We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

我们针对 Black-Merton-Scholes 模型中关于最大回撤和浮动执行的最大回撤的永久美式双重回溯看跌期权和看涨期权的定价问题提出了封闭式解决方案。结果表明，最佳行权时间是基础风险资产价格过程根据其运行最大值或最小值的当前值以及最大回撤或最大回撤的当前值而达到某些较低或较高随机边界的第一次时间。证明是基于将原始双最优停止问题简化为三维连续马尔可夫过程的单最优停止问题的适当序列。后一个问题通过最优停止边界和三维状态空间边缘的值函数的平滑拟合和法向反射条件作为等效自由边界问题求解。我们表明，最佳运动边界被确定为相关算术方程组的唯一解或适当的一阶非线性常微分方程的最小和最大解。