A financial market model with confirmation bias

Structural Change and Economic Dynamics 51 (2019) 252–259

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A financial market model with confirmation bias夽 Alessia Cafferata a , Fabio Tramontana b,∗ a b

University of Genova, Italy Catholic University of Sacred Heart (Milano), Italy

a r t i c l e

i n f o

Article history: Received 12 March 2019 Received in revised form 12 August 2019 Accepted 12 August 2019 JEL classification: G12 D83 D84 D91 C61

a b s t r a c t We develop a financial market model with heterogeneous agents who can be affected by confirmation bias. In particular we consider optimistic and pessimistic agents who adjust their beliefs giving more attention and consideration to evidences supporting their prior beliefs. These kinds of traders coexist with fundamentalists and chartists. We show that this psychological bias makes beliefs more and more distant as time passes, and permits to better explain some important stylized facts of financial markets. © 2019 Published by Elsevier B.V.

Keywords: Financial markets Heterogeneous agents Piecewise-defined maps Monte Carlo simulations

1. Introduction “Once a human intellect has adopted an opinion (either as something it likes or as something generally accepted), it draws everything else in to confirm and support it” Lord Francis Bacon (1620) This quote from Lord Francis Bacon is an excellent definition of what nowadays psychologists call Confirmation (or Confirmatory) bias. Human beings display the tendency to select the information they receive and to give more importance to new evidence confirming their prior beliefs and less (or even nothing at all) to those against them. This is an error in the updating of beliefs in the light of new evidence, that should respect the so-called Bayesian updating to be considered rational. Initial beliefs should only be a starting point, becoming less and less relevant as the number of evidence accumulates (a process known as washing-out the priors). Instead, according to the definition of confirmation bias, our

夽 The authors want to thank Gian Italo Bischi, Frank Westerhoff and two anonymous referees for their useful comments and suggestions on a previous version of this paper. ∗ Corresponding author. E-mail address: [email protected] (F. Tramontana). https://doi.org/10.1016/j.strueco.2019.08.004 0954-349X/© 2019 Published by Elsevier B.V.

first impressions will influence how we update our beliefs about some uncertain thing, distorting our reasoning process. The first attempt to collect and review evidence of confirmation bias in several contexts can be probably found in Nickerson (1998). He identified confirmation bias in medicine, judicial reasoning and other fields, but he did not talk about finance and about its influence in the formation of asset prices. Quite surprisingly, only recently confirmation bias has been empirically found in financial markets, for instance by Park et al. (2013) in the field (in South Korea) or by Bisiere et al. (2014) in the lab. The role played by such a bias in the persistence of mispricings (i.e. prices different from their fundamental values) is intuitive. The more an asset price increases, the more bullish traders are reinforced in their beliefs, making their assumptions correct with their own behavior. The same happens for bearish traders when prices drop. Confirmation bias may thus play a role in the emergence and/or other features (duration, frequency, etc.) of speculative bubbles and other important stylized facts in financial markets. Besides experiments and empirical evidence of confirmation bias, there is also a branch of research dealing with the formalization of a beliefs updating consistent with this bias. The most known attempt is the one made by Rabin and Schrag (1999), who corrected the classical Bayesian updating mechanism, proving that under some circumstances almost all agents may come to believe in a wrong hypothesis. Among the huge amount of follow-up papers,

