A passport option grants its holder the right to engage ina short/long trading strategy of his own choice, while obligating the option writer to cover any net losses on the strategy. Using appropriate shifts of numeraire, it is shown that the passport option price is the solution to a Markov control problem. The authors identify the optimal trading strategy and derive a nonlinear PDE satisfied by the asset-deflated option price. A closed-form pricing formula is derived in a special case. For the general case, the paper illustrates how a Crank-Nicholson finite-difference scheme can be used to price the option. The numerical scheme is shown to be applicable to a number of variations on the basic option contract, including American exercise and time-discrete trading strategies.