  We extend the fundamental theorem of asset pricing to a model where the risky
stock is subject to proportional transaction costs in the form of bid-ask
spreads and the bank account has different interest rates for borrowing and
lending. We show that such a model is free of arbitrage if and only if one can
embed in it a friction-free model that is itself free of arbitrage, in the
sense that there exists an artificial friction-free price for the stock between
its bid and ask prices and an artificial interest rate between the borrowing
and lending interest rates such that, if one discounts this stock price by this
interest rate, then the resulting process is a martingale under some
non-degenerate probability measure. Restricting ourselves to the simple case of
a finite number of time steps and a finite number of possible outcomes for the
stock price, the proof follows by combining classical arguments based on
finite-dimensional separation theorems with duality results from linear
optimisation.