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only recently Pouget et al. (2017) adapted and applied to financial markets the Rabin and Schrag mechanism, replicating several stylized facts such as excess volatility, excess volume and momentum. In addition to them, also Charness and Dave (2017), adapted the Rabin and Schrag mechanism to test background strategies to correct or mitigate the negative effects of the bias, and formerly Bowden (2015) inserted this mechanism to an agent-based framework, finding that the bias may have ambiguous consequences on the volatility and kurtosis. To our knowledge the only paper where the confirmation bias is modeled in an alternative way with respect to the one proposed by Rabin and Schrag is Aldashev et al. (2011), who introduce confirmation bias in the way social interaction takes place, and combine it with adaptive expectations, showing that the two biases may both increase or decrease informational efficiency. A first goal of the present work is to build a simple financial market model where traders (or a portion of them) are endowed with confirmation bias in order to better understand how this bias influences the characteristics of financial bubbles and other stylized facts. In this sense the work is similar to Bowden (2015) but differently from him, we introduce an alternative, simpler, mechanism to insert confirmation bias, that leads to a piecewise definition of the dynamical system regulating the time evolution of asset price and beliefs. The model is at least partially analytically tractable in its fully deterministic version while by introducing some noise we replicate important features of real financial markets. Our model is inserted into the so-called heterogeneous agents models (HAM) literature, that has proved to be a quite good framework to introduce and study the effects of the behavioral biases of traders (see Chiarella et al., 2009; Hommes and Wagener, 2009; Lux, 2009; Westerhoff, 2009 for completed surveys). Usually the interactions between fundamental and technical trading rules are able to reproduce the features of financial markets that are hardly reconcilable with the assumption of rationality of traders and efficiency of the markets. Both laboratory experiments (Hommes, 2011) and other empirical evidences (like those surveyed by Menkhoff and Taylor (2007)) support the hypothesis that traders rely on simple rules. Starting from the pioneering contribution of Day and Huang (1990), several papers have proved that complicated price dynamics can be obtained by simple deterministic models (see for instance Kirman, 1991; De Grauwe et al., 1993; Lux, 1995; Brock and Hommes, 1998; Chiarella et al., 2002; Westerhoff, 2004). In this work we move from the subdivision of technical traders in pessimistic (or bearish) and optimistic (or bullish), similarly to Lux (1995, 1998) and Lux and Marchesi (1999, 2000), and we introduce confirmation bias in the way they interpret and use current information. A positive trend will be considered by bullish traders and ignored by pessimistic ones (if it is not too positive) and, at the opposite, a negative price trend will be considered by pessimistic traders and ignored by optimistic traders (if it is not too negative). The dynamical systems arising in HAM are usually smooth, but nonlinear. The nonlinearity may originate from the trading rule of the traders or from the switching mechanism from one strategy to another one, in evolutive models. Only a few papers in the HAM literature deal with discontinuous (or at least not differentiable) dynamical systems. Among them we have Huang and Day (1993), Huang et al. (2010), Tramontana et al. (2010) and Tramontana et al. (2011). Our model is discontinuous because of the intrinsic dichotomy of the confirmation bias itself. Traders affected by this bias display a sort of cognitive dissonance, oscillating between a certain behavior when they face evidence supporting their current hypothesis and a totally different one when dealing with evidence against what they actually believe. In order to mimic more qualitative features of financial markets (such as excess volatility, fat tails, volatility clustering, etc.) some researchers study a stochastic version of a deterministic HAM, obtaining quite interesting results (see Lux and Marchesi, 1999;

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Westerhoff and Dieci, 2006; Gaunersdorfer and Hommes, 2007 or He and Li, 2007). In the present work we also present a stochastic version of the model, replicating some features of real financial markets. A second goal of the model is thus to try to identify the degree of confirmation bias that better permits to replicate such stylized facts. The paper is organized as follows: in Section 2 we introduce the financial market model formalizing the behavioral assumptions. In Section 3 we study the deterministic skeleton of the model, in order to understand the role of the behavioral parameters. In Section 4 we perform an analysis of a stochastic version of the model by using the Monte Carlo method, replicating some stylized facts of financial markets. Section 5 includes final considerations. 2. The model Our financial market model consists of a standard building block which formalizes the behavior of a market maker and of four different groups of traders. One novelty we have introduced in our model is that, in addition to a group of fundamentalists traders, i.e. investors who believe that the price of the asset will follow its fundamental, and to a group of chartists traders, we consider two other kinds of traders. These last two groups differ according to the way they form the expected price or, more precisely, according to their initial guess: they can be optimistic or pessimistic. While fundamentalists believe that the price of the asset will be close to its fundamental and chartists that the current price trend will persist, the other two groups of traders consider the difference between the last realized return (or log-return) and the expected return (or log-return) for the next period and they adjust their expectations through a mechanism that is affected by confirmation bias. 2.1. Our model’s building block In our model we adopt a log-linear price adjustment mechanism with market maker, in a market with only a single risky asset. In our framework the first player we have is the market maker who quotes the log of the price (p) according to the following equation:

pt+1 = pt + ˛Dt , where Dt is the total excess demand at time t made up by the excess demands of the different kinds of traders and ˛ is a positive scaling coefficient which calibrates the log-price adjustment speed. The primary function of the market maker, in fact, is to mediate transactions out of equilibrium, when demand exceeds the supply (as in this case), or vice versa. He acts by providing or absorbing liquidity, according to whether there is a positive or negative excess of demand. We consider a classic group of traders called fundamentalists, who behave like that: f

