This paper treats the problem of pricing European options in a Black-Scholes model with proportional costs on stock transactions. The authors define the option writing price as the difference between the utilities achievable by going into the market to hedge the option and by going into the market on one's own account. Without transaction costs, this definition is shown to yield the usual Black-Scholes price. To compute the option price under transaction costs, one has to solve two stochastic control problems, corresponding to the two utilities compared above. The value functions of these problems are shown to be the unique viscosity solutions of one fully nonlinear quasi-variational inequality, with two different boundary conditions. This is used to obtain a convergent discretization scheme based on the familiar binomial approximation of the stock price process. The results are illustrated by several numerical computations.