Deterministic Arbitrage, Bid–Ask Option Bounds, Localized Arbitrage Portfolios, and Arbitrage-Consistent Projection

We develop a deterministic, model-free framework for detecting and correcting static arbitrage in European call–put books. An arbitrage is a financed portfolio profit that is nonnegative in every admissible state and positive in at least one state. At a common maturity, the continuous piecewise-affine payoff geometry reduces verification over [0, ∞) to finitely many breakpoint and tail-slope inequalities. The resulting linear programmes construct arbitrage portfolios, including portfolios positive on prescribed target regions. Calls and puts enter jointly with cash and the stock, allowing mixed-instrument inconsistencies to be detected directly. Executable bid–ask bounds follow from the same payoff comparisons. Arbitrage-Consistent Projection adjusts raw or predicted quotes by minimizing a weighted correction subject to a strictly positive dual pricing certificate; under bid–ask spreads, its shadow prices lie within the corrected quotes. For several maturities, physical settlement and passive carry yield an exact finite reduction by pre-final exercise patterns and final-price breakpoints. This also gives calendar restrictions and a multi-maturity projection formulation.