2.1 Market Mood and Adaptive Behavior

Empirical evidence in foreign exchange markets (Allen and Taylor, 1990, Taylor and Allen, 1992, Menkhoff, 1998, and Cheung et al., 2004) and managing fund industrial (Menkhoff, 2010) suggests that agents have different information and/or beliefs about market processes. They use not only fundamental but also technical analyses, which are consistent with short-run momentum and long-run reversal behavior in financial markets. In addition, although some agents do not change their particular trading strategies, there are agents who may switch to more profitable strategies over time. Recent laboratory experiments in Hommes et al. (2005), Anufriev and Hommes (2012), and Hommes and in't Veld (2015) also show that agents using simple “rule of thumb” trading strategies are able to coordinate on a common prediction rule. Therefore heterogeneity in expectations and adaptive behavior are crucial to describe individual forecasting and aggregate price behavior.

Motivated by the empirical and experiment evidence, Dieci et al. (2006) introduce a simple financial market of fundamentalists and trend followers. Some agents switch between different strategies over time according to their performance, characterizing the adaptively rational behavior of agents. Others are confident and stay with their strategies over time, representing market mood. It turns out that this simple model is rich enough to illustrating the complicated price dynamics and to exploring different mechanisms in generating volatility clustering and long memory in volatility. In the following, we first outline the model, discuss calibration and empirical estimation of the model, and then provide an analysis on the two underlying mechanisms (see Dieci et al., 2006 and He and Li, 2008, 2017 for the detail).

Consider a financial market with one risky asset and one risk free asset. Let r be the constant risk free rate, pt$p_t$ the price, and $d_t$ the dividend of the risky asset at time t. Assume that there are four types of investors, fundamental traders (or fundamentalists), trend followers (or chartists) and noise traders, and one market maker. Let $n_3$ be the population fraction of the noise traders. Among $1-n_3$, the fractions of the fundamentalists and trend followers have fixed, $n_1$ and $n_2$, and switching, $n_{1,t}$ and $n_{2,t}=1-n_{1,t}$, components respectively. Denote $n_0=n_1+n_2,m_0=(n_1-n_2)/n_0$ and $m_t=n_{1,t}-n_{2,t}$. Then the market fractions $Q_{h,t}$ ($h=1,2,3$) of the fundamentalists, trend followers, and noise traders at time t can be rewritten as, respectively,

(1) $\{\begin{array}{c}{Q}_{1,t}=\frac{1}{2}(1-n_3)[n_0(1+m_0)+(1-n_0)(1+m_t)],\hfill \\ Q_{2,t}=\frac{1}{2}(1-n_3)[n_0(1-m_0)+(1-n_0)(1-m_t)],\hfill \\ Q_{3,t}=n_3.\hfill \end{array}$

Let $R_{t+1}=p_{t+1}+d_{t+1}-R{p}_t$ be the excess return and $R=1+r$. We model the order flow³ $z_{h,t}$ of type-h investors from t to $t+1$ by $z_{h,t}=E_{h,t}(R_{t+1})/(a_h{V}_{h,t}(R_{t+1}))$, where $E_{h,t}$ and $V_{h,t}$ are the conditional expectation and variance at time t and $a_h$ is the risk aversion coefficient of type h traders. The order flow of the noise traders ${\xi }_t\sim N(0,{\sigma }_{\xi }^2)$ is an i.i.d. random variable. Then the population weighted average order flow is given by $Z_{e,t}=Q_{1,t}{z}_{1,t}+Q_{2,t}{z}_{2,t}+n_3{\xi }_t.$ To determine the market price, we follow Chiarella and He (2003) and assume that the market price is determined by the market maker as follows,

(2) $p_{t+1}=p_t+\lambda Z_{e,t}=p_t+\mu z_{e,t}+{\delta }_t,$

where $z_{e,t}=q_{1,t}{z}_{1,t}+q_{2,t}{z}_{2,t}$, $q_{h,t}=Q_{h,t}/(1-n_3)$ for $h=1,2$, λ denotes the speed of price adjustment of the market maker, $\mu =(1-n_3)\lambda$ and ${\delta }_t\sim N(0,{\sigma }_{\delta }^2)$ with ${\sigma }_{\delta }=\lambda n_3{\sigma }_{\xi }$.

We now describe briefly the heterogeneous beliefs of the fundamentalists and trend followers and the adaptive switching mechanism. The conditional mean and variance for the fundamental traders are assumed to follow

(3) $E_{1,t}\left(p_{t+1}\right)=p_t+(1-\alpha )[E_t(p_{t+1}^{⁎})-p_t],V_{1,t}\left(p_{t+1}\right)={\sigma }_1^2,$

where $p_t^{⁎}$ is the fundamental value of the risky asset following a random walk,

(4) $p_{t+1}^{⁎}=p_t^{⁎}\exp (-\frac{{\sigma }_{\epsilon }^2}{2}+{\sigma }_{\epsilon }{\epsilon }_{t+1}),{\epsilon }_t\sim \mathcal{N}(0,1),{\sigma }_{\epsilon }\ge 0,p_0^{⁎}=p^{⁎}>0,$

${\epsilon }_t$ is independent of the noise process ${\delta }_t$, ${\sigma }_1^2$ is constant, and hence $E_t(p_{t+1}^{⁎})=p_t^{⁎}$. Here $(1-\alpha )$ measures the speed of price adjustment towards the fundamental price with $0<\alpha <1$. A high α indicates less confidence on the convergence to the fundamental price, leading to a slower adjustment of the market price to the fundamental. For the trend followers, we assume

(5) $E_{2,t}\left(p_{t+1}\right)=p_t+\gamma (p_t-u_t),V_{2,t}\left(p_{t+1}\right)={\sigma }_1^2+b_2{v}_t,$

where $\gamma ⩾0$ measures the extrapolation of the trend, $u_t$ and $v_t$ are sample mean and variance, respectively. We assume that $u_t=\delta u_{t-1}+(1-\delta )p_t$ and $v_t=\delta v_{t-1}+\delta (1-\delta ){(p_t-u_{t-1})}^2$, representing limiting mean and variance of the geometric decay processes when the memory lag tends to infinity. Here $\delta \in (0,1)$ measures the geometric decay rate and $b_2\ge 0$ measures the sensitivity to the sample variance. For simplicity we assume that investors share a homogeneous belief about the dividend process $d_t$, which is i.i.d. and normally distributed with mean $\overline{d}$ and variance ${\sigma }_d^2$. Denote by $p^{⁎}=p_o^{⁎}=\overline{d}/r$ the long-run fundamental price.

