  We propose a general approximation method for determining optimal trading
strategies in markets with proportional transaction costs, with a polynomial
approximation of the residual value function. The method is exemplified by
several problems from optimally tracking benchmarks, hedging the Log contract,
to maximizing utility from terminal wealth. Strategies are also approximated by
practically executable, discrete trades. We identify the necessary trade-off
between trading frequency and trade sizes to have satisfactory agreement with
the theoretically optimal, continuous strategies of infinite activity.
