The Heston–Queue-Hawkes process: A new self-exciting jump–diffusion model for options pricing, and an extension of the COS method for discrete distributions

We propose a new self-exciting jump–diffusion process, the Heston–Queue-Hawkes (HQH) model, which integrates the well-known Heston model with the recently introduced Queue-Hawkes (Q-Hawkes) jump process. Similar to the Heston–Hawkes process (HH), the HQH model effectively captures both the slow and continuous evolution of prices, and the sudden and impactful market movements due to self-excitation and contagion. But a significant advantage of the HQH model is that its characteristic function is available in closed form, allowing for the efficient application of Fourier-based fast pricing algorithms, such as the COS method (which we extend to deal with discrete distributions). We also demonstrate that, by leveraging partial integrals of the characteristic function, which are explicitly known for the HQH process, we can reduce the dimensionality of the COS method, thereby decreasing its numerical complexity. Our numerical results for pricing European and Bermudan options indicate that the HQH model provides a broader range of volatility smiles compared to the Bates model, while maintaining a substantially lower computational burden than the HH process.