間斷界限選擇權評價之模擬－以下生效買權為例

Pricing Discrete Barrier Options by Simulation: The Case of Down-and-In Call

陳俊傑(Chiun-Chieh Chen)；張曉欽(Hsiao-Chin Chang)；張育綜(Yu-Zong Zhang)

界限選擇權 ； 蒙地卡羅法 ； 變異數縮減法 ； Barrier Options ； Monte Carlo Method ； Variance Reduction

嶺東學報

39期（2016 / 06 / 01）

147 - 168

繁體中文

自從2008 年金融風暴過後，投資人意識到風險規避的重要性，而衍生性商品也更加受到矚目，同時亦有許多與選擇權相關之議題被研究。毫無疑問地，具有連續和間斷型式之界限選擇權商品，在實務上已然是眾所周知的了。間斷界限選擇權的應用要較連續型式者為廣，雖然連續界限選擇權具有某些封閉解形式，但是類似之封閉解，卻並不存在於間斷界限選擇權上。若以連續模型來估算間斷選擇權的話，則將會導致嚴重的誤差，因此，仍需依賴數值方法來模擬求解。本文利用蒙地卡羅法(MC)，進行間斷界限選擇權價格之計算，為了降低蒙地卡羅法之變異，還另外配合變異數縮減法中的反向變異(AMC)、條件期望值(CMC)以及濾波法(FMC)等來執行模擬演算。研究結果發現：(1)採用蒙地卡羅法可避免發生樹狀模型常見之“Barrier-too-close＂的問題、(2)CMC 法與FMC 法均能有效降低變異並增加模擬準確性、(3)CMC 法在執行速度上明顯優於FMC 法，且與MC 法或AMC 法者相近。

Financial crisis occurred in 2008 and investors have cognized the importance of risk aversion. Derivative financial products became more popular and many researches related to options have been investigated. Barrier options which include continuous and discrete types are no doubts, the well-known schemes in the practice. The applications of discrete barrier options are more popular than the continuous one, even if the continuous barrier options have been developed with some closed-form solutions. However, such similar closed forms are not available in the discrete barrier options. The giant biases will occur if we evaluate discrete options by continuous model. Then some numerical simulations for discrete options are necessary. This study uses the Monte Carlo simulation (MC) to evaluate the prices of discrete barrier options. In order to decrease the variances, the MC method will be also constructed in conjunction with three different approaches respectively: Antithetic variates (AMC), Conditional expectation (CMC) and Filtered Monte Carlo (FMC). Analysis results show that (1) the phenomenon of “Barrier-too-close” which existed in the conventional tree structure methods can be avoided by the Monte Carlo simulation; (2) both CMC and FMC methods effectively reduce the variances and increase the preciseness; and (3) CMC is significantly faster in the execution than FMC. Furthermore, the execution speed of CMC is similar to the MC and AMC methods.

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