Let ${\pi }_{h,t+1}$ be the realized profit between t and $t+1$ of type-h investors, ${\pi }_{h,t+1}=z_{h,t}(p_{t+1}+d_{t+1}-R{p}_t)$ for $h=1,2$. Following Brock and Hommes (1997, 1998), the market fraction of investors choosing strategy h at time $t+1$ is determined by

$n_{h,t+1}=\frac{\exp \left[\beta ({\pi }_{h,t+1}-C_h)\right]}{{\sum }_i\exp \left[\beta ({\pi }_{i,t+1}-C_i)\right]},h=1,2,$

where β measures the intensity of the choice and $C_h\ge 0$ the cost. Together with (1) the market fractions and asset price dynamics are determined by the following random dynamic system in discrete-time,

(6) $\{\begin{array}{cc}\hfill p_{t+1}& =p_t+\mu (q_{1,t}{z}_{1,t}+q_{2,t}{z}_{2,t})+{\delta }_t,{\delta }_t\sim \mathcal{N}(0,{\sigma }_{\delta }^2),\hfill \\ \hfill u_t& =\delta u_{t-1}+(1-\delta )p_t,\hfill \\ \hfill v_t& =\delta v_{t-1}+\delta (1-\delta ){(p_t-u_{t-1})}^2,\hfill \\ \hfill m_t& =\tanh [\frac{\beta }{2}(z_{1,t-1}-z_{2,t-1}-(C_1-C_2))(p_t+d_t-R{p}_{t-1})].\hfill \end{array}$

2.2 Volatility Clustering: Calibration and Mechanisms

By conducting econometric analysis via Monte Carlo simulations, He and Li (2015b, 2017) show that the autocorrelations of returns, absolute returns and squared returns of the model developed above share the same pattern as those of the DAX 30. They further characterize the power-law behavior of the DAX 30 and find that the estimates of the power-law decay indices, the (FI)GARCH parameters, and the tail index of the model closely match those of the DAX 30. In the following we first report the calibrated results of the model developed in the previous subsection and then provide some insights into investor behavior and two underlying mechanisms of the volatility clustering.

When there is no switching between the two strategies, the above model reduces to the no-switching model in He and Li (2007), showing that the no-switching model is able to replicate the power-law behavior in return volatility. Based on the daily price index data of the DAX 30 from 11 August, 1975 to 29 June, 2007, He and Li (2015b, 2017) calibrate three scenarios of the above model: the no-switching (N) model with $\beta =0$, pure-switching (S) model with $n_0=0$, and full (F) model of (6). The results are collected in Table 1 (with fixed $r=5\text{\%}p.a$. and $C_1=C_2=0$). By conducting econometric analysis via Monte Carlo simulations based on the calibrated models, He and Li (2015b, 2017) find that, for all three scenarios, the estimates of the power-law decay indices d, the (FI)GARCH parameters, and the tail index of the calibrated model closely match those of the DAX 30. By conducting a Wald test $H_o:d_{DAX}=d$ at 5% and 1% significant levels (with the critical values of 3.842 and 6.635, respectively), He and Li (2017) show that switching model fits the data better than the no-switching and pure-switching models.

Comparing the estimates of the three scenarios leads to different investor behavior. The estimated annual return volatility σ is close to the annual return volatility of the DAX 30. Higher $a_1$ than $a_2$ implies that the fundamentalists are more risk averse compared to the trend followers. For the no-switching scenario, a higher value of α indicates a slow price adjustment of the fundamentalists toward the fundamental value, while a higher value of γ indicates that the trend followers extrapolate the price trend actively. Without switching, $m_o=-0.2$ indicates that both the fundamentalists and trend followers are active in the market, which is however dominated by the trend followers (about 60%). On the full model, the market is dominated by investors (about 70%) who constantly switch between the fundamental and trend following strategies, although some investors (about 30%) never change their strategies over the time. This is consistent with the empirical findings discussed at the beginning of this section.

We now provide two mechanisms based on the underlying deterministic dynamics. The first one is on the local stability and periodic oscillation due to Hopf bifurcation, explored in He and Li (2007). Essentially, on the parameter space of the deterministic model, near the Hopf bifurcation boundary, the fundamental steady state can be locally stable but globally unstable. Due to the nature of Hopf bifurcation, such global instability leads to switching between the locally stable fundamental price and the periodic oscillations around the fundamental price. Then triggered by the fundamental and market noises, He and Li (2007) show that the interaction of the fundamental, risk-adjusted trend chasing from the trend followers, and the interplay of the noises and the underlying deterministic dynamics can be the source of power-law behavior in return volatility. Mathematically, the calibrated no-switching and switching models share the same underlying deterministic mechanism. Economically, however, they provide different behavioral mechanisms. With no-switching, it is the dominance of the trend followers (about 60%) that drives the power-law behavior. However, with both switching and no-switching investors, dominated by these traders (about 70%) who constantly switch between the two strategies. It is therefore the adaptive behavior of investors that generates the power-law behavior. This is also in line with Franke and Westerhoff (2012, 2016) who estimate various HAMs and show that herding behavior plays a key role in matching the stylized facts. More importantly, the noise traders play an important role in generating insignificant ACs on the returns, while the significantly decayed AC patterns of the absolute returns and squared returns are more influenced by the fundamental noise. As pointed out in Lux and Alfarano (2016), noise traders is probably a central ingredient of these models.

