A Degeneracy-Aligned and Discrete Maximum-Principle- Compatible Peridynamic Differential Operator for the Heston Stochastic Volatility Model
- 1School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521000, China
Abstract
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.
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BibTeX
@article{qu2026degeneracyaligned,
title = {{A Degeneracy-Aligned and Discrete Maximum-Principle- Compatible Peridynamic Differential Operator for the Heston Stochastic Volatility Model}},
author = {Dan Qu and Yanbo Ma},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193585},
url = {https://www.mdpi.com/2227-7390/14/19/3585},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Degeneracy-Aligned and Discrete Maximum-Principle- Compatible Peridynamic Differential Operator for the Heston Stochastic Volatility Model
AU - Dan Qu
AU - Yanbo Ma
PY - 2026
DA - 2026/10/02
JO - Mathematics
DO - 10.3390/math14193585
UR - https://www.mdpi.com/2227-7390/14/19/3585
AB - 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.
SN - 2227-7390
ER -