This paper addresses the problem of pricing and hedging options under the presence of transaction costs. One of the known approaches to this problem is superhedging. This paper proposes an idea of hedging sets that allows implement the superhedging approach in a new way that is more efficient in comparison with the methods present. The idea allows use of the superhedging approach to price and hedge both European and American options under the presence of transaction costs.

The superhedging approach states the problem to find the cheapest portfolio for which there exists a trading strategy with consumption such that for every possible outcome the value of the portfolio at the time of the option execution is not less than the value of the option (when the short position in the option is considered).

The notion of a hedging set is introduced in the paper to solve this problem. The discrete time economy is considered with price movements of an underline asset being modeled by a recombining tree. At the current node of the tree a hedging set is defined as a set of the portfolios that can hedge the considered option in the sense of superhedging. It’s easy to build a hedging set for every tree node at the time of the option execution. Also it is discovered that under proportional transaction costs it is possible to build hedging sets for the current nodes of the tree given hedging sets for the next nodes of the tree. Then the solution to the stated problem comes from building hedging sets backward the tree and then finding the cheapest portfolio in the first hedging set.

The superhedging approach and the idea of hedging sets are given in detail in the paper. Pricing and hedging European and American options under the presence of symmetric and asymmetric proportional transaction costs is discussed and the appropriate numerical examples are given.