{
  "abstract": "We study a forward-time formulation of the Black-Scholes equation with state-dependent volatility. In contrast to the classical terminal-value pricing problem, where the option payoff is prescribed at maturity and the price is computed backward in time, the present problem prescribes the current option-price profile and seeks to recover the option-price profile at the expiration date T. This formulation is ill-posed, since the equation evolves in the unstable direction of the parabolic operator and high-frequency perturbations in the initial data may be strongly amplified. To address this difficulty, we introduce a price-dimensional reduction based on shifted Legendre polynomials. The original Black-Scholes equation is projected onto a finite-dimensional Legendre basis in the asset-price variable, leading to a system of ordinary differential equations in time for the expansion coefficients. This reduction acts as a spectral cutoff and also relaxes the degeneracy caused by the factor S^2 at the zero-price boundary. The main reconstruction method is a dimension-reduced Legendre--Tikhonov method. We prove existence, uniqueness, data stability, and convergence for each fixed truncation level. We also include a reduced PINN solver as a secondary computational comparison after the Legendre reduction. Numerical experiments with smooth, butterfly-spread, and European put payoffs show that the Legendre--Tikhonov method recovers the terminal option-price profile from noisy initial data, while the reduced PINN solver provides a useful additional benchmark. Comparisons with the conventional physical-space quasi-reversibility method demonstrate the stabilizing effect of the Legendre reduction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Journal of Computational and Applied Mathematics, 492 (2027), 118112",
  "arxiv": "2606.12450",
  "author_total": 3,
  "authors": [
    {
      "name": "Phuong M. Nguyen"
    },
    {
      "name": "Matt Nguyen"
    },
    {
      "name": "Loc H. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a9jg10cr4pb43.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.12450",
  "fulltext_url": null,
  "id": "a9jg10cr4pb43",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.12450",
  "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"
    ],
    [
      "arxiv",
      "q-fin.CP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.12450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a9jg10cr4pb43.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2606.12450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.12450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction",
  "title_text": "Forward-Time Black-Scholes Reconstruction via Regularized Legendre Reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/a9jg10cr4pb43",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.cam.2026.118112",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}