A Degeneracy-Aligned and Discrete Maximum-Principle- Compatible Peridynamic Differential Operator for the Heston Stochastic Volatility Model

This paper develops a non-local mesh-free framework for pricing European options under the Heston stochastic volatility model, addressing the challenges of strong negative correlation and diffusion degeneracy near the zero-variance boundary. The method employs a state-dependent adaptive horizon that refines non-local interactions near the degenerate boundary while relaxing them in the interior and enforces sign-restricted coefficients to ensure a discrete maximum principle and monotonicity in option values. For time-stepping, an implicit–explicit second-order backward differentiation scheme is used, with a Rannacher startup procedure to dampen transient oscillations from the non-smooth pay-off at the strike. Numerical comparisons against a modified Craig–Sneyd alternating-direction implicit solver confirm comparable accuracy and approximately second-order convergence, with the method maintaining close agreement in stressed regimes of high negative correlation and severe variance degeneracy. Despite higher computational costs, the framework offers greater flexibility on non-uniform point sets and robust performance near degenerate boundaries, making it particularly attractive for adaptive pricing applications involving irregular cash-flow structures or localized resolution around critical exercise regions.