We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kind of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. We obtain a weak sufficient condition for strong duality and existence for the dual problem to hold, for options defined by general payoff functions. Explicit bounds are provided for the call option. Finally, we provide a numerical example.