The second mechanism proposed initially in Gaunersdorfer et al. (2008) is characterized by the coexistence of two locally stable attractors with different size, while such coexistence is not required in the previous mechanism. Dieci et al. (2006) show that the above model can display such co-existence of locally stable fundamental steady state and periodic cycle. The interaction of the coexistence of the deterministic dynamics and the noise processes then triggers the switching among the two attractors and endogenously generates volatility clustering. More recently, by applying normal form analysis and center manifold theory, He et al. (2016b) provide the following theoretical result on the coexistence of the locally stable steady state and invariant circle of the underlying deterministic model (we refer to He et al., 2016b for the details).

Note that the market fractions of the fundamentalists and trend followers at the fundamental steady state are given by $q_1=(1+m_q)/2$ and $q_2=(1-m_q)/2$ with $m_q=n_0{m}_0+(1-n_0)\overline{m}$, respectively. When the cost of the fundamental strategy $C_1$ is higher than the cost of the trend following strategy $C_2$, an increase in the switching intensity β leads to a decrease in ${\gamma }^{⁎⁎}$, meaning that the fundamental price becomes less stable when traders switch their strategies more often. This is essentially the rational routes to randomness of Brock and Hommes (1997, 1998).

Fig. 1 illustrates two different types of Neimark–Sacker bifurcation. It is the sign of the first Lyapunov coefficient $a(0)$ that determines the bifurcation direction, either forward or backward, and the stability of the bifurcated invariant circles, leading to different bifurcation dynamics. When $a(0)<0$, the bifurcation is forward and stable, meaning that the bifurcated invariant circle occurring for $\gamma >{\gamma }^{⁎⁎}$ is locally stable. In this case, as γ increases and passes ${\gamma }^{⁎⁎}$, the fundamental steady state becomes unstable and the trajectory converges to an invariant circle bifurcating from the fundamental steady state. As γ increases further, the trajectory converges to invariant circles with different sizes. This is illustrated in Fig. 1A with ${\gamma }^{⁎⁎}\approx 0.93$ where the two bifurcating curves for $\gamma >{\gamma }^{⁎⁎}$ indicate the minimum and maximum value boundaries of the bifurcating invariant circles as γ increases.

However, when $a(0)>0$, the bifurcation is backward and unstable, meaning that the bifurcated invariant circle occurring at $\gamma ={\gamma }^{⁎⁎}$ is unstable, illustrated in Fig. 1B (with ${\gamma }^{⁎⁎}\approx 0.88$). There is a continuation of the unstable bifurcated circles as γ decreases initially until it reaches a critical value $\hat{\gamma }$, which is indicated by the two (red) dotted curves of the bifurcating circles for $\hat{\gamma }<\gamma <{\gamma }^{⁎⁎}$. Then as γ increases from the critical value $\hat{\gamma }$, the bifurcated circles become forward and stable. This is illustrated by the two (blue) solid curves, which are the boundaries of the bifurcating circles, for $\gamma >\hat{\gamma }$ in Fig. 1B. Therefore, the locally stable steady state coexists with the locally stable ‘forward extended’ circles for $\hat{\gamma }<\gamma <{\gamma }^{⁎⁎}$, in between there are backward extended unstable circles. For $\hat{\gamma }<\gamma <{\gamma }^{⁎⁎}$, even when the fundamental steady state is locally stable, prices need not converge to the fundamental value, while may settle down to a stable limit circle. We call $\hat{\gamma }<\gamma <{\gamma }^{⁎⁎}$ the ‘volatility clustering region’. In addition, a Chenciner (generalized Neimark–Sacker) bifurcation takes place when $a(0)=0$. Based on the above analysis, a necessary condition on the coexistence is that $a(0)>0$. The coexistence of the locally stable steady state and invariant circle illustrated in Fig. 2 shows that the price dynamics depends on the initial values.

When buffeted with noises, the stochastic model can endogenously generate volatility clustering and long range dependence in volatility, illustrated in Fig. 3. Economically, with strong trading activities of either the fundamental investors or the trend followers, market price fluctuates around either the fundamental value with low volatility or a cyclical price movement with high volatility, depending on market conditions. When the activities of the fundamentalists and trend followers are balanced (to be in the volatility clustering region), the interaction of the fundamental noise and noise traders and the underlying co-existence dynamics then triggers an irregular switching between the two volatility regimes, leading to volatility clustering. In particular, volatility clustering becomes more significant when neither the fundamental nor the trend following traders dominate the market and traders switch their strategies more often. The results verify the endogenous mechanism on volatility clustering proposed by Gaunersdorfer et al. (2008) and provide a behavioral explanation on the volatility clustering.

2.3 Information Uncertainty and Trading Heterogeneity

Traditional finance literature mainly explore the role of asymmetric information and information uncertainty. Most HAMs however mainly focus on endogenous market mechanism through the interaction among heterogeneous agents by assuming a complete information about the fundamental value of risky assets. An integration of HAMs and asymmetric and/or uncertain information would provide a micro-foundation on behavioral heterogeneity and a more broad framework to better explaining various puzzles and anomalies in financial markets. Instead of heuristical heterogeneity assumption of agents' behavior, He and Zheng (2016) model the trading heterogeneity by introducing information uncertainty about the fundamental value to a HAM. Agents are homogeneous ex ante. Conditional on their private information about the fundamental value, agents choose optimally among different trading strategies when optimizing their expected utilities. This approach provides a micro-foundation to trading and behavioral heterogeneity among agents. It also offers a different switching behavior of agents from the current HAMs. In the following, we brief this approach.

