An American option gives the holder the right, but not the obligation, to buy/sell an underlying asset from/to the writer at an agreed strike price at any time on or before the expiry date. Options are mainly used for speculation and hedging. The pricing of options is a fundamental problem in mathematical finance. One of the attractions of options is that they can be used to construct a wide range of trading strategies characterized by different payoff functions. As a preliminary step in the valuation of American options for a variety of trading strategies, in this article the pricing of perpetual American options with general payoffs is considered, where the perpetual American call and put are special cases. Four broad classes of payoff functions are identified for which analytical pricing formulas can be derived by utilizing a Mellin transform technique and an optimization procedure. Depending on the class of payoff functions considered, free boundary problems with one or two boundaries are obtained. Illustrative examples are provided and benchmarked numerically with the binomial method. The characterization of different payoffs for perpetual American options considered in this article will be instrumental in the identification and pricing of new free boundary problems for (non-perpetual) American-style financial derivatives.

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具有一般收益的永续美式期权定价公式

美式期权赋予持有人权利，但没有义务，在到期日或之前的任何时间以约定的执行价格从/向卖方买卖标的资产。期权主要用于投机和对冲。期权定价是数学金融中的一个基本问题。期权的吸引力之一是它们可用于构建以不同收益函数为特征的广泛交易策略。作为对各种交易策略的美式期权估值的初步步骤，本文考虑了具有一般收益的永续美式期权的定价，其中永续美式看涨期权和看跌期权是特殊情况。通过利用梅林变换技术和优化程序可以推导出四大类支付函数。根据所考虑的支付函数的类别，可以获得具有一个或两个边界的自由边界问题。提供了说明性示例，并使用二项式方法进行了数值基准测试。本文考虑的永续美式期权不同收益的表征将有助于识别和定价（非永续）美式金融衍生品的新自由边界问题。提供了说明性示例，并使用二项式方法进行了数值基准测试。本文考虑的永续美式期权不同收益的表征将有助于识别和定价（非永续）美式金融衍生品的新自由边界问题。提供了说明性示例，并使用二项式方法进行了数值基准测试。本文考虑的永续美式期权不同收益的表征将有助于识别和定价（非永续）美式金融衍生品的新自由边界问题。