American Parisian options
The objective of this paper is to develop probabilistic approach for the pricing of American Parisian options. They are described and classified. Put-call parity results are presented. The authors focus purely on currency options. The price of an American Parisian option is decomposed into two components, namely the price of the corresponding European Parisian option and the American premium. The first component is known, and the second one depends on the exercise boundary. The technique of Brownian excursions is used. For perpetual options the exercise boundary is also derived, however, in the case of perpetual American Parisian up-and-out call options it is not a constant. In this case a closed-form solution cannot be obtained. Then the problem is tackled from a different angle, the analogies between American (barrier) and American Parisian barrier options are considered and closed-form expressions are derived or simplifying assumptions are made. A new result concerning the first instant when Brownian motion either hits a positive level or spends a certain time below zero, is obtained.
scientific article; zbMATH DE number 5158697