Consider a continuum $[0,1]$ of agents trading one risky asset and one risk-free asset in discrete-time. For simplicity, the risk-free rate is normalized to be zero. The fundamental value of the risky asset $\mu \sim \mathcal{N}(\overline{\mu },{\sigma }_{\mu }^2)$ is not known publicly. Denote ${\alpha }_{\mu }=1/{\sigma }_{\mu }^2$ the precision of the fundamental value μ. In each time period, agent i receives a private signal about the fundamental value μ, given by $x_{i,t}=\mu +{\epsilon }_{i,t}$, where ${\epsilon }_{i,t}\sim \mathcal{N}(0,{\sigma }_x^2)$ is i.i.d. normal across agents and over time. Let ${\alpha }_x=1/{\sigma }_x^2$ be the precision of the signal. Agents maximize CARA utility function $U\left(W_{i,t}\right)=-\exp (-a{W}_{i,t})\text{,}$ with the same risk aversion coefficient a, in which $W_{i,t}$ is the wealth of agent i at time t. Let $p_t$ be the (cum-)market price of the risky asset and denote $I_t=\{p_t,p_{t-1},\cdots \}$ the public information of historical price. Conditional on the public information $I_{t-1}$ and her private signal $x_{i,t}$, agent i seeks to maximize her expected utility, leading to the optimal demand $q_{i,t}=[E(p_t|x_{i,t},I_{t-1})-p_{t-1}]/[aVar(p_t|x_{i,t},I_{t-1})]$, conditional on the public information $I_{t-1}$ and her signal $x_{i,t}$.

Facing the information uncertainty on the fundamental value, the agent considers both fundamental and momentum trading strategies based on the public information of the history price and her private signal about the fundamental value. More explicitly, the fundamental trading strategy is based on

(7) $E^f(p_t|x_{i,t},I_{t-1})=(1-\gamma )p_{t-1}+\gamma \frac{{\alpha }_{\mu }\overline{\mu }+{\alpha }_x{x}_{i,t}}{{\alpha }_{\mu }+{\alpha }_x},$

(8) $Va{r}^f(p_t|x_{i,t},I_{t-1})={\gamma }^{2}Var(\mu |x_{i,t},I_{t-1})=\frac{{\gamma }^2}{{\alpha }_{\mu }+{\alpha }_x},$

where $\gamma \in (0,1]$ is a constant, measuring the convergence speed of the market price to the expected fundamental value. Note that $\frac{{\alpha }_{\mu }\overline{\mu }+{\alpha }_x{x}_{i,t}}{{\alpha }_{\mu }+{\alpha }_x}$ and $\frac{1}{{\alpha }_{\mu }+{\alpha }_x}$ are agent i's posterior updating of the mean and variance, respectively, of the fundamental value of the risky asset conditional on her signal $x_{i,t}$. Condition (7) means that the predicted price is a weighted average of the latest market price and the posterior updating of the fundamental value conditional on her private signal $x_{i,t}$; while (8) means that the conditional variance is proportional to the posterior variance conditional on the private signal $x_{i,t}$. In particular, when $\gamma =1$, the conditional mean and variance (7)–(8) are reduced to the posterior mean and variance, respectively. Therefore the fundamental trading strategy reflects agent's belief that the future price is expected to converge to the expected fundamental value. Though the private signals $x_{i,t}$ are i.i.d. across agents and over time, they are partially incorporated through the current market price $p_t$ and hence reflected in the prediction of the future prices. Consequently, the optimal demand based on the fundamental analysis becomes $q_{i,t}^f=[{\alpha }_{\mu }\overline{\mu }+{\alpha }_x{x}_{i,t}-({\alpha }_{\mu }+{\alpha }_x)p_{t-1}]/(a\gamma )$, which is called the fundamental trading strategy f.

The momentum trading is however independent of the private signal $x_{i,t}$, but depends on a price trend,

(9) $E^c(p_t|x_{i,t},I_{t-1})=p_{t-1}+\beta (p_{t-1}-v_t),Va{r}^c(p_t|x_{i,t},I_{t-1})={\sigma }_{t-1}^2,$

where $v_t$ is a reference price or price trend (can be a moving average, a supporting/resistance price level, or any index derived from technical analysis), β measures the extrapolation of the price deviation from the trend, and ${\sigma }_{t-1}^2$ is a heuristic prediction on the variance of the asset price. Then the optimal demand becomes $q_{i,t}^c=\beta (p_{t-1}-v_t)/(a{\sigma }_{t-1}^2)$, which is called momentum strategy c. In particular, when $v_t$ is a moving average of the historical prices and $\beta >(<)0$, strategy c is essentially a time-series momentum (contrarian) strategy (Moskowitz et al., 2012).

Given the information uncertainty, the agent compares the expected value functions based on the two optimal trading strategies and chooses the one with relative higher value. More explicitly, the agent firstly calculates the respective value functions based on strategy f and c,

$\begin{array}{cc}\hfill E_{i,t}^f\left(U\right)& =-\exp \{-A[W_{i,t-1}+\frac{{[{\alpha }_{\mu }\overline{\mu }+{\alpha }_x{x}_{i,t}-({\alpha }_{\mu }+{\alpha }_x)p_{t-1}]}^2}{2a({\alpha }_{\mu }+{\alpha }_x)}]\},\hfill \\ \hfill E_{i,t}^c\left(U\right)& =-\exp \{-A[W_{i,t-1}+\frac{{\beta }^2{(p_{t-1}-v_t)}^2}{2a{\sigma }_{t-1}^2}]\}.\hfill \end{array}$

The agent then compares the value functions and selects the one that yields a higher value. Note that $E_{i,t}^f$ is an increasing function of the absolute value of the signal $|x_{i,t}|$, while $E_i^c$ is independent of $x_{i,t}$. Therefore there exists threshold signal values ${\overline{x}}_t$ for the private signal such that $E_{i,t}^f=E_{i,t}^c$, that is,

