Some Formulae for a New Type of Path-Dependent Option

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.