In this paper we present an explicit form of the distribution function of the occupation time of a Brownian motion with a constant drift (if there is no drift, this is the well-known arc-sine law). We also define the $\alpha$-percentile of the stock price and give an explicit form of the distribution function of this random variable. Using this explicit distribution, we calculate the price of a new type of path-dependent option, called the $\alpha$-percentile option. This option was first introduced by Miura and is based on order statistics.
@article{1177004769,
author = {Akahori, Jiro},
title = {Some Formulae for a New Type of Path-Dependent Option},
journal = {Ann. Appl. Probab.},
volume = {5},
number = {4},
year = {1995},
pages = { 383-388},
language = {en},
url = {http://dml.mathdoc.fr/item/1177004769}
}
Akahori, Jiro. Some Formulae for a New Type of Path-Dependent Option. Ann. Appl. Probab., Tome 5 (1995) no. 4, pp. 383-388. http://gdmltest.u-ga.fr/item/1177004769/