In 1993, Davis, Panas and Zariphopoulou provided a detailed discussion on the problem of European option pricing with transaction costs when the price of the underlying follows a diffusion process. We consider a similar problem here and develop a new computational algorithm for the case when the underlying price follows a geometric Lévy process. Using an approach based on maximization of the expected utility of terminal wealth, we transform the option-pricing problem into stochastic optimal control problems, and argue that the value functions of these problems are the solutions of a free-boundary problem (in particular, a partial integrodifferential equation) under different boundary conditions. To solve the singular stochastic control problems associated with utility maximization and compute the value function and no-transaction boundaries, we develop a coupled backward-induction algorithm based on the connection of the free-boundary problem to an optimal stopping problem. Numerical examples of option pricing under a double exponential jump–diffusion model are also provided.