Lim, Jia Wei
Parisian excursions of Brownian motion and their
applications in mathematical finance.
PhD thesis, The London School of Economics and Political Science (LSE).
In this thesis, we study Parisian excursions, which are defined as excursions of Brownian
motion above or below a pre-determined barrier, exceeding a certain time length. Employing
a new method, a recursion formula for the densities of single barrier and double barrier Parisian
stopping times are computed. This new approach allows us to obtain a semi-closed form
solution for the density of the one-sided stopping times, and does not require any numerical
inversions of Laplace transforms. Further, it is backed by an intuitive argument which is
premised on the recursive nature of the excursions and the strong Markov property of the
Brownian motion. The same method is also employed in our computation of the two-sided and
the double barrier Parisian stopping times. In turn, the resultant densities are used to price
Parisian options. In particular, we provide numerical expressions for down-and-in Parisian
calls. Additionally, we study the tail of the distribution of the two-sided Parisian stopping
time. Based on the asymptotic properties of its distribution, we propose an approximation
for the option prices, alleviating the heavy computational load arising from the recursions.
Finally, we use the infinitesimal generator to obtain several results on other variations of
Parisian excursions. Specifically, apart from the length, we are interested in the number of
excursions and the maximum height achieved during an excursion. Using the same generator,
we derive the joint Laplace transform of the occupation times of the Brownian motion above
and below zero, but only starting the clock each time after a certain length.
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