On Hannan and Blackwell's Approachability and Options - A Game Theoretic Approach for Option Pricing

We study the link between the game theoretic notion of approachability or “regret minimization” and robust option pricing. We demonstrate how trading strategies that are based on approachability and minimize regret over finite horizon also imply robust upper bounds for the prices of European call options. These bounds are based on no arbitrage and are robust in that they require only minimal assumptions regarding the stock price process. We then focus on the optimal bounds and solve for the optimal volatility-based bounds in closed-form, which in turn implies the optimal regret-minimizing trading strategy. The bounds we obtain seem to be empirically relevant as they resemble option price patterns observed in practice.