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作者：
Neofytos Rodosthenous;M. Zervos
我们考虑一类新的衍生产品，当其标的资产的价格相对于其历史最大值福尔斯时，其收益变为严格正的。我们通过显式构造其价值函数，推导出在为其永久美国版本定价时出现的自由裁量停止问题的解决方案。特别是，我们充分考虑了自由边界函数，提供了这些真正的二维问题的最佳停止时间作为唯一的解决方案，高度非线性的一阶常微分方程，具有分界线的特点。这些自由边界函数的渐近增长可以根据参数值采取不同的形式，这是一个有趣的新特征。
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 nonlinear 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.