Option Pricing with Transaction Costs: the Superhedging

In this paper I address the problem of pricing options in the presence of transaction costs. A known approach to this problem is a so-called superhedging approach. I suggest an idea of hedging sets and, based on it, a new algorithm to implement the superhedging approach. As far as I know, only European options were discussed in the literature on pricing derivatives when there are transaction costs. The proposed algorithm, however, can be applied to price American options as well. It is also more efficient compared with the previous methods. The superhedging approach is based on the problem of finding the cheapest portfolio now for which there is a trading strategy with consumption such that for every possible outcome the value of the portfolio at the time of the option exercise is not less than the value of the option (when the short position in the option is considered). A notion of a hedging set is introduced in the article to help to solve this problem. A discrete time economy is considered with price movements of an underlying asset being modeled by a recombining tree. At the current tree node a hedging set is defined as a set of portfolios that can hedge the considered option in the sense of superhedging. It is easy to build hedging sets for every tree node at the time of the option exercise. I also show how to build hedging sets for the current tree nodes given hedging sets of the next tree nodes when there are proportional transaction costs. Thus 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. I discuss in detail the idea of hedging sets and how to price European and American options under the presence of proportional transaction costs. Appropriate numerical examples are also given. I believe that the idea of hedging sets is versatile as it allows using a tree of any type to model price movements of an underlying asset when pricing and hedging derivatives using the superhedging approach. Also, multidimensional hedging sets may allow pricing and hedging derivatives that depend on several random factors