The first new version is a call option where the option seller pays the minimum price (in inflated dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the the selling time and the delivery time (to be chosen by the seller). This option is the dual of the put option where the option buyer receives the maximum price (in discounted dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the buying time and the exercise time (to be chosen by the buyer).

We are concerned with the optimal stopping time for this problem and its relation to the the rational (fair) value of the option. It transpires that both option prices above can be obtained by solving the corresponding free-boundary (Stephan) problem.

In the second new version we assume that optimal stopping times should be selected before (default) time, when prices reach some specified small value.

Please feel free to bring this to the attention of interested colleagues.