We study standard option prices in the one-dimensional Avellaneda-Lyons model with unknown volatility and no interest rate. We show that the price is that of an american option in the Black-Scholes model at the maximum volatility. Relying on the viscosity formulation of the problem and the qualitative property of the boundary between the regions of concavity and convexity of the solution we manage to design explicit solutions in some cases. The key idea is an embedding of the initial problem in a standard european one related to another payoff.