A passport option, as introduced and marketed by Bankers Trust, is a call option on the balance of a trading account. The strategy that this account follows is chosen by the option holder, subject to position limits.
We derive a simplified form for the price of the passport option using the elegant techniques of coupling and local times. A key insight is that the Skorokhod Lemma allows us to prove that there is a direct relationship between the prices of passport and lookback options. Explicit calculations are provided in the case where the underlying is an exponential Brownian motion, but the methods apply more generally.