彩虹型雙界限選擇權之定價

The Pricing of Double Barrier Rainbow Options

邵翊哲(Yi-Zhe Shao)

指導教授 ： 白惠明

國立臺北大學/商學院/統計學系/碩士(2010年)

摘要

界限選擇權 (barrier option) 是現在市場上很受歡迎的新奇選擇權(Exotic option)之一，跟最陽春的選擇權相比，界限選擇權與之不同的部分，它是一種路徑相依的選擇權，跟以往的香草選擇權不同，不再是只看期初期末的股價，而必須關注其到期日前標的價格是否曾經大於或小於某一個界限(即為 barrier)，所以在評價上會遭遇到許多變化和困難性。因此，界限選擇權大多在幾何布朗運動之下使用反射原理以及標的價格與其極大值(或極小值)的聯合累積機率來定價。

雙界限選擇權 (double barrier option) 和彩虹型界限選擇權 (rainbow barrier option) 是從界限選擇權衍生出來的新產品，本文所要討論的除了雙界限選擇權或是彩虹型選擇權之外，再更推廣去探討彩虹型雙界限選擇權之定價。所謂的彩虹型雙界限選擇權，即在約定時間之內，觀察雙資產其中一資產資產是否曾經跨越界限 (barrier) ，如有觸及界限 (barrier) ，則此選擇權價失效。在此將利用文獻中所討論的評價雙界限選擇權方法，利用在約定時間內資產沒有觸及界限的分配函數，推導出彩虹型雙界限選擇權評價的封閉解 (Closed form)。

關鍵字

界限選擇權；新奇選擇權；彩虹型界限選擇權

並列摘要

Barrier option is one of the Exotic options which are very popular in the market. In contrast with the simplest option, barrier option is continuous time and path dependent.
Unlike the common vanilla option, to price the barrier option one not only need to check the starting and the ending prices of the stock, but also have to pay attention to whether the underlying stock price before maturity T has ever crossed the barrier. In that case more variability and difficulties arise in pricing the option.The most popular technique in pricing barrier option is to uses the reflection principle to calculate the joint distribution of the maximum( or minimum ) and the value at maturity of the underlying stock prices based on geometry Brownian motion.

Double barrier option and rainbow barrier option are two new products evolved from the barrier option. In addition to the discussion of these options, we will extend our
exploration to double barrier rainbow option .Double barrier rainbow option depends on two assets. If one of the assets has ever been crossed the barrier, this option becomes priceless.We will use the technique of pricing double barrier options proposed by the referred papers, and the distribution of underlying assets in the case that
the asset never touches barrier, to deduce a closed form of the price of the double barrier rainbow option.

並列關鍵字

barrier option ； exotic option ； rainbow barrier option； double barrier option

參考文獻

1. Antoon Pelsser (1996). Pricing Double Barrier Options:

and Corporate Liabilities. Journal of Political Economy

Motion and Stochastic Calculus (Second Edition).

Exotics. General Re Financial Products

6. Helyette Geman and Marc Yor. (1996). Pricing and Hedging