Brownian Excursions and Parisian Barrier Options

Abstract

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time

“…We will use the same definitions for the excursions as in [5]. Let S be the underlying asset following a geometric Brownian motion, and let Q denote the risk neutral probability measure.…”

“…For b ≤ 0, the Laplace transform of the density f − b (t) of the stopping time (with D = 1) is (see [5] for more detail)…”

“…Previous literature has largely focused on using Laplace transforms to price Parisian options. In [5,7,10], the problem is reduced to finding the Laplace transform of the Parisian stopping time, which is the first time the length of the excursion reaches level D. In [5], the Laplace transform of the stopping time was obtained using the Brownian meander and Azema martingale, while Dassios and Wu [7] introduced a perturbed Brownian motion and a semi-Markov model to obtain the Laplace transform. In both of these, an explicit form of the Laplace transform of the distribution of the Parisian stopping time and consequently that of the option price is found.…”

“…Introduction. Parisian options were first introduced by Chesney, Jeanblanc-Picque, and Yor [5]. They are path-dependent options whose payoff depends not only on the final value of the underlying asset, but also on the path trajectory of the underlying asset above or below a predetermined barrier L. The owner of a Parisian down-and-out call loses the option when the underlying asset price S reaches the level L and remains constantly below this level for a time interval longer than D, while for a Parisian down-and-in call, the same event gives the owner the right to exercise the option.…”

“…which are related to the total excursion time above (or below) a barrier, are studied in [5], while double-sided Parisian options are introduced in [7] and [2].…”