Abstract
We propose a financial market model with optimistic and pessimistic fundamentalists who, respectively, overestimate and underestimate the true fundamental value due to ambiguity in the stock market. We assume that agents form their beliefs about the fundamental value through an imitative process, considering the relative ability shown by optimists and pessimists in guessing the realized stock price. We also introduce an endogenous switching mechanism, allowing agents to switch to the other group of speculators if they performed better in terms of relative profits. Moreover, the stock price is determined by a nonlinear mechanism. We study, via analytical and numerical tools, the stability of the unique steady state, its bifurcations and the emergence of complex behaviors, with possible multistability phenomena. To quantify the global propensity to optimism/pessimism of the market, we introduce an index, depending on pessimists’ and optimists’ beliefs and shares, thanks to which we are able to show that the occurrence of the waves of optimism and pessimism are due to the joint effect of imitation and switching mechanism. Finally, we perform a statistical analysis of a stochastically perturbed version of the model, which high lights fat tails and excess volatility in the returns distributions, as well as bubbles and crashes for stock prices, in agreement with the empirical literature.
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Notes
^{1} We stress that such a notion of ambiguity differs from the one employed in general equilibrium theory, where agents, in making their choices in stochastic frameworks, are assumed to take into account different probability laws describing the distribution of relevant random variables. See e.g. Dow and Werlang (1992) and Ellsberg (1961).
^{2} Starting with the crucial paper by De Bondt and Thaler (1985), a wellgrounded empirical literature has arisen to show the presence of overreaction phenomena in financial markets. We recall that, as said in France et al. (1994), page 19, overreaction is “defined as a movement in price that overshoots the equilibrium value and then subsequently returns to its true value”. Subsequently, some authors have given further foundations to overreaction events through the formulation and analysis of mathematical models. See, for instance, the works by Barberis et al. (1998), by Hong and Stein (1999) and by Veronesi (1999).
We only provide sufficient conditions for the occurrence of each scenario. We remark thatwe may indeed obtain the same scenario for other parameter configurations. Moreover, wenotice that some of the stability intervals we shall derive below may in principle be empty forany parameter configuration. We will provide numerical examples in Section 4 to show thatthere exist parameter settings for which such intervals are nonempty and the correspondingscenarios do actually occur.
Actually, if \(\mu _{1}=\mu _{2}\in (\max \left \{\mu _{\ell },0\right \},\mu _{r})\) we find two disjoint adjacent stability intervals. Such limit case can however be encompassed in the mixed scenario.
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Acknowledgments
The authors thank Professor He and Professor Hommes for their valuable comments about the interpretation of the model and the consistency with real data, and all the participants to NED 2015 for helpful discussions.
The authors wish also to thank the anonymous Reviewers and Professor Dieci, Guest Editor of the Special Issue on “Nonlinear Economic Dynamics”, for the useful suggestions.
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Appendix
Appendix
Proof of Proposition 1
The expression of the steady state for the population shares follows by noticing that, in equilibrium, P(t + 1) = P(t) and thus, by Eq. 3, we have π _{ X } = π _{ Y } = 0 at the steady state, so that \(\omega ^{\ast }=\frac {1}{2}\). Moreover, from the stock price equation we find \(P^{\ast }=\omega ^{\ast } X^{\ast }+(1\omega ^{\ast })Y^{\ast }=\frac {X^{\ast }+Y^{\ast }}{2}\), so that \(X^{\ast }=F\frac {\Delta }{2}\) and \(Y^{\ast }=F+\frac {\Delta }{2}\). Inserting such expressions in P ^{∗}, we get P ^{∗} = F, as desired. □
Proof of Proposition 2
Since from Eq. 7 it follows that
then we find
as desired. □
Proof of Proposition 3
To prove (i′) − (iii′), we use the conditions in Farebrother (1973).
