Hawkes-based microstructure of rough volatility model with sharp rise

We consider the microstructure of a stochastic volatility model incorporating both market and limit orders. In our model, the volatility is driven by self-exciting arrivals of market orders as well as self-exciting arrivals of limit orders, which are modeled by Hawkes processes. The impact of market order on future order arrivals is captured by a Hawkes kernel with power law decay, and is hence persistent. The impact of limit orders on future order arrivals is temporary, yet possibly long-lived. After suitable scaling the volatility process converges to a fractional Heston model driven by an additional Poisson random measure. The random measure generates occasional spikes in the volatility process. The spikes resemble the clustering of small jumps in the volatility process that has been frequently observed in the financial economics literature. Our results are based on novel uniqueness results for stochastic Volterra equations driven by a Poisson random measure and non-linear fractional Volterra equations.