Options with multiple reset rights

The reset right embedded in a derivative refers to the feature that the holder can alter certain terms in the derivative contract according to some preset rules. In this paper, we consider options that allow the holder to reset the strike price with preset number of times at any moment during the life of the option. The determination of the optimal reset policies adopted by the holder leads to a free boundary value problem. We explore how the critical asset value at which the holder should exercise the reset right optimally depends on the number of reset rights remaining, the relative magnitude of the riskless interest rate and dividend yield, the original strike price set at initiation, etc. In particular, we examine the asymptotic behaviors of the optimal reset policies at infinite time to expiry and the existence of threshold time earlier than which the holder should never shout.