$\frac{E_{i,t}^f(U)}{E_{i,t}^c(U)}=\exp \{-[\frac{{[{\alpha }_{\mu }\overline{\mu }+{\alpha }_x{\overline{x}}_t-({\alpha }_{\mu }+{\alpha }_x)p_{t-1}]}^2}{2({\alpha }_{\mu }+{\alpha }_x)}-\frac{{\beta }^2{(p_{t-1}-v_t)}^2}{2{\sigma }_{t-1}^2}]\}=1.$

Solving for ${\overline{x}}_t$ yields

(10) $x_t^{\pm }=\frac{1}{{\alpha }_x}[({\alpha }_{\mu }+{\alpha }_x)p_{t-1}-{\alpha }_{\mu }\overline{\mu }\pm \frac{\beta \sqrt{{\alpha }_{\mu }+{\alpha }_x}}{{\sigma }_{t-1}}(p_{t-1}-v_t)].$

Therefore, when $v_t=p_{t-1}$, the agent chooses strategy f. When $v_t\ne p_{t-1}$, the agent chooses strategy c if her signal is less informative, falling into the interval $(x_t^m,x_t^M)$, and strategy f otherwise, where $x_t^m=\min (x_t^{\pm })$ and $x_t^M=\max (x_t^{\pm })$. Therefore, the optimal demand of agent i is given by $q_{i,t}=q_{i,t}^f$ for $x_{i,t}⩽x_t^m$ or $x_{i,t}⩾x_t^M$; otherwise $q_{i,t}=q_{i,t}^c$ when $x_{i,t}\in (x_t^m,x_t^M)$. Intuitively, when agent's private signal is near the mean fundamental value, the private information becomes less valuable. However, when agent's private signal is far away from the mean fundamental value, the private information becomes more valuable and hence the agent favors the fundamental trading strategy.

The choice between the two strategies due to the informativeness of the private information about the fundamental value leads to endogenous heterogeneity and switching behavior of agents' choices. More explicitly, by aggregating the demand $D_t$ in a closed form and considering noisy supply $S_t$, the market price is determined through a market maker scenario via $p_t=p_{t-1}+\lambda (D_t+S_t)$ with $\lambda >0$. He and Zheng (2016) first conduct an analysis on the underlying deterministic model when ${\sigma }_{t-1}^2={\sigma }^2$ is a constant and $v_t=p_{t-2}$ (corresponding to a simple momentum trading based on the change in the last price). They show that the fundamental price is locally stable with small precisions of the fundamental information noise. That is, the fundamental price becomes unstable when the level of the fundamental information noise is small, leading to high price volatility. Intuitively, in this case, the fundamental information become more accurate and hence less valuable. Therefore the fundamental strategy becomes less profitable, while the momentum trading strategy becomes more popular. This is consistent with the literature on coordination game with imperfect information (see Angeletos and Werning, 2006).

When the fundamental price becomes unstable, the price dynamics can become very complicated. On the stochastic model, they have shown that the market fraction of the agents choosing the momentum (fundamental) strategy decreases (increases) as the mis-pricing increases. This underlies mean-reverting of market price to its fundamental price when mis-pricing becomes significant, burst of a bubble, and recover of a recession. This mechanism, together with the destabilizing role of the momentum trading and the stabilizing role of the fundamental trading, provides an endogenous self-correction mechanism of the market. This mechanism is very different from the HAMs with complete information, in which the mean-reverting is channeled through some nonlinear assumptions on the demand or order flow of risky asset. The market stability depends exogenously on balanced activities from fundamental and momentum trading. This integrated approach of HAMs and incomplete information about the fundamental value therefore provides a micro-foundation to endogenous trading heterogeneity and switching behavior wildly characterized in HAMs. Furthermore, He and Zheng (2016) conduct a time series analysis on the stylized facts and demonstrate that the model is able to match the S&P 500 in terms of power-law distribution in returns, volatility clustering, long memory in volatility, and leverage effect.

2.4 Switching of Agents, Fund Flows, and Leverage

Similar to Dieci et al. (2006), most HAMs employ the discrete-choice framework⁴ to capture the way investors switch across different competing strategies/behavioral rules. However, since this approach models the changes of investors' proportions, not directly the flows of funds, it is not very suitable to capture the long-run performance of investment strategies (or ‘styles’) in terms of accumulated wealth, nor the impact of fund flows on the price dynamics. For this reason, LeBaron (2011) defines such forms of switching between strategies as active learning, capturing investors' tendency to adopt the best-performing rule, in contrast to passive learning, by which investors' wealth naturally accumulates on strategies that have been relatively successful. This second form of learning is closely related to the issue of survival and long-run dominance of strategies and to the evolutionary finance approach (see Blume and Easley, 1992, 2006, Sandroni, 2000, Hens and Schenk-Hoppé, 2005, as well as Evstigneev et al., 2009 for a comprehensive survey of early results and recent research in this field).⁵

LeBaron (2011) argues that the dynamics of real-world markets are likely to be affected by some combinations of active and passive learning, and that exploring their interaction may improve our understanding of the dynamics of asset prices. Moreover, LeBaron (2012) proposes a simple framework that can simultaneously account for wealth dynamics and active search for new strategies, based on performance comparison. Besides reproducing the basic stylized facts of asset returns and trading volume, the model yields some insight into the dynamics of agents' strategies and their impact on market stability.

A further recent contribution on the interplay of active and passive learning is provided by Palczewski et al. (2016). They build an evolutionary finance framework in discrete time with fundamental, trend-following and noise trading strategies. Such strategies are interpreted as portfolio managers with different investment ‘styles’. Individual investors can move (part of) their funds between portfolio managers. The total amount of freely flowing capital is a model parameter, capturing the clients' degree of impatience (similar to the proportion of switching investors in Dieci et al., 2006). Funds are reallocated based on the relative performance of competing fund managers, according to the discrete choice principle. Therefore, portfolio managers may experience an exogenous growth of their wealth, in addition to the endogenous growth due to returns on the employed capital. The model framework appears promising to investigate the market impact of the fund flows and to incorporate different types of ‘behavioral biases’ into HAMs. In particular, Palczewski et al. (2016) show that even a small amount of freely flowing capital can have a large impact on price movements if investors exhibit ‘recency bias’ in evaluating fund performance.

