has been studied separately to understand and highlight its contribution to the dynamics. Results are not surprising and in line with the related literature [1,4,12,13]: fundamental demand helps the stability of the system and keeps prices bounded; market makers satisfy their role of restoring stability, while the chartist demand component adds instability and chaotic movement to prices. However, we see that in some cases chartist demand can compensate for fundamental demand, felt in the loop, and pushes the dynamics to equilibrium. The lesson we can take from this fact is that the instability does not stay intrinsically in the nature of the demands but their combination and proportion. Indeed, markets are places where a mix of different beliefs and attitude meets. The market dynamics cannot be anything else than the outcome of such a variety of agents. In the market, the heterogeneity of actors finds an equilibrium that can show regular dynamics and smooth oscillations coherent with the efficient market hypothesis. Nevertheless, if any kind of disequilibrium occurs in the market forces, prices can oscillate widely, bear and bull periods can arrive, and bubbles can inflate and explode.

The contribution of the paper goes in three directions: (1) it considers a common, rather so far unstudied, chartist demand implying moving averages difference; (2) the model includes market maker participation, so far that it has not received so much attention; (3) the model uses a high order piecewise linear map that provides interesting bifurcation diagrams belonging to the class of border collision bifurcations but those deserve much attention for the future, also for their economic implications.

Future investigations can move in the following directions: (1) the study of border collisions bifurcations in high order systems as the one proposed here; (2) the sophistication of the market maker demand via a piecewise map; (3) the implementation of a stochastic version of the model with a mechanism of changing the weight of each investment style; (4) the calibration of a stochastic version of the model to reproduce real markets dynamics.

**Funding:** A.F. acknowledges financial support from the project HiDEA (Advanced Econometric methods for High-frequency Data) financed by the Italian Ministry of Education, University and Research (MIUR) under the program "PRIN: PROGETTI DI RICERCA DI RILEVANTE INTERESSE NAZIONALE—Bando 2017" Prot. 2017RSMPZZ.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data were generated during the study. The paper presents any information needed to replicate the study.

**Conflicts of Interest:** The author declares no conflict of interest.
