The Mathematics of the NASDAQ and NYSE Closing Auctions

Closing auctions transform heterogeneous trading instructions into a common benchmark price, but a single price does not imply that every submitted order executes. This paper develops a mathematical framework for understanding the Nasdaq and New York Stock Exchange closing auctions, separating institutional rules, a homogeneous clearing benchmark, and economic models. For unrestricted matching on a tick grid, we establish quasiconcavity of executable volume, characterize interval structure in successive price selection stages, and provide an exact algorithm with sorting complexity. We define competitive prices as those supporting complete execution of strictly better priced interest, prove existence under market order absorption conditions, and show that a competitive refinement preserves maximum volume and minimum absolute imbalance. Execution accounting separates ineligible orders, excess eligible interest, and a compatibility deficit arising from restricted counterparties. A network cut characterization identifies when aggregate demand and supply are insufficient to determine feasible execution. The institutional analysis distinguishes Nasdaq's residual order price criterion from a simpler midpoint selector and explains the coexistence of parity and time allocation within the NYSE close. Additional results characterize capped equal allocation, competitive price endpoint sensitivity, and a linear supply function equilibrium. We distinguish expanding risk capacity from increased competition, retain background imbalance in the calculation of marking incentives, and derive inventory constrained price stabilization in a quadratic DMM model. A reproducible paired experiment evaluates 20,000 synthetic books under two exact tick bands. Independent grid enumeration and event sweep implementations agree across 40,000 auction evaluations. Exhaustive checks cover 59,049 small books, while event classifications distinguish absorption failure, band exclusion, and noncompetitive selection. Reported uncertainty intervals quantify simulation variability rather than uncertainty about actual markets. The resulting framework provides mathematical guarantees and verified computational benchmarks, while preserving the distinction between theoretical mechanisms, exchange specifications, and empirical claims requiring market data. Executable code accompanies these results.