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- 作者：Neofytos Rodosthenous ; Mihail Zervos
- 关键词：Optimal stopping ; Running maximum process ; Variational inequality ; Two ; dimensional free ; boundary problem ; Separatrix
- 刊名：Finance and Stochastics
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：21
- 期：1
- 页码：157-186
- 全文大小：1238KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Quantitative Finance; Finance, general; Statistics for Business/Economics/Mathematical Finance/Insurance; Economic Theory/Quantitative Economics/Mathematical Methods; Probability Theory and Stochastic
- 出版者：Springer Berlin Heidelberg
- ISSN：1432-1122
- 卷排序：21
文摘
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.