A Limit Order Book Model for High Frequency Trading with Rough Volatility

We introduce a model for limit order book of a certain security with two main features: First, both the limit orders and market orders for the given asset are allowed to appear and interact with each other. Second, the high frequency trading activities are allowed and described by the scaling limit of nearly-unstable multi-dimensional Hawkes processes with power law decay. The model has been derived as a stochastic partial differential equation (SPDE, for short), under certain intuitive identifications. Its diffusion coefficient is determined by a Volterra integral equation driven by a Hawkes process, whose Hurst exponent is less than 1/2 (so that the relevant process is negatively correlated). As a result, the volatility path of the SPDE is rougher than that driven by a (standard) Brownian motion. The well-posedness follows from a result in literature. Hence, a foundation is laid down for further studies in this direction.