{
  "abstract": "A $(d+1)$-state lattice is the smallest complete one-step approximation of a $d$-dimensional Brownian market. Its centered quadratic increments must lie in the span of the traded innovations. We study the resulting Gamma-to-Delta aliasing through the linear operator $\\mathcal{A}_ξ(H)=\\mathbb{E}[(ξ^\\top Hξ-\\operatorname{tr} H)ξ]$, $H\\in\\operatorname{Sym}_d$, for centered isotropic innovations $ξ$. For quadratic claims this operator is exactly the difference, scaled by $\\sqrt{h}/2$, between the lattice's replicating exposure and its price-sensitivity Delta. Applying the lattice replication rule to a Brownian market adds the tracking-error variance $hT|\\mathcal{A}_ξ(H)|^2/4$ relative to ordinary discrete Delta hedging. We prove that the smallest worst-case aliasing over all minimally complete lattices is $\\sqrt{d-1}$, uniquely attained by the equal-probability regular simplex. For positive-semidefinite and rank-one unit-Frobenius Gamma classes, the sharp minimax value is instead $(d-1)/\\sqrt{d}$, with the same unique optimizer. We also describe the corresponding orthogonal quadratic residual in nonminimal markets and the leading implementation error for smooth claims.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06692",
  "author_total": 1,
  "authors": [
    {
      "name": "Akihiro Tanaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8axsw10krg7et.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06692",
  "fulltext_url": null,
  "id": "8axsw10krg7et",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06692",
  "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",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8axsw10krg7et.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Cost of Minimal Completeness: Gamma-to-Delta Aliasing in Discrete Markets",
  "title_text": "The Cost of Minimal Completeness: Gamma-to-Delta Aliasing in Discrete Markets",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/8axsw10krg7et",
  "venue": "arXiv",
  "versions": []
}