  We present an approach for pricing European call options in presence of
proportional transaction costs, when the stock price follows a general
exponential L\'{e}vy process. The model is a generalization of the celebrated
work of Davis, Panas and Zariphopoulou (1993), where the value of the option is
defined as the utility indifference price. This approach requires the solution
of two stochastic singular control problems in finite horizon, satisfying the
same Hamilton-Jacobi-Bellman equation, with different terminal conditions. We
introduce a general formulation for these portfolio selection problems, and
then we focus on the special case in which the probability of default is
ignored. We solve numerically the optimization problems using the Markov chain
approximation method and show results for diffusion, Merton and Variance Gamma
processes. Option prices are computed for both the writer and the buyer.
