This paper introduces a new class of barrier options and its variations. We call the new class of options as two-asset alternating barrier options, since we consider alternating barrier levels for two underlying assets. The alternating barrier levels are placed in the sub-periods of the option’s lifetime; each being applied to one of the two underlying assets. We also consider vertical branches of the barrier, which are termed as icicles. The alternating barrier with icicles can be often seen as an embedded form in various equity-linked financial products. To price such new options, we obtain the joint distribution of two underlying asset prices at an intermediate time point and the maturity, along with their partial maximums under the Black–Scholes model. This joint distribution plays a critical role in the derivation of the pricing formulas for alternating barrier options and their variants. As in ordinary barrier options, we consider eight types of alternating barrier options and derive their explicit option pricing formulas. To our knowledge, the pricing formulas for these options have never been obtained explicitly in the literature even under the Black–Scholes model. We also examine an autocallable equity-linked investment product to derive its explicit pricing formula. Our results are illustrated with numerical examples, showing the effect of different barrier levels and different values of correlation coefficient between two underlying asset prices.

中文翻译：

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对带有冰柱及其变体的两资产交替障碍期权定价

本文介绍了一类新的障碍选择及其变化。我们将新的期权类别称为两资产交替障碍期权，因为我们考虑了两个基础资产的交替障碍水平。交替的势垒级别位于选件寿命的子周期中。每个都应用于两个基础资产之一。我们还考虑了屏障的垂直分支，称为冰柱。冰柱交替出现的障碍通常被视为各种股票挂钩金融产品中的一种嵌入形式。为了对这些新的期权定价，我们获得了两个基本资产价格在中间时间点和到期时间的联合分布，以及在Black-Scholes模型下的部分最大收益。这种联合分布在推导交替障碍期权及其变式的定价公式中起着至关重要的作用。与普通障碍期权一样，我们考虑八种交替障碍期权，并推导出它们的显式期权定价公式。据我们所知，即使在Black-Scholes模型下，也从未在文献中明确获得这些期权的定价公式。我们还研究了一种可赎回股票挂钩投资产品，以得出其明确的定价公式。我们的结果通过数值示例进行说明，显示了两个基本资产价格之间不同的障碍水平和不同的相关系数值的影响。我们考虑了八种交替障碍期权，并推导了它们的显式期权定价公式。据我们所知，即使在Black-Scholes模型下，也从未在文献中明确获得这些期权的定价公式。我们还研究了可赎回股票挂钩投资产品，以得出其明确的定价公式。我们的结果通过数值示例进行说明，显示了两个基本资产价格之间不同的障碍水平和不同的相关系数值的影响。我们考虑了八种交替障碍期权，并推导了它们的显式期权定价公式。据我们所知，即使在Black-Scholes模型下，也从未在文献中明确获得这些期权的定价公式。我们还研究了一种可赎回股票挂钩投资产品，以得出其明确的定价公式。我们的结果通过数值示例进行说明，显示了两个基本资产价格之间不同的障碍水平和不同的相关系数值的影响。