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

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 $\mathcal{A}_\xi(H)=\mathbb{E}[(\xi^\top H\xi-\operatorname{tr} H)\xi]$, $H\in\operatorname{Sym}_d$, for centered isotropic innovations $\xi$. 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 $hT|\mathcal{A}_\xi(H)|^2/4$ 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.