Windings of planar processes, Exponential Functionals and Asian options

Abstract

Motivated by a common Mathematical Finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of an associated Brownian motion. We prove a conjecture by Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and we present a novel simple proof of De Blassie's result (1987-1988) about the asymptotic behaviour of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, for t large. Similar issues for the exponential functional of a Levy process are also discussed. We finally use the findings obtained by the windings approach in order to get results for quantities associated to the pricing of Asian options.