The super-replication problem via probabilistic methods

We study the minimal investment that is needed in order to super-replicate (i.e., hedge with certainty) continuous-time options under transaction costs. We deal with both exotic and path-independent European and American options. In all our examples we prove that the optimal strategy is the cheapest possible buy and hold. Our method is to study the problem in a discrete-time shadow market that is free of transaction costs where the options are perpetual. We also produce useful and precise estimates of potential capital gains in a transaction cost environment. We believe that our method is robust and has both theoretical and practical implications. One advantage of our approach, in contrast with the existing literature, is that we do not impose any trading strategies restrictions related to the no bankruptcy condition. Namely we allow hedging with unlimited borrowing and still the best one can do is buy and hold. Another advantage is that we do not assume that share prices are diffusions.