Dt = f (F − pt )3 , where f ≥ 0 is their speed of reaction. They believe the price of the asset must follow its fundamental value (whose log is F), so they buy an asset when it is undervalued, (the price is below its fundamental value), driven by a higher probability of a capital gain, and they sell an overvalued one (the price is higher than the fundamental), because in this case it is more likely to incur in a capital loss. The cubic function implies that fundamentalists become more and more active when the mispricing becomes large. This nonlinear formulation of the excess demand of fundamentalists permits to capture the increasing profit opportunities that become available when the price is more and more distant from the fundamental value, as suggest by Day and Huang (1990).

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Chartists behave in an opposite way with respect to fundamentalists. The buy the asset when its price is above the fundamental value while they sell it when it is below, betting on the persistence of the current trend:

where c ≥ 0 measures the reactivity of chartists. We also consider two further groups of traders, who have different expectations about the future price of the asset: they can therefore be optimistic or pessimistic. Denoting by rt the log-return characterizing the asset in the period between t and t − 1, we have that: rt = pt − pt−1 . These investors trade on the basis of their expectations about the log-return of the asset in the next period (Ert+1 ) according to this behavioral rule: opt/pes

opt/pes

= dopt/pes Er t+1



Also here 0 ≤ ˇ2 ≤ 1 measures the anchoring to the previous belief. We will assume ˇ2 not lower than ˇ1 . They realize that their expectations were too optimistic and reluctantly they reduce they expectations but not with the same strength used for making them even more optimistic. In some senso also here confirmation bias could be considered at work. By avoiding negative values of  we prevent the scenario where optimists radically change their initial expectations and become pessimists. Pessimists, of course, have an original expectation that involves a decreasing price: pes

Er t+1 ≤ 0.

.

Again, d ≥ 0 is a not negative reaction parameter. Optimists and pessimists adjust their expectation affected by confirmation bias. Thus they tend to give different importance to facts that support their initial guess rather than news that goes in the opposite direction. The group of optimist traders thinks the price is going to increase: opt

Er t+1 ≥ 0. More precisely, they think the price is going to grow at an endogenous growth rate  t+1 :

Again, they think the price is going to drop at an endogenous growth rate  t+1 < 0: pes

Er t+1 = t+1 . Similar to optimists, they adjust the expected growth rate by considering the last realized price variation, according to the following rules. Pessimists Rule 1: If the last log-return has effectively been negative and even lower than their expectations (rt <  t ), the pessimists adjust their beliefs in this way:



opt

Er t+1 = t+1 .



t+1 = max ˇ1 t + (1 − ˇ1 )rt , ¯ ,

They then adjust the expected growth rate by considering the last log-return (rt ) and according to three different rules. Optimists Rule 1: If the last log-return has been positive and larger than the expect one (rt >  t ), the optimists adjust their beliefs as follows:





t+1 = max ˇ2 t + (1 − ˇ2 )rt , 0 .

Dtc = c(pt − F),

Dt

Optimists Rule 3: Finally, if the last log-return has been negative and greater than the threshold w (rt < −w), the investors of this group reduce their growth expectations in this way:



t+1 = min ˇ1 t + (1 − ˇ1 )rt , ¯ , where 0 ≤ ˇ1 ≤ 1 measures the anchoring to the previous belief. We assume that optimistic expectations cannot excess a certain value ¯ > 0. Optimists Rule 2: When the last log-return has been positive but lower than the expected or negative, but not excessively negative (−w ≤ rt ≤ t , with w > 0), optimists basically ignore the signal and stick to the current belief: t+1 = t .

where also in this case 0 ≤ ˇ1 ≤ 1 measures the anchoring to the previous belief and, again, negative expectations cannot cross the (negative) level given by . ¯ Pessimists Rule 2: If the last log-return has been negative but less than expected or positive, but not excessively positive (t ≤ rt ≤ w, with w > 0), pessimists ignore the signal and do not adjust their beliefs at all: t+1 = t . Pessimists Rule 3: If the last log-return has been positive and greater than the threshold w, pessimists cannot totally ignore the signal and then they adjust their negative expectations according to this rule:





t+1 = min ˇ2 t + (1 − ˇ2 )rt , 0 .

