Abstract
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.
Citation
Jiro Akahori. "Some Formulae for a New Type of Path-Dependent Option." Ann. Appl. Probab. 5 (2) 383 - 388, May, 1995. https://doi.org/10.1214/aoap/1177004769
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