In this paper we consider some insurance policies related with drawdown and drawup events of log-returns for an underlying asset modeled by a spectrally negative geometric L\'evy process. We consider four contracts among which three were introduced in Zhang et al. (2013) for a geometric Brownian motion. The first one is an insurance contract where protection buyer pays a constant premium until the drawdown of fixed size of log-returns occurs. In return he/she receives certain insured amount at the drawdown epoch. Next insurance contract provides protection from any specified drawdown with a drawup contingency. This contract expires early if certain fixed drawup event occurs prior to fixed drawdown. The last two contracts are extensions of the previous ones by additional cancellable feature which allows an investor to terminate the contract earlier. We focus on two problems: calculating the fair premium $p$ for the basic contracts and identifying the optimal stopping rule for the polices with cancellable feature. To do this we solve some two-sided exit problems related with the drawdown and the drawup of spectrally negative L\'evy processes which is of own scientific interest. We also heavily rely on a theory of optimal stopping.
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