The positive parameter w measures the strength of the confirmation bias, so how much investors are anchored to their current beliefs. This parameter is very important for our model because it quantifies the role of confirmation bias on the formation of fluctuations in the price pattern. The boundary case w = 0 permits to model a classic fundamentalists/chartists model without the confirmation bias, and we will use it as a benchmark case in the study.1 As we are going to demonstrate in Section 2, the more the strength of confirmation bias increases, the more the different expectations are far.

Again, 0 ≤ ˇ1 ≤ ˇ2 ≤ 1 measures the anchoring to the previous belief. As in the previous case, it is stronger when the market signal is opposite (denoting some kind of reluctance). We avoid a positive value of  because pessimists cannot become optimists.2 If we put everything together and consider two potentially different reactivities (d1 for optimists and d2 for pessimists), we obtain the following dynamic system regulating the asset price, the return and the expectations’ dynamics:

1 Actually we could consider as benchmark an even simpler model where optimists and pessimists are not considered, that is by assuming d1 = d2 = 0. Nevertheless we prefer to not use it as reference because prefer to stay in a class of model with the same groups of traders.

2 In a more general version of the model we could consider the anchoring and the confirmation bias parameter w different for pessimists and optimists but this kind of asymmetry is out of the scope of this paper.

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⎧   [c]lpt+1 = pt + ˛ f (F − pt )3 + c(pt − F) + d1 t+1 + d2 t+1 , ⎪ ⎪ ⎪   ⎪ ⎧  ⎪ ⎪  min ˇ1 t + (1 − ˇ1 )rt , ¯ if rt > t ⎪ ⎪ ⎪ ⎨  ⎪ ⎪  ⎪ t if − w ≤ rt ≤ t  , t+1 = ⎪ ⎪ ⎪ ⎪    ⎨ ⎩  max ˇ  + (1 − ˇ )r , 0 r < −w t t t 2 2 T:   ⎧  ⎪  max ˇ1 t + (1 − ˇ1 )rt , ¯ if rt < t ⎪ ⎪ ⎪ ⎨  ⎪ ⎪  ⎪ ⎪  if  ≤ r ≤ w =  t t t , t+1 ⎪ ⎪ ⎪  ⎪   ⎩ ⎪  ⎪ rt > w min ˇ2 t + (1 − ˇ2 )rt , 0 ⎪ ⎪   ⎩ 3 rt+1 = ˛ f (F − pt ) + c(pt − F) + d1 t+1 + d2 t+1 ,

(1) that is a system of difference equations of the first order, that allows us to manage the system more easily.

whose eigenvalues are:



Map (1) regulates the dynamics of the asset price and of the beliefs of optimistic and pessimistic traders. The system admits at least three equilibria (EF , E+ and E− ), all of them characterized by a log-return equal to 0 (r* = 0) and optimists and pessimists converge in their expectations about no price movements ( * = * = 0). The equilibria differ in the equilibrium value of the log-price3 : EF −→ p∗F = F, E+ −→

p∗+



=F+



E− −→ p∗− = F −

c , f

(2)

c . f

The three different levels of the equilibrium log-price reveal that at the equilibrium the price can be correct (EF ), overvalued (E+ ) or undervalued (E− ). If the price is equal to its fundamental value then no trader is active in the market (all the excess demands are null), while in the other two equilibria, the positive excess demands of some groups of traders are perfectly compensated by the negative excess demands of the other groups. So, they are equilibria with transactions while EF is an equilibrium without any transaction. The local stability properties of the equilibria must be studied by taking into account the piecewise definition of the map. In a neighborhood of the equilibria, dynamics are regulated by the following dynamical system:

  ⎧ pt+1 = pt + ˛ f (F − pt )3 + c(pt − F) + d1 t+1 + d2 t+1 , ⎪ ⎪ ⎪ ⎪ ⎨ t+1 = t , ⎪ t+1 = t , ⎪ ⎪ ⎪  ⎩

rt+1 = ˛ f (F − pt )3 + c(pt − F) + d1 t+1 + d2 t+1

(3)



associated with the following Jacobian matrix:

⎡ ⎢ ⎢ ⎣

J:⎢



1 + ˛ −3f (F − p)2 + c 0



0

˛ −3f (F − p)2 + c

3





˛d1

˛d2

1

0

0

1

˛d1

˛d2

0

⎤

⎥ ⎥, 0⎦



.

