Law of the exponential functional of one-sided L\'evy processes and Asian options

TL;DR

This paper provides a power series representation for the distribution of the exponential functional of spectrally negative Lévy processes and derives a formula for pricing Asian options in markets modeled by such processes.

Contribution

It introduces a novel power series approach to the distribution of exponential functionals and extends the Geman-Yor formula to Asian options in this setting.

Findings

Power series representation of the exponential functional distribution

Geman-Yor type formula for Asian options prices

Application to spectrally negative Lévy processes with unbounded variation

Abstract

The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative L\'evy process \xi with unbounded variation. We also derive a Geman-Yor type formula for Asian options prices in a financial market driven by e^\xi.