美式冪次買權的不提前履約條件及其解析價格上界

The Never-Early-Exercise Condition and Analytical Price Upper Bounds of American Power Call Options

本文討論美式冪次買權永不提前履約的充份條件。在美式標準買權下，股利率q = 0 是其不提前履約的唯一條件，但對美式冪次買權而言，則存在一個特定範圍的股利率q 使得提前履約恆為非最佳。本文首先推導出不限定模型下，使其不提前履約的股利率q 之一般化條件，可適用於冪次係數n > 1 或n < 0 的情形。針對某些特定模型，此股利率q 的條件可再放寬且適用於任意的冪次係數n。當股利率q 不滿足此不提前履約條件時，本文亦推導出若干模型下美式冪次買權價格的解析上界公式，包含基本的Black-Scholes 模型，兩種跳躍擴散模型，以及variance gamma 模型，並提供數值範例以驗證以上解析公式的有效性。

This paper discusses the sufficient condition under which the American power call options should never be early exercised. Unlike in the vanilla case where the dividend yield q = 0 is the only condition, for American power call options there actually exists a range of q such that early exercise is never optimal. We start with deriving the general (model free) condition on q for American power call options with power sufficient n > 1 or n < 0. For specific models, we provide alternative conditions which lead to a wider range of q and applicable to any n. When q does not satisfy these conditions, we also give the analytical upper bounds for the American power call prices. These analytical formulas are derived for the fundamental Black-Scholes model as well as two jump-diffusion models and the variance gamma model, with numerical examples given to demonstrate their validity.