(5)

Considering the fundamental equilibrium (EF ), we have that the fourth eigenvalue is: F4 = 1 + ˛c, which is strictly positive given the positivity of the parameters ˛ and c. So, the fundamental equilibrium is always unstable. For the other two equilibria the analysis is more complicated: in fact for both the fourth eigenvalue is: ± = 1 − 2˛c, 4 and it is higher than -1 provided that: c<

3. Study of the deterministic model



(1 , 2 , 3 , 4 ) = 0, 1, 1, 1 + ˛ −3f (F − p)2 − c

255

1 . ˛

(6)

So, if chartists reacts too strongly, also the other two equilibria are unstable. Algebraically, nothing can be said whenc < ˛1 . In  < 1) fact, while the fourth eigenvalue is in the stability region (± 4 we still have two eigenvalues equal to 1, so we cannot easily talk about a locally stable equilibrium. In such cases it is quite complicated to know which kind of dynamics are going to occur, and only numerical simulations may help to understand it. Fig. 1 represents bifurcation diagrams obtained by keeping all the parameters fixed with the exception of c. We used ˛ = 1, d1 = d2 = 1, ˇ1 = 0.95. ˇ2 = 0.98, f = 200, ¯ = 0.05, ¯ = −0.05, F = 1 and w = 0, that is with no confirmation bias. We assume the same reactivities for optimists and pessimists in order to avoid that one group becomes more important than the other. The values of all the other parameters, if such that ˇ2 ≥ ˇ1 , do not affect qualitatively the shape of the bifurcation diagrams. We can see that while the value of c is lower than 1, condition (6) holds and an equilibrium (in this case E− ) is reached, even if two eigenvalues are equal to 1. After that, when c is higher than 1 and condition (6) is violated, the fourth eigenvalue becomes lower than −1, and a cascade of period doubling bifurcations occurs, leading to chaotic dynamics when c is around 1.3. In this last case endogenous fluctuations characterize the motion of the dynamic variables. This numerical result of our benchmark model without confirmation bias is not surprising because it is typical of this branch of the literature on heterogeneous agents models. The introduction of confirmation bias (that is strictly positive values of w) has not effect on the stability of the equilibria, in fact parameter w does not appear neither in the eigenvalues (5) nor in the Jacobian matrix (4). It does not imply that confirmation bias has no effects on the dynamics and in the points of the chaotic attractor more or less visited by the trajectories. In order to better investigate the role of confirmation bias in the next section we build a stochastic version of our model and we will see that same features of real financial time series can be better replicated taking into consideration this cognitive bias. 4. The stochastic model

0⎥

(4)

0

d

There exist also other equilibria, of the form: r* = 0, p* = F and  ∗ = − d2  ∗ but 1

we do not take them into consideration because dynamically structurally unstable and meaningless from an economic point of view. In fact, even if at the equilibrium the price stays fixed, optimists/pessimists do not never change their increasing/decreasing expectations, and this is not realistic.

In previous sections, we have discussed how endogenous fluctuations of prices and returns emerge as long as the reactivity of chartists becomes higher than the threshold identified in (6). Nevertheless, in order to deepen the role played by confirmation bias we should perform some complicated analysis on the features of the chaotic attractor when w is positive. It is also possible to enlighten the role of confirmation bias by introducing some noise in our model. In particular, we introduce two stochastic elements to our setting:

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Fig. 1. Bifurcation diagrams. Parameters: d1 = d2 = 1, ˇ1 = 0.95. ˇ2 = 0.98, f = 200, ¯ = 0.05, ¯ = −0.05, F = 1 and w = 0. Reactivity of chartists c varies between 0 and 1.7. The asymptotic values of log-returns, log-prices, optimists expectations and pessimists expectations are in panels (a), (b), (c) and (d), respectively.

