Abstract We find the closed form formula for the price of the perpetual American lookback spread option, whose payoff is the difference of the running maximum and minimum prices of a single asset. We solve an optimal stopping problem related to both maximum and minimum. We show that the spread option is equivalent to some fixed strike options on some domains, find the exact form of the optimal stopping region, and obtain the solution of the resulting partial differential equations. The value function is not differentiable. However, we prove the verification theorem due to the monotonicity of the maximum and minimum processes.

中文翻译：

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美国回溯价差期权的定价

摘要 我们找到了永久美式回顾价差期权价格的闭式公式，其收益是单个资产的运行最高和最低价格之差。我们解决了一个与最大值和最小值相关的最优停止问题。我们证明了价差期权等效于某些域上的某些固定执行期权，找到最佳停止区域的确切形式，并获得所得偏微分方程的解。价值函数是不可微的。然而，由于最大和最小过程的单调性，我们证明了验证定理。