  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.