1. We assume that the fundamental value is characterized by a geometric Brownian motion so, dealing with the log-fundamental value we have: Ft+1 = Ft + F,t ,

with F,t ∼N(F ,  2F );

2. the second noise we introduce involve parameter c, characterized by a random walk dynamics: ct+1 = ct + c,t ,

with c,t ∼N(c ,  2c ).

All the realizations of the noises are independent and identically distributed and also independent each other. The motion of the fundamental value is quite common and it makes the model more realistic. The noise added to the chartists parameter c can be interpreted as a rumor on the reactivity (or relative numerosity) of chartists. Obviously a similar noise could be added to the other reactivity parameters but we want to keep the model as parsimonious as it possible and, as we will see, these two stochastic elements are sufficient to obtain interesting results. From the dynamic motion of c we expect to see in the time series a typical element of financial time series, that is volatility clustering. In fact, as suggested by the bifurcation diagrams in Fig. 1, when c is lower than one the deterministic skeleton of the model is stable, and the only variability is due to the stochastic elements. At the opposite, for higher values of c, and endogenous dynamics is added to the stochastic one. When c alternates between values higher and lower than one, we expect to see clusters of volatility.

Summarizing, the stochastic model is the following:

  ⎧ pt+1 = pt + ˛ f (Ft − pt )3 + ct (pt − Ft ) + d1 t+1 + d2 t+1 , ⎪ ⎪ ⎪   ⎧  ⎪ ⎪  min ˇ1 t + (1 − ˇ1 )rt , ¯ if rt > t ⎪ ⎪ ⎪ ⎨  ⎪ ⎪  ⎪ ⎪ t if − w ≤ rt ≤ t  , t+1 = ⎪ ⎪ ⎪  ⎪   ⎩ ⎪  ⎪ ⎪ max ˇ2 t + (1 − ˇ2 )rt , 0 rt < −w ⎪   ⎪ ⎧  ⎨  max ˇ1 t + (1 − ˇ1 )rt , ¯ if rt < t ⎪ ⎨  T˜ : (7)  ⎪ if t ≤ rt ≤ w  , ⎪ t+1 = t ⎪ ⎪ ⎪    ⎪ ⎩ ⎪  ⎪ min ˇ2 t + (1 − ˇ2 )rt , 0 rt > w ⎪ ⎪ ⎪ r = ˛ f (F − p )3 + c (p − F ) + d  + d   , ⎪ t t t t t ⎪ t+1 1 t+1 2 t+1 ⎪ ⎪ ⎪ ⎪ ⎪ Ft+1 = Ft + F,t , ⎪ ⎪ ⎩ ct+1 = ct + c,t .

4.1. Some features of financial time series With our deterministic model it is already possible, under suitable parametric conditions, to replicate some qualitative features of financial markets such as the fluctuation of prices and returns and some kind of excess volatility, as we have seen in the bifurcation diagrams of Fig. 1. These features can be obtained anytime the deterministic model exhibits chaotic dynamics and it is not a novelty in the HAM literature. Moreover, in our framework the presence of confirmation bias is not strictly necessary to replicate these features. In this section we want to identify some other, more quantitative, features of financial time series, that can be better explained if we assume that some traders are affected by confirmation bias. We take as reference the behavior of two important indexes: the FTSEMIB index (Fig. 2) and the Standard and Poor index (Fig. 3). The

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Fig. 2. Time series and probability plot of the FTSEMIB index. The underlying time series runs from 2004 to 2018 and contains 3890 daily observations.

Fig. 3. Time series and probability plot of the S&P index. The underlying time series runs from 2004 to 2018 and contains 3890 daily observations.

Table 1 FTSEMIB index. We report the variance of the returns (V) the minimum and maximum return (rmin and rmax ), kurtosis (K) and skewness (S) of the returns’ distribution.

Table 2 S&P index. We report the variance of the returns (V) the minimum and maximum return (rmin and rmax ), kurtosis (K) and skewness (S) of the returns’ distribution.

