A slightly depressing jump model: intraday volatility pattern simulation

Hawkes processes have been finding more applications in diverse areas of science, engineering and quantitative finance. In multi-frequency finance various phenomena have been observed, such as shocks, crashes, volatility clustering, turbulent flows and contagion. Hawkes processes have been proposed to model those challenging phenomena appearing across asset prices in various exchanges. The original Hawkes process is an intensity-based model for series of events with path dependence and self-exciting or mutual-exciting mechanisms. This paper introduces a slightly depressing process to model the reverse phenomenon of self-exciting mechanisms. Such a process models the decline in the intensity of jumps observed in market regimes. The proposed birth-immigration-death process captures the decline in jump intensity observed at the start of a daily trading regime while the classical immigration-birth process models an increase in jump intensity towards the close of daily trading. Each of these processes can be expressed as a special case of a simple bivariate Hawkes process.