  We study optimal stopping problems related to the pricing of perpetual
American options in an extension of the Black-Merton-Scholes model in which the
dividend and volatility rates of the underlying risky asset depend on the
running values of its maximum and maximum drawdown. The optimal stopping times
of exercise are shown to be the first times at which the price of the
underlying asset exits some regions restricted by certain boundaries depending
on the running values of the associated maximum and maximum drawdown processes.
We obtain closed-form solutions to the equivalent free-boundary problems for
the value functions with smooth fit at the optimal stopping boundaries and
normal reflection at the edges of the state space of the resulting
three-dimensional Markov process. We derive first-order nonlinear ordinary
differential equations for the optimal exercise boundaries of the perpetual
American standard options.
