A Multiscale Model of High-Frequency Trading

We propose and study a stylization of high frequency trading (HFT). Our interest is an order book which consists of orders from slow liquidity traders and orders from high-frequency traders. We would like to frame a model which is amenable to the (seemingly natural) mathematical toolkit of separation of scales and which can be used to address some of the larger issues involved in HFT. The main issue to which we address our model is volatility. An important question is how volatility is affected by HFT. In our stylized model, we show how HFT increases volatility, and can quantify this effect as a function of the parameters in our model and the separation of scales.