Quantitative Models and Construction Methods 1. Model Name: Chartist-Fundamentalist Model - **Model Construction Idea**: This model integrates the behaviors of chartists and fundamentalists to explain the coexistence of constant mispricing and oscillatory mispricing in stock markets. It reconciles the views of Grossman & Stiglitz (1980) and Lo & Farmer (1999) by considering the dynamic interactions between these two types of traders and the role of market makers[4][17][20] - **Model Construction Process**: - **Market Maker's Price Adjustment**: The market maker adjusts prices based on excess demand using the equation: $$ P_{t+1} = P_{t} + \alpha(D_{t}^{C} + D_{t}^{F} + D_{t}^{R} - N) \tag{1} $$ where \( \alpha > 0 \) is the price adjustment parameter, \( D_{t}^{C} \) and \( D_{t}^{F} \) represent the demand from chartists and fundamentalists, \( D_{t}^{R} \) is non-speculative demand, and \( N \) is the total stock supply[24][26] - **Chartists' Behavior**: Chartists extrapolate past price trends into the future, formalized as: $$ D_{t}^{C} = \beta(P_{t} - P_{t-1}) \tag{3} $$ where \( \beta > 0 \) is the market reaction coefficient of chartists[27] - **Fundamentalists' Behavior**: Fundamentalists trade based on deviations from fundamental value \( F_t \), with their demand defined as: $$ D_{t}^{F} = \begin{cases} \gamma(F_{t} - P_{t}) & \text{if } P_{t} - F_{t} > h \\ 0 & \text{if } -h \leq P_{t} - F_{t} \leq h \\ \gamma(F_{t} - P_{t}) & \text{if } P_{t} - F_{t} < -h \end{cases} \tag{4} $$ where \( \gamma > 0 \) measures the market influence of fundamentalists, and \( h \) is the threshold for mispricing[27] - **Fundamental Value Dynamics**: The fundamental value follows a random walk: $$ F_{t+1} = F_{t} + \delta_{t}, \quad \delta_{t} \sim N(0, \sigma_{\delta}^2) \tag{5} $$[28] - **Price Evolution Equation**: Combining the above equations, the price evolution is expressed as: $$ P_{t+1} = \begin{cases} (1 + \alpha\beta - \alpha\gamma)P_{t} - \alpha\beta P_{t-1} + \alpha\gamma F_{t} & \text{if } P_{t} - F_{t} > h \\ (1 + \alpha\beta)P_{t} - \alpha\beta P_{t-1} & \text{if } -h \leq P_{t} - F_{t} \leq h \\ (1 + \alpha\beta - \alpha\gamma)P_{t} - \alpha\beta P_{t-1} + \alpha\gamma F_{t} & \text{if } P_{t} - F_{t} < -h \end{cases} \tag{6} $$[29] - **Model Evaluation**: The model successfully explains the coexistence of constant and oscillatory mispricing, highlighting the dynamic nature of market efficiency and the role of trader interactions[4][17][85] --- Model Backtesting Results 1. Chartist-Fundamentalist Model - **Parameter Region R1**: When both chartists' and fundamentalists' market influence are low, prices converge to a non-fundamental fixed point, resulting in constant mispricing[21][22][66] - **Parameter Region R2**: With moderate market influence, prices either converge to a non-fundamental fixed point or exhibit endogenous oscillatory dynamics[21][22][66] - **Parameter Region R3**: When fundamentalists' market influence is high, prices either converge to a non-fundamental fixed point or diverge[21][22][66] - **Parameter Region R4**: When chartists' market influence is high, prices exhibit divergent dynamics[21][22][66] - **Impact of Fundamental Shocks**: Random shocks to the fundamental value can cause transitions between fixed-point dynamics and oscillatory dynamics, with the latter becoming dominant as the parameter \( c \) increases[78][79][80]