  In this paper, we study a pricing problem of the multiple reset put option,
which allows the holder to reset several times a current strike price to obtain
an at-the-money European put option. We formulate the pricing problem as a
multiple optimal stopping problem, then reduce it to a sequence of single
optimal stopping problems and study the associated free-boundary problems. We
solve this sequence of problems by induction in the number of remaining reset
rights and exploit probabilistic arguments such as local time-space calculus on
curves. As a result, we characterize each optimal reset boundary as the unique
solution to a nonlinear integral equation and derive the reset premium
representations for the option prices. We propose that the multiple reset
options can be used as cryptocurrency derivatives and an attractive alternative
to standard European options due to the extreme volatility of underlying
cryptocurrencies.
