An analytical solution for Parisian up-and-in calls

In this work, we derive an analytical solution for the value of Parisian up-and-in calls by using the “moving window” technique developed by Zhu and Chen [15] for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an “in” barrier, the option holder cannot do or decide on anything before the option is activated, and once the option is activated it is just a plain vanilla call, which could be of American style or European style.