Option pricing in exponential Lévy models with transaction costs

We present an approach for pricing European call options in the presence of proportional transaction costs, when the stock price follows a general exponential Lévy process. The model is a generalization of the celebrated 1993 work of Davis, Panas and Zariphopoulou, in which the value of the option is defined as the utility indifference price. This approach requires the solution of two stochastic singular control problems in a finite horizon that satisfy the same Hamilton–Jacobi–Bellman equation with different terminal conditions. We introduce a general formulation for these portfolio selection problems, and then focus on a special case in which the probability of default is ignored. We solve the optimization problems numerically 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.