金融產品創新及其定價：移動平均交換選擇權

Financial Products Innovation and Their Pricing: Moving Average Exchange Options

韓千山(CHIEN-SHAN HAN)；邱嘉洲(CHIA-CHOU CHIU)；蔡鎮宇(CHEN-YU TSAI)；簡榮治(JUNG-CHIH CHIEN)

移動平均交換選擇權 ； 移動平均 ； 交換選擇權 ； 歐式選擇權 ； moving average exchange options ； moving average ； exchange options ； European options

輔仁管理評論

27卷2期（2020 / 05 / 01）

1 - 37

繁體中文

本文探討金融產品創新及其定價，討論移動平均交換選擇權。本文主要是延伸Margrabe（1978）歐式交換選擇權，將其報償－二項資產價格正差距，延伸為以算術或幾何計算二項資產價格移動平均正差距。一般選擇權分為歐式與美式。美式選擇權價值相當於歐式選擇權價值加上提前履約溢酬（early exercise premium）。尤有甚者，美式選擇權不會提前履約情況發生時，則歐式選擇權等價於美式選擇權。因此，本文提供歐式選擇權的觀點，有助於探討該選擇權特性。另外，以股票資產為例，股票移動平均價格視為交易實務上股票陸續出售或分開時段變現時投資組合之平均價值，亦或股票陸續買進或分開時段佈局時投資組合之平均建構成本。而且，在股票價差交易（spread trade）的交易實務上，移動平均交換選擇權亦提供良好的避險工具。最後，本文提供數值積分法與（解析）封閉解法等評價歐式算術、幾何移動平均交換選擇權，運用蒙地卡羅法佐證其精確性，並且提供風險衡量，例如，Delta, Gamma等，運用於該選擇權之風險管理及其投資組合避險等工作。

This article explores financial products innovation and their pricing, and discusses moving average exchange options. This article is mainly to extend Margrabe (1978) European exchange option. The payoff of this financial products is innovated by mea ns of arithmetic or geometric mean, from a positive difference of the prices of two assets c hose n to positive difference of moving average prices of two assets c hosen. However, options are either European or American (style) options, and in theory, the value of America n options are equal to the value of European options plus the early exercise premium. But, when early exercise of the opt ions is not optimal, American options are equivalent to European options. Therefore, this article provides a view of Europe an options, which helps to explore the characteristics of these options. In trading practice, taking stock assets as an example, the average moving stock price is regarded as the average value of the selling investment portfolio, or the average construction value of the buying investment portfolio, especially when the stocks are bought or sold successively or separately. In addition, in case of stock spread trade, a moving average exchange option also provides a good hedging tool. Finally, t his paper provides numerical integration and (analytic ) closed-solution to evaluate European-style arithmetic and geometric moving average exchange options, using Monte Carlo to prove its accuracy. We also provide risk measurements, for example, Delta, Gamma, etc. It is used in the risk management of this option and its hedging portfolios.

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