A fractional Hawkes process for illiquidity modeling

The Amihud illiquidity measure has proven to be very popular in the empirical literature for measuring the illiquidity process of stocks and indices. Many econometric models in discrete time have then been proposed for this Amihud measure. Such models are however not adapted for reproduc- ing peaks of illiquidity with long-memory, for risk management and for pricing liquidity-related derivatives. This paper therefore proposes a new paradigm for modeling illiquidity via a continuous- time process with jumps exhibiting long-range dependence. More precisely, we first introduce a new fractional Hawkes process in which the intensity process is ruled by a modified Mittag-Leffler excitation function. Working with a mean-reverting jump model for the (log-)Amihud measure where jumps follow this modified fractional Hawkes process then allows to easily reproduce the observed peaks of illiquidity in financial markets while introducing long-range dependence and tractability in the model. Indeed, thanks to this modified Mittag-Leffler kernel, we show that our model for the (log)-Amihud measure admits a characteristic function in semi-closed form while having a long-memory of past events, which is not achievable with the existing Hawkes processes. We can therefore use this model to perform risk management on illiquidity as well as to introduce and price illiquidity derivatives on the Amihud measure. We hence provide with this paper new tools for a better understanding and management of the illiquidity risk in financial markets.