An analytical solution for the two-sided Parisian stopping time, its asymptotics and the pricing of Parisian options

In this paper, we obtain a recursive formula for the density of the two-sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one-sided Parisian stopping time derived in Dassios and Lim [6]. However, when we study the tails of the two distributions, we find that the two-sided stopping time has an exponential tail, while the one-sided stop- ping time has a heavier tail. We derive an asymptotic result for the tail of the two-sided stopping time distribution and propose an alternative method of approximating the price of the two-sided Parisian option