DIFFUSION TRANSFORMATIONS, BLACK-SCHOLES EQUATION AND OPTIMAL STOPPING

• 成果类型: Article
• 署名作者: Cetin, Umut
• 署名单位: University of London; London School Economics & Political Science
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1385
• 发表日期: 2018
• 页码: 3102-3151
• 关键词: one-dimensional diffusions; strict local martingales; bubbles; approximation; EXIT

摘要

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [Probab. Theory Related Fields 164 (2016) 1027-1069] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation, we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.