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Some formulae for a new type of path-dependent option. (English) Zbl 0834.90026
Summary: 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 variables. 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.

##### MSC:
 91B24 Microeconomic theory (price theory and economic markets) 60H30 Applications of stochastic analysis (to PDEs, etc.) 60G44 Martingales with continuous parameter
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