We propose an actionable calibration procedure for general Quadratic Hawkes models of order book events (market orders, limit orders, cancellations). One of the main features of such models is to encode not only the influence of past events on future events but also, crucially, the influence of past price changes on such events. We show that the empirically calibrated quadratic kernel is well described by a diagonal contribution (that captures past realised volatility), plus a rank-one ‘Zumbach’ contribution (that captures the effect of past trends). We find that the Zumbach kernel is a power-law of time, as are all other feedback kernels. As in many previous studies, the rate of truly exogenous events is found to be a small fraction of the total event rate. These two features suggest that the system is close to a critical point – in the sense that slightly stronger feedback kernels would lead to endogenous liquidity crises.

我们为订单簿事件（市价单、限价单、取消）的一般二次霍克斯模型提出了可操作的校准程序。这种模型的主要特征之一是不仅编码过去事件对未来事件的影响，而且至关重要的是，编码过去价格变化的影响关于此类事件。我们表明，经验校准的二次核可以通过对角线贡献（捕获过去实现的波动性）以及一级“Zumbach”贡献（捕获过去趋势的影响）很好地描述。我们发现 Zumbach 核是时间的幂律，就像所有其他反馈核一样。与之前的许多研究一样，发现真正外源性事件的发生率仅占总事件发生率的一小部分。这两个特征表明该系统已接近临界点——从某种意义上说，稍强的反馈内核将导致内生流动性危机。