一般化組合型選擇權的封閉解

A Closed Form Approach to Valuing General Basket Options

黃保憲(Pao-Hsien Huang)

指導教授 ： 吳庭斌 ； 白惠明

國立臺北大學/商學院/統計學系/碩士(2011年)

摘要

一般化組合型選擇權為 Borovkova, Permana and Weide (BPW) 於 2007 年所提出的概念，BPW 將組合型選擇權中標的資產權重皆正的條件放寬，使權重可納入負值的考量。在評價上，一般化組合型選擇權會面臨標的資產相加減而使得分配未知的問題，為了解決這個問題，BPW 採用一般化對數常態分配來近似一般化組合型選擇權的分配，並利用動差配適法配適一般化對數常態分配中的三個參數，最後以封閉解型式進行評價，但是 BPW 所採用的方法可能在資產設定的改變後產生明顯的評價偏誤。

　　本文改採 Johnson 分配族作為近似一般化組合型選擇權分配的考量，並以動差配適法配適分配族中的四個參數，最後以封閉解型式進行數值分析。從數值分析的結果說明，採用本法所進行的評價相當近似蒙地卡羅的模擬結果，因此本文不僅修正了 BPW 法的問題，同時也提供了一個穩健且迅速的評價方法。

關鍵字

一般化組合型選擇權 ； 動差配適法 ； Johnson 分配族

並列摘要

The concept of general basket options was proposed by Borovkova, Permana and Weide (BPW) in 2007. BPW extended the definition of a basket by allowing weights to be negative.The difficulty in pricing general basket options stems from the fact that the distribution of a weighted sum of underlying assets is unknown.To tackle this problem, BPW employed the general lognormal distribution to approximate the distribution of general basket options and used the moment matching method to estimate the three parameters of lognormal distribution.Finally, BPW evaluated the general basket options by an approximate closed-form pricing formula.However, this method may create significant pricing bias after changing the setting of underlying asset.

　　In this article, we employ a family known in the statistical literatue as the Johnson family to approximate the distribution of general basket options and use the moment matching method to estimate the four parameters of the family.We also take numerical analysis by an approximate closed-form pricing formula.Numerical simulations have shown that our approach not only correct the bias of BPW approach but also provide an extremly robust and efficient method when compared with Monte Carlo simulations.

並列關鍵字

General Basket Options ； Moment matching method ； Johnson family

參考文獻

1. Borovkova, S., Permana, F.J., and Weide, H.V.D. (2007). A Closed Form Approach to the Valuation and Hedging of Basket and Spread Options.Journal of Derivatives, 15 (Summer), 8-24.

2. Elderton, W.P., and Johnson, N.L. 1969. System of Frequency Curves. 1sted.,131. Cambridge University.

3. Flamouris, D. and Giamouridis, D. (2007). Approximate Basket Option Valuation for a Simpli ed Jump Process. Journal of Futures Markets, 27, 819-837.

4. Gentle, D. (1993). Basket Weaving. Risk, June, 51-52.

5. Hill, I., Hill, R., and Holder, R. (1976). Fitting Johnson Curves by Moments.Applied Statistics, 25, 2, 180-192.