Before revision:  These settings came up as a first approach of optimally exercising an option within the so-called " stock pinning " scenario. The optimal stopping boundary for this problem isproved to be the unique solution , up to certain regularity conditions, of an integral equation, which is then numerically solved by an algorithm hereby exposed. We face the case where the volatility is unspecified by providing an estimated optimal stopping boundary that, alongside with pointwise confidence intervals, provide alternative stopping rules .
After revision:  These settings came up as a first approach of optimally exercising an option within the so-called " stock pinning " , that is, when the price of the asset approaches the strike price of a highly-traded option close to its expiration date. Our main mathematical contribution is in characterizing the solution to the optimal stopping problem when the gain function includes the discount factor. We show how to numerically compute the solution and we analyze the effect of the volatility estimation on the strategy by computing the confidence curves around the optimal stopping boundary that, alongside with pointwise confidence intervals, provide alternative stopping rules .