Pricing Derivatives Under Self-Exciting Dynamics: A Finite-Difference and Transform Approach

Battle Count: 64

The paper is primarily relevant to insurance/catastrophe risk and weather derivatives rather than traditional equity/FX quantitative trading. However, the self-exciting jump process framework and numerical PIDE techniques have direct applicability to credit risk contagion, energy market derivatives, and jump-diffusion option pricing. The transform-based dimensionality reduction technique is broadly applicable to any 2D PIDE arising in derivative pricing.

The pipeline integrates multiple sophisticated numerical components: (1) 2D-to-1D spectral reduction via Bromwich transform, (2) IMEX finite difference with upwind discretization and tridiagonal Thomas solver, (3) generalized Gauss-Laguerre quadrature for Gamma mixture kernels, (4) PCHIP shape-preserving interpolation for off-grid evaluations, (5) Bromwich contour inversion via Simpson rule, (6) Esscher transform and EM calibration. Each component requires careful implementation and tuning (delta, Y_max, grid sizes, CFL conditions). The rigorous error analysis adds theoretical complexity.

The paper provides a detailed algorithm summary (Section 5.3), full parameter tables (Table 1), and explicit discretization formulas. However, no code repository is mentioned. The numerical experiments use synthetic parameters, making reproduction feasible but requiring significant implementation effort across multiple numerical components (IMEX, Gauss-Laguerre, PCHIP, Bromwich inversion).