  We solve the pricing problem for perpetual American puts and calls on
dividend-paying assets. The dependence of a dividend process on the underlying
stochastic factor is fairly general: any non-decreasing function is admissible.
The stochastic factor follows a Levy process. This specification allows us to
consider assets that pay no dividends at all when the level of the underlying
factor (say, the assets of the firm) is too low, and assets that pay dividends
at a fixed rate when the underlying stochastic process remains in some range.
Certain dividend processes exhibiting mean-reverting features can be modelled
as appropriate increasing functions of Levy processes. The payoffs of both the
American put and call options can be represented as the expected present value
(EPV) of a certain stream of dividends, and we show that the option must be
exercised the first time the EPV of this stream with the original process being
replaced by the infimum process starting from the current level, becomes
positive. Thus, the exercise threshold depends only on the record setting bad
news. The results can be applied to the theory of real options as well; as one
of possible applications, we consider the problem of incremental capital
expansion.
