Functionals of Exponential Brownian Motion and Divided Differences

TL;DR

This paper applies classical approximation theory, specifically divided differences, to analyze exponential Brownian motion in finance, deriving new formulas for correlation and moments of the time average.

Contribution

It introduces the use of divided differences to simplify and derive new results for correlation and moments in exponential Brownian motion models.

Findings

Correlation coefficient always at least 1/√2

All moments of the time average are divided differences of the exponential function

Moments match previous complex formulas by Oshanin and Yor

Abstract

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least $1/\sqrt{2}$ and, via the Hermite--Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor.