A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process
Abstract
We develop and analyze a B-spline scaling-function Galerkin method for the tempered fractional partial differential equation governing European and American option prices under the CGMY process. The spatial generator couples exponentially tempered left- and right-sided Riemann--Liouville derivatives of order Y∈(0,2), whose nonlocality and asymmetry present discretization challenges. We derive a corrected Fourier-symbol construction of the tempered connection-coefficient matrices whose interior blocks are Toeplitz, maintain exact fractional scaling, and enable 𝒪(Nlog N) matrix--vector products via the fast Fourier transform. In the regime Y∈(1,2), we prove full-line and finite-domain Gårding inequalities, coercive whenever the risk-free rate is positive, via zero-extension to the full-line symbol. On this foundation, we establish elliptic-projection, semidiscrete, and fully discrete error estimates with spatial rate 𝒪(hp−Y/2), alongside an unconditional shifted-energy stability estimate for the forced Crank--Nicolson scheme with mesh-independent constants. For positive rates the admissible shift vanishes, governing the plain scheme exactly as implemented. A single backward solve prices an entire strike strip at fixed maturity with smooth, analytically differentiated Greeks. The same discretization prices American puts through a projected linear complementarity solve, with early-exercise premiums and exercise boundaries validated against a resolved deterministic reference and a Longstaff--Schwartz Monte Carlo benchmark. Calibration to S\&P~500 option surfaces across low-volatility, crisis, and recovery regimes yields implied-volatility errors near two vol-points on well-identified surfaces, while an accuracy-matched comparison with a stabilized Grünwald--Letnikov baseline delineates precisely when the high-order variational structure pays off.
Related
- Symbol-based residual certification of reduced and neural surrogates for tempered-fractional CGMY pricing
- On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis
- The discontinuous Galerkin method for solving the two-dimensional transport equation with distributed-order memory kernel on triangular meshes
- An implicit locking-free B-spline Material Point Method for large strain geotechnical modelling
- A Local Discontinuous Galerkin method for the modified Camassa--Holm equation
BibTeX
@misc{damircheli2026bspline,
title = {{A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process}},
author = {Davood Damircheli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07554},
eprint = {2610.07554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07554},
}
RIS
TY - UNPB
TI - A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process
AU - Davood Damircheli
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07554
UR - https://arxiv.org/abs/2610.07554
AB - We develop and analyze a B-spline scaling-function Galerkin method for the tempered fractional partial differential equation governing European and American option prices under the CGMY process. The spatial generator couples exponentially tempered left- and right-sided Riemann--Liouville derivatives of order Y∈(0,2), whose nonlocality and asymmetry present discretization challenges. We derive a corrected Fourier-symbol construction of the tempered connection-coefficient matrices whose interior blocks are Toeplitz, maintain exact fractional scaling, and enable 𝒪(Nlog N) matrix--vector products via the fast Fourier transform. In the regime Y∈(1,2), we prove full-line and finite-domain Gårding inequalities, coercive whenever the risk-free rate is positive, via zero-extension to the full-line symbol. On this foundation, we establish elliptic-projection, semidiscrete, and fully discrete error estimates with spatial rate 𝒪(h^p−Y/2), alongside an unconditional shifted-energy stability estimate for the forced Crank--Nicolson scheme with mesh-independent constants. For positive rates the admissible shift vanishes, governing the plain scheme exactly as implemented. A single backward solve prices an entire strike strip at fixed maturity with smooth, analytically differentiated Greeks. The same discretization prices American puts through a projected linear complementarity solve, with early-exercise premiums and exercise boundaries validated against a resolved deterministic reference and a Longstaff--Schwartz Monte Carlo benchmark. Calibration to S\&P~500 option surfaces across low-volatility, crisis, and recovery regimes yields implied-volatility errors near two vol-points on well-identified surfaces, while an accuracy-matched comparison with a stabilized Grünwald--Letnikov baseline delineates precisely when the high-order variational structure pays off.
ER -