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目前場外期權產品較為火爆，尤其是成為鯊魚鰭期權的產品，在一些券商的交易平臺上均有報價。其實是這是一類障礙期權。障礙期權是路徑依賴期權，其回報取決于資產價格的已實現路徑；當資產價格過高或者過低時，合約的某些條款被觸發。例如，如果期權到期之前所有時刻標的資產都沒有以或者更高的價格進行交易，那么向上敲出（knock-out）看漲期權在到期日的支付為常見的。在這個例子中，如果標的資產價格初級了這個水平（從下觸及），那么期權將被“敲出”（knock-out），變得一文不值（payoff=0）。除了像這樣的“敲出”期權，也存在“敲入”（knock-in）期權，它們只有在資產價格觸及特定水平時才有支付，否則在到期時毫無價值。

障礙期權存在的原因很多。購買者可以用它來對沖具有類似特征的特殊現金流。其購買者往往對市場未來方向有著非常精準的判斷。如果他們想獲得看漲期權式的回報，但相信在期權到期日前標的資產價格上升幅度有限，不愿意支付全部上行潛力的費用，那么他可以購買向上敲出的期權。由于上漲幅度嚴格受限，障礙期權比相同的普通期權便宜。如果他的判斷是對的并且障礙并未被觸及，他將得到他希望的支付。障礙同當前資產價格越接近，期權被敲出的可能性就越大，那么合約就越便宜。

相反地，相信某障礙水平會實現的人講購買敲入期權。同樣地，這個期權也會比相應的普通期權便宜。

障礙期權的不同類型

障礙期權主要有兩種主要類型：

• 敲出期權（knock-out option），只有在到期日前從未觸及障礙水平下才能產生支付。如果障礙觸及稱期權被敲出了（knocked-out）。
• 敲入期權（knock-in option），只有在到期日達到障礙水平才產生支付。如果障礙被觸及稱期權被敲入（knocked in）了。

根據障礙水平與標的資產初始價格的相對位置可以進一步對障礙期權進行分類。

•  如果障礙在資產初始價格以上，被稱作向上（up）期權。
•  如果障礙在資產舒適價格以下，被稱作鄉下（down）期權。

最終到期日時收到回報（payoff）如下：

回報均是常見的形式，如看漲、看跌、二值等。

障礙期權的定價方法

障礙期權是弱路徑依賴的，這和亞式期權是有區別的。這里用表示障礙觸及前期的價值，這個價值將滿足Black-Scholes方程：

障礙特征的細節將通過邊界條件的具體形式來體現。

“敲出”障礙期權

如果標的資產價格觸及了障礙，“敲出”障礙期權將一文不值。這個特征使得障礙水平為的向上敲出障礙期權有邊界條件。

，這里t<T

我們必須在的范圍求解Black-Scholes方程，方程在S=S_u的邊界條件如上，還有一個與障礙未被觸及時的關于回報的終值條件。對于看漲期權有：

如果一個有障礙為的向下敲出障礙期權，則在 范圍內求解Black-Scholes方程，其邊界條件為：

終值條件與到期日回報相關。

很容易將邊界條件修改為適用于部分補償的情形。如果障礙被觸及時有大小為R的部分補償，則：。

“敲入”障礙期權

“敲入”期權只有在障礙觸及時才有回報。如果障礙未被觸及，那么到期時期權毫無價值

“敲入”期權的價值在于觸及障礙的潛在可能性，如果期權是向上敲入合約，那么它在向上障礙上有和普通合約相同的價值

，這里t<T

向下敲出障礙期權的邊界條件與之類似。

當障礙被觸及是我們得到的合約本身是一個衍生品，因此“敲入”障礙期權是一個二階合約。然而由于“敲入”障礙期權與同樣條件的“敲出”期權自合是普通期權，實際上我們可以將定價問題簡化為一階問題。

完全使用數值方法求解“敲入”障礙期權時，必須先求解相應的普通期權的價值，然后再求解障礙期權的價值。因此，整個求解過程需要的時間大約是“敲出”障礙期權的兩倍。

當波動率為常數時的一些公式

當波動率為常數時可以求解出某些障礙合約的理論價值。這里給出部分例子。這里繼續使用表示相應的普通合約的價值。

1. 向下敲出障礙看漲期權

作為第一個例子，考慮向下敲出障礙看漲期權，障礙水平為低于執行價E的。函數是和障礙期權有相同到期日和回報的普通期權的Black-Scholes理論價格。向下敲出障礙看漲期權的價值由以下公式給出：

圖：向下敲出看漲期權的價值

2. 向下敲入障礙看漲期權

在沒有任何部分補償時，“敲入”障礙期權和“敲出”障礙期權（具有同樣的回報與障礙水平）的關系非常簡單：

敲入期權+敲出期權=普通期權

如果“敲入”障礙被觸及，“敲出”障礙同樣會被觸及，因此無論障礙被觸及與否在到期日都會有普通期權的回報。因此，向下敲入障礙看漲期權的價值是：

3. 向上敲出障礙看漲期權

向上敲出障礙看漲期權的障礙水平一定在執行價E之上（否則期權會沒有價值）。向上敲出障礙期權的價值是：

其中N(.)是標準正態分布變量的累計概率分布函數，a,b和所有d的形式在文章后面附注給出。

下圖給出了作為S的函數的期權價值，像不像一個鯊魚鰭？由下圖可以看出，該向上敲出看漲期權的價值在標的價格越接近于障礙的時候，價值逐步減小，直到0。

以上圖形均由估值神器QuantPlus Analytics制作。

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附注：

用N（*）表示標準正態分布的累積概率分布函數。股票的紅利以表示。同時：

這里是障礙的位置，

Attachments

• 向上敲出障礙期權 (ERROR)

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