本文主要在研究在”微笑”波動率(volatility smile)的架構下，障礙式選擇權之靜態避險。吾人試著以隱含二項樹(implied binomial tree)的架構來評價選擇權。在此波動率的架構下，以靜態複製的方法取代傳統的動態避險策略。此外，在評價障礙式選擇權方面，本文採用一種調整的數值方法使得評價結果能夠更快速地趨近於選擇權的真值或是其解析解。 本文中，我們將隱含波動率的架構定義為隱含波動率與選擇權執行價相關的公式，而在此吾人略去到期期間對於波動率之影響(定義隱含波動率與時間為獨立的關係)。在建構隱含二項樹模型時，”不良機率”(bad probability)是一個很大的麻煩。在維持隱含波動率架構及其函數平滑之前提下，將導致不良機率之不良節點調整替換為好的節點。再者，由於隱含波動率樹狀模型之架構下，障礙式選擇權並無解析解，吾人所採用之調整數值方法省去許多計算所需時間，並改善了未調整數值之解不夠精確的問題。 在以靜態複製投資組合來做選擇權之避險時，投資組合中選擇權的數量必須在複製的精確性與選擇權之交易成本做取捨。使用愈多之選擇權來當作複製投資組合，將會得到愈好之複製效果；相對地，選擇權之交易成本也隨之愈大。在本文之釋例中，吾人發現採用十個選擇權之靜態複製投資組合，其複製效果的誤差遠小於使用五個選擇權當作複製投資組合。在其他的例證中，我們也比較了在隱含波動率的架構與常數波動率的架構之下，靜態複製之避險效果。 This paper investigates static hedge portfolios for barrier options with volatility “smile.” We try to value the option with an implied binomial tree approach. We replace the traditional dynamic hedging strategy with a static replication under this volatility structure. Moreover, in valuing options with barriers, we use the enhanced numerical method to make the value approach the analytic result much more rapidly. We define the smile in terms of implied volatility by giving a formula relating strike price to implied volatility, assuming the smile to be time independent. In constructing the implied binomial tree, “bad probability” is a big problem. We replace the bad nodes that generate a violation of the probabilities with the good nodes, which keep the implied local volatility function smooth. In addition, barrier options valued on an implied tree have no analytic solution. The enhanced method saves computing time and provides greater accuracy than an unenhanced binomial solution. When we hedge the options with a static replicating portfolio, the number of options should be chosen to balance an inaccurate replication against the options’ cost. The more options there are in our replication portfolio, the better the replication is, and the greater the transaction costs are as well. In the example, our findings show that the replication mismatch is much smaller when using ten options to replicate the target option instead of a five-option replication portfolio. We also compare the effect of the static replication between the implied volatility approach and the constant volatility framework.
