Some combinations of Asian, Parisian, and barrier options

Abstract

This article addresses some of the valuation problems, in the Black and Scholes setting of a geometric Brownian motion for the underlying asset dynamics, for options whose pay-off is related to the terminal price of the stock and an arithmetic average of fixing and/or involves stopping times related to excursions. In all cases, we are able to provide at least the Laplace transform in time of the option price under a form whose complexity varies with the number of exotic features. We emphasize that we do not give closed form formulas for the general case, but we aim to develop a methodology which may be used in many cases