  We consider a new family of derivatives whose payoffs become strictly
positive when the price of their underlying asset falls relative to its
historical maximum. We derive the solution to the discretionary stopping
problems arising in the context of pricing their perpetual American versions by
means of an explicit construction of their value functions. In particular, we
fully characterise the free-boundary functions that provide the optimal
stopping times of these genuinely two-dimensional problems as the unique
solutions to highly non-linear first order ODEs that have the characteristics
of a separatrix. The asymptotic growth of these free-boundary functions can
take qualitatively different forms depending on parameter values, which is an
interesting new feature.
