Price matching for multiple rescindable options and European options

We study a modification of an American option such that the option holder can exercise the option early before the expiration, and he or she can revert later this decision to exercise a number of times. This feature gives additional flexibility and risk protection for the option holder. We found that, for the Black-Scholes market model, the price of call options with this feature is the same as for European call, i.e. 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 standard American options.