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We construct a game-theoretic model characterised by perfect information, no transaction costs, in which agents can borrow at the risk-free interest rate and engage in short selling. Traders can freely and instantaneously eliminate any unexploited profit opportunities through pure arbitrage just as the efficient market hypothesis postulates they should. Yet, in this work, we put forth, in a frictionless framework, a counterexample in which the Law of One Price fails in the underlying asset markets at equilibrium. At a theoretical level, this leads both the Binomial Option Pricing Model (BOPM) and in the limit, the Black–Scholes–Merton Model (BSMM), to misprice the vanilla options written on these assets. This compelling result is pregnant with far-reaching ramifications: (i) theoretically, it establishes that no-arbitrage, while necessary, is not sufficient for any of the BOPM and BSMM to yield consistent results; (ii) practically, it crystallises the need for practitioners to rely on additional, more data-adaptive, methods of option pricing. • Shapley and Shubik meet Black, Scholes, Merton, Sharpe, Cox, Ross and Rubinstein. • The BOPM and BSMM can theoretically misprice vanilla options in a frictionless context. • No-arbitrage is a necessary but not sufficient condition for the BOPM and BSMM to give meaningful results, even theoretically. [ABSTRACT FROM AUTHOR]