The paper is devoted to optimal superreplication of options under proportional transaction costs on the underlying asset. General pricing and hedging algorithms are developed. This extends previous work by many authors, which has been focused on the binomial tree model and options with specific payoffs such as calls or puts, often under certain bounds on the magnitude of transaction costs. All such restrictions are hereby removed. The results apply to European options with arbitrary payoffs in the general discrete market model with arbitrary proportional transaction costs. Numerical examples are presented to illustrate the results and their relationships to the earlier work on pricing options under transaction costs.

American options are considered in the binary tree model under small proportional transaction costs. Dynamic programming type algorithms, which extend the Snell envelope construction, are developed for computing the ask and bid prices (also known as the upper and lower hedging prices) of such options together with the corresponding optimal hedging strategies for the writer and for the seller of the option. Representations of the ask and bid prices of American options in terms risk-neutral expectations of stopped option payoffs are also established in this setting.

The paper is devoted to optimal superreplication of European options in the discrete setting under proportional transaction costs on the underlying asset. In particular, general pricing and hedging algorithms are developed. This extends previous work by many authors, which has been focused on the binomial tree model and options with specific payoffs such as calls or puts, often under certain bounds on the magnitude of transaction costs. All such restrictions are hereby removed. The results apply to options with arbitrary payoffs in the general discrete market model with arbitrary proportional transaction costs. Numerical examples are presented to illustrate the results and their relationships to the earlier work on pricing options under transaction costs.

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