In a somewhat related framework with heterogeneous investment funds using ‘value investing’, Thurner et al. (2012) explore the joint impact of wealth dynamics and the flows of capital among competing investment funds. Evolutionary pressure generated by short-run competition forces fund managers to make leveraged asset purchases with margin calls. Simulation results highlight a new mechanism to fat tails and clustered volatility, which is linked to wealth dynamics and leverage-induced crashes. Moreover, this framework appears promising to test different credit regulation policies (Poledna et al., 2014) and to investigate the impact of bank leverage management on the stability properties of the financial system (Aymanns and Farmer, 2015).

3.1 A Continuous-Time HAM with Time Delay

We now introduce the continuous-time model of He and Li (2012) and demonstrate first that the result of Brock and Hommes on rational routes to market instability in discrete-time also holds in continuous time. That is, adaptive switching behavior of agents can lead to market instability as the switching intensity increases. We then show a double edged effect of an increase in the time horizon of historical price information on market stability. An initial increase in time delay can destabilize the market, leading to price fluctuations. However, as the time delay increases further, the market is stabilized. This double edged effect is a very different feature of continuous-time HAMs from discrete-time HAMs. With noisy fundamental value and liquidity traders, the continuous-time model is able to generate long deviations of market price from the fundamental price, bubbles, crashes, and volatility clustering.

Consider a financial market with a risky asset and let $P(t)$ denote the (cum) price per share of the risky asset at time t. The market consists of fundamentalists, chartists, liquidity traders, and a market maker. The fundamentalists believe that the market price $P(t)$ is mean-reverting to the fundamental price $F(t)$, and their demand is given by $Z_f(t)={\beta }_f[F(t)-P(t)]$, with ${\beta }_f>0$ measuring the mean-reverting speed of the market price to the fundamental price. The chartists are modeled as trend followers, believing that the future market price follows a price trend $u(t)$, and their demand is given by⁸ $Z_c(t)=\tanh ({\beta }_c[P(t)-u(t)])$ with ${\beta }_c>0$ measuring the extrapolation of the trend followers to the price trend. Among various price trends used in practice, we consider $u(t)$ as a normalized exponentially decaying weighted average of historical prices over a time interval $[t-\tau ,t]$,

(11) $u(t)=\frac{k}{1-e^{-k\tau }}{\int }_{t-\tau }^t{e}^{-k(t-s)}P(s)ds,$

where time delay $\tau \in (0,\infty )$ represents time horizon of historical prices, $k>0$ measures the decay rate of the weights on the historical prices. In particular, when $k\to 0$, the weights are equal and the price trend $u(t)$ in (11) is simply given by the standard moving average (MA) with equal weights, $u(t)=\frac{1}{\tau }{\int }_{t-\tau }^{t}P(s)ds$. When $k\to \infty$, all the weights go to the current price so that $u(t)\to P(t)$. For the time delay, when $\tau \to 0$, the trend followers regard the current price as the price trend. When $\tau \to \infty$, the trend followers use all the historical prices to form the price trend, $u(t)=k{\int }_{-\infty }^t{e}^{-k(t-s)}P(s)ds$. In general, for $0<k<\infty$, Eq. (11) can be expressed as a delay differential equation with time delay τ

$du(t)=\frac{k}{1-e^{-k\tau }}[P(t)-e^{-k\tau }P(t-\tau )-(1-e^{-k\tau })u(t)]dt.$

The demand of liquidity traders is i.i.d. normally distributed with mean of zero and standard deviation of ${\sigma }_M(>0)$.

Let $n_f(t)$ and $n_c(t)$ represent the market fractions of agents who use the fundamental and trend following strategies, respectively. Their net profits over a short time interval $[t,t+dt]$ can be measured, respectively, by ${\pi }_f(t)dt=Z_f(t)dP(t)-C_{f}dt$ and ${\pi }_c(t)dt=Z_c(t)dP(t)-C_{c}dt$, where $C_f,C_c⩾0$ are constant costs of the strategies. To measure strategy performance, we introduce a cumulated profit over the time interval $[t-\tau ,t]$ by $U_i(t)=\frac{\eta }{1-e^{-\eta \tau }}{\int }_{t-\tau }^t{e}^{-\eta (t-s)}{\pi }_i(s)ds,i=f,c$, where $\eta >0$ measures the decay of the historical profits. Consequently, $d{U}_i(t)=\eta [\frac{{\pi }_i(t)-e^{-\eta \tau }{\pi }_i(t-\tau )}{1-e^{-\eta \tau }}-U_i(t)]dt$ for $i=f,c$. Following Hofbauer and Sigmund (1998, Chapter 7), the evolution dynamics of the market populations are governed by

$d{n}_i(t)=\beta n_i(t)[d{U}_i(t)-d\overline{U}(t)],\text{ for }i=f,c,$

where $d\overline{U}(t)=n_f(t)d{U}_f(t)+n_c(t)d{U}_c(t)$ is the average performance of the two strategies and $\beta >0$ measures the intensity of choice. The switching mechanism in the continuous-time setup is consistent with the one used in discrete-time HAMs. In fact, it can be verified that the dynamics of the market fraction $n_f(t)$ satisfy $d{n}_f(t)=\beta n_f(t)(1-n_f(t))[d{U}_f(t)-d{U}_c(t)]$, leading to $n_f(t)=e^{\beta U_f(t)}/(e^{\beta U_f(t)}+e^{\beta U_c(t)})$, which is the discrete choice model used in Brock and Hommes (1998).

