Functionals of Exponential Brownian Motion and Divided Differences
Provides new analytic results for a fundamental asset-pricing model, but the problem is niche and the results are theoretical.
The paper computes the correlation coefficient between exponential Brownian motion and its time average, proving it is always at least 1/√2, and derives all moments of the time average as divided differences of the exponential function, matching known results.
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.