To such aim, we need to compute the Jacobian matrix for G in correspondence to (X ^{∗}, P ^{∗}, ω ^{∗}),which reads as
The Farebrother conditions are the following:

(i)
1 − C _{1} + C _{2} − C _{3} > 0;

(ii)
1 − C _{2} + C _{1} C _{3} − (C _{3})^{2} > 0;

(iii)
3 − C _{2} > 0;

(iv)
1 + C _{1} + C _{2} + C _{3} > 0,
where C _{ i }, i ∈ {1, 2, 3}, are the coefficients of the characteristic polynomial
In our framework, we have:
and thus simple computations allow to notice that conditions (i) − (i i i) above read,respectively, as (i′) − (iii′), while (i v) reduces to \(\widetilde {\gamma }>0\), which is indeed true. □
Proof of Corollary 1
Firstly, we keep Δ ≠ 0, μ and \(\widetilde {\gamma }\) fixed and we solve conditions \((i^{\prime })(iii^{\prime })\) with respect to β.
If 2 − μΔ^{2} ≤ 0, we have that condition (i′) is not satisfied by any β > 0, which means that we are in the unconditionally unstable scenario.
Let us now consider 2 − μΔ^{2} > 0.Condition (i′) is then equivalent to
From (iii′) we have
Finally, we notice that we can rearrange condition (ii′) as k _{1} β ^{2} + k _{2} β + k _{3} > 0 with \(k_{1}= {\Delta }^{8}\widetilde {\gamma }^{2}\mu ^{2}  2{\Delta }^{6}\widetilde {\gamma }^{2}\mu ,\, k_{2}=4{\Delta }^{6} \widetilde {\gamma } \mu ^{2} + 8{\Delta }^{4}\widetilde {\gamma }^{2}\mu  16{\Delta }^{2}\widetilde {\gamma }\) and \(k_{3}=32\mu {\Delta }^{2} + 64\).
If μ > 0, then k _{1} < 0 < k _{3}, and thus condition (ii′) is fulfilled by
with \(\beta _{r_{0}}<0<\beta _{r_{1}}\) given by
where
Combining Eqs. 14–16 and recalling that β > 0,we find that system (i′) − (iii′) is equivalent to
In particular, if β _{ ℓ } ≤ 0,i.e., if \(\widetilde {\gamma }\le 2\mu {\Delta }^{2}\), and\(\min \{\beta _{r_{1}},\beta _{r_{2}}\}>0\), we have the destabilizing scenario, while when \(0<\beta _{\ell }<\min \{\beta _{r_{1}},\beta _{r_{2}}\}\)the mixed scenario occurs; in all other cases we have the unconditionally unstable scenario.
If, instead, μ = 0, then k _{1} = 0 and it is easy tosee that conditions (i′) − (iii′) are equivalent to \(\max \{\beta _{\ell },0\}<\beta <\beta _{r_{1}}\),with \(\beta _{\ell }=\frac {4}{\widetilde {\gamma }{\Delta }^{2}} \left (\frac {\widetilde {\gamma }}{2}1\right )\)and\(\beta _{r_{1}}=\frac {4}{\widetilde {\gamma }{\Delta }^{2}}\). Then, in this case we havea destabilizing scenario (\(\beta _{\ell }\le 0<\beta _{r_{1}}\))if and only if \(\widetilde {\gamma }\in (0,2]\), while a mixed scenario (\(0<\beta _{\ell }<\beta _{r_{1}}\))occurs if and only if \(\widetilde {\gamma }\in (2,4)\).
Now we keep Δ ≠ 0, β and \(\widetilde {\gamma }\) fixed and we solve conditions (i′) − (iii′) with respect to μ. Solving (i′) we find
while (iii′) provides
Condition (ii′) can be rewritten as
where \(q_{1}={\Delta }^{6}\beta \widetilde {\gamma }(4\beta \widetilde {\gamma }{\Delta }^{2}), \, q_{2}=2{\Delta }^{2}({\Delta }^{4}\beta ^{2}\widetilde {\gamma }^{2}  4{\Delta }^{2}\beta \widetilde {\gamma }^{2}  16)\)and \(q_{3}= 16(4\beta \widetilde {\gamma }{\Delta }^{2})\).
