  In this paper, we price American-style Parisian down-and-in call options
under the Black-Scholes framework. Usually, pricing an American-style option is
much more difficult than pricing its European-style counterpart because of the
appearance of the optimal exercise boundary in the former. Fortunately, the
optimal exercise boundary associated with an American-style Parisian knock-in
option only appears implicitly in its pricing partial differential equation
(PDE) systems, instead of explicitly as in the case of an American-style
Parisian knock-out option. We also recognize that the "moving window" technique
developed for pricing European-style Parisian up-and-out options can be adopted
to price American-style Parisian knock-in options as well. In particular, we
obtain a simple analytical solution for American-style Parisian down-and-in
call options and our new formula is written in terms of four double integrals,
which can be easily computed numerically.
