Pricing formulas for perpetual American options with general payoffs

Abstract

Abstract 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.