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.