From planar Brownian windings to Asian options
The main purpose of the paper is to explain the relation between the distribution of the winding number \(\theta_ t\), \(t>0\), of planar Brownian motion around 0 and the distribution of \(A_ t^{(\nu)}=\int^ t_ 0 \exp 2(B_ s+\nu s) ds\), where \(\nu \in \mathbb{R}\) and \(B\) is a linear Brownian motion. The interest in the distribution of \(A_ t^{(\nu)}\) comes from Asian options which are functions of the time average of an asset price. Specifically the author illuminates the relationship between known formulae on transforms of \(\theta\) and \(A\) in terms of Bessel functions. The techniques used include time change, change of measure and excursion theory.
scientific article; zbMATH DE number 540711
scientific article