LAUSR

Since the invention of Hawkes processes in the early 1970s many researchers, including the pioneer seismologists, have studied applications over a very wide range of topics. However, these stochastic processes with a substantial modelling advantage were only noticed in finance research from 2005. Since then there has been a rapid expansion of papers applying Hawkes processes to diverse problems in finance. But there is still great scope to make them become standard financial econometric tools and we, as the panel of guest editors, felt that it would be both valuable and timely to have a special journal issue treating a variety of financial applications. Happily, when we approached the Editorsin-Chief of Quantitative Finance they were extremely supportive, and the result is this issue with a broad range of interesting papers. The issue begins with an introduction by the author of Hawkes processes, Professor Alan Hawkes. He has been excited by the idea of this issue and has provided a brief history to describe the basic properties of Hawkes processes —essentially a class of stochastic models for series of events whose occurrence generally increases the probability of occurrence of further events, often described as a contagious effect. He concludes with a review of some recently published papers. This is followed by a collection of nine papers, addressing many contemporary topics from both the theoretical and practical viewpoints. Achab, Bacry, Muzy and Rambaldi use a 12-dimensional mutually exciting process to model interactions between different kinds of events in a high-frequency single-asset order book. A non-parametric method is used to estimate the branching ratio matrix directly without considering the exact shape of the exciting kernels. The elements of this matrix measure the connectivity between event types and the process developed is extended to study interactions between two assets. As indicated above, Hawkes processes are usually positively exciting. However, Khashanah, Chen and Hawkes introduce a type of birth-death-immigration model which turns out to be a special kind of two-dimensional mutually exciting process. In it, the occurrence of some types of events can decrease the rate at which other events occur (so that these interactions are actually depressing rather than exciting). This feature can be useful in modelling the decay of activities that are usually observed in a market with bursts of events. For example, the burst of trading activities at the start of a trading day. A number of studies of jumps in asset prices have assumed that the kernel of a self-exciting model has a simple exponential shape or perhaps a linear combination of two or three exponential components. Chen, Hawkes, Scalas and Trinh carry out a simulation study to compare the ability of various information criteria: Akaike’s information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan-Quinn criterion (HQ) to decide on the best model to choose. Calcagnile, Bormetti, Treccani, Marmi and Lillo study a portfolio with a multiplicity of assets and are concerned with the number of assets that jump simultaneously (cojumps). It is suggested that a suitable mutually exciting Hawkes process could be used to model the time-clustering of jumps. The authors also provide a novel approach to fit such a complicated model using simplifying assumptions which reduce the model estimation to involve only three parameters! Remarkably, this approximation appears to fit data moderately well. Lu and Abergel attempt to model the various events of an order book using a mutually exciting Hawkes process with exponential kernels. On finding that interactions sometimes appear inhibitory rather than exciting, they replace the usual linear model by a non-linear model. Since the usual linear form can result in a negative intensity, they simply truncate the kernel at zero in this case. Gao, Zhou and Zhu consider a self-exciting Hawkes process in which the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are i.i.d. non-negative random variables. They obtain closed-form formulas for the Laplace transform, moments and distribution of the process. The process is then used to model the clustered arrival of trades in a dark pool and for a resting dark order they analyse various pool performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate. Schneider, Lillo and Pelizzon use a peaks over threshold (PoT) method to identify abrupt liquidity drops in limit order book data. Both the self-excitation of extreme changes of liquidity in the same asset (illiquidity spirals) and cross-excitation across different assets (illiquidity spillovers) can be quantified. The application of their method to the Mercato dei Titoli di Stat (MTS) sovereign bond markets show that the proportion of shocks explained by illiquidity spillovers roughly doubles from 2011 to 2015.