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目前场外期权产品较为火爆，尤其是成为鲨鱼鳍期权的产品，在一些券商的交易平台上均有报价。其实是这是一类障碍期权。障碍期权是路径依赖期权，其回报取决于资产价格的已实现路径；当资产价格过高或者过低时，合约的某些条款被触发。例如，如果期权到期之前所有时刻标的资产都没有以
或者更高的价格进行交易，那么向上敲出（knock-out）看涨期权在到期日的支付为常见的
。在这个例子中，如果标的资产价格初级了这个水平（从下触及），那么期权将被“敲出”（knock-out），变得一文不值（payoff=0）。除了像这样的“敲出”期权，也存在“敲入”（knock-in）期权，它们只有在资产价格触及特定水平时才有支付，否则在到期时毫无价值。
障碍期权存在的原因很多。购买者可以用它来对冲具有类似特征的特殊现金流。其购买者往往对市场未来方向有着非常精准的判断。如果他们想获得看涨期权式的回报，但相信在期权到期日前标的资产价格上升幅度有限，不愿意支付全部上行潜力的费用，那么他可以购买向上敲出的期权。由于上涨幅度严格受限，障碍期权比相同的普通期权便宜。如果他的判断是对的并且障碍并未被触及，他将得到他希望的支付。障碍同当前资产价格越接近，期权被敲出的可能性就越大，那么合约就越便宜。
相反地，相信某障碍水平会实现的人讲购买敲入期权。同样地，这个期权也会比相应的普通期权便宜。
障碍期权的不同类型
障碍期权主要有两种主要类型：
- 敲出期权（knock-out option），只有在到期日前从未触及障碍水平下才能产生支付。如果障碍触及称期权被敲出了（knocked-out）。
- 敲入期权（knock-in option），只有在到期日达到障碍水平才产生支付。如果障碍被触及称期权被敲入（knocked in）了。
根据障碍水平与标的资产初始价格的相对位置可以进一步对障碍期权进行分类。
- 如果障碍在资产初始价格以上，被称作向上（up）期权。
- 如果障碍在资产舒适价格以下，被称作乡下（down）期权。
最终到期日时收到回报（payoff）如下：
回报均是常见的形式，如看涨、看跌、二值等。
障碍期权的定价方法
障碍期权是弱路径依赖的，这和亚式期权是有区别的。这里用
表示障碍触及前期的价值，这个价值将满足Black-Scholes方程：
障碍特征的细节将通过边界条件的具体形式来体现。
“敲出”障碍期权
如果标的资产价格触及了障碍，“敲出”障碍期权将一文不值。这个特征使得障碍水平为
的向上敲出障碍期权有边界条件。
，这里t<T
我们必须在的范围求解Black-Scholes方程，方程在S=S_u的边界条件如上，还有一个与障碍未被触及时的关于回报的终值条件。对于看涨期权有：
如果一个有障碍为
的向下敲出障碍期权，则在 范围内求解Black-Scholes方程，其边界条件为：
终值条件与到期日回报相关。
很容易将边界条件修改为适用于部分补偿的情形。如果障碍被触及时有大小为R的部分补偿，则：。
“敲入”障碍期权
“敲入”期权只有在障碍触及时才有回报。如果障碍未被触及，那么到期时期权毫无价值
“敲入”期权的价值在于触及障碍的潜在可能性，如果期权是向上敲入合约，那么它在向上障碍上有和普通合约相同的价值
，这里t<T
向下敲出障碍期权的边界条件与之类似。
当障碍被触及是我们得到的合约本身是一个衍生品，因此“敲入”障碍期权是一个二阶合约。然而由于“敲入”障碍期权与同样条件的“敲出”期权自合是普通期权，实际上我们可以将定价问题简化为一阶问题。
完全使用数值方法求解“敲入”障碍期权时，必须先求解相应的普通期权的价值，然后再求解障碍期权的价值。因此，整个求解过程需要的时间大约是“敲出”障碍期权的两倍。
当波动率为常数时的一些公式
当波动率为常数时可以求解出某些障碍合约的理论价值。这里给出部分例子。这里继续使用
表示相应的普通合约的价值。
1. 向下敲出障碍看涨期权
作为第一个例子，考虑向下敲出障碍看涨期权，障碍水平为低于执行价E的。函数是和障碍期权有相同到期日和回报的普通期权的Black-Scholes理论价格。向下敲出障碍看涨期权的价值由以下公式给出：
图：向下敲出看涨期权的价值
2. 向下敲入障碍看涨期权
在没有任何部分补偿时，“敲入”障碍期权和“敲出”障碍期权（具有同样的回报与障碍水平）的关系非常简单：
敲入期权+敲出期权=普通期权
如果“敲入”障碍被触及，“敲出”障碍同样会被触及，因此无论障碍被触及与否在到期日都会有普通期权的回报。因此，向下敲入障碍看涨期权的价值是：
3. 向上敲出障碍看涨期权
向上敲出障碍看涨期权的障碍水平一定在执行价E之上（否则期权会没有价值）。向上敲出障碍期权的价值是：
下图给出了作为S的函数的期权价值，像不像一个鲨鱼鳍？由下图可以看出，该向上敲出看涨期权的价值在标的价格越接近于障碍的时候，价值逐步减小，直到0。
以上图形均由估值神器QuantPlus Analytics制作。
附注：
用N（*）表示标准正态分布的累积概率分布函数。股票的红利以
表示。同时：
这里
是障碍的位置，
