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The Cost of Minimal Completeness: Gamma-to-Delta Aliasing in Discrete Markets

  • Akihiro Tanaka
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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 𝒜ξ(H)=𝔼[(ξ⊤ Hξ−tr H)ξ], H∈Symd, for centered isotropic innovations ξ. For quadratic claims this operator is exactly the difference, scaled by √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|𝒜ξ(H)|2/4 relative to ordinary discrete Delta hedging. We prove that the smallest worst-case aliasing over all minimally complete lattices is √(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)/√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.

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Cite

BibTeXRIS
BibTeX
@misc{tanaka2026cost,
  title = {{The Cost of Minimal Completeness: Gamma-to-Delta Aliasing in Discrete Markets}},
  author = {Akihiro Tanaka},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06692},
  eprint = {2611.06692},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06692},
}
RIS
TY  - UNPB
TI  - The Cost of Minimal Completeness: Gamma-to-Delta Aliasing in Discrete Markets
AU  - Akihiro Tanaka
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06692
UR  - https://arxiv.org/abs/2611.06692
AB  - 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 𝒜_ξ(H)=𝔼[(ξ^⊤ Hξ−tr H)ξ], H∈Sym_d, for centered isotropic innovations ξ. For quadratic claims this operator is exactly the difference, scaled by √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|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 √(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)/√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.
ER  -