We present an approach for pricing a European call option in presence of proportional transaction costs, when the stock price follows a general exponential L\'evy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou (1993), where the value of the option is obtained using the concept of utility indifference price. This requires to solve two stochastic singular control problems in finite time, satisfying the same Hamilton-Jacobi-Bellman equation and with different terminal conditions. We solve numerically the continuous time optimization problem using the Markov chain approximation method, and consider the underlying stock following an exponential Merton jump-diffusion process. This model takes into account the possibility of portfolio bankruptcy. We show numerical results for the simpler case of an infinitely rich investor, whose probability of default can be ignored. Option prices are obtained for both the writer and the buyer.

中文翻译：

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具有交易成本的指数Lévy模型中的期权定价

当股票价格遵循一般的指数L'evy过程时，我们提出一种在比例交易成本存在下对欧式看涨期权定价的方法。该模型是Davis，Panas和Zariphopoulou（1993）的著名著作的概括，其中期权的价值是使用效用无差别价格的概念获得的。这就需要在有限的时间内解决两个随机奇异控制问题，满足相同的Hamilton-Jacobi-Bellman方程并且具有不同的终止条件。我们使用马尔可夫链近似方法用数值方法解决连续时间优化问题，并考虑遵循指数默顿跳跃扩散过程的基础股票。该模型考虑了投资组合破产的可能性。我们显示了一个无限富裕投资者的简单情况的数值结果，其违约概率可以忽略。期权价格是同时由作者和买方获得的。