学海拾珠系列之二百六十一：虚假信息可被容忍吗？解析其对波动的影响与边界

Quantitative Models and Construction Methods 1. Model Name: Predatory Trading Game with Disinformation - Model Construction Idea: This model incorporates disinformation into a predatory trading game framework, where participants act based on distorted information, leading to deviations in equilibrium and market volatility[3][16][23] - Model Construction Process: 1. The model builds on the microstructure frameworks of Carlin et al. (2007) and Carmona & Yang (2011), introducing a victim (forced to adjust risky asset positions) and predators (seeking profit from the victim's constraints)[23] 2. The trading rate of participant $n$ is defined as: X^{n}(t) = X^{n}(0) + \int_{0}^{t}\alpha^{n}(s)\mathrm{d}s \tag{1} where $\alpha^{n}$ represents the trading rate, constrained by: $\alpha_{t}^{n} \in \mathbb{A}^{n} = \left\{\alpha_{t}^{n} \mid \mathcal{H}_{[0,T]}^{2}, X_{T}^{n} = 0 \right\}$ 3. Temporary price impact is modeled as: P_{t} - X_{t}^{0} = \lambda \sum_{i=1}^{N}\alpha_{t}^{i} \tag{4} where $\lambda$ is the elasticity factor[24] 4. Permanent price impact is expressed as: \mathrm{d}X_{t}^{0} = \gamma \sum_{i=1}^{N}a_{t}^{i}\mathrm{d}t + \sigma\mathrm{d}W_{t} \tag{5} where $\gamma$ represents market plasticity, and $\sigma$ is the volatility parameter[24] 5. Participants aim to maximize profits: J^{n}(\mathbf{\alpha}) = \mathbb{E}\left(\int_{0}^{T}\alpha^{n}\left(X_{t}^{0} + \lambda\sum_{i=1}^{N}\alpha_{t}^{i}\right)\mathrm{d}t\right) \tag{8} 6. Disinformation is introduced as a random distortion $\tilde{x}_{0,1} = x_{0,1} + \epsilon$, where $\epsilon$ represents the distortion[27] 7. The price process under disinformation is given by: $X_{t}^{0} = X^{0}(0) - \frac{1-e^{-\frac{N-1}{N+1}\frac{T_{T}}{\lambda}}}{1-e^{-\frac{N-1}{N+1}\frac{T_{T}}{\lambda}}}\gamma\left(\sum_{i=1}^{N}x_{0}^{i}+\bar{\nu}\right) + \frac{e^{\frac{T t}{\lambda}}-1}{e^{\frac{T t}{\lambda}}-1}\gamma\bar{\nu} + \sigma\left(W_{t}-W_{0}\right)$ where $\bar{\nu}$ is the error factor[30][31] - Model Evaluation: The model effectively captures the impact of disinformation on market dynamics, highlighting its role in amplifying volatility and disrupting equilibrium[16][30] --- Model Backtesting Results 1. Predatory Trading Game with Disinformation - Maximum Price Fluctuation (MPF): $MPF_{\nu}(t_{*},t^{*}) := \operatorname*{max}_{t_{1},t_{2}\in[t_{*},t^{*}]}\left|\mathbb{E}\left(X_{t_{1}}^{0}-X_{t_{2}}^{0}\right)\right|$ The model demonstrates that disinformation increases MPF, with a lower bound determined by: $MPF_{\tilde{\nu}^{*}}(0,T) \geq \operatorname*{min}_{\tilde{\nu}\in\mathbb{R}} MPF_{\tilde{\nu}}(0,T) = \gamma\sum_{i=1}^{N}x_{0}^{i}$[34][37] - Error Factor Impact: The error factor $\nu$ significantly influences price trajectories, with higher $\nu$ leading to greater volatility[30][33] - Tolerance Thresholds: The system tolerates disinformation within specific boundaries $b_{1}$ and $b_{2}$, beyond which volatility escalates[38][40] --- Quantitative Factors and Construction Methods 1. Factor Name: Error Factor ($\nu$) - Factor Construction Idea: The error factor quantifies the degree and spread of disinformation in the market, serving as a key determinant of price volatility[30][33] - Factor Construction Process: 1. Defined as: $\tilde{\nu} := \frac{N_{w}}{N}\left(\tilde{x}_{0}^{1} - x_{0}^{1}\right)$ where $N_{w}$ is the number of misinformed participants, and $\tilde{x}_{0}^{1} - x_{0}^{1}$ represents the distortion magnitude[30] 2. Generalized for multiple distortions: $\nu := \frac{1}{N}\,\sum_{l=1}^{\kappa}N_{w_{l}}\left(\bar{x}_{0,w_{l}}^{1} - x_{0}^{1}\right)$ where $\kappa$ is the number of distinct distortions[56] - Factor Evaluation: The error factor effectively captures the interplay between disinformation magnitude and its spread, providing insights into its impact on market dynamics[30][56] --- Factor Backtesting Results 1. Error Factor ($\nu$) - Maximum Price Fluctuation (MPF): Higher $\nu$ values correspond to increased MPF, with a minimum threshold determined by: $MPF_{\nu}(0,T) \geq \gamma\sum_{i=1}^{N}x_{0}^{i}$[34][37] - Tolerance Thresholds: The system tolerates $\nu$ within boundaries $b_{1}$ and $b_{2}$, with specific dependencies on market parameters and game duration[38][40] - Dynamic Evolution: The tolerance for $\nu$ increases over time, reducing the potential for disinformation to amplify volatility in the long term[90][91] --- Additional Insights - Information Updates: New information can mitigate the impact of disinformation by adjusting the error factor $\nu$, with the timing of updates being critical to minimizing volatility[84][92][95] - Randomness and Misjudgment: Random price movements can lead even informed participants to misjudge their information, complicating the detection and correction of disinformation[100][101][103] - Profit Implications: Disinformation affects profit expectations, with informed participants benefiting under certain conditions, while widespread disinformation can erode these advantages[49][51][56]