賽局選擇權評價模型

Abstract

在Kifer（2000）的文獻中，作者提出一種與歐式選擇權、美式選擇權並列的衍生性金融商品 — 賽局選擇權。賽局選擇權亦稱為以式選擇權，其放寬了以往選擇權合約賣方只有履行義務的限制，除了買方享有提早履約的權利之外，也讓賣方能夠在到期日之前終止合約。換言之，賽局選擇權的買方有權在到期日之前，以合約所制訂的價格（稱之為履約價）購買（買權）或出售（賣權）一定數量的指定標的物；而賣方若在到期日之前終止合約，除了買方當時提早履約所能得到的報酬之外，還必須付給買方一筆違約金。 目前關於賽局選擇權的相關研究文獻並不多，其中雖然有關於評價模型的探討，但尚缺乏以常見之數值方法進行評價的文獻，其自由邊界問題與對應之變分不等式亦尚未建立。有鑑於此，本篇研究將聚焦於此新式選擇權的評價模型上，我們僅考慮最原始型式的賽局選擇權，提出合理的違約金型式，並以常見的二元樹狀法進行實作；接著我們會定義其自由邊界問題，並建構出對應之變分不等式，再以有限差分法求解；最後比較前面兩種方法的結果並進行討論。

In Kifer (2000), a new derivative security called game option was introduced. Game option, also called Israeli option, is a contract which enables both its holder (buyer) and writer (seller) to stop it at any time before expiration. That is, its buyer can exercise the right to buy (for a call) or to sell (for a put) a specified underlying asset at a predetermined price, and its seller can cancel the contract by paying the buyer the early exercise payoff plus an amount of penalty. Although some literatures probed into the valuation model of this new derivative, efficient numerical methods have not been developed yet, and both its free boundary problem and the corresponding variational inequalities have not been constructed. Throughout this thesis, we only consider the most general case of game-type contingent claims for its valuation. First we propose the rules of penalty format, choose a more practical one, and apply the familiar binomial tree method. Then we construct its free boundary problem, formulate the corresponding variational inequalities, and use finite-difference method to solve it. Finally, we compare the above results and bring up some discussions.