| Summary: | We suggest a modification of an American option such that the option holder can exercise the option early before the expiration and can revert later this decision to exercise; it can be repeated a number of times. This feature gives additional flexibility and risk protection for the option holder. A classification of these options and pricing rules are given. We found that the price of some call options with this feature is the same as for the European call. This means that the additional flexibility costs nothing, similarly to the situation with American and European call options. For the market model with zero interest rate, the price of put options with this feature is also the same as for the standard European put options. Therefore, these options can be more competitive than the standard American options
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