渐进扩大滤子内部人模型中的永久美式标准期权与回望期权
Perpetual American Standard and Lookback Options in Insider Models with Progressively Enlarged Filtrations
AI总结：
本文在渐进扩大滤子的内部人信息模型下，针对永久美式标准期权和回望期权定价问题推导出闭式解，将最优停时问题约化为三维自由边界问题，并利用法向反射、法向进入及光滑贴合条件刻画最优执行边界。
摘要：
我们在对Black-Merton-Scholes模型进行渐进扩大滤子扩展的框架下，推导出与永久美式标准期权和回望看跌、看涨期权定价相关的最优停时问题的闭式解。更具体地说，内部人可获得的信息由布朗滤子建模，该滤子通过基础风险资产价格在无限时间区间上的全局最大值或最小值发生的时间进行渐进扩大，而这些时间在基础风险资产生成的滤子中并不是停时。我们证明，最优执行时间分别是资产价格过程首次触及下随机边界或上随机边界的时刻，具体取决于在全局最大值或最小值时间已发生的情况下，其运行最大值或运行最小值的当前值。证明基于将原始问题约化为必然的三维最优停时问题以及等价的自由边界问题。我们应用法向反射条件或法向进入条件以及价值函数的光滑贴合条件，将候选边界分别刻画为相关一阶非线性常微分方程和超越算术方程的最大解或最小解。
英文摘要：
We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged filtrations. More specifically, the information available to the insider is modelled by Brownian filtrations progressively enlarged with the times of either the global maximum or minimum of the underlying risky asset price over the infinite time interval, which is not a stopping time in the filtration generated by the underlying risky asset. We show that the optimal exercise times are the first times at which the asset price process reaches either lower or upper stochastic boundaries depending on the current values of its running maximum or minimum given the occurrence of times of either the global maximum or minimum, respectively. The proof is based on the reduction of the original problems into the necessarily three-dimensional optimal stopping problems and the equivalent free-boundary problems. We apply either the normal-reflection or the normal-entrance conditions as well as the smooth-fit conditions for the value functions to characterise the candidate boundaries as either the maximal or minimal solutions to the associated first-order nonlinear ordinary differential equations and the transcendental arithmetic equations, respectively.
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