Journal of Computational Finance

Risk.net

Option pricing in exponential Lévy models with transaction costs

Nicola Cantarutti, Manuel Guerra, João Guerra and Maria do Rosário Grossinho

  • The model uses the concept of indifference pricing to compute option prices in a market with proportional transaction costs.
  • The stock price is assumed to follow an exponential Lévy process. Numerical examples are provided for diffusion, Merton and Variance Gamma processes.
  • The model is formulated as a singular stochastic control problem. This problem is then discretized using the Markov chain approximation method.

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.

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