回顧選擇權的定價

Abstract

在 Cox, Ross and Rubinstein (1979) 的架構下, Cheuk and Vorst (1997) 利用轉換計價單位的方法導出回顧選擇權的單狀態變數二項模型。 由於計算效率高, 他們的模型已成為當考慮到觀察頻率時回顧選擇權最卓越的二項定價法。 但此法在離散觀察的選擇權定價上似乎有點複雜。 除了 Cheuk and Vorst 的轉換計價單位法之外, 在文獻還有一個方法能夠導出等價的模型。 Choi and Jameson (2003) 提出了一個簡單的方法建構回顧選擇權的單狀態變數二項模型, 但他們只考慮了歐式選擇權, 忽略了兩種重要的延伸 --- 美式選擇權以及觀察頻率。 而且他們也沒有嚴格清楚的證明其結果。 在本文中, 我們以 Choi and Jameson (2003) 的想法為基礎, 利用 Cox, Ross and Rubinstein (1979) 無套利定價理論導出回顧選擇權的單狀態變數二項模型。 細分以下三步驟說明: 首先, 我們建構出與 Choi and Jameson (2003) 相同的歐式選擇權的單狀態變數二項模型, 並且寫下嚴謹易懂的證明。 接著, 我們進一步以類似的想法與證明將結果推廣至美式與離散觀察的選擇權, 推導出其單狀態變數二項模型。 最後我們拿自己的模型與文獻中其他的二項模型作比較。 發現我們的模型等價於 Cheuk and Vorst 的模型, 計算效率也一樣, 但從結果相較之下卻符合直覺許多, 而且更加簡單、 易於明瞭。 因此, 就這觀點而論, 本文所述之二項定價法更優於 Cheuk and Vorst (1997) 的方法。

Working in the framework of Cox, Ross and Rubinstein (1979), Cheuk and Vorst (1997) derive one-state variable binomial models for lookback options using a change of numeraire. Due to its efficiency, their method has been the most attractive binomial approach to pricing lookback options while the observation frequency is investigated. However, it seems a bit complicated to price discretely sampled options. Other than Cheuk and Vorst's change of numeraire, there is an alternative to derive the equivalent models. Choi and Jameson (2003) propose a simplified method to develop one-state variable binomial models for European lookback options without considering two important extensions --- the American type and observation frequency. They do not also give a rigorous and clear proof. In this paper, following Cox et al. (1979) arbitrage arguments, we develop equivalent one-state variable binomial models for lookback options on the basis of the idea of Choi and Jameson (2003): First, we construct the same models for European options as those in Choi and Jameson (2003) and give a more apprehensible proof. Moreover, we extend these models to cope with American and discretely sampled options, and then derive one-state variable binomial models for these variants. Finally we compare our models with others available. Our models perform as well as Cheuk and Vorst's in terms of efficiency, but are much more intuitive and simpler, especially for discretely sampled options. For this reason, the binomial valuation method of this paper are preferred over that of Cheuk and Vorst (1997).