An evolutive financial market model with animal spirits: imitation and endogenous beliefs

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.

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

$$\begin{array}{@{}rcl@{}} F&=&X(t+1)+\frac{\Delta}{1+e^{\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2})}}\\ &=&Y(t+1)-\frac{\Delta}{1+e^{-\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2}))}}, \end{array} $$

then we find

$$\begin{array}{@{}rcl@{}} Y(t+1)&=& X(t+1)+{\Delta}\left( \frac{1}{1+e^{\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2})}}+ \frac{1}{1+e^{-\mu((X(t)-P(t))^{2}-(Y(t)-P(t))^{2}))}}\right)\\ &=& X(t+1)+{\Delta}, \end{array} $$

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

$$ \left[\begin{array}{ccc} \frac{-\mu\Delta^{2}}{2} & \frac{\mu\Delta^{2}}{2} & 0\\ \widetilde{\gamma} & 1-\widetilde{\gamma} & -\widetilde{\gamma}{\Delta}\\ \frac{-\beta\widetilde{\gamma}\Delta}{4} & \frac{\beta\widetilde{\gamma}\Delta}{4} & \quad \frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4} \end{array}\right]. $$

(13)

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

$$\lambda^{3}+C_{1} \lambda^{2}+C_{2} \lambda+C_{3}=0.$$

In our framework, we have:

$$C_{1}=\frac{\mu{\Delta}^{2}}{2}+\widetilde{\gamma}-1-\frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4}, \quad C_{2}=\frac{-\mu{\Delta}^{2}}{2}\left( 1+\frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4}\right)+ \frac{\beta\widetilde{\gamma}{\Delta}^{2}}{4},\quad C_{3}=\frac{\mu\beta\widetilde{\gamma}{\Delta}^{4}}{8}, $$

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 − μΔ² ≤ 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 − μΔ² > 0.Condition (i′) is then equivalent to

$$ \beta>\beta_{\ell}=\frac{4}{\widetilde{\gamma}{\Delta}^{2}}\left( \frac{\widetilde{\gamma}} {2-\mu{\Delta}^{2}}-1\right). $$

(14)

From (iii′) we have

$$ \beta<\beta_{r_{2}}=\frac{4}{\widetilde{\gamma}{\Delta}^{2}}\cdot\frac{6+\mu{\Delta}^{2}}{2-\mu{\Delta}^{2}}. $$

(15)

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

$$ \beta_{r_{0}}<\beta<\beta_{r_{1}}, $$

(16)

with $\beta _{r_{0}}<0<\beta _{r_{1}}$ given by

$$\beta_{r_{0}}=\frac{2}{\mu\widetilde{\gamma}{\Delta}^{4}}\cdot\frac{\mu^{2}{\Delta}^{4}+2\mu\widetilde{\gamma} {\Delta}^{2}-4-\sqrt{\xi}}{2+\mu{\Delta}^{2}},\; \beta_{r_{1}}=\frac{2}{\mu\widetilde{\gamma}{\Delta}^{4}}\cdot \frac{\mu^{2}{\Delta}^{4}+2\mu\widetilde{\gamma} {\Delta}^{2}-4+\sqrt{\xi}}{2+\mu{\Delta}^{2}}, $$

where

$$\xi=\mu^{4}{\Delta}^{8} +4\mu^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}+16+4\mu^{3}\widetilde{\gamma}{\Delta}^{6}-16\mu \widetilde{\gamma}{\Delta}^{2}+8\mu^{3}{\Delta}^{6}+24\mu^{2}{\Delta}^{4}+32\mu{\Delta}^{2}. $$

Combining Eqs. 14–16 and recalling that β > 0,we find that system (i′) − (iii′) is equivalent to

$$ \max\{\beta_{\ell},0\}<\beta<\min\{\beta_{r_{1}},\beta_{r_{2}}\}. $$

(17)

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

$$ \mu<\mu_{r}=\frac{2}{{\Delta}^{2}}\cdot\frac{\beta\widetilde{\gamma}{\Delta}^{2}+4-2\widetilde{\gamma}}{\beta \widetilde{\gamma}{\Delta}^{2}+4}, $$

(18)

while (iii′) provides

$$ \mu> \mu_{\ell}=\frac{2}{{\Delta}^{2}}\cdot\frac{\beta\widetilde{\gamma}{\Delta}^{2}-12}{\beta \widetilde{\gamma}{\Delta}^{2}+4}. $$

(19)

Condition (ii′) can be rewritten as

$$ q_{1}\mu^{2}+q_{2}\mu+q_{3}>0, $$

(20)

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

$$\mu_{1}=\frac{\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}- \sqrt{\chi}}{\beta\widetilde{\gamma}{\Delta}^{4}(4-\beta\widetilde{\gamma}{\Delta}^{2})}, $$

and

$$\mu_{2}=\frac{\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}+\sqrt{\chi}} {\beta\widetilde{\gamma}{\Delta}^{4}(4-\beta\widetilde{\gamma}{\Delta}^{2})}, $$

where

$$\begin{array}{@{}rcl@{}} \chi&=&256+16\beta^{2}{\widetilde{\gamma}}^{4}{\Delta}^{4}+\beta^{4}{\widetilde{\gamma}}^{4}{\Delta}^{8}+128 \beta{\widetilde{\gamma}}^{2}{\Delta}^{2}+96\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}\\ &&-8\beta^{3}{\widetilde{\gamma}}^{4}{\Delta}^{6}-16\beta^{3}{\widetilde{\gamma}}^{3}{\Delta}^{6}-256 \beta{\widetilde{\gamma}}{\Delta}^{2}. \end{array} $$

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 ₂if 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

$$\beta^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}<\left( \frac{4}{{\Delta}^{2} \widetilde{\gamma}}\right)^{2}{\widetilde{\gamma}}^{2}{\Delta}^{4}-16-4\beta{\widetilde{\gamma}}^{2}{\Delta}^{2}=-4 \beta{\widetilde{\gamma}}^{2}{\Delta}^{2}<0, $$

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.⁴

Finally, the stability conditions with respect to Δ when μ = 0 and β ≠ 0 read as

$$\sqrt{\frac{2\widetilde{\gamma}-4}{\widetilde{\gamma}\beta}}<{\Delta}<\frac{2}{\sqrt{\widetilde{\gamma}\beta}}\,, $$

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. □