This paper discusses the valuation of piecewise linear barrier options that generalize classical barrier options. We establish formulas for joint probabilities of the logarithmic returns of the underlying asset and its partial running maxima when the process has a piecewise constant drift. In particular, we show that our results embrace the famous reflection principle as a special case, and that our established proposition delivers useful scalability for computing desired probabilities related to various types of barriers. We derive the closed-form prices of piecewise linear barrier options under the Black–Scholes framework, which are obtainable with little effort by relying on the derived probabilities. In addition, we provide numerical examples and discuss how option prices respond to several types of piecewise linear barriers.

本文讨论了对经典线性障碍期权进行概括的分段线性障碍期权的估值。当过程具有分段恒定漂移时，我们为基础资产的对数收益及其部分运行最大值的联合概率建立公式。尤其是，我们证明了我们的结果将著名的反射原理作为一种特例，并且我们已经建立的命题为计算与各种类型障碍相关的期望概率提供了有用的可伸缩性。我们推导出Black-Scholes框架下的分段线性障碍期权的闭式价格，而该价格可以通过依靠得出的概率而毫不费力地获得。此外，我们提供了数值示例，并讨论了期权价格如何响应几种类型的分段线性障碍。