Zhang, You You
(2014) Brownian Excursions in
Mathematical Finance. PhD Thesis, LSE-UK.
The Brownian excursion is defined as a standard
Brownian motion conditioned on starting and ending at zero and staying positive
in between. The first part of the thesis deals with functional of the Brownian
excursion, including first hitting time, last passage time, maximum and the
time it is achieved. Our original contribution to knowledge is the derivation of
the joint probability of the maximum and the time it is achieved. We include a
financial application of our probabilistic results on Parisian default risk of
zero-coupon bonds. In the second part of the thesis the Parisian, occupation
and local time of a drifted Brownian motion is considered, using a two-state
semi-Markov process. New versions of Parisian options are introduced based on
the probabilistic results and explicit formulae for their prices are presented
in form of Laplace transforms. The main focus in the last part of the thesis is
on the joint probability of Parisian and hitting time of Brownian motion. The
difficulty here lies in distinguishing between different scenarios of the
sample path. Results are achieved by the use of infinitesimal generators on
perturbed Brownian motion and applied to innovative equity exotics as
generalizations of the Barrier and Parisian option with the advantage of being
highly adaptable to investors’ beliefs in the market.
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