First we notice that, if q _{1} = 0, i.e., if \(\beta =4/(\widetilde {\gamma }{\Delta }^{2})\), then also q _{3} = 0and thus (20) is satisfied for all μ > 0 if and only if \({\Delta }^{4}\beta ^{2}\widetilde {\gamma }^{2}={\Delta }^{4} \left (4/(\widetilde {\gamma } {\Delta }^{2})\right )^{2}\widetilde {\gamma }^{2}=16< 4{\Delta }^{2}\beta \widetilde {\gamma }^{2}+ 16=32\), which is indeed true. We stress that if μ = 0 = q _{1}, then condition (ii′) is again never fulfilled and thus we do not have stability for μ = 0 if q _{1} = 0. Hence, when q _{1} = 0 conditions (i′) − (iii′) simply reduce to\(\mu \in (\mu _{\ell },\mu _{r})\cap (0,+\infty )\), and this provides the destabilizing scenario for μ _{ ℓ } < 0 < μ _{ r }, the mixed scenariofor 0 < μ _{ ℓ } < μ _{ r }, and the unconditionally unstable scenario for \(\mu _{r}\le \max \{\mu _{\ell },0\}\).
Let us now assume that q _{1} ≠ 0. If \(\chi ={q_{2}^{2}}4q_{1}q_{3}>0\), we can introduce the real numbers
and
where
We then have that Eq. 20 is solved by \(\mu <\mu _{1}\,\cup \,\mu >\mu _{2}\) if q _{1} > 0 and by μ _{1} < μ < μ _{2}if q _{1} < 0. Let us examine theformer case. Since we have q _{1} > 0 and q _{3} > 0, the sign of μ _{1/2} is the same of − q _{2}. Since from q _{1} > 0 we have \(\beta <4/(\widetilde {\gamma }{\Delta }^{2})\), we obtain
so that, in such case, we have q _{2} > 0 and thus μ _{1} < μ _{2} < 0and Eq. 20 is fulfilled by any μ ≥ 0. Combining this with Eqs. 18 and 19 we obtain \(\mu \in (\mu _{\ell },\mu _{r})\cap [0,+\infty )\), which provides the destabilizing scenario if μ _{ ℓ } < 0 < μ _{ r }, the mixed scenario if 0 < μ _{ ℓ } < μ _{ r } and the unconditionally unstable scenario if \(\mu _{r}\le \max \{\mu _{\ell },0\}\).
Conversely, if q _{1} < 0, combining μ _{1} < μ < μ _{2} with Eqs. 18 and 19 we have \(\mu \in (\max \left \{\mu _{1},\mu _{\ell }\right \},\min \left \{\mu _{2},\mu _{r}\right \})\cap [0,+\infty )\), which can again give rise to either a destabilizing, a mixed or an unconditionally unstable scenario.
If \({q_{2}^{2}}4q_{1}q_{3}<0\), we have that Eq. 20 is always fulfilled if \(4\beta \widetilde {\gamma }{\Delta }^{2}>0\) and never fulfilled when \(4\beta \widetilde {\gamma }{\Delta }^{2}<0\). In the former case, recalling Eqs. 18 and 19, we obtain \(\mu \in (\mu _{\ell },\mu _{r})\cap [0,+\infty )\), and thus we can have destabilizing, mixed and unconditionally unstable scenarios. If instead \(4\beta \widetilde {\gamma }{\Delta }^{2}<0\), conditions (i′) − (iii′) can not be satisfied and we just find the unconditionally unstable scenario.
It can be easily seen that the remaining situation \({q_{2}^{2}}4q_{1}q_{3}=0\) can only provide the previous scenarios.^{Footnote 4}
Finally, the stability conditions with respect to Δ when μ = 0 and β ≠ 0 read as
when \(\widetilde {\gamma }\ge 2\), and simply as \(0\le {\Delta }<\frac {2}{\sqrt {\widetilde {\gamma }\beta }}\,\), when \(\widetilde {\gamma }< 2\). For \(\widetilde {\gamma }\ge 2\) we then find the mixed scenario and for \(\widetilde {\gamma }<2\) the destabilizing scenario. The proof is complete. □
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Cavalli, F., Naimzada, A. & Pireddu, M. An evolutive financial market model with animal spirits: imitation and endogenous beliefs. J Evol Econ 27, 1007–1040 (2017). https://doi.org/10.1007/s0019101705068
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DOI: https://doi.org/10.1007/s0019101705068