Finally, the price $P(t)$ is adjusted by the market maker according to $dP(t)=\mu [n_f(t)Z_f(t)+n_c(t)Z_c(t)]dt+{\sigma }_{M}d{W}_M(t)$, where $\mu >0$ represents the speed of the price adjustment of the market maker, $W_M(t)$ is a standard Wiener process capturing the random excess demand process either driven by unexpected market news or liquidity traders, and ${\sigma }_M>0$ is a constant. To sum up, the market price of the risky asset is determined according to the stochastic delay differential system

(12) $\{\begin{array}{c}dP(t)=\mu [n_f(t)Z_f(t)+(1-n_f(t))Z_c(t)]dt+{\sigma }_{M}d{W}_M(t),\hfill \\ du(t)=\frac{k}{1-e^{-k\tau }}[P(t)-e^{-k\tau }P(t-\tau )-(1-e^{-k\tau })u(t)]dt,\hfill \\ dU(t)=\frac{\eta }{1-e^{-\eta \tau }}[\pi (t)-e^{-\eta \tau }\pi (t-\tau )-(1-e^{-\eta \tau })U(t)]dt,\hfill \end{array}$

where $U(t)=U_f(t)-U_c(t)$, $n_f(t)=1/(1+e^{-\beta U(t)})$, $Z_f(t)={\beta }_f(F(t)-P(t))$, $Z_c(t)=\tanh [{\beta }_c(P(t)-u(t))]$, $C=C_f-C_c$, and

$\pi (t)={\pi }_f(t)-{\pi }_c(t)=\mu [n_f(t)Z_f(t)+(1-n_f(t))Z_c(t)][Z_f(t)-Z_c(t)]-C.$

By assuming that the fundamental price is a constant $F(t)\equiv \overline{F}$ and there is no market noise ${\sigma }_M=0$, system (12) becomes a deterministic delay differential system with $(P,u,U)=(\overline{F},\overline{F},-C)$ as the unique fundamental steady state. He and Li (2012) show that the steady state is locally stable for either small or large time delay τ when the market is dominated by the fundamentalists. Otherwise, the steady state becomes unstable through Hopf bifurcations as the time delay increases. This result is in line with the results obtained in discrete-time HAMs. However, different from discrete-time HAMs, the continuous-time model shows that the fundamental steady state becomes locally stable again when the time delay is large enough. This is illustrated by the bifurcation diagram of the market price with respect to τ in Fig. 4A.⁹ It shows that there are two Hopf bifurcation values $0<{\tau }_o<{\tau }_1$ occurring at $\tau ={\tau }_0(\approx 8)$ and $\tau ={\tau }_1(\approx 28)$. The fundamental steady state is locally stable when the time delay is small, $\tau \in [0,{\tau }_0)$, then becomes unstable for $\tau \in ({\tau }_o,{\tau }_1)$, and then regains the local stability for $\tau >{\tau }_1$. Due to the problem of high dimensionality, such analysis on the effect of historical price on market price in discrete-time HAMs can become very complicated, see Chiarella et al. (2006b) that examining the effect of different moving averages on market stability. It is the continuous-time model that facilitates such analysis on the stability effect of time horizon of historical prices and stability switching. The bifurcation diagram of the market price with respect to the switching intensity β is given in Fig. 4B. It shows that the fundamental steady state is locally stable when the switching intensity β is low, becoming unstable as the switching intensity increases, bifurcating to periodic price with increasing fluctuations. This is consistent with the discrete-time HAMs.

For the deterministic model, when the steady state becomes unstable, it bifurcates to stable periodic solutions through a Hopf bifurcation. The periodic fluctuations of the market prices are associated with periodic fluctuations of the market fractions, illustrated in Fig. 5A. Based on the bifurcation diagram in Fig. 4A, the steady state is unstable for $\tau =16$. Fig. 5A shows that both price and market fraction fluctuate periodically. It shows that, when the fundamental steady state becomes unstable, the market fractions tend to stay away from the steady state market fraction level most of the time and a mean of $n_f$ below 0.5 clearly indicates the dominance of the trend following strategy. To examine the effect of population evolution, we compare the case without switching $\beta =0$ to the case with switching $\beta \ne 0$. Fig. 5A clearly shows that the evolution of population increases the fluctuations in both price and market fraction.

For the stochastic model with a random walk fundamental price process, Fig. 5B demonstrates that the market price follows the fundamental price closely when $\tau =3$, while Fig. 5C illustrates that the market price fluctuates around the fundamental price in cyclical fashion for $\tau =16$. To examine the effect of population evolution, we compare the case without switching $\beta =0$ to the case with switching $\beta \ne 0$. Fig. 5B shows that the evolution of population has insignificant impact on the price dynamics when the fundamental steady state of the underlying deterministic model is locally stable for $\tau =3$. However, when the fundamental steady state becomes unstable for $\tau =16$, the fluctuations in both price and market fraction become more significant. Therefore the stochastic price behavior is underlined by the dynamics of the corresponding deterministic model. He and Li (2012) further explore the potential of the stochastic model in generating volatility clustering and long range dependence in volatility. The underlying mechanism and the interplay between the nonlinear deterministic dynamics and noises are very similar to the discrete-time HAM by He and Li (2007). The framework can be used to study the joint impact of many heterogeneous strategies based on different time horizons of historical prices on market stability.

3.2 Profitability of Momentum and Contrarian Strategies

Momentum and contrarian strategies are widely used by market practitioners to profit from momentum in the short-run and mean-reversion in the long-run in financial markets. Empirical profitability of these strategies based on moving averages with different time horizon of historical prices and different holding period has been extensively investigated in the literature (Lakonishok et al., 1994, Jegadeesh and Titman, 1993, 2001, and Moskowitz et al., 2012).