V

rmin

rmax

K

S

V

rmin

rmax

K

S

0.0002

−0.133

+0.11

9.1929

−0.25

0.00012

−0.095

+0.11

15.75

−0.383

length considered is 3890 daily observations, from 2004 to 2018. In the panel on the left on the top of Fig. 2, we can see that the returns, that are affected by high volatility (especially around day 1250, 2000 and 3200), alternating with periods of low volatility (like for the first 700 days). The figure on the right contains the normal probability plot of returns: it is evident that they are distant from what we expect from a normal distribution. In order to measure these features we have calculated the variance of the returns (V) the minimum and maximum return (rmin and rmax ), kurtosis (K) and skewness (S) of the returns’ distribution. These measures are reported in Table 1. The high value of the kurtosis and the skewness different from 0 confirm that returns are not normally distributed, and the negative value of the skewness says that the distribution is not symmetric but its left tail extends on the left. Similar results can be found by looking at the S&P index (Fig. 3). We have still turbulent periods (like the one around day 1250) alternating with calmer ones. The return distribution is again far from normal, as witnessed by the measures reported in Table 2.

Table 3 Values of the parameters for the Monte Carlo method. Parameter

˛

f

d1

d2

¯

¯

ˇ1

ˇ2

Value

1

200

1

1

0.05

−0.05

0.95

0.98

It is well known that the features we have identified for these two indexes can be found also in almost all the others. Our aim now is to find if with a proper calibration of the parameters we can get close to these measures with our stochastic model, with particular attention at the role of conformation bias. In order to do that we have performed a Monte Carlo method. 4.2. Simulations with the Monte Carlo method Let us describe how we have implemented the Monte Carlo method. First of all, we have identified, with a trial and error procedure, the values of all the parameters with the exception of w. We want to stress here the most of them (such as F and f) are just

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Table 4 Average measures and statistics of 1000 simulations (length 3980 iterations each). We report the variance of the returns (V) the minimum and maximum return (rmin and rmax ), kurtosis (K) and skewness (S) of the returns’ distribution. Scenario

V

rmin

rmax

K

S

No c.b. (w = 0) Weak c.b. (w = 0.01) Strong c.b. (w = 0.04)

0.00061 0.00056 0.0002

−0.21 −0.2 −0.147

+0.2 +0.19 +0.13

18.56 19.93 20.6

+0.016 −0.058 −0.11

scale parameters for our model. Table 3 contains all the parameters’ values we kept fixed during our simulations. We just note that we have assigned the same reactivity value to optimists and pessimists in order to avoid that one group of trader becomes more important than the other. Then we have considered three values of w: 0 (no confirmation bias), 0.01 (weak confirmation bias) and 0.04 (strong confirmation bias). In other words, in the weak confirmation bias scenario optimists (resp. pessimists) traders require a drop (resp. rise) of more than 1% of the returns to adjust downwards (resp. upwards) their return expectations. In the strong confirmation bias scenario the threshold is 4%. We avoid higher values of w because we consider them excessive and hardly realistic. For each value of w we performed 1000 runs of simulations, each one made up by 3890 iterations, similarly to the daily observations of the indexes we showed before. For the stochastic elements we used these values of initial value, average and variance: F −→ F0 = 1; c −→ c0 = 1.5;

F = 0; c = 0;

F2 = 0.005, c2 = 0.05.

When simulations produce negative values of c we consider it 0. When the simulation produced an exploding, diverging trajectory, we did not take it into consideration.4 In Table 4 we summarize the average values that we have obtained in the three scenarios (no, weak and strong confirmation bias). We can easily see that confirmation bias is quite relevant for almost all the measures. In particular, it decreases the variance and the range of variation of the returns. Its role is also important for the negative skewness of the distribution. In fact, without confirmation bias the distribution appears close to be symmetric, with a slightly positive skewness. When we introduce confirmation bias skewness becomes more and more negative as the bias is amplified. These results would be confirmed with even higher values of w, but as previously said we consider them unrealistic. The value of the kurtosis is always extremely high in all the scenarios.5 Even if these results should be confirmed through more sophisticated analysis that we plan to perform in future works, we can already say that the scenario with strong confirmation bias appears the one better reproducing the measures of real financial time series we have taken into consideration. So, the bias seems to be present and relevant. In order to give a possible interpretation of these results we show in Fig. 4 a simulation run that can be obtained with our parameters’ configuration. First of all, we can note that the value of c oscillates alternating values higher and lower than one (panel (a)). The two series of logreturns and agents expectations are obtained by using the same

