Geodesic
Laplace transforms and shout options
Acta mathematica Universitatis Comenianae, Tome 80 (2011) no. 1 
G. Alobaidi; R. Mallier; S. Mansi. Laplace transforms and shout options. Acta mathematica Universitatis Comenianae, Tome 80 (2011) no. 1. http://geodesic.mathdoc.fr/item/AMUC_2011_80_1_a6/
@article{AMUC_2011_80_1_a6,
     author = {G. Alobaidi and R. Mallier and S. Mansi},
     title = {Laplace transforms and shout options},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2011},
     volume = {80},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2011_80_1_a6/}
}
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JO  - Acta mathematica Universitatis Comenianae
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Voir la notice de l'article provenant de la source Comenius University

We use Laplace transform methods to examine the optimal exercise boundary for shout options, which give the holder the right to lock in the profit to date while retaining the right to benefit from any further upside. The result of our analysis is an integro-differential equation for the location of this optimal exercise boundary. This equation is a nonlinear Fredholm equation, or more specifically, an Urysohn equation of the first kind. Applying an inverse Laplace transform to this equation allows us to find the behavior of the free boundary close to expiry. The results are given for both call and put shout options.