It may seem counterintuitive, but valuing “perpetual” options, with infinite maturity, can be easier than pricing those that expire at a fixed date. At least Samuelson found it so in 1965. In this article, Barone extends the valuation models for perpetual options to cover American calls and puts on dividend-paying stocks, and European options written on these infinite maturity contracts. Formulas for the Greek letter risks are presented, as well as an analysis of how put-call parity works for them. Along the way, additional results are developed for first-touch digital options, both perpetual and finite maturity.