4

It happens in particular when c becomes larger than 1.7. Despite what Table 4 may suggest, there is no monotonic (increasing) relation between the kurtosis of the returns’ distribution and the value of the skewness. If we consider also intermediate values of w we would see that kurtosis oscillates in a range bounded by 17 and 21. 5

Fig. 4. In (a) it is represented a realization of the motion of c. The corresponding log-returns when w = 0 and when w = 0.04 are shown in panels (b) and (c). Finally, panels (d) and (e) show how the expectations of optimists and pessimists evolve in the two scenarios.

realization of c but different values of w. In particular, we have considered a scenario with no confirmation bias (w = 0 in panels (b) and (d)) and with strong confirmation bias (w = 0.04 in panels (c) and (e)). As expected the erratic motion of chartists’ reactivity is able to replicate the feature of volatility clustering of the returns (panels (b) and (c)). In fact the most turbulent periods are those corresponding with higher values of c. Less turbulence is present in the scenario of strong confirmation bias even if the peaks of the returns are always in correspondence of the highest values of c. Our interpretation is that when returns are extremely high and there is no confirmation bias, optimists become more and more active, while pessimists are less relevant because their expectations get close to zero. The opposite when returns are extremely low, by exchanging the role of optimists and pessimists. The combination of these behaviors exacerbates the turbulence and permits to increase the volatility of the returns. At the opposite, when there is a strong

A. Cafferata, F. Tramontana / Structural Change and Economic Dynamics 51 (2019) 252–259

confirmation bias, despite the high value of the returns, pessimists keep persist in their idea that returns are going to decrease and their action at least partially compensate the behavior of optimists, reducing the volatility of price and returns. This is confirmed by the time evolution of the expectations in the two scenarios. In panel (d), without confirmation bias, we can see that the expectations periodically seems to convergence towards zero (when c is low, that is for the deterministic skeleton an equilibrium is stable), while they are more separated in periods of turbulence. When confirmation bias is strongly present (panel (e)) the average distance between the expectation is larger and when optimists believe in a large positive return, pessimists are still convinced that returns must drop. In our opinion this is coherent with our interpretation of the results obtained with the Monte Carlo method. 5. Conclusions In this paper we have studied a simple financial market model in which four different groups of traders coexist. These groups differ in the way they form their expectations about the price of the asset. The map originated from the different expectations is piecewisedefined and is characterized by a multiplicity of equilibria where the price stays constant and returns are null. We have deepen the role played by optimists and pessimists, whose expectations differ according to the way they interpret the new realization of prices and returns to adjust their expectation for the future. In particular, we have considered traders affect by confirmation bias, that leads to (at least partially) ignore evidence against their current beliefs. This psychological phenomenon causes a failure in the Bayesian updating. Our expectations have been confirmed by the simulations of our model, since our results were that confirmation bias makes the different beliefs more distant from each other. Despite the fact that our model is simple, we found out that it is able to explain the dynamics present in financial markets, such as the amount of volatility, volatility clustering, skewness and kurtosis in the distribution of returns. We have also checked if the dynamics simulated by our model is representative of the behavior of data – in particular of the FTSEMIB index and of the Standard and Poor index – and we noticed that it provides a good representation of reality especially in the scenario where confirmation bias is strongly present. To bring our model to reality, we have introduced a stochastic component which describes some random events that take place in financial markets. Our model may be extended and improved in various ways: we can introduce switching strategies between optimists and pessimists, we can add other stochastic components also to the two reaction parameters of fundamentalists and chartists or we can assume to use a more complex demand function. Another possible extension is to think about possible policy implications to the dynamics we get from the model. About this, it is fundamental to design more precisely how financial markets work. Nevertheless what we have found confirms that psychological bias affecting investors may have important effects on financial time series and contribute in explaining some of their most important and relevant features. Acknowledgement Work developed in the framework of the research project on “Models of behavioral economics for sustainable development” financed by DESP-University of Urbino.

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