Hawkes-driven stochastic volatility models: goodness-of-fit testing of alternative intensity specifications with S &P500 data

We introduce a novel stochastic volatility model with price and volatility co-jumps driven by Hawkes processes and develop a feasible maximum-likelihood procedure to estimate the parameters driving the jump intensity. Using S&P500 high-frequency prices over the period May 2007–August 2021, we then perform a goodness-of-fit test of alternative jump intensity specifications and find that the hypothesis of the intensity being linear in the asset volatility provides the relatively best fit, thereby suggesting that jumps have a self-exciting nature.