{
  "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 $Y\\in(0,2)$, 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 $\\mathcal{O}(N\\log N)$ matrix--vector products via the fast Fourier transform. In the regime $Y\\in(1,2)$, 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 $\\mathcal{O}(h^{p-Y/2})$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07554",
  "author_total": 1,
  "authors": [
    {
      "name": "Davood Damircheli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8xvhm43369tp9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07554",
  "fulltext_url": null,
  "id": "8xvhm43369tp9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07554",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7220923433",
  "pdf_url": "https://arxiv.org/pdf/2610.07554",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8xvhm43369tp9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07554",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07554"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process",
  "title_text": "A B-Spline Galerkin Method for Option Pricing under the CGMY Tempered Stable Process",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8xvhm43369tp9",
  "venue": "arXiv",
  "versions": []
}