To explain the profitability and the underlying mechanism of time series momentum and contrarian strategies, He and Li (2015a) propose a continuous-time HAM consisting of fundamental, momentum, and contrarian traders. They develop an intuitive and parsimonious financial market model of heterogeneous agents to study the impact of different time horizons on market price and profitability of fundamental, momentum and contrarian trading strategies. They show that the performance of momentum strategy is determined by the historical time horizon, investment holding period, and market dominance of momentum trading. More specifically, due to price continuity, the price trend based on the moving average of historical prices becomes very significant (apart from over very short time horizon). Therefore, when momentum traders are more active in the market, the price trend becomes very sensitive to the shocks, which is characterized by the destabilizing role of the momentum trading to the market. This provides a profit opportunity for momentum trading with short, not long, holding time horizons. When momentum traders are less active in the market, they always loose. The results provide some insights into the profitability of time series momentum over short, not long, holding periods. We now brief the main results of He and Li (2015a).

Consider a continuous-time model with fundamentalists who trade according to fundamental analysis and momentum and contrarian traders who trade differently based on price trend calculated from moving averages of historical prices over different time horizons. Let $P(t)$ and $F(t)$ denote the log (cum dividend) price and (log) fundamental value¹⁰ of a risky asset at time t, respectively. The fundamental traders buy (sell) the stock when the current price $P(t)$ is below (above) the fundamental value $F(t)$. For simplicity, we assume that the fundamental return follows a pure white noise process $dF(t)={\sigma }_{F}d{W}_F(t)$ with $F(0)=\overline{F}$, ${\sigma }_F>0$, and $W_F(t)$ is a standard Wiener process.

Regarding the momentum and contrarian trading, as in the previous section, we assume that both momentum and contrarian traders trade based on their estimated market price trends, although they behave differently. Momentum traders believe that future market price follows a price trend $u_m(t)$, while contrarians believe that future market price goes opposite to a price trend $u_c(t)$. The price trend used for the momentum traders and contrarians can be different in general. Among various price trends used in practice, the standard moving average (MA) rules with different time horizons are the most popular ones, $u_i(t)=\frac{1}{{\tau }_i}{\int }_{t-{\tau }_i}^{t}P(s)ds$ for $i=m,c$, where the time delay ${\tau }_i\ge 0$ represents the time horizon of the MA. Assume the excess demand of the momentum traders and contrarians are given, respectively, by $Z_m(t)=g_m(P(t)-u_m(t))$ and $D_c(t)=g_c(u_c(t)-P(t))$, where $g_i(x)$ satisfies $g_i(0)=0,g_i^{\prime }(x)>0,g_i^{\prime }(0)={\beta }_i>0,x{g}_i^{″}(x)<0$ for $x\ne 0$ and $i=m,c$, and parameter ${\beta }_i$ represents the extrapolation rate when the market price deviation from the trend is small.

Assume a zero net supply in the risky asset and let ${\alpha }_f$, ${\alpha }_m$, and ${\alpha }_c$ be the fixed market population fractions of the fundamental, momentum, and contrarian traders,¹¹ respectively, with ${\alpha }_f+{\alpha }_m+{\alpha }_c=1$. Following Beja and Goldman (1980) and Farmer and Joshi (2002), the price $P(t)$ at time t is determined by

(13) $dP(t)=\mu [{\alpha }_f{Z}_f(t)+{\alpha }_m{Z}_m(t)+{\alpha }_c{Z}_c(t)]dt+{\sigma }_{M}d{W}_M(t),$

where $\mu >0$ represents the speed of the price adjustment of the market maker, $W_M(t)$ is a standard Wiener process, independent of $W_F(t)$, capturing the random demand of either noise or liquidity traders, and ${\sigma }_M\ge 0$ is constant.

By assuming a constant fundamental price $F(t)\equiv \overline{F}$ and no market noise ${\sigma }_M=0$, system (13) becomes a deterministic delay integro-differential equation,

(14) $\begin{array}{cc}\hfill \frac{dP(t)}{dt}& =\mu \left[{\alpha }_f{\beta }_f\right(\overline{F}-P(t))+{\alpha }_m\tanh ({\beta }_m(P(t)-\frac{1}{{\tau }_m}{\int }_{t-{\tau }_m}^{t}P(s)ds))\hfill \\ \hfill & +{\alpha }_c\tanh (-{\beta }_c(P(t)-\frac{1}{{\tau }_c}{\int }_{t-{\tau }_c}^{t}P(s)ds\left)\right)].\hfill \end{array}$

It is easy to see that $P(t)=\overline{F}$, the fundamental steady state, is the unique steady state price of system (14). He and Li (2015a) examine different role of the time horizon used in the MA by either the contrarians or momentum traders. When both strategies are employed in the market, the market stability of system (14) can be characterized by the following proposition.

The three conditions (1) ${\gamma }_m<{\gamma }_c+\frac{{\gamma }_f}{1+a}$, (2) ${\gamma }_c+\frac{{\gamma }_f}{1+a}\le {\gamma }_m\le {\gamma }_c+{\gamma }_f$, and (3) ${\gamma }_m>{\gamma }_c+{\gamma }_f$ in Proposition 3.1 characterize three different states of market stability, having different implications to the profitability of momentum trading. For convenience, market state k is referred to condition (k) for $k=1,2,3$ in the following discussion. Numerical analysis shows that for market state 1, the fundamental price is locally stable, independent of the time horizon; for market state 2, the fundamental price is locally stable when $\tau \in [0,{\tau }_l^{⁎})\cup ({\tau }_h^{⁎},\infty )$ and becomes unstable when $\tau \in ({\tau }_l^{⁎},{\tau }_h^{⁎})$ (the stability switches twice); while for market state 3, the first (Hopf bifurcation) value ${\tau }_l^{⁎}(\approx 0.22)$ leads to stable limit cycles for $\tau >{\tau }_l^{⁎}$ (the stability switches only once at ${\tau }_l^{⁎}$).