The Black-Scholes option pricing methodology requires that the model for the price of the underlying asset be completely specified. Often the underlying price is taken to be a geometric Brownian motion with a constant, known volatility. In practice one does not know precise values of parameters such as the volatility, and estimates from historical prices or implied volatilities must be used instead. In this paper optimal hedging strategies are constructed when the volatility of the asset price is misspecified. Optimality refers to maximizing the utility of the investor in a worst-case volatility scenario.