Bid Shading as Option Pricing: A Derivatives-Theoretic Framework for Optimal Bidding in First-Price Auctions with Applications to Programmatic Advertising

This paper establishes a formal equivalence between optimal bid shading in first-price sealedbid auctions and European option pricing in continuous-time finance. We show that the bidder's surplus-maximisation problem admits a representation as the payoff of a European call option written on the impression value, where the competing bid distribution plays the role of the risk-neutral measure and the optimal shaded bid corresponds to the certaintyequivalent strike price. We derive closed-form bid shading formulas under log-normal, gamma, and generalised Pareto competing bid distributions, express the optimal bid in terms of familiar Greeks (∆, Γ, V), and characterise the sensitivity of surplus to distributional parameters. Our main contribution is a two-stage estimation framework that separates the statistical problem of learning the competing bid distribution from the decision problem of computing the optimal bid, yielding a transparent, interpretable bidding strategy with provable optimality properties. Empirical evaluation on 15.4 million impressions across 9 advertising campaigns from the iPinYou real-time bidding benchmark demonstrates that the options-based approach with campaign-level calibration achieves 2.2% higher expected surplus than optimally-tuned linear shading while simultaneously increasing win rates from 70.6% to 73.6%, reaching 69.5% of oracle efficiency. We further identify a systematic truncation bias in competing bid estimation from won-auction-only data and develop a correction methodology grounded in truncated distribution theory. Extensions to budget-constrained sequential bidding via American option analogies and to portfolio-level campaign optimisation via multi-asset option theory are provided.