We apply methods developed in the literature initiated by Hannan and Blackwell on robust optimization, approachability and calibration, to price financial securities. Rather than focus on asymptotic performance, we show how gradient strategies developed to minimize asymptotic regret imply financial trading strategies that yield arbitrage-based bounds for option prices. These bounds are new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. They depend only on the realized quadratic variation of the price process, which can be measured and, importantly, hedged in financial markets using existing securities. Our results also apply directly to a new class of options called timer options. Finally, we argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We improve it by solving for the optimal path-independent strategy, and compare the resulting bounds with Black–Scholes.
