Option pricing in a diffusive market with rough Hawkes jumps

Details

Supervisors

Hainaut, Donatien

Faculty

Faculté des sciences

Degree label

Master [120] en sciences actuarielles, à finalité spécialisée

Abstract

enRough Hawkes processes have been extensively studied in recent years for modeling Poisson intensities. These processes introduce memory effects through self-exciting jumps, making them theoretically particularly suitable for capturing clustering phenomena observed in financial markets. This thesis provides an overview of methods for handling non-Markovian yet affine processes, of which rough Hawkes models arise as a specific instance through the appropriate choice of rough kernel. First, self-exciting jumps are incorporated into the classical diffusive dynamics of asset prices. We focus on unidimensional memory processes and compare them with the two-dimensional rough Hawkes process studied by Hainaut. Second, the empirical observation of volatility clustering motivates the introduction of self-exciting jumps into the Cox–Ingersoll–Ross (CIR) process governing the volatility in a Heston-type model. Market evidence suggests that jumps in volatility are associated with simultaneous jumps in the underlying asset, typically of opposite sign and smaller magnitude. We provide a mathematical analysis of this framework and derive expressions for the moment-generating function of the asset price in both settings. Overall, the proposed modeling approach shows potential to reproduce the volatility smile after appropriate calibration.