Windings of planar processes, Exponential Functionals and Asian options

TL;DR

This paper explores the mathematical properties of exponential functionals of planar Brownian motion and Levy processes, proving conjectures, analyzing asymptotics, and applying findings to Asian option pricing in finance.

Contribution

It provides a proof of a conjecture on infinite divisibility, offers a new proof of asymptotic behavior of Bessel clocks, and connects these results to Asian option pricing.

Findings

Proved a conjecture on infinite divisibility of the reciprocal of exit times.

Presented a simple proof of asymptotic distribution of Bessel clocks.

Applied windings approach to derive results for Asian option pricing.

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.