American Parisian options

In this article, we describe the various sorts of American Parisian options and propose valuation formulae. Although there is no closed-form valuation for these products in the non-perpetual case, we have been able to reformulate their price as a function of the exercise frontier. In the perpetual case, closed form solutions or approximations are obtained by relying on excursion theory. We derive the Laplace transform of the first instant the Brownian Motion reaches a positive level or, without interruption, spends a given amount of time below zero. We perform a detailed comparison of perpetual standard, barrier and Parisian options.