[量化金融] 美式资产看跌期权行权边界的规律性 分立股息 [推广有奖]

摘要翻译：
我们分析了美式看跌期权的最优行权边界的规律性，当标的资产在期权有效期内的一个已知时间$T_D$支付离散股利时。除权资产价格过程遵循Black-Scholes动力学，股利金额是股利日之前除权资产价格的确定性函数。相关的最优停止问题的解可以用一个最优运动边界来表征，与没有红利的情况相比，该最优运动边界可能不再是单调的。本文证明了当红利函数为正且为凹时，边界在$t_d$的左邻域内不增加，并且随着时间趋向于$t_d^-$而趋向于$0$。当股利函数在零的邻域内为线性时，我们证明了在$t_d$的左邻域内运动边界的连续性和一个高接触原理。当它全局线性时，证明了边界的右连续性和高接触原理全局成立。最后，我们展示了在这种情况下如何将前面所有的结果推广到多个股利支付日期。
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英文标题：
《Regularity of the Exercise Boundary for American Put Options on Assets
  with Discrete Dividends》
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作者：
Benjamin Jourdain (CERMICS), Michel Vellekoop
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最新提交年份：
2010
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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英文摘要：
  We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, may no longer be monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary is non-increasing in a left-hand neighbourhood of $t_d$, and tends to $0$ as time tends to $t_d^-$ with a speed that we can characterize. When the dividend function is linear in a neighbourhood of zero, then we show continuity of the exercise boundary and a high contact principle in the left-hand neighbourhood of $t_d$. When it is globally linear, then right-continuity of the boundary and the high contact principle are proved to hold globally. Finally, we show how all the previous results can be extended to multiple dividend payment dates in that case.
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PDF链接：
https://arxiv.org/pdf/0911.5117