  American options are studied in a general discrete market in the presence of
proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms
and constructions of hedging strategies, stopping times and martingale
representations are presented for short (seller's) and long (buyer's) positions
in an American option with an arbitrary payoff. This general approach extends
the special cases considered in the literature concerned primarily with
computing the prices of American puts under transaction costs by relaxing any
restrictions on the form of the payoff, the magnitude of the transaction costs
or the discrete market model itself. The largely unexplored case of pricing,
hedging and stopping for the American option buyer under transaction costs is
also covered. The pricing algorithms are computationally efficient, growing
only polynomially with the number of time steps in a recombinant tree model.
The stopping times realising the ask (seller's) and bid (buyer's) option prices
can differ from one another. The former is generally a so-called mixed
(randomised) stopping time, whereas the latter is always a pure (ordinary)
stopping time.
