We construct a price impact model between stocks in a correlated market. For the price change of a given stock induced by the short-run liquidity of this stock itself and of the information about other stocks, we introduce a self- and a cross-impact function of the time lag. We model the average cross-response functions for individual stocks employing the impact functions of the time lag, the impact functions of traded volumes and the trade-sign correlators. To quantify the self- and cross-impacts, we propose a construction to fix the parameters in the impact functions. These parameters are further corroborated by a diffusion function that measures the correlated motion of prices from different stocks. This construction is mainly ad hoc and alternative ones are not excluded. It turns out that both the sign cross- and self-correlators are connected with the cross-responses. The self- and cross-impact functions are indispensable to compensate amplification effects which are due to the sign correlators integrated over time. We further quantify and interpret the price impacts of time lag in terms of temporary and permanent components. To support our model, we also analyze empirical data, in particular the memory properties of the sign self- and average cross-correlators. The relation between the average cross-responses and the traded volumes which are smaller than their average is of power-law form.
Reaction-Diffusion-Branching Models of Stock Price Fluctuations
股票交易的几个模型[P.Bak et al，Physica A { \bf 246}，430（1997）]被分析为与一维、两物种反应扩散分支过程类似。使用启发式和缩放参数，我们表明，短期市场价格变化是次扩散的赫斯特指数$H=1/4$。偏向于市场价格的扩散和盲目复制导致了在很长一段时间内与经验观察到的随机游走行为（$H=1/2$）的交叉。计算的交叉形式和扩散常数与模拟数据吻合良好。
