非平滑障碍的实物期权问题

Subash Acharya, A. Bensoussan, D. Rachinskii, A. Rivera
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引用次数: 0

摘要

考虑一个实物期权问题,该问题被视为随机最优控制问题。投资策略具有控制作用,包括扩大(投资)项目的一次性选择和放弃(终止)项目的一次性选择。为了实现利润期望值的最大化,投资的时机和金额以及终止时间都是需要优化的参数。这个随机优化问题相当于解决一个一维的确定性变分不等式,以及相关的障碍问题。因为我们同时考虑了停止和扩张的选择,以及扩张的固定和可变成本,所以障碍并不平坦。由于缺乏平滑性,我们使用弱解的概念。然而,这样的解决方案可能不会导致直接的投资策略。因此,我们进一步考虑基于阈值的强($C^1$)解。我们给出了一类一维非光滑障碍物变分不等式解存在的充分条件。当应用于实物期权问题时,这些充分条件产生了一个简单的最优投资策略,并根据阈值定义了停止时间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Real Options Problem with Non-Smooth Obstacle
We consider a real options problem, which is posed as a stochastic optimal control problem. The investment strategy, which plays the role of control, involves a one-time option to expand (invest) and a one-time option to abandon (terminate) the project. The timing and amount of the investment and the termination time are parameters to be optimized in order to maximize the expected value of the profit. This stochastic optimization problem amounts to solving a deterministic variational inequality in dimension one, with the associated obstacle problem. Because we consider both cessation and expansion options and fixed and variable costs of expansion, the obstacle is non-smooth. Due to the lack of smoothness, we use the concept of a weak solution. However, such solutions may not lead to a straightforward investment strategy. Therefore, we further consider strong ($C^1$) solutions based on thresholds. We propose sufficient conditions for the existence of such solutions to the variational inequality with a non-smooth obstacle in dimension one. When applied to the real options problem, these sufficient conditions yield a simple optimal investment strategy with the stopping times defined in terms of the thresholds.
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