Robust Option Pricing: Hannan and Blackwell Meet Black and Scholes

Abstract

We apply gradient strategy methods, developed in the literature on robust optimization, approachability and calibration, to develop new bounds for option prices. While this literature focuses on asymptotic performance, we provide a financial interpretation of these methods by demonstrating how the gradient strategies developed by Hannan and Blackwell to minimize asymptotic regret imply trading strategies that yield arbitrage-based bounds for option prices. These bounds are both new and robust in that they do not depend on the continuity of the stock price process, complete markets, or an assumed pricing kernel. Rather, they depend only on the realized quadratic variation of the stock price process, which can be measured and, more importantly, hedged in financial markets using existing securities. We then argue that the Hannan–Blackwell strategy is path dependent and therefore suboptimal with a finite horizon. We solve for the optimal path-independent strategy, and compare the bounds achieved with Black-Scholes.

Keywords: Option Pricing, Replication, No arbitrage pricing, approachability, calibration, robust optimization, online algorithms

JEL Classification: C70, G13