The optimal pricing of goods, especially when they are new and the innovating firm is a monopolist, must proceed without precise knowledge of the demand curve. This paper provides a pricing method with a relative robustness guarantee by maximizing a performance index which amounts to a worst-case ratio of the obtained payoff to the best possible payoff. Assuming monotonicity and complementarity of demand in price and the unknown demand parameter, the performance index is fully determined by its behavior at the boundary of the parameter space. This allows for an efficient computation of an optimal robust price. In the linear case, which can also be used for nonlinear demand with bounded slope, the method provides a simple closed-form solution. A comparison with the standard worst-case payoff criterion reveals substantial improvements in both absolute and relative performance, at only a small cost relative to the maximized expected profit.