We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

我们在具有不对称信息的 Black-Merton-Scholes 模型的扩展中推导出与永久美国可提取标准和回溯看跌期权定价相关的最优止损问题的封闭式解决方案。假设合约是由他们的作者在最后命中时间撤回的，因为其在无限时间间隔内的运行最大值或最小值的潜在风险资产价格不是关于可观察过滤的停止时间。我们表明，最佳执行时间是资产价格过程根据其运行最大值或最小值的当前值首次达到某些较低或较高随机边界的时间。该证明基于通过平滑拟合和正反射条件将原始的必然二维最优停止问题简化为相关的自由边界问题及其解决方案。我们证明了最优运动边界是一些一阶非线性常微分方程的极大解和极小解。